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Recycle dynamics and control in a simulated papermachine wet end Chong Ping, Michael 2003

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R E C Y C L E D Y N A M I C S A N D C O N T R O L I N A S I M U L A T E D P A P E R M A C H I N E W E T E N D By Michael Chong Ping B. Sc. (Chemical Engineering) Queen's University, 1988 M . Sc. (Chemical Engineering) University of Waterloo, 1996 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E DEGREE OF DOCTOR OF PHILOSOPHY in T H E FACULTY OF GRADUATE STUDIES DEPARTMENT OF CHEMICAL AND BIOLOGICAL ENGINEERING We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA April 2003 © Michael Chong Ping, 2003 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Chemical and Biological Engineering The University of British Columbia 2216 Main Mall Vancouver, Canada V6T 1Z4 Date: Abstract Over the years, increased competition and stricter environmental regulations have placed additional pressure on mills to become more efficient. Chemical additives are being used to enhance mill performance and improve paper quality. The use of recycle streams has also reduced the raw material costs. These changes have made control of the wet end system a challenging problem for mill personnel. The presence of multiple recycle streams have exacerbated the problems in the wet end. To deal with these problems, the goal of this work is to provide an understanding of recycle dynamics and control that will help improve operations in the papermachine wet end. The first step was to examine recycle dynamics to identify responses that could be potential control problems. A simple recycle loop system composed of a process model in the forward path and a process model in the recycle path was used, and the study was performed using various combinations of parameter values for the process models. The recycle responses were found to fall broadly into three groups with groups two and three being identified as having responses that may cause problems for control. To determine if the wet end exhibited significant recycle behaviour, an IDEAS simu-lation model of the paper machine wet end was used. The study looked at the effect of a filler change in the recycle streams, and in the paper leaving the wire. The effect of tank sizes on the recycle dynamics was also explored. The process tanks resulted in all of the responses being first order. It was only when the tanks were removed from the process that recycle dynamics became apparent, but the extent of the dynamics depended on the number of recycle streams present. The problems in the wet end are also attributed to the effect of disturbances being returned to the process by the recycle streams. Two disturbance types were studied: (a) Quality disturbances, such as changes in temperature and stream composition, and (b) Flowrate disturbances which are basically flowrate variations. Quality disturbances require better control of the recycle process. The common PI controller was the first controller evaluated on the recycle process. The performance of the PI controller was adversely affected by the size of the time delays in the forward or recycle path. Little improvement in controller performance was obtained when dead-time compensation, in the form of a Smith Predictor, was used. A new approach using a seasonal model to represent the recycle process for model based control was presented. The seasonal model approach was compared to the following model based control solutions found in the literature: (a) the recycle compensator which removes the recycle effect and (b) the Taylor series approach which approximates the denominator ii dead time terms of the recycle process. The seasonal model approach provided comparable performance to the existing methods and was more practical. Flow disturbances are usually dampened by the surge tanks. However, the degree of disturbance attentuation is determined by tank level controller. If the controller is tuned for tight level control, the flow disturbance may pass from the input to the output with very little attenuation. When the controller is tuned for averaging level control, disturbances in the flowrate are significantly reduced. In this work, an optimal averaging level control algorithm developed by Foley et al. (2000) was selected. The algorithm is based on constrained minimum variance and guarantees the lowest possible flowrate variation. The algorithm was to handle the case of sampling delays greater than one. The algorithm was applied to a saveall simulation to reduce the flowrate variability from the saveall to the blend chest. iii Table of Contents Abstract ii List of Tables vii List of Figures viii Acknowledgements xii Chapter 1 Introduction 1 1.1 Objectives and Contributions of the work 4 Chapter 2 Recycle Dynamics and Modeling 7 2.1 Recycle Dynamics Literature Review 7 2.2 Recycle Dynamics Study 14 2.2.1 Two F O P D T Models 15 2.2.2 Lead-Lag F O P D T and F O P D T Model 18 2.2.3 Inverse Response F O P D T and F O P D T Model 21 2.2.4 Second-Order-Plus Dead-Time-Model (SOPDT) and F O P D T Model 26 2.2.5 Overall Recycle Dynamic Summary 32 2.3 Modeling of Recycle Responses 36 2.3.1 Seasonal Model Identification 40 2.4 Recycle Structures 43 2.5 Conclusions 44 Chapter 3 Recycle Control 45 3.1 Literature Review 46 3.2 PI Control of Recycle Processes 48 3.2.1 Increasing Op 52 3.2.2 Increasing OR 57 3.2.3 PI Control Summary 60 3.3 Recycle Control Strategies 60 3.3.1 Taylor Series Expansion 61 3.3.2 Recycle Compensator 63 3.3.3 Proposed Model for Controller Design - Seasonal Models 64 3.4 Simulation Results 66 3.4.1 Simulation 1: True Process Model for Control 67 iv 3.4.2 Simulation 2: Identified Process Model for Control . . . 70 3.4.3 Simulation Results Summary and Discussion 73 3.4.4 Implementation Issues 74 3.5 F O P D T Model vs Seasonal Model for Recycle Control 74 3.6 Conclusions 80 Chapter 4 Wet End Recycle Dynamics 82 4.1 The Wet End Paper Process 82 4.2 Wet End Recycle Dynamics 85 4.2.1 Wet End Recycle Dynamics with Reduced Tank Sizes . 92 4.2.2 Wet End Recycle Dynamics for the Recycle Loop Tan-kless Case 96 4.3 Simulation Issues 99 4.4 Conclusions 101 Chapter 5 Averaging Level Control in the Wet End 103 5.1 Literature Review 104 5.2 C M V Averaging Level Control I l l 5.2.1 General C M V Summary ' I l l 5.2.2 C M V : Application to Averaging Level Control 113 5.3 Wet End Application 127 5.3.1 Simulation 1: Constant Cloudy Feed to the Saveall . . . 131 5.3.2 Simulation 2: Varying Cloudy Feed to the Saveall . . . 134 5.4 Conclusions 138 Chapter 6 Conclusions and Recommendations 139 6.1 Recommendations 142 Bibliography 145 Appendix A Recycle Dynamic Figures for G F =FOPDT and G^=FOPDT 150 A . l Trend 1 Figures 151 A.2 Trend 2 Figures 152 A. 3 Trend 3 Figures 153 Appendix B Recycle Dynamic Figures for GF=Lead-Lag FOPDT and GR = FOPDT 154 B. l Trend 1 Figures 155 B.2 Trend 2 Figures 156 B.3 Trend 3 Figures 157 B. 4 Trend 4 Figures 158 Appendix C Recycle Dynamic Figures for G^Inverse Response FOPDT and G i J =FOPDT 159 C l Trend 1 Figures 160 C. 2 Trend 2 Figures 161 C.3 Trend 3 Figures 162 C.4 Trend 4 Figures 163 C. 5 Trend 5 Figures 164 Appendix D Recycle Dynamic Figures for G F =SOPDT and G ^ F O P D T 165 D. l Category 1, Trend 1 Figures 166 D.2 Category 1, Trend 2 Figures 169 D.3 Category 2, Trend 1 Figures 172 D.4 Category 2, Trend 2 Figures 175 D.5 Category 2, Trend 3 Figures 178 D.6 Category 3, Trend 1 Figures 181 D.7 Category 3, Trend 2 Figures 184 D.8 Category 4, Trend 1 Figures 187 D.9 Category 4, Trend 2 Figures 190 D.10 Category 4, Trend 3 Figures 193 Appendix E Examples of Complex Recycle Structures 196 Appendix F C M V Averaging Level Control Design for Three Units of Sam-pling Delay 198 F . l Process Model 198 F.2 Spectral Factorization 198 F.3 Diophantine Identity 199 F.4 Controller Structure . . . 201 F.5 Closed Loop Performance for Three Units of Sampling Delay . . 202 F.6 Controller Action Variance 210 vi L i s t o f T a b l e s 2.1 Recycle Response of Control Interest 33 2.2 Stability Conditions for Recycle Systems with No Delay 35 3.1 Integral of the Absolute Error for the Effect of Bp on PI Performance . . . 56 3.2 Integral of the Absolute Error for the Effect of 9p on Smith Predictor-PI (SP-PI) Performance 56 3.3 Integral of the Absolute Error for the Effect of 9R on PI Performance . . . 60 3.4 Controller Parameters and IAE Values Using the True Process Model . . . 68 3.5 Controller Parameters and IAE Values Using Model-Plant Mismatch 8™ = 5, 6% = 5 71 3.6 Seasonal-FOPDT IAE Values, Varying 0F,8R = 1 79 3.7 Seasonal-FOPDT IAE Values, Varying 9R, 6F = 1 79 4.1 Normal Tank Sizes and Reduced Tank Sizes 93 5.1 Tank Information 129 5.2 Simulation 1: Saveall and Cloudy PID Tuning Parameters 131 5.3 Simulation 1: Broughton Level Controller Parameters 131 5.4 Simulation 1, Process Variations 134 5.5 Simulation 2: Saveall and Cloudy PID Tuning Parameters 135 5.6 Simulation 2: Broughton Level Controller Parameters 135 5.7 Simulation 2, Process Variations 135 vii L i s t o f F i g u r e s 2.1 Recycle Block System 8 2.2 Root Locus for Increasing Recycle System Loop Gain, for the Case when Tp > TR 9 2.3 CSTR System with Distillation Recycle 10 2.4 Modified Recycle Block System 11 2.5 Process Response for Different Recycle Loop Time Delay 9R 12 2.6 Effect of OR on Resonant Frequencies 13 2.7 Step and Frequency Responses for a 2 F O P D T Model Recycle System 16 2.8 Effect of Recycle Fraction on Step and Frequency Responses for 2 F O P D T Model Recycle System 17 2.9 Mixing Tank Process 18 2.10 Step and Frequency Responses for a Lead-Lag F O P D T and F O P D T Model Recycle System 19 2.11 Output at Various Stages in the Recycle Process for a Leadlag F O P D T and F O P D T Recycle System 20 2.12 Step Responses for a Inverse Response F O P D T and F O P D T Model Recycle System 23 2.13 Magnified View of Trend 4 and Intermediate Responses for a Inverse Response F O P D T and F O P D T Model Recycle System ., 24 2.14 Magnified View of Trend 4 and Intermed. Responses for a Inverse Response F O P D T and F O P D T Model Recycle System with Recycle Fraction = 0.5 . 25 2.15 Magnified View of Trend 4 and Intermediate Responses for a Inverse Response F O P D T and F O P D T Model Recycle System with Recycle Fraction = 0.75 25 2.16 Category 1 Step Responses for a SOPDT and F O P D T Model Recycle System 27 2.17 Effect of Recycle Fraction on the Underdamped Second Order Response . . . . 28 2.18 Category 2: Step Responses for a SOPDT and F O P D T Model Recycle System 29 2.19 Category 3: Step Responses for a SOPDT and F O P D T Model Recycle System 30 2.20 Category 4: Step Responses for a SOPDT and F O P D T Model Recycle System 31 2.21 Totals of International Travelers. Figure Reproduced from Box et. al (1994) . 37 2.22 Comparison of Recycle Models 38 2.23 Effect of Increased Sample Time on Recycle Model Comparison 39 2.24 Comparison of Identified Model to True Process 42 3.1 Pole Zero Plot - 0F = 2,0R = 1 50 3.2 Pole Zero Plot - 0F = 10,0R = 1 50 3.3 Pole Zero Plot - 0F = 20,0R = 1 50 3.4 ' Pole Zero Plot - Op — 1, OR = 2 51 viii 3.5 Pole Zero Plot - 9F = 1,9R = 10 51 3.6 Pole Zero Plot - 9p = 1,9R = 20 51 3.7 Effect of 9F on PI Performance - Setpoint Changes 54 3.8 Effect of 9F on Sensitivity Function - Setpoint Changes 54 3.9 Effect of 9p on PI Performance - Output Disturbance 55 3.10 Effect of 9p on PI Performance - Input Disturbance 55 3.11 Effect of 9R on PI performance - Setpoint Changes 58 3.12 Effect of 9R on Sensitivity Function - Setpoint Changes 58 3.13 Effect of 9R on PI performance - Output Disturbance 59 3.14 Effect of 9R on PI performance - Input Disturbance 59 3.15 Recycle Block System for 2 F O P D T 61 3.16 Different Order Taylor Series Expansion Compared to the True Process . . . . 62 3.17 Recycle Compensator Block System 63 3.18 Recycle Block System for 2 Discrete F O P D T . . 64 3.19 Comparison of the True Recycle Process with the Seasonal Model 65 3.20 P I / G P C Using the True Process Model: Servo Control 68 3.21 P I / G P C Using the True Process Model: Regulatory Control for an Output Disturbance 69 3.22 P I / G P C Using the True Process Model: Regulatory Control for an Input Dis-turbance 69 3.23 Case la, 9f = 5,0™ = 5: Setpoint Change 71 3.24 Case la, 9f = 5,9% = 5 : Output Step Disturbance 72 3.25 Case la, 9% = 5,9% = 5: Input Step Disturbance 72 3.26 Process and Model Bode Plot 76 3.27 Process and Model Responses 76 3.28 Seasonal-FOPDT Comparison - 9F = 5 77 3.29 Seasonal-FOPDT Comparison - 9F = 10 77 3.30 Seasonal-FOPDT Comparison - 9F = 15 78 4.1 Wet End Process Diagram 83 4.2 Wet End Process Diagram - Recycle Streams 86 4.3 Recycle Dynamics for Loop RI 88 4.4 Recycle Dynamics for Loop RI, and R2 88 4.5 Recycle Dynamics for Loop RI, R2, and R3 89 4.6 Recycle Dynamics for Loop RI, R2, R3, and R4 89 4.7 Deculator Schematic 90 4.8 Recycle Dynamics for Loop R4P1, R4P2, and R4 91 4.9 Recycle Dynamics for Loops R4P1 alone and R4P2 alone 91 ix 4.10 Reduced Tank Size, Recycle Dynamics for Loop RI 94 4.11 Reduced Tank Size, Recycle Dynamics for Loop RI, and R2 94 4.12 Reduced Tank Size, Recycle Dynamics for Loop RI, R2, and R3 95 4.13 Reduced Tank Size, Recycle Dynamics for Loop R1,R2, R3, and R4 95 4.14 No Tanks, Recycle Dynamics for Loop RI . . 97 4.15 No Tanks, Recycle Dynamics for Loop RI, and R2 97 4.16 No Tanks, Recycle Dynamics for Loop RI, R2, and R3 98 4.17 No Tanks, Recycle Dynamics for Loop RI, R2, R3, and R4 98 5.1 Feedback Control System 112 5.2 Liquid Level Control Diagram 113 5.3 Saveall system 128 5.4 Output Variance aY against Input Variance o\v 130 5.5 Simulation 1: Broughton Tank PI Level Controller 132 5.6 Simulation 1: Broughton Tank Averaging Level PI Controller 133 5.7 Simulation 2: Broughton Tank PI Level Controller 136 5.8 Simulation 2: Broughton Tank Averaging Level PI Controller 137 A.I G F = G . R = F O P D T , Trend 1 Figures 151 A. 2 G F = G A = F O P D T , Trend 1 Figures 152 A-3 GF=GR=FOPDT, Trend 1 Figures 153 B. l G F=Lead-Lag F O P D T , G#=FOPDT, Trend 1 Figures 155 B.2 G F=Lead-Lag F O P D T , G#=FOPDT, Trend 2 Figures 156 B.3 G F=Lead-Lag F O P D T , Gfl=FOPDT, Trend 3 Figures 157 B. 4 G F=Lead-Lag F O P D T , G f l = F O P D T , Trend 4 Figures 158 C l GF=Inverse Response F O P D T , G ^ F O P D T , Trend 1 Figures 160 C. 2 GF=Inverse Response F O P D T , G#=FOPDT, Trend 2 Figures 161 C.3 GF=Inverse Response F O P D T , G^—FOPDT, Trend 3 Figures 162 C.4 GF=Inverse Response F O P D T , G#=FOPDT, Trend 4 Figures 163 C. 5 GF=Inverse Response F O P D T , G B = F O P D T , Trend 5 Figures 164 D. l G F = S O P D T , G#=FOPDT, Category 1, Trend 1 Figures, r=0.25 . 166 D.2 G F = S O P D T , G K = F O P D T , Category 1, TYend 1 Figures, r=0.5 167 D.3 G F = S O P D T , G R = F O P D T , Category 1, Trend 1 Figures, r=0.75 168 D.4 G F = S O P D T , G#=FOPDT, Category 1, Trend 2 Figures, r=0.25 169 D.5 G F = S O P D T , G R = F O P D T , Category 1, Trend 2 Figures, r=0.5 170 D.6 G F = S O P D T , G/j=FOPDT, Category 1, Trend 2 Figures, r=0.75 171 D.7 G F = S O P D T , Gfl=FOPDT, Category 2, Trend 1 Figures, r=0.25 172 x D.8 GF =SOPDT, GR- =FOPDT, Category 2, Trend 1 Figures, r= =0.5 D.9 GF =SOPDT, GR-- =FOPDT, Category 2, Trend 1 Figures, r= =0.75 D.10 GF =SOPDT, GR- =FOPDT, Category 2, Trend 2 Figures, r= =0.25 D . l l GF =SOPDT, GR- =FOPDT, Category 2, Trend 2 Figures, r= =0.5 . D.12 G F =SOPDT, GR- =FOPDT, Category 2, Trend 2 Figures, r= =0.75 D.13 GF =SOPDT, GR- =FOPDT, Category 2, Trend 3 Figures, r= =0.25 D.14 GF =SOPDT, GR- =FOPDT, Category 2, Trend 3 Figures, r= =0.5 D.15 GF =SOPDT, GR- =FOPDT, Category 2, Trend 3 Figures, r= =0.75 D.16 GF - S O P D T , GR- =FOPDT, Category 3, Trend 1 Figures, r= =0.25 D.17 GF =SOPDT, GR- =FOPDT, Category 3, Trend 1 Figures, r= =0.5 D.18 GF =SOPDT, GR- =FOPDT, Category 3, Trend 1 Figures, r= =0.75 D.19 GF =SOPDT, GR- =FOPDT, Category 3, Trend 2 Figures, X-=0.25 D.20 GF =SOPDT, GR- - F O P D T , Category 3, Trend 2 Figures, V-=0.5 D.21 GF =SOPDT, GR- =FOPDT, Category 3, Trend 2 Figures, r= =0.75 D.22 GF =SOPDT, GR--=FOPDT, Category 4, Trend 1 Figures, r= =0.25 D.23 GF =SOPDT, GR- =FOPDT, Category 4, Trend 1 Figures, r= =0.5 D.24 GF =SOPDT, GR- =FOPDT, Category 4, Trend 1 Figures, r= =0.75 D.25 GF =SOPDT, GR =FOPDT, Category 4, Trend 2 Figures, r= =0.25 D.26 GF =SOPDT, GR =FOPDT, Category 4, Trend 2 Figures, r= =0.5 D.27 GF =SOPDT, GR =FOPDT, Category 4, Trend 2 Figures, r= =0.75 D.28 GF =SOPDT, GR =FOPDT, Category 4, Trend 3 Figures, r= =0.25 D.29 GF =SOPDT, GR =FOPDT, Category 4, Trend 3 Figures, =0.5 D.30 GF =SOPDT, GR =FOPDT, Category 4, Trend 3 Figures, r= =0.75 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 E . l Configuration 1 196 E.2 Configuration 2 196 E.3 Configuration 3 196 E.4 Configuration 4 197 xi A c k n o w l e d g e m e n t s First, I would like to thank my supervisors Dr. Ezra Kwok and Dr. Guy Dumont for their guidance, advice, and support during my years here at U B C . The efforts of my committee members Dr. Michael Davies and Dr. Richard Branion are gratefully acknowledged. I would like to thank Dr. Michel Perrier from Ecole Polytechnique for serving as the external examiner. Also, the financial assistance of the Network Centers of Excellence, Mechanical Pulps branch was greatly appreciated. There have been a number of people in the Chemical & Biological Engineering Department and the Pulp & Paper Center who have helped me. In particular, I would like to thank Helsa Leong, Lori Tanaka, Amber Lee, Tim Patterson, Ken Wong, Brenda Dutka, and Lisa Brandly who helped with any problems that I encountered. Thanks also to Dusko Posarac and Brian McMillan for the many computer related talks. I have also learned tremendously from many control discussions with my fellow colleagues, Stevo Mijanovic, Guantien Tan, Manpreet Sidhu, Gregory Stewart, and Mihai Huzmezan. I have also been lucky to meet other graduate students who have made my time at U B C all the more memorable, Petar Knezevich, Phillip Servio, Neeraj Gupta, Sanjiv Dhan-jal, Chris Kurniawan, Shivamurthy Modgi, Rajeev Chandraghatgi, Eman Alatar, Poupak Mehrani, and Diana Lencar. My graduate experience was made all the more fulfilling by the time spent with the Van-couver Sai Baba Organization and especially the Sai Youth Group. I am very grateful to Gurinder Dayal and Amit Nath for their friendship, and the many spiritual and worldly debates. I give special thanks to my parents Aldric and Jean Chong Ping for being there for me always. I give thanks to Sathya Sai Baba and God for everything. xii C h a p t e r 1 I n t r o d u c t i o n The process of paper-making has been known for many centuries. Although the funda-mental principle of making paper has not changed, different stock preparation and pro-cessing methods have evolved to meet different product requirements. Since the discovery of paper, the use of paper materials has flourished. Paper products such as stationery, newsprint, and product packaging are commonplace today. In fact, paper products have insinuated themselves into daily life to such an extent that they have almost become a necessity. The demand for these products has always been strong. Even with the technological advances in computing facilities which people predicted would result in less paper con-sumption, the opposite effect has occurred. The computer era has resulted in computer hardware/software being easily accessible/affordable to everyone. Nowadays, a large per-centage of the population have computers in their homes coupled with a low cost high quality printer. This has resulted in a large demand for quality printer paper for personal and business uses. The global demand of paper is expected to continue to rise, especially as developing countries increase their paper usage as a natural consequence of economic growth. To meet the demand for paper, the Pulp and Paper (P & P) industry has grown. The North American P & P industry has been the world leader in terms of production and product diversity. At one point in time, they were even the world's low cost producer of paper products. However, the P & P industry has grown in other countries where labor and raw materials are cheap, countries such as Brazil, Chile, and Indonesia. Today, some off-shore paper products are competitive in terms of product cost with corresponding North American products. 1 In this day of global trade, P & P companies are facing increased competition for paper product markets. In addition, public awareness has increased considerably on environmen-tal issues. This has led to the government placing stringent regulations on effluent/waste disposal to reduce the impact on the environment. Thus, there is sufficient incentive for mills to be more efficient while maintaining prod-uct quality. Costs have been reduced by recovering raw materials such as fiber, fines, filler, and process/white water and returning them to the process via recycle streams. 1The information contained in this paragraph concerning the North American and Global Pulp & Paper Industry was obtained from Smook (1992) 1 Chapter 1. Introduction 2 This has reduced the amount of raw materials lost in effluent streams and the cost of effluent treatment. Mill performance and product quality have been improved through the use of additives. These additives can be classed into two types: • Performance aids are used to improve the properties of the paper. Typical examples are: - Mineral fillers which give the paper better opacity, smoothness, and whiteness. Fillers were originally used to replace more expensive fibers. - Dry/wet strength aids are used to improve the strength of the paper when it is dry or water saturated. - Sizing agents to give the paper resistance to aqueous liquid penetration. • Operating/processing aids are used to improve the performance ofthe papermachine. Common examples are as follows: - Retention aids are used to increase the retention of furnish materials (fibers, fines, filler, etc.) in the paper. - Formation aids improve the fiber distribution in the sheet by decreasing the fiber-fiber flocculation. - Defoamers and antifoamers are used to control and prevent the development of foam within the process. - Deposit control agents are used to reduce/prevent the deposition of organic and inorganic solids on the equipment which could eventually end up in the paper product causing defects and paper breaks. Since all of the wet end additives interact to contribute to the final paper properties and machine performance achieved, the wet end chemistry has to be kept within some desired set of conditions to reduce process variability. If the wet end chemistry goes out of control it can result in deposits, scaling, and a drop in product quality. As can be seen, the use of additives has made the wet end chemistry more complex. Since it is well known that the wet end performance greatly affects the product quality, it would be desirable to optimize the wet end performance. However, the wet end is subject to disturbances which affect the chemistry. Disturbances may result in such things as poor retention for furnish materials, increased additive cost, loss of paper properties (e.g., strength), and machine breaks. These disturbances can enter the process in many ways, for example: Chapter 1. Introduction 3 • Interfering substances can reduce the effect of process additives and adversely affect the papermaking processes. These substances may come from the wood source or from materials recycled back to the process. • Additive concentration changes may result in the additives reacting with one another reducing their efficiency. • Flowrate variations may occur in the process water flows and stock feeds because of machine breaks and increased broke feed. • Fiber, fines, and filler consistency variations are caused by poor retention aid effi-ciency and also by an increased broke feed. Unfortunately, the effect of disturbances on the process are exacerbated by the presence of recycle streams. Disturbances are recycled back to the process causing additional disturbances. It is well known that recycle streams cause interactions between process units and increase the disturbance sensitivity of the process. When raw materials are recovered and returned to the paper process via recycle streams, other materials such as salts, suspended solids, and contaminants are also returned to the process. The buildup of these materials in the system causes problems such as deposits and increased corrosion. These problems are amplified as mills move closer to full white water closure. The severity of the problems in the wet end is not caused by one recycle stream but by multiple recycle streams. A typical mill can have more than ten recycle loops, these streams may be sequential or nested within one another. Before dealing with the problems associated with recycle streams, an understanding of recycle dynamics and the effect of recycle on different processes is required. Such knowledge will reveal any inherent control limitations and provide engineers with insight on how processes need to be designed to minimize the impact of recycle streams. Additionally, it will allow for better modeling and prediction of the process dynamics which is important for good controller design. Though recycle streams have gained wide spread use, it is only within the last decade that there has been increased focus on recycle dynamics and control. Almost all of the reported work has been in the chemical industry and mostly on controlling a specific process, i.e., a chemical reactor with a distillation column to recover unused chemical for recycle to the reactor. There have been studies performed on recycle dynamics and control in the papermachine wet end. However, the results were never reported as a recycle dynamics and control issue. For example, the work on modeling the dynamics of first pass retention and the work on retention control all involve recycle systems. Because of the limited knowledge on recycle processes, this work considers expanding the knowledge of recycle dynamics and control in a systematic manner. In the first stage, Chapter 1. Introduction 4 using computer simulation, the approach will be to assume the process is composed of simple models and the dynamics studied. Then, the complexity of the models will be increased to see if the overall dynamics can be categorized. During the study, stability analysis of the process will be considered. This aspect is not only important for controller design but also during the initial engineering design. In the past, there was little connec-tion between process design and control system application; the process was first designed and then the control system specified. This procedure has led to poorly controlled and inefficient processes. It has been recognized that control system considerations should be incorporated at the design stage where modifications can be made to avoid control problems/limitations. The second aspect of this work looks at dealing with the increased sensitivity to dis-turbances caused by recycle streams. This should be beneficial to improving the overall operation of the papermachine wet end with its multiple recycle loops. According to Cui and Jacobsen (2000), the disturbances affecting the process can be classified into two types: (a) Quality disturbances such as changes in stream composition and (b) Flowrate disturbances. Each disturbance type requires a different controller strategy and this will be addressed. Quality disturbances require better control of the recycle process. Various control strategies have been proposed in the literature. These methods will be reviewed, and if they are found to be inadequate, an alternate approach will be required. The classical way of dealing with flow disturbances has been to use averaging level control in the buffer tanks. The controller applied was usually a detuned PI controller which did not account for performance issues. The use of averaging level control will be revisited by first reviewing the current literature. -Then an averaging level control that provides optimal flow smoothening will be applied and , if necessary, extended to cover a broader range of applications. The problems associated with recycle streams are not just limited to the P & P in-dustry. Practically all industries are using some degree of process integration to reduce costs. Recycle streams can be found in many industries, such as, Oil and Gas Refining, Petrochemical, Chemical, Waste Treatment, and the Food and Beverage Industry. The results of this work will be applicable to any industry that incorporates recycle steams. 1 . 1 O b j e c t i v e s a n d C o n t r i b u t i o n s o f t h e w o r k As stated earlier, the goal is to provide an understanding of recycle dynamics and control that will help improve operations in the papermachine wet end. A number of objectives have been mentioned, these will be re-summarized below in a compact form: Chapter 1. Introduction 5 • To review the current literature for issues dealing with recycle dynamics control, especially those that can be related to the P & P industry. • A systematic and comprehensive study of recycle dynamics using a simulation ap-proach to try and categorize the overall response. • Investigate the current strategies for recycle control with the idea of reducing the effect of flow and quality disturbances on the recycle process. This work produced a number of results, and the main contributions are summarized below: • By analyzing recycle dynamics using simulation, recycle responses that are derived from typical industrial processes can be categorized into 3 general groups: (a) Group 1 has the response of the forward path process, (b) Group 2 has a fast initial rise time followed by a slow rise to steady state, and (c) Group 3 has stepwise rise to steady state. • A new modeling approach was proposed for the Group 3 category described above. This group poses the greatest challenge for controller design. The approach is based on seasonal trends from time series analysis. The seasonal model was found to better model the recycle dynamics from all three groups. Hence, an alternative advanced control strategy which used the seasonal model with a model-based controller was proposed to handle recycle processes. • In terms of recycle control, PI control was evaluated for recycle processes. The performance was adequate at small time delays but degraded when the time delays, either forward and/or recycle path, were increased. • The new seasonal model with model-based controller was compared other model-based controllers on a recycle process. The seasonal model approach provided com-parable performance and was more practical. • In the area of recycle dynamics in the papermachine, a simulation model of a indus-trial papermachine wet end was used to identify areas with significant recycle effects (i.e., Group 3). The effect of tank sizes on the recycle response was investigated. The large process tanks resulted in the responses falling into the Group 1 category. It is only when the tanks are removed from the recycle process (forward and recycle paths) that significant recycle responses are seen. • For optimal averaging level control, the algorithm by Foley et al. (2000) was ex-tended to cover level controllers with longer time delays. The controller guarantees Chapter 1. Introduction 6 the lowest possible variance in flowrate changes and is easily implemented using industrial controllers. The work contained in this thesis is organized in the following manner: Chapter 2 first provides a review of the literature on recycle dynamics. Then the dynamic study of recycle processes composed of different models is given. The chapter finishes wi th modeling of the recycle response. Chapter 3 focuses on recycle control aspects. The chapter starts wi th a review of the current work in recycle control. Next, P I control of recycle processes is evaluated to show the need for advanced control. Then, the existing model based control strategies are compared with a new proposed approach. In Chapter 4, a simulation model of the wet end of a papermachine is used to determine the extent of recycle dynamics in the wet end. The study gradually incorporates recycle streams to illustrate the effect of additional recycle loops. Results are also provided for the scenario of reduced tank sizes and fewer process tanks. The application of averaging level control to reduce flow disturbances in the wet end is addressed in Chapter 5. A review of averaging level control methods is given. The algo-r i thm by Foley, Kwok, and Copeland (2000) was chosen and a summary of the derivation was presented. The algorithm was then extended for higher sampling delays and applied to a simulated saveall section. Chapter 6 concludes the thesis with conclusions and recommendations for future work. Chapter 2 Recycle Dynamics and Modeling In the past, chemical plants were designed as individual units that were linked in series with one another. Over time, as the process industries became more competitive, there was an increased demand for higher efficiency and less wastage. To accomplish these goals, one of the solutions was the implementation of recycle streams. Unused raw materials from product streams and heat from product/waste streams were recovered and returned to the process. Currently many processes are highly integrated using recycle streams. Though recycle operation is widely used in industry, it is only recently that attention has focused on the recycle dynamics and control aspects. Though recycle studies have been performed, there has been no detailed examination of the recycle dynamics reported in the literature. The objective of this chapter is to study the dynamics of a recycle system composed of different model orders. The goal is to characterize the responses and identify significant behavior from the control viewpoint. This chapter is divided into 5 sections. Section 1 provides a review of the current literature on recycle dynamics. Section 2 presents the different model structures used in the recycle dynamic study and a summary of the results. A model to represent recycle dynamics and a quick identification algorithm are presented in Section 3. Different recycle structures are briefly covered in Section 4 and conclusions are given in Section 5. 2 . 1 R e c y c l e D y n a m i c s L i t e r a t u r e R e v i e w A recycle process can be represented by the block diagram shown in Figure 2.1 in which: • GF are the units in the forward path, • GR are the units in the recycle path, • U is the load disturbance, • R is the output being recycled, • Y is the process output, As seen in Figure 2.1, recycle systems are equivalent to positive feedback systems (Gilliland et al. 1964; Denn and Lavie 1982). It is well known that a criterion for stability is that the recycle loop gain (Kp • KR) be less than one (Gilliland et al. 1964; Luyben 1993a). Studies have shown that the presence of recycle increases the overall time constant and 7 Chapter 2. Recycle Dynamics and Modeling 8 the sensitivity to disturbances (Gilliland et al. 1964; Verykios and Luyben 1978; Denn and Lavie 1982). In addition, the recycle system may exhibit under-damped behavior (Verykios and Luyben 1978). u Y +y R GR Figure 2.1: Recycle Block System Luyben (1993a) has shown mathematically the above observations in the following manner. Assuming the forward and recycle processes are first-order models with time delay, then: Gf KFe~eFS rFs + 1 GR KRe-9«s TRS+1 (2.1) The overall equation relating the input to the output for the process block diagram shown in Figure 2.1 is: Y U Gf KF(TRs + l)e-6FS (2.2) 1 - GFGR T F T R S 2 + (rF + TR)S + 1 - KFKRe-Vr+e*)> If the system is assumed to have no time delay (i.e., 6F = OR = 0), eqn. 2.2 reduces to: Y Kf(TRS + 1) U T F T R S 2 + (TF + TR)S + 1- KFKR which can be rearranged into the following form: KF \ I TRS + 1 (2.3) Y U 1 - KFKR s2 + P^-s + l \ - K F K R (2.4) \-KFKR Equation 2.4 is in the form of a standard second-order transfer function with a first-order lead term. The process gain (K), damping factor ((0), and process time constant (r 0) shown in equation 2.5 are obtained by comparing equation 2.4 to a standard second-order transfer function. Chapter 2. Recycle Dynamics and Modeling 9 K = TFTR 1/2 TF+TR 1-KFKR \ 1 - K F K R From eqn. 2.3, the characteristic equation is 2[TFTR(1 - KFKR)]V* (2.5) TFTRS2 + (TF + TR)S + 1- KFKR = 0 (2.6) Dividing through by TFTR, eqn. 2.6 reduces to: 2 , TF + TR 1 - KFKR S + S H = 0 TFTR TFTR then the following observations can be made: (2.7) 1. When the recycle system loop gain is equal to 0 (i.e., KFKR = 0), the two roots of the characteristic equation (eqn. 2.7) are at s = As KFKR increases to 1, one root moves to ~(T F + T R) and the other root moves to 0. When KFKR is larger than 1, one of the roots lies in the right half of the s-plane so the system is unstable. This progression of the roots is shown in Figure 2.2. 0.05 a 0 -0.05 Left Half Plane KcKR= 1 KFKR = 0 KrKn = 1 > • » • • • • Right Half Plane t f Increasing KFKR Increasing KFKR -0.1 Real Axis Figure 2.2: Root Locus for Increasing Recycle System Loop Gain, for the Case when TF > TR Chapter 2. Recycle Dynamics and Modeling 10 2. When the recycle system loop gain (KFKR) increases from 0 to 1, (a) The time constant (r0) increases from 1 to oo. Thus the process time constant increases with the recycle system loop gain. (b) The process gain (K) increases from Kp to oo. The presence of the recycle results in an overall gain larger than the forward path gain. As the recycle system loop gain increases, the process gain increases, thus load disturbances are amplified. (c) The damping factor (£„) is greater than 1, thus the system is over-damped. 3. When the recycle system loop gain is greater than 0 (i.e., KFKR < 0), the damping factor is less than 1 (i.e., ( 0 < 1) and the system is under-damped. According to Luyben (1993a), recycle systems with negative gains are rare but some do exist. Verykios and Luyben (1978) obtained under-damped behavior for the continuous stirred tank reactor (CSTR) system with distillation recycle (shown in Figure 2.3), however they did not calculate the recycle gain. Luyben (1993a) concluded that the behavior of recycle processes depended strongly on the recycle loop gain and less strongly on the units within the recycle loop. F,XF Figure 2.3: CSTR System with Distillation Recycle Chapter 2. Recycle Dynamics and Modeling 11 The recycle fraction can also influence the process dynamics. As the recycle fraction is increased (i.e. more recycle flow), the time constant and disturbance sensitivity increase as well (Gilliland et al. 1964; Verykios and Luyben 1978). Kwok (1998) showed that the recycle fraction must be carefully included in the block diagram representation to obtain the proper dynamics. Knezevich et al. (1999) modified the typical recycle block diagram (Fig. 2.1) to the one shown in Figure 2.4. This new representation explicitly shows the effect of the recycle fraction on the output. r F Figure 2.4: Modified Recycle Block System There are two points to note about this new configuration: • The recycle fraction (r) affects the size of the flowrates to the forward path process (GF) and the recycle path process (GR). This means there are changes to the time constants and time delays of GF and GR. AS a result, GF and GR are shown as functions of r. • In certain cases, the recycle fraction may also act as a weighting factor (i.e., become a scalar vector). If the recycle process had flows with a liquid-solid suspension with a varying particle size distribution (e.g., the streams in the wet end of a papermachine), the amount of solids returned to the process could depend on the particle size. In such a case, particles of different sizes would have different recycle fractions and r would then become a scalar vector. In addition, Gp and GR could each become a matrix of transfer functions as they are functions of the recycle fraction. Kumar and Daoutidis (2002) also looked at the effect of recycle fraction on recycle dynamics. Kumar and Daoutidis applied asymptotic analysis to a state variable recycle model to show that small recycle fractions cause a weak coupling among the individual processes. They go on to show that when the recycle fraction was large (i.e., the recycle flow was significantly larger than the feed/product flows), the overall recycle network had a time scale separation in dynamics. The individual process units in the recycle process had fast dynamics, typically on the order of the individual unit residence times, with a Chapter 2. Recycle Dynamics and Modeling 12 weak coupling caused by the recycle stream. This weak interaction became significant over long periods and the overall recycle network had slow dynamics. Time delays within the recycle system has also to be considered. Obviously the pres-ence of delay makes the process more sluggish, this can be seen in Figure 2.5 (Scali and Ferrari 1997) which shows the effect of increasing recycle time delay on the process re-sponse. When the recycle delay becomes larger than the forward path time constant, -• 1 ' Setpoint - o - eR = o - • • eR = 2.5 - • - 9 R = 5 1 1 ' / / / .... 1 / / / • * y y' y' 1 ^ 1—1—1 *' * * * 1 --X--* R = 1 U 3R=15 i / / t » . S s' y > • * • ' » * X I i it* - > 0 10 20 30 40 50 60 70 80 90 100 Time Figure 2.5: Process Response for Different Recycle Loop Time Delay OR the process response to disturbances is characterized by a step-wise trend to steady state (Scali and Ferrari 1997). The shape of the curve is caused by the recycle effect acting when the process has almost recovered from the disturbance. Most of the studies on recycle dynamics were performed in the time domain. Only two studies analyzed recycle systems in the frequency domain: • Denn and Lavie (1982) studied a reactor-separator system. They limited their sys-tem to one in which the recycle path was slower than the forward path. In addition to the usual increase in gain, time constant, and disturbance sensitivity, they state that: "The nature of the plant response is highly dependent on the dynamics of the recycle path" Chapter 2. Recycle Dynamics and Modeling 13 KF = KR = 1, TF = 1, TR = 1, 6F = 0, 6R = 10, 15, r = 0.5 I I I — I I I I I I I I I I I I I I I I I I I I I to or co T3 "5. E < 0.14 0.01 4 •eR=io •eR=i5 - 1 — i — r - i — — 0.01 I I 1 - 1 — I I I I - T 1 1—I—111 I I 1 I I—I I I I 1E-3 20 0.1 1 Frequency (radians/second) in CO S. <D T3 co 0--20 -60--80-1 -100-1 1 1 1 1 1 1 1 1 1 1 1 1 1—1 1 1 1 1 7 1 1 1 1 1 1 Vv 1 1 i—i—i—r i i | i i—i—i—ITT . . ^ " N ^ / A \ j \ *"*" • <•» \ x \ r 1 .1 ;/..,, »/t\,». i 1E-3 0.01 0.1 1 Frequency (radians/second) 10 Figure 2.6: Effect of 6R on Resonant Frequencies 100 They found that if the recycle path delay (OR) was long relative to the forward path time constant (TF), some disturbance frequencies might resonate throughout the loop (see Figure 2.6). In addition, the number of resonant frequencies and oscillation intensities increased with increasing OR. • Ohbayashi et al. (1989) studied the dynamic characteristics of an immobilized cell flash fermentor system with recycle. Recycle delay was also found to cause multiple resonant peaks at high frequencies. As the recycle fraction increased, these resonant frequencies occurred at higher frequencies but with a smaller amplitude ratio. In addition, under certain design and operating conditions, it was possible: - for the amplitude ratio to be greater than the steady state gain, — to have phase lead occurring at certain frequencies. Finally, Jacobsen (1997) looked at the effect of recycle on the plant transfer function "zeros. He made the observation that a system with recycle might contain input/output Chapter 2. Recycle Dynamics and Modeling 14 relations which are not part of the recycle loop but are affected by the recycle loop. Jacobsen then demonstrated that recycle moves zeros across the imaginary axis in to the right hand plane (RHP) causing inverse responses. 2.2 Recycle Dynamics Study Previous studies of recycle behaviour have only dealt with specific first-order processes. In addition, very little has been reported on the effect of different model orders on the dynamic responses. In this section, an extensive study on recycle dynamics is performed to fully charac-terize the recycle behaviour. Since many processes can be modeled by either a first or a second-order model with time delay, combinations of first/second-order models were used for the Gp and GR transfer functions shown in Figure 2.4. Obviously there is a large number of such model combinations; to reduce the number of possibilities, the study was limited to model combinations typically found in the chemical industry. The following model combinations decided upon were: • Two First-Order-Plus-Dead-Time (FOPDT) models • A Leadlag F O P D T model in the forward path (GF) and a F O P D T model in the recycle path (GR). • An Inverse response F O P D T model in the forward path and a F O P D T model in the recycle path. • A Second-Order-Plus-Dead-Time (SOPDT) model in the forward path and a F O P D T model in the recycle path. The structures of the first- and second-order models are shown in equation 2.8. Different values of the time constant (r), lead time constant (s), time delay (9), recycle fraction (r), and damping factor (£) were used to generate the output in the time and frequency domain. K(es + l)e~9' Ke~°' Uf*st ~ T S + 1 ^second - ^ + ^ + ± Since there was a large number of simulations for each combination, a summary of the results are contained in the following sections. Chapter 2. Recycle Dynamics and Modeling 15 2 .2 .1 Two F O P D T Models The transfer function models for the system are: GF = Kpe-°FS TFS + 1 GR = KRe-e*s TRS + 1 (2.9) From Figure 2.4, the overall transfer function is: G = (l-r)KF(TRs + l)e-dFS (2.10) TFTRS2 + (rF + rR)s + 1 - rKFKRe-^F+eR)s Example step responses and the corresponding Bode plots are shown in appendix A. The step responses for this system were categorized into the three types shown in Figure 2.7. The trends shown in Figure 2.7 are separated so they can be easily distinguished, as a result the responses do not start at the same time. • Trend 1: is the most basic response and is similar to a first order system. This can also be seen in the corresponding line in the Bode plot. The response occurs when the forward path time constant (rF) is dominant (i.e., rF > rR,6F,dR) • Trend 2: is like having a fast initial rise time. However, the time to steady state is very long. The condition for this to occur is when the recycle path time constant TR or the forward path delay (9F) is dominant (i.e., TR > TF, 6F, 6R or 6F > TF, TR, 0R). • Trend 3: is characterized by a step-wise increase to steady state. This response occurs when either the recycle path delay (OR) or the forward path delay (0F) is dominant (i.e., 0R,0F > TF,TR). From extensive simulation results, it was found that the step-wise behaviour starts to become clearly defined when 0F + 0R > 6 * TF. This stewpise behaviour is caused by the recycle effect acting after the forward path process has almost reached steady state in response to an input disturbance. The Bode plot for trends 2 and 3 show ripples in the amplitude and phase plots. Obvi-ously certain disturbance frequencies will be amplified/resonate through the process. This behaviour was also reported by Ohbayashi et al. (1989). The effect of recycle fraction on step and frequency responses is shown in Figure 2.8 using a trend 3 type system. The responses are again separated so they can be easily distin-guished. For step responses, the recycle fraction determines the magnitude of the output steps. The magnitude of the first output step is given by (1 — r)Kp • (input step size). Subsequent output steps are of magnitude rKFKR • (previous output step) up to the steady-state value of (1 — r)KF/(l — rKFKR). Chapter 2. Recycle Dynamics and Modeling 16 K F = K R = 1, TF = 5,2,1, TR = 5, 5,1, 9F = 0, 5,1, 9 R = 1,1,10, r = 0.5 150 10"' 10"1 10" Frequency (radians/second) 10' CO CD £ O) CD -100 D_ -150 _T_. |Trend2\ I \ Trend 3 -200 -i i i 11inj i i i iniij i i i iinij i i i i1111 • ' I. I I I nil 10 10" 10'2 10"1 10° Frequency (radians/second) 101 1(T Figure 2.7: Step and Frequency Responses for a 2 F O P D T Model Recycle System For example, using the curve for a recycle fraction (r) equal to 0.75, the first output step size is 0.25 x 1 = 0.25 which corresponds to one pass through the forward path process. The second output step size is 0.75 x 0.25 = 0.1875 with an overall value at the second step of 0.25 + 0.1875 = 0.4375. This corresponds to one pass through the recycle path and then again through the forward path. Chapter 2. Recycle Dynamics and Modeling 17 KF = KR = 1, TF = 1, TR = 1, 0F = 1, 9R = 10, r = 0.25, 0.5, 0.75 cu in c o CL in d> dc 0.4 0.2 -4-0.0 -0.4375 0.25 ; i / 25 50 75 Time 100 125 150 Figure 2.8: Effect of Recycle Fraction on Step and Frequency Responses for 2 F O P D T Model Recycle System In the frequency domain, increasing the recycle fraction causes an increase in the res-onant effect of some frequencies. Even though the resonant effect is larger, the amplitude at these frequencies is smaller than the corresponding amplitude at lower recycle fractions. Chapter 2. Recycle Dynamics and Modeling 18 2.2.2 Lead-Lag FOPDT and FOPDT Model The transfer function models for the system are: GF = KF(eFs + l)e~0FS rFs + 1 The overall process transfer function is: GR KRe -0Rs TRS + 1 (2-11) G = (l-r)KF{EFs + l)(TRs + l)e-6FS rFTRs2 + (TF + TR)s + 1 - rKFKR(eFs + l ) e - ( ^ + 0 « ) s (2.12) The function GF is a lead-lag system when the lead time constant is greater than 0 (i.e., eF > 0). According to Ogunnaike and Ray (1994, page 162), it is rare to find a chemical process where the behaviour is characterized by a lead-lag system. One example given by Ogunnaike and Ray (1994, pages 163-164) is the mixing tank system shown in Figure 2.9, where material A is mixed with material B. In order to change over from one mixture to the next, it is desirable to feed a fraction (p) of B to the tank outlet. The relation of the outlet B concentration (CB0) to the inlet B concentration (cui) is described by a lead-lag system. Material A JzEL Material B C D -'B-TANK •'B-OUT Figure 2.9: Mixing Tank Process For the recycle system studied, the lead time constant (eF) was chosen to have a value of 1.5 x TF, which is probably the most severe case found in industry. Example step response and Bode plots are shown in appendix B. As expected, all trends have an initial jump in the output followed by an exponential Chapter 2. Recycle Dynamics and Modeling 19 K F = K R = 1, 6F= 12, 7.5,1.5,15, xF = 8, 5.1,10, TR = 15,10,1,1, 6 F = 5, 5,10,15, 6 R = 5,10,15, 5, r = 0.5 0 25 50 75 100 125 150 175 200 225 250 Time Figure 2.10: Step and Frequency Responses for a Lead-Lag F O P D T and F O P D T Model Recycle System decay to an intermediate steady state. This initial jump is caused by the lead term in the forward path (GF)- The rate of exponential decay is determined by the ratio of the magnitude of the lead to the lag term, for this study the ratio is constant at 1.5. The step responses of this system were categorized into the 4 types shown in Figure 2.10. The Chapter 2. Recycle Dynamics and Modeling 20 trend responses are separated in time so they can be easily seen; as a result, the trends do not start at the same time. • Trend 1 : After the initial jump and exponential return to the intermediate steady state, there is a slow rise to the final steady state. This behaviour occurs when both time constants are greater than both the delays (i.e., TF,TR > Op,OR). • Trend 2 : Following the initial spike and exponential return, there is a first-order response and then a slow rise to the final steady state. Figure 2.11, which shows the output at various stages of the recycle process, is used in the following explanation. The intial lead response is caused by one pass through the forward path GF (Curve A in Figure 2.11). The first-order response is the first recycle effect passing through GF (Curve B). The slow rise to steady state is the repeated recycle effect on Gp (Curves C and D). This trend is caused when the recycle parameters (TR,0R) are greater than the forward path time constant (rp), i.e.,r#, OR > Tp. K,, = K„ = 1, E = 7.5, Tp = 5, T„ = 10, 6F = 5,.9R = 10, r = 0.5 0.8 0.6 0.2 ' i i i i D y B 1 • • / / A • • • 1 0 25 50 75 100 Time Figure 2.11: Output at Various Stages in the Recycle Process for a Leadlag F O P D T and F O P D T Recycle System • Trend 3 : is characterized by a step-wise increase to steady state. Except for the initial response, each subsequent step has a minor lead-lag response. This trend occurs when the recycle path delay (OR) is dominant and the forward path parameters (TFJ 9p) are greater than the recycle path time constant(r^), i.e., OR > (Op, Tp) > TR. Chapter 2. Recycle Dynamics and Modeling 21 Note, Tp) means that dp and TF can take on any value with respect to each other (i.e., 9p > Tp or rp > 9p). • Trend 4 : is also characterized by a step-wise rise to steady state. The shape of the steps are also a lead-lag type response. In this case, each lead-lag step occurs before the forward path reaches the intermediate steady state. The condition for this response to occur is when forward path time constant (rp) is greater than the recycle path parameters (TR,9R), i.e., Tp > TR,6R. Both trend 3 and trend 4 are characterized by a step-wise behaviour and could be a subset of one another. However, because they each appear a number of times and at different conditions, they were categorized separately. The Bode plot of the trends did not add any additional information beyond the fact that there were resonant effects caused by the recycle. As a result, the Bode plots were not presented for the last two cases. However, they are available in the appendices with the corresponding step responses for completeness. 2.2.3 Inverse Response F O P D T and F O P D T M o d e l The transfer function models shown below for this system are similar to those of the previous section, that is: J f r f r i * + ! ) « - ' " G R = ^ (2.13) TpS + 1 TRS + 1 The resulting overall transfer function is: r (l-r)Kp(eps + l)(TRS + l)e-e-s O V E R M " TFTRS* + (Tf + rR)s + 1 - rKFKR(eps + l ) e - ( « F + « * ) . [ Z ' i V The difference between the two systems is in the value ofthe lead time constant (EF). For an inverse response system, the lead time constant is less than 0 (i.e., eF < 0). An inverse response system is one in which the initial output direction is opposite to the direction of the final steady state. This behaviour may be unusual but it does occur in many real chemical processes (Ogunnaike and Ray 1994, page 225). Examples of inverse response systems are1: • Boiler Drum Level Control: the level in the drum is a combination of the liquid and bubbles/foam on top of the liquid. When cold water is introduced, the liquid level initially drops due to the collapse of the bubbles. 1The examples were taken from Ogunnaike and Ray (1994), pages 229-231. Chapter 2. Recycle Dynamics and Modeling 22 • Distillation Column Reboiler Section : An increase in steam duty causes an increase in vapor rate through the column. This causes increased frothing and overflow of liquid from the upper trays back to the reboiler section, thus the liquid level initially rises. Eventually, the liquid level falls as the increase in heating takes over. • Exothermic Tubular Catalytic Reactor : When the feed temperature is increased, there is increased conversion at the entrance. This depletes the amount of reactants reaching the exit, thus less heat is produced at the exit and the exit temperature falls. However, the catalyst bed temperature eventually rises, so eventually the reactants reaching the exit region arrive at a higher temperature promoting an increase in reaction rates and ultimately a higher exit temperature. In the current recycle system, eF was chosen to have a value —0.5 x rF which should be representative of industrial processes. Example step responses are shown in appendix C. Since GF is an inverse response system, the initial output of the recycle system will also be an inverse response. For this system, it was not possible to uniquely specify conditions for each trend. Each trend can appear at several different conditions. The step responses were loosely classed into the five trends shown in Figure 2.12. Note, the trends are again offset so they can be easily distinguished and they all reach the same steady state value after sufficient time has passed (this result is not shown in the figure). • Trend 1: this is the simplest of the responses. It is identical to a first order process with an inverse response. This trend appears when both the process time constants are dominant, i.e., (TF,TR) > (9f,9R). • Trend 2: this response has a step-wise trend to steady state. The beginning of the second step is marked by a double bump, this behaviour is explained later on in this section. This trend occurs when the time delays are dominant, i.e., (9F,9R) > (TF,TR). • Trend 3: this trend has an initial response similar to Trend 1 followed by a slow rise to steady state. This response appears when either: o The forward path time constant (TF) is dominant, and the recycle path time constant (TR) and the forward path time delay (9F) are greater than the recycle path delay (9R) i.e. TF > (TR, 9f) > 9R o The recycle path time constant (TR) is dominant, and the forward path time constant (rF) is greater than both time delays, i.e. TR> TF > (9f,9R) • Trend 4: is similar to Trend 1, however, it has a double bump like Trend 2 appearing in the middle. The double bump in this case is stretched over time. This trend can occur for either of the following three conditions: Chapter 2. Recycle Dynamics and Modeling 23 o CL ID 2 Kp = K„ = 1, 8F= -6.25, -0.5, -5, -12.5, -1.25, TF = 12.5, 1, 10, 25, 2.5, XR = 5, 1, 10, 1, 10, 6F = 1, 1,10, 5,10, 6R = 1,10, 7.5, 25,10, r = 0.5 -i—i—i—i—i—i—i i i | Time Figure 2.12: Step Responses for a Inverse Response F O P D T and F O P D T Model Recycle System o The forward path time constant is dominant, and the time delays are greater than the recycle path time constant, i.e., rp > (OF, OR) > TR-o The time delays are greater than the time constants and the forward path parameters are greater than the the recycle path parameters, i.e., 9F > TF ^ OR > TR o The recycle path delay is dominant, i.e., OR > 0p,Tp,TR • Trend 5 is similar to Trend 2, but the steps are not as well defined and it has a smaller double bump behaviour at the start of the second step. This response can occur for either of the following conditions: o The forward path delay is dominant and the recycle path time constant is greater than the recycle path delay, i.e., Op > Tp, {TR > OR) o The recycle path time constant is dominant and the recycle path delay is greater than or equal to the forward path time constant, i.e., Tr > Op, (OR ^ Tp) For this system, some of the trends appear to be very similar and could be subsets of one another. However, since they appear frequently and for differing parameter combinations, Chapter 2. Recycle Dynamics and Modeling 24 the trends were categorized separately. As noted earlier, three of the trends (Trend 2, 4, and 5) have a double bump behaviour occurring at the beginning of the second step. This double bump is a result of the inverse response system in the forward path. Figure 2.13 is a magnified view of Trend 4; the plot also shows the output from GR, and the output from Gp when only the recycle effect is passing through Gp. Figure 2.13: Magnified View of Trend 4 and Intermediate Responses for a Inverse Response F O P D T and F O P D T Model Recycle System When a positive step input is made to the system, the process output initially moves in the negative direction (referred to as F l in Figure 2.13). The process output eventually switches direction and moves in the positive direction to steady state. When the initial negative response (Fl) passes through the recycle process (GR) the output is also in the negative direction. When this negative output from GR passes through the forward process GF it causes another inverse response, the initial movement is in the positive direction (referred to as F2). It is this movement in the positive direction which causes the first bump up appearing in the process output. Chapter 2. Recycle Dynamics and Modeling 25 0 10 20 30 40 50 60 70 Time Figure 2.14: Magnified View of Trend 4 and Intermed. Responses for a Inverse Response F O P D T and F O P D T Model Recycle System with Recycle Fraction = 0.5 Figure 2.15: Magnified View of Trend 4 and Intermediate Responses for a Inverse Response F O P D T and F O P D T Model Recycle System with Recycle Fraction = 0.75 Chapter 2. Recycle Dynamics and Modeling 26 The bump down in Trend 4 is a combination of two factors: (a) The first inverse response (Fl) changing direction, which means GR now has a positive movement which in turn causes GF to generate another inverse response (referred to as F3) initially in the negative direction and (b) The second inverse response (F2) changing direction and moving in the negative direction. The final movement up is caused by the inverse response (F3) changing direction and moving positively. The size of these intermediate bumps is determined by the values of: • The recycle path time constant (TR). The larger the the recycle path time constant the smaller the bumps. • The lead time constant (eF). The larger (i.e. more negative) the lead time constant, the larger the inverse response which means larger intermediate bumps. Intuitively, one would expect the recycle fraction (r) to also affect the magnitude of the intermediate bumps. As the recycle fraction increases, more of the inverse response is passed through the recycle path which should give larger intermediate bumps. Figures 2.14 and 2.15 shows the effect, of different recycle fractions (r = 0.5,0.75) on four streams. The streams shown are the process output, the recycle process output, and the output from the GF when only the recycle effect is passed through GF before and after the recycle stream is removed. As the recycle fraction is increased, there is a larger inverse response passed to the recycle path giving a larger negative response (Point A). When this recycle output passes through GF it gives a large inverse response (marked B ) . However, since the recycle fraction is larger, more of the stream is recycled which means the effect on the output is reduced (marked C). Thus the effect of recycle fraction is masked. 2.2.4 Second-Order-Plus Dead-Time-Model ( S O P D T ) and F O P D T Model The equations used for this system are shown below: GF — rFs2 + 2TF(S + 1 KFe-0FS GR = KRe-6*1 TRS + 1 (2.15) The overall transfer function is: {l-r)KF{TRs + l)e-°*' (2.16) overall — (rp2 + 2rF(s + 1){TRS + 1) - rKFKRe-VF+0*y IS The damping factor (£) was specified to be less than 1 so that the response would be oscillatory. The damping factors were arbitrarily set at £ = 0.75, 0.5, 0.25, and 0.1. Chapter 2. Recycle Dynamics and Modeling 27 Damping factors of one or larger would give a critically/overdamped output which is very similar to a first order process and the recycle output would have results similar to those in section 2.2.1. Example step responses and the corresponding Bode plots are shown in Appendix D. The responses were divided into four categories: • Category 1: Second-order response • Category 2: Fast initial rise time, slow rise to steady state • Category 3: Fast initial rise time, oscillatory rise to steady state • Category 4: Step-wise rise to steady state Each category has at least two different trends and they are shown in the following sections. 2.2.4.1 Category 1: Second-Order Response All the responses are typical of a second-order process. The responses are shown in Figure 2.16. KF = KR=1,TF=15,10, c^  = 0.75, 0.1, TR = 2.5, 5, 9F = 0R=10, 2.5, r= 0.5,0.25 & 0.8 V : j : I • 1.1. •.. . . . . I I i \ : : : \ «* Trend 2; \ i A ; ; ; -I : 1 : 1 \ \ 1 1 M if \ A ! V / : r;S."~~.'».j -^wi..4i. • -: 1 : 1 ; i i i \ A \ . . . . i -- ; i 4 I 1 < t A 1 \ / \ Trend 1; • i: / l: / -1 ;/ 1 / ' / . . 1A 1 , , . . • I 1 1 1 1 1 1 1 1 1 1— r 0 25 50 75 100 125 150 175 200 225 250 275 300 Time Figure 2.16: Category 1 Step Responses for a SOPDT and F O P D T Model Recycle System • Trend 1: This is similar to a critically/overdamped second-order process. This be-haviour occurred when the forward path time constant (TF) was dominant (i.e., Chapter 2. Recycle Dynamics and Modeling 28 TF > TR, 9R, 6f) and the damping factor (£) was 0.5 or larger. It is interesting to note that even though the forward path was underdamped, the recycle output did not contain any oscillation. This lack of oscillation or overshoot is caused by the presence of recycle. Figure 2.17 shows the effect of increasing recycle fraction on the output, the recycle stream reduces or removes the oscillation and thus smoothes the process output. KF = KR=1,TF=10, ^  = 0.35, TR=10,eF= eR = 1, r= 0,0.3,0.5,0.75 T Time Figure 2.17: Effect of Recycle Fraction on the Underdamped Second Order Response • Trend 2: is a typical second order underdamped response. This behaviour occurs when the damping factor is less than or equal to 0.25 and either of the following conditions hold: o The forward path time constant is dominant (i.e., Tp > TR, 0R, 9p) o The time constants are dominant with the recycle path having the dominant time constant (i.e., TR> Tp > max(9p,6R)) o The forward path parameters are dominant with the forward path delay being the largest value (i.e., 9p >Tp> (TR,9R)) Chapter 2. Recycle Dynamics and Modeling 29 2.2.4.2 Category 2: Fast Initial Rise Time, Slow Rise to Steady State All trends have a fast inital rise time to an intermediate value and then slowly rise to the final steady state value. The trends are shown in Figure 2.18. Kp = K„ = 1, TF= 10, 2.5, 2.5, £F = 0.75, 0.5, 0.5, tR = 25, 12.5, 12.5, 8F = 6R = 10, 5, 10, r = 0.5 0.8 0.2 —1—1—1—1—1—1—1—1—1—1 1 1 1 1 —1—I—1—1—|—l—l—l—l— — S —i—i—i—i—|—i—i—i—i—j—i—i—i—i— / 1 ; / / / t /• / / i / / • 1 / 1 /Trend 3 i ; / A / Trend 1 / j { T r e n d 2 i i / < / < / 1 / * / * / 1! i i / , ; / / | ; / Ii / 1 / 1 / 1 : ... i ... . 1 1 1 1 1 1 1 0 50 100 150 200 250 300 350 400 Time Figure 2.18: Category 2: Step Responses for a SOPDT and F O P D T Model Recycle System • Trend 1: This response occurs when either of the following conditions apply: o The recycle path time constant is dominant and the forward path time constant is greater than both time delays (i.e., rR> rF > (6F, OR)). o The time constants are equal and dominant (i.e., TF = TR> max(0F, OR)) o The forward path time constant is dominant and the forward path delay is greater than the recycle parameters (i.e., rF > 0F > (TR,0R)) • Trend 2: This trend overshoots and then decays an to intermediate steady state before the slow rise to the final steady state occurs. The conditions for this to occur are: o The recycle path parameters are greater than the forward path parameters and the recycle path time constant is dominant overall (i.e. r R > 0 R > (0F,TF)). In addition, the damping factor is equal to or larger than 0.5. Chapter 2. Recycle Dynamics and Modeling 30 o The forward path time delay is dominant and the parameters are ordered in the following manner Op > rR > 6R > Tp. • Trend 3: This trend is similar to Trend 2 but with a very small overshoot. The output reaches an intermediate value before rising to the final steady state value. This behaviour occurs when either: o The forward path time constant is dominant and the time delays are equal (i.e., rF > (Tr,6f = 6r)). o The forward path time delay is dominant and the forward path time constant is greater than or equal to the recycle path time constant (i.e., 6F > (6R,TF > TR)). The trends shown in Figure 2.18 are separated so that they can be easily distinguished, as a result they do not start at the same time. 2.2.4.3 Category 3: Fast Initial Rise Time, Oscillatory Rise to Steady State 1^  = 1^  = 1, Tf=2.5,2.5, c^  = 0.1, TR = 25,10, 8F = eR = 0,10, r= 0.5 1.0 0.8 0.4 0.2 1.1111111 111111111 i i i i i i i i i i i i hi —T— • 1 J.I..; <" ^ — ^ ,'l : l l 1 \ 1: ' ; M » V; • 1 : 1 A 1 \ \ \ ! \ i b \ . . !. . . • 1 / 1: I II !M ' i ii! ! ...ii] ! i • 1 1:1 ! 1:1 J II ! « I : : : • • I • • 1 • > • • 1 1 I I 1 1 1 1 1 1 1 1 1 1 1 1 1 0 25 50 75 100 125 150 175 200 225 250 275 300 Time Figure 2.19: Category 3: Step Responses for a SOPDT and F O P D T Model Recycle System The responses in this system are also characterized by an initial fast rise time, however, Chapter 2. Recycle Dynamics and Modeling 31 the eventual rise to steady state has oscillations. The oscillations may disappear or they may persist until the steady state is reached (see Figure 2.19). The responses shown in the Figure 2.19 are offset so the plots can be easily identified. The conditions for these response to occur are either of the following: • The forward path time delay is dominant (i.e., Bp > TF,TR, BR) • The recycle path time delay is dominant and the forward path time constant is greater than or equal to the recycle path time constant (i.e., BR > (Bp, Tp > TR)). • The recycle path time constant is dominant (i.e., TR > max{rp,Bp,BR)) and the damping factor is 0.25 or less. 2.2.4.4 Category 4: Underdamped Step Wise Rise To Steady State The responses in this category have a step-wise rise to steady state. Each step appears as an underdamped second order response (i.e., the damping factor is less than one). The responses are shown in Figure 2.20, and again the responses are offset in time so they can be easily seen. KF = KR=1, TF=1, £^  = 0.75,0.5,0.25, TR=1, 9p=1, 6R = 40, r= 0.5 0.8 0.6 • i i i • i i i | 1 1 ' > - 1 r 1/ » / V / ! i\. i i i 1 1 • f : 1 t i i i i h — ' • • jl ! J hr • J v : • Response/ Response B Response C j • -1 1 1 I — 1 1 1 1 1 1 1 ' ' 1 1 1 • ' • ' 0 50 100 150 200 250 300 Time Figure 2.20: Category 4: Step Responses for a SOPDT and F O P D T Model Recycle System Chapter 2. Recycle Dynamics and Modeling 32 For these responses, the system contained a dominant path time delay (forward or recycle path). Additional conditions for the responses to occur are: • The recycle path delay is dominant (i.e., 9R > 9f,TF,TR). NO other parameter arrangements or specifications were apparent for this case. • The time delays are greater than or equal to the time constants, and the time constants are.equal (i.e., (9R,9F) >TR = TF). • The time delays are dominant and the recycle path parameters are greater than their forward path counterpart (i.e., 9R> 9F > rR> rF). 2.2.5 Overall Recycle Dynamic Summary The classifications ofthe responses in the preceding sections are subjective generalizations. The trend classifications were chosen based on the majority of the combinations giving a particular response. What this means is, there are some parameter combinations which may not meet a trend requirement (e.g., rF dominant) but may give that trend type response. In addition, some parameter combinations may meet the requirements of one trend but give the response of a neighboring trend. Comparing the classifications of the four recycle systems, all of the responses generally fall into three groups. The output can be: • The response of the forward path transfer function. • A fast initial rise time followed by a slow rise to steady state. The slow rise to steady state can be monotonic or oscillatory. • A step-wise rise to steady state. Thus, the responses of the two first-order-plus-dead-time recycle system define the main possible responses of the recycle system shown in Figure 2.4. The responses of the other three systems are subsets of the main responses. It was not possible to specify generally under what conditions the first two groups will appear; the conditions change according to the model structure and parameter values used. However, it was found that group 3 (i.e., the step-wise rise to steady state) will effec-tively occur when the time delays are dominant, specifically 9F + 9R > 6 x TF L. This is an extension of the condition reported by Scali and Ferrari (1997) that the recycle time delay has to be bigger than the forward path time constant to get the step-wise behaviour. As noted here, it is possible to get the step-wise behaviour even if 9R < rF provided that 1The condition for step-wise behaviour to occur was found from extensive simulation results Chapter 2. Recycle Dynamics and Modeling 33 ®F + 9R > 6 x rp. When the forward path time delay is sufficiently large, the forward path process has enough time to reach a steady state before the effect of the recycle process is felt. Table 2.1: Recycle Response of Control Interest Systems Foward Path Recycle Path Trend Page Comments F O P D T F O P D T 3 16 Step-wise trend to steady state Lead-lag F O P D T F O P D T 1,2 3,4 19 Lead jump may cause a problem Step-wise trend to steady state Inverse F O P D T F O P D T 2 5 23 Step wise trend to steady state The long delay before the slow rise to steady state may be an issue. SOPDT F O P D T Cat. 3 30 The oscillatory rise to steady state will definitely be a problem. Cat. 4 31 Step-wise trend to steady state. The main goal of this chapter was to try and classify the overall recycle dynamic responses. Also, it would allow the identification of responses that have the potential to be challenging control problems. Loosely defined, a challenging control problem is one that has a limitation in control system performance. Basically, the frequency range over which the control system is effective (i.e., the bandwidth) is limited. This limitation may be caused by a combination of the following: • An unstable system, that is, the system has right half plane (RHP) poles. Chapter 2. Recycle Dynamics and Modeling 34 • The system exhibits inverse responses (i.e., RHP zeros). • The presence of time delay, i.e., the LHP poles move to the imaginary axis, or the discrete poles move out toward the unit circle. The closer the poles to the imaginary axis or unit circle, the slower the system. As stated earlier, all of the recycle responses fall into the 3 groups. • Group 1: These responses are similar to the forward path process. The responses are usually either first order or second order with dead-time. These responses do not pose a control problem. • Group 2: The responses are characterized by a fast rise time followed by a slow rise to steady state. This group encompasses different responses, some of which can be approximated by a first order model without loss of control performance. Other responses have additional dynamics (e.g., oscillatory rise to steady state) that can cause control problems. • Group 3: The responses move to steady state in a step-wise fashion. The shape of the step is dependent on the system. The responses in this group may present a control problem. Looking at the individual recycle systems, the responses of interest are shown in Table 2.1. 2.2.5.1 Stability Though the recycle system is composed of two stable models, it is possible for the resulting overall process to be unstable. Of the four recycle systems studied, the two F O P D T system and the inverse response F O P D T / F O P D T system were always stable for the parameters tested. The remaining two systems (i.e., lead-lag F O P D T - F O P D T and the SOPDT-FOPDT) can go unstable depending on the parameter values. The overall transfer function for a recycle system with time delay always contains a denominator dead-time term. As an example, consider a recycle system composed of two F O P D T models, the overall transfer function model is shown below (eqn. 2.10, page 15): The denominator dead-time term makes finding the roots of the characteristic equation difficult. Even for the simplest recycle system composed of pure time delays (i.e., rF = TR — 0), the exponential term results in the characteristic equation having an infinite G = (l-r)KF(rRs + l)e-er (2.17) TFTRS2 + (TF + TR)S + 1 - rKFKRe-(EF+9*)S Chapt :er 2. Recycle Dynamics and Modeling 35 CO fl O O -d cd CO Pi o T J Pi O O co fl .2 fl JH H - H CO fl 03 S-l CD > O CO >> CO fl cd o CD fl cd > >> fl cd SH ,o CS CO CO IS S3 V fx" CD . fe + • I p Q P H O CO fl o T J fl O O fe, A l t. cd CO A ^ i + to ft. '2 i P C5 CO fl co fl O cu +3 ^ •TH T J . fl cu O .-H o cu o A A o -Q a cu -fl fl CU -fl +3 O co fe, 2 fl H -fl cu Q * S g O co & faO « ^ cd t- cu i-j - > • * I i CD .fl g -fl CO fl cd CD r ° "2 fa - f l bO Q D H O fa fad i T J cd CD CD cd CO CO cd O ' V fe, a" CD a cu -1-5 CO >5 CO CD -fl CD fl SH CO cd> & Id fl CO CD -fl CD + j CO fl O a A CO CD t-, cu CO &H cu > .s CD - f l A co fl T J fl O CD ;fl C M H O fl cd o «:-s ^ § ,-r >> O +5 . -H QS cd > . ^ T J CN Q fa o fa CD CO ! H CD > fl 0$ bO fl T J fl cd CD -fl fl CD - f l V fl CD S-l V -o cd cu tH cd co fl bJO cu - f l o A CO Id CO CO CD .fl co T J • SP J CO CD CO cd fl CO rH CU - f l + 2 + CD T J cd o -fl o CM i t CN + Q O H O fa Q D H o CO Chapter 2. Recycle Dynamics and Modeling 36 number of roots. The dead-time term can be replaced by a Pade approximation, but it may result in an unstable approximation (Hugo 1992). As a result, the denominator dead-time term prevents any general statement about stability. The system can be analyzed in the frequency domain to determine the process stability; however, each system with time delay has to be examined on a case by case basis. For systems without time delay, the denominator dead-time term vanishes. The gen-eral stability conditions for each system are shown in Table 2.2. The specifications and assumptions used to obtain the stability conditions are: (a) The process models (GF, GR) are always stable with a gain of 1 or less and (b) The recycle fraction only takes on values between 0 and 1 (i.e., 0 < r < 1). To determine the stability conditions, Routh stability analysis was used. The procedure for the Routh analysis can be found in Ogunnaike and Ray (1994, pages 489-494). 2.3 Modeling of Recycle Responses The step-wise response common to group 3 are the ones usually associated with recycle systems. The shape of the response is caused by the recycle effect occurring after the forward path process has almost reached a steady state. The first step is the response of the forward path process to the disturbance, in this case a step input. The subsequent steps represent each time the recycle effect acts on the forward path process. Thus, the output displays a periodic response. This periodic behaviour is very similar to that seen in seasonal time series models. According to Box et al. (1994), a seasonal time series is one that shows periodic behaviour (with period P). That is, similarities in the series occur after P basic time intervals. For example, the totals of international travelers shown in Figure 2.21 over a three year period show two seasonal patterns. Travel is the highest in the late summer months and the second highest in early spring for each year. The general structure of a seasonal model is: G { _ 1 } = frig-1-' + b2q-2-d + bPq-p-d + bP+lq-p-l-d 1 + axq-1 + a2q~2 + aPq~p + aP+iq-p-1 In difference equation format, the general structure appears as follows: y(k) = alV(k - 1) + a2y(k - 2) + a3y(k - P) + aAy(k - P - 1) + b1u(k-d-l) + b2u(k-d-2)+b3u(k-P-d) + b4u(k-d-P-l) (2.18) where d - represents a time delay. The distinguishing feature of a seasonal model is the jump or gap in the polynomial order. This gap can appear in either the numerator or denominator polynomial or both. For eqn. 2.18, the gap appears in the numerator Chapter 2. Recycle Dynamics and Modeling 37 (• • • + b2q-2-d + bPq-p-d + ) and the denominator ( h a2q 2 + aPq p + ). Thus y(k) is based on sequential past values and values P basic time intervals in the past. 300 250 200 Figure 2.21: Totals of International Travelers. Figure Reproduced from Box et. al (1994) Using the two F O P D T model recycle system (page 15) as an example, the overall transfer function is: C = (l-r)KF(rRs + l)e-0FS TFTRs2 + {TF + TR)s + l-rKFKRe-^+e^s { ' ; If the above transfer function is converted to the time domain and then discretized using the finite difference method, the resulting discrete transfer function has the following structure: y(k) = M - 1 - ' * + b2z-2~dF u(k) 1 - ciiz-1 + a2z~2 - a3z-(dF+dR) This equation has a seasonal model structure appearing in the denominator, that is, there is a sequential increase in the powers and then there is a jump in the powers to dF + dR. The seasonal model structure was also confirmed by approximating the discretized transfer function. The approximation was obtained by taking the z-transform with zero order hold (ZOH) of the forward (GF(s)) and recycle (GR(s)) processes individually. The discretized transfer functions (GF(z~l), GR{z~1)) were then placed in the configuration shown in Figure 2.4 and the overall transfer function derived. The procedure is shown Chapter 2. Recycle Dynamics and Modeling 38 below using a two FOPDT recycle system. The general discrete form of the forward and recycle path processes are: GF(z-1) bpz -\-dF GR{z~l) = bRz -\-dR 1 — CLpZ"1 " V ' 1 — CLRZ~ The resulting overall equation for the process block diagram in Figure 2.4: y(k) = (1 -r){bFz-^ - bFaRz-2-d?} u(k) 1 — (aF + aR)z~l + aFaRz~2 — rbFbRz~2~dF~dR (2.21) This equation also has a seasonal model structure. The only difference between this equation and the one derived by the finite difference method is the jump in powers. For the finite difference method the power jumps to dF + dR while in this method it jumps to 2 + dF + dR. The extra increase in power in the z-transform approach is caused by the zero order hold in both Gp{z~l) and GR(z~l). Process Parameters : Kp = KR = 1, xF= 5, xR = 5, 9F= 6R=10, r= 0.5, T s = 1 Figure 2.22: Comparison of Recycle Models To show how well the finite difference method and z-transform approach model a recycle process, the methods were applied to a recycle process composed of two FOPDT models. The FOPDT model parameters were KF = KR = 1, Tp = rR = 5, and 9F = 9R = 10. Since both methods involve discretization, a sampling time (Ts) has to be selected. Chapter 2. Recycle Dynamics and Modeling 39 According to Astrom and Wittenmark (1997), a good choice of sampling time is in the range of r/4 to r/10. In the case of the recycle process there are three choices for r, the forward path time constant (rp), the recycle path time constant (TR), and the overall process time constant (rp). To ensure proper capture of any intermediate dynamics, the smallest time constant (i.e., minirp, T#)) should be used to select the sampling time, for the current example a Ts — rp/4 = 1 was selected. The response of the finite difference approach, the z-transform approach, and the true process are shown in Figure 2.22. Both methods give a good approximation to the true process, with the z-transform approach giving the better estimate. Process Parameters : K = K = 1, T = 5, z = 5, 9 = 0 o = 10, r= 0.5, T e = 3 Time Figure 2.23: Effect of Increased Sample Time on Recycle Model Comparison There are also a few issues that have to be mentioned: • First, the accuracy of both methods depend on the sampling time. As with any discretization, the smaller the sampling time the better the model represents the true process. Figure 2.23 shows the results of using a larger sampling time (i.e., Ts = Tp/10 = 34/10 = 3). While the models do follow the general trend of the process, there is a noticeable difference in modeling accuracy when compared to the Chapter 2. Recycle Dynamics and Modeling 40 Ts = 1 case. • While the z-transform approach does provide a better approximation, the procedure has an inherent modeling error. The recycle process consists of both Gf and GR; when discretizing the process, the ZOH should appear before the process. Thus, the recycle process sees a constant input between samples. However, the individual processes see a continuous input because of the following: o The forward path process (Gp) sees a continuous input obtained by the addition of the continuous output from the recycle path process (GR) and the constant input to the process from the ZOH. o The recycle path process receives a continuous input from the forward path process output. In the z-transform approach two ZOHs are included as part of the process, that is, there is a ZOH in the forward and recycle path. By having two ZOH inside the loop two modeling errors are introduced: o The ZOH in the forward path means the input to Gp is constant instead of being continuous. o The ZOH in the recycle path means the input to GR is constant. The two ZOH included in the process model can amplify transient dynamics. This amplification can be seen if a step change is applied to the recycle system composed of an inverse response F O P D T process in the forward path and a F O P D T process in the recycle path. The initial inverse response from Gp may disappear (i.e., change direction) between sampling periods; however, the ZOH in the recycle path holds the inverse response over the sampling period. Thus, the impact of the inverse response on the recycle path is larger when compared to the true process. The z-transform modeling errors can be reduced by using a smaller sampling time for the discretization. If the discretized system model shows unusual dynamics, the sampling time used for the discretization should be reduced to re-confirm the modeling response. 2.3.1 Seasonal Mode l Identification To easily identify the seasonal model structure shown in equation 2.22, a Matlab program using least squares was written. V(k) = (blZ-l+b2Z-2)z-dr u(k) 1 - axz~l - a2z~2 - a^z'2-^-^ Chapter 2. Recycle Dynamics and Modeling 41 First, equation 2.22 is written in difference equation format shown below: y(k) =biu(k -l-dF) + b2u(k - 2 - dF) + axy(k - 1) + a2y(k - 2) + asy(k -2-dF-dR) (2.23) The general formula for least squares is ft = (XTX)~1XTY where /3 is the parameter estimate vector for the model, Y is the output data matrix, and A" is a matrix combining the input and output data in a specific order. The matrices are shown below for the case with M data points (i.e., k = 0,1, • • • , M — 1). 3/(2 + dF + dR) y(3 + dF + dR) y(4 + dF + dR) y(M-l) 0 = h b2 Cli a2 « 3 Y = u(l + dR) u(dR) y(l + dF+dR) y(dF + dR) y(0) u(2 + dR) u(dR + l) y{2 + dF + dR) y{l + dF + dR) y(l) I X = u{M -2-dF) u(M - 3 - dF) y(M - 2) y(M - 3) y(M - 3 - dp - dR)_ The precision of the parameter estimates is obtained from the covariance matrix of the parameter estimates (XTX)~1a2, where <r2is an estimate of the output variance and is calculated using the following equation: .2 = ZY*Y-ilXY n — p where n is the number of data points and p is the number of parameter estimates (Guttman et al. 1982; Bacon 1987). The lower the values in the parameter covariance matrix the better the parameter estimates. The Matlab program requires the specification of the following: • The numerator order (without time delay included) and the time delay (dp) if any. For eqn. 2.22, the numerator order is 2 and the time delay is dp. • The first denominator order (i.e. the order of the polynomial before the seasonal Chapter 2. Recycle Dynamics and Modeling 42 jump). For eqn. 2.22, the first denominator order is 2. The denominator dead-time (i.e., the length ofthe seasonal gap). For eqn. 2.22, the denominator dead-time is dF + dR . • The second denominator order (i.e., the order of the polynomial after the seasonal jump). The first term after the seasonal jump is assumed to be order 0. For eqn. 2.22, the second denominator order is 0. The values of the polynomial order (numerator and denominator) and the time delays (dF and dF + dft) are chosen by trial and error to obtain a model that fits the data. 1.0 • co 0.8 + 0.6 + 0.4 + 0.2' .—"s //.... II II 11 True Process Identified Process f 25 50 75 100 125 150 175 200 Time Figure 2.24: Comparison of Identified Model to True Process The algorithm was tested on a two F O P D T recycle system. The system was specified to give a step-wise response, and the recycle parameters used were KF = KR = 1,TF = TR = 5, BF = OR = 10. The system was excited using a pseudo random binary sequence (PRBS) and the model fitted to the input-output data using the Matlab program. The model structure obtained was: Chapter 2. Recycle Dynamics and Modeling 43 The model parameters are bi = 0.0651, a\ = 0.8802, an = 0.0546, d\ = 9, d2 = 25 and the sampling time was Ts = 1. A comparison of the model and the true process is shown in Figure 2.24, the model obtained provided a good fit to the process. The algorithm does have one potential problem. Typically when using least squares, the parameter covariance matrix is used to check the model fit. The lower the matrix values, the lower the parameter variances and the better the model fit. However, for the seasonal model identification algorithm, the parameter covariance matrix always gave low values irrespective of the model structure. The consistently low parameter covariance values were caused by biased parameter estimates, thus the covariance matrix was not sufficient to determine if the model fit was adequate. Step response tests had to be performed to check the model against the true process for each model structure tried. 2.4 Recycle Structures Recycle behaviour is also affected by the recycle configuration. The recycle structure used in this study was the simplest recycle structure. In an industrial process (e.g. a pulp and paper process), there are usually multiple recycle loops. These loops may be sequential recycle loops or they may be nested within one another. More complex structures, such as those shown in Appendix E , are not included in this study. When this recycle dynamic study was initiated, the original intention was to examine the effect of model order on the dynamic response, and then to repeat the study using the different recycle structures shown appendix E . These configurations were chosen as it was felt they best represented industrial systems; more complex configurations could be then treated as combinations of these arrangements. However, when the simple recycle structure and two F O P D T models recycle system was studied,.four parameters were chosen to be varied (i.e., TF,TR,9F,9R). This led to 77 possible combinations which were run for different recycle ratios. For more complex systems, the number of different combinations was even larger. Thus, the alternate recycle configurations were not studied. It should be mentioned that more complex recycle structures do not necessarily imply more complex dynamics. There may be a cancellation of dynamics, i.e., the dynamics of one recycle loop may cancel out the dynamics of another loop. Cancellation of process dy-namics was seen with the SOPDT-FOPDT system where the presence of recycle removed the output oscillation from the under-damped SOPDT process. The structure of the recycle system may be simplified by approximating one of the recycle loops by a simpler process, for example a first or second order model. Alternatively, the output from the complex structure could be similar to the output from a simpler Chapter 2. Recycle Dynamics and Modeling 44 configuration (e.g., the simple recycle configuration shown in Figure 2.4). Thus the model used to represent the process could be of a lower order. 2.5 Conclusions The main purpose of this chapter was to try and categorize the overall recycle dynamics, and identify significant behaviour from a control viewpoint. Four systems were selected based on process models found in the chemical industry. All of the systems had a F O P D T model in the recycle path. The four forward path models used were a F O P D T , a lead-lag F O P D T , an inverse response F O P D T , and a SOPDT model. The study was performed using various combinations of parameter values for TF,TR, 9f, and 9R. It was found that all of the recycle responses fall broadly into 3 groups: • Group 1: A forward path process response • Group 2: A fast initial rise time followed by a slow rise to steady state. • Group 3: A step-wise response to steady state. It was not possible to specify generally under what conditions the first two groups will appear, since the conditions change according to the model structure and parameter val-ues used. However, Group 3 was found to occur when the time delays are dominant, specifically 9F + 9R > 6 * rF. Two groups were identified as having responses that maybe challenging for control. Group 2 contains responses that have an oscillatory rise to steady state and Group 3 has responses that have a step-wise rise to steady state. It was also shown using the finite difference method that recycle dynamics can be represented by a seasonal model structure. A good approximation of the recycle process was obtained obtained by discretizing the forward and recycle path separately with a zero order hold, and then deriving the overall transfer function using the standard recycle block diagram. A simple algorithm using least squares for identifying seasonal model structures was also presented. Chapter 3 Recycle Control Traditionally, control of plants has been done by applying control to the individual process units. However, such an approach may not work well in plants with recycle. The presence of recycle streams alters the process dynamics when compared to the system without recycle. The process gain and time constant are larger and there is increased sensitivity to disturbances. The changes in the process dynamics and poor control performance are caused by the interactions between units. To dampen the interactions, the typical solution has been to use large surge tanks, however, this practice is expensive. In addition, large inventories of chemicals require safety and environmental considerations which can increase the cost. The obvious solution has been to use better control. As a result, there has been more interest in developing recycle control strategies. In the previous chapter, recycle dynamics were studied and two groups were identified to have responses that may present control problems. This chapter complements the previous chapter by looking into control strategies. The objectives of this chapter are: • To evaluate PI control of recycle processes • To review the current model based recycle control strategies and propose another model-based approach. • To compare existing model-based (MB) control methods and the proposed approach using a generalized predictive controller (GPC). • To compare the use of a first order model and the proposed approach for M B recycle control. This chapter is divided into five sections. The first section reviews the current work on recycle control. The next section evaluates PI control of recycle processes. Then model based recycle control strategies are covered and another model for MB control is proposed. Existing model-based methods and the proposed approach are compared using G P C with PI control used as a benchmark. Implementation issues are then discussed, and then use of a first order model for recycle control is covered. 45 Chapter 3. Recycle Control 46 3.1 Literature Review Control system design requires the specification of • control objectives • control variables • measured variables • manipulated variables • a structure connecting the measured and manipulated variables (Nishida et al. 1981). The typical approach has been to apply control to individual process units. While information is readily available on control design for units in series, it is unclear how to apply these methods to recycle systems (Belanger and Luyben 1997). According to Luyben (1993b), a practical procedure has yet to be developed for effective control system design for recycle processes. The presence of recycle introduces control problems ranging from under-damped sys-tems to inverse responses. In addition, recycle can influence control tuning and strategy. Kapoor et al. (1986) reports that distillation tower time constants obtained by lineariza-tion can be substantially in error especially for high purity towers. They attribute the error to the positive feedback produced by the tower's inherent recycle structure (i.e., the column internal flows). Papadourakis et al. (1987) show that the relative gain array (RGA) for an isolated unit may not give a correct measure of the steady-state interactions. Because recycle can have a significant effect on the R G A value, the R G A calculation for a unit should take into account the coupling of the unit with the rest of the plant. Occasionally, the effects of recycle can be beneficial. Dimian et al. (1997) gave an example of a stand-alone column that cannot be regulated by multiple single-input-single-output (SISO) controllers. However, when the column was placed in the recycle structure of the plant, control was achieved with the same multiple SISO controllers. Dimian et al. conclude that recycle loop interactions can be exploited to obtain a plant-wide control structure which would be impossible to achieve with stand-alone units placed in series. Time delay in recycle processes also causes problems with controller design. As shown in eqn. 2.2, a dead-time term appears in the denominator. As most design techniques re-quire a rational denominator, the dead-time term can be replaced using moment matching or a Pade approximation. However, these methods may result in an unstable approxima-tion (Hugo 1992; Hugo et al. 1996). Hugo (1992) suggests using a Taylor series expansion (minimum 3rd or 4th order) to approximate the original transfer function (eqn 2.2). This Chapter 3. Recycle Control 47 method gives a series expansion that shifts the denominator dead-time terms to the nu-merator and allows standard analysis techniques to be used. The approximation gives a higher-order model, but Hugo et al. (1996) believe this is a minor limitation given today's computing power. Control structures applied to recycle processes can introduce an additional problem. Some control structures may cause large changes in recycle flow-rates for small input changes, Luyben (1993c) called this behavior the "snowball effect". The snowball effect is really an indication of how much the recycle flow-rate must change to compensate for a given disturbance. The snowball effect is a steady-state property and has nothing to do with dynamics (Luyben 1994; Luyben et al. 1996). However, it is possible during a transient for recycle flows to be larger than the final steady-state value (shown in Luyben et al. 1996). Obviously, the recycle system has to be designed to handle such occurrences for the control system to be effective. Luyben (1993c) studied a more complex process with two simultaneous reactions (A —» B —> C) and two distillation columns with material recycle. He proposed the following generic rule for recycle systems: "It is important in any recycle process to fix a flow somewhere in the recycle loop. Some manipulated variable must be found to effectively set the flow." Luyben (1993d) studied an even more complex system with second order kinetics and two recycle streams. He showed the generic recycle rule prevented the snowball effect. Several control structures were tested, and configurations that violated the generic recycle rule did not work or exhibited the snowball effect. Later, Luyben used mathematical models to show that the generic recycle rule prevented snowballing compared to the other conventional structures. Wu and Yu (1996) state that there are two ways to handle disturbances, the disturbance can be absorbed by one unit or shared between units. Poor disturbance handling can lead to excess demand on the capacity of an individual unit. This may cause process variables to hit constraints or the snowball effect. As an example, one of the Luyben (1993d) configurations was analyzed. The snowball effect was removed but disturbances were absorbed by one unit. According to Wu and Yu, using only one unit to absorb disturbances can lead to limited rejection capability (i.e., a smaller disturbance handling range). They propose the use of a balanced structure in which the disturbance load is shared by all units. The resulting structure was able to handle a wider disturbance range with control performance comparable to the Luyben strategy. Kumar and Daoutidis (2002) showed that recycle systems exhibit a time scale sepa-ration in dynamics whenever the recycle flow was significantly larger than the feed and product flows. Kumar and Daoutidis then proposed a two level control framework to sys-Chapter 3. Recycle Control 48 tematically exploit this two time scale behaviour. Distributed control was used to handle the control objectives for the individual process units in the fast time scale, and super-visory control was used to meet the control objectives of the overall recycle network in the slow time scale. The controller design approach was shown using a simulation of a reactor/distillation column system with material recycle. Taiwo (1986) uses a recycle compensator to simplify and improve the process dynamics. The compensator counters the positive feedback effect and allows the controller to be designed for the forward path alone (i.e., as if the process was without recycle). If the compensator is physically realizable (i.e. proper and causal), the presence of recycle can be eliminated. When the compensator is not causal, a causal approximation can be used; however, the effectiveness of the compensator is reduced (Scali and Ferrari 1997). Because the compensator reduces/removes recycle effects, it can handle situations when the recycle time delay is larger than the forward path time constant. Obviously, the presence of uncertainty affects control performance. As uncertainty increases, the benefits of the compensator decrease. However, the use of the compensator has been shown to be robust even with fairly high uncertainty (Scali and Antonelli 1995; Scali and Ferrari 1997). Recently the compensator has been applied to multi-input, multi-output (MIMO) processes (Taiwo and Krebs 1994; Taiwo and Krebs 1996; Scali and Ferrari 1997). The recycle compensated MIMO system gave a better performance than the uncompensated system. In the literature, the methods for controlling the whole recycle process are all model based. PI control was used but only on parts of the recycle loop, i.e., either on units in the forward or recycle path, never on the recycle process as a whole. In the next section, the use of PI control on the recycle process is studied. 3.2 PI Control of Recycle Processes In the chemical industry, the most commonly used controller is the PID controller. The PID controller can handle many control problems and more than 95% of the control loops are of the PID type (Astrom and Hagglund 1998). One reason for the widespread acceptance of the PID is that the algorithm is well understood by operations personnel and control engineers. Since PID control is so prevalent, there is a good chance that it is being used on recycle processes. In the previous chapter, recycle systems were studied to identify responses of control interest. One such response was the step-wise trend to steady state which occurred when ®F + QR > 6 x TF- According to Astrom and Hagglund (1998), whenever there is a dominant time delay (i.e., 9 > r) more advanced/sophisticated control than PID should Chapter 3. Recycle Control 49 be considered. Obviously, the performance of the PID controller will be affected by the delay in the process. Since the recycle process contains two delay terms (i.e., one in the forward path and one in the recycle path), the question that arises is how significant an effect have the delays on the PI performance? To determine the effect of the individual delays on the controller performance, PID control was applied to a recycle process. The derivative term of the controller was set to zero (i.e., = 0) for two reasons: (a) derivative action does not help much for pro-cesses with dominant time delays (Astrom and Hagglund 1998) and (b) preliminary tests with derivative action did not improve controller performance significantly. The resulting controller was PI controller with the following structure: Gc(s) = Kc(l + — The recycle process had the following parameters 'Tp = TR = 5, r = 0.5, and the process was discretized using a sampling time of TS = 1. The starting point for the continuous time delays was dp = OR = 1. Two cases were studied: • Increasing Op, OR held constant. • Increasing OR, Op held constant. To help explain what happens when either of the time delays is increased, the seasonal model representation of a recycle process shown in eqn. 3.1 and discrete pole/zero plots are used. In eqn. 3.1, dp and dR are the discrete forward and recycle path time delays respectively. { ) u(k) 1 - alZ~l - a2z~2 - a3z-2-d?-dn  { j Figures 3.1 to 3.3 show the pole-zero plots for the case when OR = 1 and Op = 2,10, and 20 respectively. In the figures, the poles are shown by crosses and the zeros by circles. As Op is increased, the number of zeros at the origin increase which means the time delay of the recycle process is increasing. The number of poles also increase which means the order of the process (i.e., the denominator order of equation 3.1) has increased. In addition, the poles also move outward toward the unit circle which indicates the process is getting slower. The pole-zero plots for the case when Op = 1 and 0R = 2,10, and 20 are shown in Figures 3.4 to 3.6 respectively. For this case, the number and location of the poles are the same as the poles in the case when Op was increased. This means the process order is the Chapter 3. Recycle Control 50 X ( 3 • \ ; .j.JT\ X J -1 -0.75-0.5-0.25 0 0.25 0.5 0.75 1 Real Part 1 0.75 0.5 « 0.25 g 0 CS E -0.25 -0.5 -0.75 -1 11 -1 -0.75-0.5-0.25 0 0.25 0.5 0.75 1 Real Part Figure 3.1: Pole Zero Plot - 8F = 2,6R = 1 Figure 3.2: Pole Zero Plot - 9F = 10,0R = 1 1 0.75 0.5 re 0.25 CL ra o ' r a ra E -0.25 -0.5 -0.75 -1 1 1 1 1 ...j L,-::::i x > ; - • '• X X X x ; x \ f X: i , ; ; f 21 V X .si t • • • ^ \ >< 7 X . : \ iX ; '!•. i x X xi X / i I 1 -1 -0.75-0.5-0.25 0 0.25 0.5 0.75 1 Real Part Figure 3.3: Pole Zero Plot - 8F = 20,9R = 1 Chapter 3. Recycle Control 51 -•i 1 1 x | 12 A i 1 \ — x \ :./7^ .vi x ! - • \ | ; ••--1 - 0 . 7 5 - 0 . 5 - 0 . 2 5 0 0.25 0.5 0.75 1 Real Part 1 0.75 0.5 n 0.25 0. eg o '•> cc E -0.25 -0 .5 -0.75 -1 I 1 -1 - 0 . 7 5 - 0 . 5 - 0 . 2 5 0 0.25 0.5 0.75 1 Real Part Figure 3.4: Pole Zero Plot -0F = 1,9R = 2 Figure3.5: Pole Zero Plot-9 F = 1,9R = 10 1 0.75 0.5 « 0.25 Q. to c CC E -0.25 -0 .5 -0.75 -1 I 1 I 1 1 1 i i i 1 1 i x / . . ; X / :x i X x x \ i xi 12 rt\ X .si „ j x X KJ H— X/i X / \ i x X X - • ; \ x x x j 1 • ; -1 - 0 . 7 5 - 0 . 5 - 0 . 2 5 0 0.25 0.5 0.75 1 Real Part Figure 3.6: Pole Zero Plot - 9F = 1,9R — 20 Chapter 3. Recycle Control 52 same in both cases. The difference between the cases lies with the zeros at the origin. For this case, the number of zeros remain constant indicating that the process time delay is constant. Thus, 6F affects the process time delay and both 9F and 9R affect the process speed of response. To ensure the controllers were tuned to provide the same level of performance (i.e., robustness) among the cases, the controllers were tuned using the maximum peak criteria in the sensitivity functions. Five sensitivity functions were used: 1. Loop Transfer Function (L) - is the open loop transfer function GK where G and K are the process and controller transfer functions. The magnitude of L has to be less than 1 after the crossover frequency (UJC), the frequency where \L\ first crossesl from above (i.e., \L\ < 1, V co > uoc). 2. Sensitivity Function (S) - relates the output disturbances to the output. The maxi-mum magnitude of S has to be less than 2 (i.e., |5| < 2). 3. Complementary Sensitivity Function (T) - relates the setpoint and the measurement noise to the output. The maximum magnitude of T has to be less than 1.25 (i.e., \T\ < 1.25). 4. Input Disturbance Sensitivity (SID) - relates the input disturbances to the outputs. The maximum magnitude of SID has to be less than 1 (i.e., \SID\ < 1). 5. Control Sensitivity (CSO) - relates the output disturbance to the control action. • For setpoint tracking, the maximum magnitude of CSO has to be less than 4 (i.e., \CSO\ < 4). • For disturbance rejection, the maximum magnitude of CSO has to be less than 2 (i.e., \CSO\ < 2). It is not possible for the respective sensitivity functions to have the same value across the different cases (i.e., L\ ^ L2 ^ ... # Ln, Si ^ S2 ^ ••• ^  Sn... where n is the number of cases). Typically l^ l or ICSOI were the limiting functions and thus the controllers were tuned to keep the maximum peak of these sensitivity functions below the values provided above. Detailed explanations of the sensitivity functions are provided in Skogestad and Postlethwaite (1996) and Goodwin et al. (2001). 3.2.1 Increasing Op The values of 6F used were 9F = 2,5,10,20, and 30. The largest 6F value is sufficient to give a step-wise trend to steady state. The results for setpoint tracking, and output and Chapter 3. Recycle Control 53 input disturbance rejection are shown in Figures 3.7, 3.9, and 3.10 respectively. The sen-sitivity function plot for the servo performance case is shown in Figure 3.8; the plot shows the maximum sensitivities among the five cases studied. Since the sensitivity function plots are all similar, no additional sensitivity plots are included in this section. As the delay is increased, there is a drop in control performance. This was also confirmed by the Integral of the Absolute Error (IAE) shown in Table 3.1. The drop in the control performance is the result of two factors: • Since 6p acts as the overall process time delay, PI performance will drop as the process time delay increases. • As 6p increases, the effective time constant ofthe process gets larger (i.e., the process becomes slower). To maintain the same performance, the controller has to be more aggressive resulting in larger control actions; however, the controller will be more sensitive to noise and disturbances. Tuning using the sensitivity functions provides a compromise between performance and disturbance attenuation. The effective time constant was specified as 25% of the settling time. By definition, the time constant of a process is the time taken for the process to reach 63.2% of the final steady state value and is an indication of how fast the process responds. This definition can be misleading when applied to recycle processes. In the previous chapter, recycle responses were found to fall into three groups: - The response of the forward path transfer function. - A fast initial rise time followed by a slow rise to steady state. The slow rise to steady state can be monotonic or oscillatory. - A step-wise rise to steady state. For responses two and three, it is possible for the initial rise or step to be greater than 63.2% of the final value, but the process could take five to ten times this r to reach steady state. As a result, an alternative measure of 25% of the settling time was arbitrarily chosen to be used as the effective time constant to indicate the speed of the response. Another measure that could have been used was the time constant of the first order model approximation of the recycle process. Since 9F causes a drop in performance because of the increasing process time delay, one obvious solution is to use a Smith-Predictor to provide dead-time compensation in conjunction with the PI controller. The simulations were re-run using the Smith-Predictor-PI and tuned using the maximum peak criteria in the sensitivity functions. The IAE results are shown in Table 3.2. Chapter 3. Recycle Control 54 Time (s) Figure 3.7: Effect of dp on PI Performance - Setpoint Changes Frequency (rads/sec) Figure 3.8: Effect of Op on Sensitivity Function - Setpoint Changes Chapter 3. Recycle Control 55 1.2 1.0 0.8 % 0.6 8 0.4 CD o S 0.2 0.0 -0.2 — 1 — 1 , , ,. „ ••— 1 1 i — i — i — ' — i • • i i \ ! • l | 11 —tf- eF = 2 t 01 I ~ o - eF = 5 • * \ \ \ * \ \ \ v\ \ \ - - - - - - 8 F = 10 - A - - eF = 20 - x - eF = 30 ! • H \ \ | • 1 ) — - i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — r 60 80 100 120 140 160 180 200 220 240 Time (s) Figure 3.9: Effect of Op on PI Performance - Output Disturbance 1.0 4 3 O . o 0.4-0.2 1 ! 1 1 ! 1 • • 1 • I • —tt-- eF = 2 --£>•- eF = 5 - • - eF = io - A - eF = 20 - X - 6F = 30 I i I \ 'I if 1 III A * I 1 T j j • Q. 3 O o O Time (s) Figure 3.10: Effect of 6p on PI Performance - Input Disturbance Chapter 3. Recycle Control 56 Table 3.1: Integral of the Absolute Error for the Effect of 6p on PI Performance PI IAE PI Parameters Setpoint Tracking PI Parameters I/O Disturbance Rejection OF OR Setpoint Tracking Output Disturbance Rejection Input Disturbance Rejection KC Tl KC Tr 2 > 8.00 11.73 10.83 2.75 16 1.480 16 5 12.22 15.57 12.71 1.75 16 1.287 16 10 1 18.76 23.24 15.42 1.52 20 1.160 16 20 36.62 36.96 26.40 1.15 28 1.135 28 30 1 55.94 55.45 38.22 1.00 35 1.000 35 Table 3.2: Integral of the Absolute Error for the Effect of dp on Smith Predictor-PI (SP-PI) Performance Smith Predictor PI IAE SP-PI Param. Setpoint Tracking SP-PI Param. I/O Disturbance Rejection OF OR Setpoint Tracking Output Disturbance Rejection Input Disturbance Rejection KC Tr KC T/ 2 7.88 10.46 6.74 3.00 16 1.52 7.00 5 11.72 15.00 12.43 2.75 13 1.75 13.00 10 > 17.89 21.48 15.42 2.75 13 1.70 9.75 20 34.96 35.95 27.80 1.70 13 1.75 14.00 30 1 54.20 52.89 41.10 1.10 13 1.25 15.00 Chapter 3. Recycle Control 57 Typically, using a Smith Predictor (SP) to compensate for dead-time should result in better performance, and the SP benefits should increase with increasing dead-time. In this case, the Smith-Predictor-PI (SP-PI) was only marginally better than the PI, even for larger dead-times. The performance of the SP-PI was limited because the tuning was performed using the sensitivity functions. The limitation was caused by the loop transfer function (L) and the control sensitivity function (CSO). When the PI gain (Kc) is increased and the integral time (77) reduced to improve performance, it results in: • The magnitude of L being greater than one at some frequency after the crossover frequency (i.e., \L\ > 1 for some co > coc). This means the closed loop will be more sensitive to high frequency noise. • The maximum magnitude of CSO equal to 6 at some frequency, say cumax (i.e., max \CSO\ = 6 at comax. Thus, if a disturbance had a sinusoidal component with frequency comax, then the controller output would have a component with the same frequency but the amplitude would be six times that of the disturbance. This may result in control valve saturation. If the sensitivity functions are ignored, and the SM-PI is tuned to obtain a fast closed loop time constant, the performance is significantly better than that of PI. However, it results in a large initial control action, a magnitude of 5.5, which may saturate the control valve. Thus, the Smith Predictor is not sufficient to handle the effect of increasing Op and a more advanced control method is required. 3.2.2 Increasing OR The same values for the delay given in the preceding section were used for OR, i.e., 6R = 2,5,10, 20, and 30. The response for setpoint tracking, and output and input disturbance rejection are shown in Figures 3.11, 3.13, and 3.14; the corresponding Integral of the Absolute Error (IAE) values are shown in Table 3.3. Only the sensitivity plot for the setpoint tracking is shown in Figure 3.12 as the remaining sensitivity plots are similar. As 9R increases, the effective time constant of the process increases, and as result, there is a drop in control performance. The decrease in control performance is not as rapid as in the increasing Op case because there is no increase in the process time delay. The short forward path time delay means the PI controller is able to control the forward path well, and the recycle stream just acts like a disturbance entering the system. The larger the 6R, the more time the controller has to get the process to setpoint before the recycle stream effect is felt. Chapter 3. Recycle Control 58 Figure 3.11: Effect of 9R on PI performance - Setpoint Changes 1E-3 0.01 0.1 1 Frequency (rads/sec) Frequency (rads/sec) Figure 3.12: Effect of 9R on Sensitivity Function - Setpoint Changes Chapter 3. Recycle Control 59 200 200 Time (s) Figure 3.13: Effect of OR on PI performance - Output Disturbance 1.0 0.8 B- 0.6 o » 0.4 o it 0.2 0.0 0.2-0.0-„ -0.2. o. 1 -0.6-o J= S -0.8-o -1.0--1.2-" 1 1 . • 1 " ! ! ! • -^-e R = 2 - D ~ eR = 5 eR = lo - A - eR = 20 - x - eR = 30 i • | | f —'—I • I •«! 1 1 —'—1—'—1—'—1 — ' — i — < — i i—<—i 1 | 1 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 Time (s) 1 1 • 1 1 • • 1 — ' — 1 i—<— i i \ I i. 1 V 1 ! 71^ , : T^fcw^ , V [>*• i t— : 3 i Z~—| x — 10 20 30 40 50 60 70 80 90 Time (s) 100 110 120 130 140 150 Figure 3.14: Effect of OR on PI performance - Input Disturbance Chapter 3. Recycle Control 60 Table 3.3: Integral of the Absolute Error for the Effect of 9R on PI Performance PI IAE PI Param. Setpoint Tracking PI Param. I/O Disturbance Rejection OR Setpoint Tracking Output Disturbance Rejection Input Disturbance Rejection Kc Tl Kc Tl I 2 6.42 9.72 6.87 2.50 10 1.50 10 I 5 7.01 10.72 6.61 2.50 9 1.50 9 I 10 8.06 13.61 7.94 2.40 8 1.30 9 I 20 10.98 16.18 8.85 2.25 8 1.50 12 I 30 10.35 20.90 11.58 2.25 8 1.45 15 3.2.3 P I Control Summary Both 9F and 9R affect the PI performance. As the delays increase the process becomes slower (i.e., a larger effective time constant) resulting in a drop in control performance. The effect of 9F is more significant because 9F also acts as the process time delay, and as the process time delay increases it further decreases control performance. The use of a Smith-Predictor (SP) to compensate for the.process time delay in con-junction with the PI controller produced only a marginal improvement. The SP-PI was tuned using the sensitivity functions which limited the SP-PI performance. Obviously, a more advanced control method than a PI is needed to handle recycle processes. 3.3 Recycle Control Strategies In the literature, only four methods were found to handle recycle systems: 1. Fixing a flow-rate in the recycle loop. 2. Disturbance load sharing 3. Taylor series expansion 4. Recycle compensator Chapter 3. Recycle Control 61 The first two methods do not entail controller design and thus, do not really account for recycle dynamics. Each method is basically a strategy to improve control by changing the control configuration. The new configuration is better able to handle the problems caused by recycle streams, specifically the increased sensitivity to disturbances and the snowball effect. The last two methods use a model of the recycle process for controller design. As a result, they are able to explicitly deal with recycle responses. In the sections following, these two methods are presented and then an alternative modeling approach for control is proposed. 3.3.1 Taylor Series Expansion The process model for a recycle process with delay is known to contain a denominator dead-time term. U GF(s) 1-r Y GR(s) Figure 3.15: Recycle Block System for 2 F O P D T For example, if the recycle system, shown in Figure 3.15, is composed of two first-order-plus-dead-time models, where: KFe-6FS TFS + 1 GR = KRe-e*s TRS + 1 (3.2) The process transfer function is: Q = (l-r)KF(rRs + l)e-9FS T F T R S 2 + (TF + TR)S + 1- rKFKRe-(dF+e^s (3.3) The denominator dead-time term makes it difficult to use this form for controller analysis and design. The system can be analyzed in the frequency domain to determine the process stability. However, frequency analysis alone is insufficient for advanced controller design (Hugo et al. 1996). The dead-time term can be replaced using a Pade approximation or moment matching. However, these methods may result in an unstable approximation. In order to remove the dead-time term in the denominator, Hugo et al. (1996) propose the use of a Taylor series expansion to obtain an approximation for the overall transfer Chapter 3. Recycle Control 62 —•—1st Order —•—2nd Order —A— 3rd Order - X - 4 t h Order — f t — 5th Order 100 120 140 160 180 200 220 Time(s) Figure 3.16: Different Order Taylor Series Expansion Compared to the True Process function shown in eqn. 3.3. In the Taylor series expansion, the differentiation is with respect to e~(9F+6R^s and the expansion performed around e~^F+eR^s = 0. For example, a third-order Taylor series expansion of eqn. 3.3 is shown below: (l-r)KF(TRs + l)e-eFS TFTRS2 + (TF + TR)S + 1 - rKFKRe-(-dF+e^s (1 - r)KFe-°FS (TFS + 1) + r(l - r)KFKRe-WF+d«> r 2 ( l - r)KFKRe-WF+29*> Ag (TFS + 1)2(TRS + 1) (TFS + 1)*(TRS + 1¥ (3-4) where 0(e~49dS) represents the higher order terms. The approximation shifts the denomi-nator dead-time terms to the numerator allowing it to be easily used as a controller model. The overall transfer function becomes a combination of the rational forward-path transfer function, the forward-path transfer function plus one complete forward and recycle path, and the forward-path plus two complete forward and recycle paths. Figure 3.16 shows the increase in accuracy with expansion order. The higher the expansion order the more accurate the approximation. However, the overall function becomes more complex. In fact, even using a third order expansion makes the function quite complicated. On the other hand, using only a second order expansion will cause a significant model-plant-mismatch which deteriorates the controller performance. Chapter 3. Recycle Control 63 3.3.2 Recycle Compensator The previous approach attempts to deal with the recycle dynamics while eliminating the irrational term in the denominator for better controller design. Another method is to eliminate the recycle dynamics completely from the overall system by designing a recycle compensator (Taiwo 1986). This allows the controller to be designed for the forward path alone. The compensator representation is shown in Fig. 3.17. G R R G K Figure 3.17: Recycle Compensator Block System In the figure, GF and GR are the units in the forward and recycle path respectively, GK is the recycle compensator, and Gc is the controller. The open-loop transfer function without recycle is: Y = GFu The open loop transfer function of the system with recycle and compensator is: GF Y = Equating eqn. 3.5 to eqn.3.6: Gpu 1 — GFGR + GFGK GI u u 1 — GFGr + GpGx The compensator GK can be found as follows: 1 = 1 — GFGR + GpGx ;. GFGK — GFGR .'. GK = GR (3.5) (3.6) (3.7) (3.8) Chapter 3. Recycle Control 64 If the compensator is exactly equal to the recycle process, the effect of the recycle stream will be cancelled by GK- Thus, the controller Gc is designed for the forward path alone and the overall controller will be a combination of Gc and GK- The performance of the compensator is obviously affected by the accuracy of the plant models. 3.3.3 Proposed Model for Controller Design - Seasonal Models The previous two methods are sufficient in handling recycle systems, however, they do have potential problems. The Taylor series expansion is complicated and gives a large model plant-mismatch at low model order. The compensator is an innovative solution to remove recycle dynamics but it requires the separate identification of the forward and recycle path process which is not always possible. U Gp(z1) GR(z1) 1-r Figure 3.18: Recycle Block System for 2 Discrete F O P D T A new approach is to use the overall process model for controller design, thus, only one process model needs to be identified. In the previous chapter, recycle processes were shown to have a seasonal model structure. The general structure of a seasonal model is: r( - biq~l~d + b2q~"~d + bpq~P~d + b^~P'1~d r , n\ { Q ' l + aiq-i + a2q-2 + aPq-p + a P + l q - p ^ 1 ' where d - represents a time delay. The distinguishing feature of a seasonal model is the jump or gap in the polynomial order. This gap can appear in either the numerator or denominator polynomial or both. For eqn. 3.9, the gap appears in the numerator ( h b2q~2~d + bPq~p~d -I ) and the denominator ( h a2q'2 + aPq~p -\ ). Thus the system is based on sequential past values and values P sampling intervals in the past. The seasonal model is obtained by taking the z-transform, with a zero order hold, of the forward and recycle processes individually. The discrete transfer functions are then placed in the configuration shown in Figure 3.18, and the process transfer function derived. The procedure is shown for a recycle system composed of the two first order plus dead Chapter 3. Recycle Control 65 time processes given below: GF(s) = KFe~0FS TFS + 1 GR(s) KRe-6*s TRS + 1 (3.10) After discretizing each transfer function separately with a zero order hold, the discrete transfer functions are: GF{Z I) = — bFz-l-dF aFz - l GR{z-1) = bRz-l-d* 1 - aRz~l Deriving the process transfer function from Figure 3.18, the resulting process model has the following structure: G = (1 - r){bFq-l-dF + bFaRq-2-dF) l-{aF + aR)q-1 + aFaRq-2 + rbFbRq-2-dF-dR (3.H) where dF = 6F/TS and dR = 6R/TS. In the denominator polynomial there is a gap in the powers +aFaRq~2+ rbFbRq~2~dF~dR, this indicates the seasonal model structure. This approach gives a good approximation to the true process (see Figure 3.19 reproduced from chapter 2). Process Parameters : K p = K R =1, Tf= 5, Tr = 5, 6 f= 8 r = 10, r= 0.5, T s = 1 Figure 3.19: Comparison ofthe True Recycle Process with the Seasonal Model Chapter 3. Recycle Control 66 Since the seasonal model structure can model recycle processes, it can be used in the design of model based controllers. It will be shown in the following simulation section that this form can be readily applied to model-based controller design such as Generalized Predictive Control (GPC). 3.4 Simulation Results Two sets of closed-loop simulations using G P C were used to compare the Taylor series approach, the recycle compensator, and the seasonal model approach. In the first sim-ulation the controller model contained no model plant mismatch, i.e,, the controller was based on the true process model. For the second simulation the controller model includes some model-plant mismatch. This would normally be the case if the controller model was obtained from system identification. The G P C control law was derived from the minimization of a receding horizon quadratic cost function comprised of the sum of squares of the prediction error and the sum of squares of a weighted control incremented over an output prediction horizon (ni to n2) and a control horizon (nu) respectively (Clarke et al. 1987). The cost function is shown below: { ri2 nu ~\ [y(k + j) - w(k + j)}2 + ^ 2 X(j) [Au(k + j - l)] 2 \ (3.12) j=m j=i ) where ri\ is the minimum output prediction horizon, n2 is the maximum output prediction horizon, nu is the control horizon such that Au(t + j) = 0,V j > nu, and X(j) is a control weighting sequence. In these simulations, the forward and recycle processes used are shown below: p-lQs GF(s) = GRs) = ^ - ^ (3.13) The recycle fraction (r) was set equal to 0.5 and the sampling time (Ts) was 2 units. The initial starting point for the G P C controller were the default tuning parameters that have been found to work well in practice, i.e., n\ — b, n2 — b + 10, nu = 1, and observer polynomial C — [1 — 0.8]A (Soeterboek 1991), where b is the discrete time process delay in sampling time units. A PI controller was also included for comparison purposes. The initial PI controller parameters were obtained using a new direct synthesis method by Jung et al. (1999); the parameters were chosen to give a closed loop time constant in the range of 5 — 10 seconds. The G P C and PI tuning parameters were then adjusted using the maximum peak Chapter 3. Recycle Control 67 criteria in the sensitivity functions. The same five sensitivity functions described on page 52 were used. The true process models were used in the calculation of the sensitivity functions to see how the controllers would actually perform. Again, the limiting sensitivity function was the control sensitivity (\CSO\) function which has to be less than four for servo control and less than two for regulatory control. Since the PI was tuned using the sensitivity function with the true process models, the PI is not affected by model plant mismatch and as a result, the PI tuning parameters for servo and regulatory control were unchanged between the simulations. 3.4.1 Simulation 1: True Process Mode l for Control For this simulation, the controller model is assumed to have no model plant mismatch. Thus, the controller model is based on the the process transfer function (G) combined from the true process models (GF, GR) shown below: C - C - 6~WS C- 0-5(5*+ l)e- 1 0* For the Taylor series approach, a third order approximation was used for the controller model. Each transfer function in the approximation is multiplied by (5s + l) 5/(5s + l ) 5 to obtain the same denominator. The transfer functions are discretized using a zero order hold and sampling time of 2 seconds. Since the transfer functions have the same denominator, they can be easily combined to give one overall transfer function which is used in the controller. For the recycle compensator approach, the compensator is set equal to the recycle process GR, and this results in a complete cancellation ofthe recycle dynamics. Thus, the controller is designed for the forward path process GF alone. For the seasonal model approach, the model is derived using the method shown on page 65. A positive unit step at time t=10 seconds was used for the setpoint changes, output, and input disturbances. The simulations for setpoint tracking, output, and input distur-bance rejection are shown in Figures 3.20 to 3.22 respectively. The corresponding tuning parameters and Integral of the Absolute Error (IAE) values are-listed in Table 3.4. For servo control (fig. 3.20), the compensator provides the best control as the effects of the recycle have been completely removed and the controller is handling only the forward path process. The seasonal model gives slightly slower control because it still has to handle the recycle effects. The Taylor series approach is an approximation and inherently contains some model plant mismatch, however, it provides almost as good control as the seasonal approach and is faster than PI control. Chapter 3. Recycle Control 68 Table 3.4: Controller Parameters and IAE Values Using the True Process Model Controller ni "2 A c TI Setpoint Tracking I A E Output Disturb I A E Input Disturb I A E Servo Control Seasonal 5 15 1 1.75 [1 -0.7]A - - 9.67 Compensator 5 15 1 0.00 [1 -0.96]A - - 7.76 Taylor/Hugo 5 15 1 2.50 [1 -0.65]A - - 10.23 PI - - - - - 0.95 15 15.57 Regulatory Control Seasonal 5 15 1 0.00 [1 -0.80]A - - 13.31 13.36 Compensator 5 15 1 20.00 [1 -0.95]A - - 34.20 17.10 Taylor/Hugo 5 15 1 2.60 [1 -0.80]A - - 16.32 13.25 PI - - - - - 0.85 15 16.97 10.79 0 20 40 60 80 100 120 140 160 180 200 220 240 Time (s) T > 1 > 1 > 1 1 1 1 1 1 1 1 1 • 1 ' 1 1 1 ' 1 ' r • f , I I I I I ! I I o - r — H — i — • — i — i — i — • — i — i — i — i — i — • — I — i — i — " — i — i — i — • — i — ' — i -0 20 40 60 80 100 120 140 160 180 200 220 240 Time (s) Figure 3.20: PI/GPC Using the True Process Model: Servo Control Chapter 3. Recycle Control 69 1.2 1.0 0.8 3 f" 0.6 O 8 0.4. 8 S 0.2 a. 0.0 -0.2 - 1 1 1 • i i i i • 1 — ' — 1 1 • • \\ \ • • \\ \ >l • Compensator - - D - - Seasonal • • w 1 V $ i ; i i --ft"-Taylor PI • 1::::::^ V_ • "*— r a Dm-w^ ~ * V • - r — -' 1 i — ' — i — < — i i — < — i — • — i i ' 1 • 1 1 1 ' — 1 ' 1 Time (s) fl-o t-o ! • c o o 0.0-)»--0.4 -0.8 -1.2 -1.6 -2.0 1 • • i • • • • 1 1 • "A \ i \ • ' " ^ t / * 1 \ 1 * f | 1 0 20 40 60 80 100 120 140 160 180 200 220 240 Time (s) Figure 3.21: PI/GPC Using the True Process Model: Regulatory Control for an Output Disturbance 1.2-Q. 3 O CO CO 8 S Q. 1.04 0.8 0.6 0.4 0.2 0.0 -0.2 "1 1 ! —A- - Compensator -•o— Seasonal Taylor PI I= A » —> 1 ' 1— 20 40 60 80 —1 ' 1 • 1 • 1 • 1 • 1 • 1 >-100 120 140 160 180 200 220 24 Time (s) 0.2 0.0 -0.2 -0.4 fl-3 o jfi -0.6 p c o o -0.8 -1.0 -1.2 1 | 1 1 • • I • I • • • • 'iV'A • • "J x \ X • V \ — \ . • s v i S a u H > — s n JA — ? * * 24 0 20 40 60 80 100 120 140 160 180 200 220 Time (s) Figure 3.22: PI/GPC Using the True Process Model: Regulatory Control for an Input Disturbance Chapter 3. Recycle Control 70 For regulatory control, the performance of the controllers depended on whether the disturbance occurred at the process input or the process output. In the case of an output disturbance (fig 3.21), the seasonal model approach was the best with the Taylor series approach giving a comparable performance. The compensator's effectiveness dropped significantly and the response was slower than with the PI controller. When the system experienced an input disturbance (fig. 3.22), the PI performed the best. Both the seasonal and Taylor series approaches gave similar performances while the compensator method gave the slowest response. 3.4.2 Simulation 2: Identified Process Mode l for Control In this simulation, the true process models were unchanged from those given in equation 3.13. However, the models used for controller design were: p OpS p °RS G ? M = ^ GS(.) = ^ (3.15) Model plant mismatch was introduced by a smaller time constant and different time delays. The mismatch in the time delays were assumed to be ±50% of the true time delay (Op = 9R = 10). Four cases were studied: (a) 0™ = 0™ = 5,'(b) 6Fl = 6R% = 15, (c) 0™ = 5, 0£ = 15, and (d) 0£f = 15, 0™ = 5. While only the results from (a) are shown here, the discussion is generalized for all four cases. The Taylor series model was derived as given in the previous section but used the models given in eqn. 3.15. The recycle compensator set GK = G1^. The seasonal model used in the controller was built using the models for Gp & GR and combined as shown on page 65. The controllers were again tested using unit step changes in the setpoint, output, and input disturbance at time t—10 seconds. The resulting simulations are shown in Figures 3.23 to 3.25 for setpoint tracking, output, and input disturbance rejection. The tuning parameters and IAE values are shown in Table 3.5. For servo control (see Figure 3.23), the control performances were the same as in the previous simulation. The compensator performed the best, the seasonal approach was slower followed by the Taylor series and the PI. Application of an output disturbance (fig 3.24) shows the seasonal approach performing the best. The Taylor series approach is affected by the larger model plant mismatch and performs slower than the PI; the compensator has the slowest response of them all. For input disturbances, the PI is able to perform better than the model based con-trollers for two (50%) of the cases, for the other two cases both the seasonal model and taylor series approaches are better. The case when the seasonal model approach is better Chapter 3. Recycle Control 71 Table 3.5: Controller Parameters and IAE Values Using Model-Plant Mismatch 9™ 5, n = s Controller m 712 A C Tl Setpoint Tracking IAE Output Disturb IAE Input Disturb IAE Servo Control Seasonal 3 13 1 1.00 [1 -0.675]A - - 11.46 Compensator 3 13 1 0.00 [1 -0.93]A - - 10.50 Taylor/Hugo 3 13 1 3.00 [1 -0.25]A - - 15.20 PI - - - - - 0.95 15 15.57 Regulatory Control Seasonal 3 13 1 3.00 [1 -0.675]A - - 14.08 9.91 Compensator 3 13 1 35.00 [1 -0.80]A - - 25.87 12.82 Taylor/Hugo 3 13 1 3.00 [1 -0.80]A - - 20.67 13.34 PI - - - - - 0.85 15 16.97 13.60 Time (s) S-o i-E c o O h / A r - i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1— 0 20 40 60 80 100 120 140 160 180 200 220 240 Time (s) Figure 3.23: Case la, 0$ = 5,9% = 5: Setpoint Change Chapter 3. Recycle Control 72 Q. o 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 • ' 1 " • T ~ | 1 —i —i ' ! ' ' 1 ' 1 r '• f [— 1 1 1 — '•V ; • —A— Compensator Seasonal --ft--Taylor - - - - - - PI 1' \ 1 • V \ u "A; ;r N. \ '"A \ l * \.V --•fry i L. * —* f ••v o — * - - a - " * — — L ^ W - i r - * - J — ! 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 Time (s) 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 Time (s) Figure 3.24: Case la, 0f = 5,0% = 5 : Output Step Disturbance 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.2 - i 1 1 1—r- T — 1 — T "3 o cn in 8 8 Q_ "cQ-fti...---A-- - Compensator — 0 - - Seasonal --ft-- Taylor - • - PI — - * 4 -1 1 1 1 1 1 j 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 30 Time (s) ' — I — I - . . . , l — ' — ! — ' — l ' ! i—>—i i—i— - ! WW \ \ \\ ; '* V \ \ * —•—i—•—i i—1—i i—'—i 1 i — ' — i — i — l — ' — I — T ~ ~ f " $ ~ i—•—i — • — i — • — fl-o p o O 20 40 60 80 100 120 140 160 180 200 220 240 260 280 30 Time (s) Figure 3.25: Case la, 9™ — 5,6% = 5: Input Step Disturbance Chapter 3. Recycle Control 73 is the one shown in Table 3.5. The Taylor series and seasonal approaches give comparable performance to each other, either one or the other is better depending on the time delay mismatch. The compensator again shows the slowest performance. 3.4.3 Simulation Results Summary and Discussion The controllers perform differently depending on whether they are tuned for servo or reg-ulatory control. Obviously a two degree of freedom controller can be used to handle both servo and regulatory control. However, the purpose here is to compare the performance of the different modeling approaches. In the case of servo control, the compensator performs the best with the seasonal model giving comparable performance. The inherent model plant mismatch of the Taylor series approach gives a slower performance than the seasonal model approach but is faster than the PI. For regulatory control, the controllers responded differently depending on where the disturbance entered the process. If the system received output disturbances, the seasonal model approach was the best. The Taylor series and compensator approach were slower than the PI, with the compensator giving the slowest response. However, if the process was excited by an input disturbance, the PI was better than the model based methods for over half of the cases studied (i.e., the cases from Simulation 1 and 2). The seasonal and Taylor series approaches were comparable, but the compensator was again the slowest. The mediocre input disturbance regulator performance of the model based methods compared to the PI is the result of using the maximum peak in the sensitivity functions for tuning. Maximum peak sensitivity function tuning results in the controllers having the same robustness in the face of disturbances (e.g., input, output, and noise) and model uncertainty. Since, the performance of the model based (MB) methods are slightly slower than the PI, it suggests that the MB methods have to be detuned more to have the same robustness as the PI. Prior to using sensitivity functions for tuning, the same controller comparisons were performed using only setpoint changes. The controllers were tuned to give approximately the same rise time by adjusting the weighting factor (A). The G P C values (ni, n 2 , and nu) were set as above, but the observer polynomial was set at C = l . 1 The results of the study were later expanded to cover output disturbances as well. The Taylor series approach was found not to work well, but the poor performance was attributed to using an observer polynomial of C = 1. The compensator was found to have performed the best, with the seasonal model providing a comparable performance. Thus, even when using 1 These results were published in the journal "Industrial and Engineering Chemical Research", Vol. 40 No. 7 pgs 1633-1640, 2001 (Kwok, Chong Ping, and Dumont 2001). Chapter 3. Recycle Control 74 two different tuning methods, the seasonal model was still found to give a comparable response to the best controller obtained for each tuning method. 3.4.4 Implementation Issues Although all three methods combined with G P C give stable control, there are practical factors that should be considered. First, the Taylor series approximation used by Hugo et al. (1996) will always contain some degree of model-plant mismatch. The approximated model gain is always lower than the real process gain and the resulting controller tends to be more aggressive. Therefore, a controller weighting is needed to provide a smooth closed-loop response. In practice, one would prefer to over-estimate the model gain instead of under-estimating it. In this work a third order approximation was used in eqn 3.4, each term was discretized separately and then combined to form the overall model. Accuracy can be improved with a higher-order model (i.e., one with more terms). However, the application of a higher order model becomes more complex. As the compensator G P C controller was based on a first-order model, the implemen-tation is simple. However, the compensator requires the identification of two models, the forward and the recycle path models, thus there will be more possibility of model-plant mismatch. Since the performance of the compensator depends on the removal of the re-cycle path, the presence of model plant mismatch means complete removal of the recycle path will be unlikely and the compensator benefits will decrease. In addition, the indepen-dent identification of the forward-path and recycle-path dynamics may not be practical or possible due to operating constraints. The seasonal model approach was based on the structure shown in eqn. 3.11 and can be easily implemented in the G P C framework. The seasonal model also has the advantage that it only requires the identification of one overall model without isolating the forward and recycle path processes. 3.5 FOPDT Model vs Seasonal Model for Recycle Control In the previous chapter, recycle responses were found to fall into three groups: • Group 1: These responses are similar to the forward path process. The responses are usually either first order or second order with dead-time. • Group 2: The responses are characterized by a fast rise time followed by a slow rise Chapter 3. Recycle Control 75 to steady state. This group encompasses different responses, some of which can be approximated by a first order model without loss of control performance. • Group 3: The responses move to steady state in a step-wise fashion. The shape of the step is dependent on the system. The first two groups can potentially be controlled using a first order-plus-dead-time model while the last group would require using a seasonal model for control. The purpose of this section is to compare the use of a F O P D T model to the seasonal model for recycle control, and to determine at what point it would be better to use the seasonal model. Obviously, using the seasonal model will give better performance but the F O P D T approach can be a good alternative. Frequency analysis and step responses were used to specify the F O P D T model used in the controller design. Figure 3.26 shows the frequency response of the recycle process with the following parameters: Tp = TR = 5, Op = 10, 6R = 1, r = 0.5, and Ts = 1. Two possible F O P D T frequency responses are also shown: • Model 1 (r = 10, 0 = 10) - the frequency response encloses the process frequency response. • Model 2 (r = 25, 0 = 10)- the frequency response matches the process frequency response up to the cutoff frequency. The corresponding step responses are shown in Figure 3.27. Model 1 is significantly faster than the true process resulting in a large model-plant mismatch. Even though Model 2 does not match the process bode at the high frequency range, it provides a better representation of the true process and is the method used to specify the F O P D T model. For the FOPDT-seasonal model control comparison, the recycle process parameters used were Tp = TR = 5, r = 0.5, OR = 1 and Ts = 1. The starting value for Op was 2 and it was increased gradually until there was a significant difference between the F O P D T and the seasonal model control. The controller used was Generalized Predictive Control (GPC) and the maximum peak in the sensitivity functions was used to finalize the tuning parameters. The results for Op = 5, 10, and 15 are shown in Figures 3.28 to 3.30 for a setpoint change and the corresponding IAE values are in Table 3.6. The IAE results for output and input disturbances are also included. For Op = 5 the F O P D T performance is comparable to that of the seasonal model. Increasing Op to 10 caused a drop in the F O P D T performance; however, the F O P D T performance is still acceptable depending on the performance criteria used. When Op = 15, the F O P D T is noticeably slower than the seasonal model. The performance of the F O P D T model is still adequate but there is a definite incentive to use the seasonal model. Thus, Chapter 3. Recycle Control 76 1 * a, 0.1 3 "5. I 0.01 1E-3 0 -30 -PS -60 CD 5. -90 -(0 co -150--180-i i i i i i i i — • — Process FOPDT- model 1 —A~ FOPDT- model 2 I I I I I I I ( I i I I i i I I | • 1 0.01 0.1 Frequency (rads/sec) ; —OTSU-J i • ^ > ' ^ « c . \> • . \ \ 1 1 1 1 1 1 1 1 W \ \ \ '\ I 1 l r—i—ill) i r 1E-3 0.01 0.1 Frequency (rads/sec) Figure 3.26: Process and Model Bode Plot Figure 3.27: Process and Model Responses Chapter 3. Recycle Control 77 ' 1 ' ' i 1 •" r — T — • 1 1 i 1 1 • 1 • / —n D'i /.^ ...^ r.-i-r.ar.'* 1 J~~—~r~~i • '/", / / ;',' /'•/ I ^ - " ^ J • it i; —A—• Seasonal - o - - FOPDT Process Step Response • I....Z • V ! 1 1 ' i ' l ' 1 • ' • 10 20 30 40 50 60 Time (s) 70 80 90 100 1 i ' ! 1 1 • , ... T ! i I t I % \ 3 A n ^•-CF 10 20 30 40 50 60 70 Time (s) 80 90 100 110 120 Figure 3.28: Seasonal-FOPDT Comparison - 0F = 5 1.2-1.0-• 3 Q. 0.8-o CO 0.6-CO CD • O 0.4-0-0.2-0.0-~i—1—i—1—i—1—i—1—r - i—'—i— ' — r ' i -.. ! i ~o—-->— u - "-is —~n—-fW-*o--wn--"«o—-^n—f\i| —A~- Seasonal ~o- - FOPDT Process Step Response — i — > — i — i — i — i — i — ' — i — • — i — i — i — ' — i — i — i — i — i — • — i — ' — i — ' — i — ' 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 Time (s) 1 1 ! ' I 1 — ™ f T 1 1 | 1 1 •> r1" rf' • ";i j D -3S--H2 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 Time (s) Figure 3.29: Seasonal-FOPDT Comparison - dp = 10 Chapter 3. Recycle Control 78 Figure 3.30: Seasonal-FOPDT Comparison - Op = 15 the condition to switch from a F O P D T model to the seasonal model appears to be around (2 —> 3) * TF < 0F, which is less than the condition for the step-wise behaviour (i.e., 6 * TF < Op + 0R) to occur. The difference in the IAE values at the switch point is approximately 10. This value will be used as a benchmark to indicate approximately where the switch point is located in the simulations for setpoint changes. The performance of the F O P D T for output and input disturbances at this IAE difference may still be considered comparable to the seasonal model, so the switch point could be set at a larger IAE value for disturbances. Use of the IAE difference for setpoint changes establishes a possible lower bound on the condition at which the switch from a F O P D T model to a seasonal model could occur. Additional combinations of time delay and recycle fraction were used, but only the tabular values are presented. The simulations were run holding Op = 1 and varying 9R. The results are shown in Table 3.7. In this case, the model switch point occurred at 6 * TF < OR. Thus, Op has a greater affect than OR on the performance of the F O P D T model for control, and it sets a possible lower bound at which the switch point may occur. The use of setpoint changes in combination with a varying 0F will establish the lower bound at which the the switch to a seasonal model should be considered. Chapter 3. Recycle Control 79 Table 3.6: Seasonal-FOPDT IAE Values, Varying 6F, 6R = 1 Model 9F = 5 6F = 10 6F = 15 Setpoint Tracking IAE Seasonal 10.76 15.74 21.15 FOPDT 12.18 23.42 32.56 Output Disturbance Rejection IAE Seasonal 14.76 21.40 30.14 FOPDT 18.31 32.64 44.57 Input Disturbance Rejection IAE Seasonal 14.64 20.84 29.98 FOPDT 17.56 30.36 41.19 Table 3.7: Seasonal-FOPDT IAE Values, Varying 6R, 9F = 1 Model 0R = 10 0 f l = 2O 9R = 30 Setpoint Tracking IAE Seasonal 6.32 6.28 6.37 FOPDT 7.76 12.84 18.52 Output Disturbance Rejection IAE Seasonal 11.93 15.25 25.36 FOPDT 13.39 21.44 36.66 Input Disturbance Rejection IAE Seasonal 11.83 14.49 24.95 FOPDT 12.68 19.61 33.03 Chapter 3. Recycle Control 80 As a last check, OR was set to 8 and Op was increased starting from 3. In this case, the switch point appeared to be in the range of (3 —> 4) * 7 > < 0p-\-0R. The switch point is also dependent upon the recycle fraction. When the simulations varying Op were repeated using r = 0.25, and 0.7, the following switch points were obtained: (a) r = 0.25, 6p > 4 * rp and (b) r = 0.7, Op > (1 -»• 2) * TF. The F O P D T model does give comparable performance to the seasonal model for small time delays and recycle fraction. However, as the time delays and recycle fraction increase, the advantage of using the seasonal approach becomes more apparent. Approximate lower bound conditions for switching from the F O P D T model to the seasonal model were obtained for different recycle fractions and time delays. These conditions are lower than that required for the step-wise response to appear. 3.6 Conclusions Recycle systems do present challenging control problems such as underdamped and inverse response systems. Application of control can add to the control problems; the system may have large recycle flowrate changes for small input changes (i.e., the "snowball" effect) which could violate operating constraints. PI control was applied to a recycle process, and it was found that both Op and OR affect the PI performance. As the time delays increase, the process becomes slower (i.e., a larger effective time constant) resulting in a drop in control performance. The effect of Op on the control performance is more significant than OR because Op also represents the process time delay. The use of a Smith-Predictor (SP) to compensate for the process time delay in con-junction with a PI controller produced only a marginal improvement. The SP-PI was tuned using sensitivity functions which limited the SP-PI performance. Obviously, a more advanced control method than PI control was needed to handle recycle processes. Various control solutions have been proposed in the literature. Two of the solutions are model based, (a) the recycle compensator which removes the recycle effect and (b) the Taylor series approach which approximates the denominator dead-time terms. An alternative approach was presented in this chapter, the seasonal model approach. For a comparison, these three methods were used to apply model based predictive control to recycle processes. The Taylor series approximation was a novel way to handle the problem of denominator dead-time terms, it is more difficult to implement but does provide comparable control to the seasonal approach. The recycle compensator is an elegant solution to minimize the effect of recycle dy-namics and gives very good servo control, however, it has slow disturbance handling Chapter 3. Recycle Control 81 characteristics. The compensator also requires independent identification of the forward and recycle path systems. The seasonal model approach is an alternative to the recycle compensator. It is more practical and provides comparable servo control performance to the compensator approach ,but has significantly better disturbance rejection capability. The M B methods had input disturbance handling capability that was comparable to a PI controller. The mediocre M B C regulator performance is attributed to using the maximum peak in the sensitivity functions for tuning to ensure the controllers had the same robustness in the face of disturbances (e.g., input, output, and noise) and model uncertainty. The MB methods had to be detuned to have the same robustness as the PI resulting in similar regulator performance to the PI controller. When the M B controllers were tuned for performance rather than robustness, they outperformed the PI. The use of a F O P D T model was compared to using the seasonal model for recycle control. At low values of time delay and recycle fraction, the F O P D T model does give comparable performance to the seasonal approach. However, as the time delays and recycle fraction increase, the advantage of using the seasonal approach becomes more apparent. Approximate conditions for switching from the F O P D T model to the seasonal model were obtained for different recycle fractions and time delays. Chapter 4 Wet End Recycle Dynamics In the previous chapters, recycle dynamics and recycle control were studied to obtain a better understanding of recycle systems. The next stage involved studying the wet end of a simulated papermachine to determine the extent of recycle dynamics. The wet end process contains numerous recycle streams which are either sequential or nested within one another. The objective of this chapter is to identify areas within the wet end that have significant recycle dynamics, specifically the step-wise trend to steady state reported in chapter two. These areas are candidates for applying recycle control methods. The chapter is presented in the following manner. First, the process used in the simulation model is described. In the second section, the results from the recycle tests applied to the mill simulation are given. Since future mills may be built with smaller and fewer tanks, the study was extended to cover these scenarios. Simulation issues are then discussed in the last section. 4.1 The Wet End Paper Process The process studied is a paper machine based in Powell River. The mill produces 550 tons/day of speciality paper. The mill formerly belonged to MacMillan Bloedel Limited but is currently owned by Norske Canada. A simplified diagram of the process is shown in Figure 4.1. Four feedstocks are used to produce the paper: 1. Chemically Treated Mechanical Pulp (CTMP) 2. Ground Wood (GWD) 3. Soft Bleached Kraft (SBK) 4. Hibrite (HIB - is a set ratio of C T M P and G W D with a brightening agent). G W B , C T M P , and HIB at approximately 3% consistency are mixed in the blend tank. The 3% consistency is maintained by the addition of re-circulated white water from the papermachine. The "blended" stock is further refined to increase the pulp freeness and enhance the inter-fiber bonding. The refined stock is combined in the mixing tube with refined Kraft, 82 Chapter 4. Wet End Recycle Dynamics 83 E £ CD ra c CD & CD 3 3 o o o I— T3 05 o .c O O OD-O CO ners| CD CD CD r - o E co ° - 0 CO ffl K I o_ cn 1 i • CO 3 i i 8^ O CO CO U-3 O CD a ughton Pit ughton Pit o m S I O CO O CD CD Q CO u 4v S CD co § § a CO (D 0) Jo O O E ~ CJ sz » o 4v CD P.E to .E sz CD x o x: 5 ™ 0 CQ > > c CO O £ o O „ CD Q cn.* > ro c ° c o m o CD 00 co c x ^ CO CL i- bi2 0 * 5 I— co 2 P •*—* CO ro CD Ch <3 C O C O a) CJ O SH CU T3 S3 H cu hO Chapter 4. Wet End Recycle Dynamics 84 broke, and stock reclaim. The mixed stock is sent to the mixing-machine chests where clay and coagulant are added to form a "thick stock". The clay is used as filler to reduce the furnish costs and enhance paper properties (e.g., opacity, smoothness, and printability). The coagulant is a polymer solution used to prepare the furnish particles for flocculation, it is the first retention aid of a dual polymer retention aid system. The thick stock is diluted by white water from the silo to a consistency of 1 %, and then the primary fan pump sends the stock to the cleaners where heavy contaminants such as stones and bark are removed. The cleaner accepts are sent to the deculator. The deculator operates under vacuum and thus deaerates the stock. Entrained air has to be removed because it retards drainage and lowers wet strength. The secondary fan pump drives the furnish through the pressure screens to remove shives which appear in the paper as dirt specs. The screen accepts are sent to the headbox. Before entering the headbox, the second retention aid (flocculant) is added to the furnish. Flocculant is a polymer solution which causes the furnish to form small floes of fibers. These floes of fibers trap filler particles and fines (i.e., very small fibers) for retention on the wire. In the headbox, the stock is forced through a narrow opening (i.e., the slice) along the width of the paper machine and distributed onto the wire. By forcing the stock through the slice, it is accelerated to the speed of the wire. The wire traps solid components (i.e., fiber, fines, and fillers) forming a wet sheet and allows water to drain through the other side. The paper sheet, at approximately 17 % consistency, leaves the wire and is sent to the presses for further dewatering and drying. The water collected from the wire is recycled to the mill for stock dilution and stock reclaim in different areas of the mill. The water from the first half of the wire is collected in the silo for thick stock dilution. Excess white water from the silo overflows to the white water chest. The white water chest provides dilution streams to the feed stocks, the cleaners, and the mix tube. The broughton pit collects the remaining water from the wire and excess water from the white water chest. The broughton pit also provides water to the saveall. The saveall recovers the solid components (i.e., fillers, fibers, and fines) and clarifies the white water. The recovered furnish components are sent to the reclaim chest, and from there returned to the process via the mix tube. The clarified white water is stored in the clean and clear chests which supply dilution to the deculator. Unclarified white water from the saveall is stored in the cloudy chest for further dilution in the cleaners. Before the paper enters the presses, the paper edges are trimmed to straighten them and the trims are collected in the couch pit. The main function of the couch pit is to act Chapter 4. Wet End Recycle Dynamics 85 as a surge tank during a paper break. When the paper between the wire and the presses breaks, the paper web is sent to the couch pit until the connection is re-established. In the couch pit, showers dilute the stock to 3 % consistency which is then sent to the broke chest for re-use in the process. 4.2 Wet End Recycle Dynamics A simulation model of the process was built by Yap (1998). The model was built using IDEAS by Simons Technologies. IDEAS is a dynamic simulator designed specifically for the Pulp and Paper Industry and used for model development and dynamic simulation. To accurately reflect the process, the model by Yap (1998) incorporated pump curves, pipe and valve characteristics, and a control system. Yap (1998) validated the model with industrial data. Yap's simulation model was used in the wet end recycle study. The original simulation was built to produce a paper with a basis weight of 52gsm. Before the study was started, the process time constants and time delays were obtained from Yap's simulation. These values were then used to identify the recycle loops that could potentially show the step-wise trend to steady state. The condition for such a trend to occur is when the sum of the forward and recycle path time delays is greater than six times the forward path time constant (i.e., 9F + 9R > 6 x rF, section 2.2.5, page 32). However, it was found that the forward path time constant was greater than the sum of the time delays (i.e., TF > 9F + 9R) for all of the recycle loops. This result was caused by the large tanks within the recycle system. The fact that rF was dominant suggested the response would likely be first order. This conclusion is based on a system with a single recycle loop. Since the paper process has multiple recycle loops, the combined dynamics could be different. To see if there were really any significant dynamics present, a small section of the process was simulated. The section used is enclosed in the dotted box of Figure 4.2. The forward path (marked in light blue with circular symbols) is considered as starting just after the basis weight valve. The forward path then goes to the primary fan pump, primary cleaners, deculator, secondary fan pump, primary screen, headbox, the wire, and then to the presses. Four recycle loops were chosen: • RI starts at the headbox and ends at the deculator. The path is in orange and has a diamond marker. • R2 starts at the deculator and ends at the entrance to the primary cleaners. The path is colored in green with a square marker. Chapter 4. Wet End Recycle Dynamics 86 co & ro .E to co O x: O O o -o >,-> S CO •e ro co 0) O to o E " m <D E ro C 4 CO I ro O CO a S m o to (co) (-3 - 0 -8 to S i ? m o .JS <]) CO 5 § o Hi CM CL i co a) ->Lz E «_ 1 8 o sz g o E jg o.-0 ! 0 £ S J — -~ CD <?.E to X O JC CD S > (0 -o U a> • o u * > ro c CD eg CO 2 2 Si1 v, CO -+—• co ro cn i— Ch C O a CD S-! -4-> CO CD o o 03 tf o3 S-l bO Q 03 O o S-4 P H W J - 5 03 SH fl bO Chapter 4. Wet End Recycle Dynamics 87 • R3 starts at the wire pit, goes through the silo, and then to the entrance of the primary fan pump. This recycle path is colored in dark blue with a cross marker. • R4 has two paths which converge at the secondary cleaners before entering the suction to the primary fan pump. Conceivably R4 could be treated as two separate recycle streams. However, since they both have the same path after the secondary cleaners they were treated as one recycle stream. The path is marked in purple with a triangular symbol. The two paths are explained below: o R4P1 starts at the primary cleaners and then goes to the secondary cleaners. o R4P2 starts at the wirepit, goes through the silo, the white water chest, and then to the secondary cleaners. A step change was made in the filler composition of the stream leaving the basis weight valve. The filler compositions were then recorded in the recycle loops and in the paper leaving the wire (i.e., the wet web). The sample points are marked by a © in Figure 4.2. The test was performed in a systematic manner. First recycle loop 1 (RI) was tested, no other recycle loops were included in the simulation, i.e., streams R2-R4 were simulated as constant feed streams to the system. Then, the second recycle loop R2 was added to the system and the test repeated. After each test, an additional recycle loop was added until all four recycle streams were present. Thus, the effect of the additional recycle loops can be easily seen. Figure 4.3 shows the result with only 1 recycle stream present. Among the three lines in the figure, the solid line represents the step change in the filler composition. The other two lines represent the filler mass percentage in the recycle stream and in the stream leaving the recycle process. Since 7 > was found to be dominant, it is no surprise that the response curves are first order, i.e., the recycle dynamics are not significant compared to the forward path dynamics. The results when loops R2 and then R3 were added to the simulation are shown in Figures 4.4 and 4.5. The responses for both cases are also first order. The addition of these recycle loops did not affect the output responses nor the time to steady state. In Figure 4.4, the change in the filler mass percentage is quicker in RI than in R2, even though they both have the same time delay and R2 appears before RI. The cause of this behaviour is the deculator. As shown in Figure 4.7, the deculator is composed of two tanks (Deculator 1 and Deculator 2). The flow from the primary fan pump is sent to Deculator 1 (the larger tank) and then to the headbox. The overflow from deculator 1 is the feed to deculator 2 and the overflow from deculator 2 is recycled (R2) to the suction of the primary fan pump. Chapter 4. Wet End Recycle Dynamics 88 Figure 4.4: Recycle Dynamics for Loop RI, and R2 Chapter 4. Wet End Recycle Dynamics 89 0.24 0.115 0.12 0.075 -, 0.675 0.650 0.625 CD «J 0.600 § S. 0.575 88 co 0.550 o IT -I 0.525 -I 0.500 19500 20000 20500 21000 21500 22000 Time (s) Figure 4.5: Recycle Dynamics for Loop R I , R2, and R3 Figure 4.6: Recycle Dynamics for Loop R I , R2, R3, and R4 Chapter 4. Wet End Recycle Dynamics 90 Flow from Primary Fan Pump Deculator 2 Deculator 1 Volume 4400 ft3 To ] Primary Fan Pump Suction (R2) Volume 1070 ff T =300 sec T=95 sec To Secondary Fan Pump and Headbox (Time Delay to headbox 18 sec] Time Delay 10 sec From Headbox Time Delay 10 sec Figure 4.7: Deculator Schematic When a step change is made in the inlet filler mass percentage, it is diluted in dec-ulator 1. This dilution slows the appearance of the filler increase in recycle stream RI. The overflow from deculator 1 is further diluted in deculator 2, which further slows the appearance of the filler increase in recycle stream R2. Thus, the filler increase has to pass through two tanks before appearing in stream R2 while it only passes through one tank before appearing in stream RI. This results in the dynamics of R2 being slower than RI. The effect of the addition of recycle loop R4 is shown in Figure 4.6. The response of the intermediate recycle loops (R1-R3) are unchanged, though the R2 and R3 responses seem to be tending towards a second order over-damped response. The R4 response does contain minor recycle dynamics very similar to a Trend - 2 type (section 2.2.5, page 32), i.e., fast initial rise, followed by a slow rise to steady state. Since R4 is composed of two paths (R4P1 and R4P2), the filler mass percentage in each path was monitored to determine if one or both paths together were responsible for the R4 response. The filler mass percentage in each path is shown in Figure 4.8. Because the flowrates in each path (R4P1, R4P2, and R4) are different, the filler mass percentage in each path is also different. The shape of the R4 response is solely caused by the R4P1 path; the R4P2 path only provides a second order response. It is possible the other recycle loops-(R1-R3) could be contributing to the R4P1 response, as a result, loops RI to R3 were removed. The simulation was run with R4P1 as the only recycle loop and then the simulation was repeated with R4P2 as the only recycle loop. The recycle stream filler mass percentages for each case are shown in Figure 4.9. The results show that the R4P1 path is responsible for the minor recycle dynamics. Additional recycle loops could have been tested, however, it is highly unlikely that Chapter 4. Wet End Recycle Dynamics 91 0.135 0.130 0.125 © CO ro • c 0.120 [j & 0.115 5 4 0.110 4 0.105 2 0.1325 f2 0.1300 £ ra 0.1275 ^  ca T3 0.1250 § o <« 0.1225 Sj ra 0.1200 D J TO C 0.1175 y 0.1150 $ 0.1125 ra 2 3400 3600 3800 4000 4200 4400 4600 4800 Time (s) Figure 4.8: Recycle Dynamics for Loop R4P1, R4P2, and R4 cn ra •*—. c CD L E 0.24 0.22 0.20 0.18 0.16 0.14, BTect of pller In Sep on | R4P1 Filler Mass Percentage 0.26 0.25 ^  0.24 § ra 0.23 D. 0.22 2 (D ro "c CD B rf = I E 0.21 2475 2500 2525 2550 2575 2600 2625 2650 2675 2700 2725 2750 2775 2800 Time (s) * 0.12 a> L E 7250 7500 7750 8000 8250 8500 8750 9000 9250 9500 9750 10000 Time (s) Figure 4.9: Recycle Dynamics for Loops R4P1 alone and R4P2 alone Chapter 4. Wet End Recycle Dynamics 92 the addition of more recycle loops would cause significant recycle dynamics in the output response. Of the recycle loops tested, the R3 and R4P2 paths were the slowest. The R3 path passes through the silo and has a relatively long pipe delay, while the R4P2 path passes through the silo and white water chests. The addition of these loops did not add any recycle dynamics. The remaining recycle loops to be included in the simulation contain more tanks than the R4P2 path. The more tanks a recycle loop contains, the smaller the change in filler mass percentage in the loop. Thus, a change in the filler mass percentage in these ad-ditional loops will be smaller than the change in R4P2 and would take longer to occur. Since the R4P2 path did not cause any recycle dynamics, it is expected that the remaining recycle loops will also not cause any recycle dynamics. This section has shown that the step-wise recycle effect is not present in the wet end simulation. The large tank (i.e., the deculator) in the forward path causes a dominant 7 > which effectively dampens the variations caused by the recycle streams. The large tanks also present in the other recycle loops also contribute to the damping and further reduce the effect of the recycle on the output. 4.2.1 Wet End Recycle Dynamics with Reduced Tank Sizes It is well known that the tanks in the paper process are used mainly to smoothen/dampen out flowrate and composition variations. While they help in reducing variability, they have the effect of increasing the process time constant. As a result, the control system takes longer to bring the system to steady state. This means there are long periods during which off specification product is produced, for example, a paper grade change can take up to an hour or longer before the process reaches steady state. In an effort to make the process faster, there is interest to build future mills with smaller tanks and eventually with some of the tanks removed from the process (Lang and Nissinen 2003). Shrinking or removing some of the tanks would reduce the process time constant and allow the control system to respond faster to such things as paper grade changes and process disturbances. Of course, it would not be realistic to remove all of the tanks as some capacity would be required to handle special cases such as entrained air removal, machine breaks, startup, and shutdown. Obviously as the tank sizes are reduced, greater demands will be placed on the control system. The impacts on the process to be considered are: • As mentioned above, the process time constant will be reduced which means the process will react faster. • The damping ability of the tanks will be less which means: Chapter 4. Wet End Recycle Dynamics 93 Table 4.1: Normal Tank Sizes and Reduced Tank Sizes Tank Normal Tank Size Reduced Tank Size Volume ft3 Residence Time sec Volume ft3 Residence Time sec Deculator 1 4410 95 331 7 Deculator 2 1071 311 63 10 Silo 1326 45 142 5.5 WW Chest 7540 255 580 15 o More variability will be passed to the process. o The interactions between process units caused by the recycle streams will also increase which means the greater possibility of recycle dynamics (i.e., the step-wise trend to steady-state). Since there is the possibility of increased recycle dynamics with reduced tank sizes, the wet end recycle study was repeated using the same system but with smaller tanks. Table 4.1 shows the original tank sizes and the new reduced tank sizes used in the simulation. The tank dimensions were arbitrarily adjusted to bring the residence time to between 5-15 seconds. The procedure of making a step change in the filler composition, recording the process responses, and gradually increasing the number of recycle loops was repeated. Figure 4.10 shows the effect of recycle loop 1 on the filler mass percentage in the paper leaving the wire. The recycle stream and the filler out are showing a very minor step in the responses. Even though the tank size is reduced, the delay in the forward and recycle path is still not sufficient to cause the full step-wise trend. Figure 4.11 shows the response when R2 is added. The responses now look closer to a first order. It seems the addition of the second recycle stream has damped some of the RI dynamics. The addition of the third recycle stream R3 shown in Figure 4.12 results in a prominent step in the responses. When the fourth recycle stream (R4) is added, there appears to be a cancelling out of the step recycle dynamics (see Figure 4.13), i.e., the step definition is not as defined as in the previous figure. The fourth recycle stream appears to smooth the recycle effects Chapter 4. Wet End Recycle Dynamics 94 0.12 0.105 0.100 A 0.095 A 0.090 4 0.085 1950 2000 2050 2100 Time (s) 2150 0.080 2200 CD cn ro c rj) y rf = LC 0) 0.650 0.625 0.600 g, ro 4 0.575 c CO & ro 0.550 ~ O LE 0.525 0.500 Figure 4.10: Reduced Tank Size, Recycle Dynamics for Loop RI 0.24 0.12 0.110 A 0.105 • Filer In • -•--Recycle-I • — D - Recycle-2 • RllerOut -CD 0.100 s c A 0.095 CD L . 0.090 = 0.085 950 1000 1050 1100 1150 "lime (s) 1200 0.080 1250 I T a> o >» o & A 0.58 0.64 0.62 0.60 g , c CD 2 & 0.56 3 0.54 O = 0.52 0.50 Figure 4.11: Reduced Tank Size, Recycle Dynamics for Loop RI, and R2 Chapter 4. Wet End Recycle Dynamics 95 0.24 0.22 h OJ 0.20 0.18 0.16 0.14 * X -0.12 I i i i i i i i x; x Filler In Ftecycle-1 Ftecycle-2 - X - Ftecycle-3 —tr-Filler Out 1 • ' 1 • 1 1 • 1 * ' ' ' ' 1 I I l—L 0.110 0.105 0 0.100 c? ^—' c 0) 2 0.095 g H 0.090 ( 0 _> IT -I 0.085 -g >* u 4 0.080 0.075 1400 1500 1600 1700 1800 1900 2000 2100 2200 Time (s) 0.68 0.66 0.64 — o> 0.62 | a> 2 0.60 £ 0.58 co 0.56 O = 0.54 1 1 0.52 0.50 Figure 4.12: Reduced Tank Size, Recycle Dynamics for Loop R I , R2, and R3 Figure 4.13: Reduced Tank Size, Recycle Dynamics for Loop R1,R2, R3, and R4 Chapter 4. Wet End Recycle Dynamics 96 caused by R3. To obtain recycle dynamics, the sum of the forward and recycle path delays has to be greater than six times the forward path time constant (i.e., Op + OR > 6 x Tp). For recycle streams RI, R2, and R4, the condition that Op + OR > 6 x Tp is not satisfied even though the tank sizes have been greatly reduced. Hence, there are no recycle step-wise dynamics visible for these streams. In the case of recycle stream R3, the condition 0F + OR > 6 x rp is true and there is a step-wise response to steady state. The reason why only one step is visible is because the recycle fraction (r) is small for stream R3. A review of the recycle dynamic plots produced in Chapter 2 also confirm that when Op + 0R > 6 x rp and r is small (i.e., r < 0.25) the step-wise response to steady state only contains one step. As shown above, reducing the tank sizes does introduce some recycle dynamics. How-ever, these dynamics appear to be minor, and the responses could be treated as a first order without causing control problems. 4.2.2 Wet End Recycle Dynamics for the Recycle Loop Tankless Case At some future point in time, mills may be designed with fewer tanks present. In such a case, the forward and recycle path time constants will be very small or zero and the recycle step output condition will be easily satisfied. In this section, the simulations are again repeated but with no tanks present within the recycle loops (forward and recycle paths). This scenario is referred to as the recycle loop tankless mill. The condition for step-wise dynamics (i.e., Op + 0R > 6 x Tp) should be easily satisfied and all the responses should have a step output to steady state. Figure 4.14 shows the result when there is only one recycle stream. As expected, the response shows a step response to steady state, again only one step is visible because of the small recycle fraction. The addition of the second recycle loop, shown in Figure 4.15, results in additional steps being present in the responses. The additional steps are not caused by change in recycle fraction but by the interaction between the streams R2, the process output, and stream RI. When the third recycle loop (R3) is included (see Figure 4.16), the response looks like a stair case with platforms appearing periodically between steps. Streams RI, R2, and R3 cause the three shallow steps, but stream R3 is responsible for the long delay (i.e., the platform) before the three step pattern repeats. The results when the fourth recycle loop (R4) is added is seen in Figure 4.17. The step definition is greatly reduced, the response looks similar to a Trend 2 type response (i.e., a fast rise time followed by a slow rise to steady state). The fourth recycle loop seems to smooth all the responses. Including additional recycle loops in the simulation may amplify or dampen the recycle responses Chapter 4. Wet End Recycle Dynamics 97 0.24 h 0.22 \-g> 0.20 ra c CD I 018 ro IT 0.16 k 0.14 k i i i i i i i • I i i i I i i i I i i +• • •• & 6 -•Filer In -Rscycle-I • Filler-Out 0.105 0.100 0.095 0.085 012 1 1 1 1 1 1 1 1 ' 1 1 1 * 1 1 1 1 ' ' * * 1 1 * * 1 * 1 1 1 1 ' ' 0 080 ' 475 500 525 550 575 600 625 650 675 700' Time (s) a> cn ro c CD 8. 0.090 "Z = IT Q) s. 0.650 0.625 CD 0.600 % CD £> 0.575 & ro 0.550 = M IT A 0.525 -I 0.500 Figure 4.14: No Tanks, Recycle Dynamics for Loop RI 0.24 0.22 h o 0.20 CO 0.18 0.16 0.14 0.12 450 filler In -- • - - R s c y c l e -1 — o - Recycle-2 RllerOut • 650 700 0.110 0.105 g CD 03 CO +~* 0.100 I S. % co 0.095 ^ = IT 0) O >» o A 0.090 0.085 750 0.650 0.625 0.600 A 0.575 0.550 ~ O 0.525 0.500 Figure 4.15: No Tanks, Recycle Dynamics for Loop RI, and R2 Chapter 4. Wet End Recycle Dynamics 98 co O) CO -*—' c CO & ft CO . 0 } I E 0.24 0.22 L 0.20 r -0.18 \-0.16 h-0.14 0.12 900 1000 1100 1200 1300 "lime (s) 1400 0.080 1500 Figure 4.16: No Tanks, Recycle Dynamics for Loop RI, R2, and R3 Figure 4.17: No Tanks, Recycle Dynamics for Loop RI, R2, R3, and R4 Chapter 4. Wet End Recycle Dynamics 99 seen in this study. As expected, removing the tanks from the recycle loop does introduce recycle dynamics in the output. The extent of the dynamics depended on the number of recycle loops and the time delay within the loops. Some loops increased the recycle dynamics while other loops smoothed the response. 4.3 Simulation Issues There are a number of issues that have to be mentioned. The first issue deals with the simulation program being used, while the remaining issues cover assumptions made in the study. • The tank model in the IDEAS program assumes perfect mixing. In reality, tanks are never perfectly mixed. A study by Ein-Mozaffari et al. (2002) has shown that imperfect tank mixing can produce results that are similar to a step-wise response which suggest internal recycling within the tank. This behaviour is a function of tank design and not of the recycle system. In an industrial situation, such a response could falsely suggest the existence of recycle dynamics or it could smooth out the real effect of recycle dynamics. • When there was a compositional change in the streams, the model by Yap (1998) was not sophisticated enough to dynamically change the separation ratios in the clean-ers, screens, and wire. The results were obtained by assuming that the separation ratios were constant. Normally, the separation ratio would change with composi-tional changes and this would affect the magnitude of the recycle response. In some units, for example the wire, the separation ratios would be equivalent to the recycle fraction. Changes in the recycle fraction would affect the recycle dynamics. • The results for the normal tank size and reduced tank size simulations were obtained by making a step change in the filler mass percentage from 0.13% to 0.23% in the stream leaving the mixing-machine chest. In a real mill, such a rapid change is not possible. The filler is fed into the mixing-machine chest. When a step change in the filler feed to the tank is made, the filler in the tank outlet has a first-order response. The rate at which the filler in the tank outlet rises is determined by the tank residence time; the smaller the residence time the faster the response. It is only when the tanks are removed is such a sudden change in the filler mass percentage possible. In addition, a large change in filler mass percentage is usually done in small steps. This allows the retention aid controllers to slowly adjust the coagulant and flocculant Chapter 4. Wet End Recycle Dynamics 100 flowrates to maintain filler retention. Too large a change may result in less filler being retained on the wire (i.e., more filler in the recycle streams). This does not affect the recycle dynamic results with the original tank sizes. However, it could be a factor for mills with reduced tank sizes and/or the recycle loop tankless mill. If the retention aid controllers can only handle small sequential filler changes, then the magnitude of the recycle dynamics will be reduced and the dynamics may not be significant for the reduced tank size case. • For the simulations with no tanks present within the recycle loop, the same piping arrangement (i.e., layout and time delays) as the original mill was assumed. As mills move to systems with fewer tanks, the equipment arrangement would likely be different from the original process. For example, piping systems could be removed and the pipe lengths (i.e., time delay) could change. These factors determine whether recycle dynamics are present. The true recycle loop tankless mill may exhibit more or less recycle dynamics than was obtained with the simulation. • The control system included in the simulation model had level controllers on the two of the tank inlets and pressure control on the outlet of the cleaners. When the tank sizes were changed, the control system was not modified beyond adjusting the level setpoints. Since only composition changes were studied and no disturbances were included in the simulation, the control system had no effect on the results. However, in a real process the control system can impact on the extent of recycle dynamics seen. Typically the controllers are tuned for a specific performance (e.g., speed of response or robustness). Essentially, the control system defines how much the flow-rates, time constants, and time delays change in response to setpoint changes and disturbances. As a result, the recycle dynamics is^lso affected by the control system. If the simulations are expanded to include disturbances, the controllers should be re-tuned to give the same level of performance among the three cases (i.e., original tank sizes, reduced tank sizes, and no tanks) so the results can be properly compared. • Lastly, mill data always contains a certain degree of noise. The simulations were done without noise being included. If minor recycle dynamics were apparent, the presence of noise may mask these dynamics and suggest a first-order type response. Realistically, if the dynamics are masked by noise, then it is probably safe to assume the dynamics are not significant. Chapter 4. Wet End Recycle Dynamics 101 4.4 Conclusions The wet end of a paper machine contains numerous recycle streams. It is well known that recycle streams cause process interactions which alter process dynamics. The responses that have been typically associated with recycle systems is the step-wise trend to steady state. The condition for this to occur is when the sum ofthe forward time delay and recycle time delay is greater than six times the forward path time constant (i.e., 9F + 0R > 6 x T>). To determine if paper machines exhibit the step-wise trend to steady state, an IDEAS simulation of a paper mill developed by Yap (1998) was used. An initial analysis of the time constant and pipe delays suggested that the step-wise trend would not be present. The study was performed by making a change in the filler mass percentage in the stream leaving the basis weight valve and recording the change in filler composition in four recycle streams and in the paper leaving the wire. The large tanks in the system caused a dominant forward path time constant which resulted in all of the responses being first order. The main purpose of the tanks in the wet end is to dampen out process variations; however, these tanks slow the process and limit how fast the control system can react to setpoint changes and disturbances. Since there is an interest to reduce tank sizes in the future to make the process faster, the wet end recycle dynamic study was repeated using smaller tanks. Though the tank sizes were reduced, the condition for the step-wise trend to occur was not satisfied for the majority of the recycle streams. As a result, only minor recycle dynamics appeared during the study. It was also found that a recycle loop may dampen out the dynamics caused by other recycle loops. Since the step-wise condition was not satisfied even though the tank sizes had been greatly reduced, the dynamic study was repeated for the system without tanks in the recycle loops (forward and recycle paths). This time the condition for the step-wise trend to steady state was easily achieved. The responses had a clear step response to steady state. As in the reduced tank case, it was found that a recycle loop could smooth the dynamics caused by other recycle loops. This suggests that it may be possible to design the reduced tank size mill or recycle loop tankless mill to have minimal or no step-wise dynamics by choosing the appropriate time delays in the forward or recycle path. The results suggest that the step-wise trend behaviour of recycle systems is not a factor in this mill. Since present day mills typically have large process tanks with residence times larger than the pipe delays, the step-wise recycle dynamics are unlikely to be visible. It is not possible to generalize the results to reduced tank size mills or recycle loop tankless mills because the differing configurations (i.e., pipe layout) causing different time delays in the forward and recycle paths. For the situation with reduced tank size or recycle loop tankless mills, recycle dynamics Chapter 4. Wet End Recycle Dynamics 102 are present. The extent of the recycle dynamics depends on the number of recycle loops and the corresponding time delays. Only four recycle loops were used in the study, the addition of more recycle streams may change the results. Also, the model was not complex enough to dynamically change separation ratios in response to compositional changes. All these factors can affect the recycle response. Chapter 5 Averaging Level Control in the Wet End Inventory level control is an important control problem for many chemical processes. It is common practice in many process plants to try and achieve tight level control to prevent the tank from overflowing or emptying, for safety and environmental reasons. Tight level control may be desirable in some instances, for example in a continuous stirred tank reactor (CSTR) where the level determines the residence time and the reactant conversion. However, in the case of surge tanks, tight level control may not be necessary. The purpose of surge tanks is to dampen stream variations (e.g., flow or composition) so they are not passed to downstream units. If the tank has tight level control, the attenuation of inlet flow disturbances will depend on the disturbance frequency. In the extreme case, the inlet disturbance will simply pass to the outlet with very little attentuation thus defeating the purpose of the tank. The typical pulp and paper mill has many tanks, some are used for raw material storage, while the remainder are used for surge tanks or mixing of raw materials recycled back to the process (e.g. white water, broke, and reclaimed stock). The level control on many of these tanks maybe tuned for tight level control which means flow disturbances are passed to downstream units with little attenuation. Unfortunately, many of these tanks are found within recycle loops. As a result, these flowrate disturbances are recycled back to the process where they can have negative effect on papermachine performance. Furnish mass flowrate variability is one cause of disturbances in the wet end that can lead to paper breaks. This disturbance can be reduced by minimizing the flowrate variability from the tanks. The tank level controllers may be detuned to provide some flow smoothing, but usually the level is not allowed to deviate too far from setpoint. These controllers are not properly tuned to obtain the best flow smoothing. A more appropriate strategy for surge tank control is averaging level control; this strategy provides outlet flow smoothing which dampens the effect of inlet flow disturbances on downstream equipment. This chapter is presented in the following manner, first a literature review of averaging level control is given. The averaging level control algorithm by Foley et al. (2000) is discussed and a summary of the original derivation is provided. The controller algorithm is re-derived to handle sampling delays greater than one, and then the algorithm is tested on a saveall simulation. 103 Chapter 5. Averaging Level Control in the Wet End 104 5.1 Literature Review The objective of averaging level control is to provide outlet flow smoothing to minimize the effect of inlet flow disturbances on downstream equipment. This goal is achieved by using the spare capacity of the tank to absorb inlet flow fluctuations by shifting the disturbance as much as possible to the level, subject to level constraints, and away from the flows. Averaging level control has two conflicting objectives: • To minimize changes in the outlet flowrate (i.e., V F 0 U I should be small) by allowing the level to float between high and low level limits. • To prevent the level from deviating too long from the normal operating level so that capacity is available for future disturbances. If a large disturbance entered when the level was far from the setpoint, the tank could overflow or run dry. Obviously the controller has to achieve some compromise between these objectives. The traditional approach to the averaging level controller problem has been to use a Pro-portional only or Proportional Integral controller deliberately detuned to provide a slow outlet flow response (McDonald and McAvoy 1986) but tuned fast enough to keep the level within the high and low level constraints. Proportional only (P-only) level control allows the outlet to respond gradually to dis-turbances and provides good flow smoothing. However, it results in a steady state level offset which reduces the tank surge capacity for future disturbances (Luyben and Buckley 1977). The process used by Luyben and Buckley was a simple tank with a control valve on the outlet stream. The level was controlled by manipulating the outlet flowrate in re-sponse to inlet flow disturbances. This configuration resulted in the process model being first order and P-only control giving a steady state level offset. Proportional Integral (PI) control returns the level to setpoint, but results in less flow smoothing at frequencies near the closed loop resonant frequency of the level control system (Luyben and Buckley 1977). In addition, PI control produces a flow overshoot (i.e., the flowrate overshoots its final value) which can cause problems for a cascade of tanks. A small overshoot in the first vessel can be amplified as it works its way down the train of tanks. Cheung and Luyben (1979b) showed that the damping coefficient for a train of tanks can become less than one even though the individual tank process has a damping factor greater than one. Thus, another objective that is important for averaging level control is to minimize the outlet flow overshoot when the system is a cascade of tanks. Cheung and Luyben (1979b) presented charts to use in selecting tuning parameters for P-only and PI averaging level controllers based on specifications of maximum peak height (MPH - i.e., the highest flow overshoot) and the maximum rate of change of outlet flowrate Chapter 5. Averaging Level Control in the Wet End 105 (MRCO). Cheung and Luyben proposed a control scheme which would eliminate many of the problems of PI control in a cascade of tanks. The scheme involved conventional PI control of the first tank; the controller was assumed to be some form of averaging level control as no further information was provided on the PI tuning. The outflows of the subsequent tanks were ratioed to their inflows. The ratios were reset by P-only level controllers. Thus, if the flow out of the first tank was smooth and non-oscillatory, then all the flows would be smooth and non-oscillatory. Though the scheme assumes that the tanks are identically sized, different sized tanks could probably be compensated for in the ratio block. While the scheme proposed by Cheung and Luyben reduces the effect of the overshoot on downstream tanks, there are other potential problems. The P-only controllers adjusting the ratio block could still lead to level offset in the downstream tanks. Also, the tank capacity of the downstream tanks may not be fully used. For example, if the inlet flowrate to the first tank goes up, then all the outlet flowrates could increase at the same amount depending on how the ratio and P-only controllers are setup. No detailed information is provided on the arrangement and no simulation results are given. Luyben and Buckley (1977) proposed a Proportional-lag (PL) controller that maintains most of the flow smoothing properties of P-only control but also eliminates offset. The idea combines a feedback Proportional level controller and a feedforward element. Flow into the tank is measured and passed through a first order lag and then added to the output of the proportional controller. The system produces slow gradual changes in the outlet flowrate. Luyben and Buckley (1977) and Cheung and Luyben (1979a) compared the perfor-mances of P-only, PI, and PL control on a tank system with and without liquid recycle. Both found that the PL controller had performance characteristics in between that of P-only and PI control. PL control can handle flow disturbances with a smaller tank than P-only control and has a smaller overshoot than PI control. Luyben and Buckley showed the effectiveness of the PL control was insensitive to the value of the first order lag time constant (rj0ff). He tested the controller with lags 2 x n a g t n o m i n a i and 0.5 x Tiag,nominai without significant loss of performance. Cheung and Luyben (1979b) derived design charts, based on M P H and M R C O , for selecting tuning parameters for the PL controller. According to Cheung and Luyben, the PL controller is a viable compromise between the P-only and PI controller. If a standard P-only or PI control cannot adequately satisfy the objectives of averaging level control, then another solution is non-linear control (Shunta and Fehervari 1976). Averaging level PI control is used as long as the level stays within a safe range, small upsets are absorbed and the reset action returns the level to setpoint. Non-linear action in Chapter 5. Averaging Level Control in the Wet End 106 the controller gain (Kc), and integral time (TJ) happens when there are large disturbances or when the level approaches its limits. Shunta and Fehervari (1976) described and tested three commercially available non-linear controllers: • Non-linear PI: the controller gain is decreased for small disturbances and increased for large disturbances. The integral time (TJ) is adjusted to maintain a constant damping factor (£). The point at which the controller switches from low to high gain is selected to cope with a specific disturbance. The controller gave a satisfactory response. • Wide Range Controller : this was an algorithm by Foxboro. The controller gain (Kc) and integral time (TJ) are related by a constant damping factor. The values when the error is equal to 0 determines the overall damping factor. Kc and TJ are continuously varied in an exponential manner; they are changed only as much as necessary to keep the level within bounds. The controller performed well even in the presence of noise. • Pneumatic PI with Proportional Over-rides : when the level becomes too high or too low the high gain P-only controller takes over. The over-ride is activated by high and low signal selectors. For example, if the level is increasing, the high signal selector compares the PI and P-only control signal and implements the largest one. It is possible for the level to hang up away from setpoint until the PI resumes control. Shunta and Fehervari (1976) suggest a solution using a double differential relief valve (DDRV) to speed the reset action and cause the PI to resume control faster. The performance of the controller with DDRV was comparable to the previous non-linear controllers. Cheung and Luyben (1980) compared the wide range (WR) controller with the P-only, PI, and PL controllers. The W R controller was found to be more versatile, that is, the ability to handle more response specifications of M P H and M R C O , than the other controllers. While the W R controller returned the level to setpoint quickly, it resulted in a sharp outflow overpeak. As noted before, this outflow overpeak is undesirable in some applications. During their study, Cheung and Luyben found the non-linearity of the W R controller caused the noise filtering characteristics of the controller to be dependent upon the noise amplitude. The W R controller approximated a PI when the noise amplitude was low. However, as the noise amplitude increased, the filtering characteristics deteriorated rapidly. Cheung and Luyben went on to propose schemes designed to capture most of the advantages of non-linear control but without the problems. Chapter 5. Averaging Level Control in the Wet End 107 • Proportional Integral/Proportional (PIP) : this is similar to the Proportional Integral with Proportional over-ride proposed by Shunta and Fehervari (1976) on page 106. The difference is the controllers are reversed, that is, the P-only controller acts within the normal error band and the PI activates when the error goes out ofthe band. The advantage of this setup is that P-only control is used for small disturbances which allows for good noise filtering. For large disturbances, the integral action forces the error to return to the normal band. The PIP controller responses are very similar to that of a PI. The disadvantage is that the level will return to within the band but then the P-only control does not return the level to setpoint. • Dual Range Integral Proportional (DRIP) - in this case, PI control is used inside and outside the error band. Only the reset time is changed when the error switches between bands. A large reset time is used within the band to provide a damping factor greater than one. A small reset time is used outside the band to give a faster response. The width of the normal error band is adjusted to keep the overall process always near critical damping. The DRIP controller met the objectives of slow action at small error and fast action at large error without level offset. However, the controller has poorer noise filtering at small errors. • Limited Output Change (LOC): for this scheme the rate of change of the controller output is constrained to meet the M R C O specifications. The change in the output constraint can be used with conventional P-only or PI controllers. When used with a PI, care must be taken to prevent reset windup. While there are advantages to non-linear control, they are difficult to tune. All of the non-linear controllers presented were tuned by trial and error. As a result, it is possible for a different set of tuning parameters to give better performance (i.e. lower MRCO) without violating the level constraints. The optimal controller would be the one to achieve the lowest possible M R C O without constraint violation. Cutler (1982) developed a model based predictive controller called Dynamic Matrix Control (DMC). The approach provides a continuous prediction of future outputs over the time horizon required for the system to reach steady-state. The optimality ofthe controller was based on finding the input changes that minimize the error over the time horizon. The controller was tested on an averaging level control problem and outperformed a PI controller. McDonald and McAvoy (1986) derived two averaging level controllers, the Ramp Con-troller and the Optimal Predictive Controller, that gave the smallest possible M R C O for a given M P H specification and a known upset in the inlet flowrate. The controllers are explained below: Chapter 5. Averaging Level Control in the Wet End 108 • Ramp Controller (RC) : is a non-linear proportional feedback controller with a stan-dard integral (or PL scheme of Luyben and Buckley 1977) to eliminate offset. The controller avoids a large outflow overshoot and minimizes the effect of the integral action by setting the reset time (or PL lag time) to be large ( approximately 10 x rmax where rmax is 2x the time for the tank to overflow). The RC was compared with the P-only, PI, PL, PIP, and DRIP schemes. The RC obtained the lowest M R C O for the specified M P H of all the controllers. • Optimal Predictive Controller (OPC) : is a feedforward-feedback controller based on a predictive extension of the control law. It assumes that the inlet flowrate can be measured and there is negligible dead-time. The OPC has Proportional and Integral modes added to counter level offset and flow measurement bias. The control performance depends on the sampling frequency, a higher sampling frequency gives a smaller MRCO. O P C was compared to the best averaging level controllers, i.e., the W R controller (Shunta and Fehervari 1976) and the above RC. The O P C gave similar or better results (i.e. lower MRCO) than the other controllers. The results depended on the input disturbance magnitude. OPC was also compared to D M C , either the O P C or the D M C performed better depending on the input. However, the D M C tuning is much more involved than the OPC. If the system contains dead-time, the O P C approach becomes more complex than the D M C scheme. According to Campo and Morari (1989), OPC has two main drawbacks: (a) It requires the inlet flow measurement which is often not available and (b) The I modes are added in an ad-hoc manner to eliminate steady state offset. Campo and Morari (1989) expanded McDonald's optimal predictive strategy into a model predictive format - referred to as Model Based Optimal Predictive Control (MBOPC) in this review. The controller min-imizes the M R C O and handles constraints. By including an end point condition in the optimization objective, integral action is included in the controller. The controller includes a single adjustable parameter (i.e., horizon length) that directly affects the tradeoff between good flow filtering and rapid settling time. The inlet flow disturbance is inferred from the level at each sample time and the optimization resolved based on the new information. Only the first of the optimal future inputs is implemented. Internal stability of the M B O P C scheme is guaranteed if the plant is stable. If the plant is unstable, the process is first stabilized using an internal feedback loop. The plant and embedded stabilizing controller is then treated as the process in the M B O P C approach. Campo and Morari (1989) compared the M B O P C scheme to the O P C method. The O P C algorithm was able to achieve a lower M R C O but with significantly longer settling Chapter 5. Averaging Level Control in the Wet End 109 time. The O P C can give the same settling time as the M B O P C with a slightly lower M R C O but has a significant flow overshoot and the level and outlet flows are more oscil-latory. The OPC achieves a lower M R C O since it uses inlet flow measurements and can adjust the flow immediately, while the M B O P C requires one sampling time to infer the flow disturbance from level measurements. Khanbaghi et al. (2001) analyzed the M B O P C algorithm in the frequency domain and gave tuning recommendations. They noted that the estimate of the inlet flow was sensitive to noise. The noise sensitivity problem was solved by using a Kalman filter for flow estimation. Khanbaghi et al. successfully applied the algorithm with a Kalman filter to an industrial saveall tower in a pulp mill. The controller reduced the outflow variability without violating the high and low level constraints. The concept of averaging level control has also been applied to Kamyr digester level control. Amirthalingam and Lee (2000) developed a model based predictive control method coupled with a state estimation method that allowed the level to float within an acceptable range. The optimization objective for the algorithm was to minimize the variations in a quality measure called Kappa number (i.e., a measure of residual lignin composition). This optimization objective is what makes the algorithm different from averaging level control in which the objective is to minimize the variations in outlet or inlet flow. Friedman (1994) provided some general advice on tuning averaging level controllers. He acknowledged the use of non-linear controllers but believed them to be over-rated. However, he still provided some advice on their tuning. Even though advanced algorithms are available, there are times when they may not provide any better performance than a PI. Korchinski (1995) states that when non-linear algorithms became available they were applied to almost every vessel. However, the prob-lems (i.e., the oscillatory response) still existed and in some cases the situation was even worse than when the standard PI was in place. The poor control was caused by surge vessels that were too small. Korchinski (1995) suggested that if the inlet flow variations .were larger than half the drum volume, PI control should be used instead of non-linear control. Thus, if the tank was too small, the PI would handle the situation by passing the inlet flow disturbance to the outlet. However, if a non-linear controller was used, it would make small changes when the level was near setpoint and large changes when the level deviated from setpoint. The resulting output flow takes the form of a square wave which is more taxing on downstream equipment. Kelly (1998) revisited using PI control for averaging level control. He presented a design approach to the problem of optimal PI tuning using historical data. The process was identified using closed loop system responses. The level control problem was modeled Chapter 5. Averaging Level Control in the Wet End 110 as a simple material balance equation, and the inlet flowrate changes were represented by a random walk noise model. Kelly states that more complex stochastic models would require model identification which is not possible with closed loop data. He derived expressions for level and outlet flow variance in terms of the PI tuning parameters. The problem then was to find the solution which minimized the outlet flow variance subject to the level variance specified. From the solution, the optimal PI tuning parameters are obtained. The constrained optimization was performed in Matlab. Kelly provided PI tuning charts for systems with one, two, and three units of sampling delay. The model identification procedure and PI tuning procedure were applied to industrial data and improved results were obtained. No information was provided on the data or the process so his results could not be reproduced. Foley et al. (2000) presented another solution to the averaging level control problem. The method was based on constrained minimum variance control (CMV). C M V has the property that for a given process output variance, no other controller has a lower variance in the differenced input. Foley et al. applied the C M V method to averaging level control using the simplified process model developed by Kelly (1998) for the case of one unit of sampling delay (i.e., b=l). The resulting controller had the form of a PI controller in series with filter of order b-1. Foley et al. derived analytic expressions for the b = 1 case which allow easy calculation of the optimal PI tuning parameters using a move surpression parameter (A). A tuning algorithm was also provided which related the selection of A to the user specified output flow variance. The algorithm was tested on a computer simulated surge tank and good flow smoothing was obtained without level constraint violation. Sidhu et al. (2001) applied Foley et al.'s algorithm to an industrial caustic dilution process in a paper mill. The controller produced significant flow smoothing which resulted in more consistent caustic dilution. Ogawa et al. (2002) proposed a new approach to averaging level control. Ogawa et al. modified the control objective to minimize the tank outlet flow variance and the variance of some downstream process variable while keeping the level within the high and low level constraints. The variance of the downstream variable was included in the performance index to be minimized and the problem was solved in continuous time using the Weiner-Hopf method. The resulting controller was a PI controller cascaded with a lead-lag network (referred to as the PIL controller). The PIL controller was able to reduce the variance of a downstream variable by as much as 20% compared to the PI controller from Foley et al. (2000). It should be noted that constraints are not built into the algorithms by Kelly (1998) and Foley et al. (2000). The tuning parameters are selected based on a maximum inlet flow disturbance and keeping the level within constraints. If the maximum inlet flow Chapter 5. Averaging Level Control in the Wet End 111 disturbance increases, then the level may violate constraints. This fact was pointed out by Khanbaghi et al. (2001) who compared Foley et al.'s algorithm (referred to as CMV-PI) to the MBOPC by Campo and Morari (1989). For a specified maximum inlet flow disturbance, both the CMV-PI and the MBOPC gave the same amount of flow filtering while keeping the level within the constraints. However, when the maximum inlet flow disturbance was increased, the MBOPC kept the level within bounds but the CMV-PI had a level constraint violation. The MBOPC is a good algorithm for averaging level control especially for cases with unknown maximum inlet flow disturbances. If the maximum inlet flow disturbance is known, then the CMV-PI can provide similar performance as the MBOPC and is easier to implement. As a result, the CMV-PI algorithm is the one chosen for application in this work. In the next section, the CMV algorithm by Foley et al. (2000) is given in more detail. 5.2 C M V Averaging Level Control The averaging level control algorithm developed by Foley et al. (2000) is based on the linear quadratic or constrained minimum variance controller. Foley et al. (2000) first derived the general equations for the CMV controller and then applied it to the averaging level control problem for the case of one unit of sampling delay (i.e., b=l). A summary of the general CMV controller equations is reproduced from the detailed derivation given in the paper by Foley et al. (2000). Since, systems are not restricted to 1 unit of sampling delay, the controller algorithm is derived for the cases when there are two (b = 2) and three (6 = 3) units of sampling delay. As this is a continuation of the work by Foley et al. (2000), the same notation used in his paper is used in the derivations presented. 5.2.1 General C M V Summary The development of the Linear Quadratic controller is based upon the Box-Jenkins process model: Y, = G(,-')r/, + G D ( z - > « = 0^U,.b + ^ T ^ < (5-1) where • Yt is the measured process output • Ut is the manipulated variable Chapter 5. Averaging Level Control in the Wet End 112 • at is a zero mean normally distributed discrete white noise with variance o\ The polynomials 8(z^1), 6(z~1), and 4>(z~l) are assumed to be invertible. The number of whole periods of delay is given as b, with b > 1. The symbol V denotes the difference operator 1 — z~l. The integer d is included as a means of introducing nonstationarity into the disturbance model (Go)- The restriction d > 1 assumes the presence of process disturbances, such as unmeasured changes in the tank inlet or outlet flowrates, which are integrated as their effect upon the plant output changes with time. xy^-* ec(z') Wblz'j 11 + D, Y. Figure 5.1: Feedback Control System The feedback control diagram for the discrete time system of eqn. 5.1 is shown in Figure 5.1. For simplicity, the setpoint is assumed to be constant at its initial value so rt may be taken as zero. The optimization objective is to determine the feedback controller (Gc(z~1)) which minimizes the constrained minimum variance cost function: J = Var(Yt) + XVar(Vd-lUt) (5.2) The optimal feedback controller obtained from minimizing the above L Q objective function (eqn. 5.2) is : c [ ' 9(z~1)ry(z~1) — z~huo(z'l)Qi(z~l) f{z~l)Vd { ' } where the polynomials 7 (z _ 1 ) , Qi(z~l), and Q2(z~1) are obtained from solving the fol-lowing equations: Spectral . ^-x^ = u{z-i)uj{z) + A ( l - . " ^ - ^ ( l - z)d (5.4) Factorization Diophantine : w(z)zb9{z-x) = 7 ^ ) Q i ( « - 1 ) + zQ2{z)Vi<j>{z~1) (5.5) Chapter 5. Averaging Level Control in the Wet End 113 The polynomials Qx(z~l) and Q/2(z~1) in eqn. 5.5 are of respective orders max(q — b,p + d—l) and max(s + b,ng) — 1, where by definition q = deg(8), p = deg((f>), s = deg{ui), and ng = deg(j). Since Foley et al. (2000) showed that the factor V r f divides B(z~1)ry(z~1) — z~bco(z )Qi(z ) with finite remainder f(z), the controller can also be written in the second form shown in equation 5.3. The closed loop output and input variances are obtained from the following equations: <7v l*l=i f(z-l)f{z) dz l{z~l)l{z) z cr; V(7 Var(Vd-lUt) 27TJ J 1*1=1 Qi{z-l)Qx(z) dz l{z~l)l{z) z (5.6) (5.7) 5.2.2 C M V : Application to Averaging Level Control The constrained minimum variance algorithm can be applied to averaging level control. Figure 5.2 shows a typical liquid level control problem where the level in the tank is controlled by PI controller cascaded to an outlet flow controller (Kelly 1998; Foley et al. 2000). Inlet Flow Tank Level h. O Outlet Flow q°, Figure 5.2: Liquid Level Control Diagram The level (expressed in % scale) is modeled by the following simple material balance: Yt - ^Ft_b - —Ft_b (5.8) Chapter 5. Averaging Level Control in the Wet End 114 The scalar coQ is defined as — 100T/V where V is the tank volume and T is the controller execution interval. The tank level is assumed to be controlled by manipulation of the outlet flowrate (F™1 = Ut-b)- The unmeasured inlet flowrate is modeled as a random walk: Ft = | (5.9) where ct is a zero mean discrete white noise with variance o2, which is the variance of the differenced inlet flowrate. Substituting eqn. 5.9 into eqn. 5.8 leads to: where a2 = u%o2 and at u(z-1)=co0 5(z~1) = l e(z~1) = l (f)(z-1) = l d = 2 (5.11) The only parameter undefined is b, the number of units of sampling delay. Foley et al. (2000) derived the case for 1 unit of sampling delay; in the following section the algorithm is expanded for the case when there are two (b = 2) units of sampling delay. The derivation for the case when there are three (b = 3) units of sampling delay case follows along the same lines as the b = 2 case and is shown in appendix F. 5.2.2.1 Averaging Level Control for Two Units of Sampling Delay When the plant time delay is equal to two units of sampling delay, the process model eqn. 5.8 becomes: Equation 5.12 has the same structure as eqn 5.1 where OJ(Z-1)=UJ0 S(z-1) = l 0(z-l) = l <t>{z-1) = l ti = 2 (5.13) The above polynomials have the following degrees: deg(uj) = s = 0 deg(8) = r = 0 deg(9) — q — 0 deg(4>) = p — 0 Y, = - £ (5.10) cooCt- Equation 5.10 has the same structure as eqn 5.1 where: Chapter 5. Averaging Level Control in the Wet End 115 Spectral Factorization The spectral factorization equation (eqn. 5.4) becomes equation 5.14 when the appropriate substitutions are made for u)(z~l), 5(z~l), and d. 7 ( ^ ) 7 ( 2 ) = ul + A(l - z-'f{l - zf (5.14) The spectral factorization equation is unchanged from the b = 1 case, thus, the equations for 7o, 71, and72 taken from Foley et al. (2000) are as shown below: 7i = -— + sgn(u0)^^ (5.15) 2A . ,1 /16A2 „A . „, 72 = \-sgn{Lj0)-\ — 5 4 A ( 5 - 1 6 ) 7 i 2 V 7 f 7o = w0 + 7i + 72 (5.17) where sgnQ denotes the signum function, e.g., sgn(coo) = +1 when co0 > 0, sgn(cu0) = — 1 when u>o < 0. Diophantine Equation Again, making the appropriate substitutions for u(z~l), 9(z~l), <j)(z~l), b, and d in equa-tion 5.5, the diophantine equation becomes: co0z2 = 7 ( ^ )Q 1 ( z - 1 ) + zQ2(z)V2 (5.18) The Q polynomials in eqn. 5.18 have the following degrees: deg(Q\) — max(q — b, p + d — 1) = max(0 — 2,0 + 2 — 1) = max(—2,1) =1 deg(Q2) = max(s + b, ng) — 1 = max(0 + 2,2) — 1 = max(2,2) — 1 = 1 Thus, the Q polynomials have the following form: Qi = Qw ~ Qnz~l (5.19) Q2 = q2o - Q21 z"1 (5.20) Chapter 5. Averaging Level Control in the Wet End 116 Substituting for 7(2), Qi(z l) and Q2(z) into eqn. 5.18: u0z2 = [70 - l\z - 72^2][?io - Qnz'1} + z[q20 - 921 ^ ][1 - z~lf (5.21) ={A) =(B) Expanding the terms (A & B) on the right hand side (RHS) individually: (A) = [70 - 7is - j2Z2}[qw ~ qnz'1} = 7o9io - loQnz'1 - Jiqwz + 7 i 9 n - 7 2 9io^ 2 + 729ii^ = -72gio^ 2 + (-7i0io + 72911)2 + (7o9io + 7i9n)^° ~ 7 o 9 i i 2 - 1 (5.22) (B) = z[q20 - q21z][l - z'1}2 = z{q20 - q2lz}(l - 2z~x + z~2) = z(q20 - 2q2Qz~1 + q20z~2 - q21z + 2q21 - q2xz~l) = q20z - 2q20 + q20z~l - q2Xz2 + 2q2Xz - q2X = -Q2iz2 + (2q21 + q20)z + (-2<?20 - 921)2° + qwz'1 (5.23) Replacing (A) and (B) in eqn. 5.21 with the corresponding expansions (eqns. 5.22 and 5.23): cu0z2 = (A) + (B) = {-72910 - 92i}^ 2 + [72911 ~ 7i9io + 2921 + 920]^ + {7o9io + 7 i 9 n - 2920 - 921 }z° + [920 - 7o9n]2~ 1 (5.24) Equating LHS and RHS z polynomial coefficients of like powers gives the following equa-tions: - 729io - 921 = w0 (5.25) - Tifto + 729ii + 920 + 2921 = 0 (5.26) 7o9io + 7i0ii ~ 2920 - 921 = 0 (5.27) - 7o9n + 920 = 0 (5.28) Now, the following identities are known from the solution of the spectral factorization equation (Foley et al. 2000) for the b = 1 case: 7(1) = 70 - 71 - 72 = w0 (5.29) 7(l)Q 1 (l) = w0 (5-30) Chapter 5. Averaging Level Control in the Wet End 117 Thus, <3i(l) = 1. Using equations 5.30 and 5.19: <9i(l) = 9 i o - 9 n = l => ? i o = l + 9n (5.31) Subsituting for qw in eqn. 5.25: -72(1 + 911) - 921 = w0 =>> 921 = - w 0 - 7 2 ( 1 + 911) (5.32) From eqn. 5.28, q20 — 7o9n- Substituting for qw, q20, and 1721 in eqn. 5.27: 7o(l + 9n) + 7 i 9 n - 27o9n - (-w 0 - 72(1 + 9n)) = 0 7o + 7o9n + 7 i 9 n - 270911 + w0 + 72 + 729n = 0 7o + w0 + 72 + (70 + 71 - 2 7o + 72)911 = 0 70 + OJ0 + 72 + (-70 + 7 i + 72)911 = 0 7o + w0 + 72 - (70 - 7 i - 72) 9 n = 0 = uio (from eqn. 5.29) Thus: wp + 7o + 72 QQ\ 9n = (5-33) Wo 2u0 + 7o + 72 , . 9io = 1 + 9n = (5-34) w0 Controller Structure The general optimal feedback controller given in equation 5.3 is shown below: c[ ' 9{z-1) - z-^z-^Q^z-1) f(z~l)Vd (5.35) Now: f{z^)Vd = eiz'1)^-1) - z-^iz-^Q^z-1) (5.36) The known values are b — d — 2, and the polynomial degrees are deg(9) = 0, deg(j) = 2, deg(u) = 0, deg(Vd) = 2, and deg{Qx) = 1. The largest degree on the RHS of equation 5.36 is deg(z~bu>(z~1)Q1(z~1)) = 3, thus, the deg(f) = 1. Making the appropriate subsitutions into eqn. 5.36: Chapter 5. Averaging Level Control in the Wet End 118 (/o - fiz X)(l - z x)2 = (70 - 7i* 1 - 722 2 ) - z 2co0(qw ~ qnz x) A = 7o - 7 i2 1 ~ 722 2 - co0qwz 2 + u0qnz 3 = 70 - 7 i z _ 1 - (72 + L)0qw)z~2 + WoquZ~3 (5.37) Expanding the LHS term (A): (/o - hz-l){l - z-1)2 = (/o - fxz-l)(l - 2Z'1 + z-2) = fo- 2/02- 1 + / 0 2 - 2 - hz~l + 2flZ~2 - fxz~3 = fo- (2/o + h)z~l + (/o + 2f1)z~2 - hz~z (5.38) Equating like coefficients in powers of z on the RHS of eqn. 5.37 with like coefficients on the RHS of eqn. 5.38: 7o = fo (5.39) - 7 i = -(2/o + /i) (5-40) -(72+w o t7io) = /o + 2/ x (5.41) w 0gn = - fi (5.42) From eqn. 5.39, / 0 = 7o- Substituting for / 0 in eqn. 5.40: 7i = 27o + / i / i = 7 i - 2 7 o (5.43) Subsituting for the known polynomials into eqn. 5.35: n , _x, _ Qi{z l)S(z : ) q10 - qnz 1 ^c\Z ) f(z-i)Vd-i V f0- hz-1 2w0 + 7o + 72 w 0 + 70 + 72 1 w0 w0 / V 70 - (71 - 270)2- 1 Chapter 5. Averaging Level Control in the Wet End 119 Making the last term on the RHS of equation 5.44 monic: G c ( z - i ) = "0 + 70 + 72 w07o 1 + W Q + 70 + 72 V i _ i ( 7 l - 2 7 o ) « - i (5.45) If the following parameters are defined as: w0 + 7o + 72 T w0 WQ7O T/ 0J0 + 7o + 72 (5.46) Then, Gc has the structure of a PI controller in series with a first order filter as shown below: Kc[l + ^  - z-1) 1-^(71 - 2 7 o ) « - 1 (5.47) Closed Loop Performance for Two Units of Sampling Delay The general equation for the output variance given in equation 5.6 is shown below: „2 _ < i f{z~l)f{z) dz _ OJ 2irj z=l )j{z) z 2nj z\=l (5.48) where $ 2 f{z-l)f{z) J(Z-1)J(Z)Z (70 - 7 i z _ 1 - j2z'2h(z)z (5.49) According to Foley et al. (2000), the contour integral given in equation 5.48 may be evaluated using the Residue Theorem: J> $zdz = 27cjJ2Reszn®(z) • / . n=l \z\=l (5.50) The symbol ResZn$(z) refers to the residue of $ ( 2 ) at zn, where zn is the n'th pole of $(z) inside the integration contour \z\ = 1. Because the residue is only concerned with the poles, equation 5.49 can be rearranged, multiplied by z2/z2, and factored to give: fji^fiz) (z2 - foz - ^)j(z) (z - Zl)(z - z2h(z) (5.51) Chapter 5. Averaging Level Control in the Wet End 120 The poles of equation 5.51 are shown in equation 5.52 (Foley et al. 2000). 7 i , Vii +4727Q 7 i Vii + 4727o ( t . K O x Z l = o — h — — i — 2 : 2 = o n — i — t 5 - 5 2 ) 27o 2 | 70 | 270 2 | 70 | In the solution of the spectral factorization equation (eqn. 5.14), 70, 71, and 72 were chosen so that ^(z"1) was an invertible polynomial (i.e., the roots lie within the unit circle \z\ < 1) with real coefficients (Foley et al. 2000). Since ^(z~l) has both poles within \z\ < 1, the roots of 7(2) are in \z\ > 1 and do not contribute poles to the residue. Thus, equation 5.48 can be written as: o\ = [ResZl$(z) + ResZ2${z)] o\ (5.53) Expanding the numerator of equation 5.51: -f{z-l)f{z) = - [ /o - hz-'U - flZ] 7o 7o = -[/o*-/i][/o-/i*] 7o = - [ / ? * - / o / i ^ - Z o / i + A2*] 7o = -I-/0/1 + (/o2 + fl)z - fohz2} (5.54) 7o The individual residue terms in equation 5.53 are as follows: o * ^[-/o/i + (/o2 + / i 2 K - / o / i ^ ] ResZl$z = -Q— v T— (5-55) (zi - 22)(7o - 7i*i - 72^i) i[-/o/l + (/o2 + fl)z2 - hhzl] (zi - ^ 2 ) ( 7 o - 71^ 2 — 72^ 1) ResZ2$z = - , w L / „ u j i „ \ r i 1 : ( 5 . 5 6 ) The numerator term I/70 common to both residues in equations 5.55 and 5.56 is moved outside the square brackets of equation 5.53 to form 0^/70 . The lowest common denomi-nator for both residue terms is : (z\ — z2)(70 — 71^ 1 — 72^1)(70 — 7i32 — 72^2)- The resulting numerator residue terms are as shown: ResZl Num = [-/0/i + (/02 + /2)zi - /o/i22](7o - 71^ 2 - 72*2) = - 7o/o/i + 71/0/122 + 72/0/1*2 + 7o(/o + fi)zi ~ 7i(/o2 + fl)ziz2 - 72(/0 2 + fx)zxz\ - 70/0/1212 + lxhhz\z<i + 72/0/1^1^2 ( 5 - 5 7 ) Chapter 5. Averaging Level Control in the Wet End 121 ResZ2 Num = [-/0/i + (/02 + f\)z2 - /0/i*2](7o - 7i*i ~ 12*1) = 7o/o/i - 7i/o/i*i - 72/0/12? - 7o(/02 + /1 )*2 + 7i(/o2 + /i)*i*2 + 72(/02 + /2)*?*2 + 7o/o/l4 ~ 7l/o/l2l*2 _ 72/o/l2222 (5.58) Combining eqns. 5.57 and 5.58: = -71/0/1*1 + 7o(/02 + / 2 )2i + 71/0/1*2 - 7o(/02 + / 2 )2 2 - 72/0/1*? - 70/0/1*? + 72/o/i*| + 7o/o/i*2 + 72(/02 + /i)*i2*2 + 7i/o/i*?*2 - 72(/02 + /1)*i*2 - 7i/o/i*i*2 - -*i[7i/o/i - 7o(/02 + fi)} + * 2[7i/o/i - 7o(/02 + /1)] - *? [72/0/1 + 7o/o/i] + *22[72/o/i + 7o/o/i] + *?*2[7i/o/i + 72(/02 + A2)] - *i*22[7i/o/i + 7 2(/ 0 2 + fl)} = [71/0/1 - 7o(/02 + /i)](*2 - *i) + [72/0/1 + 7o/o/i](*22 - *?) + [71/0/1 + 7 2(/ 0 2 + /i2)](*?*2 - *i*22) (5.59) Let: Z\ = a + b\fc (5.60) where: b = 2 I 7o c = 7? + 4727o (5.61) Using equations 5.60 and 5.61, the following identities are specified: (5.62) z\ = (a - by/c)2 = a2 + b2c - 2aby/c z\ = (a + by/c)2 = a2 + b2c + 2aby/c z\- z\ = -kaby/c (5.63) Chapter 5. Averaging Level Control in the Wet End 122 z\z2 = (a2 + b2c + 2aby/c) (a - by/c) = a 3 - a2by/c + ab2c - b3cy/c + 2a2byfc - 2ab2c = a 3 + a2by/c - ab2c - b3cy/c Z\z\ = (a2 + b2c — 2ab\/c)(a + 6\/c) = o? + a2by/c + ab2c + b3cy/c - 2a2by/c - 2ab2c = a 3 - a2byfc - ab2c + 6 3 C\/c .-. z\z2 - zxz\ = 2a2<Vc - 26 3c^ (5.64) Substituting the above equations (5.62 - 5.64) into eqn. 5.59: ResZl Num + ResZ2 Num = - 26>/c[7i/o/i - 7o(/o + /i)] - 4ab^/c[-y2f0fi + 7o/o/i] + ( 2 a V c - 263cV^)[7i/o/i + 7 2(/ 0 2 + fl)] = - 26v^{[7i/o/i - 7o(/o + /i)] + 2o[ 7 2 / 0 / i + 70/0/1] + (b2c - a 2 )[ 7 i /o/i + 72(/02 + A2)]} (5-65) The denominator for the residue terms is now: Oi - 22)(7o - 7i*i - 72*?)(70 - 7i*2 - 72*2) (5-66) Since zi — z2 — 2by/c, this term cancels the corresponding numerator term in eqn. 5.65. The new numerator is shown below: ResZl Num + ResZ2 Num = - 1{[71/0/1 - 7o(/02 + fl)] + 2a[72/0/1 + 70/0/1] + (b2c - a 2 )[ 7 i /o/i + 72(/02 + A2)]} (5.67) The resulting denominator is the same as in Foley et al. (2000), the expansion of the denominator is reproduced below from the paper: (70 - 7i*i - 72*?) = 7o - 7i(o + by/c) ~ ^{a2 + 2aby/c + b2c) = 7o - 7 i« - lib^/c - 7 2 a 2 - 7262c - ^22ab\fc = [70 - 071 - (a2 + b2c)l2] - (67x + 2a& 7 2)v^ (5.68) (70 - 7i*2 - 72*2) = 7o - 7i(a - byfc) - 7 2(a 2 - laby/c + b2c) — 7o - 7 i« + 7iby/c - 7 2 a 2 - 72&2c 4- j22aby/c = [7o - 071 - (a2 + 62c)72] + (671 + 20672)^ (5.69) Chapter 5. Averaging Level Control in the Wet End 123 (To - 7i*i - 72*i2)(7o ~ 7i*2 - 72*2) = [7o - G7i - (« 2 + &2c)72]2 - (671 + 2a&72)2c 7o _ (lL + 7i2 +47072^ 27o V47o "*70 27oill - 72) - 7i2 (7o + 72) 1 7i2(7o + 72) 4 o2 J 72 7i 27! 72 27o2 2|7o| 47o|7o| 2 •(7i2 + 47072) (7i2 + 47o72) [27o(7o - 72) - 7?]2(7o + 72)2 7i (7i + 47o72) (7o + 72)5 47o4 47o4 (27 o 2 - 7i2 + 2A) 2 ( 7 o + 7 2) 2 72(7i2 - 4A)( 7 o + 7 2) 2 47o4 47o4 10 • ^10 (70 + 72)2[(27o2 - 7i2 + 2A)2 - 7 l 2 (7 2 - 4 A ) ] 47o4 (5.70) The expansion uses the identity 7072 = — A obtained during the solution of the spectral factorization equation in Foley et al. (2000). Combining the numerator (eqn. 5.67), the denominator (eqn. 5.70), and the cr^/jo term (see the bottom of page 120) that is now outside the brackets of equation 5.53: A = (-l)-(47o 4)-[71/0/1 - 7o(/02 + /1)] + 2o[ 7 2 /o/ i + 7o/o/i] +(6 2c-a 2)[7 1/o/i+72(/ 0 2 + /i2)] 7o ' - (7o + 72)2[(27o2 - ll + 2A)2 - -fitf - 4A)] Making the following subsitutions in equation 5.71: (5.71) a 7i 27o /o = 7o 2a = ^ 7o /o2 = 7o2 b = 2 I 7o I A = 7i ~ 27o c = 7! + 47072 u2 2 be - a — •(7i2 +47o72) 7i _ 72 47o 7o (5.72) b2 = 47o2 47o2 fx = (7i - 27o)2 = 7i2 - 47oTi + 4T02 Chapter 5. Averaging Level Control in the Wet End 124 The numerator of eqn. 5.71 (excluding the — (CT2/7O)47Q term) becomes: [71/0/1 - 7o/o - 7o/i] + 2a/o/i[72 + 7o] + (b2c - a 2 ) 7 i /o / i + (b2c - a 2 ) 7 2 / o 2 + (b2c - a 2) 7o/i 2 = /o/i{7i + 2a( 7 2 + 7 o ) + (b2c - a 2) 7 l} + /o2(-7o + (b2c - a 2 ) 7 2 ) + / 2 (-7o + (b2c - a 2 ) 7 2 ) = 7o(7i - 270) S 7i H (72 + 7o) + f + To ( "To + — J I To 7o J V To/ + ( 7 l - 2 7 o ) 2 (-70 + ^ -V To 2 = — (7! - 270) {7o7i + 7i72 + 7o7i + 71T2} +— (T! ~ To) To N v ' To v v ' =27071 +27172 = 271(70+72) =(72+70) ( 7 2 - 7 0 ) + - ( T i - 2 7 o ) 2 (722 -TO 2 ) ' To v v ' = ( 7 2 + 7 0 X 7 2 - 7 0 ) = —[27071(71 - 27o)(72 + 70) + 7o(T2 + To)(72 - To) + (Ti - 27o)2(72 + 7o)(72 - To)] To = ^ ^ [ 2 7 0 7 1 ( 7 1 - 27o) + (7o2 + (Ti - 27o)2)(72 - 7o)] (5.73) To Thus, the variance of the output equation 5.71 becomes: 2 =(_,A 4 ^ [ 2 T o T i ( T i ~ 27o) + (7o2 + (7i ~ 27o)2)(72 - To)] ° Y t [ } 7o ( 7 ° j (7o + 72)2[(27 o 2 - 7? + 2A)2 - 72(7i2 - 4A)] _ ( 1 U 2d-, 2EfroTiOtt ~ 2 ^°) + fro + fr1 ~ 27O)2)(T2 ~ To)] 7 , , ~ [ W 7 ° (To + T2)[(27o2-7i2 + 2A) 2 -7i 2 (7i 2 -4A)] l ^ 4 j Controller Action Variance The C M V general equation for the controller action variance is given by equation 5.75: ' 2 r Qx{z-l)Qi{z) dz l*l=i Defining the following complex function: v t ! w 2TTJ J -y(z-lh(z) z K ' a M Qxjz-^Qxjz) ®u(z) - , _ n , , (5.76) Chapter 5. Averaging Level Control in the Wet End 125 Then, the residue theorem (eqn. 5.50) can again be used to evaluate the contour integral defined in equation 5.75. <f $u{z)dz = 2nj Reszn$u(z) (5.77) "* n=l The following variables in equation 5.76 are known: r> t - i \ - i 2uJ° + 70 + 72 , _ i x - i _ 2 Qi{z ) = g i o - g i i * , 010 = , 7(* ) = 7 o - 7 i * - 7 2 * N , N Wp + 70 + 72 2 Qi{z) = 010 - 0 i i * , 011 = , 7(2) = 7o - 7 i * - 72* Wo Substituting the appropriate polynomials into eqn. 5.76, multiplying by z2/z2, and fac-toring the denominator: ~ / x (0io - 0n*~ 1 )(0io - 011*) <*u\z) — (70 - 7 i * _ 1 - 72* _ 2 )7(*)2 ^ _ (010-011* ^(010 - 0 i i ^ ) ^ z2 _ (0io*-0u)(0io-0i i*)^ u [ z ) * 2 ~ (*2 - i * - ; ) 7 ( * ) * " ( * - * i ) ( * - 22)7(2) ( j The denominator is the same as the denominator in equation 5.51, thus the poles are the same as those in equation 5.52: = 7 i Vii +4727o _ 7 i Vii + 4 727o Z l ~ 2 7 o 2 I 7 o I 2 2 ~ 2 7 o 2 I 70 I As before let: 7 i 1 a = — , 6 = — , c = 7? 4- 47270 =>• 21 = a + 6 ^ , z 2 = a - by/c 2 7o 2 I 7o I The analysis then follows the same pattern as for o\\ o%v = [ResZl$u + ResZ2<J>u} a 2 (5.79) where, _ (01Q21 ~ 01l)(010 - 0 1 l 2 i ) ^ _ (01Q22 ~ 01l)(010 ~ 011*2^ (21 - 22)(7o - 7121 - 72*i) (22 - *i)(7o - 7i*2 - 72*2) Chapter 5. Averaging Level Control in the Wet End 126 Substituting into eqn. 5.79: (9io*i - 9n)(9io ~ 9 n * i ) ^ (9io*2 - 9n)(9io - 911*2)^ (*i - *2)(7o - 7i*i - 72*2) (*i - *2)(7o - 7i*2 - 72*1) (5.80) The structure of equation 5.78 is the same as equation 5.51 for ayt, where 910 is equivalent to /o, and qu is equivalent to f\. Thus, the residue terms wi l l also have the same structure. Using the structure of Oyt for o27U but making the subsitution of §10 for / 0 , and qn for fx: [ [71010011 - 7o(9io + 9?i)] + 2a[729io9n + 7o9io9n] 2 _ , , y a t A 4n1 + ( & 2 c - a 2 ) [ 7 i 9 i o 9 u +72(9io+ 9u)] , °vu-{ D 7 o(47o) ( 7 o + 7 2 ) 2 [ ( 2 7 o 2 - 7 l 2 + 2 A ) 2 - 7 l 2 ( 7 l 2 - 4 A ) ] ( 5 " 8 1 ) Expanding the bracketed part of the numerator in eqn. 5.81 and then making the appro-priate substitutions for a, 6, c given in equation 5.72: [719io0ii - 7o(0io + 0n)] + 2a[ 7 29io9ii + 7o9io9n] + (b2c - a 2)[ 7i9io9n + 72(9^ + 9n)] = [7i9io9n - 7o9io - 7o9ii] + 2a<?i09ii[72 + To] + {b2c - a 2 ) 7 l g i 0 9 n + {b2c - o 2 ) 7 2 9 2 0 fn + (b2c - a 2 ) 7 2 9 i = 9io9ii f 7 i + ^(72 + 7o) + ^ ) + 9 2 0 ( -70 + + 9 2 i f-7o + ^ V 7o To / V To/ V 7o 2 2 = ^ ^ ( T I T O + 7i72 + 7i 7o + 7172) + — (-To + T2) + — ("To + T2) 7o 7o 7o = — [2gi0gn(T0T1 + 7i72) + (0io + 0n)(72 - To)] 7o = —[27i0io9n (72 + To) + (92o + 0ii)(T2 - 7o)(72 + To)] To = ( 7 2 + 7 o ) [ 2 7 i 9 i o 9 i i + (92o + 9?i)(T2 - To)] (5.82) To Thus ( - l ) 5 ( 4 7 o 4 ) ^ [ 2 7 i 9 i o 9 i i + (qfo + ??i)(72 - To)] (7o + 72)2[(27 o 2 - 7 2 + 2A) 2 - 7 2 ( 7 l 2 - 4A)] = ( - l ) g 2 4 7 2 [ 2 7 i g i 0 g i i + (9io + 9n)(72 - To)] R ^ (To + 72)[(27o2 - 7i2 + 2A) 2 - 7 2 ( 7 l 2 - 4A)] { b * 6 ) This is the same equation obtained in the derivation for b = 1. This is to be expected because deg(Ql) and deg(i) are the same for b = 1 and for 6 = 2. Thus, avu is the same for both cases. Chapter 5. Averaging Level Control in the Wet End 127 The derivations for the 6 = 2, and the 6 = 3 case (shown in appendix F) confirm the conclusion in Foley et al. (2000)'s paper that the optimal averaging level controller is a PI controller in series with a filter of order b-1. In the next section, the optimal averaging level controller is applied to a paper machine. 5.3 Wet End Application The averaging level control algorithm for the b=3 case was applied to a simulation of the saveall section of a paper mill. The mill and simulation package are the same ones described in chapter four. A schematic of the saveall section is shown in Figure 5.3. Water from various sources such as the white water chest, wire drainage, and fresh water is fed to the broughton pit. From the broughton pit water is sent to various areas of the mill for dilution purposes and also to the saveall. Additional water is supplied to the saveall from the clean and cloudy tanks, and fiber is obtained from the mix tube to act as sweetner (i.e., to help with the formation of a fiber mat on the saveall drum). The fiber recovered in the saveall is sent to the blend chest, and the clarified water is sent to the clear water tank. The cloudy tank receives any unclarified water from the saveall and also the overflow stream from the white water chest and clear water tank. The water from the cloudy tank is recycled back to the saveall for clarifying (i.e., recovery of furnish materials). The existing control system has the broughton tank level controlled using the saveall speed, and the saveall level controlled by adjusting the feed from the broughton tank. Thus, when the broughton level begins to rise, the saveall speed is increased which causes the saveall level to drop. This in turn causes the broughton-saveall feed valve to open increasing the flow from the broughton pit. In addition, the flow from the cloudy tank to the saveall has two control options: 1. Constant flow from the cloudy tank to the saveall. 2. Varying flow from the cloudy tank to keep a constant flow to the saveall. Al l the flows to the saveall are measured and then the cloudy feed is adjusted to maintain a constant flow to the saveall. The control objective is to maintain a constant flow to the blend chest, thus minimizing the flow variability to the process. This also means maintaining a relatively constant saveall speed which results in less wear on the saveall engine. Disturbances to the system are assumed to enter only via the feed to the broughton pit. All other feeds to the system are assumed to be constant. Chapter 5. Averaging Level Control in the Wet End 128 ClearWater and — White Water -Drainage and Other Sources" Fresh Feed Water k Return to Process Broughton Pit Saveall Speed Control PI ©••••®H _Feed From Mix-Tube Flow Calc Saveall Fiber To Blend Chest To Clear „_ Water Tank Option (A) Constant Flow (B) Varying Flow FSP White Water & Other Feeds +0? Cloudy Chest Dilution Figure 5.3: Saveall system The problem is complicated because the control system is interlinked. The broughton level is controlled by the saveall speed, and the savell speed affects the saveall level. The saveall level is adjusted by the broughton outlet flow valve which in turn affects the broughton tank level. In addition, the saveall level controller is tuned for tight level control which means the broughton outlet flow varies as the saveall level changes. If a flow disturbance was to enter the saveall via the clear water feed, the saveall level controller would back propagate the disturbance to the broughton level. Thus, the broughton tank would not just have to handle inlet flow variations, but would also have to deal with outlet flow variations as well. However, for these simulations the disturbances are assumed to only enter through the broughton tank feed. Since the broughton level controller adjusts the broughton outlet flow via the saveall level controller, there may be some variation introduced into the broughton outlet flow by the saveall level controller. This outlet flow variation could increase or reduce the effect of the inlet flow variation on the broughton level, this fact has to be considered when designing the controller. The control of the cloudy tank feed to the saveall affects the broughton outlet flow variation and also the saveall to blend chest flow variation. If the cloudy tank flow to the Chapter 5. Averaging Level Control in the Wet End 129 saveall is controlled using: • Constant cloudy flow (Option 1) : when the broughton level increases, the saveall speed will increase causing the saveall level to drop resulting in the the broughton outlet valve opening further. This scenario is unchanged from the above description. • Varying cloudy flow to keep a constant feed to the saveall (Option 2): in this instance when the broughton level increases, eventually causing the broughton outlet valve to open as described above, the cloudy flowrate will drop to keep the total flow to the saveall constant. Since, the saveall level does not rise, the broughton outlet valve opens further to compensate resulting in the broughton level being returned to setpoint faster. Thus, a smaller change in saveall speed is required to reject the broughton tank inlet flow disturbance when compared to option 1. To minimize changes in the saveall level, the broughton tank level has to be allowed to vary, and the best way to accomplish this, is to apply averaging level control. Table 5.1: Tank Information Broughton Tank Saveall Cloudy Diameter (ft) 16.25 10 Length (ft) 25 Width (ft) 13 Height (ft) 8 9 9.4 Level Setpoint - 5.6 Low Level Limit 5 • - -High Level Limit 7.5 Deadband ft 2 Two sets of simulations were performed to test the averaging level controller, one for each cloudy tank feed to saveall' control option. Data from an industrial caustic tank (Sidhu et al. 2001) was used as the broughton tank inlet flow stream variation, the data was modified to increase the differenced inlet flow variance from 0.0195 to 552.76 (USgal/min)2. The level setpoint of the broughton tank was 6 ft and the tank overflow is at 7.5 ft. Table 5.1 contains additional information on the process tanks. The simulation was set to execute every 1 second and the controller execution interval was every 5 seconds. There is a time delay of 15 sec, equivalent to a 3 sampling time delay, before the broughton level controller sees the tank level reading. Chapter 5. Averaging Level Control in the Wet End 130 The averaging level control algorithm by Foley et al. (2000) requires the setpoint to be centered in the deadband. Since the setpoint is close to the upper level limit, there is only 1.5 ft available for the level to rise above the setpoint. Thus, the averaging level controller had to be designed to allow the level to vary between a low of 5 ft and a maximum of 7 ft. A safety margin of 0.5 ft was allowed in case the level passed 7 ft, thus the deadband was set to 2 ft or 26.67% level. The tuning procedure developed by Foley et al. (2000) was used to calculate the value of the move supression parameter (A) which is used in the calculation of 70, 71, and 72 given in equations 5.15 to 5.17. The procedure equates half the deadband, in % level, to a user specified 3 sigma ( c r y ) on the broughton tank level. For the saveall process, half the deadband is 1 ft or 13.3% level. Thus, (3a y ) 2 = (13.3%)2 => aY = 19.75 % 2 . Once o\ was set, various values of A were tried until the calculated value of oY given by equation F.79 ( appendix F page 209) is equal to 19.75%2. To help in the selection of A for this system, a plot of oY vs a27U is shown in Figure 5.41. The values of A corresponding to Oyjj are shown on the y-axis appearing on the right of the graph. Once the value of A has been found, then the corresponding controller parameters are easily found. 8 10 12 14 16 18 20 22 24 26 28 30 Input Variance ( ) Figure 5.4: Output Variance aY against Input Variance a \ v 1 T h e variance of the differenced inlet flowrate (cr^), which is used to design the controller, are s imilar in bo th cases. A s a result, this figure is applicable to bo th simulations Chapter 5. Averaging Level Control in the Wet End 131 For each control option case, two simulations were run. On using the averaging level controller and the other using the PI originally specified in the simulation by Yap (1998); the PI controller was included for comparison purposes. The simulation results are shown in the following sections. 5.3.1 Simulation 1: Constant Cloudy Feed to the Saveall In this option, the cloudy feed to the saveall is held constant. The saveall level and cloudy tank flow PI controller parameters are shown in table 5.2. The broughton level PI controller was used first to get an estimate of the broughton outlet variations, the controller parameters are shown in Table 5.3. Table 5.2: Simulation 1: Saveall and Cloudy PID Tuning Parameters Saveall Level (%) Cloudy Flow (USgal/min) Kc % Valve Open. 7 0.2 Ti (min) 4 0.06 Td (min) 0.3 1.0 Table 5.3: Simulation 1: Broughton Level Controller Parameters PI Level Averaging Level PI Kc (saveall rpm/% Level) 1.75 0.7053 Ti (min) 10 5.964 rd (min) /o (% min/USgal) 0.9453 fx (% min/USgal) -0.0272 f2 (% min/USgal) -0.0276 A 4.249 Factor (rpm/USgal/min) 0.0218 The simulation results for the broughton level PI controller are shown in Figure 5.5. Two plots are shown, the top graph shows the changes in the broughton feed and the result-ing changes in the saveall to blend chest flowrate; the second graph shows the broughton tank level deviations from setpoint and the changes in the saveall speed. Because the broughton level PI controller is tightly tuned, the maximum level deviation from setpoint is less than 0.5 ft. Thus, the saveall speed has to change rapidly to ensure the level returns to setpoint quickly. The resulting stream variations are shown in Table 5.4. Chapter 5. Averaging Level Control in the Wet End 132 Chapter 5. Averaging Level Control in the Wet End 133 Chapter 5. Averaging Level Control in the Wet End 134 Table 5.4: Simulation 1, Process Variations Variance (USgal/min)2 PI Level Averaging Level PI Broughton Feed Variance 522.76 522.76 Saveall to Blend Chest Flow Variance 0.3344 0.1184 Broughton to Saveall Flow Variance 8.8015 3.8339 Typically the averaging level controller would be designed using the inlet stream vari-ance. However, since the system may also experience outlet stream variations, the con-troller was designed using the sum of the inlet and outlet stream variations. The outlet stream variation with the PI will be higher than with the averaging level controller but it should still result in a good averaging level controller design. Thus, the averaging level controller was designed to handle a variance of 561.56 (USgal/min)2. The resulting control parameters are shown in Table 5.3. The averaging level control simulation is shown in Figure 5.6. The broughton level deviates further from setpoint than when controlled by the PI, the maximum level devi-ation is approximately 0.9 ft. The saveall to blend chest flowrate appears to be slightly smoother, and the saveall speed changes are not as quick as in the PI case. However, these changes appear to be minor and are not easily seen when the figures are compared. The change in performance is more apparent when the stream variations are used, see Table 5.4. The averaging level controller produced a small magnitude reduction in the saveall to blend chest flow variance. This also means the saveall speed and broughton outlet variations have been reduced. The reduction in the saveall to blend chest flow variance is smaller than expected, one possible reason could be the use of the sum of the input and out broughton variations to design a conservative averaging level controller. However, this is only a minor contribution because the level varies across 90% of the deadband. The main reason for the small reduction is believed to be the use of a small deadband, only 2 ft, for the averaging level control design. 5.3.2 Simulation 2: Varying Cloudy Feed to the Saveall For this simulation the cloudy flow to the saveall was allowed to vary to keep the total flow to the saveall constant. The cloudy flow controller parameters were arbitrarily adjusted to obtain a stable system. The new parameters are shown in Table 5.5. The total flow to the saveall was reduced compared to the previous simulation. This was done to reduce the initial steady state saveall speed so that the PI could keep the level under 0.5 ft, thus Chapter 5. Averaging Level Control in the Wet End 135 keeping a similar performance to the previous simulation. The broughton tank PI level control parameters are shown in Table 5.6 and the simulation results are in Figure 5.7. The maximum level deviation was approximately 0.5 ft. Table 5.5: Simulation 2: Saveall and Cloudy PID Tuning Parameters Saveall Level (%) Cloudy Flow (USgal/min) Kc % Valve Open. 7 0.2 TJ (min) 4 5.0 TD (min) 0.3 0.0 Table 5.6: Simulation 2: Broughton Level Controller Parameters PI Level Averaging Level PI Kc (saveall rpm/% Level) 1.00 0.4164 TI (min) 10 5.979 TD (min) /o (% min/USgal) 0.9454 h (% min/USgal) -0.0271 h (% min/USgal) -0.0275 A 4.292 Factor (rpm/USgal/min) 0.0129 Table 5.7: Simulation 2, Process Variations Variance (USgal/min)2 PI Level Averaging Level PI Broughton Feed Variance 522.76 522.76 Saveall to Blend Chest Flow Variance 0.1324 0.0251 Broughton to Saveall Flow Variance 4.4869 1.061 As in the previous simulation, the averaging level controller was designed to handle the sum of the inlet and outlet broughton variations (557.58(USgal/min)2). The control parameters are listed in Table 5.6 and the resulting simulation is shown in Figure 5.8. In this case there is a noticeable difference between the controllers. For the averaging level PI controller the saveall to blend chest flow is definitely smoother, the broughton Chapter 5. Averaging Level Control in the Wet End 136 Chapter 5. Averaging Level Control in the Wet End 137 (UJdBsn) M°ld 1S8LIQ pU9|a 0} IIB9ABS o o o o o o o o O L O O L O O L O O _ , T - c n c n o o c o r ^ - r ^ o o o o o o o o o o o o o O LO O LO O LO CO CO CM CN t-(ujd6sn) P93d uoii|6nojg (LUdj/o/0) p99dS HB9ABS O CO CM 00 ^ CO CO CM CM O O LO p LO CD h~ CD CD LO LO (U) I9A91 Chapter 5. Averaging Level Control in the Wet End 138 level deviates to a maximum of 0.85 ft, and there are fewer saveall speed changes. The stream variations are shown alongside those from the PI simulation in Table 5.7. There is a substantial drop in the variations for the averaging level controller compared to the PI controller. 5.4 Conclusions The typical pulp and paper mill has many tanks, some are used for raw material storage, while the remainder are used for surge tanks or mixing of raw materials recycled back to the process (e.g. white water, broke, and reclaimed stock). The level control on many of these tanks maybe tuned for tight level control which means flow disturbances are passed to downstream units. If these tanks are present within recycle loops, the disturbances are recycled back to the process where they can have a negative effect on the papermachine performance. A more appropriate strategy for these tanks would be to use averaging level control which provides outlet flow smoothing thus dampening the effect of inlet or outlet flow disturbances on downstream equipment. In this chapter the averaging level control algorithm by Foley et al. (2000) was derived for the case of two (i.e., b=2) and three (i.e., b=3) units of sampling delay. The derived equations allow calculation of the optimal controller; as Foley et al. (2000) stated, the controller has the form of a PI controller in series with a filter of order b-1. The averaging level controller for the case with three units of sampling delay was applied to a simulated broughton pit - saveall system of a paper machine. The control objective was to reduce the flow variation from the saveall to the blend chest. The problem was complicated because the control system was interlinked; the broughton pit level was controlled by the saveall speed and the saveall level was controlled by the broughton pit outlet valve which fed the saveall. In addition, the cloudy tank feed to the saveall had two operating options, the flow could be constant or varied to keep the total feed to the saveall constant. The control objective was met by using averaging level control on the broughton tank. Simulations were run using the two different control options for the cloudy tank feed to the saveall, and averaging level control was compared to the original PI controller. In both cases, the averaging level controller was able to reduce the the variations compared to the original PI tuning, while keeping the level within the deadband. Chapter 6 Conclusions and Recommendations The goal of this work was to provide an understanding of recycle dynamics and control to help improve operations in the wet end of a papermachine. The work was performed in a systematic manner, and started with a comprehensive dynamic study to characterize the recycle behaviour, and possibly identify responses that could be potential control prob-lems. Four systems were selected based on process models found in the chemical industry. All of the systems had a F O P D T model in the recycle path. The four forward path models used were a F O P D T model, a lead-lag F O P D T model, an inverse response F O P D T model, and a SOPDT model. The study was performed using various combinations of parameter values for TF,Tr,9f, and 6R. It was found that all of the recycle responses fall broadly into 3 groups: • Group 1 has a forward path process response • Group 2 has a fast initial rise time followed by a slow rise to steady state • Group 3 has a step-wise response to steady state It was not possible to specify generally under what conditions the first two groups will appear since the conditions change according to the model structure and parameter values used. Group 3 was found to occur when the time delays were dominant, specifically the sum of the forward and recycle path delays was greater than six times the forward path time constant ( 9F 4- OR > 6 * rF ). Two groups were identified as having responses that may be challenging for control. Group 2 contains responses that have an oscillatory rise to steady state and Group 3 responses have a step-wise rise to steady state. A new modeling approach was proposed to represent the Group 3 responses. The method was based on seasonal trends from time series analysis. The seasonal model was shown to provide a good approximation to recycle processes using two different methods. The first method was the finite difference method. The second approach involved the discretization with a zero order hold of the forward and recycle path separately, and then deriving the overall transfer function using the standard recycle block diagram (see Figure 2.1). Since, there were no algorithms to explicitly identify seasonal models, a simple algorithm using least squares was written to identify the seasonal model parameters. 139 Chapter 6. Conclusions and Recommendations 140 The next stage involved studying the wet end of a simulated papermachine to determine the extent of recycle dynamics. An IDEAS simulation model of a papermachine wet end, developed by Yap (1998), was used to determine if significant recycle dynamics (i.e., the Group 3 type response) were present. The study was performed by making a change in the filler mass percentage in the stream leaving the basis weight valve and recording the change in filler composition in four recycle streams and in the paper leaving the wire. The large tanks in the system caused a dominant forward path time constant which resulted in all of the responses being first order. This result confirmed the initial analysis of the time constants and pipe delays which suggested that the step-wise trend would not be present. The main purpose of the tanks in the wet end is to dampen out process variations. However, large tanks slow the process response thus limiting how fast the control system can respond to setpoint changes and disturbances. As a result, there is an interest to reduce the tank sizes in new mills to make the process faster. Since, the reduction in tank sizes could result in recycle dynamics occurring, the wet end recycle dynamic study was repeated using smaller tanks, ones with a residence time under 15 sec. Minor recycle dynamics did appear during the study and it was shown that recycle streams may also dampen out the the dynamics caused by other recycle loops. Because only minor recycle dynamics appeared even though the tank sizes had been greatly reduced, the dynamic study was repeated for the system without tanks in the recycle loop (referred to as the recycle loop tankless mills). This time the step-wise response, which is usually associated with recycle systems, was easily achieved. "The responses had a clear step response to steady state. As in the reduced tank case, it was found that a recycle loop could smooth the dynamics caused by other recycle loops. The results suggest that the step-wise trend behaviour of recycle systems is not a factor in the mill studied. Since present day mills typically have large process tanks with residence times larger than the pipe delays, the step-wise recycle dynamics are unlikely to be visible. It is not possible to generalize the results to reduced tank size mills or recycle loop tankless mills because ofthe differing configurations (i.e., pipe layout) causing different time delays in the forward and recycle paths. For the studies with the reduced tank sizes or the recycle loop tankless mill, recycle dynamics are present. The extent of the recycle dynamics depends on the number of recycle loops and the corresponding time delays. The second aspect of this work dealt with the increased sensitivity to disturbances caused by recycle streams. Two types of disturbances, quality and flow, were examined. Quality disturbances require better control of the process while flow disturbances are handled with some form of averaging level control. Chapter 6. Conclusions and Recommendations 141 Quality disturbances were examined first. To start, PI control of recycle processes was evaluated. For most processes PI control is sufficient, and it is well known that PI perfor-mance drops with increasing time delay thus requiring more sophisticated techniques. PI control was applied to a recycle process and it was found that both 8F and 6R affect the PI performance. As the delays increase, the process becomes slower (i.e., a larger effective time constant) resulting in a drop in control performance. The effect of dp is more signifi-cant because 6F also represents the process time delay. The use of a Smith-Predictor (SP) to compensate for the process time delay in conjunction with the PI controller produced only a marginal improvement. Then the various control strategies proposed in the literature were reviewed. Of the four methods found, two of them only dealt with handling parts of the recycle process. The remaining two solutions were model-based and looked at controlling the total recycle process. The approaches were: (a) the recycle compensator which removed the recycle effect and (b) the Taylor series approach which approximated the denominator dead-time terms. An alternative approach which used a seasonal model to represent the recycle process was proposed. For comparison, these three methods were used to apply model based predictive con-trol to a recycle process. The Taylor series approximation is a novel way to handle the problem of denominator dead-time terms, it is more difficult to implement but does pro-vide comparable control to the seasonal approach. The recycle compensator is an elegant solution to minimize the effect of recycle dynamics and gives very good servo control, however, it has slow disturbance handling characteristics. The compensator also requires independent identification of the forward and recycle path systems. The seasonal model approach is an alternative to the recycle compensator. It is more practical, only requir-ing the identification of the overall model, and provides comparable performance to the compensator approach for servo control but with significantly better disturbance rejection capability. The M B methods had input disturbance handling capability comparable to a PI con-troller. The mediocre M B C regulator performance was attributed to using the maximum peak in the sensitivity functions for tuning to ensure the controllers had the same robust-ness in the face of disturbances (e.g., input, output, and noise) and model uncertainty. The MB methods had to be detuned to have the same robustness as the PI resulting in comparable performance to the PI. When the MB controllers were tuned for performance rather than robustness, they outperformed the PI. As mentioned earlier, the responses of the recycle process fell into three groups. Of these groups, some of the responses from Groups 1 and 2 have a first order model be-haviour. Now, most process can be approximated as a first or second order model when Chapter 6. Conclusions and Recommendations 142 designing the controller. To evaluate the effect of using such an approximation, the use of a F O P D T model was compared to using the seasonal model for recycle control. At low values of time delay and recycle fraction the F O P D T model gave comparable performance to the seasonal approach, however, as the time delays and recycle fraction increased, the advantage of using the seasonal model approach became more apparent. Approximate conditions for switching from the F O P D T model to the seasonal model were obtained for different recycle fractions and time delays. The second disturbance type to be studied was flow disturbances. Typically, such disturbances are absorbed by the buffer tanks, and the degree of disturbance attenuation is determined by the tank level controller. If the controller is tuned for tight level control, it can result in the disturbance propagating to downstream units. This problem can be compounded if the buffer tanks are located within a recycle loop, because the disturbance is recycled back to the process. If the controller is tuned for averaging level control, the flow rate variability can be substantially reduced. The current averaging level control methods were reviewed and the algorithm by Foley et al. (2000) was chosen. The approach is based on constrained minimum variance and guarantees the lowest possible flow variation. The algorithm was extended to handle sampling delays (b) greater than one. The resulting controllers were in the form of a PI controller in series with a filter of order b-1, which confirms the observation by Foley et al. (2000) The averaging level controller for the case with three units of sampling delay was applied to a simulated broughton pit - saveall system of a paper machine. The control objective was to reduce the flow variation from the saveall to the blend chest. The control system was interlinked; the broughton pit level was controlled by the saveall speed and the saveall level was controlled by the broughton pit outlet valve which fed the saveall. Since the control loops are interlinked, variations within the saveall can be propagated back to the broughton pit level. This additional variance from the saveall was also included in the broughton pit inlet flow variance when designing the averaging level controller. The averaging level controller was able to reduce the flowrate variations while keeping the level within constraints. 6.1 Recommendations The results obtained in this thesis do not provide all the answers to dealing with the recycle problems in the papermachine wet end. Instead, this work has provided a foundation on which additional study can build to achieve the final goal of improved recycle control. The following areas need to be addressed to give a better overall picture: Chapter 6. Conclusions and Recommendations 143 1. The configuration used in the recycle study was the simplest (i.e., it contained only one recycle loop). The dynamics of more complex structures such as nested and sequential loops should be studied. As mentioned before, a comprehensive range of parameters for the complex structure results in too many cases. The parameter range can be reduced by requiring that one of the recycle loops give a response from one of the three groups specified earlier. 2. The model based controllers for recycle control were designed for single recycle loop case. However, the wet end contains multiple recycle loops. Once the control en-compasses more than one loop the performance of the controllers may change. The controllers should be evaluated on multiple recycle loops. Some simplification may be achieved using the information from the previous recommendation. 3. Though the results suggests that the step-wise trend behaviour of recycle systems is not a factor in the simulated paper machine, further study is required. • The study of recycle dynamics in the papermachine wet end included only four recycle streams. Additional streams should be included to check for further addition or cancellation of recycle dynamics. This is especially relevant for the recycle loop tankless case. • The simulation model by Yap (1998) should be expanded to include wet end chemistry effects such as retention behaviour. The model should also incor-porate dynamic separation ratios for the cleaners, screens, and wire. These factor can influence the recycle dynamics especially if the wire retention is not maintained. Model improvement along these lines is required before the recycle control strategies can be applied to the simulation. 4. The averaging level control algorithm by Foley, Kwok, and Copeland (2000) requires the setpoint to be centered in the deadband. However, this may not always be possible and may lead to the use of a smaller deadband than is actually available. This will result in the control performance not being optimal. The averaging level control algorithms that are not model-based should be reviewed to see if they can handle the non-centered deadband setpoint case and then compared with the Foley et al. (2000) algorithm to see if there is an improvement in performance. Otherwise a model based algorithm, such as the model-based optimal predictive control algorithm by Campo and Morari (1989), will be necessary. 5. Obviously there will be cases where inherent control limitations reduce the effec-tiveness of the controller. In such cases, the disturbances have to be handled by Chapter 6. Conclusions and Recommendations 144 the buffer tanks. Cui and Jacobsen (2000) provide information on specifying the re-quired buffer tank size. This work should be examined with a view to applying it to the papermachine wet end. The extent of controller limitations can be evaluated to determine the necessary buffer size required. This will give an indication on whether the levels in the process tanks are too low. If the levels are higher than required, it may be possible to reduce these levels accordingly to speed the process response. Bibliography Amirthalingam, R. and J. H. Lee (2000, May). 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A. and W. H. Ray (1994). Process Dynamics, Modeling, and Control, Chapter 5,7,14, pp. 162-164,225,229-231,489-494. 198 Madison Avenue New York, N.Y. 10016-4314: Oxford University Press. Ohbayashi, S., K. Shimizu, and M . Matsubara (1989). Dynamics ofthe Flash Fermentor System with Recycle. Industrial and Engineering Chemistry Research 28(8), 1202-1210. Papadourakis, A., M. F. Doherty, and J. M . Douglas (1987, June). Relative gain array for units in plants with recycle. Industrial and Engineering Chemistry Re-search 26(6), 1259-1262. Scali, C. and R. Antonelli (1995). Performance of different regulators for plants with recycle. Computers and Chemical Engineering iP(Suppl), S409-S414. Scali, C. and F. Ferrari (1997). Control of systems with recycle by means of compen-sators. Computers and Chemical Engineering 21 (Suppl), S267-S272. Shunta, J. P. and W. Fehervari (1976). Nonlinear Control of Liquid Level. Instrumen-tation Technology 23, 43-48. Bibliography 149 Sidhu, M . S., K. E . 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Krebs (1994). Robust Control Systems Design for Plants with Recycle. In Proceedings of the IEEE Conference on Control Applications, Piscataway, NJ USA, pp. 1567-1571. volume 3. Taiwo, O. and V. Krebs (1996). Robust Control Systems Design for Plants with Recycle. The Chemical Engineering Journal 61, 1-6. Verykios, X. E . and W. L. Luyben (1978). Steady-state Sensitivity and Dynamics of a Reactor/Distillation Column System with Recycle. ISA Transactions 17(2), 31-41. Wu, K. L. and C. C. Yu (1996, July). Reactor/separator processes with recycle - 1. Candidate control structure for operability. Computers and Chemical Engineer-ing 20(11), 1291-1316. Yap, E . F. P. (1998, September). Dynamic Modeling of a Paper Machine Wet End. Master's thesis, The University of British Columbia, Department of Chemical En-gineering, 2216 Main Mall, Vancouver B.C. Canada V6T 1Z4. Appendix A Recycle Dynamic Figures for CrV=FOPDT and G#=FOPDT 150 A. Recycle Dynamic Figures for GF=FOPDT and GR=FOPDT 151 A.I Trend 1 Figures TF > 8? = 8R> TR K F = KR = 1, T ! _ F = 15, N _ R = 2.5, Of = eR = 5, 7.5, 10, 12.5, r = 0.25 10 10 Frequency (radians/seconds) TF > 6F = 6R > TR K F = K R = 1, T i _ F = 15, N _ R = 2.5, % = 0 R = 5, 7.5, 10, 12.5, r = 0.75 1 0 " 10"' 10 Frequency (radians/seconds) A = 5 • e F = 7.5 O 9 F = 10 * 9 F = 12.5 Tf > eF = 6R > TR K F = KR = 1, N _ F = 15, N _ R = 2.5, iT = eR = 5, 7.5, 10, 12.5, r = 0.5 10 10 10 Frequency (radians/seconds) TF > df = 0R> TR K F = KR = 1, n_ F = 15, n _ R = 2.5, 0F = »R = 5, 7.5, 10, 12.5, r = 0.95 1 0.8 a 10.6 2500 3000 10 10 10 Frequency (radians/seconds) 10 10 10 10 Frequency (radians/seconds) 10' - 5 0 -100 : -150 10" A 9 F = 5 O 9 P = 7.5 O 9 P = 10 * 8 P = 12.5 H A V,\\ = 10 10 Frequency (radians/seconds) Figure A.I: GF=G R=FOPDT, Trend 1 Figures A. Recycle Dynamic Figures for GF=FOPDT and GR=FOPDT 152 A.2 Trend 2 Figures TR> eF = rF> OR KF = KR = 1, r ! _ F = 20, 7 i _ R = 25, 0 F = 20, eR = 0, 5, 10, 15, r = 0.25 10 10 Frequency (radians/seconds) - 5 0 -100 -150 10' TR > Of = Tf > 6R KF = KR = 1, T i _ F = 20, TJ_R = 25, 6E = 20, eR = 0, 5, 10, 15, r = 0.5 A 8 R = 0 • 9 R = 5 O 9 R = 10 * 6 R = 15 10 10" 10 Frequency (radians/seconds) CD -50 CD °--1S0 10" TR > OF = T F > 0jj KF = KR = 1, n _ F = 20, n _ R = 25, flF = 20, 0R = 0, 5, 10, 15, r = 0.75 10' 10 10 10 Frequency (radians/seconds) 10 i ^ ^ ^ c j Ni!!!! A 9 R = 0 : ^ " ^ U j . : :::::: • «R = 5 O 6 R = 10 * 9 R = 15 10 10 10 10 Frequency (radiansVseconds) 10" 0 In !S - 5 0 O) CD ? - 1 0 0 CO « a--150 10' TR > OF = TF > OR KF = K„ = 1, n _ F = 20, T ! _ R = 25, 8F = 20, SR = 0, 5, 10, 15, r = 0.95 10 10 10 Frequency (radians/seconds) L i l ^ ^ . L ; . . i . i ; i L L i A 9 R = 0 • S R = 5 O 9 R = 10 * 8 R = 15 \ • 10"" 10 10 10 Frequency (radians/seconds) Figure A.2: GF=GR=FOPDT, Trend 1 Figures A. Recycle Dynamic Figures for GF=FOPDT and GR=FOPDT 153 A.3 Trend 3 Figures 6R> 6F = Tp > TR 9R > 6F = Tf > TR K F = KR = 1, N _ F = 5, T!_R = 1, K F = KR = 1, N _ F = 5, N _ R = 1, % = 5, 0 R = 10, 15, 25, 50, r = 0.25 0 F = 5, 6R = 10, 15, 25, 50, r = 0.5 Frequency (radians/seconds) Frequency (radians/seconds) 6R> 8F = TF > TR 6R>6F = T F > TR K F = KR = 1, TX_R = 5, T i _ R = 1 , K F = KR = 1, 71_F = 5, T!_R = 1, 0 F = 5, 9 R = 10, 15, 25, 50, r = 0.75 Of = 5, 0 R = 10, 15, 25, 50, r = 0.95 Figure A.3: G F = G ^ = F O P D T , Trend 1 Figures Appendix B Recycle Dynamic Figures for GV=Lead-Lag FOPDT and GR = FOPDT 154 B. Recycle Dynamic Figures for GF=Lead-Lag FOPDT and GR = FOPDT 155 B . l Trend 1 Figures TF > TR, 0F = 8R = Q K F = K R = 1, £ j _ F = 11.25, 15, 18.75, 22.5, T l _ R = 5, T i _ F = 7.5, 10, 12.5, 15, t?F = 6R = 0, r = 0.25 1.5, CC0.5 40 60 Time 100 10 10 10 Frequency (radians/seconds) 10 50 ,|k|, - : * — 0 - 5 0 -100 --150 10 10 10 Frequency (radians/seconds) 10 TF > TR, eF = eR = o K F = K R = 1, £ j _ F = 11.25, 15, 18.75, 22.5, T ^ R = 5, T i _ F = 7.5, 10, 12.5, 15, « p = 0R = 0, r = 0.5 1.5 0:0.5 10 o CO CC ® • 110 E < 10" 10"" » 5 ° i CD 2> 0' o> CD S . - 5 0 CO I-100 Q_ -150 10 • 20 40 60 Time 10 10" 10 Frequency (radians/seconds) A -F = 11.25 • « i -F = 15 O -F = 18.75 • « -F = 22.5 10 10" • 10 Frequency (radians/seconds) jiiiiijiiiijiijijipil;!;;;^ 10 10 TF > TR, eF = eR = o Kp = KR = 1, f j _ F = 11.25, 15, 18.75, 22.5, T i _ R = 5, n_F = 7.5, 10, 12.5, 15, Sp = « R = 0, r = 0.75 10" CD 3> 0 Ol CD S - - 5 0 -CD | - 1 0 0 : Q. -150 10' 10 10 10 Frequency (radians/seconds) A _F = 11.25 • *i -F = 15 O -F = 18.75 * -F = 22.5 10 10 10 Frequency (radians/seconds) 10 TF > TR, eF = eR = o K F = K R = 1, ca_F = 11 25, 15, 18.75, 22.5, T J _ r = 5, T ! _ F = 7.5, 10, 12.5, 15, 0F = 0R = 0, r = 0.95 10" 10" J - ! . ^ - ' . : , . - . O - - . , . " .N X t r t - . - « B - » n * . s A A _n - w-*»TL \ \ • \ \ 50 100 150 Time 200 250 A -F = 11.25 • -F = 15 O -F = 18.75 * -F = 22.5 10 10" 10 Frequency (radians/seconds) 10 10 10 10 Frequency (radians/seconds) Figure B . l : G F=Lead-Lag F O P D T , G R = F O P D T , Trend 1 Figures B. Recycle Dynamic Figures for GF=Lead-Lag FOPDT and GR = FOPDT 156 B.2 Trend 2 Figures eF>eR>rR> TF K F = K R = i, e!_F = 1.5, 71_F = 1 , n-R = 5, 0F = 25, t?R = 10, 15, 20, r = 0.25 10 10 10 Frequency (radians/seconds) 200 A 6 R = 10 • 9 R = 15 O 9 R = 20 0f>OR>TR> rF K F = K R = 1, £ ! _ F = 1.5, N _ F = 1, N _ R = 5, t% = 25, 6R = 10, 15, 20, r = 0.5 10 10 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) eF>en>rR> TF K F = KR = 1 , h_f = 1.5, n_F = 1, n . R = 5, % = 25, 0 R = 10, 15, 20, r = 0.75 50 100 150 200 250 300 350 400 Time 10 10 10 10 Frequency (radians/seconds) CD - 5 0 ra a> tfi CO °--150 10" A 8 R = 10 • 9 R = 15 \ O 9 R = 20 iT ;; ; i l ; 10 10 10 Frequency (radians/seconds) 10 8f>9R>TR> TF K F = K R = 1 , «1 -F = 1 5 , n - F = 1 , n -R = 6f = 25, t9R = 10, 15, 20, r = 0.95 1 0.8 o 2 0.6 a OA rr 0.2 oU&G^ 400 10 10 10 Frequency (radians/seconds) A 9 R = 1 0 : • 9 R = 1 5 O e R = 20 10"' 10"' 10 Frequency (radians/seconds) Figure B.2: G F=Lead-Lag F O P D T , G R = F O P D T , Trend 2 Figures B. Recycle Dynamic Figures for GF=Lead-Lag FOPDT and GR = FOPDT 157 B.3 Trend 3 Figures OF > OR > Tp = TR K F = KR = 1, <h_F = 3.75, T i _ F = 2.5, T J . R = 2.5, 0F = 50, 63R = 5, 10, 15, 25, 45, r = 0.25 10 10 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) 10 A 9 R = 5 • 9 R = 10 O 6 R = 15 * 6 R = 25 0 6 R = 45 10 OF > OR > Tp = TR K F = KR = 1, & _ F = 3.75, n _ F = 2.5, n _ R = 2.5, 8E = 50, 0R = 5, 10, 15, 25, 45, r = 0.5 i i > ^•o»0' Time 10 10 10 Frequency (radians/seconds) A 9 R = 5 • 9 R = 10 O 9 R = 15 * 9 R = 25 0 9 R = 45 10 10 10 Frequency (radians/seconds) 10 10 0F > vR> TF = TR K F = KR = 1, <a_F = 3.75, r i _ F = 2.5, n _ R = 2.5, 8F = 50, BR = 5, 10, 15, 25, 45, r = 0.75 10 10 10 Frequency (radians/seconds) A 9 R = 5 • 9 R = 10 O 9 R = 15 • 9 R = 25 9 R = 45 10 10"' 10" Frequency (radians/seconds) 400 10 OF > 0R> TF = TR K F = KR = 1, & _ F = 3.75, T I _ f = 2.5, T l _ R = 2.5, 9F = 50, eR = 5, 10, 15 , 25, 45, r = 0.95 10" 0 § -so: CD ? - 1 0 0 CO CO °--150 10 10 10" 10 Frequency (radians/seconds) 400 10 A 9 R = 5 • 9 R = 1" O 9 R = 15 • 9 R = 25 > £ \ <> 8 R = 45 : : : : : : : ; i • • .• 10 10 10 10 10 Frequency (radians/seconds) Figure B.3: G F=Lead-Lag F O P D T , G R = F O P D T , Trend 3 Figures B. Recycle Dynamic Figures for GF=Lead-Lag FOPDT and GR = FOPDT 158 B.4 Trend 4 Figures TF>TR>Br> eF K F = K R = 1 , € I _ F = 3 7 5 > 3 7 5 > 7 5 . 7 5 > 7 5 > T l - F = 25, 25, 50, 50, 50. 7I_R = 10, 15, 40, 40, 40, % = 1, 0R = 5, 10, 10, 25, 35, r = 0.25 1.5 « 1 c & </) 0:0.5 E < 10" S 5 0 CD I §> 0 CD 2- -50 CD | - 100 D_ -150 10' . . . . 50 100 150 Time )BO*0 A D O * y 10 10 10 10 Frequency (radians/seconds) A _F = 37.5 • «, - F = 37.5 O «1 - F = 75 * 6, - F = 75 0 *, - F = 75 I 10 10 10 Frequency (radians/seconds) TF>TR>0R> eF K F = KR = 1, Ci -F = 37.5, 37.5, 75, 75, 75, T w = 25, 25, 50, 50, 50, r j _ R = 10, 15, 40, 40, 40, flF = 1, 6K = 5, 10, 10, 25, 35, r = 0.5 1.5r o 5 0 CD §> 0 cn <u 3 - 5 0 <u | - 1 0 0 CL -150 10" 10 10 10 Frequency (radians/seconds) A _F 37.5 • - F ~ 37.5 O - F = 75 * - F = 75 0 « - F = 75 10 10 10 Frequency (radians/seconds) TF>Tr>6r> eF K F = KR = 1, (a_F = 37.5, 37.5, 75, 75, 75, T i _ F = 25, 25, 50, 50, 50, T!_ R = 10, 15, 40, 40, 40, fff = 1, SR = 5, 10, 10, 25, 35, r = 0.75 1.5r 101 o co cc •°10° E < _ _ -— . 200 Time A 5 _F = 37.5 • -F = 37.5 O -F = 75 • -F = 75 0 -F = 75 10 10 10 Frequency (radians/seconds) 10 10" 10 Frequency (radians/seconds) TF>Tr>6R> eF K F = K R = 1, £ I _F = 37.5, 37.5, 75, 75, 75, r ^ p = 25, 25, 50, 50, 50, Ti_R = 10, 15, 40, 40, 40, flF = 1, SR = 5, 10, 10, 25, 35, r = 0.95 1.5r DC 0.5 A 'T* — A — _ | Q-: J3--" .-•o : - - - o ^ T ^ j ^ i * ^ - - - " t - e 10 10" 10 Frequency (radians/seconds) A 4 -F = 37.5 • -F = 37.5 O -F = 75 * -F = 75 0 -F = 75 \ 10 10 10 Frequency (radians/seconds) Figure B.4: G F=Lead-Lag F O P D T , G R = F O P D T , Trend 4 Figures Appendix C Recycle Dynamic Figures for GV=Inverse Response FOPDT and G#=FOPDT 159 C. Recycle Dynamic Figures for GF=Inverse Response FOPDT and GR—FOPDT 160 C . l Trend 1 Figures TF > TR >oF = oR K F = K R = 1, d_ F = - 3 . 7 5 , - 5 , - 6 . 2 5 , - 7 . 5 , T ^ R = 5, T ! _ F = 7.5, 10, 12.5, 15, 0 F = 0R = 2.5, r = 0.25 0 . 5 / / ; / - ' It" 10? 1 0 ' 17 5 0 CD 2 0 TO -100 0-- 1 5 0 1 0 1 0 1 0 Frequency (radians/seconds) A -F - - 3 . 7 5 • -F = - 5 • • • • * J S H L o -F = - 6 . 2 5 . . • -F " - 7 . 5 TF > TR > OF = 0R K F = K R = 1, £ i _ F = - 3 - 7 5 , - 5 , - 6 . 2 5 , - 7 . 5 , TJ . R = 5, T ! _ F = 7.5, 10, 12.5, 15, Of = 0R = 2.5, r = 0.5 1|—i 1 : — I / I J I M u j u i B II II" fl— 1 0 1 0 1 0 Frequency (radians/seconds) 1 0 1 0 1 0 Frequency (radians/seconds) 1 0 " " 1 0 1 0 Frequency (radians/seconds) 1 0 TF > Tit > 0F = oR K F = K R = 1, (? I_F= - 3 . 7 5 , - 5 , - 6 . 2 5 , - 7 . 5 , 7 i _ R = 5, T i _ F = 7.5, 10, 12.5, 15, Of = 0R = 2.5, r = 0.75 TF > TR> Op = 9R K F = KR = 1, ex_f = - 3 . 7 5 , - 5 , - 6 . 2 5 , - 7 . 5 , T i _ R = 5, n_ F = 7.5, 10, 12.5, 15, Of = 0R = 2.5, r = 0.95 10 o "co rr o> _ "R10 E < 10" 1 5 0 0 1 0 1 0 1 0 1 0 1 0 Frequency (radians/seconds) 1 0 1 0 " 1 0 Frequency (radians/seconds) A -F - - 3 . 7 5 • - F = - 5 O -F = - 6 . 2 5 * - F = - 7 . 5 1 0 " " 1 0 1 0 1 0 Frequency (radians/seconds) 1 0 Figure C . l : GF=Inverse Response F O P D T , G R = F O P D T , Trend 1 Figures C. Recycle Dynamic Figures for GF=Inverse Response FOPDT and GR=FOPDT 161 C.2 Trend 2 Figures TR > rr > 6F — 8R K F = K R = 1, e!_ F= -2.5, T l _ ¥ = 5, T ! _ R = 7.5, 10, 12.5, 15, 6F = 6R = 2.5, r= 0.25 11 1 ! 1 ni>i' n 50 o 0) 0 O) -50 CO --100 CL -150 10 10 10 10 Frequency (radians/seconds) A V R = 7.5 • V R = 10 O V R = 12.5 * V R = 15 10 10 10 Frequency (radians/seconds) TR > TF > 6F = 9R K F = KR = 1, d_ F = - 2 . 5 , N _ F = 5, T J _ R = 7.5, 10, 12.5, 15, (5F = t?R = 2.5, r= 0.5 m y . B . A o 10 10"' 10" Frequency (radians/seconds) A x - R = 7.5 • - R = 10 O T - R = 12.5 * 1 - R = 15 10 10" 10 Frequency (radians/seconds) Figure C.2: GF=Inverse Response FOPDT, G#=FOPDT, Trend 2 Figures C. Recycle Dynamic Figures for GF=Inverse Response FOPDT and GR=FOPDT 162 C.3 Trend 3 Figures OR > Op = Tp > TR K F = K R = 1, & _ F = -2 .5 , 7 i _ F = 5, T ! _ R = 1, t?F = 5, 0R = 10, 15, 25, 50, r = 0.25 > u - j a B : - i A . - ^ 1 ' B - A - - 0 - -« - i o o CL -150 10 10 10 Frequency (radians/seconds) -A 6 D = 10 . . . . . . V s j ^ ^ i • R 6 R = 15 O 6 R = 25 - * » R = 50 • i : • — ill!!! OR > Op — Tp > TR K F = K R = 1, C 1 _ F = -2 .5 , T I _ f = 5, r i _ R = 1, 0e = 5, 0R = 10, 15, 25, 50, r = 0.5 In 1 ^&-M_U-&> ° -Ag = ftfL=4 10 10 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) 0R> 0p= Tp > TR K F = KR = 1, fc.F = -2 .5 , T L _ F = 5, TJ_R = 1, 0r = 5, t?R = 10, 15, 25, 50, r = 0.75 10" 500 10 10 10 10 Frequency (radians/seconds) 10 A 8 R = 10 • E R = 15 O 8 R = 25 * E R = 50 10 10"' 10" 10" Frequency (radians/seconds) 10 OR > 0F = Tp > TR KF = KR = 1, ? I _ F = -2 .5 , n _ F = 5, n_R = 1, % = 5, 0R = 10, 15, 25, 50, r = 0.95 0.5 —T^ST^. - -—© $gr^-^_. -& • ^ *-^ 4c~~rr'- --He" 500 Time 1000 1 0 " 10 10" 10 Frequency (radians/seconds) 10 A = 10 • 8 R = 15 O 8 R = 25 • 9 R = 50 10 10 10 Frequency (radians/seconds) 10 Figure C .3 : GF=Inverse Response F O P D T , G#=FOPDT, Trend 3 Figures C. Recycle Dynamic Figures for GF=Inverse Response FOPDT and GR=FOPDT 163 C.4 Trend 4 Figures 8f > Tp> 6R> TR 8F >TF>eR>TR K F = K R = 1, < f 1 _ F = -5, -10, Ti_r = 10, 20, K F = KR = 1, <n_F = -5 , -10, T i _ F = 10, 20, Ti_R = 1, 5, 0F = 15, 30, 8R = 5, 10, r = 0.25 T : _ R = 1, 5, 0F = 15, 30, 8R = 5, 10, r = 0.5 Frequency (radians/seconds) Frequency (radians/seconds) VF>TF>VR> TR K F = K R = 1, f i -F = -5 , -10, T I _ f = 10, 20, T I_R = 1, 5, 6f = 15, 30, 6R = 5, 10, r = 0.75 10 tr CD •R10 E < 10" 100 200 300 400 Time 600 10 10 10" 10 Frequency (radians/seconds) 10 10 10" 10 Frequency (radians/seconds) 8F>TF>8R> TR K F = KR = 1, c l i - p = -5, -10, T i _ F = 10, 20, T!_ R = 1, 5, « F = 15, 30, 8R = 5, 10, r = 0.95 ^ - ^ " ^ - E J -_ . . - B ^yZ-^r . . B . - ° — " 10 10 10" 10 Frequency (radians/seconds) Figure C.4: GF=Inverse Response F O P D T , G f l = F O P D T , Trend 4 Figures C. Recycle Dynamic Figures for GF=Inverse Response FOPDT and GR=FOPDT 164 C.5 Trend 5 Figures TR> OR> Tf = Of K F = K R = 1, f j _ F = -0.5, T ! _ F = 1, 7 l _ R = 50, Of = 1, 0R = 5, 10, 25, 45, r = 0.25 E < 10" A = 5 • 9 R = 10 O E R = 25 * 9 R = 45 10 10 10 10 Frequency (radians/seconds) 10 10 10 10 Frequency (radians/seconds) 200 10 TR > OR > Tf = Of K F = KR = 1, f!_F = -0.5, n _ F = 1, = 50, Of = 1, 9R = 5, 10, 25, 45, r = 0.5 10 10 10" 10 Frequency (radians/seconds) 10 10 10" 10 Frequency (radians/seconds) TR> 0R> Tf = Of K F = K R = 1, £ , _ F = -0.5, n _ F = 1, n . R = 50, 0F = l , t?R = 5, 10, 25, 45, r = 0.75 0.5 100 10 6 < A E R = 5 • E R = 10 O 6 R = 25 * OR = 45 10" 10 10 10 10 Frequency (radians/seconds) 10 10 10" 10 Frequency (radians/seconds) 600 10 TR > OR > Tf = Of K F = K R = 1, f i _ F = -0.5, N _ F = 1, T ! _ R = 50, Of = 1, 0R = 5, 10, 25, 45, r = 0.95 -0 .5 . 10 CC •a 10 E < 10" 10 500 1000 Time 1500 A OR = 5 ! • OR = 1 " f o OR = 2 5 \ • OR " 4 5 I 10 10 10" 10 Frequency (radians/seconds) 10 10 10"' 10"' 10" Frequency (radians/seconds) Figure C.5: GF=Inverse Response F O P D T , (?*=FOPDT, Trend 5 Figures Appendix D Recycle Dynamic Figures for CrV=SOPDT and G^=FOPDT 165 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 166 D . l Category 1, Trend 1 Figures 8R=TF>BF = TR K F = KR = 1, T j . p = 10, 15, 25, 50, CF = 0.75, T l - R = 1, » F = 5, 6>R = 10, 15, 25, 50, r = 0.25 _ .A . q - A . i D — o _p__ 100 150 200 250 300 350 400 Time 10" 10 10 10 Frequency (radians/seconds) 10 10"' 10" Frequency (radians/seconds) 10 A T F = 10 • T F = 15 O X F = 25 • T F = 50 8R = Tp > 6F = TR K F = K R = 1, ' T i _ F = 10, 15, 25, 50, CF = 0.5, n_R = 1, t% = 5, 0R = 10, 15, 25, 50, r = 0.25 1.5 rr0.5 /.' / . 11 / / ' / -10 10 10 Frequency (radians/seconds) A * F = 10 • \ = 15 ":"i'T'M'!"r O \ = 25 :..U.Ui * xr = 50 10 10 10 Frequency (radians/seconds) 10 VR = TF > 8p = TB Kp = K R = 1, 7 I _ F = 10, 15, 25, 50, CF = 0.25, T i _ R = 1, 9f = 5, (5R = 10, 15, 25, 50, r = 0.25 IT 0.5 W \ 50 100 150 200 250 300 350 400 Time 10 10 10 Frequency (radians/seconds) 9R = TF>SF = TR K F = K R = 1, T ! _ F = 10, 15, 25, 50, CF = 0.1, T J _ R = 1, 0p = 5, t9R = 10, 15, 25, 50, r = 0.25 rr0.5i . .. Ill — ti'' 10 10 10 Frequency (radians/seconds) V  50 CD S 0 <? S - 5 0 1-100 a. -150 10 A , F = 10 • t F = 15 -• O T p = 25 * T p = 50 : > v j A : : \ : \ 10 10 10 Frequency (radians/seconds) 10 Figure D . l : G F = S O P D T , G#=FOPDT, Category 1, Trend 1 Figures, r=0.25 D. Recycle Dynamic Figures for GF=SOPDT and GR—FOPDT 167 OR = Tf > tif = TR Kf = KR = 1, T ! _ F = 10, 15, 25, 50, CF = 0.75, T i _ R = 1, 9 F = 5, t?R = 10, 15, 25, 50, r = 0.5 • - A — i y - A 50 100 150 200 250 300 350 400 Time 10 rr 10 Q.10 < 10 10 10 Frequency (radians/seconds) A = 10 • \ = 15 o = 25 •!!•: * T F = 50 10 10 10 Frequency (radians/seconds) OR — Tp > Of = TR K F = KR = 1, r j _ F = 10, 15, 25, 50, C F = 0.5, T!_ R = 1, Of = 5, 0R = 10, 15, 25, 50, r = 0.5 10 10 10 Frequency (radians/seconds) 10 A * F = 10 • V = 15 • O V = 25 * T F = 50 10 10 10 Frequency (radians/seconds) 0R = Tf> Of = Tp 1, n . p = 10, 15, 25, 50, C F : rx0.5! 0R = Tf>0f = TR K F = KR = 1, T X _ F = 10, 15, 25, 50, Cr- = 0.1, n _ R = 1, t?F = 5, (9R = 10, 15, 25, 50, r = 0.5 A / / Ii / _ / / . . : . / , /•''/ 10 10 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) A = 10 • • T F = 15 O X F = 25 * X F = 50 1 0 " 10 10" Frequency (radians/seconds) 10 Figure D.2: G F = S O P D T , G f l = F O P D T , Category 1, Trend 1 Figures, r=0.5 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 168 $ R = Tp > Bp = TR BR = Tp > 6p= TR K F = K R = 1, T i _ F = 10, 15, 25, 50, CF = 0-75, K F = KR = 1, r ^ p = 10, 15, 25, 50, CF = 0.5, T ! _ R = 1, % = 5, OR = 10, 15, 25, 50, r = 0.75 T L _ R = 1, % = 5, 6R = 10, 15, 25, 50, r = 0.75 Frequency (radians/seconds) Frequency (radians/seconds) BR = Tp > Bp = TR K F = K R = 1, Ti_F = 10, 15, 25, 50, CF = 0.25, T l ^ R = 1, 6f = 5, t?R = 10, 15, 25, 50, r = 0.75 400 10 10 10" Frequency (radians/seconds) A = 10 • = 15 O = 25 • = 50 LU.H 10 10"" 10" 10 Frequency (radians/seconds) 10 8R-Tp>Bp = TR K F = K R = 1, r ! _ F = 10, 15, 25, 50, CF = 0.1, TJ_R = 1, tVp = 5, (9R = 10, 15, 25, 50, r = 0.75 ^ i . - y - - a : ' - 8 - - ^ ' 10"' 10" 10 Frequency (radians/seconds) A = 10 • X F = 15 O \ = 25 • • T F = 50 10"' 10" 10 Frequency (radians/seconds) 10 Figure D.3: G F = S O P D T , G#=FOPDT, Category 1, Trend 1 Figures, r=0.75 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 169 D.2 Category 1, Trend 2 Figures « 5 0 CD B o1 cn 0) S . - 5 0 to | - 1 0 0 Q--150 10" Tp = Tp = tip > Hp K F = K R = 1, T ! _ F = 10, CF = 0.75, = 10, f?F = 10, tiR = 0, 2.5, 5, 7.5, r = 0.25 10 10 10 Frequency (radians/seconds) A 9 R = 0 • 8 R = 2.5 O 9 R = 5 * 9 R = 7.5 \ -Tp = Tp K F = K R = 1, n . io, % = io, eR 10 = Sp > Op . F = 10, CF = 0.5, = 0, 2.5, 5, 7.5, r = 0.25 6&—Ai • A 10 10 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) 10 10 A 9 R = 0 • 9 R = 2.5 O 9 R = 5 * 9 R = 7.5 10 10 10 Frequency (radians/seconds) = 10, Tf — Tf — tip > tig •- K R = 1, TJ_F = io, CF = 0.25, tif = 10, 0R = 0, 2.5, 5, 7.5, r: 0.25 r r 0 . 5 r * 5 0 CD I £ o| cn 0) 3 -50! <D "> to -100 CL -150; 10" 10 10 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) - -A e_ = 0 _ • R 9 R = 2.5 O 9 R = 5 \ * 9 R = 7.5 -\ -10 Tf = Tp = tip > tip K F = K R = 1, T L _ F = 10, CF = 0.1, TJ_R = 10, tiF = 10, 0R = 0, 2.5, 5, 7.5, r: DC 0.5 rx <D o •O10 E < 10" <n 5 0 CD : £ 0 cn <o S - 5 0 0) to . jS -100 0--150 10' 0.25 50 100 Time 10 10" 10 Frequency (radians/seconds) 10 10" 10 Frequency (radians/seconds) 10 A 9 R = 0 • 9 R = 2.5 O 9 R = 5 * 9 R = 7.5 10 Figure D.4: G F = S O P D T , G f l = F O P D T , Category 1, Trend 2 Figures, r=0.25 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 170 TF = T R = 8f> eIt K F = K R = 1, 7 i_p = 10, CF = 0.75, = 10, 6»F = 10, 6R = 0, 2.5, 5, 7.5, r = 0.5 •Q^t>—B A - i 10 10 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) 10 A 6 R = 0 • 9 R = 2.5 -j- |-hh O 9 R = 5 * 9 R = 7.5 10 TF = TR = 9F> 8R Kp = KR = 1, T i_p = 10, CF = 0.5, 7 l _ R = 10, flp = 10, 9R = 0, 2.5, 5, 7.5, r = 0.5 1n 1 ! ix~*»+*'1t A • O A » • 400 10 10 10 Frequency (radians/seconds) A 9 R = 0 • 9 R = 2.5 O 9 R = 5 9 R = 7.5 10 10 10 Frequency (radians/seconds) TF = TF = eF> 9 a K F = K R = 1, 7 i_p = 10, CF = 0.25, T i _ R = 10, % = 10, 9R = 0, 2.5, 5, 7.5, r= 0.5 I n ! ! ^ A m i f r t ^ ! • OA*. * 5 0 CD S 0 cn CD S -50 CD I-100| C L -1501 400 10 10 10 Frequency (radians/seconds) n A 9 R = 0 : • 9 R = 2.5 O 9 R = 5 * 9 R = 7.5 10 10 10 Frequency (radians/seconds) 10 TF = TR = 9F> 9p K F = K R = 1, n _ F = 10, CF = 0.1, T!_ R = 10, t?F = 10, <?R = 0, 2.5, 5, 7.5, r = 0.5 1.5: rxO.5 -10 10 10 Frequency (radians/seconds) A 9 R = 0 o 9 R = 2.5 o 9 R = 5 * 9 R = 7.5 10 10 10 Frequency (radians/seconds) 10 Figure D.5: G F = S O P D T , G R = F O P D T , Category 1, Trend 2 Figures, r=0.5 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 171 n - R Tf = Tp, = eF>eR Kp — KR — 1, T I _ F = 10, CF = 0.75, = 10, t% = 10, SR = 0, 2.5, 5, 7.5, r = 0.75 S 50 CD £ 0 cn CD S -50 CD Jj-100 CL -150h 10" 10 10 10 10 Frequency (radians/seconds) A 6 R = 0 • 6 R = 2.5 O 8 R = 5 ii! 9 R = 7.5 A • 10 10 Frequency (radians/seconds) n - R : TF = T„ = eF > eR K F = K R = 1, 7 i _ F = 10, CF = 0.5, 10, t9F = 10, 6H = 0, 2.5, 5, 7.5, r = 0.75 1 0.8 CD w n e C 0.6 O CL CD 0.4 tr 0.2 0, » 5 0 <D I ° S. -50 CD 8-100, CL -150 10 10"" 10 10 10" Frequency (radians/seconds) A = 0 • 6 R = 2.5 O 6 R = 5 * 6 R = 7.5 10~- 10 10 Frequency (radians/seconds) 10 10 TF=TB = bf> UR K F = K R = 1, T L F = 10, CF = 0.25, n - R = 10, t% = 10, OR = 0, 2.5, 5, 7.5, r = 0.75 1 0.8 CD |0.6| § 0 . 4 tr 0.2 0' «• 5 0 CD £ 0 & S- -50 CD J-100 CL -150j 10 P 500 600 10 10 Frequency (radians/seconds) A eR = o • eR = 2.5 o eR = 5 n - R : TF = TR = dp > 6R K F = KR = 1, T L _ F = 10, CF = 0.1, 10, 8f = 10, flR = 0, 2.5, 5, 7.5, r = 0.75 jvrtr >A • 1 O'£k p 10 10 10 10 Frequency (radians/seconds) 10 10 10 10 Frequency (radians/seconds) Figure D.6: G F = S O P D T , G f l = F O P D T , Category 1, Trend 2 Figures, r=0.75 D. Recycle Dynamic Figures for GF=SOPDT and GR—FOPDT 172 D.3 Category 2, Trend 1 Figures Tf < TR, 8f = 0R = 0 Tf < TR, 8f = 8R = 0 K F = K R = 1, T ! _ F = 2.5, CF = 0.75, K F = K R = 1, T L _ F = 2.5, C F = 0.5, 71_R = 5, 10, 25, % = r?R = 0, r = 0.25 TJ .R = 5, 10, 25, 9p = 8R = 0, r = 0.25 Frequency (radians/seconds) Frequency (radians/seconds) Tf < TR, 6F = BR = 0 TF < T„, 8F = 6R = Q K F = KR = 1, n.p = 2.5, CF = 0.25, K F = KR = 1, TX_f = 2.5, CF = 0.1, T ! _ R = 5, 10, 25, SF = 0R = 0, r = 0.25 T J . R = 5, 10, 25, 0p = 0R = 0, r = 0.25 Frequency (radians/seconds) Frequency (radians/seconds) Figure D.7: G F = S O P D T , G f l = F O P D T , Category 2, Trend 1 Figures, r=0.25 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 173 1 0.8 0) W ft e l c 0.6 o $0.4 0.2 OJ K F = K R = 1, r !_ F = 2.5, CF = 0.75, T ! _ R = 5, 10, 25, 6r = 6R = 0, r = 0.5 ... *CcF A • - j A g . - - - E 100 150 Time 250 1 0.8 CD c 0.6 o SO.4 rr 0.2 0. Tf < TR, t?f = 0R = 0 K F = K R = 1, N _ F = 2.5, CF = 0.5, TJ_R = 5, 10, 25, % = 0R = 0, r = 0.5 ... S^JX -/ , ' , - e - ' ' /•' -^;' .q-. -r> A, •-• J A Q . - E ... 10 10 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) 1 0.8; CD S0.6 ! o § 0 . 4 CC 0.2 0, 10 ~ 50 CO CD £ 0 CO CD S -50 CD co-100 0--150 10 Tr < TB, eF = eR = o K F = K R = 1, T L F = 2.5, CF = 0.25, T J _ r = 5, 10, 25, Of = I?R = 0, r = 0.5 . - o - — ..JQ-'. A D u . A o - E ... /V,-' i \ i 100 150 Time 250 10 10 10 Frequency (radians/seconds) = 10 10"' 10"' 10 Frequency (radians/seconds) 10 1.5, CC 0.5 Tf < TR, 8f = 6R = 0 K F = KR = 1, T ! _ F = 2.5, CF = 0.1, T ! _ R = 5, 10, 25, 8f = SR = 0, r = 0.5 50 100 150 Time 200 250 10 10 10 Frequency (radians/seconds) Figure D.8: G i r=SOPDT, Gfl=FOPDT, Category 2, Trend 1 Figures, r=0.5 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 174 TF < TR, OF K F = K R = 1, T ! _ F = n_R = 5, 10, 25, 0F = 2.5, CF = 0.75, r = 0.75 1 0.8 CD m n o C 0.6 O § 0 . 4 rr 0.2 0, • ft ^ _ - A ^ ^-'"°J ':~.......„y„j&-±~~.~. —a—i - © - - - -K F Tf < TR, 6F = 6R = 0 - K R = l , n_F = 2.5, cF = 0.5, 1 0.8 co S 0 . 6 -o Q . I rr 0.2, 0, T ! _ R = 5, 10, 25, c9F = (9R = 0, r = 0.75 -~ . * B • " • /-' 1 1 1 1 1 10 10 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) Tf < TR, eF = eF = o K F = KR = 1, T I ^ F = 2.5, CF = 0.25, n_R = 5, io, 25, er = eR = o, r = 0.75 Tf < TR, SF = 6R = 0 K F = K R = 1, T i „ F = 2.5, CF= 0.1, n_R = 5, 10, 25, % = (9R = 0, r = 0.75 10 10 10 Frequency (radians/seconds) 10"' 10" 10 Frequency (radians/seconds) Figure D.9: C F = S O P D T , Gfl=FOPDT, Category 2, Trend 1 Figures, r=0.75 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 175 D.4 Category 2, Trend 2 Figures TR = 0R> Tf = eF K F = K R = 1, r j . p = 1, CF = 0.75, T ! _ R = 5, 10, 25, 50, Of = 1, t?R = 5, 10, 25, 50, r = 0.25 1 0.8 CD S0.6 o Q. W ft A CD 0.4 CC 0.2 0' 50 Time 10 10 10 Frequency (radians/seconds) n-R 1 i — TR = On > TF = 0F K F = K R = 1, n _ F = 1, cF = 0.5, 5, 10, 25, 50, Of = 1, 0R = 5, 10, 25, 50, r = 0.8 CD 20.6 o I W 0.21 °( 10° tx10 | 1 0 " 2 < 10 "O = 0.25 50 Time 100 A x - R = 5 • x - R = 10 O - R = 25 * z - R = 50 10 10 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) 150 10 RR = &R > Tf = Of Kf = K R = 1, T I _ F = 1, CF = 0.25, T X _ R = 5, 10, 25, 50, Of = 1, BR = 5, 10, 25, 50, r = 0.25 1.5 CC 0.5 pi .Ag ^-OA u • Time TR = OR > Tf = 0f Kf = K R = 1, T i _ F = 1, CF = 0.1, Ti_ R = 5, 10, 25, 50, Of = 1, 0R = 5, 10, 25, 50, r = 0.25 10 10 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) Figure D.10: G F = S O P D T , G i ? = F O P D T , Category 2, Trend 2 Figures, r=0.25 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 176 TR = OR>TF = 0F K F = K R = 1, T ! _ F = 1, CF = 0.75, r i _ R = 5, 10, 25, 50, Of = 1, 0R = 5, 10, 25, 50, r = 0.5 1 0.8 CD 2o.6| o Q . CD 0.4 rr 0.2 0' -> . .A . , t i u . - . . . A — i _ _ - 0 _q.-4 L . - J . - A 300 = > = oF K F = K R = 1, r i _ F = 1, C F = 0.5, n.R = 5, 10, 25, 50, flF = 1, 0R = 5, 10, 25, 50, r = 0.5 1r 0.8 r CD 2 0.6 0.2 r 0. •• Ul..'' | . . . A . . g - r — — A — | - a — _ A , i X L , -A -© r " ~ ~ ;..••+ 300 10 10 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) TR = OR > TF = Of K F = KR = 1, T L _ F = 1, CF = 0.25, T ! _ R = 5, 10, 25, 50, Of = 1, c9R = 5, 10, 25, 50, r = 1 0.8 ?0.6r 0.5 CD 0.4 rr If .. * TR = OR > Tf = Of K F = K R = 1, T^F = 1, CF = 0.1, T ! _ R = 5, 10, 25, 50, Of = 1, 0R = 5, 10, 25, 50, r = 0.5 ,.A_ o-r A — ! _ 0 _ _ i J L - A 10 10 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) Figure D . l l : G F = S O P D T , G R = F O P D T , Category 2, Trend 2 Figures, r=0.5 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 177 TR = OR > Tf = Of K F = K R = 1, N _ F = 1, CF = 0.75, n_R = 5, 10, 25, 50, Of = 1, 0 R = 5, 10, 25, 50, r = 0.75 1 o.8; CD S0.61 o Q . <D0.4 CC 0.2 0. 10° .9 I lO-ci) T3 < -t*-—i— - - O f " /yn'' _ ,_ -©-"" ...............4....^, - ' '" \ 300 A -R = 5 • T -R = 10 O T1 -R = 25 * T1 -R = 50 10 10 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) 10 TR = OR > Tf = Of K F = KR = 1, T ! _ F = 1, CF = 0.5, n_R = 5, 10, 25, 50, Op = 1, 0R = 5, 10, 25, 50, r = 0.75 10 10 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) 10 Figure D.12: G F = S O P D T , G « = F O P D T , Category 2, Trend 2 Figures, r=0.75 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 178 D.5 Category 2, Trend 3 Figures Of >0R = TF>TR K F = K R = 1, T I _ f = 5, 10, 15, 20, CF = 0.75, T J _ r = 2.5, Of = 25, 0R = 5, 10, 15, 20, r = 0.25 250 10 10 10 Frequency (radians/seconds) 10 A * F = 5 ............... • V = 10 ..:..:..L.ii.:L O = 15 * \ = 20 10 10 10 Frequency (radians/seconds) 10 Bp > OR = Tf > TR K F = K R = 1, T ! _ F = 5, 10, 15, 20, CF = 0.5, n_R = 2.5, Of = 25, e?R = 5, 10, 15, 20, r = 0.25 CC 110° E < 10"" 100 150 Time 10 10 10 Frequency (radians/seconds) A T F = 5 • V = 10 O T F = 15 • T F = 20 10 10" 10 Frequency (radians/seconds) 10 Of > 0R = Tf > TR K F = K R = 1, T ! _ p = 5, 10, 15, 20, CF = 0.25, 1.5 rr0.5 T I _ R = 2.5, Of = 25, 9R = 5, 10 15, 20, r = 0.25 -Ann f i ' f i . / v_. / ~ //// ' //'•: 10"' 10"' 10 Frequency (radians/seconds) A = 5 • V = 10 O \ = 15 • = 20 10 10" 10 Frequency (radians/seconds) Of>0R = Tf> TR Kp = K R = 1, T I _ F = 5, 10, 15, 20, CF = 0.1, 7 I _ R = 2.5, Of = 25, 0R = 5, 10, 15, 20, r = 0.25 10 310" E < 10" 10"" 10 10" 10 Frequency (radians/seconds) A T F = 5 • XF = 10 O T F = 15 • ZF = 20 10 10" 10 Frequency (radians/seconds) 10 Figure D.13: G F = S O P D T , G H = F O P D T , Category 2, Trend 3 Figures, r=0.25 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 179 3)0.4 tr eF > oR — TF > TR - KR = 1, T ! _ F = 5, 10, 15, 20, CF = 0.75, 2.5, 0r = 25, 0R = 5, 10, 15, 20, r = 0.5 10 10 10 Frequency (radians/seconds) A \ = 5 • \ = 10 • O T F = 15 . . . . . * V = 20 10 10 10 Frequency (radians/seconds) 10 9F > eR = TF > TR K F = KR = 1, 7 i _ F = 5, 10, 15, 20, CF = 0.5, 7 i _ R = 2.5, Of = 25, 6R = 5, 10, 15, 20, r = 0.5 1.5 tr 0.5 10 1 • 1' If/-' L1' 200 300 Time 400 500 10 10 10 Frequency (radians/seconds) 10 A T F = 5 • XF = 10 O X F = 15 * *F = 20 10 10 10 Frequency (radians/seconds) Of > VR = Tf > TR K F = KR = 1, T I _ F = 5, 10, 15, 20, CF = 0.25, TJ_R = 2.5, t?F = 25, 6R = 5, 10, 15, 20, r = 0.5 tf 0.5 10 tr fl) o • g i o E < <? 5 0 CD s> o cn <D S -50 CD Jj-100 CL -150 10"" j i Ay 100 200 300 Time 400 500 10 10" 10 Frequency (radians/seconds) 10 A Tp = 5 *~~~~~»*atv • x F = 10 O T f = 15 -* ^ = 2 ° ^ > : : ^ A I !' 1' ;";:!';: 10 10" 10 Frequency (radians/seconds) 10 , Of > 0R = Tf > TR K F = KR = 1, rj-p = 5, 10, 15, 20, Cp = 0.1, TI_R = 2.5, 0e = 25, <?R = 5, 10, 15, 20, r = 0.5 10" 250 10 10" 10 Frequency (radians/seconds) 10 A = 5 : r ; • T F = 10 pi i O = 15 ::: * \ = 20 ||.^ 10 10" 10 Frequency (radians/seconds) Figure D.14: G F = S O P D T , G i i = F O P D T , Category 2, Trend 3 Figures, r=0.5 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 180 6F>6p = T F > Tp K F = K R = 1, ri^p = 5, 10, 15, 20, CF = 0.75, T I _ r = 2.5, % = 25, 6R = 5, 10, 15, 20, r = 0.75 1 0.8 CD c>0.6| o 1 SO.4 CC 0.2 0' • • I/Z' I-'' 600 9F > VR = Tf > Tp K F = K R = 1, T I _ F = 5, 10, 15, 20, CF = 0.5, T i _ R = 2.5, % = 25, 0 R = 5, 10, 15, 20, r = 0.75 10 10 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) A = 5 • T F = 10 O X F = 15 * T F = 20 10 10 10 Frequency (radians/seconds) 10 » f > « H = Tf > Tp K F = K R = 1, Ti_f = 5, 10, 15, 20, CF = 0.25, 7 i _ R = 2.5, 9F = 25, 0R = 5, 10, 15, 20, r = 0.75 cc 0 rr 10 |-10" _ < 10 10"' r\ : yJA / \ L\ j \ In ' 100 200 300 Time 10 10 10 Frequency (radians/seconds) 10 10 10 10 Frequency (radians/seconds) Vf>t>p = Tf> TR Kp = K R = 1, T I _ F = 5, 10, 15, 20, CF = 0.1, Ti_ R = 2.5, Op = 25, t?R = 5, 10, 15, 20, r = 0.75 3| . 2 W c CL 1 « CD CC 10 10 10 Frequency (radians/seconds) A = 5 Irr • T F = 10 .• L O T F = 15 * T F = 20 10 10 10 Frequency (radiansfeeconds) Figure D.15: G F = S O P D T , Gtf=FOPDT, Category 2, Trend 3 Figures, r=0.75 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 181 D.6 Category 3, Trend 1 Figures @F > Tf - TR > 9R K F = K R = 1, T J _ F = 5, 10, 15, 20, CF = 0.75, T i _ R = 5, 10, 15, 20, % = 25, (9R = 2.5, r = 0.25 10' 10 10 10 Frequency (radians/seconds) A \ = 5 • = 10 iHHB O X F = 15 • T F = 20 10 10 10 Frequency (radians/seconds) 0 F > T F = T R > 9 R K F = K R = 1, T j . p = 5, 10, 15, 20, CF = 0.5, 7 i _ R = 5, 10, 15, 20, 9 F = 25, 0 R = 2.5, r = 0.25 CC0.5 10 10 10 Frequency (radians/seconds) A X F = 5 • T F = 10 O T F = 15 • T F = 20 ...:..:..L.L;jL 10 10 10 Frequency (radians/seconds) 9F > TF = TR > 9R K F = K R = 1, r!_ F = 5, 10, 15, 20, CF = 0.25, 7 l _ R = 5, 10, 15, 20, % = 25, OR = 2.5, r = 0.25 10 10 10 10 Frequency (radians/seconds) 10 A 5 • T F = 10 O T F = 15 * \ = 20 .............. 10"' 10 10 Frequency (radians/seconds) 10 9F > TF = TB > 9R K F = K R = 1, T I _ F = 5, 10, 15, 20, CF = 0.10, TI_R = 5, 10, 15, 20, 9F = 25, 9R = 2.5, r = 0.25 1.5, tr 0.5 "VOL /XTftii-j^ , \; t \4 \y X/ "v-'v :/y v M ; 10 10" 10 Frequency (radians/seconds) A * F = 5 • T F = 10 -O T F = 15 * T F = 20 10"' 10"' 10 Frequency (radians/seconds) 10 Figure D.16: G F = S O P D T , G#=FOPDT, Category 3, Trend 1 Figures, r=0.25 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 182 K F = K R = 1, rj .p = 5, 10, 15, 20, CF = 0.75, n - R = 5, 10, 15, 20, df = 25, OR = 2.5, r = 0.5 A^--Lfc.'^ .'-^ T-B.F.l.!!rf>l 10 10 10 Frequency (radians/seconds) A T F = 5 • XF = 10 • O \ = 15 • T F = 20 Bp > Tp = Tp > Bp K F = K R = 1, T J _ F = 5, 10, 15, 20, CF = 0.5, T ! _ R = 5, 10, 15, 20, 0 F = 25, 6R = 2.5, r = 0.5 500 10 10 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) Bp > Tp = TK> Bp K F = K R = 1, T ! _ F = 5, 10, 15, 20, CF = 0.25, T J . R = 5, 10, 15, 20, (?F = 25, r?R = 2.5, r = 0.5 rr 0.5 jirJ />"'•' I'l: 100 200 300 Time 400 500 Bp > Tp = Tp > Bp K F = K R = 1, T i _ F = 5, 10, 15, 20, CF = 0.1, T J _ R = 5, 10, 15, 20, Bf = 25, 6K = 2.5, r = 0.5 10" 10"' 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) Figure D.17: G F = S O P D T , G*=FOPDT, Category 3, Trend 1 Figures, r=0.5 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 183 6f>Tf = TR> 8p K F = K R = 1, n _ F = 5, 10, 15, 20, CF = 0.75, T i _ R = 5, 10, 15, 20, % = 25, 0 R = 2.5, r = 0.75 1 0.8 CD S 0 . 6 o Q . 0)0.4 CC 0.2 0, Tf::::::::.::::... lit/ 0 100 200 300 400 500 600 700 800 Time 10"' 10"' 10 Frequency (radians/seconds) 10"' 10" 10 Frequency (radians/seconds) 10 A T F = 5 • T F = 10 O X F = 15 • T F = 20 eF>TF = rR> eR K F = K R = 1, T ! _ F = 5, 10, 15, 20, CF = 0.5, T j . R = 5, 10, 15, 20, 0 F = 25, 0 R = 2.5, r = 0.75 1 0.8 CD g 0.6 o a 0)0.4 CC 0.21 0 : ft!* 10' 0 100 200 300 400 500 600 700 800 Time 10" 10" 10" Frequency (radians/seconds) A X F = 5 • X F = 10 O T F = 15 * X F = 20 10 10 10 Frequency (radians/seconds) K F = K R = 1, T j _ F = 5, 10, 15, 20, CF = 0.25, r i _ R = 5, 10, 15, 20, t?F = 25, 6K = 2.5, r = 0.75 "0 100 200 300 400 500 600 700 800 Time 10 10" 10 Frequency (radians/seconds) A T F = 5 • T F = 10 -O T F = 15 • X F = 20 10" 10" 10 Frequency (radians/seconds) 10 Of > TF = TR> e„ Kp = K R = 1, T ! _ F = 5, 10, 15, 20, CF = 0.1, T J . R = 5, 10, 15, 20, % = 25, t?R = 2.5, r = 0.75 2 1.5| I 1 3 0.5 0 -0.5, «• 5 0 CD e o 8> S -50 CD ra-100 a. -150] 10 .... 200 10 10" 10 Frequency (radians/seconds) 10 A T p = 5 • T F = 10 O T p = 15 * t p = 20 10 10 10 Frequency (radians/seconds) 10 Figure D.18: G F = S O P D T , G*=FOPDT, Category 3, Trend 1 Figures, r=0.75 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 184 D.7 Category 3, Trend 2 Figures Op > TF = Tp > Of Kp = KR = 1, T i _ p = 10, CF = 0.75, T i _ R = 10, 0 F = 5, BR = 15, 25, 50, 75, r = 0.25 1 0.8 o -.0.6 D SO.4 0.2 r ° 0 . . . '•.•rSr.r."^ . . . 50 100 150 Time 200 10 10 10 Frequency (radians/seconds) 10 10" 10 Frequency (radians/seconds) 250 10 A 8 „ = 15 • » R = 25 •H'f O 6 R = 50 * E R = 75 10 0p>Tp=Tp> Of K F = K R = 1, T i _ F = 10, Cp = 0.5, T : _ r = 10, tip = 5, t?R = 15, 25, 50, 75, r = 0.25 1|-1 p - A - -m J B - A O n f l 10 10" 10 Frequency (radians/seconds) 0P > Tf = Tp>DF Kp = K R = 1, r !_p = 10, CF = 0.25, T i _ R = 10, t9p = 5, eR = 15, 25, 50, 75, r = 0.25 1.5 rf 0.5 10= tr CD o •0 10° E < 10" 10"' » 5 0 CD £ 0 <? S -50 CD n - 1 0 0 f -150h 10' -3 50 100 150 Time 200 10 10" 10 Frequency (radians/seconds) A 9 R = 15 • 6 R = 25 O 6 R = 50 • 6 R = 75 10 10 10 Frequency (radians/seconds) 250 10 10 OR > Tf = R R > Of Kp = KR = 1, 7 - ! _ F = 10, CF = 0.1, T i _ R = 10, Sp = 5, 0R = 15, 25, 50, 75, r = 0.25 tr CD o • ° 1 0 ° 10" 10"° 100 150 Time 200 10 10" 10 Frequency (radians/seconds) 10 10" 10 Frequency (radians/seconds) 250 10 Figure D.19: G F = S O P D T , G#=FOPDT, Category 3, Trend 2 Figures, r=0.25 D. Recycle Dynamic Figures for GF=SOPDT and GR—FOPDT 185 10' 10 10 10 Frequency (radians/seconds) A OR = 15 • OR = 25 i'ff O 6 R = 50 * E R = 75 BR > TF = TR > BF K F = K R = 1, T j . p = 10, < F = 0.5, T I _ R = 10, Bf = 5, « R = 15, 25, 50, 75, r = 0.5 1 n ! „ Ai"B_\ >. A - -P-\-$.... 10 10 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) A 6 R = 15 • E R = 25 O 6 R = 50 * OR = 75 10 10 10" Frequency (radians/seconds) n O . s h BR > rF = TR > 6F K F = K R = 1, T i _ F = 10, C F = 0.25, = 10, BF = 5, 8R = 15, 25, 50, 75, r = 0.5 100 200 300 Time 400 10 10 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) 500 A OR = 15 • OR = 25 O OR = 50 • OR = 75 BR> TF = TR > BF K F = K R = 1, T i _ F = 10, < F = 0.1, T ! _ R = 10, Bf = 5, t ? R = 15, 25, 50, 75, r = 0.5 2 0 i 1 0 g "0.5 0' 200 300 Time 400 500 10 10 10 Frequency (radians/seconds) Figure D.20: G F = S O P D T , G R = F O P D T , Category 3, Trend 2 Figures, r=0.5 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 186 6 R > TF = TR> 6f K F = K R = 1, 7 i _ F = 10, <F = 0.75, T ! _ R = 10, er = 5, 6R = 15, 25, 50, 75, r = 0.75 . A . . . . . . J ^ M $ S 1 0.8 CD S 0 . 6 o CL 0)0.4 rr 0.2 0, ~ 200 CD CD . OI 100 CD 3 , CD 0 CO CO £ - 1 0 0 10 10 10 Frequency (radians/seconds) 10"" 10 10" 10 Frequency (radians/seconds) 0 100 200 300 400 500 600 700 800 Time • -H:r A 8 R = 15 • 9 R = 25 O 8 R = 50 * 8 R = 75 "*WL • 10 eR>TF = TR > eF K F = K R = 1, T ! _ F = 10, CF = 0.5, T ! _ R = 10, flF = 5, 8R = 15, 25, 50, 75, r = 0.75 0 100 200 300 400 500 600 700 800 Time 10 10" 10 Frequency (radians/seconds) A 8 R = 15 • 8 R = 25 O 6 R = 50 * « R = 75 10 10"' 10" Frequency (radians/seconds) Figure D.21: G F = S O P D T , G ^ F O P D T , Category 3, Trend 2 Figures, r=0.75 D. Recycle Dynamic Figures for GF=SOPDT and GR—FOPDT 187 D.8 Category 4, Trend 1 Figures TF = T R < e F = Op K F = K R = 1, T i _ F = 5, C F = 0.75, T J _ R = 5, Of = 0R = 10, 15, 25, 50, r = 0.25 1 0.8 CD go.6 o § 0 . 4 tr 0.2 0. -Ii l : /.' ;' : Tf = Tp < Of = Op K F = K R = 1, n _ F = 5, CF = 0.5, r l - R = 5, 9F = 9R = 10, 15, 25, 50, r = 0.25 1 0.8 CD £ 0 . 6 o § 0 . 4 rr 0.2 o; -=s--—o-I;; . 10 10" 10 Frequency (radians/seconds) 10 10" 10 Frequency (radians/seconds) 300 A 9F = 10 • 9 F = 15 O 9 F = 25 * 9 F = 50 10 10" 10 Frequency (radians/seconds) tf 0.5 Tf=Tp<0p = Op Kp = K R = 1, T±_f = 5, CF = 0.25, T i _ R = 5, 0T = t?R = 10, 15, 25, 50, r = 0.25 •S 5 0 <U £ 0 ct> CD S -50 CD « - 1 0 0 CL -150 10' 1 1 1 1 • Wyr P>-'--10 10" 10 Frequency (radians/seconds) A 6 F = 10 • 9 F = 15 O 9 F = 25 • 9 F = 50 10"' 10" 10" Frequency (radians/seconds) 300 10 Tf = Tp < Of = Op = K R = 1, 7 i _ F = 5, C F = 0.1, « 5 0 <D S> 0 CO CD s -so! CD CO-100; i f -150j 10" 10 10" 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) 300 A 9 F = 10 rM'ir • 6 F = 15 O 9 F = 25 * 6 F = 50 Figure D.22: G F = S O P D T , G f l = F O P D T , Category 4, Trend 1 Figures, r=0.25 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 188 Tf = T„<6f = 8R K F = K R = 1, TX_f = 5, Cr = 0.75, T i _ R = 5, 6f = 6K = 10, 15, 25, 50, r = 0.5 iij^lfl, - IJ - A n . a Q A 10 10 10 Frequency (radians/seconds) A 6 F = 10 • E F = 15 ••H'HT! O 9 F = 25 E F = 50 10 10 10 Frequency (radians/seconds) 10 Tf =TR<6F = 8R Kf = K R = 1, n _ F = 5, CF = 0.5, n.R = 5, % = 6R = 10, 15, 25, 50, r = 0.5 10' 10 10 10 Frequency (radians/seconds) I A 9 F = 10 • 9 F = 15 j O 9 F = 25 * 9 F = 50 10 10 10 Frequency (radians/seconds) 10 10 ccO.5 Tf = TR<i)f = i)R K F = K R = 1, TX_f = 5, CF = 0.25, TI_R = 5, Of = flR = 10, 15, 25, 50, r = Ii • /- i 100 200 300 Time 400 10 10 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) 500 A 9 F = 10 • 9 F = 15 O 9 F = •25 * 9 F = 50 10 Tf = TR <6f = 6R Kf = K R = 1, n _ F = 5, CF = 0.1, n-R = 5> eF = *R = 1°. I 5 . 25, 50, r = 0.5 10 10 10 Frequency (radians/seconds) Figure D.23: G F = S O P D T , G R = F O P D T , Category 4, Trend 1 Figures, r=0.5 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 189 T f = Tg < Bf = eR K F = K R = 1, r , _ F = 5, CF = 0.75, T j . R = 5, Bf = <?R = 10, 15, 25, 50, r = 0.75 1 0.8 CD £ 0 . 6 o § 0 . 4 rr 0.2 10" . . . . • • i ^ i i ~ . . . . - • ST/T^S <• - O - ' . Q / / ' / / ' / 10 10 10 Frequency (radians/seconds) 10 10 10 Frequency (radians/seconds) 200 10 A = 10 • 6 F = 15 O 6 F = 25 * 9 F = 50 T f = Tp < Bp = Bp K F = K R = 1, n _ F = 5, CP = 0.5, T ! _ R = 5, 6e = 8R = 10, 15, 25 , 50, r = 0.75 0 100 200 300 400 500 600 700 800 Time 10" 10" 10 10" Frequency (radians/seconds) A 9 F = 10 • 9 F = 15 O 9 F = 25 * 9 F = 50 10 10 10 Frequency (radians/seconds) 10 T f = Tp <0p = Bp K F = K R = 1, n _ F = 5, CP = 0.25, = 5, BF = 0R = 10, 15, 25, 50, r = 0.75 10' 10 10 10 Frequency (radians/seconds) 10 A 9 F = 10 • 9 F = 15 ': r O 9 F = 25 • 9 F = 50 10 10 10 Frequency (radians/seconds) 10 cc 0 T f = Tp < Bp = Bp Kp = K R = 1, T!_p= 5, C F = 0.1, TJ_R = 5, flp = 6R = 10, 15, 25, 50, r = 0.75 -ASA A y 1 " V A©' 100 Time 150 200 cc 10 | -10" 2 < : : : : : : : : t . . . 10"" 10 10 10 Frequency (radians/seconds) 10 10 10 10 Frequency (radians/seconds) 10 Figure D.24: G F = S O P D T , G H = F O P D T , Category 4, Trend 1 Figures, r=0.75 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 190 D.9 Category 4, Trend 2 Figures 8R > Bp > T F = T B K F = K R = 1, 7 i _ F = 1, CF = 0.75, 1, % = 5, BR = 10, 15, 20, 25, 50, r = 0.25 n -R = VR> Op > Tp = TR K F = K R = 1, 7 I _ F = 1, CF = 0.5, 1, 0F = 5, 6R = 10, 15, 20, 25, 50, r = 0.25 10' 10 10 10 Frequency (radians/seconds) W 5 0 CD £ 0 L? I i -50 CD m-100 a. -150 10" 10 10 10 Frequency (radians/seconds) A e R = 10 • E R = 15 O E R = 20 • 8 R = 25 0 8 R = 50 200 10 10 10 Frequency (radians/seconds) 10 1 > Bp > TF = TR i, n_F = i, CF = 0.25, 10, 15, 20, 25, 50, r = 0.25 10 10 10 Frequency (radians/seconds) 200 A 9 R = 10 • 9 R = 15 O 6 R = 20 * E R = 25 NB 6 R = 50 BR > Bp > Tp = TR K F = K R = l , T x _ f = l , CF = 0.1, T ! _ R = 1, 0F = 5, 8R = 10, 15, 20, 25, 50, r = 0.25 10" 10"' 10 Frequency (radians/seconds) 10"" 10" 10 Frequency (radians/seconds) 10 A OR = 10 • OR = 15 O OR = 20 * OR = 25 OR = 50 10 10" 10 Frequency (radians/seconds) Figure D.25: G F = S O P D T , G « = F O P D T , Category 4, Trend 2 Figures, r=0.25 D. Recycle Dynamic Figures for GF=SOPDT and GR—FOPDT 191 BR > 6F > TF = TR BR > 6F > TF = TR K F = K R = 1, n _ F = 1, CF = 0.75, K F = K R = 1, n _ F = 1, CF = 0.5, T J . R = 1, 0F = 5, BR = 10, 15, 20, 25, 50, r = 0.5 T J _ R = 1, 0F = 5, OR = 10, 15, 20, 25, 50, r = 0.5 Frequency (radians/seconds) Frequency (radians/seconds) BR> 8 F > TF = TR K F = K R = 1, T I _ F = 1, CF = 0.25, 1, BR = 5, OR = 10, 15, 20, 25, 50, r= 0.5 W 5 0 CO %. 0 CO CD S -50 01 8-100 Q_ -150 10 300 10 10 10 Frequency (radians/seconds) 10 A 8 R = 10 ':_ • 9 R = 15 I O 8 R = 20 f * 9 R = 25 [ I ' M V ' ' O 9 R = 50 j : V'lW ' : : nJYl: : • 10 10 10 Frequency (radians/seconds) 10 BR> BF> TF — TR KF = KR = i, n_F = l, CF = o.i, _ R = 1, BE = 5, OR = 10, 15, 20, 25, 50, 0.5 2 1.5 o (0 I 1 CO CD CC 0.5 0'. 10 2 o CC 10 Q-10" < 20 40 Time 60 10" A 6 R = 10 • 9 R = 15 O 6 R = 20 * 6 R = 25 9 R = 50 10"" 10 10" 10 Frequency (radians/seconds) 10"' 10"' 10 Frequency (radians/seconds) A A A *A 'i\ 4v 100 10 10 Figure D.26: G F = S O P D T , G#=FOPDT, Category 4, Trend 2 Figures, r=0.5 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 192 Og > 6V > Tf = Tg Bp > 8F > TF = Tg K F = K R = 1, T X _ f = 1, CF = 0-25, K F = K R = 1, T ! _ F = 1, CF = oi", r l - R = 1, sf = 5 , #R = 10, 15, 20, 25, 50, r = 0.75 TJ_R = 1, % = 5, 0R = 10, 15, 20, 25, 50, r = 0.75 Frequency (radians/seconds) Frequency (radians/seconds) Figure D.27: G F = S O P D T , G K = F O P D T , Category 4, Trend 2 Figures, r=0.75 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 193 D.10 Category 4, Trend 3 Figures OR > TR > T F = 0f K F = K R = 1, T ! _ F = 1, CF = 0.75, r j _ R = 5, t?F = 1, 0R = 10, 15, 25, 50, r = 0.25 1 I 1 t j U G < | I I . . ^ . . . B P C Y . I . . f e n A n OR > TR > TF = 6F K F = K R = 1, 7 i _ F = 1, CF = 0.5, T l - R = 5, « F = !. ER = 1 0 , 15, 25, 50, r = 0.25 -uAu ~ - B CA ' » . ' O i A O 10"' 10" 10 Frequency (radians/seconds) « 5 0 CD 8> 0 8" S -50 CD j5-100 Q. -150 10" 10 10" 10 Frequency (radians/seconds) A 8 R = 10 • 8 R = 15 O 8 R = 25 8 R = 50 10 10" 10 Frequency (radians/seconds) OR > Tp > TF = 0F K F = K R = 1, N _ F = 1, CF = 0.25, T i _ R = 5, c% = 1, 0R = 10, 15, 25, 50, r = 0.25 tf 0.5 A » ^ > , ^ T 200 OR > TR > Tf - Bf K F = K R = 1, T ! _ F = i, C F = 0.10, T j . R = 5, 6f = 1, 9R = 10, 15, 25, 50, r = 0.25 CD , CO ' C & CD CC0.5 10 10" 10 Frequency (radians/seconds) 10 10" 10 Frequency (radians/seconds) Figure D.28: G F = S O P D T , G R = F O P D T , Category 4, Trend 3 Figures, r=0.25 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 194 9 S > T B > T F = eF eR> TB> TF = sF K F = K R = 1, T ! _ F = 1, CF = 0.75, K F = K R = 1, T ! _ F = 1, CF = 0.5, n - R = 5, % = 1, » R = 10, 15, 25, 50, r= 0.5 n - R = 5 , 9F = 1, flR = 10, 15, 25, 50, r= 0.5 Frequency (radians/seconds) Frequency (radians/seconds) #i? > TR > TF = tV 6R>TR>TF = 6F K F = K R = 1, T ! _ F = 1, CF = 0.25, K F = K R = 1, T i _ F = 1, CF = 0.10, T J . R = 5, 1% = 1, t?R = 10, 15, 25, 50, r = 0.5 T i _ R = 5, % = 1, (9R = 10, 15, 25, 50, r = 0.5 Frequency (radians/seconds) Frequency (radiansfeeconds) Figure D.29: G F = S O P D T , G i J = F O P D T , Category 4, Trend 3 Figures, r=0.5 D. Recycle Dynamic Figures for GF=SOPDT and GR=FOPDT 195 Figure D.30: G F = S O P D T , G*=FOPDT, Category 4, Trend 3 Figures, r=0.75 Appendix E Examples of Complex Recycle Structures R, Figure E . l : Configuration 1 r ) Figure E.2: Configuration 2 LHiJ Figure E.3: Configuration 3 196 E. Examples of Complex Recycle Structures 197 i Figure E.4: Configuration 4 Appendix F C M V Averaging Level Control Design for Three Units of Sampling Delay F . l Process Model When the plant time delay 6 = 3 the process model eqn. 5.1 becomes: _ u)(z-1) 6{z-1) _ UJ0 at Y t ~ J(z^)vUt-b + W W d t ~ v U t ~ * + where: UJ{Z~1) = UJQ Siz'1) = 1 d(z-1) = 1 <j>(z~l) = 1 d = 2 (F.2) The polynomials in eqn. F.2 have the following degrees: degioj) = s = 0 deg(5) = r = 0 deg{6) = q = 0 deg((j>) = p = 0 F.2 Spectral Factorization When the appropriate subsititutions are made in the spectral factorization equation (eqn. 5.4), the following is obtained: 7(^- 1 )7(^)=^ + A ( l - ^ 1 ) 2 ( l - z ) 2 (F.3) The spectral factorization equation is unchanged from the 6 = 1 case, thus, the equations for 7o,7i, and 72 taken from Foley et al. (2000) are as shown below: 7i = - y + sgn(uj0) (F.4) 2A . .1 /16A2 ^ ._, . 72 = + 8gn(u0)-*—r-4\ (F.5) 7i 2 V 7i 70 = Lo0 + 71 + 72 (F.6) where sgnQ denotes the signum function, e.g., sgn(uo) — +1 when cu0 > 0, sgn^o) = —1 when CUQ < 0. 198 F. CMV Averaging Level Control Design for Three Units of Sampling Delay 199 F.3 Diophantine Identity Again, making the appropriate substitutions for w(z - 1), 9(z~l), </>(z-1), 6, and d in equa-tion 5.5, the diophantine equation becomes: OJ0Z3 = ~{(z)Qi(z~l) + zQ2{z)V2 (F.7) The Q polynomials in eqn. F.7 have the following degrees: deg(Qi) = max(q — b,p + d — 1) = max(0 — 3,0 + 2 — 1) = max(—3,1) =1 deg(Q2) = max(s + b, ng) — 1 = max(0 + 3,2) — 1 = max(3,2) — 1 = 2 Thus, the Q polynomials have the following form: Qi{z~l) = qio-qnz-1 (F.8) Q2(z~l) = q20 - q21z~l - q22z~2 (F.9) Substituting for 7(2), Q\(z~l), and <32("Z) into eqn. F.7: uoz3 = [70 - liz - 72-22][?io ~ Quz"1} + z[q2Q - q2Xz - q22z2][l - z~1]2 (F.10) v v ' " v ' (A) (B) Expanding the terms (A & B) on the RHS individually: (A) = 7o<?io - loqnz'1 - 7i<?io2 + 7i<?n - Ww*2 + l2q\\Z = -l2q\QZ2 + (-7i9io + 729n)« + (7o9io + 7i9n)^° - 7o9ii^ _ 1 (F- 1 1 ) (5) = z{q20 - q21z - q22z2}(l - 2z~l + z~2) = z(q20 - 2q20z~l + q20z~2 - q21z + 2q2l - q2\Z~X - q22z2 + 2q22z - q22) = q20z - 2q20 + q20z~x - q21z2 + 2q2lz - q2l - q22zz + 2q22z2 - q22z = ~q22Z3 + (2q22 - q2i)z2 + (2q2i + q2Q - q22)z + (~2q20 - q2i)z° + q20z'x (F.12) Replacing (A) and (B) in eqn. F.10 with the corresponding expansions (eqns. F . l l and F.12): UJQZ3 = (A) + (B) = -q22zz + {2q22 - q2l - 72<?io}z2 + [q20 + 2q21 - q22 - 71910 + 72911]* + {7o9io + 7i0ii ~ 2?20 - 921} + feo - 7o9n]* - 1 (F.13) F. CMV Averaging Level Control Design for Three Units of Sampling Delay 200 Equating LHS and RHS z polynomial coefficients of like powers gives the following equa-t i o n s : -<?22 = W 0 (F.14) -729io -921+2922 = 0 (F.15) ~7i0io+720ii +920+2921 -922 = 0 (F.16) 7o9io+7i9n-292o -921 = 0 (F.17) - 7 o 9 n +920 = 0 (F.18) The following identities are known from the solution of the spectral factorization equation (Foley et al. 2000) for the b = 1 case: 7(1) = 70 - 71 - 72 = coo (F.19) 7 ( l )Qi ( l )=w 0 (F.20) Thus, Qi( l) = 1. Using equations F.20 and F .8: Qi( l) = 9 i o - 9 n = l 9io = l + 9 n (F.21) From equations F.14 and F.18, q22 = — o^ a n d 920 = 7o0n- Subsituting for qi0, and q22 in eqn. F .15: - 72(1 + 9n) - 92i - 2wo = 0 .-. ?2i = -2w 0 - 72(l + 9n) (F.22) Then, substitute for qw, q2o and q22 in eqn. F.17: 7o(l + 9n) + 7i0n - 2 7 o 9 n + 2o;0 + 72(1 + 9n) = 0 7o + 7o9n + 7i0ii - 2 7 o 9 n + 2w0 + 72 + 729n = 0 7o0ii - 2 7o9n + 7i0ii + 729n = ~2w 0 - 7o - 72 -7o0ii + 7i0ii + 720ii = -l(2u0 + 70 + 72) - 0 i i (7o - 7 i - 72) = - l(2w 0 + 7o + 72) v v ' = OJO from eqn.F. 19 Thus: 2tJ 0 + 7o + 72 , „ , 0n = (F.23) A . 3cj 0 + 7Q + 72 (I?OA\ qw = 1 + 9n = (F.24) F. CMV Averaging Level Control Design for Three Units of Sampling Delay 201 F.4 Controller Structure The general optimal feedback controller given in equation 5.3 is shown below: c{ ' 9(z-i)j{z-1) - z-^iz-^Qiiz-i) f(z^)Vd (F.25) Now: / ( ^ - 1 ) V d = 6{z-l)^(z-1) - z-^z-^Qxiz-1) (F.26) The known values are b = 3, d = 2, and the polynomial degrees are deg(6) = 0, deg(^) = 2, deg(uj) = 0, deg(Vd) = 2, and deg(Q\) = 1. The largest degree on the RHS of equation F.26 is deg(z~buj(z~1)Q1(z~1)) = 4, thus, the deg(f) = 2. Making the appropriate subsi-tutions into eqn F.26: ••• (fo - fiz~l - hz~2)(\ - z~1)2 = (7o - jxz'1 - 7 2 £ - 2 ) - z-3LU0(qw ~ quz~l) " v ' A = 7o - 7 i * _ 1 - l2Z~2 ~ u0qwz-3 + uQqxxZ~A (F.27) Expanding the LHS term (A): (fo - hz-1 - f2z-2)(l - z-1)2 = ( / 0 - fxz-1 - f2z-2)(l - 2 Z ' 1 + z-2) =/o - 2/0-2"1 + / o * - 2 - hz~l + 2flZ~2 - hz~3 - f2z~2 + 2f2z-3 - hz'4 =/o - (2/ 0 + h)z~l + (fo + 2fx - h)z~2 - (h ~ V2)z~3 - hz~' (F.28) Equating like coefficients in powers of z on the RHS of eqn. F.27 with like coefficients on the RHS of eqn. F.28: fo = 7o (F.29) -(2/o + /i) = - 7 i (F-30) fo + 2/i -h = - 7 2 (F.31) - ( / i - 2 / 2 ) = -o;o9io (F-32) -h = wo9n (F.33) From eqn. F.29, / 0 = 7o- Subsituting into eqn. F.30: 2 7 o + / i = 7 i / i = 7 i - 2 7 o (F.34) F. CMV Averaging Level Control Design for Three Units of Sampling Delay 202 Using equation F.33 and equation F.23: f2 = (-l)(2a;o + 7o + 72) Subsituting for the known polynomials into eqn. F.25: Q^z-^Siz-1) qw-qnz-1 1 (F.35) Gc(z-') f(z-i)Vd~l ' fo - fiz'1 - f2z~2 3a>o + 70 + 72 _ 2a; 0 + 70 + 72 _ A 1 z I • 2LO0 + 70 + 72 UJ0 1 + LO0 L00 V To - (7i - 27o)z~1 - (27! - 370 + 7 2 K 2cu0 + 7o + 72 — z -1 1 V 7o - (71 - 27o)^_1 - (271 - 370 + 72)z~ 2 Making the denominator of the last term on the RHS monic: x = 2^ o + 7o + 72 o^7o 1 + LO0 - 1 2CJ 0 + 7o + 72 1 V (F.36) 1 - i ( 7 l - 2y0)z^ - i ( 2 7 i - 3 7 o + 7 2 ) ^ " 2 (F.37) If the following parameters are defined as: Kr 2co0 + 70 + 72 LU0 (F.38) coojo 2a;0 + 70 + 72 Then, Gc has the structure of a PI controller with a second order filter as shown below: T _ „ -n Kc[l + z-1} V ' 1 - i ( 7 l - 27o)z-1 - i ( 2 7 l - 3 7 o + l2)z~2 (F.39) F.5 Closed Loop Performance for Three Units of Sam-pling Delay The general equation for the output variance given in equation 5.6 is shown in equation F.40: F. CMV Averaging Level Control Design for Three Units of Sampling Delay 203 l*l=i l*l=i Defining: ^ = — / ( f )/w ( F. 4 1 ) (7o - 7 i * 72* 2J7(*)* The contour integral given in equation F.40 can be evaluated using the Residue Theorem (Foley et al. 2000): j $zdz = 2TTj ^2 ReSzM*) (F-42) The symbol ResZn$(z) refers to the residue of $(z) at zn, where zn is the n'th pole of $(z) inside the integration contour \z\ = 1. Because the residue is only concerned with the poles, equation F.41 can be rearranged, multiplied by z2/z2, and factored to give: $ = loll J J K ' = 2PJL: N K ' cp 43) * ^ i ^ z ) (z ~ z ^ z ~ Z2)<z) ' The poles of equation F.43 are shown in equation F.44 (Foley et al. 2000): 7i . \/7i 2 + 4727o 7i ^7? + 47270 , „ A A , Z l = 2 ^ + 2 | 7 0 | 2 | 7 0 | ( F ' 4 4 ) As explained in Foley et al. (2000), in the solution of the spectral factorization equation 7i, and 72 were chosen so that the roots of 7(* _ 1) would be inside the unit circle. Since 7(z - 1 ) has roots in the unit circle, 7(2) has roots outside the unit circle and does not contribute poles to the residue. Thus, equation F.40 can be written as: a\ = \Restl${z) + ResZ2$(z)]a2a (F.45) Expanding the numerator of equation F.43 -f(z-1)f(z) = -[fo - hz'1 - hz-2][f0 - hz - hz2} 7o 7o = - [ / o * - / i - / 2 * - 1 ] [ / o - / i * - / 2 * 2 ] 7o F. CMV Averaging Level Control Design for Three Units of Sampling Delay 204 = -[/o* - /o/i*2 - M2Z3 - foil + ftz + /1/222 - hf2z'1 + hf2 + f2z] 7o = -h/0/22 3 + A/222 - Ufiz2 + f%z + fiz + fiz - /0/1 + /1/2 - M2Z-1] 7o = - [ - / 0 / 2 2 3 + (/2 - f0)hz2 + (/o2 + fl + f22)z + A(/2 - f0) - M2Z-1} (F.46) 7o The individual residue terms in equation F.45 are as follows: i[-hf2z! + (f2 - /o)/j2 2 + (/ 2 + f2 + fi)zX + / l ( / 2 ~ /Q) - /p/2^- 1] (ZI - z2)(7o - 1\Z\ - J2z2) (F.47) ResZl$z = ^[-/o/2^23 + (/2 ~ /o)/l^ 22 + (/I + fl + / j)** + /l(/2 ~ /o) - A W ] (*i ~ z2) (70 - 7122 - 72^ 2) (F.48) The numerator term l / 7 o common to both residues in equations F.47 and F.48 is moved outside the square brackets of equation F.45 to form cr2/7o . The lowest common denomi-nator for both residue terms is : (zi -22X70 -7^1 -72<z2)(7o - 7 i 2 2 -72^2)• The resulting numerator residue terms are as shown: ResZl Num = [-f0f2z3 + (f2 - / 0 ) / i2 2 + (f2 + fl + f\)zx + A(/ 2 - /0) - A M " 1 ] . (7o - 71^ 2 - 72^2) = - 7 o / o M 3 + 7o(/2 - /o )M 2 + 7o(/o + fl + fl)zi + 7o(/2 - /o)/i - 7 o / o M _ 1 + 71/0/22?z2 - 7i( / 2 - U)hz\z2 - 7i(/o + fl + fl)z\Z2 - 7i(A - fo)hz2 + 7i/oM _ 1 22 + I2hf2z\z\ - 72(/2 - h)hzlz2 - 72(/02 + A2 + fl)zlZ2 - 72(/2 - / 0 )/l2 2 2 + l2hf2Z^Z2 (F.49) For equation F.48, the -1 outside the bracket is moved inside the bracket of the numerator: RezZ2 Num = [f0f2z32 - (f2 - f0)flZ2 - (f2 + fl + fl)z2 - A(/ 2 - /0) + f0f2z^1}. (70 - 1\Z\ - 722?) = 7o/o/222 - 7o(/2 - /o)/i22 - 7o(/o + Zi + A2)22 - 7o(A - /o)A + 70/0/222"1 - 7l/o/22l4 + 7l(A - fo)flZlZ2 + 7 i ( / 2 + A2 + f\)zxZ2 + 7l(/2 - / 0 ) / l2 i - 71/0/22i22_1 - I2hf2z\z\ + 72(/2 - h)hz2Z2 + 72(/o2 + /l + fl)z\z2 + J2(f2 ~ /o)/l22 - l2hhzW (F.50) F. CMV Averaging Level Control Design for Three Units of Sampling Delay 205 Combining eqns. F.49 and F.50: Num 1 + Num 2 = -70/0/2*? + 7o/o/2*23 +7o(/2 - /o)/i*2 - 7o(/2 - /o)/i*22 v-A B +7o(/o + /1 + ftU - 7o(/02 + fl + fl)z2 +7o(/2 - /o)/i - 7o(/2 - /o)/i -70/0/2*11 + 7o/o/2*2 1 +71/0/2*1*2 - 71/0/2*1*: 3 D E - 7 i ( / 2 - /o)/i*2*2 + 7i( / 2 - /o)/i*i*22 -7i(fo + /1 + / 2 2 )^2 + 7i(/o + fl + / 2 2 )^2 -7 i ( /2 - /o)/i*2 + 7i(/2 - /o)/i*i +7i/o/2*i 1z2 - 71/0/2*1*: =0 -1 2 G H +72/o/2*l*2 - 72/0/2*l2*2 -72(/2 - /o)/l*l*2 + 72(/2 - /o)/l*l*I > v ' v v ' / =0 -72(/o2 + fl + /l)*i*22 + 7 2(/ 0 2 + fl + /22)*2*2 - 7 2 ( / 2 - /o)/i*22 + 72(/2 - /o)/i*i2 J K v-l~2 „, r F -1 +72/0/2*1 x * 2 - 72/0/2*1*2 1 (F.51) v v , L The terms on the RHS of equation F.51 are then expanded individually: A => - 70/0/2 (*? ~* 2 ) 5 7o(/2-/o)/i(*1 2-*2 2) C* => 7o(/02 + /i +/ 2 2)(^i-^) D => - 7o/o/2(*i~1 - *2_1) = - 7o/o/2*r1*2_1(*2 - *l) = 7o/o/2*i"1*2"1(*l - *2) £ 7l/o/2*l*2(*l - *2) ^ =>• - 7l(/2 - /o)/l*l*2(*l - Z2) G => 7i(/2 -/o)/i(*i-*2) # 7i/o/2(*f1*2 - *i*2~2) = 7i/o/2*r1*2_1(*2 - *?) = -l\fohz7xz2x(z\-X I => 72/o/2*?*2(*1 - *2) </ => 72(/o + /l + /|)*1*2(*1 - *2) # 72(/2-/o)/l(* 2-*2 2) £ - l2hh(z\z2y - *1"1*2) =-72/o/2*r 1* 2" 1(^i -^2) Then the terms with like powers of 2 are combined: A + L => - 70/0/2(*i3 - *2) - 72/0/2*1 lz2 1 - 4) = -/o/2[7o + 72*r1*2-1](*?-*23) (F.52) F. CMV Averaging Level Control Design for Three Units of Sampling Delay 206 B + E + H + K =• 7 o ( / 2 - / o ) / i ( 2 2 ~z22) + 71/0/2^2(4 -4) - T i / o M - 1 ^ " 1 ^ 2 - 4) + 7 2 ( / 2 - fo)fi(zf - z\) = [(70+72X/2 - / 0 ) / i+71/0/2(2^2 - *rV)K*i - 4) (F.53) C + D + F + G + I + J => 7o(/o2 + / 2 + / 2 2 ) ( 2 i - 2 2 ) + 7o/o/22r 1 2 2 - 1 (2i ~ 2 2 ) - 7i(/2 - fo)fiZiZ2(z1 - z2) + 7i( / 2 - / 0 ) / l ( 2 i - z2) + 72/o/22 22|(2l - Z2) + 72(/o + / l + / 2 2 )2 l2 2 (2 l - Z2) = [(70 + 722l2 2 )(/ 0 2 + / 2 + f2) + ( 7 0 2 r 1 2 2 - 1 + l2z\z\)Uf2 + 7i( / 2 - /o)/i(l - 2i22)](zi - z2) (F.54) The total numerator is then sum of eqns. F.52, F.53, and F.54: Num 1 + Num 2 = - / 0 / 2 [7o + 722^2 2~ 1](2 1 3 - z\) + [(70 + 72)(/2 ~ / 0 ) / l + 7l/o/2(2l22 - 2 1 - 1 2 2 - 1 )](2 2 - Z\) + [(70 + 722l2 2 )(/ 0 2 + fl + /22) + ( 7 0 2 r 1 2 2 - 1 + 722 2 2 2 ) /o / 2 + 7i( / 2 - /o)/i(l - 2i^ 2)](2l - Z2) (F.55) Let: Z\ = a + 6\/c 2:2 = a — &\/c (F.56) Thus: a = ^ 7 6=^IVT C = 7I2 +4 727O (F.57) 27o 2 I 70 I Using equations F.56 and F.57, the following identities are specified: zx - z2 = a + by/c - a + bx/c = 2by/c (F.58) z\ = (a + by/c)2 = a2 + b2c + 2abx/c z2 = (a- by/c)2 = a2 + b2c - 2abyfc Zl 2 2 4aby/c (F.59) F. CMV Averaging Level Control Design for Three Units of Sampling Delay 207 zxz2 = (a + by/c)(a-by/c) = a2-b2c (F.60) z\ = (a2 -r-b2c + 2aby/c)(a + by/c) = a 3 + a2by/c + ab2c + b3cyfc + 2a2by/c + 2ab2c (F.61) z\ = (a2 + b2c - 2aby/c)(a - by/c) = a3 - a2by/c + ab2c - b3cy/c - 2a2by/c + 2ab2c (F.62) z3~z^ = 2a2by/c + 2b3cy/c + 4a2by/c = 6a2by/c + 2b3cy/c = 2by/c(3a2 + b2c) (F.63) Substitute for a,b, and c in the following identities: 2 . 2 7i 1 , 2 , A \ 7i 7? - 47o72 -47o72 -72 f „ a A . Zlz2 = a -b c = —2-7--2(71 + 47o72 = 7-2 = , 2 = ( F - 6 4 ) 47o 4 7 2 y 47^ 4 7 2 7 o 1 —l—i -i -7o , . To 72 + 7o Cz\ 1 - zi z2 = 1 = H = (F.65) 72 72 72 4 - z\ = 2bV~c{3a2 + b2c] = 2byfc + 7 * + ^ ° 7 2 } = 2 6 ^ = 2^ {^ P} (F.66) 47i + 47072 47o2 zxz2 - *r V = ^  + ^  = = ( 7 0 + 7 2 ) ( 7 ° ~ 7 2 ) (F.67) 7o 72 7o72 7o72 , -72 7o -72 7o + 72) (7o -72) , „ „ Q v 7o + 72*1*2 = 7o + 72 = = (F.68) 7o 7o 7o F. CMV Averaging Level Control Design for Three Units of Sampling Delay 208 -1 - l , 22 -To , ll -To , 72 "To + T2 7o*i * 2 + 72^*2 = To + T 2 ^ = + — = 2 T2 To T2 To T0T2 = -l(7o4 ~ T24) ^ -l(To2 ~ T22)(To2 + T22) = -1(70 ~ 72)(TO + T2)(T0 2 + T22) ( p 6 g ) T0T2 T0T2 T0T2 1 - = 1 + ^ = 2 * ± ^ (F.70) To TO To + 72*i~1*2~1 = To + T2 = To - To = 0 (F.71) T2 Substituting the above equations (F.58 - F.63) into eqn. F.55 Num 1 + Num2 = -/0/2[TO + 72*i_1 *2~2]2<bv/c{3a2 + b2c} + [(To + 72X/2 - /o)/i + 71/0/2(31*2 - z7l z2l)}Aaby/c + [(7o + 72*i*2)(/02 + / 2 + /I) + (7o*{ * 2~ 1 + l2z\z2)hh + 7i( / 2 - /o)/i(l - zlZ2)]2bV~c (F.72) The 2b\[c term can be factored out: Num 1 + Num 2 = 2by/c {-/ 0/ 2[7o + 72*r1*2~1]{3a2 + b2c}+ [(70 + 72)(h ~ /o)/i + 71/0/2(^ 1*2 - *r1*2~1)]2a + (70 + 72*1*2) (fo + / i 2 + /22) + (7o*r1*2~1 + 72*2*22)/o/2 +7i(/2-/o)/i(l-*i* 2)} (F.73) Substitute in for the rest of z, a, b, and c using equations F.64 and F.71: (70 + 72) (7o - 72) 2 ^ { - / „ / 2 ( 0 ) (l±™l)+H I V 7o / 7o (7o + 72)(/2 - /o)/i + (71/0/2)- 7o72 , (7o+ 72) (7o -72) f , 2 ,2 , -2x (7o ~ 72)(7o +72) (7o2 + 722) , , H • Uo + /1 + h) o /0/2 7o72 7o72 + 7 i ( / 2 - / o ) / i ( : Z ^ ) } (F.74) The term ^J°t^ can be factored out of the equation: 2 ^ i ^ f ) ( ° + 7o727i(/2 - /o)/i + 7i2/o/2(7o - 72) + 7o72(7o - 7 2)(/ 0 2 + fl + /22) -(7o - 72)(7o + 72)/o/2 + 7o727i(/2 - /o)/i} F. CMV Averaging Level Control Design for Three Units of Sampling Delay 209 =• 2bV~c ( ^ P 1 ) {0 + 27o727i(h ~ /o)/i + 7i2/o/2(7o - 7a) + 7o72(7o - 7 2)(/ 0 2 + fl + fl) \ 7o72 / -(7o-72)(7o+72 2)/o/2} 2bV~c f 1 ^ ) {7i2/o/2(7o - 72) + 27o727i(/2 - /o)/i + (70 - 72)[7o72(/02 + fl + fl) V 7o72 / - (7o+ 722)/o/2]} (F.75) The denominator is: (zi - z2) (70 - 7121 - 72^ 1) (7o - 7i^ 2 - 72^ 2) Recall from equation F.58, zx — z2 = 2by/c, this term cancels the corresponding numerator term in eqn. F.75. The new numerator is shown below: Num 1 + Num 2 = (^^) {72/o/2(7o - 72) + 27o727i(/2 - /o)/i V 7o72 / +(7o - 72)[7o72(/02 + fl + fl) ~ (7o2 + 722)/o/2]} (F.76) The resulting denominator is shown below: (70 - 7i*i - 7222)(7o - 7i^ 2 - liz\) (F.77) The denominator is the same as in the b=l case (Foley et al. 2000) and the 6 = 2 case. The expansion of the denominator was reproduced in the 6 = 2 derivation shown in Chapter 5, the final expansion is shown in equation F.78 below: (70 + 72)2[(27o2 - ll + 2A)2 - 7^ (72 _ 4A)] Thus, combining the numerator (eqn. F.76) and the denominator (eqn. F.78): : (4 74) 2^2+72"| / 7i2/o/2(7o - 72) + 27o727i(/2 - /o)/i (F.78) cr _ 7 0 I 7 0 7 2 ' \ +(7o - 72)[7o72(/o2 + fl + /22) ~ (7o2 + 722)/o/23 (7o + 72)2[(27o2 - 7i2 + 2A)2 - _ 4 A ) ] 47o^2 2 } 72/o/2(7o - 72) + 2707271 (h ~ /o)/i 2 , +(7o - 72)[7o72(/o2 + fl + fl) - (7o2 + 722)/o/2] } 72(70 + 72)[(27o2 - 7i2 + 2A)2 - - 4A)] ^ ™> Where f0 = 70, /1 = 71 - 270, and f2 = 2~jX - 3^Q + 72. F. CMV Averaging Level Control Design for Three Units of Sampling Delay 210 F.6 Controller Action Variance The C M V general equation for the controller action variance is given by equation F.80 1*1=1 Defining the following complex function: 7(2 l)-y{z)z Then the residue theorem (eqn. shown below) can be again used to evaluate the contour integral defined in equation F.80. r k j <j>u(z)dz = 2TT; Reszn$u{z) (F.82) n=l The following variables in equation F.81 are known: n 1 -i\ -1 3o;o +7o + 72 , - i s -1 -2 Qi{z )=qio-qnZ , qw = , 7(2 ) = 70 - Jiz - 7 2 2 r. f \ 2uJ° + 70 + 72 , v 2 <5i(^) = 910 - 9n2 , 9n = , 7(2) = 70 - 7i2 - 722 CJo Q(2) and 7(2) have the same order as in the derivations for 6 = 1, and 6 = 2. Thus the structure of o\v is the same. Thus 2 _ (-l)^47o[27igiogn + (gio + 9n)(72 - 7o)] ( F R ^ W " (7o + 72)[(27o2 - 7? + 2A)2 - 7 (^72 _ 4A)] [ t - * 6 ) Where 70, 71, and 72 are as defined on page 198 and qw, and qu are as defined above. 

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