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The design of a direct digital controller for sampled-data systems 1982

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THE DESIGN OF A DIRECT DIGITAL CONTROLLER FOR SAMPLED-DATA SYSTEMS by OGBONNA CHARLES OKORAFOR M.A.Sc, The Uni v e r s i t y of B r i t i s h Columbia, Vancouver, Canada, 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Chemical Engineering Department) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1982 © Ogbonna Charles Okorafor, 1982 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Ctfem I C A L iroee/ii <QQ The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date p Sr 106 I & 3 - DE-6 (3/81) Leaf i i missed in numbering i i i ABSTRACT This study is made up of three parts viz: 1. For a process which can adequately be modelled as second-order overdamped with pure delay, design techniques are presented for choosing the loop gain and sampling rate of the proportional, feedback, sampled- data controller. Control of an experimental higher-order system is used to verify these suggested designs. 2. Discrete control algorithms, suitable for programming in a direct d i g i t a l control computer, are presented. Digital compensation algorithms are derived to yield theoretically a response with f i n i t e settling time, when the system is step forced in either set point or load. The u t i l i t y of the proposed designs is experimentally verified by application to a higher order (heater-heat exchange) process whose dy- namics can be described as fourth order overdamped with pure dead time. 3. Finally, this study is concerned with the problem of designing an adaptive controller for a class of single-input single-output time- invariant linear discrete systems modeled as second-order overdamped with pure delay. In each case the effect of using either a zero-order hold or half-order hold as the smoothing device was considered. In every case the system with half-order hold gave better transient responses than systems with zero-order hold and better s t a b i l i t y conditions. i v TABLE OF CONTENTS page AUTHORIZATION i i ABSTRACT i i i TABLE OF CONTENTS i v LIST OF TABLES v i i LIST OF FIGURES v i i i ACKNOWLEDGEMENT x i v CHAPTER ^ 1. INTRODUCTION , . 1 2. LITERATURE REVIEW 8 2.1 Adaptive D i r e c t D i g i t a l C o n t r o l . . . ... 12 3. RESEARCH OBJECTIVES 14 4. SAMPLED-DATA PROPORTIONAL CONTROL OF A CLASS OF STABLE PROCESSES ................ 20 4.1 Analysis of System... 20 4.1.1 Overdamped Second-Order System with Zero- Order Hold . V.. 20 4.1.2 Second-Order Overdamped System with H a l f - Order Hold 34 4.1.3 Control System with Dead Time 40 4.1.3a Control System with Dead Time for Zero- Order Hold C i r c u i t 41 4.1.3b Analysis of System with Dead Time for H a l f - Order Hold 46 4.2 Transient Response of System 49 4.2.1 Transient Response of Second-Order Over- damped with Zero-Order Hold 52 4.2.2 Transient Response of Second-Order Over- damped with Half-Order Hold 60 V Chapter page 4 . 2 . 3 Second-Order Overdamped P l u s Dead Time 65 4 . 2 . 4 C o n t r o l System w i t h Zero-Order H o l d 68 4 . 2 . 5 C o n t r o l System w i t h H a l f - O r d e r H o l d 73 4 .3 E x p e r i m e n t a l Equipment 81 4.4 System I d e n t i f i c a t i o n and I n i t i a l i z a t i o n 84 4 . 4 . 1 I d e n t i f i c a t i o n by G r a p h i c a l Method 84 4 . 4 . 2 Q u a s i l i n e a r i z a t i o n Method • 86 4 . 5 E x p e r i m e n t a l R e s u l t . . . 87 5. DIGITAL COMPENSATION DESIGN 97 5.1 Deadbeat ( M i n i m a l P r o t o t y p e ) Performance D e s i g n 97 5 .1 .1 Development of A l g o r i t h m 100 5.1 .2 Compensator Design f o r System w i t h Z e r o - Order H o l d 101 5 . 1 . 3 Compensator Des ign f o r System w i t h H a l f - Order Hold 113 5.2 A p p l i c a t i o n of Combined Optimum C o n t o l and P r e d i c t i o n Theory to D i r e c t D i g i t a l C o n t r o l 135 5 . 2 . 1 A n a l y s i s and Design of C o n t r o l System w i t h Zero-Order H o l d 136 5 . 2 . 2 A n a l y s i s and Design of C o n t r o l System w i t h H a l f - O r d e r H o l d . . . . . . . . . . . . . 140 5.3 Improved P r o p o r t i o n a l C o n t r o l l e r 154 5 . 3 . 1 Improved P r o p o r t i o n a l C o n t r o l l e r of System w i t h . Zero-Order Hold 155 5 . 3 . 2 Improved P r o p o r t i o n a l C o n t r o l l e r of System w i t h H a l f - O r d e r Hold 157 6. ADAPTIVE CONTROL 166 6.1 Parameter E s t i m a t i o n by a M o d i f i e d Moments Method 170 6.2 Compensator D e s i g n 174 6.3 Implementat ion and E x p e r i m e n t a l R e s u l t 175 v i page 7. DISCUSSION AND CONCLUSION 181 7.1 Discussion 181 7.1.1 Analysis of P r o p o r t i o n a l Control 181 7.1.2 Compensator Design 183 7.1.3 Adaptive Control 184 7.2 Conclusion 185 7.3 Recommendation 186 NOMENCLATURE 188 BIBLIOGRAPHY 191 APPENDICES 197 1. HALF-ORDER HOLD TRANSFER FUNCTION DERIVATION 197 2. STATE VARIABLE DERIVATION AND PARAMETER DEFINITION FOR CONTROL SYSTEM WITH HALF-ORDER HOLD 201 3. RELATIONSHIP BETWEEN <j> AND DECAY RATIO INDEX . 212 4. PARAMETER DEFINITIONS FOR CONTROL SYSTEM WITH ZERO- ORDER HOLD . . 214 5. SYSTEM IDENTIFICATION AND INITIALIZATION IDENTIFICATION BY GRAPHICAL METHOD . 216 6. IDENTIFICATION BY QUASILINEARIZATION METHOD 229 7. , THEORY OF VARIABLE GAIN METHOD OF DESIGN 238 8. DEADBEAT COMPENSATOR DESIGN FOR CONTROL SYSTEM WITH ZERO ORDER HOLD 244 9. DEADBEAT COMPENSATOR DESIGN FOR CONTROL SYSTEM WITH WITH HALF—ORDER HOLD 258 10. OPTIMUM DESIGN OF CONTROL SYSTEM BY DYNAMIC PROGRAMMING .. 277 11. ZERO-ORDER HOLD PARAMETER DEFINITION 289 12. NOISE SENSITIVITY ANALYSIS 293 13. HEAT EXCHANGER DIAGRAM 299 v i i LIST OF TABLES page 4.1 Output Response of Process with Dead Time Component 40 4.2 Loop Gain as a Function of Sampling Time for the Two Performance Indices: (Control System with Zero-Order Hold) 59 4.3 Loop Gain as a Function of Sampling Time for the Two Performance Indices: (Control System with Half-Order Hold) 66 4.4 Loop Gain as a Function of Sampling Intervals Used for Various Performance Index Values and Sampling Time (Zero-Order Hold) 69 4.5 Loop Gain as a Function of Sampling Intervals Used for Various Performance Index Values and Sampling Time (Half-Order Hold) 74 A5.1 Transient Response Data for Process Reaction Curve........ 218 A5.2 Table of Coefficients (V. Strejc) .222 v i i i LIST OF FIGURES page 4.1 Block Diagram of Sampled-Data Feedback C o n t r o l System 21 4.2 Input and Output of Sample and Zero-Order Hold 22 4.3 T y p i c a l S i g n a l Flow Graph 25 4.4 S i g n a l Flow Diagram of Equation (4.8a) 25 4.5 S i g n a l Flow Diagram of Equation (4.8b) 25 4.6 S i g n a l Flow Diagram of Equations (4.8a) and 4.8b) Combined 27 4.7 C o n t r o l System S i g n a l Flow Diagram 27 4.8 S t a b i l i t y C o n s t r a i n t of Sampled-Data Second Order System with no Dead Time as a Function of Sampling Rate f o r Various Ratios of the Time Constants 33 4.9 Input and Output of Sample and Hold of Half-Order... 36 4.10 S i g n a l Flow Graph of Equation (4.33) 37 4.11 S t a b i l i t y C o n s t r a i n t of Sampled-Data Second Order System with no Dead Time as a Function of Sampling Rate for Various. R a t i o s of Time Constant>2 to Time Constant 1 ..... 39 4.12 The S t a b i l i t y Boundary of a Sampled-Data System w i t h Zero-Order Hold as a Function of Dead Time 45 4.13 The S t a b i l i t y Boundary of a Sampled-Data System with Half-Order Hold as a Function of Dead Time 50 4.14 S i g n a l Flow Graph of Equation (4.74) 55 4.15 T r a n s i e n t Response of Uncompensated Sampled-Data Second-Order Process with no Dead Time f o r a Unit Step Change f o r D i f f e r e n t Sampling Rates 53 4.16 C o n t r o l System with P r o p o r t i o n a l C o n t r o l l e r 55 4.17 Closed Loop T r a n s i e n t Response of Compensated Sampled- Data Second-Order Process with Zero-Order Hold f o r a Unit Step Change f o r D i f f e r e n t Sampling Rates 57 i x page 4.18 C l o s e d Loop T r a n s i e n t Response of Compensated Sampled- Data Second-Order Over Damped Process w i t h Zero-Order H o l d and no Dead Time f o r a U n i t Step Change f o r D i f - f e r e n t Sampling Rates 58 4.19 Open Loop T r a n s i e n t Response of Sampled-Data Second- Order Process w i t h no Dead Time f o r D i f f e r e n t Sampling Rates ( H a l f - O r d e r H o l d ) 61 4 .20 C l o s e d Loop T r a n s i e n t Response of Sampled-Data Second- Order Process w i t h no Dead Time f o r D i f f e r e n t Sampling Rates ( H a l f - O r d e r H o l d ) 63 4.21 C l o s e d Loop T r a n s i e n t Response of Sampled-Data Second- Order P r o c e s s w i t h no Dead Time f o r D i f f e r e n t Sampling Rates ( H a l f - O r d e r Hold) 67 4.22 T r a n s i e n t Response of Uncompensated Zero-Order H o l d C o n t r o l System 72 4 .23 T r a n s i e n t Response of Compensated Zero-Order Hold C o n t r o l System w i t h Performance Index as Parameter 77 4.24 T r a n s i e n t Response of Uncompensated H a l f - O r d e r H o l d C o n t r o l System 78 4 .25 T r a n s i e n t Response of Compensated H a l f - O r d e r Hold C o n t r o l System w i t h Performance Index as P a r a m e t e r . . . . . . . . . 79 4.26 Schematic Diagram of Equipment 82 4.27 Process R e a c t i o n C u r v e . . . . . . . . . . . . . . . . . . . . . . . . ; . . . . ; . . . . . . . . . 85 4 . 2 8 a E x p e r i m e n t a l C losed-Loop T r a n s i e n t Response of Sampled-Data System f o r a 2% Step Change i n Load ( H a l f - O r d e r Hold) 88 4.28b M a n i p u l a t i v e V a r i a b l e Response of Sampled-Data System f o r a 2% Step Change i n Load ( H a l f - O r d e r Hold) 89 4 . 2 9 a E x p e r i m e n t a l Closed-Loop T r a n s i e n t Response of Sampled-Data System for a 3°C Step Change i n Set P o i n t ( H a l f - O r d e r H o l d ) . 90 4 .29b M a n i p u l a t i v e V a r i a b l e Response of C losed-Loop Sampled- Data System for 3°C Step Change i n Set P o i n t ( H a l f - Order Hold) 91 4 . 3 0 a E x p e r i m e n t a l Closed-Loop T r a n s i e n t Response of Sampled-Data System f o r a 2% Step Change i n Load ( Z e r o - Order Hold) . 93 X page 4 .30b M a n i p u l a t i v e V a r i a b l e T r a n s i e n t Response of C l o s e d - Loop Sampled-Data System f o r a 2% Step Change i n Load ( Z e r o - O r d e r ) 94 4 .31a E x p e r i m e n t a l Closed-Loop T r a n s i e n t Response of Sampled-Data System f o r a 3°C Step Change i n Set P o i n t (Zero-Order Hold) 95 4.31b M a n i p u l a t i v e V a r i a b l e T r a n s i e n t Response of C l o s e d - Loop Sampled-Data System f o r a 3°C Step Change i n Set P o i n t (Zero-Order Hold) • 96 5.1 S t a t e - V a r i a b l e Diagram of C o n t r o l System w i t h Z e r o - Order H o l d ) 102 5.2 Schematic B l o c k Diagram of C o n t r o l System w i t h D i g i t a l C o n t r o l l e r . . . . . . . 110 5 .3 S i g n a l Flow Graph of C o n t r o l System w i t h D i g i t a l C o n t r o l l e r (Zero-Order Hold) ' 110 5.4 T r a n s i e n t Response of Compensated C o n t r o l System w i t h Z e r o - O r d e r Hold I l l 5 .5 T r a n s i e n t Response df Compensated C o n t r o l System w i t h Zero-Order H o l d . . . . 112 5.6 S t a t e - V a r i a b l e Diagram by Cascade Programming M e t h o d . . . . . . 114 5.7 Schematic B lock Diagram of System w i t h D i g i t a l ' . C o n t r o l l e r . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 2 5 .8 S i g n a l Flow Graph of C o n t r o l System w i t h D i g i t a l C o n t r o l l e r ( H a l f - O r d e r ) . . . . 122 5.9 T r a n s i e n t Response of Uncompensated C o n t r o l l e d System w i t h H a l f - O r d e r H o l d . . . . 123 5.10 T r a n s i e n t Response of Compensated C o n t r o l l e d System w i t h H a l f - O r d e r Hold . . 124 5.11a T r a n s i e n t Response of D i g i t a l C o n t r o l l e d Sampled-Data System w i t h Zero-Order Hold f o r a 2% Step Change In Load V a r i a b l e (Steam P r e s s u r e ) 127 5.11b M a n i p u l a t e d V a r i a b l e Response of System ( Z e r o - O r d e r Hold) f o r a 2% Step Change i n Load V a r i a b l e (Steam P r e s s u r e ) 128 x i page 5.12a T r a n s i e n t Response of a D i g i t a l C o n t r o l l e r Sampled- Data System w i t h Zero-Order Hold f o r 4% Step Change i n Set P o i n t . 129 5.12b M a n i p u l a t e d V a r i a b l e Response of System (Zero-Order Hold) f o r a 4% Step Change i n Set P o i n t 130 5.13a, T r a n s i e n t Response of a D i g i t a l C o n t r o l l e d Sampled- Data System w i t h H a l f - O r d e r Hold f o r 2% Step Change i n Load V a r i a b l e (Steam P r e s s u r e ) 131 5.13b M a n i p u l a t e d V a r i a b l e Response of System f o r 2% Step Change i n Load V a r i a b l e (Steam P r e s s u r e ) . . . . 132 5.14a T r a n s i e n t Response of a D i g i t a l C o n t r o l l e d Sampled- Data System w i t h H a l f - O r d e r Hold f o r 4% Step Change i n Set P o i n t . . . 133 5.14b M a n i p u l a t e d V a r i a b l e Response of System f o r a 4°C Step Change i n Set P o i n t 134 5.15 S t a t e - V a r i a b l e Diagram of C o n t r o l System by I t e r a t i v e Programming Method 138 5.16 . S t a t e - V a r i a b l e Diagram of C o n t r o l System w i t h A n a l y t i c a l P r e d i c t o r 138 . 5 . 1 7 S t a t e - V a r i a b l e Diagram of C o n t r o l S y s t e m . . . . . . . . . •. . . . . . . . . 141 5.18 S t a t e - V a r i a b l e Diagram of C o n t r o l System w i t h A n a l y t i c a l P r e d i c t o r ." 144 5.19a Optimum C o n t r o l of Sampled-Data System w i t h Zero-Order Hold for, a 5°C Step Change i n Set P o i n t . . . 146 5.19b M a n i p u l a t e d V a r i a b l e Response of Sampled-Data System w i t h Zero-Order Hold f o r a 5°C Step Change i n Set P o i n t 147 5.20a Optimum C o n t o l of Sampled-Data System w i t h Z e r o - O r d e r Hold f o r a 2% Step Change i n Load V a r i a b l e (Steam P r e s s u r e ) 148 5.20b M a n i p u l a t e d V a r i a b l e Response of Sampled-Data System With Zero-Order Hold f o r a 2% Step Change i n Load V a r i a b l e (Steam P r e s s u r e ) ; 149 5.21a Optimum C o n t r o l of Sampled-Data System w i t h H a l f - Order Hold for 5°C Step Change i n Set P o i n t 150 x i i page 5.21b M a n i p u l a t e d v a r i a b l e response of sampled-data system w i t h h a l f - o r d e r h o l d f o r 5°C step change i n set p o i n t 151 5 .22a Optimum c o n t r o l of sampled-data system w i t h h a l f - o r d e r - h o l d f o r a 2% step change i n load v a r i a b l e (steam p r e s s u r e ) 152 5.22b M a n i p u l a t e d V a r i a b l e Response of Sampled-Data System w i t h H a l f - O r d e r Hold f o r a 2% Step Change i n Load V a r i a b l e (Steam P r e s s u r e ) 153 5.23 S t a t e - V a r i a b l e Diagram of C o n t r o l System and P r e d i c t o r . . . . 156 5.24 S t a t e - V a r i a b l e Diagram of C o n t r o l System w i t h A n a l y t i c a l P r e d i c t o r 159 5 . 2 5 a E x p e r i m e n t a l . C l o s e d - L o o p Response of Improved P r o p o r t i o n a l C o n t r o l of a Sampled-Data System w i t h Z e r o - O r d e r Hold f o r a 3°C Step Change i n Set P o i n t 162 5.25b M a n i p u l a t e d V a r i a b l e Response of C losed-Loop Improved P r o p o r t i o n a l C o n t r o l of Sampled-Data System w i t h Z e r o - Order Hold f o r 3% Step Change i n Set P o i n t 163 5 .26a E x p e r i m e n t a l Closed-Loop Response of Improved P r o p o r t i o n a l C o n t r o l l e r of Sampled-Data System w i t h H a l f - O r d e r Hold f o r a 3°C Step Change i n Set P o i n t . . . . 1 6 4 5.26b M a n i p u l a t e d V a r i a b l e Response of Improved P r o p o r t i o n a l C o n t r o l of Sampled-Data System w i t h H a l f - O r d e r Hold f o r a 3°C Step Change i n Set P o i n t . 165 6.1 F lowsheet of O n - l i n e Parameter I d e n t i f i c a t i o n . 167 6.2 N o i s e S e n s i t i v i t y as a F u n c t i o n of L a p l a c e Transform O p e r a t o r 173 6 .3a A d a p t i v e C o n t r o l Response of a Sampled-Data System w i t h Zero-Order Hold 176 6.3b M a n i p u l a t e d V a r i a b l e Response of Sampled-Data System w i t h Zero-Order Hold A d a p t i v e C o n t r o l l e r 177 6 .4a A d a p t i v e C o n t r o l l e r Response of a Sampled-Data w i t h H a l f - O r d e r Hold 179 6 . 4 b * M a n i p u l a t e d V a r i a b l e Response of Sampled-Data A d a p t i v e C o n t r o l l e r 180 x i i i page A l . l Impulse Response of Half-Order Hold 198 A2.1 Signal Flow Diagram...! 200 A5.1 Process Reaction Curve for a 10% Step Change in Steam Pressure , 219 A5.2 Approximate Estimation of Transfer Function Parameters Strejc Method 220 A5.3 F i t t i n g Transient Response to a Second-Order with Dead Time Model by Oldensbourg an Sartorius Method...... 225 A5.4 Oldenbourg and Sartorius Diagram for Equivalent Time Constants from Process Reaction Curve for Second-Order Process • 226 A5'.5 Determination of Time Constants for the System Modelled as a Second-Order Plus Dead Time Process 228 A7.1 A Digital Control System........ . 239 A8.1 State-Variable Diagram by Iterative (Cascade) Programming Method . 245 A8.2 The State-Variable Diagram of the Digital Controller 256 A8.3 Control System with Digital Controller and State-Space ' , Representation....... 256 A9.1 State-Variable Diagram by Iteration (Cascade) Programming Method 259 A9.2 Control System with Digital Controller and State-Space Representation... 268 xiv ACKNOWLEDGEMENTS I wish to place on record my sincere gratitude to Dr. K.L. Pinder, under whom this i n v e s t i g a t i o n was conducted, for his guidance i n helping to carry out this project. I wish to thank Mrs. H. Pinder for her motherly love and care f o r me during the course of my stay i n Canada. I would l i k e to express my deep appreciation for the assistance of a l l those men i n Chemical Engi- neering Department workshop and stores during the equipment design and f a b r i c a t i o n . My thanks to Marlene and Kathrine, s e c r e t a r i e s to Chemical Engineering Department and also to Laurie and E l i z a b e t h Truant for t h e i r concern during my course of study. My g r a t e f u l acknowledgement i s also due to my mother, Mrs. F.N. Okorafor for her love and also to my guardian, Mr. H. Skeokoro, who has been more of a father to me. I dedicate this, work to the memory of Chineme Onyekere (Nee Nkere) who unfortunately l e f t t his world a month before the completion of this work. Praises and adoration to God and my Saviour Jesus C h r i s t for the guidance and protection I have received a l l through my l i f e t i l l now. CHAPTER 1 INTRODUCTION Control may be defined as the organisation of activity for a purpose. If one defines a system as an identifiable entity and auto- matic as meaning self-acting, then an automatic control system is a self-acting indentifiable entity in which the activity is organised for some purpose. According to this definition, the first automatic control systems were living organisms, but today in addition to living things there are organisations of living things and systems devised by living things, a l l of which are automatic systems. Thus an amoeba is an auto- matic control system and so is a modern political state or an industrial corporation. Regardless of their great diversity a l l control systems have five characteristics in common, viz: a l l control systems are time varying systems, and their temporal behaviour is a measure of their performance; they a l l ingest signals, digest signals, manipulate and generate signals; their design and analysis must involve a holistic or system approach; most control systems must involve signal feedback and finally any control system containing feedback circuits has the possibi- li t y of becoming unstable even though each of the elements comprising the system is itself quite stable and incapable of runaway. Control systems for the Chemical Engineer can be broken into many parts to simplify their presentation. The most common classifications are open loop, closed loop, set point, averaging, cascade, and optimising. In open loop control, there is no feedback from output to input. The ratioing of two flows is often accomplished by an open-loop system. The most common kind of closed-loop control is regulatory control, where one is primarily interested in holding a particular process variable within narrow limits. Most flow and'temperature control systems f a l l within this group, as do endpoint control systems, which make use of analyzers. Averaging level control is one of the most common types. In these control systems the level in a tank between process units is allowed to vary in order to make up for differences in flow between one unit and the next. In most cases, the control simply prevents the tanks from flowing over, or running dry. Cascade control systems are very popular in the industries. The primary purpose of control is to eliminate the effects of minor disturbances. Optimising (adaptive) control systems vary the set points, flow ratios, etc. as the conditions in the plant vary. The process to be controlled must supply one with a variable that is either directly or indirectly indicative of the quality of the process that one wishes to control. It must also supply a quantity that can be varied in order to effect the desired control. Both of these, the measurement of the measured variable, and the change of manipulated variable, have certain speed requirements. They must both be capable of being accomplished fast enough to effect the desired control. Without this, the controller w i l l be incapable of doing i t s job regardless of how expensive, exotic and complex i t may be. Essentially, automatic (process) control can be divided into two major classes, — the continuous (analog) system and the sampled-data 3 ( d i s c r e t e ) s y s tem. A s a m p l e d - d a t a c o n t r o l system i s one i n w h i c h the c o n t r o l s i g n a l i n a c e r t a i n p o r t i o n of the system i s s u p p l i e d i n t e r m i t t e n t l y a t a c o n s t a n t r a t e . I n t h i s c o n t r o l system the d a t a s i g n a l a t one or more p o i n t s i s a sequence of p u l s e s w h i c h a r e m o d u l a t e d i n a c c o r d a n c e w i t h the c o n t i n u o u s f u n c t i o n of the s i g n a l from w h i c h t h e samples a r e t a k e n . I t i s assumed t h a t t h e s e p u l s e s convey a d e q u a t e l y a l l t he e s s e n t i a l i n f o r m a t i o n c o n t a i n e d i n the c o n t i n u o u s f u n c t i o n . The d e s i g n of a p r o c e s s c o n t r o l s y s t e m g e n e r a l l y i n v o l v e s t h r e e s t e p s : i d e n t i f i c a t i o n of the p r o c e s s , s t o c h a s t i c a n a l y s i s , and compen- s a t o r d e s i g n . The i d e n t i f i c a t i o n r e q u i r e s a s t r i c t d e l i m i t a t i o n of t h a t p a r t of the p h y s i c a l u n i v e r s e w h i c h i s under c o n s i d e r a t i o n , and t h e r e l e v a n t c o n c e p t i s the thermodynamic p r i n c i p l e of s t a t e s , a c c o r d i n g t o w h i c h a l l p r o p e r t i e s of a s y s t e m a r e f i x e d when a c e r t a i n few p r o p e r t i e s of the system a r e f i x e d . " I n most p r o c e s s e s a c c u r a t e i d e n t i f i c a t i o n i s r a r e l y p o s s i b l e due to u n c e r t a i n t y of the p r o c e s s measurements. I n a d d i t i o n , many f e e d b a c k q u a n t i t i e s a r e based on sampled d a t a systems ( a n a l y z e r s , c h r o m a t o g r a p h s , e t c ) and l a b o r a t o r y t e s t s . The s a m p l i n g systems have t h e m s e l v e s i n h e r e n t e r r o r . M o r e o v e r , i d e n t i f i c a t i o n t h r o u g h a d e t e r m i n i s t i c model does not t a k e i n t o a c c o u n t d i s t u r b a n c e i n p u t s . These i n p u t s are h e u r i s t i c i n n a t u r e and d i r e c t l y a f f e c t t h e a c c u r a c y of the c o n t r o l s y s t e m . " I f f o r no o t h e r r e a s o n , t h e s e u n p r e d i c t a b l e l o a d changes r e q u i r e some form of f e e d b a c k a c t i o n or model a d a p t a t i o n o r b o t h . To model t h e s e u n c e r t a i n t i e s , the use of s t o c h a s t i c e s t i m a t i o n a n a l y s i s i s always s u g g e s t e d . The r e a l m of the a p p r o a c h i s based upon the t h e o r y of p r o b a b i l i t y and s t a t i s t i c s . I t i s assumed t h a t the d i s t u r b a n c e i n p u t s as w e l l as the s e n s o r e r r o r s can be a p p r o x i m a t e d A by the random process known as Gaussian white noise. This means that, mathematically the random processes have a mean value of zero, they are independent of each other, and they have known covariance matrices. In the wider concept of compensator design, six c r i t e r i a must be 2 6 satisfied for effective control: (i) A b i l i t y to maintain the controlled variable at a given set point. This most essential requirement of process control is often the most d i f f i c u l t to f u l f i l l as i t creates mathematical d i f f i c u l t i e s for most of the optimization algorithms proposed thus far. ( i i ) Set-point changes should be fast and smooth. As the over- a l l system may be slow and complex, i t is' important for the operator to be able to perform individual set-point changes as fast as possible. However, minimum time res- ponse often leads to large excursions in the system tran- sient response, which is in contrast to a smooth response (or low overshoot) which has a significant advantage, ( i i i ) Asymptotic s t a b i l i t y and satisfactory performance for a wide range of frequencies: The total system (not neces- sarily the controller) should obviously be asymptotically stable to be suitable for operator control. This condi- tion should be achieved even though the process parameters may change within a range of system parameter values. Furthermore, the closed-loop transfer function frequency response should not have peaks indicating strong amplification of certain input signals. This means^that 5 the maximum amplification in the transfer function from disturbance input to process output should be low. (iv) The controller should be designable with a minimum of information with respect to the nature of the input and * the structure of the system. In many cases of process control, a rather imprecise knowledge of the nature of the disturbances and their variation with time is known. It is being suggested that care should be taken that the control action achieved in theory is not strongly depen- dent on that part of the model which is inaccurate. Consider, for example, a distributed-parameter system (as for example, a heat exchanger), which features both mixing processes and transport delays. For mixing studies, i t can be successfully modeled as a series of three stirred tanks. However, a design of an optimal controller for three stirred tanks might lead to a controller which combines derivative action with a very high gain. While this w i l l function well in three stirred tanks, i t w i l l lead to i n s t a b i l i t y in the real system due to the f i n i t e time lags involved, (v) The controller should be insensitive to change in system parameters: In a real control situation the parameters of the system and noise parameters are not accurately known and, in addition, often change with time. The controller must be able to handle reasonable changes, with a s u f f i - cient s t a b i l i t y margin. The reason for this requirement 6 is twofold. F i r s t , the throughput through process equip- ment changes due to varying overall needs of the plants. That means in a process with a time lag the controller i must be able to perform while the actual time constants of the system change, and these changes are in no way negli- gible. The second reason is due to the assumption made of linear system equations which are often a linearization around a steady state and when the steady state set-point is changed these linearized system parameters may change significantly. (vi) Excessive control action should be avoided: There are two main reasons for limiting the control effort. The f i r s t i s mathematical. When dealing with a linear problem i t is often common to neglect One important non-linearity, the f i n i t e limits on the magnitude of allowed control signals. To avoid errors, reasonable limits on magnitude of control must be placed or the nonlinearity should be accounted for in the design. During the past three decades, attention has been placed on the design of controllers that can operate at varying process conditions giving rise to an optimum result. This modern control theory (optimal control) has been mostly applied to adaptive control. Adaptive control implies the a b i l i t y of a control system to change i t s own parameters in response to a measured change in operating conditions. These control systems are distinguished by their a b i l i t y to compensate automatically 7 for either changes in the system input, such as a change in the signal- to-noise ratio, or changes in the system parameters, such as a change due to environmental variations. In recent years, a number of methods for adaptive control system design have been suggested. According to the way that adaptive behaviour is achieved, adaptive control systems may be divided into input-sensing adaptation, plant-sensing adaptation, and performance-criterion-sensing adaptation; alternately, they may be classified mainly as passive adaptation, system-parameter adaptation, and system characteristic adaptation. Control systems with passive adaptation achieve adaptive behaviour without system parameter changes, but rather through design for operation over wide variations in environ- ment. Examples of control systems of this nature are the conventional feedback systems and the conditional feedback systems. Control systems with system-parameter adaptation adjust their parameters in accordance with input-signal characteristics or measurements of the system var i - ables. Control systems with system-characteristic adaptation achieve adaptive behaviour through measurement of transfer characteristics. A useful approach to the design of adaptive control systems generally involve three basic principles: (i) provision of a means for continuous measurement of system dynamic performance; ( i i ) continuous evaluation of the dynamic performance on the basis of some predetermined criterion; and ( i i i ) continuous re-adjustment of system parameters for optimum operation by using the measured and evaluated results. 8 CHAPTER 2 LITERATURE REVIEW In the evaluation of a co n t r o l system, two questions have to be considered: whether the con t r o l system i s stable, and whether or not the q u a l i t y of con t r o l attained i s good. Quality of con t r o l involves the a b i l i t y of the con t r o l system to damp out quickly the e f f e c t of a disturbance on the plant. Unlike s t a b i l i t y , q u a l i t y i s not a we l l defined concept and many d i f f e r e n t c r i t e r i a have been suggested and used for' i t by con t r o l system designers. The c o n t r o l l e r s e t t i n g that causes deadbeat performance a f t e r stepforcing has been widely used i n the 7 7 5 l i t e r a t u r e (Callander et_ a l ; Z i e g l e r and Nichols; Oldenbourg and 52 72 Sa r t o r i u s ; and Wolfe .) as one c r i t e r i o n f o r optimum q u a l i t y c o n t r o l . Deadbeat return (performance), sometimes known as c r i t i c a l damping, i s the f a s t e s t possible response of the c o n t r o l l e d v a r i a b l e which involves no undershoot and/or overshoot of the steady state value. Deadbeat performance i s not r e s t r i c t e d to stepforcing inputs but includes the response to parabolic inputs with minimum-squared error r e s t r i c t i o n s on ramp and step responses (Pokoski and P i e r r e ) . ^ S Yih-Shuh 7 4 used time polynomial f o r c i n g inputs i n his design. Chien et_ ad. considered this c r i t e r i o n along with one which requires 20% undershoot. Cohen and 9 5 8 Coon, and Ream found c o n t r o l l e r settings by s p e c i f y i n g the subsidence r a t i o of the fundamental component in the closed-loop transient response. The minimization bf the i n t e g r a l square of the c o n t r o l l e d v a r i a b l e from zero to i n f i n i t y as a function of the c o n t r o l l e r para- 22 6 9 meters was suggested by Hazebroek and van der Waerden, and Wescott. 9 Wills postulated that the integral of either the absolute value of the controlled variable or the absolute value of the controlled variable multiplied by time should be minimized as a function of the controller parameters. It is worth noting that a l l the integral c r i t e r i a can only be used in cases where integral control is involved, otherwise the c r i - teria w i l l give rise to divergent controller modes which may result in unstable control systems. In a l l of the above mentioned studies, with the exception of that of Wills, the plant step (transient) response was simulated by either a delayed ramp function or the response of a first-order transfer stage plus a deadtime. McAvoy and Johnson used an underdamped second-order stage plus deadtime, which according to them, is more r e a l i s t i c than the other two, since i t accounts for the inertia present in physical systems and i t allows a more flexible matching of the plant's characteristics. 3 3 Latour et a l . , used an overdamped second-order model plus deadtime. This model has been used to represent the dynamic response of l i q u i d - liquid and gas-liquid extractors (Biery and Boylan; Gray and Prados ), mixing in agitated vessels (Marr and Johnson ), some heat exchangers 2 3 3 5 (Hougen) , d i s t i l l a t i o n columns (Lupfer and Parsons; Moczeck et 4 2 6 2 3 2 a l . , Sproul and Gerster ), and some chemical reactors (Lapse, 3h 39 59 Lupfer and Oglesby; Mayer and Rippel; Roquemore and Eddey ). A l l the aforementioned models have been applied to continuous (analog) control systems. With the increase in the use of d i g i t a l comp- uters for controlling process systems, study of sarapled-data control systems and design of direct d i g i t a l control — which means putting the computer and process together so that the process reports to the 10 computer and the computer issues commands to the process — algorithms have presented interesting and challenging problems. Several authors have presented direct d i g i t a l algorithams for lumped-parameter systems (Mosler et a l . , W Moore e_t a l . , h h Luyben., 3 6 D a h l i n 1 3 ) . Most of these published control algorithms used a first-order plus deadtime transfer stage model and a sampler plus a zero-order hold as the smoothing device. A lot of papers have appeared on continuous feedback control of 27 distributed parameter systems. Koppel considered continuous nonlinear feedback control of tubular chemical reactors and heat exchangers. 28 Koppel et_ al_., reported theoretical and experimental results on two- point linear control of a flow-forced heat exchanger and extended the principle to other parametrically forced distributed parameter systems. 55 Paraskos: et a l . , reported on an algorithm which they considered su- perior to conventional Ziegler-Nichols settings from experimental study of feed forward computer control of a flow-forced heat exchanger. In their paper Seinfeld et_ a l . ^ showed useful results on offset and s t a b i l i t y of a flow-forced isothermal tubular reactor system under proportional feedback, feedforward, and optimal controls. They stated that the system is stable, irrespective of the value of the proportional gain. Oscillations in outlet concentration increased as the propor- tional gain was increased; however, there was an upper limit on the gain because of the physical requirement that the velocity should be greater than zero. Very few researchers have worked on feedback sampled-data control of distributed-parameter systems (Palas; 5 1 + Hasson et a l . , 2 1 49 Mutharasan, et al. ). 11 The s t a b i l i t y of sampled-data systems containing delay time has been v e r i f i e d by T o u , 6 4 T r u x a l , 6 7 T s y p k i n . 6 8 They showed that, " f o r a given system and sampling rate, the ultimate loop gain i s observed to increase i n i t i a l l y as delay time i s added to the system. This ultimate gain passes through a maximum and then decreases as the amount of delay time i n the system i s further increased." These i n v e s t i g a t o r s examined systems containing no hold c i r c u i t a f t e r the sampler, but t h i s unexpec- ted phenomenon has been shown by Mosler et a l . 1 * 7 to exist i n the presence of hold (zero-order). It has been proposed that, below the maximum gain, a d d i t i o n of delay time s t a b i l i z e s the loop, since the 1 8 ultimate gain i s increased. This proposal has been proved to be i n 47 6 29 error by Mosler et_ al_. Buckley and Kou examined systems for which the delay time i s equal to an i n t e g r a l number of sampling periods and then designed a d i g i t a l compensator such that the s e t t l i n g time for the output sequence at the sampling instants i s minimized for a given c l a s s of disturbances. Several a l t e r n a t i v e methods for tuning continuous c o n t r o l l e r s of processes characterized by a single time constant and delay have been 4 7 published by some authors (9, 10, 11, 39, 76). Mosler et_ al_. extended this work into the sampled-data domain by o f f e r i n g a systematic proce- dure for choosing the gain and sampling rate of a sampled-data, propor- t i o n a l c o n t r o l l e r , using a zero-order hold. Their method i s l i m i t e d to processes which may be adequately described as f i r s t - o r d e r with delay. They also showed that in the absence of load disturbances extremely "slow" sampling i n t e r v a l s can be used and a decay r a t i o of four to one can s t i l l be obtained. A l l e n , P.1 confirmed this f i n d i n g . Soliman and 12 A l - S h a i k h O A showed that by u s i n g a f i r s t - o r d e r w i t h d e l a y , that i t i s p o s s i b l e to e s t i m a t e the bounds on the va lues of the c o n t r o l l e r c o n - s t a n t s . I t i s worth ment ion ing that the f requency and d u r a t i o n of samp- l i n g i s no longer as c r i t i c a l as i t used to be s i n c e the i n t r o d u c t i o n of microcomputers and m i c r o p r o c e s s o r s i n process c o n t r o l . Choosing a s u i t a b l e sampl ing i n t e r v a l f o r d i r e c t d i g i t a l c o n t r o l i s an important a d d i t i o n a l v a r i a b l e . A s imple r u l e to f o l l o w would be to sample s u f f i c i e n t l y q u i c k l y to ensure that the sampled par t of the 7 3 c o n t r o l loop behaves l i k e a cont inuous sys tem. Y e t t e r and Saunders s t u d i e d a number of systems and found that f o r the cont inuous case the c l o s e d loop c y c l e d w i t h a p e r i o d between 10 seconds and 640 seconds f o r 95% of the types of loops g e n e r a l l y found i n chemica l p r o c e s s e s . They showed that to s a t i s f y the requirement mentioned above the sampl ing p e r i o d must be one e i g h t h of the loop p e r i o d . Th is means that the a v a i l a b l e sampl ing per iods must be In the range of one second to 80 seconds . The a p p l i c a t i o n of t h i s procedure f o r d e t e r m i n i n g the r e q u i r e d sampl ing r a t e , i n v o l v e s some i n f o r m a t i o n on the dynamics as w e l l as o f f - l i n e s i m u l a t i o n t e s t s . Eckraan, B u b l i t z and H o l b e n , 1 ^ c a r r i e d out s i m u l a t i o n s t u d i e s on four d i f f e r e n t c o n t r o l loops which had time cons tants cons idered to be t y p i c a l of f l o w , p r e s s u r e , temperature , and c o m p o s i t i o n l o o p s . They recommended the f o l l o w i n g sampl ing r a t e s : Flow l o o p s : 0 . 1 s ; P r e s s u r e l o o p s : 1 . 0 s ; Temperature : 10s; C o m p o s i t i o n : 6 0 s . 2 . 1 A d a p t i v e D i r e c t D i g i t a l C o n t r o l The c o n t r o l of an unknown l i n e a r , t i m e - i n v a r i a n t p l a n t has remained an open q u e s t i o n f o r a long t ime and i n recent years many 3 0 attempts have been made to r e s o l v e i t by Landau, Astrom and 13 Wittenmark; Monopoli; Feuer and Morse; Narendra and Valavani. The methods used for the resolution of the adaptive control problem can be broadly classified as: (i) indirect control and ( i i ) direct control methods. In the f i r s t group, the parameters and/or state variables of the unknown process are estimated and in turn, used to adjust controller para- meters. These control systems are sometimes referred to as self-tuning regulators in the literature. In the direct control there is no expli- cit' identification of the plant but the control parameters are adjusted so that the error between the process output and that of a reference model (known as the desired output) tends to zero asymptotically. Di- rect control systems have also been called model reference adaptive con- t r o l . The algebraic and analytical d i f f i c u l t i e s associated with the control problem are common to both approaches and have been discussed by Narendra and Valavani. 5 1 In the indirect control problem, the observer plays a central and important role. The process parameters are continu- ously estimated and used to determine the control parameters of the sys- tem. The rationale behind such an adjustment is that, when the identi- fication parameters tend to their true values the control parameters w i l l approach the desired values, for which the transfer function of the feedback loop w i l l match that of a specified reference model. Narendra • and Valavani 5 1 have shown that the above approach, in general, leads to non-linear st a b i l i t y problems which are intractable. The principal d i f - ficulty in such cases arises when attempting to relate estimates of the identification parameters to those of the control parameters. 14 CHAPTER 3 RESEARCH OBJECTIVES Modern con t r o l systems often include i n the loop a d i g i t a l compu- ter f or processing the output measurements of the process, and s y n t h e s i - zing the optimal control law. Development of a mathematical model for the plant i s often the f i r s t step undertaken i n the design of the c o n t r o l l e r . The mathematical model i s u s u a l l y obtained a f t e r a c a r e f u l study and thorough understanding of the underlying p h y s i c a l phenomena, and, i n many cases, turns out to be high order, nonlinear and/or st o c h a s t i c i n nature. Many i n d u s t r i a l c o n t r o l systems must e f f e c t i v e l y cope with systems whose operating c h a r a c t e r i s t i c s change with operating l e v e l (they are nonlinear) and i n most cases i t i s often very d i f f i c u l t to determine the actual nature of the non-linear function. Since u s u a l l y the o r i g i n a l mathematical model of the plant i s complex, or of high order, the requirements on the memory size and the speed of the c o n t r o l computer can be very demanding. Consequently, attempts are often made to obtain a low order model which represents the plant with some accuracy. In p a r t i c u l a r i t has been found that high order overdamped systems, as often encountered i n chemical process c o n t r o l , can be represented to a f a i r accuracy by a second order model containing dead 11 12 19 time or transportation lag (Coughanour; Cox; G a l l i e r ). The simple p r i n c i p l e behind this structure, i s that a portion of the phase lag i n the system due to large numbers of poles can be lumped into a s i n g l e pure time delay. It i s worth noting that t h i s time delay gives an extra 15 degree of freedom without increasing the order of the model. Conse- quently, computation and synthesis of the optimal control law for this second order model is a relatively simple task. Furthermore, i f such a model can be determined and updated on line as the process evolves, an adaptive controller can be easily synthesized and interfaced between the plant output and the control input. This model has been used with analog controllers. The desire of this study is to extend the use of the model into the sampled-data domain of d i g i t a l adaptive control where the model parameters are d i g i t a l l y updated at extended time periods. Usually a smoothing device follows a d i g i t a l computer in the control loop and the most popular of these devices is the zero-order 29 hold. According to Kuo, B.C. , the amplitude of a zero-order hold drops off rapidly at low frequencies and the amplitude characteristics of a first-order hold exhibits an overshoot which greatly enhances i n s t a b i l i t y of the system. , In the work reported in this thesis a frac- tional order hold (1/2 order) which has an amplitude characteristic that f a l l s between the zero-order and the first-order characteristics and is believed to come close to approximating an Ideal f i l t e r response is used. A comparison of the response with zero-order hold and half-order hold w i l l be carried out to i l l u s t r a t e the characteristic of each. This work is divided into three parts viz (i) Analysis of propor- tional control for sampled-data control of a class of stable processes. (Ii) Design of d i g i t a l compensators for the control system and ( i i i ) Adaptive control of the system. In a l l the three parts, the control systems with zero-order and half-order holds are considered. Also experimental verification of the theoretical results is carried out. 16 ( i ) A nalysis of proportional c o n t r o l of the sampled-data system: The a n a l y s i s of proportional c o n t r o l for sampled-data c o n t r o l of a c l a s s of stable processes using a second-order overdamped plus dead time stage function model requires the determination of the s t a b i l i t y range of the pro p o r t i o n a l c o n t r o l l e r . Since no known work has been published on t h i s area, the e f f i c i e n c y of using a half-order hold instead of a zero-order hold, — which i s easier to apply — , as determined on the r e s u l t of the performance c r i t e r i o n , i s v e r i f i e d . The r e l a t i v e e f f e c t s of the process time constants and dead time on the s t a b i l i t y of the process are also determined. C r i t e r i a which are often used for judging good closed loop per- formance are, maximum overshoot, decay r a t i o , s e t t l i n g time and the i n t e g r a l of some function of the er r o r . Maximum allowable overshoot i s not p a r t i c u l a r l y u s e f u l c r i t e r i o n for automatic processes since i t always involves e x c i t i n g the system up to the threshold of s t a b i l i t y . The decay r a t i o and the i n t e g r a l of some function of the error have been 4 5 11 used by some workers. , The sum of the modulus of the erro r or the sum squared can e a s i l y be determined, because the error i s c a l c u l a t e d during the normal con t r o l c a l c u l a t i o n s . The error squared gives more weight to the larger deviations than the modulus of the er r o r , though small deviations can more r e a d i l y be to l e r a t e d . The decay r a t i o which i s commonly used i n c o n t r o l designs i s not, at least t h e o r e t i c a l l y s u i t - able for second-order overdamped models since overdamped systems t h e o r e t i c a l l y do not overshoot. A new performance index defined as the r a t i o of erro r f i r s t moment to error second moment i s used to derive an e f f e c t i v e c o n t r o l algorithm for a s p e c i f i e d response. This c r i t e r i o n Is mathematically 17 l e s s complex to apply than minimisation of an error i n t e g r a l . As i s shown l a t e r on, there e x i s t s a r e l a t i o n s h i p between t h i s performance measure and the one-quarter decay r a t i o index. Also t h i s new perfomance c r i t e r i o n i s a generalised algorithm index for any second-order system. For an underdamped second-order system, the performance index N-1 Z e ( i T ) • - ^ N-1 Z e (IT) 1=0 i s negative, while i t i s p o s i t i v e for an overdamped system. At steady state o s c i l l a t i o n the performance index <|> i s equal to zero. The error i s defined as the deviation between the desired setpoint and the a c t u a l value at any i n s t a n t of sampling. This error summation i s performed from zero time to N sampling times, where N i s s u f f i c i e n t l y large to allow the system to a t t a i n steady state conditions. The performance, index computed i n t h i s way i s probably greater than the true d e v i a t i o n , — since the error i s only evaluated at the sample i n t e r v a l — , i f the system i s s t a b l e . The use of t h i s performance index, to estimate loop gain w i l l be shown for proportional d i r e c t d i g i t a l c o n t r o l of a heat exchange process. In t h i s study the modern con t r o l theory method (state v a r i a b l e approach) i s used and synchronous sampling i s assumed. In the a c t u a l c o n t r o l loops t h i s may only be an approximation because the output from the computer i s delayed by the computing time, but as t h i s i s small compared to the smallest possible time used, the assumption i s reasonable. 18 ( i i ) D i g i t a l Control Compensators: This deals with the development of discrete control algorithms, which are suitable for programming in a direct d i g i t a l computer. In a l l , three algorithms are developed, two of them are optimum controls, — one is derived on the basis of dead beat performance, while the other is formulated from optimum state feedback control law with inaccessible states; the inaccessible states are deter- mined from an analytical predictor algorithm and not calculated from estimates of measured output — , the last is an improved form of proportional control algorithm where the predicted state values are used instead of the actual measured values for the control. Each of these control algorithms was tested on a heat transfer process with zero-order and half-order holds as smoothing devices. ( i i i ) Adaptive Control: Model reference adaptive control (MRAC) is chosen as the basis for the adaptive procedure, because the control system to be adapted is of high order while a second-order model is used in the algorithm formulation. This adaptive control scheme is consi- dered to be an especially efficient method that has been widely noted in reviews (Donaldson, D.D. et_al. ; l l t Landau, I.D.;31 Beck, M.S.3). There are many variations within the 'MRAC1 category which can be investiga- ted. The model reference characterization is appealing with respect to the unit operations type of process involving flowing fluids, such as the heater-heat exchanger system. Detailed knowledge about the struc- ture of signal flow and the time constants is usually lacking for these processes. Consequently, adaptive control methods could be useful in offsetting the effects of inaccurate process information. In addition, extensions of the model reference concept to the estimation of state variables for more advanced control schemes could also be possible. 19 Since i n Chemical Engineering c o n t r o l of slow time-constant vary- ing high order processes i s common, an algorithm was sought which could have general a p p l i c a t i o n . To handle s i g n a l degradation caused by large sample times a half-order hold was tested. To match process of high order a model made of an overdamped second order lag element and trans- port lag where the delay time i s any multiple of sampling t i m e — was used. As f a r as i s known t h i s model has not been tested for d i r e c t d i g i t a l adaptive c o n t r o l . I t i s worth noting that some workers have used t h i s model but r e s t r i c t e d themselves to using a dead time value equal to an i n t e g r a l multiple of the sampling time. The f i n a l part of the study involved the experimental v e r i f i c a - t i o n of these techniques and algorithms on a heater-heat exchanger c o n t r o l system. The co n t r o l e f f e c t i v e n e s s r e s u l t i n g from the loop gain and sampling rate of the proportional, feedback, sampled-data c o n t r o l l e r selected from the performance c r i t e r i o n of the transient response was inv e s t i g a t e d . For each of the remaining three c o n t r o l l e r s , operating conditions for the heater-heat exchanger c o n t r o l system were changed to simulate two d i f f e r e n t process dynamic changes. These cases included: ( i ) step change i n outlet set point and ( i i ) step change i n load v a r i a b l e (steam pressure). . The c r i t e r i a to be used i n the comparison of the two c o n t r o l sys- tems, — zero-order hold and half-order — , for good con t r o l q u a l i t y are: ( i ) the speed at which steady state i s attained and ( i i ) the behaviour at steady state conditions. The co n t r o l system that a t t a i n s steady state condition f a s t e r and with less steady state o s c i l l a t i o n w i l l be deemed as the better c o n t r o l system. I t should be noted that no attempt w i l l be made to s p e c i a l l y tune any of the co n t r o l systems thereby e l i m i n a t i n g any bias towards a s p e c i f i c c o n t r o l . 20 CHAPTER 4 SAMPLED ~ DATA PROPORTIONAL CONTROL OF A CLASS OF STABLE PROCESSES 4 . 1 Analysis of System Consider a simplified sampled-data feedback control system as shown in Fig. 4 . 1 , where Gc(s) is the Laplace transform of the propor- tional controller transfer function; Gp(s) is the process transfer func- tion and H(s) is the hold (smoothing device) transfer function. Two conditions w i l l be invesigated viz (i) when the process transfer function is second-order overdamped plus a hold (zero or half-order) and ( i i ) when i t is second-order overdamped plus dead time plus a hold (either a zero-order or half-order). 4 . 1 . 1 Overdamped Second-Order System With Zero-Order Hold Consider a sampled-data feedback control system as shown in Fig. 4 . 1 . It is assumed that the measured variable is sampled every T units of time, and that the resulting values appear at the output of a zero- order hold c i r c u i t , such that the input, C(t), and output, Cc(t), of the sample-and-hold device are related as shown in Fig 4 . 2 . The transfer function of the hold circuit is generally given as V s ) = L T ^ T S ( 4 ' x ) Note that the impulse-modulated sample and the hold c i r c u i t H ( g ) , shown in Fig. 4 . 1 , are merely a mathematically convenient repre- sentation of the input-output relation of Fig. 4 . 2 . T H<s> G c<s> F i g . 4.1 - Block diagram of sampled-data feedback c o n t r o l system. It JL, » : 1—: 1 1— 0 T 2T 3 T 4 T Time Fig. 4.2 - Input and output of sample and zero-order hold. 23 I t i s assumed that the con t r o l system i s synchronously sampled, and that the value of the cal c u l a t e d p r o p o r t i o n a l gain i s i n d i c a t i v e of the degree of s t a b i l i t y of the system. The o v e r a l l t r a n s f e r function i s given as -TC „, . K (1 - e )6 O N G ( s ) = s (s+e^s+e^ <4'2> where 6̂  = l / x ^ ; 62 = 1/^25 8 = ^\^z a n < * T l > T2 a r e the f i r s t and second time constants of the c o n t r o l l e d system r e s p e c t i v e l y . As has been stated e a r l i e r on, state v a r i a b l e approach w i l l be used i n the c a l c u l a t i o n s . The f i r s t step towards obtaining a set of f i r s t - o r d e r d i f f e r e n t i a l equations to describe the dynamics of the system and Hence the state v a r i a b l e formulation i s drawing of a s i g n a l flow graph. This diagram ( s i g n a l flow graph) i s made up of nodes and di r e c t e d l i n e s s i g n i f y i n g d i r e c t i o n of information flow. An example i s shown i n F i g . 4.3. The r e l a t i o n s h i p e x i s t i n g between an input node and an output node i s derived by the a p p l i c a t i o n of Mason's gain formula. : "This formula gives d i r e c t l y the o v e r a l l transmittance from an input node to an output node. That i s , Y ET.A. i n Where T^ i s the gain (transmittance) of the i th forward path from the input node Y. to the output node Y , and A i s the in out determinant of the graph, which i s defined as A = 1 - (sum of a l l i n d i - v i d u a l loop gains) + (sum of gain products of a l l combinations of two 24 non touching loops) - (sum of gain products of a l l combinations of three non touching loops) + .... A^ = Determinant of graph i n which a l l loops that touch the i th forward path are set equal to zero. A forward path i s any path which goes from the input node to the output node along which no node i s passed through more than once. A loop i s any path which or i g i n a t e s and terminates at the same node along which no node i s passed through more than once. Touching loops are loops which have one or more nodes i n common. S i m i l a r l y , a loop which touches the i th forward path i s one that has one or more nodes i n common with the path. Transform the plant's transfer function into a second-order d i f f e r e n t i a l equation and from t h i s point reduce the system to a set of f i r s t - o r d e r d i f f e r e n t i a l equations. That i s , £.+ (e.+e.) — + e.e.c = KS (4.4) dt I f the hold [ h ( t ) ] i s introduced into equation (4.4), the second-order d i f f e r e n t i a l equation becomes + e 3 dC dt + 0C = K6h(t) (4.5) where 8.̂  = 8 1 + Q^, where 8̂^ = 1/^, &2 = 1/T2 and 8 = 9 ^ Let C = X 1 and C = X = X, (4.6) Therefore equation (4.5) reduces to 6 3X 2 - 6XX + K6h(t) (4.7) 25 Input Node Output Node F i g . 4.3 - T y p i c a l s i g n a l flow graph. .-I X 2 ( S ) <?X1(kT) Ys*' F i g . 4.4 - Signal flow diagram of Equation (4.8a) Q X 2 C K T ) H(s), X2Cs) ^ P X ^ s ) F i g . 4.5 - Signal flow diagram of Equation (4.8b) 26 Laplace transforming equations (A.6) and (4.7) gives sX 1(s) - X 1 ( t Q ) = X 2(s) (4.8a) sX 2(s) - X 2 ( t Q ) = -e 3X 2(s) - 6X1(s) + K6H(s) (4.8b) where X^(tg) and X^tg) are the i n i t i a l state values of state variables X 1 and X2> The signal flow diagram for equation (4.8a) is given in Fig. 4.4 The signal flow diagram for equation (4.8b) is as shown in Fig. 4.5. Combining equations (4.8a) and (4.8b) hence their respective signal graphs gives the signal flow diagram of Fig. 4.6. The error signal at time t = kT is e(kT) = r(kT) - C(kT) r(kT) - X^(kT) (4.9) Combining equations (4.8a), (4.8b) and (4.9) gives the overall control system signal flow diagram as shown in f i g . 4.7. Mason's gain formula is then applied to give the set of f i r s t order di f f e r e n t i a l equations in Laplace transform form. There are two loops in the system and are given as : 2 L l = ~Q3^S a n d L2 = ~ Q^ S Since there are no non touching loops, the determinant A of signal graph is A = 1 - (L 1+L 2) 27 QXJCKT) qxT(KT) x,(s) Fig. 4.6 - Signal flow diagram of equations (4.8a) and (4.8b) combined. f(KT) e(KT) H(S) X.CKT) X,(s) F i g . 4.7 - C o n t r o l system s i g n a l f l o w d i a g r a m . 28 The transfer function r e l a t i n g the input X^(kT) to the output X^(s) i s <|>̂ (s) and consists of two parts: ( i ) Tj = 1 / g ( i i ) T j 1 = - £ | (4.12) s The forward path T^ i s touched by loop L^' The loop i s then set equal to zero. 1 S + 93 Thus, A. = 1 + L_ i e . (4.13) l i s The forward path T^\ i s touched by loops and L^, thus = = 0 and = 1 - 0 . Hence, the transfer r e l a t i n g the input X^(kT) and the output X^(s) i s T i A i ( S + V K9 *UKS) A A (s+O^s+C^) 3(3+0^(3+62) The transfer function r e l a t i n g the input X2(kT) and the output X i ( s ) i s <J>12(S)* This i s given as: the transmittance i s T2 = 1/s and A2 = 1, since the path i s touched by loops and L.2- The transfer function r e l a t i n g the input X2(kT) and the output X^(s) i s then given by * 1 2.Cs) = . _ _ l - _ . (4.15) ( s + e 1 ) ( s + e 2 ) The transfer function r e l a t i n g the input Xi(kT) and the output T3 A3 T3 A3 X 2 ( s ) i s 4 ,2^(s)> that i s <p^(s) i s made up of — ^ 1 ^— , -K6 whei re T̂  = —2" D u t the path i s touched by and , hence   = 1; s T3 = - —2~ and  ' = 1 since the path i s touched by and s 29 K6 9 Thus * 2 1 ( . s ; - ( s + Q + e - ( s + Q + Q (4.16) The t r a n s f e r function r e l a t i n g the input X2(kT) to the output X2(s) i s given as T 4 A 4 s * 2 2 ( s ) A (s+0 1)(s+9.) (4.17) 2' where the transmittance T^ = 1/s and A 4 = 1; since L^ = L2 = 0. The t r a n s f e r function r e l a t i n g the input r(kT) to the output X^(s) i s T A . . . K6 5 5 V s ; 8(3+6^(3+62) A (4.18) K0 where the transmittance T^ = — j and A,. = 1; = = 0 s The t r a n s f e r function r e l a t i n g the input r(kT) to the output X2(s) i s K6 T 6 A 6 * 2 ( s ) (s+e1)(s+e2) A (4.19) K6 where the transmittance T, — —=- and hr = 1. 6 2 6 s Therefore the set of f i r s t - o r d e r d i f f e r e n t i a l equations i n Laplace transform i s " X(s) = s + K6 ( 3 + 6 ^ ( 3 + 8 2 ) 5(5+6^(5+62) ( 5 + 6 ^ ( 5 + 6 2 ) - K6 ( 5 + 6 ^ ( 3 + 6 2 ) ( 3 + 6 ^ ( 3 + 6 2 ) ( 3 + 8 ^ ( 3 + 6 2 ) X(kT)+ K0 5 ( 5 + 6 ^ ( 5 + 6 2 ) K6 . ( 5 + 6 ^ ( 3 + 6 2 ) V(kT) Due to the time delay to = kT which ex i s t s i n the c o n t r o l system because of the presence of the sample and hold, a f t e r obtaining the inverse Laplace transform of equation (4.20), the time t i s replaced by t - kT. 30 Inverse Laplace transforming gives x ( t ) = r 1 * * * « * 2 _ - e t - e t - e l t - e 2 t +a e - K ( l + a e + c^e ) J a 5 ( e -e * ^ * - B , t - e „ t * * -8 t -6-t - a 6 ( e 1 - e )(1+K) X X(KT) + X 2(KT) * * -e t . -e t K[l+a 3e + a 4 e 1 -6 t -6 t Ka,(e - e ) o r(KT) (4.21) ; 2 _ 1 • _ • • ... • where a, = . n a ; , a 0 - ^ — — \ - a-> ~ a /a _a \ > a/, _ i e ^ e ~ ' a2 " 3 e 1 ( 9 1 - o 2 ) ' "4 V V V 1 0 " 1 = _2_ J2 u l " 2 u l '. 2 • 1 " 1 2 t* = t - KT 31 Thus X(K+1T)= ~ -e1T -82T -Q^T -e4T -e1T -e2T~l [a^e +a 2e -K(l+a 3e +a 4(e )] a,.(e -e ) -8..T -6„T - a 6 ( e -e )(1+K) - Q ^ -e 2T a^e + a 8 e X(KT)+ -9 T -8 T K[l+cue + a,e ] -9 T -9 T' K a 6 ( e - e ) r(KT) (4.22) C(t) = [1 0] x (KT) The s t a b i l i t y conditions are determined from the r e l a t i o n det[zI-Q] = 0 where where Q= •i-8 T -9 2T -9 1T -9 4T - 8 ^ -8 T [a^e + a 2 e -K(l+a^e + a ^ e )] a 5 ( e -e ) -9 T -9 T - a 6 ( e - e )(1+K) ~ 9 1 T ~ 9 2 T a^e + a 8 e (4.23) That i s ~~ -8 T -8 T -8 T -9 T -8 T -6 T Z - [ a i e + a 2 e - K(l+a 3e +a^e ' )] - a 5 ( e -e ) -8 T -8 T a 6 ( e - e )(1+K) - 8 i -e 2T Z - [ a ? e + a ge ] = 0 (4.24) -6 T -8 2T Let = e and P 2 = e . The determinant and hence the character- i s t i c equation of the c o n t r o l system becomes 32 Z - [ a 1 P 1 + a 2 P 2 - K(l+a 3P 1+a 4P 2)] - a ^ P ^ ) a 6 ( P r P 2 ) ( l + K ) Z _ [ a 7 P l + a 8 P 2 J = 0 (4.25) That i s , applying Jury's s t a b i l i t y c r i t e r i o n gives Condition I: a 2 + a l + a0 ^ 0 K > - [ l - ( a 1 + a 7 ) P 1 - ( a 2 + a 8 ) P 2 + ( a 7 P 1 + a 8 P 2 ) ( a 1 P 1 + a 2 P 2 ) + a 5 a 6 ( P 1 - P 2 ) [ ( l + a 3 P 1 + a 4 P 2 ) ( l - a 7 P 1 - a 8 P 2 ) + ^ ( P - P ^ l ' (4.26) Condition I I : ai - a\ + an > 0 K > [ l + ( V a 7 ) P 1 + ( a 2 + a 8 ) P 2 + ( a 7 P 1 - r a 8 P 2 ) ( a 1 P 1 + a 2 P 2 ) + a 5 a 6 ( P 1 - P 2 ) ^ [ ( l + a 3 P 1 + a 4 P 2 ) ( l + a 7 P 1 + a 8 P 2 ) - « 5« 6(P ] L-P 2) ] (4.27) Condition I I I : a 0 - a 2 < 0- K < [ l - ( a 7 P 1 + a 8 P 2 ) ( a i P 1 + a 2 P 2 ) - a 5 a 6 ( P r P 2 ) ^ [ a 5 a 6 ( P 1 - P 2 ) - ( a 7 P 1 + a 8 P 2 ) ( l + a 3 P 1 + a 4 P 2 ) ] (4.28) Figure 4.8 i s the s t a b i l i t y c onstraint of the co n t r o l system as a func- t i o n of sampling time for four d i f f e r e n t time constant r a t i o s , that i s the r a t i o of second time constant to f i r s t time constant. The e f f e c t i v e or l i m i t i n g s t a b i l i t y constraint shown i n Figure 4.8 i s equation (4.27). The s t a b i l i t y range of the co n t r o l system increases with increase i n time constant r a t i o but decreases with increased sampling period, F i g . 4.8.. This trend has been suggested by other workers. Increase i n sampling time introduces large i n s t a b i l i t y to the system but with a smaller sampling time the sampled data system approaches that of 33 Sampling Time, Sec- Fig. 4.8 - Stability constraint of sampled-data second order system with no dead time as a function of sampling rate for various ratios of the time constants (zero-order hold). 34 continuous (analog) control system. Although i t is a well known fact that a l l proportionally controlled f i r s t and second-order systems are stable in. the continuous time domain, regardless of loop, this is not true for second-order systems in the sampled data domain. 4.1.2 Second-Order Overdamped System With Half-Order Hold Consider now the process with a half-order hold as the smoothing device. From discussions given in the control literature, the amplitude characteristic of a zero-order hold drops off rapidly at low frequencies and amplitude characteristic of first-order hold exhibits an overshoot 2 9 before cutting off sharply. An amplitude characteristic which f a l l s between the zero-order and the first-order characteristics is being suggested to come close to approximating an ideal f i l t e r response. This f i l t e r charateristic could be realized by use of a fractional-order hold, in this study a half-order hold is used. Fig. 4.9 is the input and output of sampler and holdV. for the half-order hold c i r c u i t . The transfer function of half-order hold is given (see Appendix 1 for derivation) as H. / 7(s) = (1 - l/2e i S ) ( - J ) + 1/2T(- - ) Z (4.29) 1 /1 s s The impulse-modulated sampling and hold circuit H(s), shown in Fig. 4.1 is merely a mathematically convenient representation of the input-output response of Fig. 4.9. Almost a l l the dig i t a l computers used in industrial control systems have b u i l t - i n zero-order hold c i r c u i t s . A half-order hold may be expressed as a function of a zero-order hold. Equation (4.29) can be expressed as = [-Ts + (Ts - f l ) ( l - e - T S ) ] , ( 4 > 3 ( ) ) 1 2Ts 35 1 - ~ T s where H 0(s) = — ^ (zero-order hold) Expanding the e T s in the paranthesis up to T 2 s 2 term, assuming T 3 s 3 i s negligible, reduces equation (4.30) to -Ts „ , . ,4 + 5 T s w l - e . H l / 2 ( s ) = ( ) ( > 4 + 4Ts s The overall transfer function of the control system i s now given as -Ts (4.31) G(s) = H(s)G (s)G (s) = K ( 4 + 5 T S ) c p. e ( — ) (4.32) (4+4Ts) (s+61)(s+62) s where Qi = l/rlt 62 = /T 2 and 9 = 9^; also T 1 , T 2 are as before. The signal flow diagram of equation (4.32) Is shown in Fig. 4.10. Application of Mason's gain formula gives X 1(s) X 2(s) * i i ( s ) < f ) i 2 ( s ) * 2 1 ( s ) ^ 2 ( s ) X :(KT) X 2(KT) y (KT) (4.33) Derivation of equation (4.33) and parameter definitions are shown in Appendix 2. There exists a time delay to = kT due to the zero-order hold present in the control system. After the inverse Laplace transform is obtained, t is replaced by t-KT. r(KT) (4.34) where t* = t - KT. The value of the output at the sampling instants is obtained by letting t = (K+1)T, in which case t* = t - KT = (K+l) T - K T = T. Therefore equation (4.34) becomes Thus, x x(t) ^ ( t * ) •i2<t*> X X(KT) + *{(t*) x 2(t) • 2 1<t*) • 2 2<t*) X 2(KT) ^ ( t * ) TIME Fig. 4.9 - Input and output of sample and hold of hal f - o r d e r . - F i g . 4.10 - Signal flow graph of equation (4.33). 38 X 1(K+1) *i l < T > *{ 2(T) X 1(KT) = + X 2(K+1) • 2 1<T> * 2 2 ( T ) X 2(KT) ^ ( T ) _____ ___ r(KT) (4.35) The c h a r a c t e r i s t i c equation of the co n t r o l system i s given as -«P{2(T) Z - *21<T> Z " <022(T) = 0 (4.36) S t a b i l i t y conditions are determined by applying Jury's s t a b i l i t y test and are given as: K > K < d - Q 3 - Q4> (Q 5 " Q 6 " Q 7 ) (1 + Q 3 + Q 4) « 5 + Q 6 " Q 7 > K < q - Q 4> (Q 6 - Q 7) (4.37) (4.38) (4.39) F i g . 4.11 i s the s t a b i l i t y constraint of the c o n t r o l system as a func- t i o n of sampling time for four d i f f e r e n t time constant r a t i o s . The representation i s that of equation (4.39) since i t i s the e f f e c t i v e s t a b i l i t y constraint on the system. Just as i n the case of zero-hold, the s t a b i l i t y of the co n t r o l system increases with increase i n time constant r a t i o s while i t decreases with increase i n sampling time. In a l l conditions (increase i n time constant r a t i o and increase sampling time) the value of the proportional gain for the system with h a l f - o r d e r hold i n the c i r c u i t i s greater than that with zero-order hold i n the c i r c u i t . 39 Sampling Time F i g . 4.11 - S t a b i l i t y constraint of sampled-data second order system with no dead time as a function of sampling rate f o r various r a t i o s of time constant 2 to time constant 1 (Half-order hold). 40 4.1.3 Control System With Dead Time When a delay time i s included i n the system, the process t r a n s f e r f u n c t i o n becomes Time delay i s a dynamic c h a r a c t e r i s t i c that can be represented, when i t occurs i n a path containing sampling and holding, by adding t h i s time delay to the e x i s t i n g delay caused by the presence of a hold i n the c i r c u i t . In other words, there w i l l r e s u l t a forward s h i f t of x = AT, where T i s the sampling time and A i s any number, i n the output response; that Is, at any instant of sampling, the output response w i l l be equal to the, response at (i-A)T. Table 4.1 i s a t y p i c a l response con d i t i o n . Sampling instants T 2T 3T . . . NT Response equivalent to sampling at (l-A)T (2-A)T (3-A)T • • • (N-A)T Table 4.1 - Output response of process with dead time component It i s observable from Table 4.1, that many s i t u a t i o n s e x i s t . I f A i s an i n t e g r a l multiple of the sampling time, the output response w i l l be zero for a l l sampling instants less than or equal to A ( i f the i n i t i a l system condition i s zero), but i f A i s a f r a c t i o n of the sampling time, the value at the f i r s t sampling time i s equal to that of (l-A)T sampling i n s t a n t . The t h i r d case i s when A i s both an i n t e g r a l and f r a c t i o n a l multiple of the sampling rate; the output response i s a combination of the above two conditions. For s i m p l i c i t y of a n a l y s i s , the dead time i s added to the delay due to the hold. The t o t a l delay 41 time in the control system is t = (K+A)T, where KT is the delay caused by the hold in the c i r c u i t . The process equations describing the control system w i l l be same as those for the c i r c u i t with no dead time but the inverse Laplace transform w i l l be different, since the delay time w i l l be added in the overall time delay. 4.1.3a Control System with Dead Time f o r Zero-Order Hold C i r c u i t The o v e r a l l t r a n s f e r function for the c o n t r o l system i s given as G(s) = K e e ~ T S ( i - e ~ T s ) / s ( s + e 1 ) ( s + e 2 ) (4.41) The state d i f f e r e n t i a l equations i n Laplace transform i s S + 6 3 K6 1 X(s) = (s+ e l ) ( s+ e 2 ) 8 ( s+ e l ) ( s+ e 2 ) (s+ e l)(s+ e 2 ) -K6 e s (s+YXs+e^) " (s+ e i ) ( s+ e 2 ) (s+e i)( s+8 2) x ( t 0 ) + K9 s(s+9 1)(s+6 2) K0 (s+ e 1)(s+ e 2 ) r ( t 0 ) (4.42) Taking the inverse Laplace transform of equation (4.42) gives X(t) = -e.t + -6„t + - e , t + -e,t + [a e 1 +c*2e 2 -K(l+a 3e 1 +a^e 4 )] -9 t + ~ e ? t + -a 6(e 1 - e 2 )(1 + K) « 5(e -e ) -9 t + -9 t + 1 2 a ?e +age \ 42 X(t 0) + -9 t + -6 t + K[l+a 3e 1 + a 4e 2 ] Ka 6(e - e ) r ( t 0 ) (4.43) where t + = t - (K+A)T and T is the sampling time. The process dead time A can be broken down Into A = (j+6)T where j is the integral multiple of the sampling time part of the process dead time and <5 is the fractional part. Thus, for the condition t = (K+j+l)T, t + = (1-6)T = V. Hence the sampled-data state equations in the transformed state space is x(K + j + 1) - X(i + 1) = [ v 1 -8.V -9 V -9 V -9 V -9 V -9 V +a2e -K(l+a 3e +c<4e )] o (e' -e ) -6 V -9 V -a 6(e - e )(1 + K) --e1v -9 2v a^e + a 8 e X(KT)+ -0 V -8 7 K(l + a 3e + a 4e ) -0 V -0 V Kcx (e 1 - e 2 ) o r(KT) (4.44) 43 The c h a r a c t e r i s t i c equation i s D e t [ Z j + 1 I - <KV)] = 0 (4.45) where <K^) = * n < v > * n < ' > *22<V> See Appendix 4 for parameter d e f i n i t i o n s . That i s , z 2 a + l ) _ z j + l [ ( j ) v i ( v ) + ^ ( v ) ] + [ ^ ( V ) ̂ ( V ) - ^ ( V ) ^ ( V ) ] = 0 (4.46) K22 v 11 22v '12' Equation (4.46) shows that there are an i n f i n i t e number of cases that can e x i s t depending on the, value of the integer j . The constra i n t s on. the loop gain of th i s control sampled system are determined a n a l y t i c a l l y below for two.cases j = 0, 1-, i . e . 0 < T < 2T. Thus adding dead time to the system and/or increasing the sampling rate increases the order of the c h a r a c t e r i s t i c equation. The s t a b i l i t y analysis becomes a l g e b r a i c a l l y more involved as j increases. However, the sampled data s t a b i l i t y l i m i t approaches the continuous s t a b i l i t y l i m i t for systems with large amounts of delay time r e l a t i v e to the sampling period. Case 1: j = 0 (0 < T < T), equation (4.46) becomes 12 T21 = 0 (4.47) 44 Applying Jury's s t a b i l i t y criterion on equation (4.47) (1 - A' + Al) K > ( A 3 - A 4 ) 2 ^ (i + A: K < 1 ... (4.49) (^5 " Ap -(A: - i ) K > 0 (4.51) (See Appendix 4 for parameter definitions) A l l the above four conditions must be satisfied to ensure the st a b i l i t y of the system. A typical case of these conditions is shown in Fig. 4.12, where the ultimate st a b i l i t y limit is plotted against delay time (0 < T < T) for constant sampling time. The amount of delay which maximizes the ultimate stability limit for constant sampling rate T is defined as t . For T < T , equation (4.49) places the severest max max constraint on the ultimate stability limit; for x > T , equation max (4.50) constrains. The correct value of x is determined by the max intersection of these two constraints. Case II: j = 1 (T < x < 2T) equation (4.46) becomes Z 4 - Z 2 [ ^ + V22 ] + ^ 4>22 " * i 2 *21 - ° < 4' 5 2 ) The s t a b i l i t y constraints are 1 " A 2 K < -= V (4.53) A* + A 6 4 45 Fig. 4.12 - The s t a b i l i t y boundary of a sampled-data system with zero- order hold as a function of dead time. 46 K < A2 - 1 A6 + A4 (4.54) The other two constraints involve higher orders of K. 4.1.3b Analysis of System with Deadtime f o r Half-Order Hold Time delay i s a dynamic c h a r a c t e r i s t i c that can be represented when i t occurs i n a path containing sampling and hold i n the c o n t r o l c i r c u i t . For s i m p l i c i t y i n a n a l y s i s , the dead time i s added to the delay due to the zero-order hold. The o v e r a l l c o n t r o l system transf e r f unction nr \ , Q / 4 + 5 T s w l - e G ( S ) = k 6 ( 4 4- 4 T s ) ( -Ts -TS (s + 8 1 ) ( s + 6 2) (4.55) The s i g n a l flow graph of equation (4.55) i s as shown i n F i g . 4.10. The state d i f f e r e n t i a l equations i n Laplace transform are: X(s) *il<S> • { 2 ( 8 ) • ' (s) 21 4>' (s) 22 X(t 0) + 2 r ( t 0 ) (4.56) The inverse Laplace transform is X(t) *il<tV> * 2 1 ( t V ) 4» 2 2(t v) x ( t 0 ) + * { ( t v ) r2(tv) r ( t 0 ) (4.57) 47 where t V = t - (K + A)T and T i s the sampling time. Assuming that the process dead time A can be broken down into A = (j+6)T, where j i s the integer multiple of sampling times of the process dead time and 6 i s the f r a c t i o n a l part. Since the output response at times less than or equal to jT i s zero, the c o n d i t i o n t = (K + 1 + j)T i s used. Thus for t h i s case t = (K+j+l)T, t = (1-6)T = VT (4.58) Hence the sampled-data state equations i n the transformed state space are X(K+j+l) = ^ ( V T ) <P{2(VT) ^ 2 ( V T ) X(KT) + ^ ( V T ) * 2 ( V T ) r(KT) (4.59) The c h a r a c t e r i s t i c equation i s set to zero, i . e . Det[Z J I - <KVT)] = 0 where (4.60) *(VT) = , } , i i ( V T ) * i 2 ( V T ) > 2 1(VT) <P22(VT) Therefore, the c h a r a c t e r i s t i c equation i s 2 ( i + l ) i+1 Z - Z J [4>n(VT)+<(,22(VT)] + • 1 1(VT)<|) 2 2(VT)-* 2 1(VT)4) 1 2(VT) = 0 (4.61) 48 As can be seen i n Equation (4.61) there are an i n f i n i t e number of cases that can e x i s t depending on the value of integer j . The constraints on the loop gain of th i s sampled data c o n t r o l system are determined analy- t i c a l l y below for the two cases j = 0, 1, i . e . 0 < x < 2T. Thus adding dead time to the system and/or increasing the sampling rate increases the order of the c h a r a c t e r i s t i c equation. The s t a b i l i t y a n a l y s i s becomes a l g e b r a i c a l l y more involved as j increases. However, the sampled data s t a b i l i t y l i m i t approaches the continuous s t a b i l i t y l i m i t f o r systems with large amounts of delay time r e l a t i v e to the sampling period. Case I; j = 0 (0 < x < T); Z 2 - Z N f ^ V T ) + <^ 2(VT)] + •{ 1(VT)^ 1(VT) - ^ 1(VT)r>' 2(VT) = 0 Applying Jury's s t a b i l i t y c r i t e r i o n to equation (4.62)'gives the s t a b i - l i t y l i m i t s as: (i - Q; - % K><Q;-Q;-QP (4-63) -(1 + Q' + Q! (1 T % • > K < T5np (4-65) In a d d i t i o n , for s t a b i l i t y K > 0 (4.66) A l l the above four conditions must be s a t i s f i e d to ensure the s t a b i l i t y of the system. A t y p i c a l case of these conditions i s shown i n F i g . 4.13, where the proportional gain i s plo t t e d against delay time 49 (0 < T < T) for constant sampling time. The amount of delay which maxi mizes the l i m i t i n g proportional gain for constant sampling rate T Is defined as x . For x < x , equation (4.64) places the severest max max con s t r a i n t s on the l i m i t i n g proportional gain; f o r x > x , equation max (4.65) c o n s t r a i n t . The correct value of x i s determined by the max i n t e r s e c t i o n of these two c o n s t r a i n t s . As with the case of zero-order hold, the l i m i t i n g proportional gain increases with increase i n dead time u n t i l the x i s reached, a f t e r which the proportional gain max decreases, increase i n sampling time also decreases the pro p o r t i o n a l gain. In a l l the conditions,, i n v e s t i g a t e d , the c o n t r o l system with h a l f - o r d e r hold gave higher values of the pro p o r t i o n a l gain than that of co n t r o l system with zero-order hold. Case 2: j = 1 (T < x < 2T) The c h a r a c t e r i s t i c equation becomes Z 4 - Z 2[4.[ 1(VT)+** 2(VT)] + • 2 1(vT)4>' 1(VT) - • 2 1 ( V T ) * [ 2 ( V T ) - 0 (4.67) Using Jury's s t a b i l i t y a n a l y s i s , the constraints on K are K < (1 (R (4.68) K < (1 + Q9) (4.69) « 6 " V 4.2 Transient Response of System The response of the second-order system to a step change i n set point was in v e s t i g a t e d . C r i t e r i a which are often used for judging good 50 20! 1 0-3 x « 0 ;6 09 ~V2 T5 Dead Time Sec ; "c Fig. 4.13 - The s t a b i l i t y boundary of a sampled-data system with h a l f - order hold as a function of dead time. 51 closed loop performance have been discussed i n Chapter 3. A new performance index defined as N Z e ( i T ) • = (4-70) 2 E e (IT) i=0 i s used to estimate an optimum loop gain value f o r the pr o p o r t i o n a l c o n t r o l l e r . For a minimum steady state error response, the performance index <J>, should be greater than 1. The transient response of the c o n t r o l system i s derived as a s o l u t i o n to the state d i f f e r e n c e equations i n matrix form. For a system with a set of f i r s t - o r d e r d i f f e r e n c e equations i n matrix-form which i s given as X(n+1) = AX(n) + br(n) (4.71) y = C TX(n) X where the sampling time T has been dropped for convenience, and C i s the c o e f f i c i e n t of the output. The s o l u t i o n to the model, equation (4.71), i s given i n matrix form as N-1 . X(n) = A nX(0) + E A br(n) (4.72) i=0 where A n = Z - 1{Z(ZI-A)" 1} The transient response i s n-1 . y(n) = C [A nX(0) + E A b] (4.73) i=0 52 The e r r o r response i s the d i f f e r e n c e between the desired response and the ac t u a l response and i s given as e(n) = r(n) - y(n) (4.73a) The r e l a t i o n s h i p between t h i s performance index and the one-quarter decay r a t i o c r i t e r i o n i s shown i n Appendix 3. 4.2.1 Transient Response of Second-Order Overdamped with Zero-Order Hold Consider the process shown i n F i g . 4.1, but with a process dynamics of a second-order overdamped t r a n s f e r function and a zero-order hold i n the c i r c u i t . The o v e r a l l t r a n s f e r function i s r v o = 9(1 ~ e " T s ) 3(3+6^(3+62) Figure 4.14 i s the s i g n a l flow graph of the c o n t r o l system. The set of f i r s t - o r d e r d i f f e r e n c e equations Is (4.74) X(K+1T)= -6 T- -6„T — 6 T — 6 T -0 T - e„T r 1 2 1 2 i 1 2 [c^e +a 2e Z - ( l + c y )] a . ( e -e ) -6 T - 8 ? T -2a,(e -e ) o -8 T -8 Tl l+a_e + a.e 3 4 -01T -02T a 6(e - e ) r(KT) -6 T -0 2T a^e +a0e / o X(KT)+ (4.75) C(K+1T) = [1 0] x (KT) Parameters are as defined for equation (4.22). The general s o l u t i o n to the matrix d i f f e r e n c e equation (Equation 4.75) for step input change i s 53 T=0-5s Sampling Interval, N F i g . 4.15 - Open loop t r a n s i e n t response of uncompensated sampled-data second-order process with no dead time for a u n i t step change for d i f f e r e n t sampling rates (zero-order hold). 54 X(nT)'= 9llV 612 Y2 615^ + 616 Y2 613Y1+814^2 917^1+618Y2 X(0) + N-1 Z 1=0 l l ' l 12T2 13'l 14T2 9 yN-^i+g yN"1"1 9 y^ 1" 1-^ y ^ 1 " 1 15 1 16 2 17 1 18 2 1 + 0 t 3 P l + a 4 P 2 a fp -p ) 6K 1 2} (4.76) See parameter definitions in Appendix 4. If the states are initially at rest, then Equation (4.76) reduces to X(nT) = (l-H, 3P 1 +a 4P 2)(e i lyf 1- i +e i 2Yr 1" i) + ( 1 l + a 3 P 1 + a 4 P 2 ) ( 6 1 5 y f 1 - 1 + 8 1 6 y f 1 - i ) + /T, „ ... N - l - i ^ Q N - l - i N a 6(P 1-P 2)(6 1 3y 1 + 9 u y 2 ) „ . , • N-l-i, . N-l-i. a (P -P )(8 y +9 y ) 6 1 2 17 1 18 2 (4.77) Therefore, the transient response is given as N-1 C(nT) = Z {([l+ajPj+a^] e 1 1 + a 6 e 1 3 [ P 1 - P 2 I ) Y l " 1 1 + 1-0 ([l-ra 3P 1 +o t 4P 2]9 1 2 +a 60 1 4[P 1-P 2])yf 1- 1} (4.78) Fig. 4.16 - Control system with proportional controller. 56 F i g . 4 .15 i s the c l o s e d loop t r a n s i e n t response of the c o n t r o l system as a f u n c t i o n of sampl ing I n t e r v a l f o r v a r i o u s sampl ing t i m e s . I n t r o d u c - t i o n of a p r o p o r t i o n a l c o n t r o l l e r i n the feedback loop r e s u l t s i n F i g . 4 . 1 6 , and the e r r o r response i s r (nT) - C (nT ) . The e r r o r response of the c o n t r o l system w i t h the a d d i t i o n of the p r o p o r t i o n a l c o n t r o l l e r f o r a u n i t s tep change i n s e t p o i n t Is N-1 e(nT) - 1 - K S {( [ l + a ^ + c ^ ] 8 n + a 6 8 ^ [ P ^ ]) ] vj x+ ( [ l + a 3 P 1 + a 4 P 2 ] e i 2 + a 6 e i 4 [ P 1 - P 2 ] ) Y f 1 - i } ( 4 . 7 9 ) The amount of loop g a i n K i s es t imated from the performance c r i t e r i o n . That i s , N-1 Z e ( j T ) 3=0 N-1 S e Z ( j T ) 3=0 •I [ 1 " K c " V { C C 1 + < ^ P l + 0 l 4 P 2 1 ^ l " ^ 6 13 t P l " P 2 1 J=l 1=0 ? [ i - K c V { ( [ i + a 3 P 1 + a 4 P 2 ] e 1 1 + a 6 e 1 3 t P 1 - P 2 ] ) Y f 1 - i + 3=1 i=0 ( [ i + a 3 P 1 + a 4 P 2 ] e 1 2 + a 6 e 1 4 [ P 1 - P 2 ] ) Y f 1 - i } ] 2 T h e r e f o r e , the optimum va lue of the p r o p o r t i o n a l c o n t r o l l e r , shou ld s a t i s f y the c o n d i t i o n 2 2 D 2 " ^ D 2 + 4 D 1 D 3 < K c < D 2 + / D 2 + 4 D 1 D 3 2D^ ° 2 D L ( 4 / 8 1 ) 57 S a m p l i n g I n t e r v a l , N F i g . 4.17 Closed loop transient response of proportionally controlled sample-data second-order process with zero-order hold for a unit step change for d i f f e r e n t sampling. 58 S a m p l i n g I n t e r v a l , N F i g . 4.18 - Closed-loop t r a n s i e n t response of p r o p o r t i o n a l l y c o n t r o l l e d sampled-data second-order overdamped process w i t h zero-order h o l d and no dead time f o r a u n i t step change f o r d i f f e r e n t sampling r a t e s . Table 4.2 - Loop gain as a function of sampling time for the two performance indices: (control system with zero-order hold) Time Constant 1 = 1.3; Time Constant 2 = 3.1 Sampling Time • = 3 ... ° <f> = = 1/4 decay r a t i o Loop Gain Response at 12th Sampling Loop Gain Response at 12th Sampling 0.5 sees 0.01921 -0-.-98136 0.0.2310 1.18033 0.7 sees 0.03401 1.01366 0.0409 1.21910 0.9 sees 0.07132 .1.06803 0.08579 1.28438 1.1 sees 0.16183 1.16441 0.19458 1.40005 60 See Appendix 4 f o r parameter d e f i n i t i o n s . The transient response of the c o n t r o l system with p r o p o r t i o n a l c o n t r o l l e r i s shown i n F i g . 4.17 f o r a performance index <J> = 3 and various sampling periods. An equivalent performance index to one-quarter decay r a t i o i s used for the transient response of F i g . 4.18. For a l l the tested sampling rates, the one-quarter decay r a t i o equivalence gave a poorer response than that of performance index <f = 3. For both performance i n d i c e s , an increase i n sampling time r e s u l t s i n an increase i n loop gain and hence less s t a b i l i t y margin. Table 4.2 l i s t the values of loop gain and response a f t e r 12 sampling times f o r the two cases shown i n Figs 4.17 and 4.18. 4.2.2 Transient Response of Second-Order Overdamped With Half-Order Hold Consider the process shown i n F i g 4.1 but with a process dynamics of a second-order overdamped transfer function and a half-order i n the c i r c u i t . The o v e r a l l t r a n s f e r function i s G(s) = 4 + 5Ts 4 + 4Ts ( 1 - e ) ( s + ^ X s+e^ (4.82) and the set of f i r s t - o r d e r d i f f e r e n c e equations i s x 1(k+l) "•!>) +_V(T> x 2(k+l) • £ < T > •£CT> X 2(k) + r(k) (4.83) c(k) = [ l 0] x(k) 61 15CM in c o a (/> S10CH c a> w c. ro: I— - r- T= 0-5S T= 0-7S T= 0-9 s T= M s T j = 1 3 s x a=3-1s 2 4 § $ S a m p l i n g I n t e r v a l , N 10 F i g . 4.19 - Open loop t r a n s i e n t response of sampled-data second-order p r o c e s s w i t h no dead time f o r d i f f e r e n t sampl ing r a t e s ( h a l f - o r d e r h o l d ) . 62 The general s o l u t i o n to equation (4.83) i s x(nT) = N , N N N a l l T l l + d 1 2 T l l a l 3 Y l l + a l 4 Y 1 2 N N N N a y + a y a Y + a Y 1 5 r l l 16 r12 17 T11 18 Y12 N-1 x(0) + I i=0 a y N - l - i a N - l - i N - l - i y N - l - i 11 11 12 12 13 11 14 12 15 11 16 12 17 11 18 12 * J V ( T ) * 2 1 V ( T ) (4.84) I f the states are i n i t i a l l y at rest, the p a r t i c u l a r s o l u t i o n becomes N-1 x(nT) = I i=0 N - l - i , N - l - i a Y + a Y l l r l l 12 T12 N - l - i ' N - l - i a 1 5 Y l l + a l 6 T 1 2 N - l - i , N - l - f a ! 3 T l l + a l 4 T 1 2 N - l - i ^ N - l - i a l 7 Y l l + a l 8 Y 1 2 ^ V ( T ) t 2 1 V ( T ) and the transient -response i s N-1 C(nT) = I i=0 { a n t 1 V ( T ) + ^ ^ T ^ - M a ^ t ^ (4.86) The transient response of the system i s as shown i n F i g . 4.19 for various sampling rates. Introduction of a proportional c o n t r o l l e r i n the feedback loop, gives an error response at any instant of N=l ' . . . . . . e(nT) = l-K E { [a^JVHa^V) ] Y^"1" ̂ a ^ * VHa^V) ] n^1"*} 1 = 0 (4.87) 63 3.0769 x 10" = 1.68954 x 1.0 1.35124 x 10" -3 = 1.0781 x 10 - l 2 4 6 8 S a m p l i n g I n t e r v a l , N 10 F i g . 4.20 C l o s e d l o o p t r a n s i e n t r e s p o n s e o f a p r o p o r t i o n a l l y c o n t r o l l e d s a m p l e d - d a t a s e c o n d - o r d e r overdamped p r o c e s s w i t h no dead t i m e f o r d i f f e r e n t s a m p l i n g t i m e ( h a l f - o r d e r h o l d ) . 64 The amount of the loop gain K i s estimated from the performance c r i t e r i o n . That i s » N-1 • = Z e(jT) j=0 N-1 Z e (jT) j=0 Hence, N N-1 Z [1-k S { [ a n ^ V ( T ) + a 1 3 ^ V ( T ) ] Y ^ 1 " 1 + [ a 1 2 ^ V ( T ) + a 1 4 ^ j=l j=0 : 1 N N-1 I [1-k E { [ a 1 1 ^ ( T ) + a 1 3 ^ ( T ) ] y ^ 1 ^ [ a 1 2 t ^ T ) + a 1 4 t 2 l v ( T ) ] T f ; - 1 - i } ] 2 j = 1 J = ° (4.88) Therefore the loop gain i s given by D21 + / D 2 1 + 4 D 1 1 D 3 1 K - — f i (4.89) 11 See appendix 2 for parameter d e f i n i t i o n . Hence, -the sampling rate i s a free parameter, and for any sampling period, there e x i s t s a loop gain such that the performance index i s <|>. The transient responses of the c o n t r o l system with the corresponding loop gains i s shown In F i g . 4.20 f o r performance index a) = 3 and for various sampling times. The trend of the transient response for the various sampling rates i n d i c a t e s that a decrease i n the sampling time re s u l t s i n a reduced deviation from set point before steady state i s attained. Thus, increasing sampling time decreases the s t a b i l i t y margin of the system since high loop gain values mean low 65 s t a b i l i t y margin. For a l l conditions considered, the c o n t r o l system with h a l f - o r d e r hold gave better transient response and attained steady state conditions f a s t e r than did the c o n t r o l system with zero-order hold. This suggests that the half-order hold Is a better i d e a l f i l t e r approximation than i s the zero-order hold. The equivalent one-quarter decay r a t i o performance index gave a poorer transient response as shown i n F i g 4.21. Also the co n t r o l system with h a l f - o r d e r hold i s more stable than the system with zero-order hold with both performance i n d i c e s as measured by the values of the loop gain. Table 4.3 shows a t y p i c a l loop gain v a r i a t i o n for the c o n t r o l system with h a l f - o r d e r hold and f o r the two performance i n d i c e s . 4.2.3 Second-Order Overdamped Plus Dead Time A d d i t i o n of delay time to the second-order dynamics gives a process t r a n s f e r function V s ) = (s+vcs+v (4-90) where jT < x _< (j+l)T J = 0,1,2,3,.... Once again the type of response i s second-order overdamped, but the outputs are delayed by the dead time and occur at sampling instants plus the dead time. To analyse the outputs of the t r a n s i e n t response, the same approach used i n the case of second-order overdamped with no delay i s u t i l i z e d , with the minor mod i f i c a t i o n of adding the dead time to the hold delay. The output s i g n a l w i l l be delayed by an amount x such that the outputs w i l l occur at the Instants of sampling the delayed output s i g n a l . From equations (4.46) and (4.61) the order of the Table 4.3 - Loop gain as a function of sampling time for the two performance indices: (control system with half-order hold) Time Constant 1 = 1.3; Time Constant 2 = 3.1 Sampling Time $ = 3 <f> = l ./4 decay r a t i o Loop Gain Response at 12th Sampling Loop Gain Response at 12th Sampling 0.5 sees . 3.0769 x 10" 4 0.99249 3.7318 x 10" 4 1.08243 0.7 sees 1.68954 x 10" 3 0.99371 2.04911 x 10~ 3 1.08391 0.9 sees 1.35124 x 1 0 - 2 0.99550 1.6388 x 10~ 2 1.08608 1.1 sees 1.0781 x 10 - 1 0.99823 1.26744 x 10" 1 1.08938 67 2 4 S a m p l i n g ^ I n t e r v a l , N 10 Fig. 4.21 - Closed loop transient response of a proportionally controlled sampled-data second order overdamped process with no dead time for different sampling rates (half-order hold). 68 characterist ic equations is seen to be 2(j+l) . Thus for j > 0 — i . e . , x > T — the analysis w i l l be extremely d i f f i c u l t by analytical techniques. 4.2.4 Control System With Zero-Order Hold The overall transfer function is -Ts G(s) = - T S - (' " 6 ) ( 3+6^(5+6,,) (4.91) The set of f i r s t -order difference equations of the above transfer function is x[(k+j=l)T] = * 2 1 ( V ) < | ,22 I V ) x 1 (kT) x 2 (kT) " ^ ( V ) t 2 v i ( v ) (4.92) C[(k+J)T] = [ l 0] x [(k+j)T] where V = (1-6)T. The transient response is given as (assume i n i t i a l states are at rest) N-1 C(k+jT) = H K ^ C V ) + a 3 ^ 1 ( V ) } Y ^ - 1 + { a 2 ^ ( V ) + V) j y ^ 1 " 1 ] (4.93) Parameter definitions in appendix 4. When a proportional controller is added to the feedback loop, and the performance cr i ter ion is applied, the design loop gain is determined. Table 4.4 shows the loop gain as a function of number of samplings used for various values of performance index and sampling rates for the control system. As can be seen from the table, an increase in the performance index results in a decrease in s t a b i l i t y . Also as the number of sampling intervals used in the performance index, .yi,„xi N - l - i , v i , , v i / r 7 X i N - l - i - 69 Table 4.4 - Loop gain as a function of number of sampling intervals used for various performance index values and sampling time (zero-order hold) Sampling Time =1.5 sees; Process Time Constants = 5.027 Desired Steady State Value = 1; Process Dead Time =7.4 Performance Index Loop Gain No. of Sampling Intervals Used Transient Response at Steady State 1.0 0.3827 6 0.32207 0.19791 8 0.25146 0.11460 10 0.20395 0.7126 12 0.16985 1.5 1.17431 6 0.98804 0.76279 8 0.96917 0.53336 10 0.94921 0.39002 12 0.92967 2.0 . 1.41954 6 1.19437 Or.92370 8 1.17361 0.64683 10 1.15115 0.47357 12 1.12882 2.5 1.54883 6 1.30315 1.0084 8 1.28122 0.70647 10 1.25728 0.51743 12 1.23335 3.0 1.62974 6 1.37123 1.06134 8 1.34849 0.74371 10 1.32356 0.54480 12 1.29859 70 Sampling Time = Desired Steady 2 sees; State Value = 1 Process Time ; Process Dead Constants = 5.027 Time =7.4 Performance Index Loop Gain No. of Sampling Intervals Used Transient Response at Steady State 0.43035 6 0.31606 1.0 0.21978 8 0.24679 0.12601 10 0.20020 0.07764 12 0.16671 1.33077 6 0.97733 1 5 0.85394 8 0.95890 0.59129 10 0.93938 0.42855 12 0.92014 1.60876 6 1.18149 2.0 • 1.03414 8 1.16125 0.71713 10 1.13930 0.52038 12 1.11731 1.75532 6 1.28913 2 5 1.12899 8 1.26776 0.78327 10 1.24437 0.56858 12 1.22080 1.84703 6 1 .35648 3.0 . 1.18828 8 1.33433 0.82457 10 1.30999 0.59866 12 1.28538 7 1 Sampling Time = Desired Steady 3 sees; ' State Value = 1 Process Time ; Process Dead Constants = 5.027 Time =7.4 Performance Index Loop Gain No. of Sampling Intervals Used Transient Response at Steady State 0.50224 6 0.31037 1.0 0.25410 8 0.24265 0.14490 10 0.19715 0.08895 12 0.16441 1.56500 6 0.96712 1.5 0.99457 8 0.94973 0.68455 10 0.93137 0.49391 . 12 0.91312 1.89203 6 1.16921 2.0 , '': 1.20452 8 1.15021 0.83028 10 1.12965 0.59978 12 1.10884 2.06443 6 1.27575 2.5 ' 1.31502 . 8 1.25573 0.90686 10 1.23385 0.65534 12 1.21156 2.17230 6 1.34241 3.0 • 1.38408 8 1.32168 0.95468 10 1.29891 0.69001 12 1.27566 1 f l Sampling Interval, N 4.22 - Transient response of uncompensated zero-order hold control system. 73 N, increases the system becomes more stable as measured by the low loop gain values but the error (deviation from desired value of 1) decrease. F i g . 4.22 i s a typical transient response (equation 4.93) of the uncompensated control system as a function of sampling i n t e r v a l . The proportional controlled system transient response is shown i n F i g . 4.23 as a function of the sampling interval with performance index as parameter. Increase in performance index reduces steady state error u n t i l a point is reached after which the error increases. Also increase i n performance index decreases the s t a b i l i t y of the control system. 4.2.5 Control System With Half-Order Hold With the addition of the dead time, the overall process transfer function with half order hold becomes G(s) = + 5Ts1 (1_ + 4Ts_ -Ts . -TS - e ) 9 e |4  s| s (s+61)(s+92) The set of f i rs t -order difference equations of equation (4.94) is x[(k+j+DT; (See Appendix 2 for details and parameter definit ion) c[(k+j]T) = [l 0] x [(k+j)T] ; where V = (1-6)T The transient response is given as (assume i n i t i a l states are zero) N-1 (4.94) •ii<7> d> +' (V) *12K ; x(kT) + r(kT) (4.95) _ * 2 i ( v ) d>+ (V) C [ ( k + J ) T]=T [ivI(V) + V l ^ K l 1 1 + ' { V l ( V ) + b 4 ^ 2 ( V ) ^ 3 2 1 " i l (4.96) When a proportional controller is added to the feedback loop, and the 74 Table 4.5 - Loop gain as a function of number of sampling i n t e r v a l s used for various performance index values and sampling time (half-order hold) Sampling Time Desired Steady = 1.5 sees State Value = 1; Process Time Process Dead Constants = 5.027 Time =7.4 Performance Index Loop Gain No. of Sampling Intervals Used Transient Response at Steady State 1.49164 x 1 0 - 2 6 , 0.17013 1.0 4.07075 x 1 0 - 2 8 0.12699 1.21437 x 1 0 - 3 10 0.10128 3.808 x 1 0 - 4 12 0.08423 6.01111 x 10~ 2 6 0.68561 1.5 2.11693 x 10" 3 8 0.66040 7.72897 x 1 0 - 4 10 0.64463 2.8658 x 10~ 4 12 0.63386 7.28172 x 1 0 - 2 6 0.83053 2.0 2.56944 x 1 0 - 3 8 0.80157 9.39432 x 1 0 - 4 10 0.78353 3.48692 x 1 0 - 4 12 0.77124 7.95029 x 1 0 - 2 6 0.90679 2.5 2.80706 x 1 0 - 2 8 0.87570 1.02675 x K T 2 10 0.85636, 3.81223 x 1 0 - 4 12 0.84319 8.36811 x 10" 2 6 0.95444 3.0 2.95539 x 10~ 2 8 0.92197 1.08121 x 10" 2 10 0.90178 4.01498 x 10~ 4 12 0.88804 9.4321 x 1 0 - 2 6 1.2131 4 .0 3.0187 x 10" 2 8 1.1042 1.63432 x 10~ 2 10 1.0346 5.19831 x 10 - 1 + 12 1.0021 75 Sampling Time = 2 sees; Process Time Constants = 5.027 Desired Steady State Value = 1; Process Dead Time =7.4 Performance Index Loop Gain No. of Sampling Intervals Used Transient Response at Steady State 1.0 7.66176 x 1 0 - 2 6 0.17133 3.64853 x 1 0 - 2 8 0.12768 1.90363 x 10 - 3 10 0.10173 1.04544 x 10~3 12 0.08453 1.5 3.07823 x 10 - 1 6 0.68833 1.8928 x 10~2 8 0.66239 1.20918 x 10~3 10 0.64617 7.8545 x I0~k 12 0.63511 2.0 3.72881 x 10"1 6 0.83381 2.29736 x l 'O" 2 8 0.80397 1.46971 x 10 - 3 10 0.78539 9.55674 x 10 _ l t 12 0.77275 2.5 4.07114 x 10 _ 1 6 0.91036 2.50981 x 10 - 1 8 0.87831 1.60631 x 10~2 10 0.85839 1.04483 x 10 - 3 12 0.84485 3.0 4.28508 x 10"1 6 0.95820 2.64241 x 10 - 1 8 0.92472 1.6915 x 10~2 10 0.90391 1.1004 x 10~3 12 0.88978 4.0 4.68201 x 10 _ 1 6 1.2321 2.8434 x 10"1 8 1.1356 1.9003 - x 10~2 10 1.0632 1.16831 x l O " 3 12 1.0042 76 Sampling Time Desired Steady = 3 sees; State Value = 1; Process Time Process Dead Constants = 5.027 Time =7.4 Performance Index Loop Gain No. of Sampling Intervals Used Transient Response at Steady State 1.88962 x 10 - 1 6 0.17217 1.0 1.22913 x 10 - 1 8 0.12818 8.82331 x 10 - 2 10 0.10205 6.69777 x 10 - 3 12 0.08476 7.57575 x 10 _ 1 6 0.69024 1.5 6.36551 x 10"2 8 0.66382 5.59645 x 10~3 10 0.64730 5.02585 x 10"3 12 0.63604 9.17672 x 10 - 1 6 0.83611 2.0 7.72597 x 10 - 2 8 0.80569 6.80216 x 10 - 3 10 0.78675 6.11502 x 10 - 3 12 0.77388 1.00192 6 0.91287 2.5 8.44039 x 10 _ 1 8 0.88019 7.43438 x 10 - 2 " 10 0.85988 6.68548 x 10"3 12 0.84607 1.05456 6 0.96084 3.0 8.88633 x 10"1 8 0.92670 7.82866 x 10"2- 10 0.90548 7.04104 x lO" 3 12 0.89107 1.13627 6 1.3675 4.0 8.99741 x 10"1 8 1.2003 7.97865 x 10 - 2 10 1.0856 7.15603 x 10 - 3 12 1.0473 Sampling Interval, N Fig. 4.23 - Transient response of Compensated zero-order hold control system with performance index as parameter. X 1 0 T I = T 2 = 5.0271 sees 4 ~6 ~ 8 10 12 Sampling Interval, N F i g . 4.24 - Transient response of uncompensated half-order hold control system. 79 -* Sampling Interval, N Fig. 4.25 - Transient response of compensated half-order hold control system with performance index as parameter. 80 performance index applied, the design loop gain is calculated. Table 4.5 shows the loop gain as a function of number of samplings used for various values of performance index and sampling rates for the control system. Increase in performance index decreases s t a b i l i t y as measured by the value of the loop gain, — small loop gain values implies more s t a b i l i t y — . A l l the trends observed in the case of the control system with zero-order are repeated here. F i g . 4.24 is a typical response [equation (4.96)] of the uncompensated system as a function of sampling i n t e r v a l . The proportional controlled system transient response is shown in F i g . 4.25 as a function of the sampling interval with performance index as a parameter. Increase In performance index reduces steady state error and also decreases the s t a b i l i t y margin of the control system. Common to both control systems (system with zero-order hold and system with half-order hold) is the increase in loop gain and hence less s t a b i l i t y as the value of the performance index is increased. This is also true for increase in sampling time. Also the error, — d e v i a t i o n from the desired steady state value of 1 — , decreases with increased value of performance index but increases with increased number of samplings used and sampling time- In a l l the conditions tested, the control system with half-order hold gave better transient response and is more stable than the system with zero-order hold. An interesting feature observed from the analysis of the transient response of the two control systems is their behaviour with different performance index values. The best transient response and hence minimum error response occurs at a performance index of 1.5 ± 0.25 for the control system with 81 zero-order hold, while for the system with half-order hold, the best response occurs at <j> = 3.0 ± 0.25. Also the system with half-order hold attains steady state conditions faster than that of the zero-order hold system. 4.3 Experimental Equipment The designs suggested above were tested experimentally using the equipment shown schematically in F i g . 4.26. The system consists of a heating tank of about 0.08327m (22 gals.) capacity connected through 1.9cm (3/4 inch) pipe of length 0.762m (2 1/2 feet) to a U-tube shel l and tube exchanger. The heat exchanger shel l is 0.914m (3 feet) long and 20.32cm ( 8 inches) in diameter and is made of 6 inch schedule 40 iron pipe. There are 18 — 1.27cm (1/2 inch) outside diameter copper tubes of length 76.2cm (30 inches) in the tube compartment of the heat exchanger (see appendix 13 for heat exchange diagram). The heat exchanger is connected as a feedback loop to the heating tank through a 1.27cm (1/2 inch) copper pipe and a recirculating pump. _ Also on this feedback loop is a by pass that is controlled manually through a gate-valve. The heating tank (drum) has a copper heating c o i l through which steam from the main l ine in the laboratory is used to heat the water in the tank. The steam flow rate is controlled by a gate valve that is manually controlled. Five copper-constantan 'ungrounded' thermocouples are placed as shown in the diagram. Water is heated in the drum by steam and flows through the connecting pipe, where one of the thermocouples is located, into the shel l side of the heat exchanger. It is assumed that the water temperature at this connecting pipe is the same as that in the tank. The hot water in the shel l is used to heat the coolant water in the tubes. The outlet shell water is returned to the tank through the recirculating pump and an 'MK 315' L E G E N D AID A / D A / D - O r Cold Later — x > a — I Steam gate valve pleating c o i l (£" copper c o i l ) LL) O r IM1 D/A Pipe l i n e Cable l i n e s Flow sensor 'ungrounded' Thermocouple A n a l o g - D i g i t a l converter D i g i t a l - A n a l o g Converter Heat tank r~ HEAT EXCHANGER 8 Bypass va.lve A/D A / D | I I HP Pump ' (Constant f l o w r a t e at 31.13 f t 3 / s ) D / A A / D A / D DIGITAL COMPUTER I F i g . 4.26 - Schematic Diagram of Equipment 83 paddle wheel flow sensor which measures the flow rate and transmits a flow s i g n a l to an 'MK 314' s i g n a l c onditioner. This conditioner converts the s i g n a l to voltages and transmits i t to the d i g i t a l computer through a 'Miniac' analog computer f o r voltage scale down. The PDP8 d i g i t a l computer reads t h i s voltage i n machine units which i n turn i s converted to flow rate values by a co n t r o l program l o g i c . Another 'MK315' paddle wheel flow sensor i s placed at the tube water i n l e t p o s i t i o n . This sensor measures the flow rate of water which flows through the co n t r o l valve. The thermocouples transmit temperature readings i n voltage to f i v e (one for each thermocouple) 'Model 199 Omega d i g i t a l temperature i n d i c a t o r s ' that are mounted on a v e r t i c a l panel to enable a v i s u a l inspection of the temperature p r o f i l e i n the c o n t r o l system. Voltages proportional to temperature are sent from the temperature i n d i c a t o r to the d i g i t a l computer (PDP8) through the Miniac analog computer. In the analog computer, the voltages are magnified ten times to reduce the error i n the A/D converter. The c o n t r o l system consists of a PDP8 d i g i t a l computer which samples the i n l e t - o u t l e t water temperatures, and water flow rates and manipulates the control valve to obtain the desired o u t l e t water temperature. The computer i s in t e r f a c e d to the co n t r o l valve through an operational a m p l i f i e r . The voltage s i g n a l i s 'power amplified' to 24 vo l t s and sent out i n square wave form to a power-current converter, t h i s then transmits the current s i g n a l to a current to a i r pressure converter which then drives an a i r - t o - c l o s e 'Foxboro' c o n t r o l valve positioned at the water i n l e t tube of the heat exchanger. A l l through t h i s study, i t has been assumed that the dynamics of the co n t r o l valve, thermocouples and flow sensors are n e g l i g i b l e compared to the process dynamics. 84 4.4 System Identification and I n i t i a l i z a t i o n 4.4.1 Identif icat ion By Graphical Methods The control system as described above was used in the ident i f i ca t ion and i n i t i a l i z a t i o n process. In this stage of the study the air that controls the valve was cut off making the loop an open one. Under this condition, water was allowed to flow through the tube and out to the drain continuously, while the heating tank was f i l l e d and the recirculating pump was used to circulate the water from the drum through the heat exchanger shell and back to the heating tank. This situation was allowed to continue u n t i l steady state in temperature as observed from the d i g i t a l temperature indicators was attained. Then a 10% increase in steam pressure, manually set by turning the steam valve on the mainline was effected. A sampling time of one second was used to datalog the temperature profile of the outlet tube water. Due to the excessive noise in the system, the temperature response was f i l t e r e d . This was done by a program which averaged the temperature from f i f teen measurements taken at equal times calculated for each sampling rate. In this case fifteen measurements were averaged in one second, the average was f i l t e r e d by multiplying i t by a weighting factor and added to a weighted value of the previous f i l t e r e d response. The relationship used in this algorithm (temperature response datalog) is given as T ( J ) = c x ^ C J ) + ( l - a f ) T ( J - l ) (4.97) where T ( J ) is the jth f i l tered response; T^(J) is the averaged temperature, T ( J - l ) is the previous ( J - l ) t h f i l tered response a.f is the weighting factor. , --f— _ , . , ... 100 200 3 0 0 Time Sec- Fig. 4. 27 - Process Reaction Curve. 86 The used in this work was 0.4. Both the number of samplings summed up and averaged and the weighting factor were determined by t r i a l and error, comparing the printed responses with that observed on the d i g i t a l temperature indicator . The process reaction curve is shown in F i g . 4.27. Since there Is no prior knowledge of the control system dynamics and hence transfer function, an approximate transfer function was obtained by the method of Strejc. The control system approximate transfer function was determined to be -0.8734s G (S) = * v- - (4.98) P (4.2s+l) This was then modified to a second-order system with a transfer function of G (s) = e" 7 * 4 s / (6 .8s+l ) 2 (4.99) P See Appendix 5 for details 4.4.2 Quasilinearization Method This method has been known to give better parameter values than graphical method 1 6 . A better approximation of the second-order transfer function parameters was calculated by quasil inearization method. The basic assumptions necessary for the formulation of the ident i f i ca t ion algorithm used in this study are constant dead time (or negligible variation in i t ) , constant values for sampling time, f i l t e r i n g time and weighting factor for f i l t e r i n g the measured temperature response. The quasilinearization method (Eveleigh, V . W . 1 6 ) ident i f ies x̂  in the second-order overdamped plus dead time transfer function by solving for 87 s u c c e s s i v e s o l u t i o n s of the t r a n s f e r f u n c t i o n l i n e a r i z e d w i t h r e s p e c t t o v a r i a t i o n s i n the unknown p a r a m e t e r s . The above a l g o r i t h m was used w i t h R u n g e - k u t t a 4 t h o r d e r f o r m u l a t o e s t i m a t e the time c o n s t a n t o f the p r o c e s s ( s e e a p p e n d i x 6 f o r d e t a i l s ) and i t was found t o be 5.0271. The same dead time as d e t e r m i n e d by t h e g r a p h i c a l method was used a g a i n s i n c e the l i n e a r i z a t i o n method employed h e r e r e q u i r e s the c o m p u t a t i o n o f the d e r i v a t i v e | T ( t - T ) = " ( t - T ) The p r o c e s s r e a c t i o n r e s p o n s e used i n t h i s d e t e r m i n a t i o n was g e n e r a t e d by a s t e p i n p u t w h i c h does not y i e l d s u f f i c i e n t i n f o r m a t i o n t o c a l c u l a t e t h e d e l a y t i m e . 4.5 E x p e r i m e n t a l R e s u l t The s u g g e s t e d d e s i g n , u s i n g the new p e r f o r m a n c e i n d e x - d e f i n i t i o n , was t e s t e d . A 50% p r o p o r t i o n a l band about the s e t p o i n t was imposed on t h e c o n t r o l l e r . A p r o p o r t i o n a l c o n t r o l a l g o r i t h m f o r the h a l f - o r d e r h o l d c i r c u i t was programmed i n t o the PDP8 d i g i t a l computer. The s w i t c h e s f o r the c i r c u l a t i n g pump, c o n t r o l v a l v e and d i g i t a l t e m p e r a t u r e i n d i c a t o r s were s e t on. The c o l d w a t e r from the tap was a l l o w e d to f l o w t h r o u g h the v a l v e and i n t o the heat e x c h a n g e r t u b e . The whole system was l e f t at t h i s c o n d i t i o n f o r about f i v e m i n utes i n o r d e r to a t t a i n s t e a d y s t a t e . A s t e p change i n the l o a d v a r i a b l e (steam p r e s s u r e ) was m a n u a l l y imposed- on the h e a t i n g drum-heat ex c h a n g e r c o n t r o l s y s tem. Due to e x c e s s i v e n o i s e p r e s e n t i n the s y s t e m , the s i n g l e - e x p o n e n t i a l f i l t e r i n g e q u a t i o n was a g a i n used to smoothen the measured o u t l e t t e m p e r a t u r e r e s p o n s e . The s i n g l e - e x p o n e n t i a l f i l t e r i n g e q u a t i o n i s g i v e n as 1 2 3 4 5 6 7 Time M ins- Fig. 4.28a - Experimental closed-loop transient response of proportionally controlled sampled-data system for a 2% step change in loan (half-order hold) . Fig. 4.28b - Manipulated variable response of proportionally controlled sampled-data system for a 2% step change in loan (half-order hold) . — i 1 3 — 4 5 — r ~ — f Time Min- Fig. 4.29a - Experimental closed-loop transient response of proportionally controlled sampled-data system for a 3°C step change in set point (half-order hold) . i 5 5 4~ 5 £ T 7 Time Min- Fig. A.29b - Manipulated variable response of proportionally controlled closed-loop sampled-data system for a, 3°C step change in set point (half-order hold). 92 DP(J) = ot fDl(J) + ( l - a f ) DP(J-l) (4.100) where DP(J) is the smoothened temperature at instant J , DP(J-l) is the previous smoothened temperature and D1(J) i s the average actual tempera- ture after f i f teen sampling times. The is the f i l t e r factor and i s equal to 0.4. The control algorithm is written in such a way that the valve i s only activated or moved after f i f t e e n sampling measurements. For the control system with half-order hold the actual temperature printout i s calculated from the relation DK(J) = DP(J-l) + 0.5[DP(J-1) + DP(J-2)]*(t-T)/T (4.101) where DK(J) is the calculated output response at instant J , DP(J-l) and DP(J-2) are the actual smoothened output temperature for the previous and penultimate periods response respectively. A half-order hold uses the two, previous responses to determine the new response. F i g . 4.28a,b and4 .29a,b are the transient responses and manipulated variable responses respectively for the control system with half-order hold for i two different values of performance index. F igs . 4.30a,b and 4.31a,b are the same conditions for the control system with zero-order hold. These results confirm what has been shown theoretically to be true that the half-order hold c i rcui t always results in better responses than that of zero-order hold. The c r i te r ion used to arrive at this conclusion is the less osci l la tory nature of the temperature and manipulated variable responses of the half-order hold control 1system than that of zero-order hold control system. Hence the p o s s i b i l i t y of the half-order hold control system exceeding threshold s t a b i l i t y condition is greatly minimised. Also the system with h a l f o r d e r hold attained steady state conditions faster than those of zero~order hold. Fig. A.30a - Experimental closed-loop transient response of proportionally controlled sampled-data system for a 2% step change in load (zero-order hold). 1 3 Time 4 Min. Fig. 4.30b - Manipulated variable transient response of proportionally controlled closed-lodp sampled-data system f o r a 2% step change in load (zero-order hold). 1 Time 4 Min. Fig. 4.31a - Experimental closed-loop transient response of a proportionally controlled sampled-data system for a 3°C step change in set point (zero-order hold) . 2 " 3 4 § & f Time Min. F i g . 4.31b - Manipulated variable response of proportionally controlled closed-loop sampled-data system for a 3°C step change in set point (xero-order hold). 9 7 CHAPTER 5 DIGITAL COMPENSATION DESIGN Discrete control algorithms, suitable for programming in a direct d i g i t a l control computer are now derived. Three compensation design algorithms viz: deadbeat performance or minimal prototype design (Bergen and Ragazzini); 4 improved proportional controller (Moore et a l ) 4 * 4 and optimum feedback control (Tou, J . T . ) 5 6 are formulated and experimentally verified. 5.1 Deadbeat (Minimal Prototype) Performance Design Many special purpose algorithms, both continuous and discrete, for lumped parameter systems have been published in the control l i t e r a - ture.Most of these works have dealt with f i r s t order plus dead time model systems. In an earlier paper, Mosler et a l 4 6 reported on minimal 6 0a prototype algorithms for this type of system. Shunta and Luyben gave minimal prototype and minimum squared error designs for a process with inverse response behaviour. Also Luyben, W.L.363 presented damping coefficient design charts for sampled data control of a first-order process with dead time. Several workers, Gupta, S.C. and C.W. Ross; 2 0 3 Hartwigsen, C.C. et a l ; 2 0 b Morley, R.A. and CM. C u n d e l l 4 4 3 and 6 3a Thompson,A have reported the use of- discrete versions of conventional control algorithms. The performance of these -systems under computer control is of course limited to that which is obtainable from their continuous-data analogs. Although the responses of these special purpose algorithms are excellent for the specific tasks for which they 98 are designed, their performance often deteriorates under undesigned load condition or parameter shifts. In this study a generalized, — single algorithm that can apply rJo setpoint and load changes — , direct d i g i t a l control algorithm Is designed for a second-order overdamped process with dead time using either a zero-order hold or half-order hold as smoothing device. A direct d i g i t a l control computer is normally used to control a number of process loops on a time-shared basis. In this study a typical loop is considered, and other loops in the overall control system can be treated in a similar manner. At the end of each sampling period for this particular loop, the computer samples the output of the loop and compares i t with the desired setpoint value to form a value for the error. The computer then calculates a new value for the manipulated variable. The manipulated variable of the loop is held constant at the value calculated by. the computer u n t i l the loop is sampled again. The computer memory is used to store sequentially past values of the error and manipulated variable. Note that only a small number of the most recent values, as defined by the algorithm, are retained in the computer. The control algorithm u t i l i z e s a linear combination of the past history of the system in forming a new value for the manipulated va r i - able. The absolute position u(t) of the f i n a l control element is deter- mined from the formula k P u(nT) = I g e [(n-i)T] - J h u[(n-j)T] (5.1) i=0 j=l 3 Equation (5.1) gives the value at which u(t) is to be held constant during the entire (n+l)st sampling period, that i s , u(t) = u(nT) for 99 nT < t < (n+l)T. T i s the sampling time and the g's and h's i n equation (5.1) are a l l constants. In this algorithm only the (K+l) most recent values of the error and the p most recent values of the manipulated va r i a b l e need be stored. The design objective i s to determine s u i t a b l e va lues of {g i}, and {h.}. The deadbeat performance index design by the method of t r a n s i t i o n state matrix i s used. The requirements of the deadbeat performance c r i t e r i o n for the control system are: ( i ) The compensation algorithm must be p h y s i c a l l y r e a l i z a b l e , which implies that the order of the numerator should be l e s s than or equal to that of the denominator, ( i i ) The output of the system should have zero steady state e r r o r at the sampling points, ( i i i ) The f i n a l output should equal the input i n a minimum number of sampling periods. However, for app l i c a t i o n s of d i g i t a l compensation to r e a l . systems-, several a d d i t i o n a l constraints are included: ( i v ) The d i g i t a l compensation algorithm should be open-loop sta b l e . (v) Unstable or nearly unstable pole-zero cancellations should be avoided, since exact c a n c e l l a t i o n i n r e a l processes i s impossible, and the r e s u l t i n g closed-loop system may be unstable or excessively o s c i l l a t o r y , ( v i ) The design should consider the entire response of the system to eliminate hidden o s c i l l a t i o n s (intersampling r i p p l e ) . 100 ( v i i ) In addition to the system responding optimally to a given test input, i t should perform satisfactorily for other possible inputs and disturbances. These extra constraints are required inorder to ensure that the proposed compensation algorithms perform satisfactorily on real systems. To meet these requirements, the resulting control system may respond with a settling time longer than the deadbeat performance settling time. However, the idea of f i n i t e settling time is used only «' as a theoretical performance criterion. In real systems, as with the case in this study, where, modeling error, noise, and momentary disturb- ances are present, i t is not possible to bring the state of the system completely to rest. This does not negate the value of the theoretical concept of f i n i t e settling time, because systems designed to meet this theoretical requirement give satisfactory performance in real tests as is observed in this study. 5.1.1 Development of Algorithm The compensator design procedure is known as the variable-gain approach due to Tou, J.T. 6 5 The basic principle underlying this approach i s the assumption that the desired d i g i t a l controller can be treated as a variable-gain element Kn, which w i l l have different values during different sampling periods. The input to the variable-gain element Kn i s the control signal u, and the output is assumed to be uj. At any instant t = nT*",' the input and output of the variable-gain element are related through a constant multiplying factor Kv, that is u (nT+) = Kv u(nT+) (5.2) 101 where Kv is the gain constant of the variable-gain element during the (n+l)st sampling period. See Appendix 7 for theory. Let the deadtime x be any multiple of the sampling time. The design of the required d i g i t a l compensation w i l l depend only on the response of the system at instants of sampling plus deadtime. Therefore, i t w i l l be necessary to ver ify that the system does have satisfactory intersampling behaviour. At least two methods exist for determining the presence of hidden osc i l la t ions (that i s , intersample r i p p l e ) . One c l a s s i c a l tech- nique is to analyze the system by the modified Z-transform after the compensator D(Z) has been designed. The entire response can then be v e r i f i e d to have f i n i t e set t l ing time. A second method is to determine the corresponding response of the manipulated variable M(Z). For l inear , time-invariant, overdamped processes, i f the response of the closed-loop system has zero steady state error at sampling plus deadtime instants, and i f the manipulated variable also has a f i n i t e se t t l ing time, then i t is assured that no hidden osc i l la t ions exist because the system is receiving constant input. It is not necessary to use the modified Z-transform on the manipulated variable, because i t is a piece- wise constant signal and i t s values at sampling plus delay time instants completely describe i ts reponse. If the same system responds with f i n i t e set t l ing time, but the manipulated variable continues to o s c i l - late , the response of the system must obviously have an intersampling r i p p l e . 5.1.2 Compensator Design for System with Zero-Order Hold The f i r s t step towards obtaining a set of f i rs t -order d i f feren- t i a l equations to describe the dynamics of the system and hence the r + Or Hold u -8. ->' o Fig. 5.1 - State-variable diagram of control system with zero-order hold. 103 state variable formulation is to draw a block diagram for the control system. This diagram is made up of integrators and constants. Consider the control system of Fig. 4.1 and remove the controller, the overall transfer function becomes -ts G(s) = (s + 8 1)(s + e2) ( L j l 4 — > (5.3) The state-variable diagram of Equation (5.3) is shown in Fig. 5.1 for a unit step change. The dotted line represents the future position of the i compensator. The state vector V is defined as V = r X l X 2 u (5.4a) and the i n i t i a l state vectors are V(A) 1 0. 0 0 (5.4b) while after the step change the state vectors become V(A +) 1 0 0~ 1 (5.4c) where A is the deadtime. From Fig. 5.1 the first-order differential equations are: dV — = AV dt (5.5) 104 where A = 0 0 0 0 0 -Qi 1 e 0 0 -6 2 9 0 0 0 0 and the state transition difference equations are where B V(n + AT""") = BV(n + AT) (5.6) 1 0 0 0 0 1 0 0 0 0 1 0 1 - 1 0 0 The solution to the differential equations by state transition matrix method is V(t) = 4(A) V(A+) (5.7) where X = t - (n + A)T Note that 4>(X) is the overall transition matrix and is given as $(X) = L~ 1[SI-A]~ 1 Thus, I- -1 0 0 0 - e x - e x -e x - e x - e x 0 e b'(e -e ) b^+b^e 4-b̂ e 4>(X) = 0 0 0 e 0 - v 0^1 - e ) 1 (5.8) 105 See Appendix 8 for details of derivation and parameter definition. If i t is assumed that the d i g i t a l compensator is a variable-gain element K , which implies that the value of K varies from one period to n n another, and let this compensator be introduced into the control loop as shown in Fig. 5.1; then at any instant t = (n + A)T +, the input and output of the variable-gain element are related through a constant multiplying factor K, that i s , ui[(n + A)T +] = Knu[(n + AT +]. For the condition t = n + j + l T , the X in the transition matrix becomes X = (n + j + 1)T - (n + j + 6)T = (1 - 6)T = V. (where j is the integral multiple of sampling time part of the process delay). Hence the transition matrix is given as (V) = -0 V -0 V -0 V 1 1 2 0 e b^(e -e ) 0 e -02V -0 V -0 V [bl+ble 1 +b!e 2 ]K L 5 4 n -0 V i l ( l - e )K n (5.9) Thus, equation (5.7) can be written as V[(n + j + 1)T] = <j>n(V)BV[(n + j)T] (5.10) 106 From equation (5.10) when n=0 V(l+jT) = 4>o(V)BV(J) - -e v -e v [b' + b'e + b'e ] K - 0 2 v 0 l ( l - e ) K Q (5.11) and when n=l V[(2 + j)T] = * 1(V)BV(l+jT) = - 0 . V - 0 V - 0 V -e v - 0 v - 0 v {e 1 (bj+bje 1 +bje 2 ) K q + b ^ l - e 2 )(e 1 -e 2 ) K Q + -9 7 -6 7 - 0 7 - 0 7 - 9 7 - 0 7 K l ( b 2 + b 3 e + b 4 e )tl-(b 2 ,+b^e +bje 1 ^ j l ^ e 2 (1-e 2 ) K £ - 0 7 -6 7 - 9 V - 9 7 - 0 7 { 0 i e (1-e * ) K Q + 6 1 ( l - e * ) [ l - ( b ^ e +bje 1 ) K Q ] K } . - 0 7 - 0 7 1 - (b^+b'e +b£e ) K Q (5.12) Since the process has been assumed to be a second-order system, the following condition must be sat isf ied for dead beat performance, i . e . system responds to a stepwise input in the quickest manner without overshoot. - 0 7 - 0 7 - 0 V - 9 V -6 7 -6 V X 1 (2+jT) = e (b^+b^e +b£e ) K q + b j e ^ l - e )(e -e ) K Q - 0 7 -6 7 -6 7 -6 7 +(b'+b'e +b'e )[l-(b'+b'e +b'e ) K ] K - 6 2 (5.13) 107 -0 V -0 7 -0 7 -0 7 -0 7 X 2(2 + jT) = 6 i e L (1-e Z )K o +9 1 (l-e Z ) [ l - (b^b'e +b£e Z ) K 0 1 K 1 = 6 1 (5.14) Equations (5.13) and (5.14) are solved simultaneously for K q and -0 7 -0 7 -0 7 [8 (1-e ) - (bl+ble +b'e )] K = - — (5.15) ° -0«V -8 7 -8 7 -0 7 -0 7 -8 7 (1-e )(e -e ) [ (b^+b»e +b£e )+b{8 (1-e )] -8 7 -0 7 [1-e (1-e ) K ] K = (5.16) -8 7 -0 7 -0 7 (1-e * )[l-(b'+b'e +b£e )K } For simplicity in analysis; i t is assumed that a l l the delay effects in the control system are encountered in the compensator such that the output from i t , i s a delayed s ignal . Thus, instead of having the output signal be ui(nT), an output signal of ui(n+jT) is derived. The relationship between the input and output signals to and from the compensator is ui[(n+j)] = K n u(nT) (5.17) Also required for deadbeat performance is that the output from the variable-gain element after the second sampling plus deadtime instant should be held constant at 1/0^ Equation (5.11) gives u(0+) = 1, thus ui(jT+) = K 1 0 8 From Equation ( 5 . 1 2 ) , + - V -e2v u(T ) = 1 - (bl+ble +b.'e ) K = X, ( 5 . 1 8 ) i. J 4 O 1 and U l [ ( l + j ) T + ] = K l U ( T + ) = K 1 X 1 ( 5 . 1 9 ) Thus, the Z-transform of the output sequence from the d i g i t a l compensator ( v a r i a b l e - g a i n element K ) may be expressed as n u x ( Z ) = Z ~ J [ K Q + K 1 X 1 Z ~ 1 + X 2Z~ 2 + X 2Z~ 3 + ] ( 5 . 2 0 ) which reduces to Z ' ^ K ' + ( K . X . - K ) Z _ 1 + (X. - K.X.)Z~ 2] /r,\ 1 o 1 1 o 2 1 1' ] ,_ u.(Z) = : — ( 5 . 2 1 ) ( 1 - Z l ) But the Z-transform of the input s i g n a l to K i s n ; u(Z) = 1 + X j Z - 1 ( 5 . 2 2 ) Thus, the pulse transfer function of the desired d i g i t a l c o n t r o l l e r i s given by u.(Z) Z _ J [ K + ( K X - K ) Z _ 1 + (X -K.X. )Z" 2] M(Z) D(Z) = — 1 1 0 _ 1 1 = ( 5 . 2 3 ) u(Z) ( 1 + \Z~ ) ( 1 - Z ) E(Z) Equation ( 5 . 2 3 ) i s a generalised compensator algorithm for the c o n t r o l system with zero-order hold i r r e s p e c t i v e of the value of the deadtime. The three important cases are as follows: Case I: No dead time (x = 0 ) In t h i s case j and 6 are zero, thus V = T. 109 TRANSIENT RESPONSE OF COMPENSATED SYSTEM: Therefore, the compensator tra n s f e r function becomes D(Z) = [ K q + Y z" 1 + Y 2Z* 2] [1 + Y 3 Z - 1 - X ^ " 2 ] (5.24) where Yj_ = K X - K Q ; Y 2 = * 2 - K 1X 1; Y3 = ^ - 1 A schematic diagram of the system c o n t r o l l e d by the d i g i t a l computer i s shown i n F i g . 5.2 At each sampling instant, the d i g i t a l c o n t r o l l e r samples the error s i g n a l e ( t ) . The c o n t r o l l e r operates on t h i s sampled value e*(t) and the previous sampled values to obtain an output m*(t). This value of m*(t) i s then retained u n t i l a new value i s computed at the next sampling i n s t a n t . The s i g n a l flow diagram of the con t r o l system i s shown i n F i g . 5.3. The state d i f f e r e n t i a l equations i n matrix form are given as: X 2 ( s ) X 3(K+1) X 4 ( K+1) * i i ( s ) < t > i 2 ( s ) * i 3 ( 8 ) 0 4 > 2 1 ( s ) < t > 2 2 ( s ) < J > 2 3 ( s ) 0 -K„ -K, h Y3 X 1 ( K T ) X 2(KT) + X 3(KT) K2 X 4(KT) K3 _ _ _ _ r(KT) where K = Y. " K T ; K - y + K Y, - Y 0K„. (See Appendix 8 for I. 1 O J J / o l J z parameter d e f i n i t i o n s ) 110 Fig. 5.2 - Schematic block diagram of control system with d i g i t a l controller. r(t) e(kj Fig. 5.3 - Signal flow graph of control system with d i g i t a l controller (zero-order hold). I l l F i g . 5 . 4 - Open loop t r a n s i e n t of uncompensated con t r o l system with zero-order hold. 112 Fig. 5.5 - Open loop transient response of compensated control system zero-order hold. 113 The transient response of the d i g i t a l l y compensated second-order overdamped process with zero-order hold and no dead time is derived from, the solution of Equation (5.25), noting that the output C(nT) is equal to Xi (nT) . F igs . 5.4 and 5.5 are the transient responses of both the uncompensated and compensated system respectively.Case I I : Dead time x = mT, where 0 < m < 1. This condition results in j = 0 and V = V; and K + (K.X - K ) Z _ 1 + (X0 - K. X. ) Z ~ 2 D(Z) = -2 1 2 ^ 1 (5.26) ( i - z A ) ( i .+ x 1 z " i ) Case I I I : Dead time x = (j + m)T where 0 < m < 1; and V = V. This results in Z~ J [K + (K.X. - K ) Z - 1 + (X_ - K.X . )Z~ 2 ] D ( Z ) = —!-2 L J _ o 2 ( 5 . 2 7 ) ( i - z x ) ( i . + x x z L ) 5.1.3 Compensator Design for System with Half-Order Hold Consider the control system of F i g . 4.1 and remove the controller , the overall transfer function becomes: A + i _ p Ts o p " T S G<s> " (VT^"—I—> (s * v<s + e2) <5-28> The state-variable diagram of Equation (5.28) is as shown in F i g . 5.6 for a unit step change. The dotted l ine represent the future position L _ _ J t > - W - ^ H > - ¥ H > 9 i M Fig. 5.6 - State-variable diagram by cascade programming method. i of the compensator. r X, V = 115 The state vector V i s defined as (5.29a) and the i n i t i a l state vectors are V(A) = 1 0 0 0 0 (5.29b) while a f t e r the step change the state vectors become V(A+) 1 0 0 0 1 (5.29c) From F i g . 5.6 the f i r s t - o r d e r d i f f e r e n t i a l equations are dV dt = AV (5.30) 116 where A = 0 0 0 -6- 0 0 0 0 0 0 0 0 0 0 -1/5T (5/4)8 >2 -1/5T (5/4)9 -1/T (5/4)9 0 and the s t a t e t r a n s i t i o n e q u a t i o n s a r e (5.31) V[(n + A)+] = BV[(h + A)] where B. = 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 -1 0 0 0 and A i s the t o t a l p r o c e s s dead t i m e . As has been s t a t e d e a r l i e r on, the s o l u t i o n to the d i f f e r e n t i a l e q u a t i o n s (5.30) by s t a t e t r a n s i t i o n m a t r i x method i s V ( t ) =-<{>( X) V ( A + ) ( 5 . 7 ) . 117 Thus, 1 0 0 e 4>(X) = 0 0 0 0 1 - e x - e x <*{(e -e ) * 2 4(X) * 2 5 ( A ) - e 2 x -aX - e x -«'(e -e ) ^ ( X ) -aX ^ 2 ( l - e " a X ) 0 0 0 0 1 (5.32) where a = 1/T; a n =.1/5T; a 2 i = (5/4)6. See Appendix 9 for parameter definitions. Applying the same procedure used in the zero-order hold case gives (V) = 1 0 0 0 e a^(e - e ) 0 * 2 4 ( V ) 0 *25 k; 0 0 -e v 2 _ av _e v -a'(e -e 2 * K' 6 ) 35 n 0 0 0 0 0 0 a' Cl-e ")K' 22> 6 ; n 118 Thus, equation (5.7) can be written as V[(n + 1 + j)T] = <j> (V)BV[(n + j)T] (5.10) From equation (5.10), when n = 0 V[(l + j)T] = <|)o(V)BV(J) = 25 o 35 o a 1 (1 22v - a v e )K' o (5.34) when n = 1 'V'." V[(2 + j)T] = * 1(V)BV[(1 + j)T] -8 7 -0 V -9 7 1 - av ^ 2 5 K ' o e + a i ( e " e ) * 3 5 K ; - * 2 4 a 2 2 ( 1 - e >Ko +*25 ( 1-*25Ko ) Ki> -0 7 -aV. - a7 -9 7 e *35 K;- a6 a22 ( 1- e ) ( e " e > Ko^35 ( 1-*25 Ko ) K i _a7 - a7 - a7 a 22 e ( 1 " 6 ) K ; + a 2 2 ( 1 - e ) ( 1 - * 2 5 K o ) K i 1 " +7.K 25 o (5.35) 119 Since the process i s a second-order system, the following conditions must be s a t i s f i e d for dead-beat performance, i . e . system responds to a stepwise input i n the quickest manner without overshoot. -6 V -0 7 6 7 -a7 X1[(2+J)T] = » 2 5K;e + a i * 3 5 ( e -e ^ - ^ a ' . a - e )KM- * 2 5 ( 1 - * 2 5 K o ) K i = 1 ( 5 ' 3 6 ) -9 7 -97 9 7 - aV X1[(2+j)T] = • 3 5K'e + a 6 a 2 2 ( e " e ) ( 1 " e ) K Q + • 3 5 ( 1 - * 2 5 K o ) K l = ° ( 5 ' 3 7 ) Equations (5.36) and (5.37) are solved simultaneously for K' and K'. ^ o i . -9 7 -9 7 -a7 -0 7 K; = * 3 5 / [ ( e -e ) ( * 2 5 * 3 5 + a i * 3 5 ) + * 2 5 a 6 a 2 2 ( 1 - e ><e > " -9 7 • <t»24<f)35a22(1"e ) ] (5.38) K i - [ a 6 a 2 2 k ; ( e ~ a V - e " e V> " ̂ o 6 ' 9 71 /*35 ( 1-*25 K; ) ( 5 ' 3 9 ) For s i m p l i c i t y i n analysis, i t i s being assumed that a l l the delay effects i n the control system are concentrated i n the compensator such that the output from i t i s a delayed s i g n a l . Thus, instead of having the output signal be uj(nT), an output signal of ui[(n+j)T] i s derived. The relationship between the input and output signals to and from the compensator i s given as u i t ( n + j)T] = r u(nT) (5.40) 120 Equation (5.34) gives u(0 +) = 1, thus u i ( j T + ) = K1 o From Equation (5.35), u(T+) = 1 - <f.25K^ = ^ (5.41) and U l [ ( l + j ) T + ] = K[u(T +) = K| (l-<t>25K^) = K j f ^ (5.42) —aV - a V -aV X 3(2+j)T) = e a a 2 2 ( l - e a )K^ + <* 2 2(l-e a ) ( l - * ^ ) ^ = B 2 (5.43) It should be noted that deadbeat performance requires zero input to the t h i r d integrator for t > (2+j)T. To s a t i s f y this requirement on the third integrator, the output of the variable-gain element K' must be n maintained at 0 2 after the second sampling plus deadtime period. Thus the Z-transform of the output sequence from the d i g i t a l compensator (variable-gain element K') may be expressed as n u T(Z) = Z j [ K ^ + K ^ Z 1 + 32Z 2 + 32Z 3 + ,...] (5.44) which reduces to _ _ _ [K» + (K g -K )Z + (3 -K'3 )Z~ 2] u. (Z) = Z"J -2 Li_^ _ ?_J_1 ( 5 > 4 5 ) (1 - z l ) But the Z-transform of the input signal to K' i s n u(Z) = 1 + B 1Z~ 1 , (5.46) Thus, the pulse transfer function of the desired d i g i t a l c o n t r o l l e r i s given by M z > -n tKA + ( K 1 B 1 " K A ) Z + (B 9-K'B )Z 2 ] D ( Z ) = n T 7 ^ = Z — — I i _ l (5.47) 121 Equation (5.47) i s the generalised compensator algorithm i r r e s p e c t i v e of the value of the deadtime. The three prominent cases are as follows: Case I: No dead time (T = 0) In this case j and 6 are zero, thus V = T TRANSIENT RESPONSE OF COMPENSATED SYSTEM: Therefore the compensator transfer function becomes [K' + g.Z - 1 + B,Z~ 2] D(Z) = — ^ (5.48) [1 + 6 5Z - BjZ z ] where 3„ = K.'B. - K'; g. = 6 - K'B1 and g = 3 - 1 3 1 1 o 4 2 1 1 5 1 A schematic diagram of the system c o n t r o l l e d by the d i g i t a l computer i s shown i n F i g . 5.7. At each sampling Instant, the d i g i t a l c o n t r o l l e r samples the error sign a l e ( t ) . The c o n t r o l l e r operates on this sampled e*(t) and the previous sampled values to obtain an output m*(t). This value of m*(t) i s then retained u n t i l a new value i s computed at the next sampling instant. The s i g n a l flow diagram of the con t r o l system i s shown i n F i g . 5.8. The state d i f f e r e n t i a l equations i n matrix form are given as (see Appendix 9 for parameter d e f i n i t i o n ) . 122 r(t) + ^ em, e*ct) m*ft) H o l d m(t) {4+5TS)0 C ( t ) , jr Fig. 5.7 - Schematic block diagram of system with d i g i t a l controller. 123 Sampling Interval, N F i g . 5.9 - Open loop t r a n s i e n t response of uncompensated c o n t r o l system w i t h h a l f - o r d e r h o l d . 124 _ i , t , f — 2 4 6 8 10 Sampling Interval, N F i g . 5.10 - Open l o o p t r a n s i e n t r e s p o n s e of compensated c o n t r o l s y s t e m w i t h h a l f - o r d e r h o l d . 125 X x ( s ) X 2 ( s ) X 3(K+ 1 ) X 4(K+ 1 ) _ _ > u(s) * 1 2 ( s ) * 1 3 ( s ) 0 4» 2 1(s) <(>22(s) <()23(s) 0 -h 1 0 0 -h„ "3: X 1(KT) X 2(KT) X 3(KT) X 4(KT) * 2 ( s ) Y(KT) where h = B „ - K'0 • h = g, + K'g - B h . 1 2 o 5 2 4 o l 5 1 The transient response of the d i g i t a l l y compensated second-order overdamped process with no. deadtime is derived from the solution of Equation (5.49), noting that the output C(nT) is equal to Xi(nT). Figs. 5.9 and 5.10 are the transient responses of the uncompensated and compensated system respectively. Case II: Dead T = XT, where 0 < X < 1. This condition leads to j = 0, and V = V, and D(Z) = [ K 0  + ( K i e r K 0 ) z 1 + (VK{ 3i ) z 2 ] (1- - Z _ 1 ) ( l + B 1Z _ 1) (5.50) Case III: Dead time T = (J+X)T where 0 < X < 1 This results in D(Z) = Z [K^ + (K ' B^K^Z 1 + (0 2-K'B 1)Z 2] (1 - Z _ 1 ) ( l + B^" 1) (5.50) 126 EXPERIMENTAL RESULT: In evaluating any proposed control algorithm, the closed loop system should perform sa t i s fac tor i ly in the presence of modeling error and noise which inevitably occur in real systems. The d i g i t a l compensation of Equations (5.23) and (5.51) for control system with zero-order hold and half-order hold respectively were tested experimentally. Due to the excessive noise present in the system, the output temperature response was averaged after f i f teen measurements and f i l t e r e d using the single-exponential f i l t e r equation as has been explained in Chapter 4. F igs . 5.11a,b; 5.12a,b, 5.13a,b and 5.14a,b are the transient and manipulated variable responses for step changes In the load variable (steam pressure) and set-point respectively for control systems with.zero-order hold and half-order hold. Though, i t was not possible to bring the state of the system completely to rest after two sampling plus dead time periods; this does not negate the value of the theoretical concept of f i n i t e set t l ing time, because systems designed to meet this requirement theoretically as observed in this work, give satisfactory performance in real tests. In the two conditions tested the system with half-order hold gave better responses than those of system with zero-order hold. This is in agreement with what has been 29 suggested in the l i terature since the half-order hold is a better approximation to an ideal f i l t e r than zero-order hold. What looks l ike ripples in the manipulated variable responses of the two systems may be due to the process noise in the control system. Time Min- Fig. 5.11a - Transient response of a di g i t a l l y controlled closed loop sampled-data system with zero-order hold for a 2% step change in load variable (steam pressure) . Fig. 5.11b - Manipulated variable response of a d i g i t a l l y controlled closed-loop sampled-data system with zero-order hold for a 2% step change in load variable (steam pressure). to 1 — 1 — i i A—~5 r i Time Min- F i g . 5..12a - Transient response of a d i g i t a l l y controlled closed-loop sampled-data system with zero-order hold for a 3°C step change in set point. 1 2 3~ 4 5 6~ 7 Time Min. - Manipulated variable response of a d i g i t a l l y controlled closed loop sampled-data with zero-order hold for a 3°C step change in set point. Fig. 5.13a - Transient response of a di g i t a l l y controlled closed-loop sampled-data system with half-order hold for a 2% step change in load variable (steam pressure). Time Min • 5.13b - Manipulated variable response of d i g i t a l l y controlled closed-loop sampled-data system with half-order hold for a 2% step change in load variable (steam pressure). F i g . 5.14a - Transient response of a d i g i t a l l y controlled closed-loop sampled-data system with half-order hold for a 3°C step change in set point. 5.14b - Manipulated variable response of a d i g i t a l l y controlled closed-loop sampled-data system with half-order hold for a 3°C step change in set point. 135 5.2 Application of Combined Optimum Control and Prediction Theory to Direct D i g i t a l Control The dynamics of chemical process, as distinguished from those of mechanical or e l e c t r i c a l systems, are characterized by large time con- stants, distributed parameters, and often time sluggish response. Although the c r i te r ion for optimum control in the chemical industry is generally maximum p r o f i t , a c r i te r ion of minimum steady state error would seem to be almost equivalent and may clearly be more convenient and simplier for analysis . The optimum feedback control law gives the manipulated variable as a function of state output. In the control of many industr ia l processes transport lag has a s ignificant effect on the performance of the control algorithm. Common also, is the fact that the state variables of these systems are not per- fectly known, but instead noisy measurements of a subset of them are available . Furthermore, there is often substantial process noise pre- sent. This part.of the study develops a methodology for the combined optimum control and prediction of a class of these systems using either a zero-hold order or half-order hold as the smoothing device. Combined control and prediction theory is applied to second-order plus dead time approximations of higher order overdamped systems. For example, in a d i s t i l l a t i o n column, the transfer function between feedrate and overhead composition can accurately be represented by this approximation. Also, a heater-heat exchanger system can be approximated by this model. For these systems, the combined control and prediction algorithm may be used as a direct replacement for conventional direct d i g i t a l control . Using combined control and prediction, optimum control of noisy systems can be 136 achieved within r e a l i s t i c operating constraints. This section describes the implementation of combined control and prediction to single input - single output systems which may be approximated with the second-order plus deadtime representation. The dynamic programming me t h o d 6 6 is employed for the derivation of optimum feedback control law. See Appendix 10 for details of theory.5.2.1 Analysis and Design of Control System with Zero-Order Hold The state-variable diagram of the control system for a step change i s given in F i g . 5.15. The existence of transportation lag and process noise makes the measurement of an accurate value of the one state variable which is accessible for direct measurement very questionable. Therefore an analytical predictor is introduced into a feedback loop, such that the predicted state variable values at time t + (0.5 + j + 6)T in the future is used in the minimization process instead of actual values. The time used in the prediction includes, t, the future time; 0.5T which is the time suggested by M u r r i l , P.W. , to represent the dynamic effect of the interface between the discrete and continuous parts of the control system; plus (j + 6)T the process deadtime. The control system overall transfer function is G(s) = (1 - e" T s ) 6e" T S /s(s + 6^(3 + 6^ (5.52) The set of f i rs t -order d i f f e r e n t i a l equations is X' = u (5.53) where X = [C XL X r ] ; D' = [0 0 1 0] 137 and A' = 0 0 0 0 0 0 1 1 0 0 0 0 0 It is assumed here that the effect of the process deadtime is eliminated by using the predicted state-variable values in future time which includes the deadtime. The performance criterion requires that N 2 . J = Z . [C(K) - r(K ) r K=l be minimised. Equation (5.53) i s solved by state transition matrix method to give X(t) = <f>'(T)X(to) + / T <j)'(T,A)DV u(A)d\ o •where § '(T) = L 1[SI - A±] 1 The performance index can be expressed as N Min J = E X'(K) Q'X(K) N K=l (5.54) (5.55) where Q1 1. 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 1 is a weighting factor, positive definite symmetric matrix chosen in such a way as to give more significance to the measurable state variables. 138 u 9r e F i g . 5 . 1 5 - S t a t e - v a r i a b l e d-iagram of c o n t r o l system by i t e r a t i v e programming method. Predictor F i g . 5 .16 - S t a t e - v a r i a b l e diagram of c o n t r o l system w i t h a n a l y t i c a l p r e d i c t o r . 139 The choice of a positive definite matrix guarantees the uniqueness and l inear i ty of the control law and the asymptotic s t a b i l i t y of the control system for a controllable process. The state transition equation in discrete form describing the control process is X[(K + 1)T] = <^(T)X(K) + G'(T)u(K) (5.56) where G'(T) = [g|(T) g^T) g^(T) 0] is estimated from J>J(T, X)DdX, since u(KT) i s assumed to be a piecewise constant input. See Appendix 11 for parameter d e f i n i t i o n . By a dynamic programming method and for a control system with accessible state variables for direct measurement, the optimum law is given as u°(K) = B'X(K) (5.57) where B' = [G*(T)Q'G(T)] _ 1 G ' ( T ) Q ' ( T ) (5.57a) Since X^(K) and X^(K) are not direct ly measurable, the solution above, Equation (5.57) is not complete, and a method for the estimation of these state-variables must be applied. Normally the states can be determined from the values of the direc t ly measurable state but due to the existence of the transport lag and excessive process noise, a predictor is used instead of an estimator. The difference between a predictor and an estimator is that the former predicts future values of the state variables while the estimator uses the past measurements to calculate the values of the state variables . With the addition of the analyt ical predictor in the feedback loop, the state-variable diagram of the control system is shown in F i g . 140 The predictor algorithm must therefore c a l c u l a t e a c o n t r o l a c t i o n at time, t, on the basis of an output X , predicted for time P t + T(0.5 + j + 6). Equation (5.53) i s solved to give the predicted state v a r i a b l e values. The output state v a r i a b l e s predicted at time t + T(0.5 + j + 6) are given as j i-1 X p = (l - A p D'u t + A - ^ l - B p D ' E (Bp u ^ ) + A ^ B ^ t d - C ' p D ' u ^ ^ +CpCt] (5.58) (See Appendix 11 f o r parameter d e f i n i t i o n ) 5.2.2 Analysis and Design of Control System with Half-Order Hold Control of single manipulated input- s i n g l e c o n t r o l l e d output processes i s considered, where dynamics may be represented by a t r a n s f e r f u n c t i o n of the form u(s) (s + 6 ^ ( 3 + 8 2) . ^ ' ^ > A unity process gain with no loss of g e n e r a l i t y i s assumed. C(s) and u(s) are the normalised, transformed process output and input v a r i - ables. The c o n t r o l system o v e r a l l t r a n s f e r function i s approximated by 4 + 5Ts 6 e ~ T S 1 - e ~ T s G<s> = <TT1& (s + 8 ^ ( 8 + e 2 ) 'Hr—* (5'60) The s t a t e - v a r i a b l e diagram of the c o n t r o l system for step change i s given i n F i g . 5.17. Because of the a v a i l a b i l i t y of only one s t a t e - v a r i a b l e for d i r e c t measurement, process deadtime and presence of process noise which cannot be e f f e c t i v e l y determined or eliminated, an a n a l y t i c a l predictor i s Fig. 5.17 - State-variable diagram of control system. 142 introduced into a feedback loop. The predicted state variable values at time t + (0.5 + j + 6)T in the future are used i n the minimization process instead of actual values. The set of f i rs t -order d i f f e r e n t i a l equations is X - AjX + Du (5.61) where X = [C X x X 2 X 3 r ] ; D = [0 0 0 1 0] and - a n 0 A = 0 0 1 0 - & 1 1 0 0 -Q, -a 0 . 11 - a i l 0 0 0 0 - a 2 i 0 0 0 0 0 0 a = (5/4)9; a n = 1/5T; a ^ = 1/T The performance index requires that N • 2 J N = Z [C(K) - r (K) ] Z . K=l be minimised. Equation (5.61) i s solved by a state transit ion matrix method. The performance cr i ter ion can be expressed as N Min J = Z X*(K)QX(K) K=l (5.62) where Q = 1 0 0 0 -1 0 0 " 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 1 143 The state transit ion equation, in discrete form, describing the control process is X(K + 1) = <KT)X(K) + G(T )u(K) (5 .63) where G(T) = [gi(T) g 2 (T) g3(T) gi+(T) 0] is estimated from /«|)(T,X)DdX, since u(K) is assumed to be a piece wise constant input. See Appendix 11 for parameter d e f i n i t i o n . For a control system with accessible state variable for direct measurement and by a dynamic programming method, the optimum feedback control law is given as u°(K) = BX(K) (5.64) where B = [G'(T)QG(T)j" 1 G'(T)Q(KT) ( 5 .64a) Equation (5 .64) is not a complete solution of the optimum control problem, since Xi(K) , X 2(K) and X3(K), the state-variables, are not accessible for direct measurement. An analytical predictor is Introduced into a feedback loop to' estimate the inaccessible state variables. The state-variable diagram of the control system is given i n F i g . 5.18.The predictor algorithm is given as j i - l i X = (l-A)Du +A(1-B1)D E (B u t _ i T )+AB 3 [ ( l -C)Du _T+CXfc] (5.65) (See Appendix 11 for parameter def ini t ion) . IMPLEMENTATION: The implementation of combined optimum feedback control and prediction is as follows: ( i ) Calculation of optimum Gain: The optimum gain is precompu- ted for both control systems; Equations (5.57a) and (5.64a). Equations (5.57a) and (5.64a) show that the optimum gain is a function of the i t 5 T -Q- 2k 50 4 T ft predictor V Fig. 5.18 - State-variable diagram of control system with analytical predictor. 145 weighting matrix Q' and Q. The necessary requirements for the selection of the weighting matrix are that, i t should be positive definite symmetric so that the control law is unique and linear. This condition also guarantees the asymptotic s t a b i l i t y of the control system. the second requirement is that the values of Q1 or Q should be such that more weight is given to a l l directly measurable state variables. ( i i ) Prediction Matrices: The prediction matrices are computed of f - l i n e . The process deadtime should be broken down into its integral and fractional components with respect to the sampling time. With these off-line calculated values, the on-line prediction equation (5.58) or (5.<65), is used to estimate the states. Note that the states are assumed to be i n i t i a l l y at rest. ( i i i ) Control Equation: The optimal feedback control (Equations (5.57) and (5.64) is applied at the present time t = KT and stored as f i r s t element in the manipulative 'u' vector. EXPERIMENTAL RESULTS: the combined optimal and prediction algorithms, of' Equations (5.57) and (5.58), or (5.64) and (5.64) for control system with either a zero-order hold or half-order hold respectively were tested experimentally. Due to the excessive noise present in the system, the output temperature response was averaged after fifteen measurements and filtered using the single-exponential equation as has been described in Chapter 4. Figs. 5.19a,b; 5.20a,b; 5.21a,b and 5.22a,b are the transient and manipulated variable responses for step changes in the load variable (steam pressure) and setpoint respectively for control system with zero-order hold or half-order hold. A compari- son of the two control systems shows that the system with half-order    F i g . 5.20b - Manipulated variable response of optimum controlled sampled-data system with zero-order hold for a 2% step change in load variable . 2 3 4 5 6~ 7 Time Min. 5.21a - Optimum control of closed-loop sampled-data system with half-order hold for a 3°C step change in set point. Time Min- g. 5.21b - Manipulated variable response of optimum controlled sampled-data system with half-order for a 3°C step change in set point. -» 1 :—• 1 1 — +- 4- 1 2 3 5 6 7 Time Min- Fig. 5.22a - Optimum control of closed-loop sampled-data system with half-order hold for a 2% step change in load variable (steam pressure) .  154 hold has the best transient response and less oscillation or variation in the response of the manipulated variable. 5.3 Improved Proportional Controller In the process industry the commonest control equipment is the analog computing element which exerts continuous control action (u) based on the instantaneous difference between the desired condition and the actual condition. Electronic or pneumatic controllers, using Proportional, Proportional-Integral, or Proportional-Intergral- Derivative algorithms are standard instruments in virtually a l l process pl'ants. Efforts in d i g i t a l control s t i l l rely heavily on numerical approximations of analog algorithms. This practice may result in degraded performance. Control degradation in a sampled-data system can be understood by considering the Interaction between a d i g i t a l computer and a continuous process. The dynamic effect of the interface between . the discrete and continuous systems is similar to that of pure deadtime 4 ft or transportation lag, equal to half the sampling time. A d i g i t a l algorithm which eliminates the effect of deadtime is not penalized by sampling. This can be accomplished by including an analytical predictor in the control process, to estimate the value of the process output at time equal to half the sampling time plus deadtime in the future. Corrective action is then based on the predicted rather 4 4 than the actual ,output. This approach suggested by Moore et a l . is used to derive a simple proportional control algorithm for the system. 155 5.3.1 Improved Proportional Controller of System with Zero-Order Hold The overall transfer function is -Ts G(s) - (r—rr ) 6e -TS (s + e1)(s + e2) (5.66) The state-variable diagram by the iterative programming method is given in Fig. 5.23. The set of first-order differential equations describing the control system is X = FX + Eu (5.67) C = Xx -9! 1 where F = 0 -0; and E 0 0 The effect of the total delay time in the control system,— delay due to the hold and process transportation lag — , is eliminated by using a proportional controller that operates on the error between the desired value and the predicted value from the analytical predictor. That is, u = K (r - c ) (5.68) c p The solution to equation (5.67) by state-transition matrix method is x(t) = K t t n) x (t.) + fi <|.(t X)EudX' (5.69) » u u to , If the assumption that u(t) is approximately constant from one sampling period to another, that Is jT K T K. (j+l)T, Is made, then u(t) can be brought out of the integral sign. Thus, equation (5.69) becomes j i 1 i x = (1-P^EU + F l ( l - F 2 ) E I (F 2 o ) + F 1Fj[(l-F 3)Eu • + F 3x t ] i=l Predictor Fig. 5.23 - State-variable diagram of control system and predictor 157 where F l " -e T/2 , -eLT/2 -e2T/2 e a^(e -e -82T/2 : a = l/f6 -6 ) 1 / K 2 r F2 " -9 T -9 T -9 T e a ( e -e ) 0 -9 T 2 ' F3 " -9 6T -8 5T -9 6T e a (e -e ) 0 -9 26 T Note that jT is the integral multiple of the sampling time part of the process dead time and <5T is the fractional component. But C = [ 1 0 ]X (5.71) P P Thus, the proportional controller algorithm is given as K t 1 + EK (1-F ') c I r t - : F I X I - F 2 ) E M F ^ u ^ ) 1=1 - F,F 3[ E(l-F„)u + F„x 1 1 2L V 3 • t-jT-T 3 t. J (5.72) 5.3.2 Improved Proportional Controller Of System With Half-Order Hold The control system overall transfer function is -Ts „ - T s G ( s ) - (A ... /.T„)(—:— ) 4̂ •+ 4Ts / v s (s+e^Cs+e^ (5.73) 158 The state-variable diagram is as shown in Fig 5.24. The set of first-order differential sequations for the control system i s x = Hx + qu c = [ 1 0 0 ]X (5.74) where H = 0 0 -a. '2 " a l and q = [ a a a ] a = (5/4 )9 ; & l = 1/5T; ^ = 1/T As in the case of the control system with zero-order hold, the effect of the delay is eliminated by using a proportional controller that operates on the error between desired value and the predicted value from the predictor. The predicted states are X = (1-H l ) qu t + V l - H 2 ) q J ( H ^ u ^ ) , + :I1HJ[(1-H3)qut _ T + ^ X ] 1=1 J (5.75) where H l = a^e - 92j2 , - e ) -6 2/2 A_ 9 l j 2 . _A_ a2?2 , „A 62?2 V + a 3e + a e 4 a 5 ( e - e ) *2?2 -a 2/2 a l = A. a3 = A a r = -a i / ( e 2 - e l ) ; a 2 = -a(8 2-8 1+l)/a 2-G 1)(e 2-8 1); -a(e 2-a 2+l)/(6 1-a 2)(8 2-a 2) a 4 = - a / ( 8 ^ ) ( a ^ ) ; l / ( a 2 - 6 2 ) X + K. u H(s| 5T Sh1 H -a Xt+'05+j+S)T Predictor X Fig. 5.24 - State-variable diagram of control system with analytical predictor, i 160 H 2 " "-V A , "V - V i a 1 ( e - e ) -82T r A " ° 1 T + A " a 2 T + A - V ; [a e + a e + a e J A, 2 2 . <*5(e - e ) - a 2 T and H 3 " 1 A 1 2 1 e a^(e - e _e 6 T 2 ) , - 8 6x . -a <5x . - 6 6T A l A 2 A ? • x e + a e + a e 2 3 4 A - e 2 6 T " a 2 6 T « 5 ( e - e ) - a 2 6 T But u = K ( r - c ) c p S u b s t i t u t i n g the va lue of and r e a r r a n g i n g g i v e s the c o n t r o l a l g o r i t h m as ' u - [i + q y i - H l ) ] K - H i ( i - H 2 ^ .^"r^t-iTV - H ^ 3 [ q ( l - H 3 ) u t _ . T _ T + H 3 x t ] } ( 5 . 7 6 ) 5.3 E x p e r i m e n t a l R e s u l t s The improved p r o p o r t i o n a l c o n t r o l l e r equat ions (5 .72) and (5 .76) for the two c o n t r o l systems were e x p e r i m e n t a l l y v e r i f i e d . Due to the n o i s e i n the p r o c e s s , the output temperature response was averaged a f t e r f i f t e e n ' measurements and f i l t e r e d us ing the s i n g l e - e x p o n e n t i a l e q u a t i o n as has been d e s c r i b e d i n Chapter 4 . The same p r o p o r t i o n a l g a i n va lues f o r the normal or c o n v e n t i o n a l p r o p o r t i o n a l c o n t r o l l e r s (see Chapter 4) were used- F i g s . 5 .25a,b and 5.26 a,b are t y p i c a l t r a n s i e n t and 161 manipulated variable responses respectively for the two control systems. As is seen there is a marked improvement over the results i n f i g s . 4.31a,b and 4.29a,b. Fig. 5.25a - Experimental closed-loop response of proportional control with predictor of a sampled- data system with zero-order hold for a 3°C step change in set point. ' F i g . 5.25a - Experimental closed-loop response of proportional c o n t r o l with pr e d i c t o r of a sampled- data system with zero-order hold for a 3°C step change i n set point. Time Min- Fig. 5.25b - Manipulated variable response of closed-loop proportional control with predictor of a sampled-data system with zero-order hold for a 3°C step change in set point. K\ = 1.81971 x 10 - 3 Time Min- F i g . 5.26a - Experimental closed-loop response of proportional controller with predictor of sampled-data system with half-order hold for a 3°C step change in set point. •01 1 • Experimental T i = T2 = 5.0271 sees T = 7.4 sees T = 0.5 sees N = 8 = 3 Kx = 1.81971 x 10~3 Time Min- Fig . 5.26b Manipulated variable response of proportional control with predictor of sampled-data system with half-order hold for a 3°C step change in set point. 166 CHAPTER 6 ADAPTIVE CONTROL Controllers are linear elements that are often required to operate in non-linear systems. Thus they cannot be expected to provide optimum performance over a wide range of system operating conditions. However, through a linear representation of the non-linear system a controller can be designed with adaptive features that do provide opti- mum compensation for the transient system requirements. In general terms, an ideal adaptive controller would, based on measurements of only the input and output variables of a totally unknown plant, ensure that the plant's output converges to a desired value as specified by the operator. This controller would imply good servo control, that i s , response to changes in plant's setpoint, plus good regulatory control, that i s , rejection or compensation for the effect of external disturb- ances. The adaptive algorithms in most of the adaptive controllers developed recently are based on one of the search strategies or a stabi- l i t y analysis that guarantees global asymptotic s t a b i l i t y of the complete closed loop system. Also in most adaptive algorithms the controller is required to continuously test and update the system para- meters. This has the disadvantage of requiring large memory storage capacity and thus increased cost of operation. Since the majority of system dynamics found in chemical, petroleum, and other continuous process industries are slowly time varying, i t is unnecessary to have continuous updating of system parameters. In this study the system parameters are updated periodically. D DC Computed f ft) Process input pulse ^Actuator Process Process pulse response Sensor Fig. 6.1 - Flowsheet of on-line parameter identification. 168 A direct d i g i t a l control computer can periodically test plant dynamics and tune parameters of the control algorithm. F i g . 6.1 shows an approach to accomplishing self-tuning of the system. In the control algorithm, the computer Is expected to internally disconnect the feed- back, thereby making the process open looped. It then carries out the following steps: (1) It pulses the process. In a pulse test, the principal re - quirements are that.the system be driven s u f f i c i e n t l y hard so that the dynamics of the system are excited but not so hard that the capacity of the system to respond is exceed- ed. In applying the pulse method compromises are made, part icular ly in selecting the pulse height and width. For example, i f the width of the input pulse is long compared with the response, the dynamics of the system are only mod- erately excited; hence, the high frequency responses are suppressed, obscured, or non-existent. Ideally, i t would appear that the smaller the input pulse the better, for then perturbations of the output would be a minimum and the - system would tend towards linear behaviour. However, the presence of noise necessitates the production of a response which is discernible from the interferences. While the disturbing pulse excites the system with a l l frequencies at once, the amplitude of the exciting frequencies contained i n a pulse are not necessarily constant. In fact, except for an impulse 169 function (one that dif fers from zero for only an infinitesimal period of time), the amplitude of the harmonic content diminishes monotonically with frequency. Depending upon the shape of the pulse, amplitude functions may or may not diminish to zero. Those which do may increase again and exhibit another zero at a higher frequency. Pulse shapes and the location of their f i r s t zeros are important c r i t e r i a for evaluating their usefulness as pulsing functions. For a given pulse width T , a rectangular pulse has the smallest useful harmonic content. In this study a smooth pulse given in Equation (6.1) is used. f( t ) = K[l - cos(2-t/T )] (6.1) P This type of pulse has been suggested by Hougen et a l . 2 4 to extend the useful harmonic content considerably. ( i i ) The computer ident i f ies the process In the form of a second- order plus deadtime f i t , and f i n a l l y ( i i i ) calculates the controller settings for the deadbeat perform- ance cr i ter ion compensator already designed in Chapter 5. The direct d i g i t a l control computer identif ies overall process dynamics through the same actuator and sensor dynamics that i ts control action sees. This is a dist inct advantage over attaching special sensors to the control loop for performing dynamic analysis and control synthesis. It is assumed that the pulsing inputs f^(t) t o t n e process are noise free since these values are internally computed and applied. Only the process response, the outlet temperature, contains noise. Some 169 function (one that differs from zero for only an infini tesimal period of time), the amplitude of the harmonic content diminishes monotonically with frequency. Depending upon the shape of the pulse, amplitude functions may or may not diminish to zero. Those which do may increase again and exhibit another zero at a higher frequency. Pulse shapes and the location of their f i r s t zeros are important c r i t e r i a for evaluating their usefulness as pulsing functions. For a given pulse width T , P a rectangular pulse has the smallest useful harmonic content. In this study a smooth pulse given in Equation (6.1) is used. f( t ) = K [ l - cos(2jtt/T )] (6.1) P 24 This type of pulse has been suggested by Hougen ê t al_. to extend the useful harmonic content considerably. ( i i ) The computer ident if ies the process in the form of a second- order plus deadtime f i t , and f i n a l l y ( i i i ) calculates the controller settings for the deadbeat perform- ance cr i ter ion compensator already designed in Chapter 5. The direct d i g i t a l control computer identif ies overall process dynamics through the same actuator and sensor, dynamics that i ts control action sees. This is a dist inct advantage over attaching special sensors to the control loop for performing dynamic analysis and control synthesis. It is assumed that the pulsing inputs f^(t) to the process are noise Eree since these values are Internally computed and applied. Only the process response, the outlet temperature, contains noise. Some 170 other possible noise sources include: other process input, internally generated noise, and measurement noise. The problem then is to charac- terize the process from these input-output sequences. A moment method as suggested by Michelsen et al.** 1 in which noise effect on the process characterization i s small was used. 6.1 Parameter Estimation by a Modified Moments Method The transfer function of any stable linear, any dimensional sys- tem can be evaluated by numerical integration of i t s experimental trans- ient response to an arbitrary pulse forcing function. The experiment- a l l y determined, normalised transient response C^(t) and C Q(t) are converted into moments of the form Mn'S = /" C(t)exp(-st).t ndt = ( - l ) n (C(s)) (6.2) ds where C (s) = /' C(t)exp(-st)dt . The Laplace transforms and their derivatives are related to the system transfer function, G, through the relations: C (s) G = -2 '- (6.3) C.(s) i GJ_ = C'(s) G C(s) o (6.4) i 21 _ ( G ' 2 ) = c"(s) _ ( C C s ) ^ G G C(s) C(s) o i < 6' 5 171 or, in general ds n G ds n C(s) i The right hand sides of Equations (6.3 to (6.6) are evaluated by o (6.6) TI S computing the moments M ' , and G and i t s derivatives are thus determined for an arbitrary number of s-values. The model transfer function has been chosen as ,. - T S G(s) = - j r (6.7) ( x l S + i r where x is the deadtime and x^ the process time constant. The parameters x and t\ are determined from the following relations: (Note: that the moments are calculated using a fixed s^value). In G = -xs - 2 l n ( x l S +1) , (6.8) Z j _ = T + 7 * (6.9) G (x^s +1) 2 2x 2 (£1) _ ( £ 1 _ ) L _ ( 6 . 1 0 ) G G (x l S+l) -G' . ,G". ,G\2 Let u[ = — and u 2 = (—) - (—) Solving equations (6.9) and (6.10) with these substitutions gives u l / 2 x " 2 1 / 2 (6.1D (1.414 - s u 2 / Z ) j7 172 and U'(T s+1) - 2T x = L ( 6 . i 2 ) (T^S+1) Evaluation of T and thus requires the calculation of at least two n s moments M ' for each transient response. Parameter evaluation for a model containing r parameters requires calculation of at least r moments. It is normally advantageous to compute a larger number of moments and evaluate the parameters by s t a t i s t i c a l analysis, because the v a l i d i t y of the system model may thereby be assessed. This is not necessary in this case since the model has been proved earlier on to be correct. The main advantages of the outlined method compared with the normally used method of central moments are: (i) the validity of the model may easily be assessed, and ( i i ) The sensitivity to experimental errors, in the determination of the transient responses is greatly .reduced, provided suitable s-values are used. The optimum s-value is determined from the noise sensitivity function. Fig. 6.2 is the noise sensitivity as a function of s for = 5.027, T = 7.4 and unit step input. See appendix 12 for details of derivation. In the central moment method, the s-value is always taken to be zero but as the noise sensitivity analysis shows, (Fig. 6.2); as the s-value is increased from 0 to 1, the noise intensity decreases un t i l a minimum is reached at an s-value of one. Since this mimimum occurs for both time constant and dead time i t is advisable to use a s-value of 1. Noise has a great degrading effect on processes, more over high noise effects leads to greater model error which w i l l result in poor control of the system. Fig. 6.2 - Noise s e n s i t i v i t y as a function of laplace transform operator. 174 6.2 Compensator Design The control algorithm u t i l i z e s a linear combination of the past history of the system in forming a new value for the manipulated variable . The absolute position of the f i n a l control element i s determined from the formula t k p u(nT) = I g .e[(n-i)T] - J h u[(n-j)T] (6.13) i=0 1 j=l J Equation (6.13) gives the value at which u(t) is to be held constant during the entire (n+l)st sampling period, that i s , u(t) = u(nT) for n/T _< t < (n+l)T. T is the sampling period and the g i ' s and h ' s in equation (6.13) are a l l constants. In this algorithm only the (k+1) most recent values of the error and the p most recent values of the manipulated variable need be stored. The design objective is to determine suitable values of {g^}, and {hj}* These constants have already been calculated in Chapter 5 and are given as: (a) For control system with zero-order hold D f , uco M(z> - j t k o +. (Vrv*" 1 + (yvx^"2] D ( z ) = uTzT ~ ETZT ~ z r i + x rhn 7h ~~~~~ ' ( } (l+X1z ) ( l -z ) (b) For control isystem with half-order hold { Z ' (I-z )(l+3 L z ) See chapter 5 for deta i l s . 175 6.3 Implementation and experimental r e s u l t ( i ) The f i r s t step i n the implementation of the adaptive c o n t r o l i s i n i t i a l i z a t i o n of the model parameters o f f - l i n e . With some mod i f i c a t i o n i n the programming of the algorithm i t i s possible to i n i t i a l i z e the model parameters on-line using the estimation subroutine of the algorithm. ( i i ) Program the manipulated v a r i a b l e p o s i t i o n algorthm of equations (5.27) and (5.51) for the two co n t r o l systems r e s p e c t i v e l y i n t h e i r d i s c r e t e time form. ' ( i i i ) The model parameter estimation subprogram should include the pulsing function given i n equation (6.1). When the computer c o n t r o l l e r i s i n t e r n a l l y disconnected, the manipulated valve i s made to track equation (6.1). During t h i s p u lsing time, the out l e t water temperature i s i n t e r n a l l y datalogged and used In the modified moments method as, has been discussed e a r l i e r on to estimate the parameter values. This parameter estimation may be performed continuously, that i s , a f t e r every sampling and manipulated valve move or a f t e r some time i n t e r v a l that may be held constant as i n t h i s study, decreased or increased as the operator seems necessary. The above adaptive designs were experimentally tested on the heater-hear exchanger c o n t r o l system described i n chapter 4. A 50% proportional band about the set point was imposed on the c o n t r o l l e r . Due to the noise present i n the system, the single-exponential f i l t e r i n g equation was used to smoothen the measured outlet temperature response. No f i l t e r i n g was used i n the pulse temperature datalog program since i t Time Min. F i g . 6.3a - Adaptive control response of a sampled-data system with zero-order hold. Time Min F i g . 6.3b - Manipulated variable response of sampled-data system with zero-order hold adaptive-controller. 178 is assumed that the process noise is negligible because i t is interna generated. Figs. 6.3a,b and 6.4a,b are typical adaptive control responses for the controlled temperature and manipulated variable for control system with zero-order hold and half-order hold respectively. Time Min- Fig. 6.4a - Adaptive controller response of a sampled-data system with half-order hold. • Updating Point ,Experimental 1 2 - Manipulated variable response of sampled-data system with half-order hold adaptive controller. + 181 CHAPTER 7 DISCUSSION AND CONCLUSION 7.1 Discussion 7.1.1 Analysis of proportional control In the stability analysis of the control system under investigation, the value of the limiting proportional gain is a measure of the degree of stability of the system. In the analysis of the second-order overdamped system, — that is, model with no transport lag — , increase in sampling time results in decreased stability of the control system irrespective of the smoothing device employed. This is expected since longer sampling periods imply a greater deviation from the continuous system state. This observation has been made by many 2 9 H7 6̂ other workers. , , In both cases (control system with zero-order hold and control system with half-order hold) increase in the ratio of the time constants results in increased stability. This trend is expected because, as has been reported in control literatures, 1 1, 6 I + multiplicity of poles or zeros always introduces greater instability to a control system.. It Is a common practice in the design of compensators for control systems to place the poles far apart from each other. This will definitely lead to increased difference between the time constants and hence increase their ratio. As has been stated in the main study, although a l l proportionally controlled first and second-order systems are stable in the continuous domain, regardless of the value of the loop gain. This is not true for a second-order system in the sampled data domain, irrespective of the smoothing device used. 182 For a second-order plus dead time overdamped system with e i t h e r a zero-order hold or half-order hold, the s t a b i l i t y increases with increased dead time u n t i l a maximum i s reached and thereafter the s t a b i l i t y decreases. In a l l the cases considered, — F i g . 4.12 and 4.13 are t y p i c a l s t a b i l i t y boundary conditions for the two c o n t r o l systems — , the points of maximum s t a b i l i t y for the two c o n t r o l systems occur at approximately the same dead time value. This i s expected from equation (4.55) where the half-order c i r c u i t gain i s just that of a zero-order hold c i r c u i t gain m u l t i p l i e d by. a p o s i t i v e f a c t o r greater than one. This a m p l i f i c a t i o n r e s u l t s i n greater operating proportional gain • range. Hence, for a l l the conditions tested the c o n t r o l system with half-order hold i s more stable than the c o n t r o l system with zero-order hold. In the transient response analysis of the two con t r o l systems, the new performance c r i t e r i o n gave a more stable and better response f o r 75 the system than the one-quarter decay r a t i o index. This i s expected, since a second-order overdamped system, at l e a s t t h e o r e t i c a l l y , does not overshoot. The existence of an overshoot i s the underlying assumption of the one-quarter decay r a t i o c r i t e r i o n . I r r e s p e c t i v e of the performance c r i t e r i a used, increase i n sampling time Introduces greater i n s t a b i l i t y to the c o n t r o l systems. For the new performance index used, increase i n the number of sampling Intervals used Increases the s t a b i l i t y of the c o n t r o l systems. This i s expected since for any sampling time, the amount of time given fo r the c o n t r o l system to a t t a i n steady state conditions i s dependent on the number of sampling i n t e r v a l s employed. A shorter time w i l l impose greater constraint on the c o n t r o l system and hence w i l l introduce 183 i n s t a b i l i t y . A detrimental aspect of a larger imposed sett l ing time is the production of a sluggish response which results in greater steady state error. Thus an optimum number of sampling intervals of 8 was used in the experimental tests. For comparison purposes, the performance index <|> was increased from 1 to 3, while in one case <|> was increased to 4. Increase in <j> brings about greater i n s t a b i l i t y to the systems. Also the error response decreases with increased <j> u n t i l a minimum is reached; — for control system with zero-order hold, <p = 1-5 and for the half-order system <b = 3 — , after which the error response increases with increase <J>. (See Tables 4.4 and 4.5). In a l l the conditions tested, the control system with half-order hold gave better responses for both theoretical and experimental verif icat ions than the control system with zero-order hold. (See Figs. 4.28 to 4.31). 7.1.2 Compensator Design Of the three compensators, the algorithm derived from the deadbeat performance principle gave the best transient responses. This may be due to the constraints of s t a b i l i t y , fastest response and set t l ing time, and zero steady state error used in the derivation. Although the s t a b i l i t y of the compensated control system is a prerequisite for the application of any compensator to a control system and must be used in the derivation of the algorithm; the other constraint of error minimisation imposed on the combined optimum control and predictor compensator does not always guarantee zero steady state error, since the minimum may not be zero. This may explain why the dead beat performance compensator gave the best response. Also the 184 I n a c c e s s i b i l i t y of some of the s t a t e s i n the optimum c o n t r o l which l e d to the use of a p r e d i c t o r may c o n t r i b u t e to i t s poorer response as compared to that of deadbeat per formance. The worst response was g i v e n by the improved p r o p o r t i o n a l c o n t r o l l e r which made use of p r e d i c t e d output response i n s t e a d of a c t u a l measured v a l u e s i n i t s c o r r e c t i v e r e s p o n s e . Th is i s expected ; a f t e r a l l a zero s teady s t a t e e r r o r response i s not a c o n s t r a i n t on i t s f o r m u l a t i o n but an o b j e c t i v e . D e s p i t e i t s poor response when compared to the o ther two, i t s t i l l gave a b e t t e r response than the c o n v e n t i o n a l p r o p o r t i o n a l c o n t r o l l e r ( F i g s . 5 . 2 5 to 5 .26 and 4 .28 to 4 . 3 1 ) . Even though, i t was not p o s s i b l e to b r i n g the s t a t e of the sys tems, — f o r the deadbeat performance compensator — , c o m p l e t e l y to r e s t a f t e r two sampl ing p l u s dead time p e r i o d s , t h i s does not negate the v a l u e of the t h e o r e t i c a l concept of f i n i t e s e t t l i n g t i m e , because systems designed to meet t h i s requi rement t h e o r e t i c a l l y , as shown i n t h i s s t u d y , g ive s a t i s f a c t o r y performance i n r e a l t e s t s . 7 . 1 . 3 Adapt i ve C o n t r o l The good response of the a d a p t i v e c o n t r o l system used i n t h i s s tudy has demonstrated that i t i s not necessary to c o n t i n u o u s l y update model parameter va lues i n most equipment found i n the chemica l and p e t r o c h e m i c a l i n d u s t r i e s . As has been s t a t e d i n t h i s s tudy , the p e r i o d i c updat ing of parameters has the added advantage of r e q u i r i n g l e s s computer memory. A l s o , t h i s study has i l l u s t r a t e d the p o s s i b i l i t y of u s i n g pu lse t e s t s f o r o n - l i n e parameter e s t i m a t i o n . The t rend of the e x p e r i m e n t a l t r a n s i e n t responses a l l through t h i s study conf i rms the assumption that any h i g h order system can be 185 approximated by a second-order overdamped plus dead time model. During the experimental v e r i f i c a t i o n a sample averaging method was applied to reduce and i n some cases eliminate the excessive noise i n the process. Each of the f i f t e e n sampling measurements for any p a r t i c u l a r sampling time, i s summed up and the average i s taken, and the single-exponential f i l t e r i s applied to i t to give the response. 7.2 Conclusion • For sampled-data systems which can be adequately modeled as over- damped second-order plus dead time with e i t h e r a zero-order hold or half-order hold as the smoothing device, a systematic design procedure has been given f o r choosing the loop gain and sampling rate of a sampled-data feedback c o n t r o l l e r using a new performance Index. This type of c o n t r o l l e r i s simple to set and implement. A comparative study was c a r r i e d out on the r e l a t i v e e f f i c i e n c i e s of the new performance index and the normally used one of, one-quarter decay r a t i o index. The new performance index gave better responses than the one-quarter decay r a t i o c r i t e r i o n . This may be due to the assumption of an overdamped second-order model for t h i s work. I f the recommended settings are used, a r e l a t i v e l y small amount of information i s needed f o r s a t i s f a c t o r y c o n t r o l . This i s of s i g n i f i c a n t importance for co n t r o l when the measurement i s d i f f i c u l t and /or expensive, and when information channels may be l i m i t e d , as, i n t h i s case when a small d i g i t a l computer i s used. S a t i s f a c t o r y performance of the proposed algorithms has been demonstrated when applied to the heater-heat exchanger system with higher order dynamics. The r e s u l t i n g model error may be the reason why 186 the c o n t r o l system d i d not have f i n i t e s e t t l i n g time f o r the deadbeat performance c r i t e r i o n compensator. However, f i n i t e s e t t l i n g time i s o n l y a t h e o r e t i c a l c r i t e r i o n . Of a l l the three compensator des igns the deadbeat performance c r i t e r i o n gave the best response w h i l e the s i m p l e feedback p r o p o r t i o n a l c o n t r o l l e r gave the w o r s t . An a d a p t i v e c o n t r o l scheme has been developed which can be a p p l i e d to a wide c l a s s of s i n g l e i n p u t - s i n g l e output p l a n t s . Any of the c o n t r o l a l g o r i t h m s can be used but i n the e x p e r i m e n t a l v e r i f i c a t i o n o n l y the deadbeat performance c r i t e r i o n was t e s t e d . The a d a p t i v e c o n t r o l scheme used here depends on a l i n e a r second-order overdamped model w i t h a time delay i n the p r o c e s s . The method f o r d e t e r m i n i n g such a model from f i n i t e time i n p u t - o u t p u t o p e r a t i n g data of the p l a n t was d i s c u s s e d . A p u l s e method was developed f o r on l i n e u p d a t i n g of the model parameters . I t i s a l s o observed that i n c r e a s e d sampl ing p e r i o d s degrades the response of the c o n t r o l system. 7.3 Recommendation - . The g i v e n responses i n t h i s work are those of compensators de- s igned as second-order p lus dead time model c o n t r o l l i n g a f o u r t h o r d e r p l u s dead time model . I t i s recommended that an a c t u a l second-order p lus dead time c o n t r o l system be t r i e d to v e r i f y how e f f e c t i v e the com- pensators a r e . A comparat ive study of these compensators w i t h c o n t i n u - ous or analog c o n t r o l s v i z p r o p o r t i o n a l , p r o p o r t i o n a l - i n t e r g r a l , and p r o p o r t i o n a l - i n t e r g r a l - d e r i v a t i v e compensators should be c a r r i e d o u t . The equipment has been b u i l t such that m u l t i v a r i a t e c o n t r o l i s p o s s i b l e , so , s i n c e t h i s work d e a l t w i t h only s i n g l e i n p u t - s i n g l e output system; 187 i t is suggested that a multivariate control compensator be tried out on the equipment. The v e r s a t i l i t y of the compensators should be investiga- ted with other than temperature response loops. And f i n a l l y the effect of varying the coefficient of the zero-order hold to give different holds should be investigated. « 188 NOMENCLATURE C(t) Assumed process response with no hold C (t) Assumed process response with hold o DP(J-l), DP(J-2) Actual smoothened output temperature for the previous and penultimate periods respectively for control system with half-order hold DK(J) Calculated output temperature at J-th instant for control system with half-order hold G,G',G" Model transfer function and it s f i r s t and C second derivatives Error coefficents of deadbeat performance compensator h. Manipulated response coefficient of deadbeat 2 performance compensator H (s) Laplace transform of zero-order hold o \ H^^Cs) Laplace transform of half-order hold J Performance criterion for optimum control N j Integral multiple of sampling time part of model dead time K Proportional gain c K , K , K , K' K', K', Variable-gain elements of deadbeat performance o 1 o 1 n v compensators Q,Q' Weighting matrices for optimum control s Laplace transform operator T Sampling time T Pulse function period P T(J) J-th smoothened temperature 189 T ( J - l ) ( J - l ) - t h smoothened temperature T^(J) J - th averaged temperature u, u^ Input> output of variable-gain compensator respectively <|> Performance Index $» <t>(̂ )> <KT), <t>'(T) Transition-state matrices F i l t e r weighting factor a Decay ratio 1 6 Fractional multiple of sampling time component of process dead time. Xa,b,c ,d Ratio of model f i r s t time constant to second time constant. A, A^ Determinants used in Mason's formula V = (1-6)T Effective time used in the transition matrices x, (j+6)T, A Model dead time , M o d e l time constants The following parameters are defined i n 5.3 E, F, F 1 , F 2 , F 3 , H, H 1 , H 2 > H 3 > q, 6̂ , & 2 , a^, , &5 The following parameters are defined in appendix 2: V V V V D l l ' D 21' ° 3 1 ' V V V V Q 7 ' Q3 • %> Q 5 ' %' Q7 Q9> V " l l ' a i 2 ' ° 1 3 ' °15» "16' ° 1 7 ' " i g ' Y l l > Y 12' T 3 l ' Y32' • l l ( s ) ' 4 » i 2 ( s ) ' W'* * 2 2 ( s ) ' * 1 1 ( V ) » * 1 2 ( 7 ) ' * 2 1 ( V ) ' < f , 22 ( V ) ' • l l ( T ) » *12 ( T )> * 2 1 ( T ) ' * 2 2 ( T ) ' * 1 ( V ) » * 2 ( V ) ' ^ l V ( T ) > * 2 V ( T ) 190 The following parameters are defined i n appendix 4: a l ' a 2 ' a 3 ' a4' ̂ 1' ̂ 2' ^3' ̂ 4' ^5' ̂ 6' ̂ 1' ̂ 2' ̂ 3* ̂ 1' ̂ 2* a l ' a 2 ' a 3 ' V °5» V V V 811' 612> V 614« 915' 616' 617> V V Y2> Y21 Y 2 2, V s ) , V S ) « * 1 ( V ) » * 2 ( V ) » *l"W> * 2 i ( V ) > • n ( s ) , <f>12(s), * 2 1 ( s ) ' * 2 2 ( s ) ' * 1 1 ( V ) ' • l 2 < 7 ) » * 2 1 ( 7 ) » * 2 2 ( 7 ) ' * 1 1 ( V ) 4 • l 2 ( V ) » * 2 1 ( V ) » * 2 2 ( V ) The following parameters are defined i n appendix 8: b i ' b2> b3» b4> W V V V V V * i l ( s ) ' * 1 2 ( S ) ' *13 ( s )> * 2 i ( 8 ) » * 2 2 ( s ) ' •iV' 0 , * i ' ( s ) ' ^2 ( S ) " The following parameters are defined i n appendix 9: a, a l x , a 2 1 , h^, h 2 , K Q> K{» «{»"02.' a3> al> a'5> a D ' a7» a8» a9» a i 0 ' ail> a i 2 ' ai3> ai4> °15' ai 6> "I?' °18'' a i 9 ' a20' a2i> a22' B l ' B2' *3' 0 4, B 5,*£, +2..*{1, • { 2 ' * 1 3 ' Hi' *22' *23 The following parameters are defined i n appendix 11: A, A v A|, A'1, B, B L, B \ B J , C, C'\ D, D', <|>'(T) 191 BIBLIOGRAPHY 1. 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ISA J . 11, 77 (June 1964); 75 (July 1964); 63 (Aug. 1964). 197 APPENDIX I HALF-ORDER HOLD TRANSFER FUNCTION DERIVATION The output waveform of a zero-order hold has a zero slope between two consecutive sampling periods. Also i t s frequency characteristic shows a rapid attenuation for low-frequency signals. In other words, the zero-order-hold cir c u i t holds the measured response of the system at any sampling instant at that level u n t i l the next sampling instant. A f i r s t order hold exhibits an impulse response that has a constant slope between two consecutive sampling instants, which is determined by the values of the two preceding samples. Thus, the first-order hold estimates the response over the sampling interval from K to K+l as a ramp with the slope determined by the signal values at time K and K-1. It is conceivable that better response characteristics may be obtained from a half-order hold; which has an impulse response with constant slope between two consecutive sampling instants, and which lie s midway between the impulse responses of and as shown in Fig. A l . l . The estimation of the new response measurements is as described for first-order hold but with a ramp slope of 1/2. The impulse response of the half-order hold can be represented as a series of step and ramp functions. That is H 1 / 2 = (1 + I T - M O - j u(t-T) - i (t-T) u(t-T) + \ u(t-2T) + |Y (t-2T) Vu(t-2T) (Al.l) 198 First Order Fig. Al.1 - Impulse response of half-order hold. 199 The Laplace transform gives „ f v I s .3 N -sT 1 -sT , 1 -2Ts , 1 -2Ts H 1 / 2 ( s ) = - 4 r - - (—) e e + — e + e ' s 2Ts 2s Ts 2s 2Ts / A 1 O N (Al .Z) Rearranging gives -Ts H . (s) = (1 - ( l / 2 ) e " s T ) ( ^ - £ ) + _ 1 ( 1 _ e- TS)2 ( A 1 > 3 ) 1 S 2TsZ Fig. A2.1 - Signal flow diagram. 201 APPENDIX 2 STATE VARIABLE DERIVATION AND PARAMETER DEFINITIONS FOR CONTROL SYSTEM WITH HALF-ORDER HOLD The overall transfer function is -Ts „, . ,4 + 5 T s w l - e . K0 , . n 1 N G ( s ) - ( 4 - T ^ T S - ) ( s — } ( S + e l ) ( s + e 2) ( A 2 - 1 } To determine the state equations, Mason's gain formula is applied. There are three loops in the diagram and are given as: ' L i L2 L3 ~=k (a2'2) s Loops L j , L3 and L 2, L3 are non touching loops, thus the determinant of the signal flow graph is given as: A = 1 - ( L 1 + L 2 + L 3 ) + ( I n L 3 + L 2 L 3 ) That i s , [Ts 3+s 2(T6 +1) •+ s(T9+8 ) + 9] . A = — - - ' (A2..3) Ts The transfer function relation the input X^(KT) to the output X^(s) i s (j)^^(s) and is composed of three parts derived from three forwardj paths (a), (b), (c). The transmittance of forward path (a) from X^(KT) to X^(s) is T' = 1/s. Path (a) is touched by loop L 2 > therefore the determinant of the process becomes Aj = 1 - (L x + L 3) = ( T S + l l + l) .(A2.4) 202 Thus, [Ts"+s"(T93+l) + s(T8+63) + 6] 1 1 s(Ts + T9 + 1) , ^ , „ , . — = 3 — 5 " = *,,(s) (A2.4a) The transmittance (path (b)) from X 1(KT) to X (s)•is -K8 T" = 1 _ 4 Ts this path is touched by a l l loops, thus   = 1. Therefore, T"A'' J _ i KG ; = 4>" (s) (A2.4b) s[Ts +s (T9.3+l) + s(T9+93)+9] The transmittance from X^(KT) to X^(s) (path c) is -5K9 T 1 1 1 _ 1 , 3 4 s The path is touched by a l l loops, thus A|'1 = 1 Therefore, rjn I I I ^ I I I 1 A 1 = o—-5 = +1'1,,(s) (A2.4c) 4[Ts +s (T8 3+l) + _sCT8+8 )+6]. Hence, the overall transfer function relating the input X^(KT) to the output X (s) is ^ p s ) which is the sum of equations (A2.4a), (A2.4b) and (A2.4c). That i s , *11 ( S ) = *11 ( S ) + * l l ( s ) + * i i ' ( s ) ( A 2 - 5 ) The transfer function relating the input X^(KT) to the output X^(s) 203 is <}>,',•, (s). There is only one forward path (d) in this case. The trans- 2 mittance of the forward path is T^2 = 1/s . This path is touched by two Ts + 1 loops , L 2; thus, A 12 Ts T A Therefore 1 2 . U = =—= ( T s + 1 ) '=+1,,(s) (A2.6) [Ts 3+s 2(T0 3+l) + s(TG+03) + 0] 1 2 The transfer function relating the input xi(kT) to the output X2(s) is <f>2̂ (s) and is made up of three paths b, c and d. The transmittance of -K0 path b is T' = —5- . This path is touched by a l l the loops hence S Ts '  = 1. Therefore T* A * 21 21 -K0 . . , . , o N T— = 5 o = •*,(«) (A7,2a) [Ts +s (T63+l)1 + s(T0+03) + 0] 5K0 The transmittance of path c is = 2~ ; the path is touched by a l l 4s the loops, thus A ^ = 1. T " A " Therefore 2 1 2 1 = =- ~ 5 K T 9 s : = <J>''(s) (A2.7b) 4[Ts +s (T03+1) + s(T0+93) + e] —0 The transmittance of path d is T^j' = — ; the path is touched by loops s" Li and L 2, thus A-' = Igl 204 That i s , [Ts 3+s 2(T9 +1) + s(T9+9 ) + 9] A = — : - (A2.3) Ts The transfer function relation the input X^(KT) to the output X^(s) is (fi^^(s) and is composed of three parts derived from three forward paths (a), (b), (c). The transmittance of forward path (a) from X^(KT) to X 1(s) i s T| = 1/s. Path (a) i s touched by loop , therefore the determinant of the process becomes A; = 1 - ( L l + L 3). = ( T S + H + l) (A2.4) Therefore, J k ^ . :. 3 - 2 - e ( T . H - i ) - - - . , . , . ( s ) ( A 2 . 7 c ) [.Ts +s (T9 +1) f s(T8+e3)+6] - The overall transfer function relating the input X^(KT) to the output X 2(s) is *21 ( s ) = *21 ( s ) + * 2 l ' ( s ) + < f ) 2 l ' ( s ) (A2.8) The transfer function relating the input X 2(KT) to the output ^ ( s ) * s <j>22(s) and the transmittance is given as T = 1/s and the path is touched by loops and L ?, thus „ Ts + 1 A__ = — . Therefore, Z z is 205 T 2 2 * 2 2 = s C T s ^ t l ) ( A 2 > 9 ) [Ts +s (T63+l) + s(T6+e3)+e] The transfer function relating the input r(KT) to the output X ^ s ) i s t|^(s) and is made up of two paths: T ' A ' JL1 „ 5! = ̂*(S) (A2.10a) s[Ts +s (T83+l) + s(T6+e3)+9] and T"A" A-2 — ^ — = f"(s) (A2.10b) 4[Ts +s (T93+l) + s(T8+93)+9] The overall transfer function is *J(s) = **(s) + *J(s) ( A 2 . l l ) The transfer function relating the input r(KT) to the output X 2 (s ) i s composed of two paths v i z : ^ K9 _ (A2,12a) [Ts 3+s 2(T9 3+l) + s(T9+63)+9] 2 and TJ^L = -  5*™°— — = K ( s ) (A2.12b) 4[Ts +s (T93+l) + s(T9+93)+9] The overall trartsfer function relating the input r(KT) to the output X 2 (s) i s *2'(s) = +2'(s) + ijT(s) (A2.13) 206 Therefore, the set of f i r s t order d i f f e r e n t i a l equations expressed i n matrix form is X(s) = *il<S> 4>' ( ) .•{2<s> • J 2 ( ) XL(KT) X 2(KT) r(KT) (A2.14) Due to the presence of sample and hold, there is a time delay t o = KT i n the control system; after obtaining the inverse Laplace transform, the time t i s replaced by t - KT. The f i r s t step i n determining the inverse Laplace transform of equation (A2.14) is to find the factors of the main determinant. That i s , T s 3 + s 2 (T8 3 ' + 1) + s(T8 + 83) + 0 '(' (A2.15) or 3 2 s + e.s + e cs + ê  4 5 6 where 8, = (T0O + 1)/T; 8C = (T6 + 6^)/ T; 6, = 6/T 4 3 5 3 6 (A2.15a) The cubic function Equation (A2.15a) is reduced to the form y 3 + Vy + w by performing the substitution s = (y - 6 ^^ 3 ) * T n e three roots of the reduced cubic function as given by Cardan are yi = ( Q 1 + Q 2 ) ; Y2 = " <1/2>H<*i + V + ̂  ( Q 1 " V1^ y 3 = - (1/2)[-(Q 1 + Q2) - / I (Q x - Q 2 ) i ] where 207 n = \— + (^: + X_V/2l1/3- y l L 2 ^ A 2 7 ; J 2 3 r-w /W , V . l / 2 i l / 3 ,,N Q2 = ["2 ~ (-4 + 27) J 5 (A2.16) and w = [ 2 6 3 - 9 e 4 6 5 + 278 ] / 2 7 ; V = (39 5 - 8 2)/3 Since the system is assumed to be overdamped, the roots should be real and hence the condition 2 v3 2- + — < 0 4 27 ^ should be satisfied. Therefore, the solutions to the unreduced cubic Equation (A2.15a) are - S l " y l " 94/3 ; S2 = y2 " 94/3 ; S3 = y3 ~ 94/3 ( A 2 ' 1 7 ) The major determinant is then given as (s + s )(s + s )(s + s ) (A2.17a) -s 1(-Ts 1+T8+l)/(s 2-s l)(s 3-s 1); B ' ^ -s 2(-Ts 2 + T e ?+l ^ ( s ^ X s ^ ) CL1 = - S 3 ( - T s 3 + T 9 + 1 ) / ( S 1 - S 3 ) ( s 2 " s 3 ) ; D * = ~ 6 / s l s 2 S 3 208 A l l = 8 / s 1 ( s 2 _ s i ^ s 3 ~ s i ) ; B i i = 0 / s 2 ( s 1 - s 2 ) ( s 3 - s 2 ) C i ' l = e / , s 3 ^ s l ~ s 3 ^ s 2 ~ S 3 ^ ; A12 = ^ ~ T s 1 ) / ( s 2 - s 1 ) ( s 3 - s 1 ) B 1 2 - (1 " T s 2 ) / ( s 1 - s 2 ) ( s 3 - s 2 ) ; B J 2 = (1 - T s 3 ) / ( ) ( s 2 - s 3 ) A j j ' = - 5 1 8 / 4 ( 8 2 - 8 ^ ( 8 3 - 8 ^ ; B-' = -5T6/4 ( S ; L - s 2 ) ( s 3 - s 2 ) C-'=-5T9/4 ( S l - 8 3 ) ( 8 2 - s 3 ) ; A^=-0/(s 2-8 1)(s 3-8 2); B' - e y ^ X s ^ ) C^ 1=-8/(s 1-s 3)(s 2-s 3); A 2 1=5esl/4(s 2-s 1)(s 3-s 1); 8^5882/4(3^32 ) C 2 1 = 56s3/4 ( s 1 - s 3 ) ( s 2 - s 3 ) ; A^'' =.0(18^1 ) ( s 3 - s 2 ) B 2 l ' = 6 ( T s 2 " 1 ) / ( s r s 2 ) ( s 3 " s 2 ) ; C2l ~ 8(Ts 3~l )(s 2~s 3) A22 = s l ( _ T s l + l ) / ( s 2 - s 1 ) ( s 3 - s 1 ) ; B 2 2 = - s 2 ( - T s 2 + l ) / ( s 1 - s 2 ) ( s 3 - s 2 ) ' C22 =-s3(-T'S3+ l)/C8 1 - 8 3 ) ( 3 2 - 8 3 ) ; D u l - e / S l 3 2 83 ; A ^ - 8 ^ ( 8 2 - 8 l ) (s^s} B n i = - 6 / s 2 ( s r S 2 ) ( s 3 - S 2 ) ; C i l l = - 0 / s 3 ( s r S 3 ) ( s 2 - S 3 ) ; A m = 5 T 9 / 4 ( V 8 ! > (VS1 > B l U = 5 T 9 / 4 ( s r S 2 ) ( s 3 " S 2 ) ; C l U = 5 T 9 / 4 ( s r S 3 ) ( s 2 " S 3 ) ; A2 = 0/(s ?-s 1 ) ( s 3 ~ s 1 ) B̂  = e / ( S l - s 2 ) ( s 3 - s 2 ) ; C2' = 0 / ( s 1 - s 3 ) ( s 2 - s 3 ) ; A 2 =-5T8sl/4(s 2 ,-s 1)(s 3-s 1) 209 B2" = - 5 T 9 s 2 M ( s 1 - s 2 ) ( s 3 - s 2 ) ; C 2 = - 5 T 6 s 3 / 4 ( S l - s 3 ) ( s 2 - s 3 ) Pj = exp(-s 1T); P^ = exp ( - s 2 T ) ; ?^ = exp ( - s 3 T ) • 1 1 ( T ) = K T ^ A j p K A j p i ^ ^ P j + C B j p K B ^ p r a ^ • 1 2 ( T ) = A 1 2 P ; + B 1 2 P 2 + C 1 2 P 3 ; * 2 1 ( T ) K A 2 r + K A 2 1 « A 2 p p ; + ( B 2 r + K B 2 1 ^ B 2 1 ) P 2 ( C 2 { ' + K C 2 1 + K C 2 r ) P 3 ; * 2 2 ( T ) = A ^ P J + B ^ P ' + C ^ P ' ^ ( T ) = K D 1 1 1 + K ( A I l l + A 1 ' 1 1 ) P i + K ( B i u + B 1 - 1 1 ) P ' + K C C ^ + C ^ ^ tp2(T) = K(A^+A 2 ,)P'+ K(B^+B 2 )P 2 + K(C^+C 2')P 3 Q 3 - A ^ P J + B ' ^ + C ^ P ' +.* 2 2(t)[l " ( A p P ' + B ^ P ' + C p p p ] Q4=*;i2 (T) [ A - ' P[+B2{ ' P ' + C y ' P' ] ; Q5=D+(A' , 1+A|' * )P|+(Bp+B[|' yP^+CC^+C'{ ' )P Q 6 = * 2 2 ( T ) Q 5 ; Q7 = * 1 2 ( T ) [ ( A 2 p A 2 p p p ( B 2 p B 2 1 ) P 2 + ( C 2 1 + C 2 1 ) P 3 ] P| x = exp ( - S l V T ) ; P|2 = exp ( - s 2 VT ) ; P ^ = exp ( -s 2 VT) ^ ( o T ) = K D * - K A | 1 « A p K A | p ) P | p ( B | 1 4 K B p + K B ' ' ^ P p + C C p + K C p + K C l p ) ? ^ • i 2 ( v T ) = A 1 2 P ( 1 + B 1 2 P ; 2 ^ 1 2 P { 3 ; •. 2 (7T) = A ^ P ^ + B ^ P ^ - K ^ P ^ *21 ( V T ) = ( A 2 i , + K A 2 1 + K A 2 1 ) P i l + ( B 2 i * * * * h ™ " 2 l ) P i 2 + ( C 2 i ' + K C 2 1 K C 2 1 ) P i 3 -ft; *{(VT) = K D 1 1 p K ( A [ l p A p p p ; i + K ( B l l l + B 1 - u ) P ; 2 + K ( C l u - f C p i ) P ; 3 210 ^(VT) = K(A, Â"̂P'1|K(B,JB, P̂,124C(C'JC"̂P,13 Q' = A^P^+B^P^P- Q4 = ̂ 2(VT)[Q'-1] Q5 = ( 1 - * 2 2 ( V T ) [ D * + ( A i i + A ' , ' ) p i i + ( B n + B ' , , ) p i 2 + ( c i i ^ ' ' , ) p i ^ % " * i 2 ( V T ) t ( A 2 1 + A 2 \ ) P l l + ( B 2 1 + B 2 V P l 2 + ( S l ^ n ) P ; 3 ^ = * 2^(VT)[Q 3+1]; = <I»22(VT)Q3 Q8 ' = ( 1 + * 2 2 ( V T ) ) [ D * + ( A i \ + A i i , ) p i i + ( B n + B i i , ) p i 2 + ( c n ^ ' ' , ) p n RL - • «2(VT)[D*4-(A-1+Ai')?i1+(B1-1+Bi;• )Pi2+(c--1+c;;• >p i 3] *J V(T) =A1?Pi+Bl2P2+Cl2P3; 1 V *2l( T) = ( A21 + A2i + A2i ' ) P i +( B21 + B2; + B2r) P2 ( C 2 1 + C 2 1 C 2 i , ) P 3 • ̂ ( T ) = W-(Ail+Aii+Aii')P1 + (B̂+Bi+B-̂P- + (Ĉ  i+C{ {')P' *\V2iV. = * 2 2 ( T ) ; *J V(T) = ̂ m+(A[11+Aiil)Pi+(Blll+Biil)P2+(Clll+Ciil)P 4>2V(T) = ( + PJ + (B̂B̂P' +_(C2,+C2,,)P31 r i v i v ' / Tv i v 2 Tv i v i v fv~ T Y 1 1=[(<|) 1 1(T)+* 2 2(T))+/(<(. 1 1(T)+(|) 2 2(T)) -4(<t, U(T )4> 2 2(T)-^ 1 2(T)<() 2 1(T))]/2 r i v iv / iv Tv 2 i v fv ' Tv Tv i T 1 2=[(<t> 1 1(T)+4» 2 2(T))-/(<j) 1 1(T)+<j) 2 2(T)) -4(<t, U(T)4> 2 2(T)-<}> 1 7(T)^ 1(T))]/2 211 °ir(Yir*22(T))/(YirY12)j ^ 2 - < Y 1 2 " * 2 2 ( T ) ) / ( Y 1 2 - Y l l ) IV IV IV a 1 3 = * 1 2 ( T ) / ( Y n - Y 1 2 ) ; " i A - ^ ^ ^ ^ ^ l l ^ a 1 5 " * 2 1 ( T ) / ( Y i r Y 1 2 ) IV IV IV IV a 1 6=0» 2 1(T)/(Y 1 2-y 1 1); « 1 7-(.Y 1 1-* 1 1(T)/(Y 1 1-Y 1 2); " i s " ^ - * ! ! ™ ^ ^ ! ^ N N-1 j=l 1=0 x ̂  /Vfr ,1V,^,, , l V / m N , N - l - i i r ,1V,,,,, rlV,,,,. N - l - i i . 2 11 . - . n 11 1 ( T ) +°13*2 ( T ) l Y l l +I-°12*1- ( T ) + a14*2 ( T ) ] Y 1 2 *>j j=l i=0 J D 3 1 = N(<j> - 1) V) = D+CAp+Ap+Ap ' )P[[+(B|1+B-+B ' ' ' ) P ( ' + ( C ' ' ' + C • ' ' )P{̂ * ! 2 ^ ) = A 1 2 P i i + B 1 2 P i 2 + C 1 2 P i 3 ' <P2lCV) = < M 1+A- +A' r )P i i +(B^ 1 +B- +B-«)P- + * 2 2 ( V ) = A 2 2 P 1 1 + B 2 2 P 1 2 + C 2 2 P 1 3 ; ^ 1 P{ 2=exp(-s 2V); P-=exp(-s 3V) Y31= [(*{pv)+<f,22(V))W(f^ Y32= [(^(VyH^CV^ + Z ^ ^ b l - ( Y 3 r 4 2 ( V )> / ( Y3r Y32 ) ; b2 = ( Y 3 2 - * 2 2 ( V ) ) / ( Y 3 2 - Y 3 1 ) b, = <2(V)/(Y31-Y32); b, - <2(V)/(Y32-Y31) 212 APPENDIX 3 RELATIONSHIP BETWEEN j> AND DECAY RATIO INDEX Let a be the decay ratio, defined as the ratio of the overshoot/undershoot at one sampling instant to the overshoot/undershoot in the succeeding sampling time. That is _ C(1T) - CO") C(«Q - C(iT) . a , n C(nT) - C(») C(°°) - C(nT) (.AJ.I; where C(iT) is the output response value at the sampling instant where an overshoot or undershoot occurred and C(nT) is the next sampling point where an overshoot or undershoot resulted. C(°°) is the system output response at an i n f i n i t e sampling point. For a unit step input change, C(°°). tends to •unity. If i t is assumed that there is overshoot or undershoot at each sampling time then for N sampling points, the total error expressed in terms of the decay, ratio and the fi n a l overshoot or undershoot is given as ~ N-1 - _ _? _M_? I e(iT) = [c{(N-l)T} - C(NT)][l + a + a + + a ] (A3.2) i=0 Equation A3.2 is a geometric series with f i r s t term C(N-1T) - C(N) and geometric progression ratio of a. Expressing in short form gives N-1 r -U-li I e (iT) = [c{(N-l)T} - C(NT)] ^ (A3.3) i=0. 1 - a Also for the condition N-1 _ r -2(N-l)i I e (iT) = [C(N-1T) - C(NT) ] [- J i=0 , - 2 213 Therefore, the ratio becomes N-1 I e(iT) i=0 = (1+g) Y e 2(iT) [clCN^l) 1} " C(NT)] ( l + a N _ 1 ) i=0 But since N has been taken to be a function of the settling time, the overshoot or undershoot [c(N-lT) - C(NT)] has already been specified. Thus '<)> - • <A3-5> (1+a ) is ,the relationship between <|> and the decay ratio index. 214 APPENDIX 4 11 PARAMETER DEFINITIONS FOR CONTROL SYSTEM WITH ZERO-ORDER HOLD (BJ+8J) + /(3{+3^) 2 - 4(B'B^+B2'ep Y, = 2 — (8£+8^) - A8{+8^) 2 - 4 ( 0 ^ + 3 ^ ) Y 2 = — 2 -9 T -9 T -9 T -9 T -9 T -0 T S{ = c^e 1 + a 2e 2 - (l+c^e 1 + c^e 2 ); 8̂  = a.5(e 1 -e 2 ) -9 T -9 T -9 T -9 T -9 T -6,1 $j = 2a 6(e 1 -e 2 ); 8̂  = a ? e 1 + a ge 2 ; P L = e 1 ; P 2 = e Y, - 8. Y„ - 8. 80 89 1 • 4 . „ 2 4 • „ _ 2 -. Q _ 2 . Q _ T l - T 2 ' 12 Y 2 " Y L ' 13 YL " Y 2 ' 14 Y 2 - Y^' 15 Y 1 T l - B l , Y2 " \ "16 Y? - T^' "17 Y ; " Y?_' "18 Y2 " YL N N-1 v N - l - i Dx = * I [ I U ( 1 + a 3 P l + a 4 P 2 ) e i l + - a 6 8 1 3 ( p - r P . 2 ) ^ l j=l i=0 .+ [ < 1 + V l + V 2 ) 9 1 2 + a 6 6 1 4 ( p r P 2 ) ] T 2 " 1 " 1 N N-1 N - 1 . D 2 = (24--1) [ [ [ {[(l+a 3p 1+a 4p 2>e i l + a & e ^ C p ^ ) ]y\ j=l i=0 + [ ( l + a 3 p 1 + a 4 p 2 ) 0 1 2 + a 6 0 u ( P l - p 2 ) ] T ^ 1 " 1 } -0 V -9 V -8 V - 9 ^ D3 = N(cf)-l); ^ ( v ) = c^e 1 + c^e 2 - K(l+a 3e + c^e ) -0 V -6 V -0 V -8 V 4 > i 2 ( V ) = a 5 ( e 1 - e ( * ) 2 i ( V ) = " V 1 + k ) ( e " e • ) 215 -e v -e2v 6 2 8i e ^ ( V ) = a ? e 1 + a 8e ; ̂  = °<2 = ̂  _ a 3 = Tff=B^ e _ L _ e - 9 i "G2 4 - e 2 ( e 2 - e 1 ) — 5 e 2 - V " 6 . e 2 - e p ~7 e 2 - e p 8 ^ - e 2 -ep -e2v Let P 1 ; L = e ; p 1 2 = e . . A i = V l l + a2P12 + a7 P l l + a8P12 ! A2 = ( V l l + V l 2 ) ( a i P l l + V l 2 ) A3 = ( 1 + a3 p l l + a4 P 1 2 ) ( a7 p l l + a8 P 1 2 " 1 ) 5 A4 = a6°5 ( p l l _ p 1 2 ^ A' = a + V i i + V i 2 ) ( V i i + V i 2 + 1 ) ; A- = («7P11+«8P12)(I+«3P11+«4P12) -6 V -6 v -e V • -6 V , v i . 1 2 , 1 A . 4> U(V) = a e + a 2e - ( l + ^ e + a^e ) .- -e v -e v . -9 v -o v < f >12 ( V ) = a 5 ( e " e ) ••; <t»2J(V) = -2a 6(e - e ) -G v -o v -9 v. -e v •' 4>22(V) = a?'e 1 + a ge 2 ; * v i ( V ) = (1 + ^ e 1 + ^ e 2 ) -9 V -0 V ^J X ( V ) = a 6(e - e ) r21 -'[ • i J ( V ) + ^ ( V ) + V { ^ ( V ) + ^ ( V ) } 2 - 4{<ov1i(V)^(v) - 4.^(V)^( T 2 2 = [(<t>Vj(V)+<t>V2(V)- / { ^ ( V ) + J ^ ( V ) } 2 ' - 4 { ^ 1 i ( V ) ^ V 2 i ( V ) - 4> V 2(V)^j(V)}]/2 a l - [Y2r*22(7^ 1 <Y2rY22) 5 32 = [Y22-*22(V)^ 7 <Y22-Y21> a3 = *12(V) ' (Y2rT22) ; a4 ° *12(7) 7 (Y22-Y21) 216 APPENDIX 5 SYSTEM IDEDNTIFICATION AND INITIALIZATION IDENTIFICATION BY GRAPHICAL METHOD The control system as described in section 4.3 is used in the identification and i n i t i a l i z a t i o n process. In this stage of the study the air that controls the valve is cut off so that the valve is no more manipulated. At this condition, water is allowed to flow through the tube and out to the drain continuously, while the heating tank is f i l l e d and the recirculating pump used to circulate the water from the drum through the heat exchanger shell and back to the heating tank. This situation is allowed to continue until steady state in temperature as observed from the di g i t a l temperature indicator is attained. Then a ten percentage increase In steam pressure, manually set by turning the steam valve on the main line, is effected. A sampling time of 1 second is used to datalog the temperature pro f i l e . Due to the excessive noise in the system, the temperature response is f i l t e r e d . The datalog program requires that the temperature response, (which is the temperature of water at the outlet of the heat exchager tube), be summed up for fifteen samplings and the average used. This averaged value is fil t e r e d by multiplying with.a weighting factor and added to a weighted value of the previous filtered response. The relationship used in this algorithm (temperature response datalog) is given as T (J ) = a ^ C J ) •+ (l - a f ) T ( J-l) (A5.1) where T ( J ) is the J-th filtered response T , ( J ) is the J-th (present) averaged temperature 217 T ( J - l ) is the previous (J-1) f i l t e r e d response temperature cx̂  is the weighting factor The used i n this work is 0.4. Both the number of samplings summed up and averaged and the weighting factor are determined by t r i a l and error and comparing the printed responses with that observed-on.the d i g i t a l temperature indicator . The response of the control system for an open loop te percentage steam pressure change Is shown in Table A5.1. The "process reaction curve" is shown in F i g . A5.1. Since there Is no prior knowledge of the control system's transfer function, an approximate transfer function is obtained by the method of 6 3 " Strejc . This method is based on the fact that the step change response of a system comprising n time constants can be val idly approximated by a transfer function containing n times the same time constant. In most cases, the expression of the approximate transfer function is combined with a delay time (distance velocity lag) in order to increase the accuracy of the approximation. The general form of the transfer function is Ke" T S - G (s) = — - (A5.2) P ( T l S + l ) n The transient response of the control system is represented in F i g . A5.2. The method of analysis requires the knowledge of the point of inf lexion P. A tangent to the response curve is drawn through this point and extrapolated to both the horizontal axis and the horizontal l ine joining through 13 C as shown in the figure. Thus by setting the response deviation 218 Table A5.1 - "Transient Response" Time i n Seconds Temperature °C Deviation From I n i t i a l Point 0 12 • 0 15 12.4378 0.4378 30 12.8209. 0.8209 60 12.99320 0.99320 75 13.10150 1.10150 90 13.27170 1.27170 105 13.3787 1.3787 135 13.51770 1.51770 150 13.69870 1.6987 13.903 1.903 ' '; . 195 . 14.1001 2.1001 210 14.2684 2.2684 240 14.49980 2.4998 255 14.66620 2.6662 300 14.81210 2.81210 Fig. A5.1 - Process reaction curve for a 10% step change in steam pressure. Fig. A5.-2 - Approximate estimation of transfer function parameters St're'jd method. 221 from the i n i t i a l temperature (12 C) to zero, the value of T is obtained. u By increasing the deviation f to 1 (13 C), the value of T is determined, i b then T is calculated as T = T, - T . The table drawn up by V. Strejc a a b u (Table A5J.2), directly affords the order 'n' of the transfer function as dependent upon the value of T /T . This result can also be verified by using the deviation response (iK) value or the ratio T /T . For the value of 'n' i e a thus obtained. Table A5.2 gives pairs of values which enable a direct calculation to be made of the time constant r , by using the ratio T /T and T /T, or vice versa. If the T /T value gives a ratio that does not correspond to an u a integral ' n' value but f a l l between two consecutive values, take the lower integral n of these two values. This simplification is taken into account by introducing a dead time into the expression of the transfer function. The actual value of the ratio T /T ,. corresponding to the process reaction curve was calculated, but the approximation caused a lower value of this ratio to be selected. Knowledge of these two ratios permits the calculation of x. . The absolute value of T is not affected by the dead time. It is thus possible to write T T +x . (/-) T (-£-) a real a table (A5.3) T T (^) = C^> + f a real a table a 222 TABLE A5.2 'TABLE OF COEFFICIENTS" (V. STREJC) 6 3 n To/T / T /T u T /Ta u Td/T * i T /T e T /T e a 1 1 0 0 0 0 1 1 2 2.718 0.282 0,104 - 1 0.264 2.00 0.736 3 3.695 0.805 0.218 2 0.323 2.500 0.677 4 4.463 1.425 0.319 3 0.353 2.888 0.647 5 5.119 2.100 0.410 4 0.371 3.219 0.629 6 5.699 2.811 0.493 5 0.384 3.510 0.616 7 . 6.226 3.549 0.570 6 0.394 3.775 0.606 8 . 6.711 4.307 0.642 7 0.401 4.018 0.599 9 ; 7.164 5.081 0.709 8 0.407 4.245 0.593 10 7.590 5.869 0.773 9 0.413 4.458 0.587 223 It is thus sufficient to multiply the difference between the two ratios by T in order to obtain the value of x. If the system possesses a a natural dead time, which was neglected when the origin of the curve for the analysis was chosen, the value of the natural dead time is added to the calculated value of x to give the effective dead time. A l l the coefficients of the approximate transfer function are then determined.(the value of constant K can always be reduced to Unity by choosing a suitable unit). For this study T = 6.33; T = 18.67; T = 12 u a e . . V r The ratio T , ~„ T .. ~ T^ " I 8 T 6 T = ° ' 3 3 9 ' " 1FT67 " ° ' 6 4 2 7 a n d *1 3 t P " ° ' 3 5 7 a a - From Table A5.2," these ratios f a l l between n = 4 and n = 5. Choosing n = 4, the dead time is calculated from equation A5.3. That is A real a table a 0.339 - 0.319 = Y~ a But T =. 18.67. Therefore x = 0.02(18.67) = 0.3734 a T Also from Table 6; the ratio — for n = 4 is 4.463 T Therefore T = .. . a_ = 4.183248 ^4.2 4.463 . From Fig. A5.2, K = .1 . 224 Since i t is not possible to determine the natural dead time due to the method formulated for datalogging the response, a dead time of 0.5 sees, is assumed from previous observation. Thus the effective dead time is (0.3734+0.5) = 0.8734 sees. The approximate transfer function of control process is -0.8734s G(s) = - 7- (A5.4) (4.2s+l) A useful graphical method of analyzing the response of a system 53 having two time constants is the Oldenbourg and Sartorius method. This method also depends on locating the inflexion point and the slope of the curve at this point. More convenient for analysis are the quantities T^ and T^ shown in Fig. A5.3. In the Oldenbourg and Sartorius diagram Fig. A5.4 the ratio T /T. is used as the intercept on each axis of the straight line, c A The straight line intersects the curve at two points, if the ratio T /T is C A greater than 0.73, either of the intersection points can be use to calculate the two time constants T̂  and . Their graph Fig. A5.4 covers the whole range of possible ratios of T̂  to T̂  from infinity to unity. Their response curve was derived from an actual second-order process. When the ratio of T /T is 0.73, the straight line T /T = T /T + T /T is tangential to 3 C C A i. A Z. A their curve and thus T^ = T^. As is observed from Fig. A5.3, the ratio of T /T. for the control system is less than 0.73 (0.6427) which also is C A expected for a higher order process as shown earlier on. In order to model the control process to a second-order case, the two time constants are set Time Sec • Fig. A5.3 - Fitting transient response to a second-order with dead time model by Oldensbourg and Sartorius method. 226 F i g . A5.4 Oldenbourg and s a r t o r i u s diagram f o r e q u i v a l e n t time constants from process r e a c t i o n curve f o r second-order process. 227 equal to each other. The method of calculating the time constant ignores that part of the reaction curve which precedes point A in F i g . A5.3. An appreciable time has actually elapsed before point A is reached. This is shown in the smoothed reaction curve of F i g . A5.5. The f lat section of F i g . A5.5 is not necessarily caused by dead time alone. To determine process parameters, the following steps are performed: ( i ) The time constant T 1 , T 0 are set equal to each other and equal to where T = 18.67 (see F i g . A5.3) thus T = T 2 = 0.365(18.67) = 6.81455 6.8 ( i i ) Set T F i g . A5.5 equal to 0.365T = 6.8 ( i i i ) Measure T from F i g . A5.5. That i s , T = 14.2 P ... P ( iv)- The dead time T is calculated, as T. = T - T 14.2 - 6.8 = 7.4 D P i Thus, the transfer function of the process as a second-order plus dead time model is -7.4s F i g . A5.5 - Determination of time constants f o r the system modelled as a second-order plus dead time process. 229 APPENDIX 6 IDENTIFICATION BY QUASILINEARIZATION METHOD The basic assumptions necessary for the formulation of the identification algorithm used in this study are constant dead time (or negligible variation in i t ) , constant values for sampling time, f i l t e r i n g . time and weighting factor for f i l t e r i n g the measured temperature response The quasilinearization method (Eveleigh, V.W)16 identifies in the second-order overdamped plus dead time transfer function by solving for successive solutions of the transfer function linearized with respect to variation in the unknown parameter. For a given input [r(k)j the model output is forced to f i t the observed output in a least square error sense The dif f e r e n t i a l form of the transfer function is given as 2 V + 2 T1 of + C ( t ) = r ( t _ T ) ' (A6.1) dt where = process time constant x = dead time c(t) = response or output • This can be expressed in the form of a set of first-order linear d i f f e r e n t i a l equations with c = x1 as 230 Let T j be treated as an additional state variable with the linear d i f ferenta l equation (A6.2c) Expressing equations (A6.2a), (A6.2b) and (A6.2c) i n matrix form gives x 3 = xl = 0 x = Ax + B r ( t - T ) where x = (x^ x 2 x^); x( t o ) = x̂  (A6.3) "0 l cr ~0~ A = -e -e^ 0 ; B --e 0 . 0 0 0 / — — 0 = - T 4 T Equation A6.3 can be represented as k = f ^ x , r, t) 1 = 1, 2, . . . , N (A6.3a) Let i t be assumed that f^ and i ts derivatives relative to x and r are continuous functions of x and r . Also let Y(t) and x(t) denote a nominal control input and the corresponding output response, respectively and known over the interval [tQ , Expanding equation (A6.3a) i n Taylor series about the measured X(t) provides the equations defining changes from the nominal trajectory in terms of changes in i n i t i a l state and control over the interval as N 3f. N Zf± 6 x * J-, 3x^ 6 x j + J 3rT 6 r k + R i ( x ' r> c> J=l J k=l k (A6.3b) where p A j . - , A r • • • fix. = x, - x. ; or. = r. - r, ; ox. = x, - x, 3 3 j ' k k k ' i i i R^(x, r, t) includes a l l terms involving higher-order derivatives and are made negligible by taking s u f f i c i e n t l y small values of 6x and Sr. The p a r t i a l derivatives in equation (A6 .̂3b) are evaluated along the nominal 231 trajectory and are thus generally dependent upon t. Disregarding the R^(x, r, t) term in (A6.3b) and recognizing that the input is a step change, the l inearized equation of (A6.3) about the past trajectory i s 6x = H6x ; 6r = 0 (A6.4) where H is defined later on. . The computational procedure involves the following steps: ( i ) r ( t ) and x(t) are recorded over a time period T. Note that r( t ) and x(t) are the measured input and output responses, ( i i ) The model equations, — equations (A6.2a), (A6.2b) and (A6.2c) are programmed i n the computer. ( i i i ) Guess starting conditions x ° as near the true value as possible, otherwise convergence becomes a problem. These i n i t i a l values can be obtained from least square estimates, or through physical knowledge of the process. In this study the i n i t i a l value used is that obtained earl ier on by graphical analysis , ( iv) With these i n i t i a l values, numerically integrate equation A6.3, use this value and the input r ( t - x ) to minimize the function JN = ^ t + T ViA* d t ( A 6 , 5 ) o (v) The model equations of (A6.2) are l inearized about the trajectory obtained in step ( i v ) . (vi) The effects of change in x° upon system responses are obtained by solving the l inearized equations derived in step (v) computationally. 232 ( v i i ) The linearized solutions are weighted by an arbitrary constant matrix Q and is expressed as a general function of Q. ( v i i i ) J N is minimized relative to Q, to determine the desired parameter changes on the next iteration. (It should be remembered that these changes are based upon a first-order system linearization and are not , in general, the exact changes required), (ix) The process is repeated, i f necessary, until successive adjustments provide negligible improvements in J^. (x) The resulting model parameters are read out as the desired plant identification . i The linearization about the past trajectory gave equation (A6.4), that i s 6x = H6x ; 6r = 0 (A6.4) where the elements of matrix H are h. . = 0* ) and "*" means that the state i j 6 X j variables are evaluated on the past trajectory. The component of the function H are h = x 2 ; h 2 = -6 2 X ; L - 6 4x 2 + 6 2 r ( t - T ) ; h~3 = 0 (A6.6) 2 1 2 But 6 = — r - ; 9. = — in terms of the state vector, x. Therefore the 2 4 x_ X3 3 elements of H are A "fc A A 2 A h,, = 0 ; h 1 0 = 1 ; h n, = 0 ; h 0 1 = -6 . ; h 9 9 = -6 4 ; 233 23 2 0 3 X l + 26 2 x 2 - 29 3 r(t-x) = y (A6.7) '31 = 0 h32 = ° h33 = 0 Thus H = 0 1 0 -e 2 -e^ ^ 0 0 0 Solving equation (A6.4) by state transit ion method gives 5x(t) = <))(t, t Q)6x ; 6r = 0 (A6.8) where <|)(t, tQ) is determined by solving the relat ion K t , t Q ) = H ( t ) * ( t , t Q ) ; - * ( t Q , t Q ) = I (A6.8a) Since equation (A6.8a) shows that H is a function of time, the equation can be solved correctly by making the approximation <j>(t, t ) = <|>(t-t ); where o o <|>(t-t ) is the transition matrix for the constant coefficient l inear o d i f f e r e n t i a l equation over the interval t . , - t = At and h . . , are n+1 n i j ' s evaluated at x(t ). Hence n 4>(t) = H-<Kt) ; <Kt ) = 1 (A6.8b) Laplace transforming (A6.8b) gives <J>(s) = (SI-H) * <|>(o). Inverting results into the relation <j> = T <(> ; (> = ())(o) = 1 (A6.8c) n+1 n n o 234 where T = L - 1[(SI-H and H = H(t ) n L v n J n v n That is (SI-Hn) -1 (A6.8d) Thus T = n e- 9 t + 8te- 9 t - e 2 t e - 0 t -8t te - e t _ - e t e - 8te 1 ( 1 _ e - e t + e t e - 6 t n ) e -8tn (A6.8e) It should be borne in mind that T^ varies as t changes, since the x's will be assuming new values for each change in t. The values <J>(t ) serve as n the i n i t i a l conditions for the computation of over the interval n+l At = t^ +^ - t^. This process is repeated over the interval of integration to obtain the trajectory. The response of the model to i n i t i a l conditions o , r o x + ox is x . ( x +6x ) = x . ( x ) + J 6x. d>.; I I • i J i J j=l (A6.9) where the "t^jtg a r e from equation (A6.8c). 235 Let the performance index be, minimize t +T 3 3 JN = U° I + I - \]Z dt (A6.10) o i=l j=l J J Where t and T are the start and duration, respectively, of the observation interval; the 4̂ 's are weighting factors—generally held constant and assumed known; the x^ are model responses to observed input r(t-x); and x̂  are observed values of the system state. The weighting factors ¥^ corresponding to unavailable (or unmeasured) elements of state are set equal to zero. Partial differentiating equation (A6.10) with respect to 5x°j and 9 JN setting -= 0; gives a set of 3 algebraic equations of the form 3 3 0 = f[+T 2 l V j x ^ x 0 ) +; I ' ^ • i k . - . x i ] * l j d t (A6.ll) But since there is only one output, (A6.11) reduces to t + T 3 0 = / ° 2 [ X ; L(x 0) + I 6x^*lk " x j * dt (A6.1U) o k=l J That is, by setting 4̂  = 1 and ̂  = = 0 because the states are no directly measurable. x^(x°) is the model response and is given as x L = x 2 (A6.11b) from equation (A6.3). 236 Numerically integrating (A6.11a) by Runge-Kutta 4-th order formula given as Ah X _LI = x + T~<K i + 2 K o + 2 K o + K /) (A6.11c) n+l n 6 nl n2 n3 n4 where Ah is the integration grid size and K = f ( t , x ) = x.(x°) nl n n 2 n K = f ( t + \ Ah, x + I Ah) = x_(x°) + ^- k n2 n 2 n 2 2 n 2 nl Kn3 = f ̂ n + \ A h> X n + \ ^ = ̂  + ^ Kl K . = f ( t + Ah, x + Ah) = x„(x°) + Ah K „ n4 n n 2 n n3 (A6.12) Thus at any time t, x , = x„(x°) + T^(k , + 2 k „ + 2 k 0 + k . ) (A6.12a) n+l 2 n 6 nl n2 n3 . n4 where the K 's are as defined in equation (A6.12) substituting equation n (A6.11c) for•x- ' in equation (A6.11a), a l l the terms in the equation are. known except ^xj^. Thus, (A6.11a) is a linear equation with one unknown, which is solved for 6x.° as k t +T • t +T - J t ° [{x^x0') - x^}* ]dt•= \ ° * u • 6x°df (A6.13) and -/ t° [ { ^ ( x 0 ) - xt} • ]dt 6x° =• - _ _ _ _ _ _ _ _ (A6.13a) 237 The next computer i t e r a t i o n i s made based upon the revised i n i t i a l conditions, or x° = x° + 6x° (A6.13b) new old The e n t i r e process i s repeated based upon these new i n i t i a l c onditions. The i t e r a t i v e process i s terminated when improvements become n e g l i g i b l e . Using i n i t i a l state values of x^ = 0; x^ = 0 and x^ = 6.0, the above algorithm was programmed and run i n the PDP8 d i g i t a l computer. This method gave a time constant of 5.02710 with an i n t e g r a t i o n gr i d size of 1. The same dead time value as determined i n the graphical method was used since the l i n e a r i z a t i o n used i n this method requires the computation of the d e r i v a t i v e 4~̂ t T ^ = - r ( t - T ) . The process reaction response used i n this oT determination was generated by a step input which does not y i e l d s u f f i c i e n t information to c a l c u l a t e the delay time. 238 APPENDIX 7 THEORY OF VARIABLE GAIN METHOD OF DESIGN65 L e t the d e s i r e d d i g i t a l c o n t r o l l e r as shown i n F i g . A7.1 be t r e a t e d as a v a r i a b l e - g a i n element K^, which takes on d i f f e r e n t va lues from one sampl ing time to a n o t h e r . The i n p u t to the v a r i a b l e - g a i n element K ^ i s the c o n t r o l s i g n a l u , and the output Is assumed to be u ^ . At any s a m p l i n g i n s t a n t t - n T + , the i n p u t and output of the v a r i a b l e - g a i n element are r e l a t e d through a constant m u l t i p l y i n g f a c t o r K^; that i s u 1 ( n T + ) = K n u ( n T + ) (A7.1) where k i s the g a i n constant of the v a r i a b l e - g a i n element d u r i n g the n s t (n+1) sampling p e r i o d . ' Based upon the above s u g g e s t i o n , the t r a n s i t i o n m a t r i x of the system i s expressed as a f u n c t i o n of the v a r i a b l e - g a i n K and has d i f f e r e n t n v a l u e s at d i f f e r e n t sampling i n s t a n t s . I t has been shown by Tou, J . T , 6 5 t h a t the s t a t e - t r a n s i t i o n equat ions for a l i n e a r system are g i v e n by v ( n T + ) = BV(nT) (A7 .2) v [ ( n + l ) T ] = <j>(T) V ( n T + ) .. (A7 .3) v[ (n+l)T] , = <)>(T) BV(nT) (A7.4) Thus, when n = 0 v ( 0 + ) = BV(0) (A7.5) HOLD U DIGITAL COMPENSATOR PROCESS Fig. A 7 . 1 , - A d i g i t a l control system. 240 and at t = T V(T) = <|>0(T) BV(o) (A7.6) where V(o) is the given (or derived from state variable diagram) i n i t i a l state vector. Since the transition matrix <)>g(T) is a function of the gain constant Kn of the variable-gain element during the f i r s t sampling period, the state vector V(T) at t =.T is also a function of K0. Once V(T) is determined from equation (A7.6) in terms of Kg, the state-vector VCT*") can readily be found from the state-variable diagram of the system or by use of equation (A7.2). It follows from equation (A7.3) that the state vector V(2T) is given by V(2T) = ^ ( T ) V(T+) (A7.7) where the transition matrix ^ ( T ) is a function of the gain constant Kj of the variable-gain element during the second sampling period, and the state vector V(T +)' is a function of..-Kg. Thus, the state vector V(2T) is a func- tion of both Ko and K i . Once V(2T) is found, the state vector V(2T +) follows from equation (A7.2). The state vector V(2T +) is also a function of both Ko and K\ . With the same reasoning, at t = jT, the state vector is given by V ( J T ) = > ( ( j - l ) T +) ^ (A7.8) where the transition matrix <t>j_̂ (T) is a function of the gain constant of the variable-gain element during the j-th sampling period, and the state vector v [ ( j - l ) T + ] is a function of the gain constants K^, , K̂ , K-_2* 241 Hence the state vector v(jT) i s a function of the gain constants K Q , , IC, • • •, K.j 2 and ^ • The pulse-transfer function D(z) of the d i g i t a l controller can be expressed in terms of the various gain constants of the variable-gain elements as follows: ' - Beginning with ,-• then s i m i l a r l y , u(o + ) = r (o + ) U l ( o + ) = K Q u(o + ) = K Q r (o + ) U l ( T + ) = K l U ( T + ) + + + where u(T ) is obtained from v(T ) = Bv(T) = Bd> (T) v(o ) o (A7.9) (A7.10) ( A 7 . U ) (A7.12) Since u(T ) is defined as an element of V, where V = S i m i l a r l y , U l ( 2 T + ) = K 1 u(2T + ) where u(2T +) is derived from V(2T+) = B^CT) V(T + ) V(2T+) = B* 1 (T) B<J>0(T) V(0+) In general, U l (jT+) = K j U ( j T + ) r x (A7.13) (A7.14) (A7.14a) (A7.15) 242 where uCjT**") is determined from V(jT+) = B* (T) V[(j-1)T+] (A7.16) V(jT+) = B* (T) B<J>_._2(T) B<t>0(T) V(0+) (A7.17) The z-transform for the control sequence u(jT +) is n _ . u(z) = I u(jT+)z 2 (A7.18) j=0 and also the z-transform of the sequence ui(JT +) is given as n _ . u (z) = I K u(jT+)z J (A7.19) j=0 J the pulse-transfer function D(z) is the ratio u l ( z ) ^u(^)«"J D ( z ) = - i y - = ^ . - (A7.20) I u(jT+)z J j=0 Thus, the design reduces to the determination of the various gain constants K of the variable-gain element. Once the gain constants K are .] J found, the desired d i g i t a l controller is derived. The gain constants k are j evaluated from the performance specifications. For a deadbeat performance, the following conditions must be satisfied. The output response is always less than the input signal for t < pT, where T is the sampling period. The system error is zero for t > pT. These conditions are satisfied i f x L( PT) = r ( P T ) (A7.21) x 2 ( P T ) = x 3 ( P T ) = = x p ( P T ) = 0 (A7.22) where p denotes the order of the control process, and the state variables 243 x (pT), x (pT) , x (pT) are functions of the gain constants K , K 2 3 p 1 0 1 K K , which are derived from (A7.8). Equation (A7.22) implie 2 p-1 that the inputs to the various integrators are equal to zero for t >_ pT. The successive gain constants K_. can be determined by solving equations (A7.21) and (A7.22) simultaneously. 244 APPENDIX 8 DEADBEAT COMPENSATOR DESIGN FOR CONTROL SYSTEM WITH ZERO-ORDER HOLD The overall transfer function for the control system is - - T S , . -Ts. h { s ) 8 ( 8 + 6 ^ ( 5 + 6 2 ) (A8.1) For simplicity in state-variable diagram, the zero-order hold is treated as a clamp (cl) or hold. The state-variable diagram of (A8.1) is as shown in Fig. A8.1. where 8 1 T ] L' "2 = 8^8^. and T 0 are the control system time constants. The state vector V is defines as V = (A8.2) The i n i t i a l state vectors are V(0) 1 0 0 0 (A8.2a) Since there is process delay of A in the system, there shift of A of the origin. After a unit step change in i n i t i a l state vectors, become w i l l be a set point, rightward the F i g . A8.1. - S t a t e - v a r i a b l e diagram by i t e r a t i v e (cascade) programming method. 246 v(A) 1 0 0 1 (A8.2b) From F i g . A 8 . 1 , the set of d i f f e r e n t i a l equat ions i s f = 0 or i 2 = -e 2x 2 + e u x l = ~ 6 1 X 1 + X 2 + 8 u (A8.3) V = AV A = .0 0 0 0 - 0] 0 0 0 0 1 9 -e 2 e o o The t r a n s i t i o n d i f f e r e n c e equat ions fOr the c o n d i t i o n t = nT + A, where T i s the sampling p e r i o d and A i s the process dead t i m e , are r [ ( n T+A)+] = r[(nT+A).] ; u[(nT+A)+] = r [ ( n T+A)].- x [(nT+A)] x 2 ( n T + A) = x 2 ( n T+A) ; X ] L [ ( n T + A ) + ] = x ^ n T+A) The t o t a l delay time i n the c o n t r o l system i s nT + A, where nT i s due to the h o l d and A which can be broken down to (j+,6)T ( i n t e g r a l and f r a c t i o n a l ( A 8 . 4 ) 247 components) is the dead time. The solution to equation (A8.3) is v(t) = 4>(A) V(A +) where X = t - (n+j+S)T and <t>(A) = ^-""[SI-A]"1. But [sI-A] (s + O 1)(s+0 2) 1 s + B 3(s+8 2+l) j(s+0 1)(s+° 2) s(s+« 2) (A8.5) and <j)(X) = L _ 1[SI-A] -1 0 0 -V o 0 - e x - e x b{(e -e ) • - e 2 x \ - e x e ? x bl +. ble + b!e 2 j 4 - e x e. d-e V ) where (A8.6) b2 = e 2 + i 6(8-8+1) b' = • b' 6 1 ( e 1 - e 2 ) • " 4 6 ^ - 6 ^ 2 1 Introduce the di g i t a l compensator at the dotted line position in Fig. A8.1 Assume that i t is a variable-grain element K , which means that the value of n 248 varies from one sampling period to another. The input to the variable- gain element K is the control signal u, and the output is u . At any n 1 instant t = (n+j+6)T+, the input and output of the variable-gain element are related through a constant multiplying factor k , ie. n u1[(n+j+6)T+] = knu(nT+) (A8.7) (By assuming that the whole process delay is encountered in the compensator). With the addition of the d i g i t a l compensator, the transition matrix becomes 4 (X) = 4> (K ) = n n n 0 - e x - e x - e x - e x - e x 0 e b.'(e -e ) (bl+bie +b!e )K l Z J 4 n 0 0 o ". '• 0 - e 2 x \ ^ >Kn 0 - •; •.. . • • : i v ; (A8.8) For the condition t = (n+j+l)T, X becomes (n+j+l)T - (n+j+6)T = (1-5)T = V. Thus for n = 0, the transition matrix is V V ) - • o ( K o ) 1 0 0 -e v _e v -e v 0 e 1 b|(e 1 -e 2 ) 0 0 0 0 -e v 2 0 -0 7 -0 V b 2 + b 3 e 2 + b 4 e 2 ] K0 -0 7 ) K Q 1 (A8.9) 249 But the c o e f f i c i e n t of the t r a n s i t i o n difference equation i s B = 1 0 0 0 0 1 0 0 0 0 1 0 1 -1 0 0 (A8.10) Thus, v[(l+j)T] = <t>Q(V) BV(0); but BV(O') = V(A+) 0 0 Therefore V [ ( I + J ) T ; • -e v - e v . b' + b 3e + b 4e ]K 0 -e 2v ^ d - e )K Q (A8.ll) 250 Also, v[(l+j)T+] = BV[(1+J)T] = -6 V -9 V [b' + b 3e + bje ] K Q -9 V \a-* 2 ) K 0 -9 V -9 V 1 - [b£ + b^e 1 + b^e 2 kQ (A8.12) For n = 1 0 0 -9. V -9 V -9, V 0 e -9 V -9 V b'(e 1 -e 1 ) [b^+b^e +b^e 2 ] ^ 0 0 0 0 -e 2v -e 2v 9 1(l-e )K r (A8.13) 2 5 1 Therefore, V ( ( 2+j)T) = • 1(V)V((l+j)T +) = 1 -e v -e v -e v -e v -e v -e v -e v -e v e ( B 2 + B 3 E +b^e )K +b'Q ( 1-e )(e -e )K +(b2+b3e e Z ) [ 1 " ( (I))]K1 - 9 V - 9 V - 9 V - 9 V - 9 V \e 2 ( 1-e 2 ) K ^ p i - e 2 )[l-(b 2+b 3e 1 +b'e 2 )K Q]K 1 - 6 V - 9 V 1 - (b 2 + b 3e 1 '+ b 4e 2 )K Q ••; - 9 V - 9 V •  • '-. * = ( B 2 + B 3 E + B 4 G > K 0 ( A 8 . 1 4 ) Designing for deadbeat performance requires that the inputs to a l l integrators be zero for t >_ [ ( 2 + j ) T J . From the state-variable diagram Fig. A 8 . 1 , the following conditions hold: ' X J ( 2 + J ) T ] = e 2 ( A 8 . 1 5 ) X 2 [ ( 2 + J ) T ] = ep and u 1 [ ( 2+j)T] =' k n u ( 2 T ) = |— for n > ( 2 + j ) . That i s , after the second sampling plus dead time instant, the output from the variable gain element should be kept constant at , consequently deadbeat performance requires 252 that -9 V -9 V -9 V -9 V -9 V -9 V X l [ ( 2 + j ) T ] = e (b^+b 3e +b£e )K Q + b' O ^ l - e )(e -e )K Q -9 V -9 V -9 V -9 V + (b^+b 3e +b^e 1 )[l-(b^+b 3e +b^e ) K 0 J K 1 = e 2 (A8.16) x 0 -9 v -e v -9 v -9 v -e v (2+jT) = 6 i e 2 (1-e 2 ) K Q + 8^1-e 2 ) [ l - (b'+b^e 1 +b^e ) K Q ] K ^ (A8.17) Solving equations (A8.16) and (A8.17) simultaneously gives -9 V -8 V -9 V [ 6 2 ( l - e ) - (b'+b^e +b^e )] K0 ' =9TV -8 V -9 V : ^97V =97V -8 7 ( A 8* 1 8) (1-e 2 )(e 1 -e )[(b^+b«e +b^e ) + b j e ^ l - e )] . and -0 V . .-9 V [l - e (1-e )K j _ - - K I -9 v / . . -• ; - 9 l v , -8 2v... -•• • •••••, <^'i9r (1-e ) [ l - (b^+b'e 1 +b£e )K Q] . : * -6 V -8 V From (A8.14) u(T +) = 1 - (b^+b^e +b^e )K Q = ^ Also, . U l ( j T + ) = K Qu(0 +) = K Q ; since u(0 +) = 1 and u j ( l + j ) + ] = K l U(T+) = K 1X 1 Thus, the input signal to K (variable-gain element) has the z-transform n . _ -8V -8 V u(z) = 1 + \ z'1 where X = 1 - (b2+b^e +bje )K Q (A8.20) 253 and the z-transform of the output signal from K is n u 1 ( z ) z J = K Q + K^X^z + X2z + X2z + which reduces to t . Z " J [ K 0 + ( K l X 1 ^ 0 ) r l + <X2-K1X1>Z"21 u (z) = (A8.21) 1 (1-z ') Therefore the pulse transfer function of the desired d i g i t a l controller is given by u.(z) z J[K + (K X -K )z + (X -K X )z ] D(z) = - J — - = — ^ 1 (A8.22) U ( Z ) (l - z ' S d + ^ z " 1 ) For the case when the process dead time is zero; j = 0 and 6=0; therefore V = T. Hence, n , , M(z) [KQ + ( K . X ^ ^ z - 1 + ( X ^ X ^ z ' 2 : _ D ( Z ) = E ( 7 ) - " n -I' . +. -K " (A8.23) (1-z )(1+X1z ) Equation (A8.23) can be expressed as - 1 -2 w/ v Kn• + Y, z + Y 0z where Y, - - K Q ; Y, = * 2 - ; h = \ ~ l Rearranging equation (A8.24) gives M(z)[l + Y 3 z _ 1 - X l Z" 2] = [KQ + Y L z _ 1 + Y 2z" 2]E(z) (A8.24a) 254 Expressing in difference equation form gives M(k+2) + T3M(k+l) - ^MCk) = KQe(k+2) + y e(k+l) + Y 2e(k) Let M(k) = x 3 (k) + KQe(k) Performing f i r s t and second differencing on (A8,24c) gives M(k+1) = x 3(k+l) + kQe(k+l) M(k+2) = x3(k+2) + kQe(k+2) (A8.24b) (A8.24c) (A8.24d) (A8.24e) Substituting equations (A8.24c), (A8.24d) and (A8.24e) into equation (A8.24b) gives x3(k+2) +Y 3x 3(k+1) - X ^ k ) - (Y 1 -k 0 Y 3 )e(k+l) + ( Y ^ K ^ ) e ( k ) (A8.25) Expressing equation A8.25 in the form x 3(k+l) xA(k+l) "0 X. 1 -Y, x 3 (k) x 4(k) + e(k) (A8.26) That is , x 3(k+l) = x 4(k) + k 2 e(k) x 4(k+l) = X l X 3 ( k ) - Y 3 x 4 (k) + k 3 e(k) : Solving equation (A8.27) for 4 x (k) gives x 4(k) = x 3(k+l) - k 2e(k) Differencing equation (A8.29) gives x 4(k) = x3(k+2) - k 2e(k+l) Substituting equation (A8.30) and (8.29) in (A8.28) gives (A8.27) (A8.28) (A8.29) (A8.30) x3(k+2) + Y3x3(k+1) - X L x 3 (k) = k 2e(k+l) + (Y 3k 2+k 3)e(k) (A8.31) 255 Compare equations (A8.31) and (A8.25) gives the following relations k2 " Y l K0 Y3 k3 = ' W r W (A8.32) The state diagram for the d i g i t a l compensator controller i s as shown in Fig. (A8.2). The overall transfer function of the compensator control system l i s -Ts, , _,_ -1. . -2 G(s) = 9(j-e " ) f O '2- i > ~ S ( s + e l ) ( s + e 2 ) -x + ^-v _ ^-2 J (A8.33) The signal flow graph of equation (A8.33) i s as given in Fig. (A8.3). The output nodes for this system are x^(s), X2(s), x^Ck+l) and x^(k+l). The input nodes are x^(k), x 2 (k), x^Ck), x^(k) and r(k). There are two loops -6. L = — 1 s and L2 2* s The state differential equations in matrix form is given as x 1(s) •iV8> •;2(s) • l 3 ( s ) 0 x x(kT) x 2(s) * 2 1 ( s ) <l>22(s) *23 ( S ) 0 x 2(k) +2'(s) = + x 3(k+l) " k2 0 0 1 • x 3(k) k2 x 4(k+l) _" k3 0 \ V k ) _ k3 r(k) (A8.34) 256 F i g . A8.3 - C o n t r o l system w i t h d i g i t a l c o n t r o l l e r and s t a t e - s p a c e r e p r e s e n t a t i o n . 257 where ( s + e 3 ) (8+9^(8+62). V . , 1 sCs+9, ) ( s + 9 . ) ' *12 (s+9, )(s+9„) i l ^ s ) s(s+9 1)(s+9 2) * *2iK8) (s+e 1)( s+9 2) s(s+e 1)(s+e 2) 4>o,(s) = 4 > 2 2 ( s ) s ( s+ 9 1 ) ( s+ 9 2 ) ' Y 2 3 ^ ' (s+9, )(s+ 9 0 ) ' s( s+9, ) ( s+9„ ) V 2' K 0 6 V s ) ( 3+9^ ( 8+02) Inverse transforming equation A8.34 gives x1(k+l) •iVT> •lYT> * I 3(T) x 2(k+l) •21( T > * 2 2 ( T ) * 2 3(T) x 3(k+l) " V 0 0 x 4(k+l) " k3 0 0 0 1 x L(k) x 2(k) *2(T) x 3(k) k2 x 4(k) k3 _ _ _ _ r(k) (A8.35) the transient 258 APPENDIX 9 DEADBEAT COMPENSATOR DESIGN FOR CONTROL SYSTEM WITH HALF-ORDER HOLD The overall transfer function for the control system is K s ) = 9 ^ 4 + 5 T s ^ -TS H (s) [4 + 4Ts] (s + 6 )(ar+ 8 ) (A9.1) The state-variable diagram of Equation (A9.1) is as shown in Fig. A9.1 for unit step change. NB: The H represents the zero-order hold or inbuilt delay in d i g i t a l ' o control computers. The state vector V is defined as V (A9..2) and the i n i t i a l state vector is 1 0 V(0) = 0 0 (A9.2a) U I •a. F i g . A9.1 - State-variable diagram by i t e r a t i o n (Gascade) programming method. 260 while the state vector just after the step change is made is 1 0 V(A+) = 0 0 1 From Fig. A9.1, the first-order d i f f e r e n t i a l equations are r = 0; u = 0 (A9.21 X 3 = (1/T) X3 + (5/4) 8u X2 = " 62 X2 ~ ( 1 / 5 T > X3 + ( 5 A ) 6 u X l = ~ 91 X1 + X2 " ( 1 / 5 T ) X3 + ( 5/4)6u or V = AV where A 0 0 0 0 0 0 -°1 1 -1/5T (5/4)6 0 0 -e 2 -1/5T (5/4)6 0 0 0 -1/T (5/4)6 0 0 0 0 0 The transition difference equations for the condition T = nT + where T is the sampling period and A is the process dead time, are r((nT+A) +) = r((nT+A)); u((nT+A)+) = r((nT+A)) - X (nT+A); 261 X3((nT+A)+) = X3((nT+A)); X2((nT+A)+) = X2((nT+A)) X ( ( n T+A)+) = X1(nT+A). The total delay time in the control system is nT+<f>, where nT is due to the zero-order hold and A, - which can be broken down into (j+S)T (integral and fractional components) is the dead time. The solution to Equation (A9.3) - 1 - 1 is V(t) = <()(X)(A+)> where X = t-(n+j+6)T and <KX) = L [sI-A] Thus 4>(A) = 0 0 0 e -0 ,X 1 A n<r0-ul —8 iA —69A a,'(e 1 -e z ) -<f>24(X) 0 e -e x 2 0 0 0 0 whe re - a X * 2 5(X) - a ' ( e - a X - e 9 2 ) ^ ( X ) 2 2 U 6 ) a = 1/T; a n = 1/5T; a 2 1 = (5/4)6 a' = l / ( 8 2 - 9 1 ) ; a' = a n / ( 6 ^ 3 ) (e^a) ; a' = a ^ C a - S ^ e ^ ) 0̂  = a 1 1 / ( a - e 2 ) ( 6 1 - e 2 ) ; <x» = a 1 1(8 1-a); ci£ = a 1 1 / ( 6 2 - a ) ; a} = a 2 1/6; a8 " a 2 1 / O l ( ° r 0 2 ) ; a 9 = a 2 1 / 0 2 ( 9 2 - 9 l ) ; °<io = a 2 1 / 9 l ; a i l = a l l a 2 1 / a 9 = a \ i - - ^ n ^ v ^ V ^ «;3 = - a u a 2 i / 9 i ( a - V ( V V ; 262 a' = -a a /Q (a-0 )(9 -9 ); a' = a a /a9 : o1 = a a /a(a-9 ); 14 11 21 2 2 K 1 2'' 15 11 21 1 16 11 21 1 ' °22 = 3 2 1 / a ' -aX -9 X .-0X -aX -9 X Y 2 4 ( X ) = a 2 e + a 3 e + a 4 e + a 5 ( e ~ e 1 } -9 X ' -9 X -6X -aX -9 X -9 X <t>25(X) = a7+a^e +^e +a{0(l-e ) - {o^+a^e +a^e \+«J4e } - -aX - e x {a' 5 +a i 6e + a ^ e } - e 2 x -ax - e 2 x • 3 5 ( x ) = a i 8 ^ 1 - e J ~ ^ a l 9 + a 2 0 e + a 2 1 e ' • Assume that the compensator is variable-gain element K1 , which means that n the value of K1 varies from one sampling period to another, and let i t be n introduced at the dotted rectangle on Fig. A9.1. The input to the variable-gain element is the control signal u, and the output is u^. At any instant t = (n+j+6)T+, the input and output of the variable-gain element are related through a constant multiplying factor K , i.e. n u, ((n + j + 6)T+) = K u(nT+) (A9.6) 1 n (By assuming that the process delay is encountered in the compensator for simplicity case). With the addition of the d i g i t a l compensator, the transition matrix becomes 263 4 (X) = $ ( K 1 ) = n n 1 0 -9 X „ „ _9 X - e x 0 e 1 <x'(e -e ) - * 2 4 ( X ) • 2 5 ( X ) K ; 0 0 e 0 0 -v -aX -9 X -a.(e -e ) ^ ( X V K ; -aX 0 0 -aX « 2 2 d-e ) K ; 1 (A9.7) For the condition t = (n+ j + 1)T, X becomes (n + j + 1)T - (n + j + 6)T (i - 6)T = v... , . , .• . : ; . . .. • .'/.._ • Thus for n = 0, the transition matrix is 1 0 -9 7. -0 7 . -9 V 0 e 1 a-( e 1 -e 2 ) -+ 24< V ) * 2 5 ( V ) K ' 0 0 e 0 0 - e 27 - a7 -0 7 -a'(e -e ) 4>35(V)K'_ -aX -aV a 2 2 ( l - e )K' 0 0 264 But the c o e f f i c i e n t of the t r a n s i t i o n d i f f e r e n c e equation i s 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 -1 0 0 0 (A9.8) (A9.9) Thus, 1 0 V ( ( l + j ) T ) = * (V)BV(A); but BV(A) = V(A+) = 0 0 1 Therefore, 1 V ( ( l + j)T) = 22 (1-e 1 265 Also, V((l+j)T ) = BV ((l+j)T) = For n = 1 * 2 5(V)K Q *35 ( V ) K6 -aV «' (1-e )K' 22K ' 0 1 - * 2 5 < 7 ) K , 0 (A9.ll) -e v -0 7 -6 7 0 e o'(e -e ) -*24<7> *25<7>Ki 0 0 e 0 0 •e27 -aV -0 7 -a»(e. -e ) ^ ( V * ' - aV a 2 2 ( l - e - a V ) K i 0 0 266 Therefore, V((2+j)T) = * i ( V ) V ( ( l + j ) T + ) _0 V -6 V -9 V - aV *25 K0 e 1 + a i * 3 5 K 0 ( e 1 " 6 2 ) ~ " k W 1 " 6 > +*25< 1^25 K0> Kl -9 V - a7 " - aV -8 7 o t 2 2 e " a V ( l - e a V ) K Q + a 2 2 ( 1 " e " a V ) ( 1 " < t ) 3 5 K i ) K l 1 - 4> K' W25 0 (A9.13) Since the process has been assumed to be a second-order system, the follo w i n g conditions must be s a t i s f i e d for deadbeat performance: .X1((2+J)T) = 4»25K0e Va'*3 5K'(e ^ V 2 V a ' ^ K ' U - e 1 - ^ ) ^ 1 (A9.14) -9 7 -a7 -a 7 -8 7 X 1((2 + j)T) = 4>35K0e a ' a ' 2 K 0 ( l - e )(e -e ) +. • 3 5 ( l - t 2 5 ^ ) K { = 0 (A9.15) Equations (A9.14) and (A9.15) are solved simultaneously to give Ko - *35 -8 7 -9 7 -a7 -9 7 a 7 -a7 [(e -e )(*25 < ( ,35 + < ! )35 ai ) + < , ,25 a6 C t22 ( e ~ e ) ( 1 _ e )"*24*35°22 ( 1 _ e ) ] (A9.16) 267 -aV -6 V -6 7 -• \ 1 tr I -* ., _ [ « D « 2 2 ^ ( e -e ) - ] and K' — — — (A9.17) •35 ( 1 "*25 K 0 ) Also u 1 (jT)+ = K^u(0+) = K^; since u(0+) - 1 u(T+) - 1 - ^ = B l . u 1((l+j)T+) = Kju(T+) = K J ( 1 - * 2 5 K J ) = K J 3 X and X 3 « 2 + j ) T ) = a 2 ^ e " a 7 ( l - e " a V ) K ^ + ( l - e " a V ) ( l - c ^ K ^ K j = ^ (A9.18) It 'should be noted that deadbeat performance requires zero input to the third integrator for t > (2 + j )T. To satisfy this requirement on the third integrator, the output of the variable-gain element K 1 must be maintained at n $2 after the second and deadtime instant. Thus the Z-transform of the . output sequence from the d i g i t a l compensator (variable-gain element K^) may be expressed as u 1 ( z ) z J = KQ + K 1 e 1 + 3 2 z ~ 2 + e 2 z ~ 2 + ... which reduces to Z - J [K^ 4- ( K ^ - K ^ z ' 1 + (B 2 - K {g )Z- 2 ] u (Z) = — (A9.19) (1 - z l ) But the Z-transform of the input signal to K' i s n U ( z ) = i + e 1 z ~ 1  269 the pulse transfer function of the desired d i g i t a l controller is given by .-1 . -2 D(Z) u^Z) Z~J[K- + (K|B1-K')Z + (g^K'B^Z Z ] u(Z) (1 - Z _ 1 ) ( l + ! ) For the case when process deadtime is zero; j = 0 and 6 = 0 : Therefore V = T. Thus, Equation (A9.20) becomes D(Z) [K2 + (K|<(>1 - K')Z 1 + (g 2-K|0 1)Z _ 2 ( l - z _ 1 ) ( i + f^z" 1) Equation (A9.21) can be expressed as M(Z) D(Z) = + 6 3Z 1 + 3 4Z - 2 E(Z) 1 + g 5Z 1 - B Z 2 where B3 = 1 ^ - K'; 3, = .B, - K'B - ^ = ^ - 1. .Rearranging Equation (A9.22) gives -2. (A9.20) (A9.21) (A9.22) M(Z) [1 .+ B5Z 1 - 3̂ Z 2] = [K£ + 32Z 1 +' B^Z.'] E(Z) (A9.23) After a bit of differencing and collecting of terras as has been shown in Appendix 8 the d i g i t a l controller difference equations are X 3(K+l) — 0 1 X 3(K) + h l " X 4(K+l) " B5_ X 4(K) _ h2_ e(K) (A9.24) where 1̂  = 3 3 - K'B5; h 2 = B4 + K ' B ^ f ^ 270 The overall transfer function of the control system with d i g i t a l controller is 4 + 5Ts ( ) K' + a z" 1 + 8 Az 2 t — £ Z i Z T — ] Hn(s) (A9.25) 4 + 4Ts (s + e ) ( S + e ) i + 3 5 z _ 1 -.3 1z" 2 ° The signal flow graph of Equation (A9.25) is shown in Figure A9.2. The output nodes for the control system are X (s), X 2(s), X^CK+l) and X^(K+1). The input nodes are X (K), X^K), X 3(K), X^(K), and r(K). There are 3 2 loops: L x = -Q3/s, L 2 = -9/s , L 3 = -1/Ts. Also M(K) = X (K) + K Qe(K) The state differential equations in matrix form is given as X 1 ( s ) - ^ ( S ) •{2(8) •{3(8) 0 X X(K) X 2(s) _ *21<8> * 2 2 ( s ) . • 2 3 (8) 0 X 2(K) + ; ^ ( s ) X 3(K+l) " h l 0 0 1 X 3(K) h l X 4(K+l) ; h2 0 \ ~\ X 4(K) _ h2 T(K) where, 3 2 C| = Ts + s (T6 +1) + s(T0+63) + s(Ts +'T03+1) K^e 5TKQ0 °1 (Ts + I) 0 5T9 1 1 1 271 4 2 1 ( s ) -K^O 5TKQ0 8(TS+1) c l Y 2 2 ( s ) s(Ts + 1) 6 5T9 ' 9 2 3 ^ S ; sCj 4 C J ' K^6 5TKQ0 K^0 5TK^s6 4C[ Inverse transforming gives X1(K+1) X2'(K+1) X3(K+1) X4(K+1) The solution t X(nT) Y l l ( T ) Y 1 2 ( T ) Y 1 3 ( T ) 0 • 2 1.(T) -h. * 2 2<T) -h. * 2 3 (T) X 1(K) X 2(K) * 2(T) + X 3(K) h l X 4(K) _ h2 _ r(K) n-1 i=0 n - l - i . (A9.28) for unit step change where <j>n -.Z" 1{Z(ZI - A)"1} 272 But (ZI - A) -1 611 612 612 21 J31 ) 42 6 22 332 42 '23 333 3 43 '14 324 334 344 (A9.29) where, = [Z(Z+B 5)-B 1](Z-4 2 2); 9 ^ = <f[2 [Z(Z+B,. )-B1 ] ; II 5' "1- 313 - (Z+B5 ) [n3(Z-*22 ) +n2*23]; V = •13(Z"*22) + *'l2*23> 321 " • 2 2 I' Z ( Z + B5 )- 31 1 " Y23 [ V Z + V + V 322 = ( z- <t >} 1)[ z( z +e 5)-e i] - * 1 3 [ h l < : Z + B 5 ) + V 3 2 3 = { ( z - 4 [ 1 ) ( z + 3 5 ) [ 4 ; 3 ( z - * 2 2 ) + 4 ; 2 4 2 3 i - 4 ; 3 ( z + s 5 ) t ( z - 4 [ 1 ) ( z - 4 ; 9 ) - 4 1 , 9 4 ; 1 i }/4 '24 r22' Y12T21J = {(z-4[ 1)[4| 3(z.-4 2 2) + 4 { 2 4 2 3 ] -4' 3(z-< r; 1)[(z-< r; 9) - "i^*;,]}/*;. 22' '21Y12J Y12 e 3 i = -[(z+P 5)+h 2](z-4 2 2); e 3 2 = -4[ 2[h 1(z+B 5)+h 2]; Q 3 3 - (Z+B 5)[Z-4 i l)(Z-4 2 2) - • { 2 * 2 1 1 - 034 " [ ( Z - * l l ) ( Z - Y 2 2 ) - n 2 Y 2 1 ] ; 941 = VZ(Z+35> " e i ] ( Z " Y 2 2 ) " z [ n 1 ( Z + e 5 ) + h 2 ] ( Z " < * ) 2 2 ) 942 = V i 2 [ z ( Z 4 V ~ V ' Y i 2 z t v z + y + h 2 ] 643 - ̂ K Z - ^ . X Z - ^ ) -4; 24 2 1] -VYl3(z-*22) + Y1 2*23 ] V " W u ^ - W +*i2*23 ] + fCZ- n i)(Z-4> 2 2) -A 2 14^ 2] 273 A = { [ ( Z - ^ ^ C Z - ^ ^ ^ H Z C Z ^ ) - ^ ] + [h l (Z+B 5) -H»2][*{3(Z-*i2)+*12*23]} Expanding A and grouping terras gives A = Z 4 + AjZ 3 + k^Z1 + A3Z + A 4 (A9..30) where A l " -<*il +*22- B 5> 5 A2 - I * i l * 2 2 - * 1 2 * 2 r e 5 * i r P 5 * 2 2 - 0 l + A3 " [ V I l Y 2 3 - V 2 1 Y l 2 + V n + V ^ A = fh 3 6' 6' - h 6* 4' +h V A' - h g A' A1 - 3 A' A ' + B A' A' 1 4 1 1 5Y12Y22 2*13*22 2Y12Y23 1 5Y12Y22 1Y11Y22 1Y12Y21 J The general quartic, Equation (A9.30) is reducible (substitute Z = X - Al /4) to the form 4 2 X + uX + VX + W = 0 (A9.31) The four roots of the reduced quartic for positive V are X, = -/Z\ - /Z„ - /z„; X„ = -/z~ + /Z-+ /z„; X • = V z - /z~ + 7 z „ ; 1 1 2 3 2 1. 2 3 3 2 3 X^ = /z^ ^2 - ' w b e r e ^> ^ 2 > a n ^ ^ 3 a r e t n e r o o t s of the unreduced cubic Equation 2 2 Y 3 + (y)Y 2 + ( — - |T; = 0 (A9.32) 3 2 or Y + A,Y '+. A,Y + A, = 0 (A9.32a) J O / . where A c = u/2; A, = (u 2 - 4w)/16; A 7 = -V 2 /64 274 and 2 2 3 3 6 A 3A 3  4A A.A. u = [ ( - f ) - - T L + A ]; V = [ — i i- - A ] 4 4 2 A1 A r A2A1 3̂̂ 1 W = " ~3 + 2 " " + A4 ] 4 4J 4 4 Equation (A9.32a) can be reduced to . Y l + V1 Y1 + W l = ° (A9.32b) 2 3 where VL = (3A6~A5)/3 and W1 = (2A5 - 9A5A& + 27A?)727 The three roots are r = B + B ; r = -l/2[-(B +B ) + /3 (B -B ) i 1 1 2 2 L 1 2 1 2 arid r 3 = -1/2 [ - ( B ^ ) - V T ( B ^ B ^ l ] where .' •. '. \.' • - \. • •'.' ' B _ r"wi + r w i + v? a/2,1/3. B _ r _ w i A + v u 1/2.1/3 B l _ [~2~ + ("T + 27 } ] ' B2 " [ ~ •" ("T + 27 ) ] Thus A = (Z - X L ) ( Z - X 2)(X - X 3 ) ( Z - X 4) (A9.32c) If the i n i t i a l states are assumed to be zero, then the transient response 275 the system is n-1 C(nT) = Z [* * * * ] i=0 *2 (A9.33) where [ x 1 ( y 3 5 ) - 3 1 K x 1 - ^ 2 2 ) *11 = — — X i •: • ( x 1 - x 2 ) ( x 1 - x 3 ) ( x 1 - x 4 ) [ x 2 ( x 2 + B 5 ) - e 1 ] ( x 2 - ^ 2 ] v n _ 1 _ , : x2 ( x 2 - x 1 ) ( x 2 - x 3 ) ( x 2 - x 4 ) [ X 3 ( X 3 + 3 5 ) - 3 1 ] ( X 3 - ^ 2 2 ) ^ _ 1 _ . + [ X 4 ( X 4 + 3 5 ) - 3 1 ] ( X 4 - ^ 2 ] ^ _ 1 _ , ~ 3 X4 ( x 3 - x 1 ) ( x 3 - x 2 ) ( x 3 - x 4 ) ( x 4 - x 1 ) ( x 4 - x 2 ) ( x 4 - x 3 ) [ X 1 ( X 1 + S 5 ) - 3 1 ] r n - l - i 1 2 ' ( x 1 - x 2 ) ( x 1 - x 3 ) ( x 1 - x 4 ) l '12 [X 2(X 2+3 5)-3 1] ( x 2 - x 1 ) ( x 2 - x 3 ) ( x 2 - x 4 ) + +1' 12 ( x 3 - x 1 ) ( x 3 - x 2 ) ( x 3 - x 4 ) n - l - i , x 3 + <D12 ( x 4 - x 1 ) ( x 4 - x 2 ) ( x 4 - x 3 ) . n - l - i 13 (X +3 ) U ' (X -4' +<J>' cb' 1 (X +3 )fd>' (X-cb' ) + * ' cb' ] ^ 1 5 n v 1 3 ^ 1 T22 V12*23J v n - l - i . v 2 5 ; L +13 V 2^22; l12 v23 J n - l - i X T  + ' — — ; x 2 ( x 1 - x 2 ) ( x 1 - x 3 - ) ( x 1 - x 4 ) ( x 2 - x 1 ) ( x 2 - x 3 ) ( x 2 - x 4 ) 276 (X +B Hcb' (X -cb' +<b' <b' 1 (X +g )r<b' (X-cb' )+<b' cb' 1 ^ 3 5 M y 1 3 k 3 ̂ 22 912 V23 J v n - l - i , 4 5nvl3K 2*22J y12 y23 J X3 + ( x 3 - x 1 ) ( x 3 - x 2 ) ( x 3 - x 4 ) ( x 4 - x 1 ) ( x 4 - x 2 ) ( x A - x 3 ) ^13 ( Xr*22 ) +*12*23 3 n - l - i ^ "^VW*^^1 n - l - i + x + X + ( x 1 - x 2 ) ( x 1 - x 3 ) ( x 1 - x 4 ) ( x 2 - x 1 ) ( x 2 - x 3 ) ( x 2 - x 4 ) [ • { 3 ( X 3 - * 2 2 ) + * { 2 * 2 3 l I * 1 3 ( V*22 ) +*12*23 3 n - l - i X ( x 3 - x 1 ) ( x 3 - x 2 ) ( x 3 - x 4 ) ( x 4 - X l ) ( x 4 - x 2 ) ( x 4 - x 3 ) 277 APPENDIX 10 OPTIMUM DESIGN OF CONTROL SYSTEM BY DYNAMIC PROGRAMMING (66) Consider a m u l t i v a r i a b l e process governed by the following f i r s t - o r d e r vector-matrix d i f f e r e n t i a l equation x = Ax + Du (A10.1) where A i s the c o e f f i c i e n t matrix of the process; D i s the d r i v i n g matrix; i s the state vector and u i s the cont r o l vector. The s o l u t i o n of equation (A10.1) by s t a t e - t r a n s i t i o n matrix method i s given as x(t) = 6(t , t Q ) x ( t Q ) + f i Ut, T)DudT (A10.2) where <(>(t, t^) i s the o v e r a l l t r a n s i t i o n matrix and s a t i s f i e s the condi t i o n <Kt, t Q ) = A6(t, t Q ) and ( r ( t Q , t Q ) = I (A10..3) Since this i s a d i g i t a l c o n t r o l system, u ( x ) = u(kT) for kT £ T < (k+l)T, and the so l u t i o n i n d i s c r e t e form i s given as x(k+l) = <})(k) x (k) + G(k) u(k) (A10.4) where - <f>(k) = *(k+-lT, kT) (A10.4a) G(k) = / ^ 1 T 4(k+lT, T) D ( T ) dT (Al0.4b) where sampling time T has been dropped i n (A10.4) for convenience. Applying a quadratic performance index of the form N J« = I {[x°(k) - x(k)]'Q(k)[x°(k) - x(k)] + Xu'(k-l) H(k-l) u(k-l)} k=l (A10.5) 278 for the control system optimization, in which Q and H are positive definite symmetric matrices. The selection of positive definite matrix assures the uniqueness and linearity of the control law and asymptotic s t a b i l i t y of the contol system for the controllable process. The expansion of the above equation leads to a weighted sum of squares of (x°-x) and u, with the weighting determined by the elements of matrices Q and H. The f i r s t terra in the right hand side of equation (A10.5) could be used to specify the deviation of the process from the desired condition at any time KT, and the second term provides an energy constraint on'the control signals. The f i r s t terra in the right hand side can be considered as a representation of economic penalties caused by response errors and the second term may be viewed as the cost of control. The multiplier X is a penalty factor and can be determined directly from engineering considerations. The addition of,the second term, provides a mathematically convenient way to ensure the avoidance of saturating the control signal,^ thus the multiplier can be chosen so that the square of the control signal is less than a certain limit where saturation occurs. The optimum control problem can be formulated as follows: Find the control law (u(i)}, i = 0, 1, 2, N-1, which minimizes the. expected value of the performance index of equation (A10.5) subject to the relationship of equation (A10.4), for any arbitrary i n i t i a l state x(0). If the optimum di g i t a l control problem is viewed as an N-stage decision process, the determination of the optimum control law can best be carried out by means of dynamic programming technique. Let the minimum of 279 the expected value of the performance index be denoted by f N [x(0)j = mn±) E J N (A10.6) N I k=l IN = Min E { {[x (k) - x(k)]' Q(k)[x°(k) - x(k)] + Au'(k-l) H(k-l) u(k-l}} u(0) u(l) u(n-l) Expressing in a more general form gives f N _ j [ x ° ( j ) " X ( J ) ] = M i n E J N _ j (A10.7) If the optimum d i g i t a l control problem is viewed as an N-stage decision process, the determination of the optimum control law can best be carried out by means of dynamic programming technique. Let the minimum of the expected value of the performance index be denoted by f M [x(0)] = Min. E j . (A10.6) N . u (1) N • =• Min E { I I[x (k) - x(k)] ' Q(k)[x (k) - x(k)] + Xu'(k-l) H(k-l) u(k-l}} k=l u(0) u(l) u(n-l) Expressing in a more general form gives f N _ j f * 0 ( j ) " x(j)] = Min E J N - j (A10.7) N = Min E { I {[x°(k) - x(k)]* Q(k)[x°(k) - x(k)]+X u'(k-l) H(k-l) u(k-l}} k=j+l u(j) u(j+l) u(n-l) j = 0, 1, 2, ...... N when j = 0, equation (A10.7) reduces to (A10.6) and i t is apparent that fg = 0. 280 Suppose that the return from the f i r s t ( j - l ) stages is optimum. Then the output of the remaining N-j stages is equal to output from the j-th stage plus optimum output from the remaining [N-(J+1)] stages; that is {[x°(j+l) - x(j+l)]'Q(j+l)[x°(j+l)-x(j+l)] + Xu'(j) H(j) u(j)] (A10.8) + f N _ j + l t x ° ( J + 1 ) - x ( j + 1 ^ Applying the principle of optiraality f N _ j [ x ° ( j ) " x ( j ) ] = M i n E {([x°(J+l)-x(J+l)],Q(j+l)[x0(j+l)-x(j+l)] (A10.9) + Xu'(j) H(j) u(j)] + f N_ j + 1[x°(j+l)-x(j+l)]} Since the function f is quadratic in [x^ - x], i t can be assumed that f N _ j [ x 0 ( j ) " X ( J ) ] = [X°(J) - X ( J ) ] ' P(N"j)[x°(j) - x(j)] (A10.10) and f N _ j + L t X ° ( j + 1 ) " x ( J + 1 ) i = [ x°(i +l) - x(j+l)]' P(N-j+l)[x°(j+l) x(j+l)] (A10.ll . . •• • • J ' This assumption is readily justified by mathematical induction. The P's are positive definite, symmetrical matrices which put these two expression in quadratic form. Substituting equation (A10.10) in equation (A10.9) gives [x°(j)'- x(j) ] ' P(N-j)[x°(j) - x(j)] = Min E {([x°(j+l) - x(j+l)]' S(N-j+l) u(j) (A10.12) [x°(j+l) - x(j+l)]) - Xu'(j) H(j) u(j)} where S(N-j+l) = Q(j+1) + P(N-j+l) (A10.12a) 281 Let I = E{([x°(j+1) - x(j+l)]' S(N-j+-l)[x°(j+l) - x(j+l)]) + Xu'(j) H(j) u(j)} (A10.13) In light of equation (A10.10), I n _ j ' " E^tx°^> " X < J > ] ' K H [ N _ (J+D][x°(j) ~ x(j)]) +u'(j) L G +(n-(j+l)) [x°(j) = x(j)]'•>L^G(N-j+i)u(-j)-+ u'(j) ]LGG(S-j+i)-+-X H(j)] u(j)} (A10.14) where L [N - (j+l)] = S[N - (j+l)] cj>(j) (A10.15) L G G[N - (j+l)] - G'(j) S[N - (j+l)] G(j) (A10.16) L [N - (j+l)] = G'(j) S[N - (j+l)] <j>(j) (A10.17) L + ( J[N - (j+D] = 4>'(j) S[N - (j+l)] G(j) (A10.18) The minimization procedure is readily carried out through ordinary differentiation, since the N-stage decision process has been reduced to a sequence of single-stage decision process. Differentiating equation (A10.14) with respect to u(j) yields L G +[H - (J+D][x°(j) - x(j)] + [x°(j) - x(j)]' -.L^ [ N _ ( + [L G G[N - (j+l)] + XH(j)] u(j) + u'(j)[L G G[N - (j+l)] + XH(j)] (A10.19) In view of the symmetry of the matrices, = 2LG<f)[N - (j+D][x°(j) - x(j)] + 2[L G G[N - (j+l)] + XH(j)] u(j) (A10.20) At the minimum the derivative is zero, and thus the optimum control law is 282 u ° ( j ) = B(N-j) [ x ° ( j ) - x(j)] (A10.21) where the feedback matrix B is given as B(N-j) = - [ L G G [ N - (j+1)] + XH(j ) ] " 1 LG ( ( )[N - (j+1)] (A10.22) It is noted that the optimum control law is a function of the state variables of the system. Since the feedback matrix B Involves the unknown matrix P, the optimum control law is s t i l l undefined u n t i l the matrix P is determined. Substituting equation (A10.17) into equation (A10 .12) yields the minimum [ x ° ( j ) - x ( j ) ] ' P(N - j ) [ x ° ( j ) - x(j)] = [ x ° ( j ) - - x ( j ) ] ' L H [ N - (J+D] [ x ° ( j ) - x(j)] + ( B(N - j ) [ x ° ( j ) - x(j)]} ' L G ( ( )[N - ( j + l ) ] [ x ° ( j ) • - x( j ) ] + [ x ° ( j ) - x( j)]'L ( J ) G[N - ( j + l ) ] { B(N - j ) [ x ° ( j ) - x(j)]}+{B(N-j)[x° (j) " x(j)]} ' [ L G G [ N - (j+1)] + XH(j)] { B(N - j ) [ x ° ( j ) - x(j)]} = [ x ° ( j ) - x ( j ) ] ' L H [ N - ( j+D ] [ x ° ( j ) - x(j)] [ x ° ( j ) - x(j)] + [ x ° ( j ) - x(j)]}'B'(N-j) L G + [ N - ( j+D ] [ x ° ( j ) -" x(j ) ] + [ x ° ( j ) - x ( j ) ] ' L ^ [ N - (j+1)] B(N - j ) [ x ° ( j ) - x(j)] + [ x ° ( j ) - : « ( j ) ] } ' B'(N - j ) [ L G G [ N - (j+1)] + XH(j)] ( B(N - j ) [ x ° ( j ) - x(j)] ] (A10.23) Applying equation (A10.18) to equation (A10.23) gives • [ x ° ( j ) - x ( j ) ] ' P(N - j ) [ x ° ( j ) - x(j)] = [ x ° ( j ) - x ( j ) ] ' [ L ^ [ N • - (J+D] 283 + L ^ N - (j+l)] B ( N - j ) ] [ x ° ( j ) - x(j)] (A10.24) Since Q and P are positive definite matrices, then by equation (A10.12a), S i s also def ini te . Comparing both sides of equation (A10.24) yields P(N-j) = L H [ N - (j+l)] + L [N_- (j+l)] B(N-j) . (A10.25) Starting with P(0) = 0 for j = N-1; equation (A10.18) gives B(l) = ~[LG G(0) + XH(N-l) ]" 1 LG ( j )(0) (A10.26) and equation (A10.25) gives P(l) = L (0) + L . „ ( 0 ) B(l) (A10.27) cpcp <pG Thus, combining equations (A10.22) and (A10.25) gives the recurrence relat ion and hence the control law is defined. It is being assumed that the control system in this study is of f i n i t e stage process and there is no , constraint on the control s ignal . Equation (A10.25) reduces to P(N-j) = L H [ N - (j+l)] - LjjQ[N - (j+l)] L ^ N - (j+l)] LG- +[N - (j+l)] (A10.28) Using equations (A10.15) to (A10.18) and for simplici ty dropping the arguments gives p = <j>'s<j> - <(>,SG[G,SG]~1G'S4> = (b'Scb - cb'SGG~ 1S~ 1[G'] _ 1G'Scb = cb'Scb - <b*Scb = 0 (A10.29) Provided that the inverse matrices exist; that i s , i f there are as many 284 control signals as there are state variables. The feedback matrix is then given by B(N-j) = - [ G ' ( j ) Q(J+1) G ( j ) ] " 1 G'(j ) Q(j+1) 4<j) (A10.30) It should be noted that, when the performance index is not time weighted, the process is time invariant, and matrix G is non singular, the feedback matrix is a constant matrix given by B = - [G' (T) QG(T)]" 1 G'(T) Q4(T) (A10.31) The solution as given by equation (A10.21) is not complete since in some cases the states are unaccessible for direct measurement. Due to the dead time present in the process a predictor is used to estimate the state variables. Consider a linear control system with the following equations x = Ax(k) 6(k) (A10.32) y(k) = Mx(k) + n(k) (A10.33) x(0) = x 0 + e 0 . (A10.34) Where x is an n vector of states, y is an i vector of discrete time outputs, e(k) is an n vector of random process,h(k) is an I vector of random measurement error, A and M are n x n and £ x n constant matrices, XQ is an estimate of the i n i t i a l state, and e 0 is i ts random error. The variables x(k), y(k) are discrete stochastic random variables having some probability distr ibution at any instant of time k, and equation (A10.32) is a stochastic 5 7 d i f f e r e n t i a l equation. Using the weighted least squares performance index along with the system equations (A10.32) to (A10.34) given as 285 J M = j[*<0)'- X Q ] ' P Q 1 [ X ( 0 ) - xQ] + |/ k T {(x-Ax)'R(x-Ax) + [y - Mx]q[y -Mx]}dt (A10.35) where the f i r s t term minimizes the squared error of i n i t i a l condition estimates, the second term minimizes the integral squared modeling error, and the third term minimizes the integral squared measurement error. The weighting factors P o - 1 . R, Q are chosen based on the s ta t i s t i cs of the problem. Assuming that the noise process e(k), n(k) in equations (A10.32) and (A10.33) to be Gaussian and uncorrelated in time ( ie . white noise) as well as uncorrelated with the i n i t i a l state. Also, assume that the expected value relations E[e(k)] = 0; E[n(k)] =0 ; E[e(k) e(x) '] = R _ 1 ( k ) 6(kT-x) E[n(k) x'(0)] = 0; E[e(k) x'(0)J = 0; E[e(k) n'(x)] . o E[x(0)] = xQ; E { [ X q - x(0)][x 0 x(0)]'} = PQ E[n(k) n ( T ) 1 ] = Q _ 1 ( k ) 6(kT-T) hold, where PQ is the covariance of the i n i t i a l state errors, R - 1 ( k ) is the covariance of the process noise, and Q~ 1 is the covariance of the measurement errors. Equation (A10.35) can be reformulated by defining u(k) = x - Ax, and rewriting the objective relation (equation A10.35), that i s , - J M = |-[x(0) - X Q ] ' P Q 1 [ X ( 0 ) - xQ] + y / k T {u ' (k) Ru(k) + [y - Mx(k)]' Q[y - Mx(k)]}dt (A10.36) Thus, the estimation problem can be posed as a deterministic optimal control problem; i e . , select the control u(k) such that J in equation (A10.36) is 286 minimized subject to the constraints x = Ax + u (A10.37) x(0) unspecified (A10.38) Applying the maximum principle to this problem gives H = y[u'Ru + (y-Mx)' Q(y-Mx)] + X'(Ax+u) (A10.39) 8H and the condition ~— = yields 3u J u(k) = - R _ 1 X (A10.40) where X' = —— - [M'Q(y-Mx) - A ' X J ' (A10.41) or X = -M'QMx - A ' X + M'Qy (A10.42) Because both x(0), x(k) are free, there are then two boundary conditions on • x i • •"" , : X(k) =0 (A10.43) and x(0) = xQ - P Q X ( O ) (A10.44) Substituting equation (A10.40) into equation (A10.37) gives x = Ax - R _ 1 X (A10.45) Equations (A10.42) to (A10.45) form a two-point boundary value problem which can be solved for x(k), X(k) and thus produce the optimal estimates.^ Let xCkVjj), u(k7kf ), — where k^ is the f i n a l time — , denote the optimal estimates and controls at time kT, with data y(k) up to time k^. Thus 287 A(k/kj) i s the estimate found from the two-point boundary value problem of of equations (A10.42) to (A10.45). Make the transformation x(k/k ) = W(k) - P(k) X(k) (A10.46) where the n vector W(k) and nxn matrix P(k) are to be determined. Substitute equation (A10.46) into equation (A10.45) gives for each side of the equation RHS = A(W-PX) - R _ 1 X (A10.47) LHS = W - PA - PA = W - PX + P[M'QM(W-PX) + A'X - M'Qy] (A10.48) Collecting terms gives W - PM'Q(y-MW) - AW = (f-PA'-AP-R~1PM'QMP)X (A10.49) It is now possible to choose to define W(k), P(k) such that the coefficients in equation (A10.49) vanishes and choose the boundary conditions to sat isfy equations (A10.43) and (A10.44). Thus, W = AW + PM1Q(y-MW W(0) = xQ (A10.50) $ = PA' + AP' + R - 1 - PM'QMP P(0) = PQ (A10.51) Note that the estimates may be found f i r s t by equations (A10.50) and (A10.51)forward in time to produce W(k), P(k), then solving backward in time using equations (A10.43), (A10.45) and (A10.46) to find the optimal estimates x(k/k j . 288 From equation (A10.43) i t could be seen that at the end of a data period KT = kT , ^(k ) |lways vanishes. Thus for any kT , equation A10.46 yields the result x ( k f / k f ) = W(k f) (A10.52) Thus, the f i l t e r e d estimates are determined from the sequential real-time equation (A10.50), where k is always the current time. x(k/k) = Ax(k/k) + P(k) M'Q[y(k) - Mx(k/k)] x(0/0) = xQ (A10.53) The n x n matrix function P(k) is given by equation (A10.51). Prediction estimates x(k/kp) for the control system are estimates at time k > k^ for a system having no data after time k^. These arises direct ly from the f i l t e r i n g equations i f Q is set equal to 0 for k > k^ i n equations (A10.51)and (A10.53). Thus the prediction equations are x(k/k Q ) = Ax(k/kQ) (A10.54) where x(k/kQ) is the f i l t e r estimate at time k^. The prediction covariances PCk/kg) are given by P(k/k Q ) = P(k/k Q ) A' + AP(k/kQ) + R _ 1 P ( k 0 / k Q ) = P(kQ) (A10.55) Equation (A10.54) i s solved by numerical integration, in this study the Runge-Kutta fourth order formula is used. 289 APPENDIX 11 ZERO-ORDER HOLD PARAMETER DEFINITION M , + " 9 l T / 2 + " 6 2 T / 2 (1-e ) (a1  a2e + ĉe -9 T/ -9 T/ -8 T/ 1 '2 , 2 I 7 2 l I X 2 N e a2(e - e ) -9 T/ n 2 2 0 e 0 0 -9 T -9 T (l-e-V) ai + V + V - 6 i T , -v - e i \ e a2(e -e ) - 9 2 T 0 e 0 0 0 1 - 9 L 6 T - 9 1 6 T - 9 2 6 T (1-e ) + a2e + ôe -9 6T -9 6T -9 6T 1 2 1 e a2(e -e ) 0 0 -e 26 T e 290 2. i ( 2 9rV, °1 ~ V ' a2 (\-*2> ' ° 3 W V 2 V 2 g{(T) = a 2 _ e i T a 3 _ e 2 T <_ - 9 1 T 0 ^ °1 0 6, e 0, 1 ^ g 2 ( T ) = a 2 ( Q - e - Q - e ) - a 2 ( - e - g - e ) , -8,Tn -0„T 8iCD-|-(. 2 ° - e 2 > HALF-ORDER HOLD PARAMETERS g r ( T ) 9 >: e -e T 10 1 — - e a - 0 T 11 2 a. 12 "Vi — e J " t V o ~ 0- e a io " V o °11 "VO g 2 ( T ) r a l 3 "V °14 °15 - a2 T! _ r a ! 3 " V o ^ a ! 4 6 2 T 0 + a i 5 + [-T— e + ^ e + •0. 0. F ° " " 6 2 T + " l e " a 2 T 1 r °16 - 6 2 T 0 + °16 g 4 ( T ) - - l / a 2 ( e - e ) 291 "4 =  a / 6 l ; "5 = <8i+a)/8ie2; "6 = a / 0 l ( 9 r e 2 ) 5 a 7 = ( er e 2 + a ) / 6 2 ( 82 " 8 l ) ; °8 = 1 / ( 6 2 - 9 1 ) ; a 9 = a ^ a S ^ l - e ^ S ^ / e ^ a . ^ ; a 1 Q = -a . 1 (e 2 -e 1 +i )/e 1 (8 2 -e 1 ) (a 2 -e 1 ) ; « n = - a 1 ( a 2 - e 1 - e 2 ) / 8 2 ( e 1 - 6 2 ) ( V e 2 ) a l 2 - - [ a a 1 ( e 2 - a 2 + 1 ) "WV " al ( 6r a 2 ) ( e 2 " a 2 ^ / a 2 ( eir a 2 ) ( 9 2 " a 2 ) a i 3 = -i.(92-fll+1)/(V8l)(VV; a H = « 1/( f l 1- 82 ) CV 82 ) a ! 5 = a l ( 8 2 - a 2 + 1 ) / ( 8 r a 2 ) ( 8 2 - a 2 ) ; a !6 = - a l / ( a 2 - 8 2 ) -8 T/2 - S T / 2 - 8 1 / 2 1 a4(l-e ) [a5+a6e +a?e ] : \ -8 1T /2 -8 2T /2 0 e • a g(e -e :-; . A = | -8 T/2 0 0 e 0 0 0 0 0 0 -8 T/2 -8 2 T/2 ~a 2 T/2 [a 9+a 1 ( )e +ctne + « 1 2 e ] 0 -9 T/2 -6 2 T/2 -a 2 T/2 [a 1 3 e +cxue + « 1 5 e ] 0 -8 T/2 -a T/2 a 1 6 ( e -e ) 0 -a T/2 e 0 1 292 B l = -e , T — 6 , T - 6 „ T 9,T - 9„T -a„T 1 — 1 2 1 2 2 i 1 o 4 ( l - e ) [a 5+a 6e +aye ] [a g +a 1 ( ) e + a n e +a 1 2e J 0 - e 1 T 0 0 0 0 -e T - e 2 T a g(e -e ) - e 2 T o o - 0 1 T - 0 2 T - a 2 T [« 1 3e +« 1 4e +a 1 se ] 0 - 8 2 T - a 2 T a 1 6 ( e " -e ) - a 2 T C = ':- -e 6 T -e 6 T -e 6 T - -e 6 T - B J T - a 2 6 T 1 a 4 ( l - e ) [a 5+a 6e +a?e ] [ V alO e + a n e + a l 2 e J . ° -6 - 6 T 0 e 0 0 - 6 6 T - 6 2 6 T - 6 1 . 5 T ~ 9 2 6 T ~ A 2 6 T a g(e -e ) [ o 1 3 e + a i A e + a i s e ] 0 0 0 - 6 2 6 T 0 0 - 9 6 T - a 2 6 T a 1 6 ( e -e ) -a26T 2 9 3 APPENDIX 12 NOISE SENSITIVITY ANALYSIS The f o l l o w i n g assumption i s made: ( 1 ) the e x p e r i m e n t a l l y determined t r a n s i e n t responses can be rep resented as C ( t ) , = C ( t ) + V f t ) ( A 1 2 . 1 ) 1 measured i exact 1 and C ( t ) . = C ( t ) + V ( t ) ( A 1 2 . 2 ) o measured o exact o where /°°C(t) dt = 1 o and J~V(t) e x p ( - s t ) t n d t « /"cCt) e x p ( - s t ) t n d t f o r the v a l u e s of n and s i n c o n s i d e r a t i o n . The n o i s e f u n c t i o n s V ( t ) are d e f i n e d as the d i f f e r e n c e between the measured responses and the exact r e s p o n s e s . The r e l a t i v e e r r o r on dead t ime x , that i s Ax = - ^ i ( A 1 2 . 3 ) r , o T V / . due to e x p e r i m e n t a l and measuring e r r o r s on C Q ( t ) , i s expressed as A T = jTv ( t ) F ( t ) dt ( A 1 2 . 4 ) r , o ' o o o where F ( t ) i s the n o i s e s e n s i t i v i t y f u n c t i o n , o The assumption J"v(t) e x p ( - s t ) t n d t « /°°C(t) e x p ( - s t ) t ° d t 294 means that the total error on the calculated dead time is equal to the sum of the individual errors due to noise at inf ini tes imal time intervals . The error on the dead time due to noise on, say, the outlet response at times t = T - , T„, T_ for time intervals of duration At can thus be written as 1 2' 3 n Ax = V(T ) F (T ) • At + V(T ) • F (T_) • A t . + . . . (A12.5) r ,o 1 1 1 L 2. I or in the l imit AT = f v ft.) F (t) dt r , o o o o If V(t) = A • 6(t-T), where A « 1, then T T T „ • r V J t ) ' F (t) dt = A • F (T) (A12.6) r, o o o o o That is F (T) = lm - AT - / A%o A r ° The value of the weighting function F ( t ) for t = T, F ( T ) , is thus the ratio between the relative dead time error due to noise at t = T and the noise intensity at t = T. The objective is to determine the function F ( t ) and hence calculate the optimum s-value to be used. In order to determine the noise function F Q ( T ) , T is calculated using the relation C (t) * = C (t) = C (t) + A • 6(t-T) (A12.7) 0 measured o o exact and C f t ) , = C (t) = G\(t) 1 measured i i exact n s F i r s t the perturbated moments M ' are calculated. 295 The normalisation factor for C q is now changed, that i s , and J^C o(t) dt = /~C o(t) dt + fy • 6(t-T) dt = 1 + A (A12.8) "I'* = ro7T=' £ C o ( t ) e x a c t + A ' *W1 d t 1 + A —k- ( M ° ' S + A e " S t ) • (A12.9) 1 + A i M l , s = j » _ l _ . [ c ( t ) + A • 6( t -T ) l t e ~ S t dt o o , . - T - 1 o exact v ' J 1 + A - — • ( M 1 , S + A • T • e~S t) 1 + A (A12.10) 1 + A - • . ..• . — ^ - 3 ( M 2 ' S + A • T 2 • e~ S t ) (A12.l l ) 1 + A ° M n,s = n,s (A12.12) l i Equation (A12.12) comes from the assumption that the input pulse is noise —st free. Expanding the e term in both equations (A12.10) and (A12.ll) and second order and higher terms in A, gives Au, = uJ* - u' = -Ae [T - — 1 = -A • R,(T) (A12.13) 1 1 1 L M o , s J 1 M o 296 and -r -sT f / m o x2 r o _ / o .2 = -A • R (T) (A12.14) But . , 1/2 1/2 , „ 1/2 2su ~ 1 2 T f / T T 172 172 u2 S U2 ~ S (A12.15) D i f f e r e n t i a t i n g equation (A12.15) and s e t t i n g dx = x ^ - x; A\xy = A U ; L and d u 2 = A u 2 * That i s r 1/2 A/2, -T . " T [2su - 2 J : = F.(T) = —TT^ m wTT A u [ u l / 2 + - u l / 2 _ _ 2 l / 2 ] . 1 [ ( u l / 2 + s u l / 2 . 2 l / 2 ) ( s - l / 2 u , . - l / 2 ) . ( 2 s u , u l ^ 2 (A12.16) 2 , 1/2A 1/2 1/2 2  ( u 2 +su 2 -2 ) S u b s t i t u t i n g for uJ, u 2 > Au 1 and Au 2 gives. M 1 ' 3 F T(T) = - A i e - S T ( T - -g—) - A 2 A 3 e " s T + A ^ e " ^ (A12.17) M ' o 297 where A l * 2i 2S(- l _ ^ l / 2 _ 2 l/2 2t ( T J S + I ) ' ) 1 / 2 ( l + s ) - 2 1 / 2 [(-^L-y)1'2 ( l + s ) - 2 1 / 2 ] ( ^ ^ ) - 1 / 2 [ s ( r + T ! l l r ) - 1] (T 1S+1) ( T l S + l ) ' A. = {(T M 1,8 0 ) 2 M o, s 2x [ ( ^ _ ) 1 / 2 ( 1 + S ) - 2 1 / 2 ] 2 ( V + i ) ' M2'S M1'5 [ — - ( — ) 2 ] } M°> S M°' S ^ ( - ^ ^ { s C x * ^ A, = ( T l S + l ) ' 2x, 1' (^8+1)' 2x ) 1 / 2 ( l + s ) - 2 i / Z ] (T 1 S+1) ' l / 2 i 2 A p p l y i n g the same approach f o r the n o i s e s e n s i t i v i t y f u n c t i o n of the time c o n s t a n t , the time constant x i s g i ven as u l / 2 V 1/2 2 1/2 <A 1 2 ' 1 8> s u 2 - 2 298 Different ia t ing equation (A12.18) and setting dx, = x - x • du„ = Au 1 l e a l 1 2 2 gives - ( l / 2 ) ( 2 1 / 2 u - 1 / 2 > _ T l c l l T , 1/2 ,1/2,2 A u2 (A12.19) (su 2 -2 ) but 2 2 (x l S +l ) Z substituting gives (2i«[_!!L ]̂-irt) - 1 ^ 1 L = F T ( T ) = T - i : A 3 e S T (A12.20) (x l S +l ) Z The noise sensi t ivi ty functions, equations (A12.17) and (A12.19) are esperimentally ver i f ied by pulsing testing and the resulting analysis gives the plot of F i g . 6.2. The noise sensi t iv i ty function can be expressed as -s t 2 F(T) = e (at +bt+c), which means that F(.T) approaches zero for t approaching i n f i n i t y . Using the normal central moment method, i e . s = 0, the damping influence of the exponential factor is lost , and the noise s e n s i t i v i t y increases unbounded for high t-values, thus making the calculations extremely sensitive to errors in the ' t a i l s ' of the transient responses. 299 Appendix 13 1 " 5 \ 8 ' — — * i Heat Exchanger Shell 300 Heat Exchanger Tube

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