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Dynamic modelling and control of a paper machine headbox Tuladhar, Anil 1995

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D Y N A M I C M O D E L L I N G A N D C O N T R O L O F A P A P E R M A C H I N E H E A D B O X By Anil Tuladhar B. E. (Electrical Engineering) University of Rajasthan A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF M A S T E R OF A P P L I E D SCIENCE in T H E FACULTY OF G R A D U A T E STUDIES E L E C T R I C A L ENGINEERING We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA December 1995 © Ariil Tuladhar, 1995 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for refer-ence and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Electrical Engineering The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: ~ 29* fcc /ftsr Abstract Many paper mills have replaced their paper machine airpad headboxes with high speed hydraulic headboxes to increase production rate and also to improve the operational stability. However high speed hydraulic headboxes can contribute to higher Machine Direction (MD) basis weight variation. The wider bandwidth of the headbox may allow pulsations generated in the approach system to pass through it and to influence the jet velocity. This thesis is devoted to the analysis of one of such case where the replacement has resulted higher MD basis weight variation. There could be several reasons for the MD basis weight variation. The most likely sources are hydrodynamic pressure pulsations generated in the approach system and hydrodynamic disturbances produced within the headbox. A comprehensive study of the approach system has been carried out by recording pressure data from the major pieces of wet end equipment simultaneously. The thesis explains the experiments in detail highlighting the features of the sensors and data acquisition system. To study the possible generation of pulsations within the headbox^ and also to analyze the effectiveness of the headbox in suppressing pulsations from the approach system, a theoreti-cal nonlinear dynamic model representing a typical Sym-Flo headbox has been derived. The headbox is a multivariable system, and so a multivariable Linear Quadratic Gaussian (LQG) controller has been used to control the level and the total head optimally. As there are con-straints that should be imposed on the inputs and outputs, LQG has drawbacks, therefore a Model Predictive Controller (MPC) is investigated to accommodate input/output constraints. The resulting controllers are simulated for various situations and results are included in the thesis. The headbox model includes the slice hp opening as one of its manipulated variables and so it is possible to simulate grade change events. From the simulation results it is quite clear that the ii grade change can be achieved smoothly using a multivariate controller. The decoupling effect and marked reduction in interaction among the variables are achieved through multivariate control and are illustrated through various simulations. iii Table of Contents Abstract ii List of Tables viii List of Figures ix Acknowledgement xiii 1 Introduction 1 1.1 Background 1 1.1.1 Wet-end Pulsations and Paper Quality 2 1.1.2 Headbox Design and Paper Quality 3 1.1.3 Motivation 4 1.1.4 Contribution . 5 1.1.5 Thesis Outline 6 2 Sym-Flo Headbox Modelling 7 2.1 Introduction 7 2.1.1 General Description 7 2.1.2 Headbox Details 8 2.2 Nonlinear Headbox Model 11 2.2.1 Assumptions: 11 2.2.2 Attenuator Modelling 13 2.3 Turbulence Generator Modelling 18 2.4 Consistency Dynamics 21 iv 2.5 Overall Model 23 2.6 Linearization 25 2.6.1 A Numerical Example 25 2.7 Invertibility: 28 2.8 Controllability and Observability . . 29 2.9 Conclusions 29 3 Mill Experiments: Analysis of Wet End Pressure Pulsations 30 3.1 Introduction 30 3.2 Equipment Review 31 3.2.1 Transducers 32 3.3 Data Processing 32 3.3.1 Decimation 33 3.3.2 Spectrum Estimation and Analysis 33 3.3.3 Scalloping Loss or Picket-fence Effect . 33 3.3.4 Spectral Leakage and Spectral Smearing 34 3.4 Known Sources of Pressure Fluctuations 34 3.5 Experiments and Results 36 3.5.1 Experiment No 1 36 3.5.2 Experiment No 2 40 3.5.3 Experiment 3 46 3.6 Conclusions 51 3.7 Recommendations 54 4 Multivariable Control Simulations 56 4.1 Sensitivity Analysis 56 4.2 Frequency Responses 58 4.3 Open Loop Response 58 v 4.4 Process Control • • 62 4.5 Linear Quadratic Control Design . 65 4.5.1 Introduction 65 4.5.2 Regulation Problem Formulation . 66 4.5.3 Servo Response 68 4.5.4 Linear Quadratic Gaussian Control . 69 4.5.5 Limitations . . 69 4.6 Model Predictive Control 70 4.6.1 Introduction 70 4.6.2 Basic theory of MPC 72 4.7 Application to the Headbox Problem 74 4.7.1 Interaction Analysis 75 4.7.2 Disturbance Review: . 76 4.7.3 Sensors and Actuators 77 4.7.4 Simulation Results with LQG: . . ."•, 84 4.7.5 Simulation Results with MPC . . . 92 4.7.6 Discussion 101 5 Conclusions 104 Nomenclature 107 Bibliography 110 A Equipment used in Wet end Pressure Pulsation Analysis. 117 A . l Portable Validyne Pressure Transducers: 117 A.1.1 Sensor (Transducer): . 117 A.1.2 Carrier Demodulator . .119 A.1.3 Specifications: • • • 119 vi A.1.4 Calibration: . . 119 A.2 Signal conditioning devices: 120 A.2.1 The plug-on A / D board: 121 A.2.2 The DaqBook200: 121 A.3 Data Storing Device: . 122 A.4 Practical Hints 122 A.4.1 Sensor Locations 122 A.4.2 Configuration 123 A.4.3 Grounding 124 A.4.4 File Format 124 B M-Files for the Analysis and Simulations of the Headbox Model 126 C M-Files to Implement M P C Controllers. 135 vii List of Tables 2.1 Specifications of a Sym-Flo Headbox 11 2.2 Hydraulic Radius for Various Cross-sections 20 4.1 Input-output Pairing 76 4.2 Parameters for MPC Controller 85 4.3 Parameters for MPC Controller for Grade-change Event 85 4.4 45 lb Grade of Paper 91 4.5 54 lb Grade of Paper 91 A . l Pressure Transducers 119 viii List of Figures 2.1 White Water System . . 7 2.2 Sym-Flo Headbox Showing Different Parts . 9 2.3 Schematic Diagram of a Sym-Flo Headbox . . . . . 12 2.4 Schematic Diagram of the Attenuator Cross-section . 14 2.5 Gas-sizing Coefficient 17 2.6 Turbulence Generator Section 18 2.7 Attenuator Plate Loss 19 2.8 Sym-Flo Tube Array ; 20 2.9 Frictional Factor . . : . . . 21 2.10 Nonlinear Simulation Results . . . . . . . . . . . . . i 27 3.1 Schematic Diagram of a Data Acqusition System 32 3.2 Power Spectrum of MD Basis Weight Signal 37 3.3 Power Spectrum of Total Head Signal. . 38 3.4 Power Spectrum of Ash Sensor Signal. 39 3.5 Results of Experiment No. 1. . 40 3.6 Schematic Diagram of an Approach System Showing Sensor Locations . 41 3.7 Power Spectrum of Fan Pump Signal. . . 42 3.8 Power Spectrum of Screen Signal. • 43 3.9 Power Spectrum of Cleaner Signal 43 3.10 Power Spectrum of Tapered Header Signal. . 44 3.11 Power Spectrum of Total Head Signal 44 3.12 Results of Experiment No. 2b. 45 ix 3.13 Power Spectrum of Fan Pump Signal 47 3.14 Power Spectrum of Cleaner Signal 48 3.15 Power Spectrum of Screen Signal 48 3.16 Power Spectrum of Tapered Header Signal 49 3.17 Power Spectrum of Total Head Signal 49 3.18 Results of Experiment No 3a 50 3.19 Comparison of Wet end Disturbances Before and After Modification 51 3.20 Power Spectrum of Stuff-box Level 52 3.21 Power Spectrum of Stuff-box Consistency 52 3.22 Power Spectrum of Total Head 53 3.23 Power Spectrum of By-pass Valve Position . 53 3.24 Results of Experiment No. 3b 54 4.1 Sensitivity Towards Frictional Factor 57 4.2 Step Response for Different Air Flows (Bleed Valve Opening) 57 4.3 Frequency Response (45 lb grade paper) 58 4.4 Frequency Response (54 lb grade paper) 59 4.5 Frequency Response (72 lb grade paper) 59 4.6 Step Change in Stock Flow 60 4.7 Step Change in Air Flow 60 4.8 Step Change in Slice Opening 61 4.9 Robust Servo Control Structure 69 4.10 Block Diagram Representation of Model Predictive Control 72 4.11 Model Predictive Control 73 4.12 Bleed Valve Characteristics 79 4.13 Thin Stock Valve Characteristics 79 4.14 Simulink Model of Headbox Control System using LQG 81 4.15 Simulink Representation of Valve Nonlinearity. 82 x 4.16 Response to a Sinusoidal Disturbance : 86 4.17 Other Two Inputs. . 87 4.18 Response for a Major Disturbance in Level 87 4.19 Other Inputs During the Disturbance. . 88 4.20 Change in Total Head Set Point. , 89 4.21 Change in Inputs During the Set Point Change 89 4.22 Change in Stock Level Set Point. . . [ 90 4.23 Change in Inputs During the Set Point Change. . . . 90 4.24 Grade-change Simulation. . . 93 4.25 Inputs During the Grade-change. . 93 4.26 Slice Opening During the Grade-change • • . 9 4 4.27 Simulink Implementation of MPC for Nonlinear Simulations. 94 4.28 Response to a Sinusoidal Disturbance (MPC). 95 4.29 Controller Outputs During the Disturbance (MPC). 95 4.30 A Major Disturbance in Level (MPC). 96 4.31 Inputs During the Disturbance (MPC) 97 4.32 Set point Change in Total Head(MPC). 97 4.33 Set point Change in Stock Level (MPC). . . 98 4.34 Inputs During the Set point Change in Level (MPC). 98 4.35 Grade-change Simulation (MPC) 99 4.36 Inputs During Grade-change (MPC) 99 4.37 Constrained MPC with DMC estimator 100 4.38 Constrained MPC with a Different Estimator. 101 4.39 Constrained (Clipped) MPC with DMC Estimator. 102 4.40 Constrained MPC with DMC estimator 102 4.41 Grade-change with a Constrained (clipped) MPC. 103 4.42 Grade-change with a Truly Constrained MPC. 103 xi A. l Transducer Bridge Circuit 118 B. 2 Simulink Model of Headbox Control System Using LQG 132 C. 3 Simulink Implementation of MPC for Nonlinear Simulations 143 Acknowledgement I wish to express my gratitude to Dr. Michael S. Davies for his guidance and encouragement. I would like to thank the co-reader of my thesis, Dr; Guy A. Dumont for his advice and help. I acknowledge the invaluable support of Mr.. Glen Woods. I would also like to thank Mr. Carl Yim, Mr. Howard Robinson, Mr. Robert Gilmaine, Mr. Rollie, Mr. Joe Emery, and Mr. Jeorge for their assistance in carrying out the industrial experiments. I appreciate the advice I obtained from the colleagues at the Pulp and Paper Center. My thanks to the Librarian Rita Penco for her help in literature survey. I would like to acknowledge Dr. Gunnar Berg and Nepal Engineering Education Project for providing financial support. I am grateful to my wife Jasmin (Chameli) for her invaluable help and understanding. This thesis would have never been completed without her inspirations. xiii Chapter 1 Introduction 1.1 Background The headbox is a critical paper machine component. It is here that the fibre/water suspension is finally mixed and delivered on to the moving wire or fourdrinier. Any pressure fluctuations that are transmitted through the headbox will cause irregularities in the flow to the wire and consequent lack of uniformity in paper sheet properties. The pressure pulsations in the approach system and the hydrodynamic properties of the paper machine headbox have become even more important with the introduction of high speed headboxes. Many paper mills have replaced their airpad headboxes with hydraulic ones to obtain better jet stability and higher machine speeds. The hydraulic headboxes use stationary elements to influence flow and turbulence generation at much higher flow velocities. The higher velocities and the use of stationary elements have radically increased the turbulence intensity level and reduced turbulence scale, so improving deflocculation. On the other hand, these hydraulic headboxes have a wider bandwidth than airpad headboxes, and therefore may allow pulsations generated in the approach system to influence the jet velocity. The present work is concerned with the wet-end pressure pulsation analysis and the deriva-tion of a theoretical model of a Sym-Flo headbox to analyze its dynamic behavior. The work was motivated by the observed oscillations in the slice lip head of a fine paper mill. The mill acquired a Sym-Flo headbox manufactured by Valmet, Finland to replace its previous airpad headbox. This replacement has apparently increased the Machine direction (MD) basis weight variations in the paper sheet. The mill is producing fine coated papers, so that the variations -in the MD basis weight is also adversely affecting the coating process. There can be many 1 Chapter 1. Introduction 2 reasons for the MD basis weight variations, however, it is possible to locate some of the major sources of basis weight variations by conducting frequency response analysis on the wet-end pressure data. The effectiveness of the headbox in suppressing the disturbances coming from the approach system can be studied in detail by deriving its dynamic model and carrying out extensive simulations. This thesis explores both of these aspects in detail, and also looks for advanced multivariable control systems that may give better control of the headbox. A literature survey on the subject indicated that only a few dynamic models of headboxes are available. Most of these are highly simplified and are based on airpad headboxes. Natarajan et al in [44] derive a theoretical dynamic model for a hydraulic headbox using first principles. They however do not consider the level control. In their case the attenuator is just a surge tube with no bleed valve and no air input. Lebeau et al in [34] derive a nonlinear model for a pressurized headbox using first principles. The nonlinear model considers cross section of the attenuator (headbox) to be constant irrespective of the level. This assumption is not true for a Sym-Flo headbox. Similar simplified theoretical dynamic models for airpad headboxes can also be found in [1], [18] and [66]. Simplified linear models will not be enough to study the realistic behavior of the system in response to some disturbances. A comprehensive nonlinear model is desirable which accounts for the geometric nonlinearity of the attenuator and the nonlinearities of the valves. The valve nonlinearity can drastically change the dynamics of the system. The present work considers these factors. 1.1.1 Wet-end Pulsations and Paper Quality Pressure pulsations in the approach system to the headbox can affect the paper machine runnability and sheet quality and so should be analyzed properly. The approach system (wet-end) pulsations can be categorized into three classes[29]: • long term (frequency < 0.005Hz), generally due to system stability and slow control loops. Chapter 1. Introduction 3 • medium term (0.005 to 1 Hz), generally due to blending, flow stability, fast control loops. • short term (1 to 20 Hz) generally due to rotating machines, equipment vibrations. Pantaleo [49] points out that the design of the stuff box, the locations of basis weight valve, the entry position of the heavy stock line in the fan pump suction and the air entrainment in the stock have profound effects on long term variability of the paper. Medium term variability of the paper is generally due to the consistency dynamics. Koivo discusses the problem in [32] along with some remedies. Short term variations are due the the wet-end equipment. The fan pump is one potential source of pulsations. The individual impeller vanes of a fan pump create pulsations as they pass from the high pressure discharge zone to the low pressure volute. If the impellers are not properly balanced, pulsations can be very high. Another source of short term variations is the screen. The frequency of pulsations produced by the screen is the product of screen speed and the number of foils. Other sources of noise, both random and periodic, in the piping system include resonate motion of a pipe, valve cavitation, and structural or equipment vibrations [49]. Thus it is evident that the pulsations generated in the approach system are likely to be present and could cause basis weight variability. It is therefore desirable to study the pulsations present in the wet-end and to identify the causes of variability. It is possible to collect pressure data from various crucial points in the wet-end by using multiple pressure transducers and a multichannel data acquisition system. A detailed account of pressure pulsations analysis performed on a fine paper mill is included in the thesis. 1.1.2 Headbox Design and Paper Quality The headbox is perhaps the most important part of the paper machine. It is the key determinant of the basis weight profile and a major determining factor of the sheet formation [63]. The functions of headboxes are : Chapter 1. Introduction 4 • to provide a uniform and stable jet over the width of the machine, without lateral veloc-ities, random components, or streaks. • to provide a geometrically stable and defined slice lip geometry independent of tempera-ture, pressure and slice opening. • to produce a well dispersed fibre suspension with a minimum of floes. • to provide necessary controls of basis weight profile, impact line and angle, and jet velocity. • to stay clean and be easy to maintain and operate. It is therefore essential to know the design features of the headbox to ascertain whether it will be able to meet the requirements listed above. A detailed nonlinear dynamic model is desirable in order to study the characteristics of the headbox through simulation. This thesis therefore includes a detailed description and derivation of a dynamic model of a Sym-Flo headbox using the design constants and mass balance and energy balance constraints. 1.1.3 Motivation The initial motivation for present work was the observed wide MD basis weight variations and occasional wave on the Fourdrinier table of a fine paper mill after the replacement of an airpad headbox with the Sym-Flo headbox. Wet-end pressure pulsation analysis also indicated many periodic disturbances in the system. In order to study the ability of the headbox to suppress these pulsations, a proper physical model must be derived. The model will also be helpful to study the control problems. For the proper control of crucial headbox variables such as total head and level, the multivariate interactions between the control loops should also be analyzed. The nonlinear nature of the model demands a robust control structure. The physical lim-itations of actuators and process outputs demands a constrained control system. This is the motivation for assessing a constrained version of Model Predictive Control algorithm. Chapter 1. Introduction 5 1.1.4 Contribution The thesis makes three contributions. First a novel nonlinear dynamic model of a Sym-Flo headbox is derived. Secondly, tests are carried out on an operating paper machine in order to identify substantial pressure fluctuations. Finally , the model and known pressure fluctuations are combined to simulate the effectiveness of novel multivariable constrained control methods for regulation about a set point and for control of a coordinated grade change. A comprehensive nonlinear dynamic model for a typical Sym-Flo headbox has been de-veloped using first principles. The model includes geometric nonlinearity of the attenuator cross-section and also accounts for the effect of turbulence generation on the frictional head loss. The static nonlinearities of the stock flow valve and air bleed valve are also included. In order to identify the causes of MD basis weight variations, extensive mill experiments have been conducted using portable pressure transducers and a multichannel data acquisition system. To evaluate the effect of adding a pressure screen in parallel to the existing one, the pressure pulsation analysis was carried out before and after the wet end modification. The results were compared and discussed with the mill technical staff. The slice lip opening is included as one of the manipulated variables. This makes it pos-sible to analyze the system for any perturbations in this parameter (treating it as a measured disturbance). This also makes the implementation of automatic grade change possible. For the steady state regulation problem, the slice lip is not allowed to change as it disturbs the cross directional profile of the paper. But for the grade change implementation all the inputs are manipulated in concert to track the desired new set points. As the tracking is done under active control, the stability and quality of the operation remains superior. The grade change problem will then reduce to selecting the tracking rate and input output weighting factors. In order to make the simulations realistic, the nonlinearities of the stock and bleed valves are taken into account. The coupling effect among the inputs and outputs are also analyzed. In order to get an optimal control performance, multivariable Linear Quadratic Gaussian (LQG) and Model Predictive Controller (MPC) are investigated. Chapter 1. Introduction 6 The saturation of the actuators and output constraints are taken into account by using constrained version of MPC algorithms. 1.1.5 T h e s i s O u t l i n e A detailed model of a Sym-Flo headbox is derived in Chapter 2. The model is linearized around an operating point representing conditions during the manufacture of a 45 lb. grade of paper. The linearized model is analyzed for various mathematical properties. Chapter 3 is devoted to mill experiments and the wet-end pressure pulsation analysis. Chapter 4 discusses the control problems and simulation results, starting with a brief literature review and then after establishing the desired objectives of the controller, LQG and MPC controllers are selected. The underlying theory of these controllers is briefly discussed and the simulation results are presented. Chapter 5 concludes the thesis and proposes future studies in this area. Chapter 2 Sym-Flo Headbox Modelling 2.1 Introduction Before starting the technical details of headbox modelling, it is important to understand the various components of a paper machine wet end and the constructional details of the headbox properly. The following sections will introduce these components. 2.1.1 General Description Figure 2.1 depicts a typical white water system of a fine paper mill. rejects > f ^ J Figure 2.1: White Water System 7 Chapter 2. Sym-Flo Headbox Modelling 8 The thick stock consistency from the stuff box mixes with the white water silo consistency through the action of the fan pump and the resulting suspension is pumped through the five stage centricleaner system. The function of the centricleaner system is to concentrate and remove the heavier contaminants such as grit, sand, dirt, etc. from the thin stock. The primary cleaner accept is then passed through a nonpulsating pressure screen where unwanted long fibres are removed. Filler clay and cationic starch are added to the stock in the screen. The thin stock is then piped to the Sym-Flo headbox through the tapered header and on to the fourdrinier where formation and drainage take place. White water draining from the fourdrinier forming zone flows through a flume to the white water silo. This silo provides rich white water dilution for the thick stock from the stuff box. Thus a cycle is established. 2.1.2 Headbox Details A brief description of a typical Valmet's Sym-Flo headbox is presented here. The Sym-Flo Headbox is basically a hydraulic headbox with an integral airpad attenuator tank. Valmet, Rautpohja works Finland, has been building this headbox since June, 1984 [62]. The purpose of the headbox is to deliver a stable jet flow profile and a uniform fibre dis-tribution in both cross and machine directions. To meet these requirements under fluctuating operating conditions, the headbox and its associated flow system must be a complicated hydro-dynamic device. Figure 2.2 depicts a cross section of a Sym-Flo headbox. The headbox can be divided into four parts: the distributor, the attenuator section, the turbulence generator, and the slice channel. The distributor deals with the initial distribution of the flow from the pipe to the full width of the headbox. The manifold tube bank section connected to the attenuator takes the flow from the distributor and stabilizes it into a uniform flow. The attenuator dampens out pressure pulsations in the incoming stock. The turbulence generator generates small scale, high intensity turbulence providing a uniform distribution of fibres and fibre orientation. The slice channel finally accelerates the flow to a speed close to that of the wire and controls the slice Chapter 2. Sym-Flo Headbox Modelling TURBULENCE GENERATOR OUTLET • • • • • • 4 TURBULENCE GENERATOR INLET @ ® @ $ < .» @ © @ € MANIFOLD TUBE SANK OUTLET 1. Air cushion 2. Overflow weir 3. Equalizing chamber 4. Inlet header 5. Manifold tube bank 6. Profile microadjustors 7. Worm gear jacks for vertical adj. of top slice 8. Worm gears for horizontal adj. of top slice 9. Top slice beam 10. Slice channel 11. Slice bar 12. Apron 13. Turbulence generator Figure 2.2: Sym-Flo Headbox Showing Different Parts opening and the impact conditions. The distributor of the Sym-Flo headbox uses the one-sided tapered rectangular header [40]. This gives an even flow distribution to the manifold tubes. The flow enters from the side and perpendicular to the machine. This header has been designed to allow up to 10% recirculation. More details on the design aspects of the tapered header could be found in [61]. The manifold tube bunch system has been designed to give an even stock distribution and to generate pre-turbulence. The flow first passes through precision-made inserts at the inlet side and continues in larger diameter tubes to an equalizing chamber. This step change in diameter generates pre-turbulence and also helps in removing flow disturbances that might occur in the header from deflecting the flow through 90deg [24]. Stock then flows into an equalizing chamber, which is a full width space across headbox (3 in Figure 2.2) where the tubular flows are united. Above the equalizing chamber there is Chapter 2. Sym-Flo Headbox Modelling 10 a pressure attenuator which is filled with compressed air and is designed to dampen pressure pulsations. The air supply, in a typical industrial headbox, will be stepped down from the mill air pressure of 90 psi and regulated to a constant 46 psi. The infeed air line will have a control valve interlocked with the fan pump. When the fan pump is started, the solenoid on the control valve opens to provide air pressure for liquid level control. The attenuator chamber also has overflow lines on the front and back in order to skim any foam, etc. from the surface. The overflow volume should not exceed 300 gpm in the typical headbox under consideration. There will be a control valve in the overflow route too[3]. From the equalizing chamber the stock flows through a perforated attenuator plate, contin-ues in the converging array of tubes, known as turbulence generators, into the slice channel over the heated apron and finally on to the wire. The round tubes gradually become rectangular so that the open area at the outlet is high. It is known [54] that when stock flows in a round tube, the consistency is at its maximum in the centre line of the tube and diminishes towards the edges. To eliminate any consistency streaks in the Sym-Flo headbox, the tubes must overlap each other. This staggered configuration of the tubes, however, affects the frictional headloss in the turbulence generator and must be taken into account while developing the mathematical model. The turbulence generator is equipped with edge flow pipes, fed from the tapered header directly, to aid in fibre orientation and stability of sheet edge. Resistance to flow caused by the turbulence generator directs the pressure pulsations back into the attenuator tank. The top slice of the headbox can be moved in both vertical and horizontal directions. However, due to the placement of the top slice pivot, horizontal movement will result in a change of slice opening. The jet flow profile across the machine is controlled with the top slice lip which can be adjusted by means of micro-adjusters. All the headbox surfaces coming into contact with the stock are made of stainless (acid-proof) steel. All the flow surfaces are polished electrolytically except for the manifold tubes Chapter 2. Sym-Flo Headbox Modelling 11 Design Speed Jet Width Flow to wire (Max.) Recirculation (Max.) Slice Opening Range Horizontal adjustment Top Slice Microadjusters Number Spacing Microadjuster Range Table 2.1: Specifications of a Sym-Flo Headbox and the turbulence generator. The specification of a Sym-Flo headbox is in Table 2.1: Developing a comprehensive model for the study of the turbulence, streaks and the forma-tion is beyond the scope of this work. Although some turbulence related disturbances affect Machine Direction (MD) basis weight variation, these effects are most significant in changing the cross directional basis weight. For MD basis weight analysis, the hydrodynamic disturbances generated within the headbox and in the approach system are more important. The present work is concerned with the derivation of a nonlinear dynamic model of a Sym-Flo headbox to study the hydrodynamic behavior of the attenuator and the turbulence generator for different flow disturbances in the approach piping system. 2.2 Nonlinear Headbox Model 2.2.1 Assumptions: In developing a dynamic headbox model, the following assumptions have been made: 1. The stock consistency in the headbox is low. This is justified since the consistency is normally about 1% by weight. At this level the stock flow behavior is equivalent to that of water. 3,940 fpm 174.8 inches 16275 gpm 10% 0.31 to 5.91 inch 0.32 to 1.3 inch 36 rods 4.87 inch 0.012 inch Chapter 2. Sym-Flo Headbox Modelling 12 Air Supply from compressor Air Bleed Valve Stock supply from pressure screen Breast Rol W I R E Figure 2.3: Schematic Diagram of a Sym-Flo Headbox 2. The stock is incompressible. This assumption is also justified by the fact that the behavior of the stock at a very low consistency level is essentially same as that of water. 3. Mass density distribution in the stock is uniform. This assumption is valid as the Sym-Flo headbox is physically designed to obtain good homogenization in the suspension. This assumption helps to relate the mass balance with the flow balance. 4. The air in the attenuator behaves like an ideal gas. This assumption is not strictly true. However the airpad pressure is not much higher than the atmospheric pressure (max 1.6 atm.), and so the assumption could be considered as reasonable. 5. A i r flow through the air valve restrictions is subsonic. This assumption is also valid because the airpad pressure does not go much higher than atmospheric pressure. 6. A i r flow through the valves behaves adiabatically. This assumption is valid since the heat transfer between the stock and the air is negligibly small. This assumption is required to establish the relationship between the pressure and density of air in the airpad. 7. There is no mass exchange between the stock and the air system. This assumption is valid as the solubility of air in the stock at the given temperature and pressure is negligibly small. This is essential for applying the mass balances to the stock and air systems. Chapter 2. Sym-Flo Headbox Modelling 13 8. The momentum terms in the energy balances (Bernoulli's equations) are neglected. This assumption is legitimate since the momentum terms give rise to dynamics of the order of magnitude of the effective flow time through the system [1]. 9. The outlet areas are considered small in comparison with the cross section of the tank. Hence the velocity of the fluid is uniform throughout the cross section. This assumption is required to calculate the velocity of the fluid at different points. 2.2.2 Attenuator Modelling The dynamics of the attenuator are those of an interconnected liquid and gas-flow system. The accumulation of stock and air in the attenuator gives rise to time varying quantities. The stock and air systems are treated separately below: The Stock Flow System A mass balance for the stock in the attenuator gives d(pVs) , 9 1 , —-£— : Pu>q\ - PwQs - PwQo (^ -1) with the notation described in the nomenclature and Figure 2.3. Since the stock is assumed to be incompressible, P-Pw hence equation 2.1 becomes = Ql - Qs - qo (2.2) and dV, dh Because the attenuator has a complicated cross section (see Figure 2.2), the area of the free surface A and the attenuator air volume Va are functions of h. Chapter 2. Sym-Flo Headbox Modelling 14 0.48 Figure 2.4: Schematic Diagram of the Attenuator Cross-section For the headbox under consideration, see Figure 2.4. If h < 0.48 metres 2(0.48-fr)\ A = W 1^ 0.610 TTR2 v/3 ai = a2 = I arcsm . (0.655 - 0.48) \ d 2 R R Z a3 (0.G55 - 0.48) 0.G10 V = W j ai + a 2 + a 3 + (0.48 - h) Uh> 0.48 ^ = ^ ( 2 i ? c o s ( a r c s i n ^ | ^ ) ) (0.610 + 4)' „ „.'irR2 . ((0.655 -h)\ » (0.655 -fc)i4' i / = W I —^- + arcsin ( J i j 2 + -v ' 2W From equation 2.2 and equation 2.3 Adh ~dl = q i ~ q s ~ q ° (2.4) (2.5) (2.6) (2.7) (2.8) The stock flow out of the headbox can be found by applying energy balance (Bernoulli's theorem). Assuming point A to be inside the attenuator and point C to be outside in the overflow tube (Figure 2.3). Chapter 2. Sym-Flo Headbox Modelling 15 For the overflow, fc dv [vl-v\] fc dp • ' Neglecting the momentum term and the gravitational potential differences which are small compared to the kinetic energy term, gives „ * = 2 < ^ L ™ > ( 2 . 1 0 ) Pw Knowing the velocity, the volume rate of overflow can be calculated as follows: q0 = 2S0vc (2.11) The stock going out through the slice, is determined by the jet velocity. This can be found by writing down the energy balance equation for the hydraulic section (turbulence generator and slice channel). This will be discussed later in detail under the heading of turbulence generator modelling. At this stage the volume of stock forced out through the slice lip can be modelled as follows: qs = CdsSjyf2g~hj (2.12) Using equation 2.8, equation 2.11 and equation 2.12 dh _ ?l - CdsSjy/2ghj - 2S0^2^j^ ^ The Air Flow System A mass-balance for the air in the attenuator gives d(PaVa) dt qi-Qe (2.14) Using energy balance (Bernoulli's theorem), the mass of air leaving the attenuator airpad can be determined. Chapter 2. Sym-Flo Headbox Modelling 16 Neglecting the momentum term and the potential of the external forces, and assuming the velocity at the point A (see Figure 2.3) to be small enough to be neglected, it follows that: rB dp __B + f__ = 2 JA Pa 0 (2.15) To integrate this equation it is necessary to establish a relationship between the pressure P and the density pa-in, adiabatic state change, Po \poJ (2.16) Differentiating both sides, __ __ k f^Y f c _ 1 *E± Po \poJ po (2.17) Hence rB dP L (2.18) Using equation (2.15) and equation (2.18), the velocity of air at point B is given by vaB = 2k Pp fpa \l k - 1 po [\po. ^Jfc-l (2.19) The mass out flow through the bleed valve is given by: qe = poSbVas Rewriting <7e = Sbt 2k PoPo 'or- (2.20) The air-bleed rate may also be calculated using the flow characteristics of the bleed valve. This is, perhaps, a more practical way than using equation 2.20 since it avoids the valve opening area Sb which cannot be. measured directly. It is the valve travel that is readily available. The universal gas sizing equation establishes a relationship between the valve travel and the gas flow through the valve. The universal gas sizing equation includes a coefficient C\ to account for differences in flow geometry among valves and C9 to account for critical flow capacities. C\ Chapter 2. Sym-Flo Headbox Modelling 17 2 to 1 (Line to Body - Size Ratio) Body Size Inches Class Valve Rotaion, Degrees 10 20 30 40 50 .60 70 80 90 1 150-600 0 12.1 75.8 167 275 395 513 621 675 1-1/2 150-600 10.5 127 293 500 734 995 1280 1560 1670 Cg 2 150-600 6.90 162 429 743 1090 1480 1920 2370 2800 3 150-600 44.6 253 755 1480 2220 3050 4000 3130 6120 4 150-600 75.8 518 1300 2310 3540 4850 6270 7860 9420 6 150-600 136 1070 2810 4660 8860 9500 12,100 14,800 17,900 8 150-600 194 1680 4040 7050 10,700 14800 19,100 22,100 27,100 1 150-600 37.1 33.4 31.4 29.8 28.8 27.4 26.7 26.5 1-1/2 150-600 25.7 38.6 36.6 32.3 29.5 28.2 27.2 23.2 23.9 2 150-600 21.2 34.2 33.8 32.3 28.8 27.3 26.7 25.2 25.2 C1 3 150-600 29.9 34.0 31.4 33.0 30.6 28.2 27.2 26.2 24.6 4 150-600 32.9 29.9 32.3 31.4 30.0 28.7 27.9 25.7 23.9 • 6 150-600 29.3 35.4 33.1 31.7 30.5 29.7 28.0 25.6 18.6 8 150-600 34.3 35.1 33.7 31.5 30.7 29.8 27.2 21.7 20.4 Figure 2.5: Gas-sizing Coefficient. is defined as the ratio of the gas sizing coefficient Cg and the liquid sizing coefficient Cv and provides numerical indicator of the recovery capabilities of the valve. Two sizing coefficients are needed to size valves accurately for gas flow. n - r ™ Hscfh — y QJ, C9P sin (2.21) Critical flow occurs when the sine angle reaches 90 deg. Cg and C\ values are selected from the chart shown in Figure 2.5. More instructions on gas sizing using nomographs are found in [26]. Rewriting equation 2.15 dpa dVa Va~dT + pa-dT =qi ~qe (2.22) Or dpa _ qj- qe PaA^ dt Va Va (2.23) Chapter 2. Sym-Flo Headbox Modelling 18 __F=aiwat M = pAeL Figure 2.6: Turbulence Generator Section Using the pressure density relationship, above equation is written in terms of pressure as follows: dP _ kPO [PJX^ ~dt ~ ~p~o~ \Po) Qi ~ qe + P0A(%)*9' Va (2.24) 2.3 Turbulence Generator Modelling The presence of the turbulence generator makes the dynamics of Sym-Flo headbox quite dif-ferent from the dynamics of an airpad headbox. The dynamics of the turbulence generator are completely hydraulic. This section can be modelled by considering a control volume and writing a momentum balance equation for the system. The head at the inlet to the turbulence generator can be established algebraically in terms of the attenuator level, airpad pressure and head-loss occurring in the attenuator plate. hi = h + (P- PO) - ho (2.25) The head-loss due to sudden enlargement in the attenuator plate is determined by using both energy and momentum equations [56]. Assuming uniform velocity over the flow cross section which is approached in turbulent flow, applying momentum balance equation gives, PlA2 -P2A2 = pwV2(V2A2) + pwVii-ViAt) Applying energy balance equation, Vi , Pi , V2 p2 — + 1- — + h/io 2g 7 2g 7 (2.26) (2.27) Chapter 2. Sym-Flo Headbox Modelling 19 VI :V_-: A1 + p1A2 • V1~ A2 p2A2 -V2 Figure 2.7: Attenuator Plate Loss Eliminating pi, P2 and using V\Ai = V2A2 gives r2 / A 1 \ 2 ho 2</ V A2j (2.28) It is now possible to write the equation which governs the rate of change of the head at the slice lip hj due to the forcing head hi in the attenuator. A control volume coinciding with the walls of the turbulence generator and the slice channel is considered (see Figure 2.6). It is also assume that the effective cross sectional area is Ae. Writing down the force balance equation, gives pwghtAe — pwghjAe = but dpwLAeVj (2.29) Vj = ^2ghj (2.30) Differentiating both sides with respect to time dvj / 2g dhj dt Ahj dt (2.31) Rearranging gives: dhj y/2ghj [hi — hj — hf] (2.32) dt L Now the frictional headloss hf occurring in the turbulence generator is to be found. Tur-bulence generator tubes have complex shapes which do not fit to any standard sections. Here Chapter 2. Sym-Flo Headbox Modelling 20 area of stream cross section II,! = ; h y d r a u l i c d i a m e t e r = 4H:l wetted perimeter Siiajx; of cross section fin Pipes and ducts, running full Circle, diameter = D D/4 Annulus, inner diameter = d; outer (D — d)/4 diameter = D Square, side = D D/4 Rectangle, sides a. b ' ab/[2{a -t b)] Equilateral triangle, side = a a/4\fo> Ellipse, major axis = la, minor axis ab/[K{c -i- ii]" = 2b Table 2.2: Hydraulic Radius for Various Cross-sections O D O L O O O v Flow Staggered arrangement Figure 2.8: Sym-Flo Tube Array an approximation will be found to the hydraulic radii at the both ends of the tubes and the average will be taken. Table 2.2 is taken from [46]. For the circular section the hydraulic radius is given by r? . — Df.ir I l cir — 4 For the rectangular section Tf heightxwidth •"^ ec* — 2(height+width) The mean hydraulic radius is _ Raj* I I^vcct / \ •Ti-mean ' ' ^ yZ.oo) To even out the consistency profiles, the tubes in the turbulence generator are arranged in 2. Sym-Flo Headbox Modelling 21 ' < ^ N R . = 8 - 0 0 0 0 . 0 4 ~ ~ N R . = 2 0 , 0 0 0 V ^ Use transverse flow area ——- ; ^ Dotted tines indicate arrangement in which diagonal tlow area = tranverse tlow area Figure 2.9: Frictional Factor a staggered fashion as shown in Figure 2.8 This configuration also affects the frictional loss. A formula to calculate the frictional head-loss for turbulent flow is found in [46] which is reproduced below. The equation is recommended to predict pressure drop across staggered tube banks for tube spacings [(a/Dt), (b/Dt)] ranging from 1.25 to 3.0. „,2 hf =4fNrPw7±-^9 (2.34) where / = friction factor to be read from the curves given in Figure 2.9. Nr = number of rows of tubes in the direction of flow. vt = fluid velocity through the minimum area available for flow. There are different sets of curves for different Reynolds numbers. Reynold number can be calculated as follows: NR A* (2.35) 2.4 Consistency Dynamics The accumulation of fibres within the headbox gives rise to consistency dynamics. In the past, the difficulty of obtaining reliable consistency measurements made the modelling of consistency Chapter 2. Sym-Flo Headbox Modelling 22 variations less useful. Although measurements are now possible, only a few mills have on-line consistency sensors. Consistency dynamics are modelled here for completeness, but wi l l not be considered in control. From a fibre mass balance in the headbox stock, ^Ml = q i C i - qsCa - q0Cs (2.36) where Mf = VSC3 Differentiating both sides *£-°-%™% ™ Using 2.2, 2.36 and 2.37, (2.38) For the detailed modelling of consistency dynamics, all the recirculations and consistencies of different mixing and diverting flows in the approach system must be considered. Some issues which makes the modelling of consistency complicated are given below. The input stock flow to the headbox can be expressed in terms of the fan-pump outflow as follows: q\ = 0.9Bfractqf where 0.9 accounts for the recirculation of stock i n the tapered header and the factor Bfrac accounts for the by-pass flow. By-Pass valves are provided in the mills where constant speed fan-pumps are used. To adjust heads for different grades of paper production, the by-pass valve is adjusted. This diverts a fraction of the fan-pump outlet, which is then recirculated to the white water silo. . Ci=Cf(t-T) Chapter 2. Sym-Flo Headbox Modelling 23 where Cf is the stock consistency at the fan pump. This consistency is the result of mixing two streams namely the thick stock from the stuff box and the dilution water from the white water silo. Since consistency is denned as the weight ratio of total solids to the solid plus water, n _ CdQd + CthQth r 9 „ Q x C f ~ (Cd + l)qd + (Ctk + 1 ] Obviously <lf = Qd + qth The consistency at the inlet to the headbox is thus a complex function of many factors. Further there is a time delay between the thick stock consistency variation and the thin stock consistency at the headbox inlet. Deriving a detailed model for the consistency dynamics is therefore reserved for the future work. 2.5 Overall Model Equations 2.13, 2.24, 2.32 and 2.38 describing the headbox dynamics are reproduced below: dh «i ~ CdsSj^h-- 2 S j 2 ^ & (k-1) dP kPQfP\k dt po t , - » , ^ ( * ) , » (2.41) f = | ( C-C.) (2.43) Here the stock level h in the headbox, the airpad pressure P, the head at the slice lip hj and the consistency at the slice flow are chosen as the state variables. The stock in-flow qi, the valve sizing for gas flow Cg, the slice lip area Sj and the consistency of the stock at the headbox inlet C,- are treated as inputs. The headbox dynamics can thus be represented by a fourth order nonlinear dynamical sys-tem. The model is characterized by 11 parameters. The parameters A, Va, S0, f and Cds are Chapter 2. Sym-Flo Headbox Modelling 24 construction dependent parameters while k,Po,po,pw and g are fundamental physical param-eters. The values of q\, S_ or Cg/C\ and Sj depends upon the operating point. Knowing the design details of the headbox and the operating point, all parameters have been calculated. The first three equations are highly coupled. Equation 2.13 shows that if the input stock flow rate increases, the liquid level increases. This in turn increases the airpad pressure, the total head and so the out going stock flow. There is thus a self regulating effect in the sys-tem. Equation 2.24 shows that the airpad pressure will return to its original value when the equilibrium point is reached. Any deviation in the level will thus contribute to the total head. Changing the bleed valve area changes the air outflow affecting both the level and the pressure inside the airpad. It is seen clearly from equation 2.25 that the head at the inlet of the turbulence generator remains unchanged in steady state. Opposite changes in the level and the airpad pressure cancel each other. An increase in the slice opening will produce a decrease in the level, airpad pressure and the total head. The most important criteria for a headbox is to maintain constant jet velocity, remove pressure fluctuations and to have a good dynamic behavior when changing the grades. To get good dynamic behavior, the level has to be maintained constant. The attenuator level is subjected to variations from the changes in stockflow and airflow. Variations in fan-pump speed, headbox showers, cleaner throughput, screen throughput, white water silo level, machine chest level etc. are the sources of stockflow variations. The variations in the compressor flow, the air content in the stock and the temperature variations are some of the sources of airpad pressure variations. The consistency variations also affect the dynamics. Astrom in [1] quotes Lindstrom stating that it is reasonable to require a peak to peak variation of jet speed less than 1% for the machine speed above 8m/sec. It is also desirable to keep the level variations within 2-20 mm. Measurements by Lindstrom on many headbox systems in Sweden have shown that difficulties have been observed from pressure variations in the frequency range of 0.05 - 1 Hz [1], Similar results obtained from several experiments of pressure pulsation measurements around Chapter 2. Sym-Flo Headbox Modelling 25 the wet-end components including headbox are discussed in the next chapter. 2.6 Linearization The nonlinear model obtained in the last section can be used to study the dynamic behavior of the headbox. Start-up behavior and other nonlinear simulations can be carried out. However to design control strategies, it is always convenient to have linearized models which approximate the nonlinear dynamics for small perturbations around a steady state point. For a given steady state point, the inputs required for maintaining the steady state levels can be calculated easily by setting all time derivatives to zero. Once the inputs and the states are known, the model can be linearized either by using a Taylor series expansion or by using Matlab commands. The Matlab m-files to find the linearized model are provided in the Appendix A. 2.6.1 A Numerical Example The numerical values of the parameters for producing 45# paper are listed below: Construction Parameters A = 2.1 m 2 V_=1.7 m 3 Cd5=0.98 5o=0.002 m 2 f = 0.13; Physical Parameters k = 1.4 po=1.29 k g / m 3 pw=1000 k g / m 3 Po=10.35 m of water g = 9.81 m / s e c 2 Chapter 2. Sym-Flo Headbox Modelling 26 Operating data h = 0.365 m P = 16.44 m hj = 5.53 m Ca = 0.6 % The inputs required for this steady state point are u(l) = qi = 0.8422m3/sec u(2) = C9 = 483.5783 u(3) = Sj = 0.0804m2/sec and u(4) = d = 0.6 % Knowing the operating point and the inputs, the linearized model can be found either by using a Taylor series expansion and taking the first linear terms only, or by using MATLAB. The linearized state-space model obtained for the 45# grade is given below: A = 0.0000 -0.0008 0.0000 -0.0897 9.1371 9.1371 B = C = 0 0.4760 13.9243 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 -0.0016 0 0 -0.0353 -1.0330 -10.6820 0 -4.8587 -142.1382 -212.5876 0 0 0 0 -7.3053 0 0 0 7.3053 Chapter 2. Sym-Flo Headbox Modelling 27 Nonlinear tknutatkn results 0.4 8 £0.2 Airpad Prsuum 500 1000 Tkrw In seconds Figure 2.10: Nonlinear Simulation Results D = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Eigenvalues of the system matrix A are -0.0020 -9.6626 -1.1072 -7.3053 Note that the system has a very slow mode with a time constant of about 500 seconds and three fast modes with time constants of less than a second. These are due to the strong coupling present in the system between the level and pressure dynamics. The area of air bleed valve opening is responsible for the coupling effect. Had there been no restriction on the air flow, the level dynamics would have been quite faster. But the narrow exit area of the bleed valve is responsible for the slower dynamics. Figure 2.10 shows a nonlinear simulation of the headbox where stock flow and air bleed valves are manipulated by 1% at different points of time. Chapter 2. Sym-Flo Headbox Modelling 28 The simulations started with the steady state inputs for 45# grade paper. Then after 500 seconds of simulation, a step change of 1% in stock flow is introduced. After 1000 seconds a step change of 1% in air bleed valve is introduced. Figure 2.10 clearly shows the sluggish response of the stock level and fast responses of total head and airpad pressure. It is also interesting to note the large change in stock level and pressures for 1% change in stock flow. Thus it is clear that the headbox can not function well without any controller. It is also interesting to note that the values of the airpad pressure and the total head are quite different. It is because of the loss occurring in the attenuator plate and the turbulence generator. 2.7 Invertibility: The study of regulation in linear multivariable systems led to the concept of invertibility [53]. This concept is useful in understanding and solving many fundamental multivariable problems such as decoupling, model matching, minimal design etc.. It is shown in [53] that a system will have a stable inverse if the rank of the following matrix is r(n-l). Where r is the number of inputs and n is the system order. D CB CAB CA2B CAZB CA4B CA5B CA6B CA7B 0 D CB CAB CA2B CA3B CA4B CA5B CA6B 0 0 D CB CAB CA2B CA3B CA4B CAhB 0 0 0 D CB CAB CA2B CA3B CAAB 0 0 0 0 D CB CAB CA2B CA3B Here r = 4 n = 4 Calculation of the rank of the above matrix for this particular case clearly showed that the system has a stable inverse. The concept of inversion is particularly important when a model based controller is used. In model predictive control, if all the controller parameters are selected as default values (in MPC toolbox), a model inverse controller is resulted. If the model is not invertible, the controller will not work. Chapter 2. Sym-Flo Headbox Modelling 29 2.8 Controllability and Observability The linearized system is completely observable as all the four states are available as outputs. Controllability of the system is, however, not so evident. This can be checked by calculating a controllability matrix and finding its rank. The controllability matrix for all four inputs can be found as follows: Cm = [B AB A2B A3B] (2.44) For the given numerical values, the rank of the controllability matrix was found to be 4 with all four inputs, which shows that the system is controllable. It is evident from the linear model of the headbox that the fourth state, the consistency, is completely decoupled from the first three states so it is impossible to affect this state through any of the first three inputs. As the fourth state is completely decoupled, the model can be divided into two; a third order model represents the hydrodynamic system, and a first order model represents the consistency dynamics. The third order model is found to be fully controllable from any one of the inputs. It is, however, more convenient for control design and simulations to treat all the four equations together. It is again stressed here that the consistency dynamics deserves a separate study, so will not be treated rigourously. 2.9 Conclusions A nonlinear dynamic model of the headbox has been derived. The model is fourth order, but can be treated as a third order hydrodynamic model separate from first order consistency dynamics. For the headbox under consideration, physical parameters may be obtained to be entered in the model equations. For control design, a linear model may be obtained representing dynamics in the neighbour-hood of an operating point representing production conditions for a specified grade of paper. It is found that the linearized model is invertible, controllable and observable. Chapter 3 Mill Experiments: Analysis of Wet End Pressure Pulsations 3.1 Introduction Among the many possible sources of machine direction basis weight variations, the most likely sources are pressure pulsations originating in the approach system, and hydrodynamic distur-bances produced within the headbox. The nonlinear model developed in the previous chapter describes the dynamics of the headbox. In order to assess the effect of the hydrodynamic dis-turbances originating in the approach system, measurements are taken to determine pressure fluctuations around the major wet end components. In this chapter, the machine direction basis weight and wet end pressure pulsation analysis carried out at a fine paper mill will be discussed in detail. The use of a multichannel data acquisition system and frequency response techniques to detect the sources of disturbances are explained. Computer programs, based on the Fast Fourier Transform, have been written in MATLAB code to calculate power spectra, cross power spectra, and correlations. The results of the analysis are presented for tests made on three occasions before and after wet end modifications. During the test period, the mill modified the wet end configuration by adding a new pressure screen in parallel to the existing screen. The experiments were performed before and after this modification so that the effect of the modification could be evaluated. There are many possible sources of MD basis weight variations. It is always very difficult to track down the causes using simple techniques. When the variations due to several different sources are brought together, it becomes almost impossible to see the periodic nature of the variations in the time domain signal. It is, however, possible to separate the variations according to the frequency at which they are taking place by using frequency response techniques such as 30 Chapter 3. Mill Experiments: Analysis of Wet End Pressure Pulsations 31 the power spectral density. Knowing the frequencies of the disturbances, it often becomes easy to identify the most likely sources. It is sometimes also possible to determine the time delay between two signals, if they are significantly correlated, by using the cross correlation. The severity of disturbances is partly set by their amplitude so that the standard deviation represents the strength but fluctuations are also relative to the mean value. To obtain an index independent of mean value the normalized quantity known as 'coefficient of variations'(CV) is used. The percentage standard deviation (CV) in comparison to the mean is used as a measure of the severity of the disturbance. For a comprehensive study, pressure data from all the major pieces of wet end equipment should be recorded simultaneously. For this reason a multichannel data acquisition system is essential to carry out such experiments. The features of the transducers, signal conditioning devices and the data storing devices are discussed briefly in the next section. The theoretical aspects of the work are explained under the heading of Data Processing. The following assumptions are made for the analysis of the data: 1. The period when the analysis is done is typical of normal process operation. 2. The noise present in the system is Gaussian. 3. The residual variations are small compared to the true MD variations. 4. The Coefficient of Variation (CV) can be used to represent the severity of disturbances. 3.2 Equipment Review The principal components being used to carry out the experiments are the transducers, signal conditioning devices, and data storing devices. The schematic arrangement is shown in Figure 3.1. Chapter 3. Mill Experiments: Analysis of Wet. End Pressure Pulsations 32 Figure 3.1: Schematic Diagram of a Data Acqusition System. 3.2.1 Transducers For obtaining meaningful data, high sensitivity transducers are used. For measuring the pres-sures over a wide frequency range, the transducers should have very good dynamic character-istics. Although a set of high quality pressure transducers was installed as part of the project, some measurements were made using standard mill equipment. Standard Mill Equipment Sheet basis weight, total head in the headbox, thick stock consistency, stuffbox level, and by-pass basis weight valve opening are measured using standard mill sensors. Validyhe Variable Reluctance Pressure Transducers The portable pressure transducers manufactured by Validyne are used to measure the pressure pulsations around the wet end components. The technical/operational details and calibration of the pressure transducers are provided in Appendix A. 3.3 Data Processing A sampling rate of 1000 Hz has been selected for the data collection. An anti-aliasing filter with a cutoff frequency of 280 Hz has been used. The filter being used is a passive RC filter in order to avoid active filters. A higher sampling rate has been selected to avoid the aliasing problem. With 1000Hz, the Nyquist frequency becomes 500Hz, so that signals having a frequency up to Chapter 3. Mill Experiments: Analysis of Wet End Pressure Pulsations 33 500 Hz axe faithfully represented. The filter will block all the signals with frequencies much higher than 500 Hz effectively. Higher sampling rate gives rise to large volumes of data. However, since there is no signifi-cant information of higher frequencies, decimation is used to reduce the sampling rate digitally. 3.3.1 Decimation The process of decimation allows the sampling rate to be decreased without significant unde-sirable effects of aliasing and quantization errors. Decimation consists of a digital anti-aliasing filter and a sample rate compressor. The rate compressor reduces the sampling rate from Fs to F s / M where M is the compression factor. To prevent aliasing, a digital filter is used to ban-dlimit the input signal to 80% of F s / 2 M beforehand. The sampling rate reduction is achieved by discarding M - l samples for every M samples of the filtered signal. Decimation by a factor of 10 was used in the data analysis of the experiments. Thus the new sampling rate is 100 Hz. 3.3.2 Spectrum Estimation and Analysis As previously mentioned, information on the nature of disturbances present in the system can be obtained through frequency domain analysis. There are two common methods for spectrum estimation and analysis : nonparametric and parametric. The nonparametric method uses the fast fourier transform to obtain the spectrum, but with a short data length, the frequency resolution becomes quite poor. Parametric methods can provide high resolution but need an accurate model of the process. The Welch modified periodogram method is a non parametric method which provides meaningful result by using suitable windowing and overlapping [23]. 3.3.3 Scalloping Loss or Picket-fence Effect When a Discrete Fourier Transform (DFT) is performed, the D F T components wi l l be harmon-ically related with the first harmonic frequency / given by Chapter 3. Mill Experiments: Analysis of Wet End Pressure Pulsations 34 J NT where N is the number of data T is the sampling interval. Thus the spacing of the DFT components depends on the number of data samples and sampling rate. Both these factors are restricted by the data acquisition hardware and the storage device. If, therefore, there is a signal component which falls between two adjacent harmonic frequency components in the spectrum then it cannot be properly represented. This problem can be solved by adding data. 3.3.4 Spectral Leakage and Spectral Smearing The DFT of the sampled data is not the true DFT of the process because the process data is truncated. This is equivalent to multiplying or windowing the signal by a rectangular pulse of width equal to the data length. This time domain product is equivalent to the convolution in the frequency domain so the DFT consists of the true spectrum convolved with that of the window function. The effect of window manifests itself as spurious peaks or side lobes in the spectrum. These lobes distort the amplitude spectrum. To avoid this it is necessary to modify the data by multiplying them by a window shaped to reduce the side lobe effects. In the Welch method, a Hanning window is used as a default window. 3.4 Known Sources of Pressure Fluctuations Constructional features of the fan pump and screen can introduce pressure pulsations in the approach system. The fan pump and the screen are coupled with headbox through the stock medium. If the pulsations generated by these wet end components are strong, they will disturb the headbox operation. The amplitude and frequency of pulsations generated by the fan pump depend upon speed and number of blades on the pump impeller. Speed decides the fundamental component of the Chapter 3. Mill Experiments: Analysis of Wet End Pressure Pulsations 35 pulsation while number of blades decides the harmonic frequencies. Power distribution among the harmonics depends on the characteristic curve of the pump. The specifications for the fan pump are reproduced below: Pumpsize 18 x 20 inches Total Head 131 feet RPM 1180 Power 800 HP Efficiency 85% The frequency of pulsation generated by the screen depends on its speed of rotation and the number of vanes (foils). The specification of the screen is given below: Type Black Clawson Rotor RPM 287 Number of foils 4 Type of drive V-Belt Motor 75 HP The following the first set of tests, a new screen was added in parallel with the existing screen. The screen added in parallel to the existing one had the same specifications to the first one, described above. The new screen has a motor of 100 HP. Thus it can be speculated that the fan pump can introduce a fundamental pulsation of 19.7 Hz. The screen can introduce a fundamental pulsations of 4.8 Hz. The next harmonic that the screen can introduce is expected to be 4 x 4.8 which is 19.2 Hz. Chapter 3. Mill Experiments: Analysis of Wet End Pressure Pulsations 36 3.5 Experiments and Results Three experiments were performed in the mill to investigate the causes of basis weight variations. First two experiments were done prior to the wet-end modification(the addition of new pressure screen). The third one was carried out just after the modification. A brief account of each experiment is given below: 3.5.1 Experiment No 1 The first experiment considered the disturbances present in the MD basis weight signal and also studied whether the pulsations in the total head correlate with the basis weight and/or ash variations. The data collection device then used was a plug-in data acquisition board with four differential analog inputs installed into a 386 - Toshiba laptop. The sensors being used were the standard mill sensors. To capture the MD variations in the paer sheet, the basis weight and ash sensors(scanners) were held stationary at the middle of the sheet. The experiment was done on 29th of March 1995. The grade under production was 54#. The data was collected for ten minutes. Then off-sheet data from the basis weight and ash sensors were also collected to identify the sensor noise levels. Some of the details of the grade are presented below: 1.91 V 4.59 V 52.2 lb/3000 sq.ft 0.45 lb/3000 sq.ft 0.86 Since the voltage bears inverse relationship with the basis weight, the basis weight profile can be generated by inverting the voltage profile. Basis weight = 52.2 - (volt-mean)/mean*52.2 The mean value of voltage on sheet The mean value of voltage off-sheet The mean basis weight The standard deviation The coefficient of variance Chapter 3. Mill Experiments: Analysis of Wet End Pressure Pulsations 37 The Measurex system does the calibration using a nonlinear curve referred to as transmis-sion curve. Since interests are in the frequency content of the signal rather than its absolute amplitudes, a simple linear relationship as given above is sufficient. The power spectral density is calculated using the Welch periodogram method [23]. Param-eters used are given below: NFFT 48000; WINDOW 24000; NOVERLAP 20000; DFLAG 'mean' (detrending mode) Fs 100; The data was first decimated by a factor of 10. Figure 3.2 shows the very low frequency components. Much of the power is concentrated in the low frequency range. It is calculated that 65.6% of the total power in the signal is below 0.2 Hz. Powor Spectral Density plot of BW signal O 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Frequency In Hz Figure 3.2: Power Spectrum of MD Basis Weight Signal. Here the sampling rate is 100 Hz and the NFFT for Fast Fourier Transform was taken to be 48000. Thus the resolution is 100/48000 Hz = 0.0021 Hz. Chapter 3. Mill Experiments: Analysis of Wet End Pressure Pulsations 38 The peaks are found around 0.0021 Hz, 0.0125 Hz, and 0.0312 Hz. The frequency resolution hides effects below 0.0021 Hz, so that the first peak is misleading. Other two frequencies are more dependable. A similar analysis of Total Head signal gives following results: Mean value of voltage Mean total head Standard deviation of the voltage signal Standard deviation of the head signal Coefficient of variations 3.26 V 177.2 in of H20 0.0445 2.42 1.37 The power spectral density plot is given in Figure 3.3 x io4 Power Spectral Density plot of Total Head signal & 5 I 3 2 1 1 1 1 1 1 1 1 : 1 1 ! ! ! ! ! ! ! i n i L .. I .. i . : : : A [ y : : :; ; : 0 0.02 0.04 0.06 0.O8 0.1 0.12 0.14 0.16 0.18 02 Frequency in Hz Figure 3.3: Power Spectrum of Total Head Signal. The signal has a peak at 0.0417 Hz, and most of the power is concentrated around this frequency. The headbox is not attenuating this pulsation, which may have originated inside the headbox itself. This frequency corresponds to a period of 24 seconds and may be due to a slow control loop or recirculation effects. The signal may also originate from the approach system or air compressor. Note that the basis weight signal, however, did not display the frequency component at 0.0417Hz that was present in the total head signal. Turbulence created on the table will filter Chapter 3. Mill Experiments: Analysis of Wet End Pressure Pulsations 39 away some oscillations, but, is unlikely to influence such a low frequency. It is to be noted that the basis weight shows a peak at 0.0315 Hz. The analysis showed that the basis weight is dominated by low frequency disturbances. To catch high frequency disturbances, off-line basis weight analysis will have to be done. The basis weight sensor has significant nuclear noise and filtering, so high frequency disturbances could be lost in that back ground. Since the ash sensor uses an X-ray signal, it is comparatively less noisy. A single point ash sensor data was also collected, but the analysis showed that, the low frequency disturbances also appeared in the ash sensor data but to a smaller extent only. The Power spectral density plot is shown in Figure 3.4. It is to be expected that ash content will track basis weight. Power Spectral Density plot of Ash Sensor signal 0.25 -02 -•^ 0.15 -0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.00 0.1 Frequency in Hz Figure 3.4: Power Spectrum of Ash Sensor Signal. Two peaks are observed in Figure 3.4 at 0.0143 Hz and 0.0357 Hz. Allowing for the dispersing effect, these peaks correspond to peaks appearing in the basis weight signal. High frequency disturbances are even less apparent in this signal. It was observed that both the total head sensor and basis weight sensor were unable to detect high frequency variations, which may give a false impression that the headbox is doing well in suppressing them. From the experience, however, it is known that the variance of basis weight is far more greater than the one calculated from the analysis of the basis weight signal Chapter 3. Mill Experiments: Analysis of Wet End Pressure Pulsations 40 here (Std. Dev. = 0.45). The results of the first experiment axe summarized in Figure 3.5 Basis weight/Total head/Ash analysis Results Quantity Mean Std Coeff. ofVar Peaks at Basis weight 52.21 0.451 0.86 0.0021 Hz, 0.0125 Hz, 0.0312 Hz Total Head 177.2 inch 2.422 inch 1.3668 0.0417 Hz Ash • 11.2% 0.0704% 0.6286 0.0143 Hz, 0.0357 Hz Figure 3.5: Results of Experiment No. 1. 3.5.2 Experiment No 2 In this experiment, pressure data from various wet end points were measured using newly installed sensors. Again the plug-in type Data Acquisition Card was used with the same 386-Toshiba laptop. The test was performed on 3rd April 1995. The data acquisition system had only four differential channels, and so only four pressure data points could be recorded simul-taneously . So the experiment was done in two parts. In the first part, the pressures at fan pump, cleaner accept, screen inlet and tapered header inlet were recorded. In the second part, the total head replaced the cleaner accept line pressure measurement. The sensors were located at the following locations: 1st sensor after the fan pump 2nd sensor after the headbox (total head) 3rd sensor before the primary screen 4th sensor near the tapered header inlet Chapter 3. Mill Experiments: Analysis of Wet End Pressure Pulsations 41 Figure 3.6: Schematic Diagram of an Approach System Showing Sensor Locations. Figure 3.6 depicts the white water system of the miU. The locations of pressure measurement are indicated by the thick arrows. Both parts of the experiment gave similar results. To avoid redundant descriptions only the results of the second part of the experiment are presented below. Note that the noise spectrum of the cleaner pressure is similar to the screen pressure in the first part of the experiment. Both these signals show broad band noise and peaks around 4.8 Hz, and 19.8 Hz. The data from the sensors were filtered through an anti-aliasing filter with a cut-off fre-quency of 280 Hz. Then they were sampled at 1000 Hz. The grade under production was 45#. Some relevant parameter values are reproduced below: Chapter 3. Mill Experiments: Analysis of Wet End Pressure Pulsations 42 Basis weight Thick stock flow Main stream valve opening By pass valve opening Consistency Ash Caliper Total head: Headbox flow PCC Rush-drag 43.1 lb/3000 sq.ft 1410 gpm 70% 53.7% 0.5 14% 3.36 mills 227.2 inch water 12600 gpm 8.2 gpm 2 fpm Power Spectral Density of Fan Pump Pressure 1 i 1 , 20 25 30 Frequency In Hz Figure 3.7: Power Spectrum of Fan Pump Signal. Figure 3.7 shows that the fan pump signal does have a significant disturbance component. The peak at 19.78 Hz is very prominent, and the frequency matches with the rotational speed of the pump. The coefficient of variations of the cleaner pressure signal was found to be 1.158. There were pulsations present at 0.122 Hz, 4.8 Hz, and 19.78 Hz. As pointed out earlier the signal of Chapter 3. Mill Experiments: Analysis of Wet End Pressure Pulsations 43 19.78 Hz matches with the possible pulsations from the fan pump. Figure 3.8 shows the screen pressure spectrum in the first part of the experiment. The spectrum is much similar to that of the cleaner spectrum shown in Figure 3.9. Power Spectral Density of Screen pressure 20 18 16 14 1 g 12 •& | 10 S. .3 8 £ 6 4 2 0. LJIUIU^^ i^Jml .LL.I 4 5 6 Frequency in Hz Figure 3.8: Power Spectrum of Screen Signal. Power Spectral Density of Cleaner Accept pressure S i s s $4 1 1 1 L . i i A 10 15 20 25 30 35 40 46 50 Frequency in Hz Figure 3.9: Power Spectrum of Cleaner Signal. The screen pressure signal also shows broad band noise. The screen pressure spectrum includes very low frequency variations. In the second part of the experiment, the channel collecting the cleaner accept pressure is disconnected and total head signal is instead connected to that channel. The pressure signal at Chapter 3. Mill Experiments: Analysis of Wet End Pressure Pulsations 44 Power Spectral Density of Header Inlet pressure 14 1 1 1 1 , • -• I , , . , — J i j 19 19.1 19.2 19.3 19.4 19.5 19.6 19.7 19.B 19.9 20 Frequency In Hz Figure 3.10: Power Spectrum of Tapered Header Signal. Power Spectral Density of Total Head pressure Frequency -resoKition-s OjOOSHz- - -Max pea* appears-to- be at0:006Hz-Figure 3.11: Power Spectrum of Total Head Signal. the tapered header shows peaks around 19.2 Hz and 19.8 Hz. The signal at 19.2 Hz matches with the screen pulsation and 19.8 matches with the fan pump. Figure 3.10 shows the header signal. Figure 3.11 shows low frequency disturbances in total head signal. The results of this test are tabulated in Figure 3.12 Discussions From Figure 3.12, it can be seen that the frequency of the prominent disturbances present in the system vary from 0.006 Hz to 19.78 Hz. Knowing the frequencies, the sources of disturbances Chapter 3. Mill Experiments: Analysis of Wet End Pressure Pulsations 45 Pressure Pulsation Analysis Results Location Mean PSI Std. Coeff. of Var Peaks Freq. Hz Power Fan Pump 50.508 0.458 0.907 19.78 32 0.006 27 0.02 28 Screen 21.591 0.318 1.472 0.042 12 4.8 2 19.2 2 HeadBox Inlet 12.696 0.141 1.111 19.2 3.4 0.006 14 Total Head 8.036 0.0896 1.115 0.02 2 0.042 1 Figure 3.12: Results of Experiment No. 2b. could be located. The short term variations (1 to 20 Hz) are mainly due to equipment vibrations. Peaks at 19.78 Hz, 19.2 Hz, and 4.8 Hz can be related to the motor speeds or rotor speeds. As indicated before, the 19.78 Hz matches with the speed of the fan pump (1180 rpm). This may show that the fan pump motor is poorly balanced or is misaligned with the pump. The 19.2 Hz disturbance can be related to the screen itself. The speed of screen rotor is 287 rpm. This matches with 4.8 Hz signal. Since there are 4 foils in the screen, or 4*4.8 = 19.2 Hz vibration is possible. It is to be noted that 0.02 Hz signal seems to be the most prominent one in the screen disturbance and the corresponding period is 50 sees. In the low frequency disturbances, 0.006Hz, 0.02Hz and 0.042 Hz seem to be prominent. These represent 167, 50 and 24 second periods. These medium term variations (0.005 to 1 Hz) are generally due to blending problems, flow stability problems or fast control loops [29]. From Figure 3.12 it is evident that the screen pressure has got the highest CV . It is also clearly seen in Figure 3.9 and Figure 3.8 that the noise has broad frequency content. Turbulence-induced disturbances generally exhibit such characteristics. It follows that there may be turbulence generated between the fan pump and the screen. Possible sources are the Chapter 3. Mill Experiments: Analysis of Wet End Pressure Pulsations * 46 cavitation in the main and the by-pass stream valves or the cavitation in the pipe bendings. It is also seen that the headbox control system is not successful in suppressing low frequency disturbances. 3.5.3 Experiment 3 The third experiment was performed on 30th May 1995 after the addition of a new pressure screen in parallel to the existing pressure screen. The new pressure screen is identical to the old one. Stock coming from the cleaner accept is divided between these two screens. The accepts of these screens are then re-united and fed to the headbox. The experiment was also performed in two parts. In the first part, pressure pulsations in the various parts of the wet end components were recorded. In the second part, data from stuff box level, stuff box consistency, by-pass valve position (basis weight) and total head from the mill sensors were collected. This two stage data gathering was made necessary by distance constraints at the mill. The grade under production was 45#. Some relevant details are reproduced below: Basis weight 43.2 lb/3000 sq.ft Thick stock flow 1440 gpm Main stream valve opening 68.1% By pass stream valve opening 37.3 % By pass thick stock valve opening 53.7% Consistency 0.46 Ash 13.9% Caliper 3.52 mills Total head: 252.7 inches of water Headbox flow 14444 gpm PCC 14.8 gpm Chapter 3. Mill Experiments: Analysis of Wet End Pressure Pulsations 47 Rush-drag 2 fpm Slice opening 0.901" Wire speed 2203 fpm Part a The power strectra of the pressure signals measured around the wet end components are shown in Figure 3.13 - 3.17. Power Spectral Density of Fan pump pressure I10 I I I ' - ' I ,M^mikj*ammimfiiiiuh,t , u i I O 6 10 16 20 25 30 36 40 46 50 Frequency in Hz Figure 3.13: Power Spectrum of Fan Pump Signal. The results of the experiment No. 3a are presented in Figure 3.18. Comparison of coefficient of variations before and after the wet end modification are done in Figure 3.19. Discussions It is clear from Figure 3.19 that the coefficient of variance of different pressures disturbing the wet end system performance have definitely been reduced. Comparing Figure 3.12 and Figure 3.18, it is found that the coefficient of variance of the screen pressure has dropped from 1.472 to 1.071. Most of the disturbance frequencies are unchanged. The frequency of 0.042 Hz has Chapter 3. Mill Experiments: Analysis of Wet End Pressure Pulsations 48 Power Spectral Density of Cleaner Accept pressure 15 20 25 30 Frequency In Hz Figure 3.14: Power Spectrum of Cleaner Signal. Power Spectral Density of Screen pressure Frequency resolution = 0j002Hz 20 25 30 Frequency in Hz Figure 3.15: Power Spectrum of Screen Signal. disappeared while 6 Hz signal has appeared. It is very interesting to note that the relative strength of 0.02 Hz signal has been drastically diminished. The speed of the fan pump has apparently dropped slightly(as the suspected pulsation due to fan pump now has slightly lower value than the one before). The disturbance due to the fan pump pulsation seems to be more powerful than before as is the 4.8 Hz pulsation. But the lower frequencies are attenuated, improving overall variations. The total head signal is improved compared to the previous results. It is clear that the modification to the wet end has resulted in a better performance. Chapter 3. Mill Experiments: Analysis of Wet End Pressure Pulsations 49 Power Spectral Density of Header Inlet pressure 8| 1 1 , n , , , — 7 - i :• :• : : : : e ! \ \ I! \ 2h 0 l i J-LJ Ii- i ii i i —J i I 0 6 10 16 20 25 30 35 40 45 50 Frequency In Hz Figure 3.16: Power Spectrum of Tapered Header Signal. Power Spectral Density of Total Head pressure 0.251- T 1 1 1 1 1 1 Frequency In Hz Figure 3.17: Power Spectrum of Total Head Signal. Part b In the second part of the experiment, signals from the stuff box consistency sensor, the stuffbox level sensor, by-pass valve position sensor and the total head sensor were recorded simultane-ously. The operating conditions were: Stock flow 1442 gpm Stuffbox consistency 3.04 % Stuffbox level 51 inches Chapter 3. Mill Experiments: Analysis of Wet End Pressure Pulsations 50 Pressure Pulsation Analysis Results Location Mean PSI Std. Coeff. of Var Peaks Freq. Hz Power Fan Pump 57.919 0.339 0.585 19.66 275 Cleaner Accept 20.797. 0.160 0.770 19.66 2 Screen Outlet 23.644 0.253 1.071 0.006 27 0.02 2 4.8 3.5 6 2 19.66 2 HeadBox Inlet 15.999 0.081 0.506 19.17 6 19.66 10 Total Head 9.109 . 0.0113 0.124 .0.006 0.06 0.02 0.08 0.042 0.22 Figure 3.18: Results of Experiment No 3a. By-pass valve position(opening) 57.1% The power spectra of the signals are shown in Figure 3.20-3.23. The results obtained from the analysis of the signals are tabulated in Figure 3.24. Discussions Very low frequency variations were present in the stuffbox level and consistency variations. The by-pass basis weight valve was under active control during the experiment and its position signal was taken to represent the basis weight variations. Total head signal was collected again to confirm the link between this variable with the approach system pressure pulsations. From the Figure 3.24, it is clearly seen that the variance of consistency in the stuff box is high. There seems to be very long term pulsations in the system with 0.03 Hz frequency prominent The corresponding period is 33 seconds. The total head signal was taken from the mill control loop itself. It is evident that the signal noise is comparatively less than that obtained previously from the separately installed Chapter 3. Mill Experiments: Analysis of Wet End Pressure Pulsations 51 Comparison of CV before and alter wet-end modification * » £ I I \ \\ j V ! \ | s ! s l l 1 I 1-1 ] j 1 ] r 1 Fan Pump Cleaner Screen Header Total Head . G Before Wet-end Moaifcation 153 After Wet-end Modifhalion Figure 3.19: Comparison of Wet end Disturbances Before and After Modification. smart sensor. This indicates substantial filtering in the mill control and display system. The frequency component of 0.044 Hz is still visible. The by-pass valve signal shows very low frequency signals only. The stuff box level is found to be smooth with a very small frequency variations at 0.005Hz. 3.6 Conclusions Frequency response techniques have been found to be very useful in tracing source of MD problems. From the first experiment, it is found that the basis weight sensor signal is very noisy. Due to the low S/N ratio, high frequency pulsations are buried in the noisy background. The single point machine direction measurement of basis weight gives a combination of the true MD variations and the residual variations of the headbox and fourdrinier table. From the second and third experimental results, it can be safely concluded that the addition of a new pressure screen has definitely improved the performance of the wet end system. The Chapter 3. Mill Experiments: Analysis of Wet End Pressure Pulsations 52 2 1 . 8 1 . 6 1 . 4 IP | 1 . 2 I 1 .<£ 0 . 8 0 . 6 0 . 4 0 . 2 O P o w e r S p e c t r a l D e n s i t y o f S t u f f b o x l e v e l s i g n a l P e a k e l 0 . 0 0 6 H z Figure 3.20: Power Spectrum of Stuff-box Level. P o w e r S p e c t r a l D e n s i t y o f S t u f f b o x C o n s i s t e n c y i 5 . .as o. o.< o.: ( I t ! ! ! : : P e a k s a t 0 . O 3 , O . O 6 8 ; 0 . O 9 H j \ A i fi i ^ O 0 . 0 2 0 . 0 4 0 . 0 6 O . 0 8 0 . 1 0 . 1 2 0 . 1 4 0 . 1 6 0 . 1 8 0 . 2 F r e q u e n c y i n H z Figure 3.21: Power Spectrum of Stuff-box Consistency. noise levels in the cleaner pressure and screen pressure have been reduced significantly. The total head variations has markedly reduced. No pulsations related to fan pump and screen appear in the total head signal. But it is to be noted here that the total head sensor used is a mill transducer. The response of such a sensor (on line) is generally slow. The same is true for the Measurex basis weight sensor. Typically the response range of basis weight sensor is less than O.lfTz. High frequencies present in the total head and basis weight signals will not be evident from the outputs of these slow sensors. To detect the high frequency pulsations in MD basis weight, the analysis should be carried out off-fine. Chapter 3. Mill Experiments: Analysis of Wet End Pressure Pulsations 53 P o w e r S p e c t r a l D e n s i t y o f T o t a l H e a d p r e s s u r e P e a k a t 0 . 4 3 H z Figure 3.22: Power Spectrum of Total Head. P o w e r S p e c t r a l D e n s i t y o f S t u f f b o x l e v e l s i g n a l Figure 3.23: Power Spectrum of By-pass Valve Position. In the low frequency range, three major pulsations at 0.043 Hz, 0.006 Hz, and 0.02 Hz were detected. 0.043 Hz is present in the total head signal of all of the experiments, and it seems that this frequency originates in the headbox. The cause may be from the level fluctuation or the airpad pressure fluctuation. The 0.02 Hz component may be due to the recirculation of the primary cleaner accept in the white water silo. The period of 50 seconds indicates recirculation in the white water. The 0.006 Hz fluctuation may be from the consistency dynamics within white water silo or the stuffbox level variations. The data from the experiments were collected for 15 minutes only which is not enough to identify these low frequency sources precisely. Chapter 3. Mill Experiments: Analysis of Wet End Pressure Pulsations 54 Pulsation Analysis Results Location Mean Value Std. Coeff. of Var Peaks Freq. Hz Power Total head 252.7 inch 0.253 0.100 0.044 0.33 Stuffbox Consistency 3.04% 0.04 1.3151 0.03 Hz 1.2 0.068 Hz 1 0.09 Hz 1 Stuffbox level 51 inch 0.031 0.06 . 0.005 2 Bypass Valve 57.1% 0.2713 0.475 < 0.01 Hz 288 Figure 3.24: Results of Experiment No. 3b. It is a common claim that low frequency variations are mainly caused by consistency dis-turbances. Consistency variations are mainly caused by non-ideal mixing in the wire pit or machine chest, feed or dilution disturbances of the thick -stock or separation or precipitation problems in the wet end [32]. The improvement in wet end pulsations with addition of a new screen in parallel to the old one also supports the hypothesis that the pulsations originate within blending problem. Koivo et al in [32] suggest two ways to tackle the problem; actively with control or passively by means of wet-end modification (using delay line concept to average-out the consistency variations by proper branching and re-combining stock pipes). 3.7 Recommendations In order to obtain more information about the source of the disturbances the following steps are recommended: 1. Because the basis weight sensor is found to be too slow to respond to the high frequency variations, an off-line analysis of the sample paper is recommended to provide more in-formation about the high frequency contents. 2. To make sure that the residual variations in the paper are small, it is necessary to measure basis weight simultaneously at two or more places across the web. Measuring the coherence Chapter 3. Mill Experiments: Analysis of Wet End Pressure Pulsations 55 between these signals the true and residual MD variations could be distinguished. 3. The possibility that the disturbance of 0.044 Hz being originated in the headbox can be as-certained by performing some experiments on the headbox. Measuring the headbox level, airpad pressure, and total head simultaneously would establish the source. The headbox total'head control can also introduce the cyclic variations. It is therefore recommended that the control loop be retuned. 4. The performance of the headbox has to be studied meticulously. The pressure measure-ments on the front and back sides of the manifold provide the base data for evaluating the performance of the headbox. These pressure signals indicate how well balanced the man-ifold is and also gives the estimates of the frequency content of the pressure fluctuations entering the headbox [37]. 5. The medium term variations ( 0.005 to 0.1 Hz) may indicate a problem with the consis-tency dynamics. The consistency dynamics in the stock loops could be detected by using on line consistency gauge. 6. With conventional control methods only a small frequency range can be controlled. If the process delay is long, (relatively) fast disturbances in the process cannot be affected by any conventional feedback control strategy. Developing a comprehensive model, advanced multivariable control schemes such as Model Predictive Control could be used. Chapter 4 Multivariate Control Simulations In this chapter, the model developed in Chapter 2 is used to assess multivariable controllers that are driven by disturbances of class identified in Chapter 3. Two design methods, the linear quadratic gaussian (LQG) and the Model Predictive Control (MPC) are investigated. Tests are made of the performance of the two methods for regulators, and also during the transitions between operating points that are necessary in grade changes. 4.1 Sensitivity Analysis Before investigating the control aspects of the problem, the sensitivity of the model to some of its design and operating parameters is considered here. For many of the design parameters the values are set so there is no scope for perturbations. Others for example, the frictional coefficient of the turbulence generator surface, are not exactly known. This parameter is a function of the Reynold's number so it is essential to study the sensitivity of the model towards the perturbation in this parameter. Figure 4.1 shows the step response of the system to a change in stock flow for different frictional factors. It is clear from the figure that the system is not seriously sensitive to this parameter. Figures 4.2 shows the step response to a change in stock flow for different values of air flow. Clearly the system is much more sensitive to this parameter. Similar analysis for different levels and slice openings also indicate large sensitivities towards those operating parameters. These observations are important as the operating parameters change with different grades of paper production. It is thus important that the control algorithm be adjusted for various grade 56 Chapter 4. Multivariable Control Simulations 57 Step response 1o stock flow for different frictional factors rji i i i i i i i i 0 200 400 600 800 1000 1200 1400 1600 Time In seconds Figure 4.1: Sensitivity Towards Frictional Factor. Step response to stock flow for different Bleed valve Openings 16r Time In seconds Figure 4.2: Step Response for Different Air Flows (Bleed Valve Opening). Chapter 4. Multivariable Control Simulations 58 operating conditions. 4.2 Frequency Responses The frequency response of the headbox model linearized around three major grades are shown in Figures 4.3-4.5. Frequency response to stock flow for 45# grade paper m 20 0 TJ f-20 E < -40 -60 10' - - + -: ! ! ! ! ! ! ! ! ! '. ! ! ! ! ! ! !~r-... ... =L,evei ... :•::—=Air Pressure: ; I : ; ii i I I I I i Ii i i i ~ =Slice Pressurla • IO"' IO"' IO'' 10° Frequency response to air flow for 45# grade paper 10 10" 10" Frequency in rad. Figure 4.3: Frequency Response (45 lb grade paper). It is clear from Figures 4.3-4.4 that the dynamics of the system in the low frequency range are affected by the operating point. Increase in lower corner frequency is observed as the operating point changes from 45 lb paper to 72 lb paper. It is also interesting to note the wider bandwidth of the headbox compared to the conventional airpad headboxes. 4.3 Open Loop Response The open loop responses of the system to step changes in the inputs are shown in Figures 4.6-4.8. Figure 4.6 shows that the stock level rises very slowly but the total head and the airpad Chapter 4. Multivariable Control Simulations 59 Figure 4.4: Frequency Response (54 lb grade paper). Figure 4.5: Frequency Response (72 lb grade paper). Chapter 4. Multivariable Control Simulations 60 14 12 E10 c 0) § 8 1 !e 3 8-o Step input in stock flow , / i r— i 1' ii .u ( I .( = Stock level i t 1 1 ; >— = Airpad pressure 1 i = Totaj head f i , i i j 20 30 40 Time in seconds 50 Figure 4.6: Step Change in Stock Flow. Figure 4.7: Step Change in Air Flow. Chapter 4. Multivariable Control Simulations 61 0 -20 E -40 .£ CO I "60 • ? 1 3 -80 3 3 -100 -120 -140 0 200 400 600 800 1000 1200 1400 Time in seconds Figure 4.8: Step Change in Slice Opening. pressure rise very fast to reach a maximum value. The air pressure then decays slowly to the steady value (note the different time scales). The total head seems to approach its steady state value quite fast. It is interesting to note the higher slope of the stock level curve at the beginning (see Figure 4.6). Initially all the dynamics are less coupled and faster but later the dynamics are coupled and become sluggish in nature. K.J. Astvom in [1] interprets the headbox dynamics as the dynamics of a coupled water tank and air tank system. If not for the coupling effect, the dynamics will be governed by the time constants of the stock system and the air system only. As the volume of Sym-Flo headbox is comparatively small, one should expect the dynamics to be quite fast. But due to the coupling effect, the dynamics are quite slow. Nonlinear simulation results indicate that the speed of response could be increased by increasing the airflow (see Figure 4.2). The small bleed valve (1" in diameter) is responsible for the strong coupling effect and Figure 4.2 clearly shows the speed of the system as a function of the bleed valve opening. The system becomes quite faster for higher airflows. Svein Hem in [57] has done extensive simulations with different shapes and sizes of bleed area arid airflows. Hem also concludes that the bigger hole and higher airflow results in a faster and more stable Chapter 4. Multivariable Control Simulations 62 system, but this may also result a larger overshoot in the total head before settling. Figure 4.7 shows that a step change in air flow will force the air pressure to increase rapidly during the first few seconds and then continue to increase slowly until a new steady state is reached. The liquid level decays very slowly. The total head seems to deviate slightly only. Figure 4.8 shows that a step change in slice lip opening greatly affects the airpad pressure and the level. Changing the slice lip by 0.1 mm results in the reduction in the level by 56 mm, the airpad pressure by 3.4 mm and the total head at slice by 59.6 mm. It is therefore clear that any perturbations in the slice lip opening is a source of major process upset. Including the slice lip area either as a measurable disturbance or as a manipulated variable helps to account for these perturbations. From the open-loop responses it is clear that the headbox is a highly interacting, multivari-able system. For optimal control, a multivariable design is desirable. Initially a multivariable linear quadratic controller will be considered. To accommodate the input-output constraints optimally, a Model Predictive Control will then be investigated. The results obtained by LQ-control and MPC will be compared. Control valve nonlinearity will also be included in the simulations. Finally the controllers will be applied to the nonlinear model to simulate a "Grade change" operation. This simulation will also prove the robustness of the controllers for the chosen set of tuning parameters. 4.4 Process Control The control of paper machine headbox has become more difficult with the introduction of high speed headboxes. The wider bandwidth of the high speed headbox allows more external dis-turbances to influence the outputs. Further, to maximize production, the off-grade production during grade changes and start-ups should be minimized. These requirements demand very good headbox control during the dynamic changes. Variations in the head at the slice lip and the level in the attenuator can have significant influence upon basis weight uniformity. Because of their importance, much work has been done Chapter 4. Multivariable Control Simulations 63 on the automatic control of these variables and now there are many complex headbox control systems available. During the last two decades, many experimental results have been published on the advanced control of paper machines [14]. Fjeld dealt with quadratic optimal control of the paper machine in [18]. Foulard and coworkers suggested an adaptive multivariable controller in [45] based on model reference adaptive control theory. Starting with a linear quadratic criterion minimization controller, then pointing out some of the shortcomings of the method, it was proposed that an internal model receding horizon quadratic criterion linear controller be used. To accommodate the nonlinearity of the headbox during large set point changes, an adaptive controller is proposed. Lebeau et al compares the performance of the multivariable headbox control with classical loops in [35]. They propose a control structure which eliminates integrator action to avoid possible actuator saturation due to input constraints. Lebeau and Bornard obtained US Patent for the above elaborate multivariable control system for headbox operation in 1983. The invention in [34] claims non-interacting control. Borisson discusses a self tuning multivariable controller applied to a simulated paper headbox. Borisson suggests decoupled control as substitute for complex multivariable controllers. F.M.D'Hulster and co-workers discussed several parameter-adaptive controllers as alternatives to the multivariable controllers in [14]. A comparison is made of the control simulation results of the parameter adaptive controller with the results of other controllers in [15]. Xia Qi-jun et al proposes an optimal decoupling controller with application to the control of paper machine headboxes in [64]. T.T. Jussila, Tanttu and Koivo suggest an adaptive multivariable PI controller for headbox control in [28]. The simplicity of the method and simple tuning are the attractive features of this multivariable PI controller. Juha T. Tanttu et al (1991) explore tuning methods for multivariable PI controllers in [60]. Q.J. Xia, M. Rao, Y.X. Sun and Y. Q. Ying propose a new decoupling control approach for a headbox in [39]. Koivo et al propose a new self tuning controller for a paper machine headbox with time varying delay in [33]. Some recent papers propose the use of predictive techniques in multivariable control prob-lems such as headboxes. R. Scattolini suggests a multi-rate self tuning predictive controller Chapter 4. Multivariable Control Simulations 64 for multivariable systems [55]. A simulation example on a headbox nicely shows the strength of the multi-rate predictive self tuning controller in tracking the set-point changes despite of the change in system gains. Clarke et al present the application of multiple model GPC and adaptive GPC to a SIMULINK model of a paper machine [4]. The capability of GPC to incor-porate inequality constraints highlighted. The capability of Adaptive GPC to update the gains while changing the set points is also shown. Clarke outlines new developments in Model Based Predictive Control(MBPC) in [10]. All the good features such as simplicity, flexibilty, practical-ity etc. of MBPC are systematically presented. The ability of MBPC to include the practical constraints on actuators (inputs) and outputs through Quadratic Programming (QP) is also shown. In [58] D.G. Sbarbaro and R.W. Jones propose the nonlinear control of a multivariable paper machine headbox with a Model Predictive Control (MPC) approach. The ability of the MPC approach to handle constraints in a systematic way is illustrated. A conventional airpad headbox is studied in the nonlinear control frame work. Before choosing any controller, the desirable properties and complexity of the control system are to be established. The multivariable nonlinear nature of the Sym-Flo headbox was shown in Chapter 2. The stiffness of the system is also evident as the system has very fast and very slow modes. Yet another point to be noted is the large operating point transition required for every grade change. Further there are practical constraints to be imposed on the inputs and outputs. The controller should thus achieve following objectives with the hard constraints on its inputs and outputs: • Decoupling the multivariable system to avoid interactions. • Regulating process perturbations. • Tracking operating point transitions during grade changes. From the headbox characteristics and the desired control objectives, it is clear that a ver-satile controller is needed. Chapter 4. Multivariable Control Simulations 65 In Chapter 2 it was shown that the headbox dynamics can be represented by a fourth order system with four inputs and four outputs. The model included simplified consistency dynamics. In the conventional design, only two inputs and two outputs are considered. The two outputs conventionally considered are the total head at the lip, arid the stock level in the attenuator. The total head is controlled to obtain the desired rush/drag ratio, which in turn decides the formation and hence the quality of the paper. The level is controlled to avoid filling or emptying of the headbox during the "start-up" or "grade-change" processes. It is to be noted here that in the conventional airpad headbox, the head at the slice lip is directly related to the airpad pressure so that the airpad pressure and the total head are not two independent states. In Sym-Flo headbox these two are separate states since the turbulence generator section introduces one additional order of dynamics to give three distinct states in the Sym-Flo headbox. In principle it is possible to control all three states in the Sym-Flo headbox; this is a distinct advantage over conventional airpad headboxes, But three inputs are needed to achieve this control. The slice lip opening can be treated as the third input. It is not possible to control the slice lip to regulate process perturbations, but including the slice lip as the third input makes the implementation of an automatic grade-change process possible. This will be discussed in detail later in connection with the simulation results. The control of the linearized model of Sym-Flo headbox obtained in Chapter 2 will now be discussed. Though consistency dynamics have been included, emphasis will be upon hydraulic dynamics. 4.5 Linear Quadratic Control Design 4.5.1 Introduction The Sym-Flo headbox being a truly multivariable system, the conventional way of designing the controllers based on single loop concepts will not be optimal. Further the strong interactions among the inputs may produce oscillations. With a multivariable controller, the interaction among the inputs will greatly be reduced. Chapter 4. Multivariable Control Simulations 66 The Linear Quadratic Regulator (LGR) is the standard state-space method available for designing multivariable control systems. This method reduces the problem of design to the choice of suitable-weighting matrices in the quadratic performance index. These matrices can be formed from the information concerning the desired performance and physical constraints imposed on the system states and inputs. The resulting system is always stable. Moreover the LQR has some desirable sensitivity and robustness properties. The tracking(servo) property can be imparted to the LQR by using a servo compensator. 4.5.2 Regulation Problem Formulation The design problem can be specified by giving the process model, the criterion, and the admis-sible control signals. The process model: The process to be controlled here is a Sym-Flo headbox. The continuous-time state space model is given below: x(t) = Ax(t) + Bu(t) x(t0)=.x0 y(t) = Cx(t)-r Du(t) (4.45) Where A and B are system matrices. x(t) is the state vector and u(t) is the input vector. It is to be noted here that LQR design can be realized in both continuous and discrete domain. We will use the continuous version here. The aim is to find the control law which will give a closed loop system with desirable responses. In pole placement design the desired rate of response decides the locations of the poles. In order to achieve faster response, higher input energy is required. In quadratic regulator problem, attempt is to strike a compromise between the input energy and the speed of response. This is done by choosing the input u(t) so as to minimize the following quadratic cost criterion: J=ffyT(t)Qx(t) + uT(t)Ru(t)]dt + xT(tf)Qfx(tf) (4.46) JtQ The weighting matrices R, Q and Qy are assumed to be symmetrical with R positive defi-nite and Q and Qj positive semidefinite. These matrices are chosen according to the relative Chapter 4. Multivariable Control Simulations 67 importance of the cost of control and the cost of deviations in the states. It is to be noted here that the job of selecting proper weighting matrices is not a trivial one. It is however fairly easy to see how the weighting matrices influence the properties of the closed loop system. Large weight correspond to small responses. One rule of thumb to decide the weighting matrices is to choose the diagonal elements as the inverse value of the square of the allowed deviations [2]. For a controllable and observable system, the optimal control which minimizes Equation 4.46 is given by linear time-invariant state feedback i.e. «(*) = -Koptx(t) t>t0 tf -> oo (4.47) This gives a closed-loop matrix which is always stable. The matrix Kopt depends on A, B, Q and R only. This time invariant or steady state LQR is the only one which yields the optimal control as a linear time-invariant state feedback. The rigorous derivations and proofs are to be found in any standard control literature e. g. [53]. Important aspects and properties of a LQR problem will, however, be discussed in some details here. To obtain the optimal feedback gain matrix Kopt, P is first obtained by solving the following Algebraic Riccati Equation (ARE) ATP + PA- PBR~lBTP + Q = 0 where P is a n X n symmetric matrix which is the unique positive semidefinite solution of the ARE. Q is a positive semidefinite matrix which can be expressed as Q = MTM and K0pt = R~^BT P The optimal feedback gain matrix K, the steady state solution P and the close loop eigen-values E can be found using standard control design software. Chapter 4. Multivariable Control Simulations 68 4 . 5 . 3 Servo Response It is desirable that the multivariable controller being designed for the headbox control be robust. This is because the model obtained by linearizing around an operating point does not represent the exact behaviour of the system at other operating points. The operating point may deviate due to disturbances in the system or may be changed deliberately during the grade changes. It is therefore important to take into account the effect of perturbations in the linear model 4.45. Let the problem be defined again with perturbations below: x = Ax(t) + Bu{t) + Ev{t) y{t) = Cx(t) + Du{t) + Fw(t) e(t) = y(t) - yr(t) (4.48) where x(t) is the state vector, u(t) is the input vector, y(t) is the output vector which is to be regulated, v(t) and w(t) are the disturbance vectors which are not measurable in our case. yr{t) is the reference signal vector and e(t) is a vector denoting the error. It is assumed here that the disturbances w(t), v(t) and the setpoint vector yr(t) are step functions. The objective here is to find a controller for the above system such that the resulting con-trolled system is stable and the setpoint error vector goes to zero asymptotically. In addition the controller requires to be robust. These objectives can be achieved by using a servo compensator along with a LQR design. The structure of the resulted controller is given below: A set of simple integrators are considered as the servo compensator. Then the compensator is defined by: C(t) = e(t) (4.49) The control signal u(t) is now given by u(t) = fciC(t) + k2r}(t) • - ' • ' (4.50) where n(t) is the output of the stabilizing compensator LQR. k\ and ki are obtained by applying LQR law to the augmented system consisting of the headbox and the servo compen-sator. For stochastic case, w(t) and v(t) are stochastic disturbances. Then it will be a general Chapter 4. Multivariable Control Simulations 69 Figure 4.9: Robust Servo Control Structure, quadratic stochastic optimization problem. 4.5.4 Linear Quadratic Gaussian Control Practical systems wil l be subject to uncertainty but this need not affect the control system design. The criteria in this case wil l be to minimize the expected value of performance index. This leads to a general quadratic stochastic problem. If the stochastic disturbances are of Gaussian type, the optimal control comprises a Kalman filter to estimate the states together with the linear state variable feedback design for the noise free case [2]. 4.5.5 Limitations Some of the limitations are listed below. A n important drawback of L Q control is the lack of proper method to select the weighting matrices. It is often fairly easy to select one by trial and error method. A second drawback of L Q design is the requirement of all the states for feedback. For Chapter 4. Multivariable Control Simulations 70 the headbox problem under consideration, all the states are available as outputs but are con-taminated by the measurement, process and output disturbances. A Kalman filter is used to estimate the states. Note that, in general, the choice of covariance matrices in the Kalman filter design will have influence on the robustness of the design. These effects are not considered here. LQ control does not include inherent integral action. It is always possible to increase the state variable representation to include integrators. With constraints on the inputs, the integrators may drift to saturation. Lebeau et al in [35] suggest that the input signal be split into two parts, one will be determined by the optimal law and the second calculated by using inverse gain matrix of the system. This method may suffer from the problem of model mismatch and numerical errors involved with the matrix inversion. One possible way to avoid the integral wind up is to weigh the inputs heavily. The integrators are desirable as they are quite robust and force the steady state error to zero despite model mismatch or other perturbations. LQ control does not permit direct inclusion of input-output constraints. For the headbox under consideration, all the inputs, states and outputs are constrained by physical limits. The stock flow can not be increased beyond 16575 usgpm (1.05m3/sec). The slice lip opening cannot be set below 0.31 inch. The bleed valve cannot be opened beyond 90 degrees, as is true of the thin stock valve and the thick stock valve. Furthermore, limits on the rate change of inputs are also desirable. For instance, the bleed valve has a closing time of 3.5 seconds and so cannot be adjusted any faster than 25.7 degrees per second. The last two limitations of LQG, the.lack of inherent integral control and lack of hard constraint imposition are very serious from a practical point of view. To overcome some of these shortcomings, model predictive control is. also considered. 4.6 Model Predictive Control 4.6.1 Introduction Model Predictive Control (MPC) was conceived in the 1970s primarily by industry. In [8] Cutler and Ramaker explains the Dynamic Matrix Control algorithm. A series of papers was published Chapter 4. Multivariable Control Simulations 71 on MPC with different model structures. Several names have been associated with MPC in-cluding Dynamic Matrix Control (DMC), Quadratic Dynamic Matrix Control (QDMC), Model Algorithmic Control (MAC), Model Predictive Heuristic Control(MPHC), Internal Model Con-trol (IMC) and Multivariable, Optimum Constrained Control Algorithm (MOCCA) [36]. Each of these algorithms differs in detail but the main ideas behind them are very similar. Some key features of MPC are outlined below [36]: • Future outputs are predicted using a set of discrete step or impulse response coefficients rather than a typical state space or transfer function model. • A "correction" for each element of the predicted output is usually calculated to account for the difference between the estimated value and the measured plant output. The correction can be calculated using known disturbance response data, by identifying parameters on-line to permit forecasting of future values, or by estimation techniques such as a Kalman filter. • A predictive control strategy is used to calculate the control action Su(k + i) for i=0, 1 M-l ; M < P which minimizes a user-specified performance index, e.g., which minimizes the square of the difference between the desired trajectory Yd(k) and predicted trajectory Y(k) that would result if no further control action were taken. Here P is the prediction horizon and M is the control horizon. • The control calculation is usually formulated as an optimization problem: linear or non-linear; weighted or unweighted; constrained or unconstrained. Depending upon the char-acteristics of the optimization problem, the solution may require anything from simple off-line calculations to on-line, constrained, nonlinear optimization. Morari and Ricker have developed a MPC MATLAB Toolbox [38] for the analysis and design of model predictive control systems. The toolbox uses a step response model description or a state-space model description. Chapter 4. Multivariable Control Simulations 72 In [6] - [7] Cutler et al described Dynamic Matrix Control (DMC) and its variant in an industrial perspective. In [36] Sifu et al showed how the step or impulse response model can be put into state space form thus reducing computation time and permitting the use of state space theorems and techniques. In [41] Kenneth et al discussed the implementation of MPC techniques with open-loop unstable plants. In [50] Ricker dealt with a challenge problem using MPC. 4.6.2 Basic theory of M P C The conceptual structure of a generic MPC approach is given in Figure 4.10 below. Reference W Design Parameters Constrains Figure 4.10: Block Diagram Representation of Model Predictive Control. In Figure 4.10 u denotes one of the inputs to the process, and y denotes the process output corresponding to that input, w denotes the desired process output [59]. In Figure 4.11, the current time is denoted by sample k and u(k), y(k) and w(k) are the controller output, the process output and the desired process target at sample k, respectively. The MPC control law can be derived as follows: The predictive controller MPC calculates present and future incremental control moves 6u(k), 6u(k + 1 ) , 6 u ( k + m — 1) in such a way Chapter 4. Multivariable Control Simulations 73 Future k k+1 k+2 Figure 4.11: Model Predictive Control. that the predicted future outputs y(k + l\k),y(k + 2\k), ...,y(k + p\k) are close to the desired process target w. The m control moves (m < p) are computed to minimize the quadratic objective function: p J = E » r ? iy(k+w - w(k+£)l l l 2 + £ H r ? + 1 - !)l I I s e=i " .-. • e=\ (4.51) Here and T% are weighting matrices to penalize particular components of y and u at certain future time intervals. w(k + l) is the vector of future reference values (set points). Out of m calculated moves, only the first one (6u(k) is implemented. At the next sampling interval, new values of the measured output are obtained and the computations are repeated. Thus a "receding horizon" control law is used in MPC. The predicted process outputs depend on the current measurement of output and the as-sumptions made about the unmeasured disturbances and measurement noise affecting the sys-tem. The unmeasured disturbances can be assumed to be a white noise passed through an integrator. The measurement noise is assumed to be white. Chapter 4. Multivariable Control Simulations 7 4 For the unconstrained case, the linear time invariant control law is given by: 8u(k) = KMPCEp(k + l\k) ( 4 . 5 2 ) where Ep(k + l\k) is the vector of predicted future errors over the horizon p which would result if all present and future input moves were equal to zero. The noise assumption does not affect the response of the system to setpoints changes or mea-sured disturbances. But it affects the robustness and the response to unmeasured disturbances [38] . The control law can also be computed when hard constraints on the manipulated variables and outputs are present. Manipulated variable constraints: « m i n ( € ) < u(k +1) < umax(£) ( 4 . 5 3 ) Manipulated variable rate constraints: \6u(k + £)\ < 6umax(l) ( 4 . 5 4 ) Output variable constraints: Vminii) < V(k + £)< ymax(t) ( 4 . 5 5 ) For the constrained case, a quadratic program has to be solved at each time step to determine the control action and the resulting control law is generally nonlinear. The details of theory behind the DMC and quadratic programming problem may be found in [7] and [8]. 4.7 Application to the Headbox Problem The linear model derived in Chapter 2 and the control theories outlined in the previous section will now be put together to form a closed loop system. The interactions among the inputs and outputs will first be studied. This analysis also helps to pair the inputs and outputs if Chapter 4. Multivariable Control Simulations 75 single loop control design is to be done. Then the possible disturbances will be reviewed. The sensors and actuators used in the system are briefly introduced. Finally the simulation results are presented. K = 4.7.1 Interaction Analysis The A, B, C and D matrices of the linearized system can be used to calculate the static gain matrix and hence the relative gain array in order to assess coupling present in the system. This can easily be done by first finding the transfer function matrix for the system and then equating 's' to zero. The static gain matrix thus obtained is reproduced below. 15.7582 0.0252 -137.5927 0.0000 0.0000 -0.0245 0.0000 0.0000 13.4791 0.0006 -137.5940 0.0000 0 0 0 1.0000 The inverse system matrix is found to be 0.4387 0.4405 -0.4387 0.0000 0.0000 -40.7750 0.0000 0.0000 0.0430 0.0430 -0.0502 0.0000 0 0 0 1.0000 The relative gain matrix A is then given by [16] X = K.*MT A = M = 6.9138 0.0000 -5.9138 0 0.0000 1.0000 0.0000 0 -5.9138 0.0000 6.9138 0 0 0 0 1.0000 The relative gain array clearly shows that the input/output pairing should be done as shown in Table 4.1 if SISO controllers are to be used. Chapter 4. Multivariable Control Simulations 76 Stock flow (input 1) to Stock level (state 1) Bleed Valve opening (input 2) to Airpad pressure (state 2) Slice opening (input 3) to Total head (state 3) Input consistency (input 4) to Output consistency (state 4) Table 4.1: Input-output Pairing. Here it is to be noted that the slice lip should not be manipulated for control purposes; the slice hp opening is changed during the grade change process only. As mentioned earlier, including the slice lip as one of the manipulated variables, however, makes the automatic implementation of grade change process possible. To make the simulation results more realistic, the slice lip is constrained from moving when the regulation properties are studied. The headbox is then essentially a two by two system from regulation point of view if the consistency dynamics are also neglected. The interaction analysis for the reduced system shows that the total head can be paired with the stock flow and the level can be paired with the air bleed valve opening for SISO controllers. 4.7.2 Disturbance Review: The major disturbances are the fluctuations in stock flow and air flow. The mill experimental results presented in Chapter 3 clearly show the presence of many periodic disturbances in the approach system. These pressure disturbances introduce variations in the stock flow at the corresponding frequencies. Using the steady state gain matrix, the uncontrolled change in level and total head for 1% change in the stock flow can easily be found. The calculations show a d.c. variation of 36.36% in level and 2.05% in the total head. In [1], Astrom suggests 4% variation in the stock level and 0.4% variation in the total head for a variation of 1% in stock flow as a reasonable criterion. In the variations in stock and airflow, the slice lip vibration could also introduce distur-bances. 1% change in slice opening changes the stock level by 30% and the total head by 2% in the absence of control. Chapter 4. Multivariable Control Simulations 77 4.7.3 Sensors and Actuators The pressure sensor at the slice measures the total head. The total head set point is determined by sensing the wire speed and using the desired rush/drag set point. The total head set point is then sent to the controller. The stock level in the attenuator chamber is measured by a wetted capacitance probe and is sent to the controller. Both the horizontal and vertical position of the slice lip are measured and displayed on the front of the headbox. There are four valves controlling four inputs to the headbox. Typical specifications of the valves are listed below: Air Bleed Valve: Manufacturer : Fisher Model no. : V-100 Type : V-ball Size : 1" Line size : 2 Max. flow : 140 scfm Closing time : 3.5 sec. Valve travel : 0 - 90 degrees. Stock Flow Valve: (by-pass) Manufacturer : Fisher Model no. : V-100 Type : V-ball Size : 8" Line size : 12" Max. flow : 2431.33 usgpm Chapter 4. Multivariable Control Simulations 78 Closing time : 3.2 sec. Valve travel : 0 - 90 degrees. Basis Weight Valve: (by-pass) Manufacturer : Fisher Model no. : V-100 Type : V-ball Size : 4" Line size : 6" Max. flow : 80 usgpm Closing time : 60 sec. Valve travel : 0 - 90 degrees. The headbox slice lip has a pneumatic loading tube with pressure control valve. The valves dynamics are represented by a first order lag with time constant of 1 second. Slice lip pressure control valve Manufacturer : Fisher Minimum Pressure: 25 psi The sizing coefficients of all other valves are available with the exception of the pressure control valve of slice lip mechanism. The valve sizing coefficient Cg vs valve travel for the bleed valve and Cv vs valve travel for the thin stock flow valve are plotted in Figures 4.12 and 4.13 respectively. It is to be noted here that the stock flow valve (by-pass) handles only a fraction of the input stock flow. This valve is connected in parallel to the main stream valve. The Measurex system monitors the position of this valve on the Honeywell T D C and adjusts the mainstream valve to keep the stock valve in its mid-range. It can thus be assumed that the steady state valve Chapter 4. Multivariable Control Simulations 79 Nonlinear characteristics of Bleed valve 700 j 1 1 1 1 1 i T ——r _100' 1 J 1 _i i i L_i i . I 0 10 20 30 40 50 60 70 80 90 Valve Travel In degrees Figure 4.12: Bleed Valve Characteristics. Nonlinear characteristics of thin Stock Valve 16001 1 1 1 1 1 1 -Valve Travel In degrees Figure 4.13: Thin Stock Valve Characteristics. Chapter 4. Multivariable Control Simulations 80 position is 45 deg. Knowing the valve position (valve travel), the swaged flow coefficient Cv can be found from the valve curve. Thus deviations in the valve travel can easily be converted into the deviations in volume flow. The deviation can then be applied to the headbox system thus including the nonlinearity of the valves in the simulation. Similar reasoning can be carried out for the thick stock valve. However, consistency dynam-ics are not a major concern, the consistency at the inlet of the headbox itself can be treated as the input. The slice lip opening is also taken directly, not modelling the valve travel of the air control valve. In the case of air bleed valve, the calculation is quite straightforward. From the valve travel (input signal), the Cg is directly found by using a lookup table. The operating target value of Cg is then subtracted. The deviation is applied to the linear model of the headbox for simulation. For the nonlinear simulation, the absolute value is used. The constraints on the valve inputs are clearly 0-90 degrees for the first two inputs. The slice lip can be closed down to 8 mm and opened up to 150 mm. Converting them into areas gives 0.0348m2 to 0.55m2. For 45# paper the steady state value is 0.0804 so the limits are -0.0456 < u(4) < 0.55 Note that these constraints are caused by the actuators themselves. The constraints on the states are imposed by the process itself. The stock level in the attenuator can not fall below 0.230 m because the cross section of the attenuator becomes very narrow below this point and even a slight imbalance between the inflow and outflow of the stock can cause instability. Similarly the stock level cannot go beyond 1 m. The total head is also limited by the speed of the wire and rush/drag ratio. Figure 4.14 shows the SIMULINK implementation of the LQG controller with servo com-pensation. The LQR block calculates the state feedback gain matrix K by using the augmented matrices. The states are estimated by the Kalman filter block. Here the attenuator level and the total head are servo compensated while the airpad pressure is not. This leaves the airpad pressure to respond actively to changing conditions. The speed Chapter 4. Multivariable Control Simulations 81 Figure 4.14: Simulink Model of Headbox Control System using LQG. of response of bleed valve is faster than that of the stock flow valve, and so it is the bleed valve which is actively controlled. This has added advantage that the rapid movement of stock valve is avoided, preventing the fluctuations in the consistency of the stock in the headbox. In the simulation the nonlinearity of the air bleed valve and the stock flow valve are also included, The block named headbox &; actuator houses valve elements and are shown in Figure 4.15. The augmented matrices used by LQR to calculate the feedback gain matrix including valve Chapter 4. Multivariable Control Simulations 82 Absolute Inputs Figure 4.15: Simulink Representation of Valve Nonlinearity. nonlinearities are in given below: Baug — -0.0001 -0.0008 -0.0353 0 00 0.0003 -0.0897 -1.0330 0 00 9.1371 9.1371 -10.6820 0 00 0 0 0 -7.3064 00 -1.0000 0 0 0 00 0 0 -1.0000 0 00 0.0006 0 -4.8587 0 0.0181 -0.0203 -142.1382 0 0 0 -212.5876 0 0 0 0 7.3064 0 0 0 0 0 0 0 0 Chapter 4. Multivariable Control Simulations 83 -'aug D, aug 1.0000 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 As a first guess, the diagonal elements of the weighting matrices Q and R are taken as the reciprocal of allowed deviations in states and inputs squared. The allowed deviations in states and inputs are taken as 10% of their steady state values. Q and R are retuned using optimal root locus method. The third input, the slice lip opening, is heavily penalyzed by putting R(3,3) = 2000000. Two added integral states modify the Q matrix. The Q and R used in the program are given below: The output weighting matrix Q is also changed to QaUg 7.5061 0 0 0 0 0 0 0.0257 0 0 0 0 0 0 10.0 0 0 0 0 0 0 280 0 0 0 0 0 0 50 0 0 0 0 0 0 50 Qaug — Chapter 4. Multivariable Control Simulations 84 5 0 0 0 0 0.1 0 0 0 0 2000000 0 0 0 0 3 Raug — The covariance matrices for the estimator are selected as 0.0238 0.6972 1.0329 0 0.6972 20.3971 30.2168 0 1.0329 30.2168 45.1935 0 0 0 0 0.0534 The measurement noise is assumed same for each output: 0.01 0 0 0 0 0.01 0 0 0 0 0.01 0 0 0 0 0.01 Measurecov = The noise model does not affect the control design but it affects the robustness and response to the unmeasured disturbances. Selecting bigger values of these matrices makes the system more robust at the expense of response to the unmeasured disturbances. The parameters used for the controller design using MPC to study the regulation and servo properties of the closed loop system are given in Table 4.7.3 and Table 4.3. 4.7.4 Simulation Results with LQG: The controller performance is investigated by examining the response of the headbox level, airpad pressure and total head to different disturbances and set point changes. Initially the control is examined on a linear model. Then the valve nonlinearities are in-cluded. Finally the same controller is evaluated on the nonlinear model. Regulation: Chapter 4. Multivariable Control Simulations 85 Simulation time tend = 200 Process Sampling Time T = 1 Output weights ywt = [200 0 500 100] Input weights uwt = [0.1 0.01 15000 1] Prediction Horizon P = 20 Manipulated Variable moves 2 Input Constraints U £ [Umin'U'Tnaxfi'U'Tnax] Umin [-45 -67.7 -le-12 -0.2] umax [45 22.3 le-12 0.2] [20 27 0 0.02] Output Constraints [2/mtnJ/max] Umin [-0.155 -4 -4 -0.2] Umax [0.5 4 4 0.2] Table 4.2: Parameters for M P C Controller. Simulation time tend = 600 Process Sampling Time T = 1 Outp9t weights ywt =[2 2 2 2] Input weights uwt = [0.1 0.01 150 1] Prediction Horizon P = 50 Manipulated Variable moves [2 3 4 5] Input Constraints U e [uminUmax6Umax\ [-45 -67.7 -0.0075 -0.2] Umax [45 22.3 0.0075 0.2] fiUmax [20 27 0.001 0.02] Output Constraints [yminymax] Vmin [-0.135 -4 -4-0.2] Vmax [0.5 4 4 0.2] Table 4.3: Parameters for M P C Controller for Grade-change Event. Chapter 4. Multivariable Control Simulations 86 Figure 4.16 and Figure 4.17 show the responses of the headbox to a sinusoidal variation of 0.048 Hz in stock flow. The controller seems to perform well as only 0.134% variation in total head is observed for 1% variation in the stock flow. It is to be remembered here that 1% variation in the stock flow causes 2% variation in the total head if no controller is used. Output responses to 1% deviation in stock flow 1 1 ! % std. in slice head = 0.134% \ / j i ] " & . \ { f i k „ k,~ L i *J JS&^ mk J>K •• V V u*K #%\ w _=Level ; —=Air pressure -.=Slice pressure 50 100 150 200 Inputl (stock valve opening) 80 »60 £ !?40 Q 20 °0 50 100 150 200 Time in seconds Figure 4.16: Response to a Sinusoidal Disturbance. It is to be noted here that the responses shown in Figure 4.16 and 4.17 are when the nonlinearity of the valves are included but the model nonlinearities are not included in the simulations. It was found that the same controller worked well on a nonlinear model too. Plots are not included to avoid redundancy. The same controller seems to work well to cope with small step disturbances in level and airpad pressure. The controller however failed when a large output disturbance occurred in the level. Figure 4.18 shows the responses to a step disturbance of 0.2 m in level. Figure 4.19 shows other two inputs during the disturbance. It is clearly seen that the inputs are driven beyond their limits. As there is no mechanism in LQG to accommodate the input/output constraints optimally, the controller performance is not optimal for such cases. Servo Performance: 0.02 o c g-0.02 A f\A Chapter 4. Multivariable Control Simulations 87 80 $60 CD t u O 20 0.1 i0.05|-Input2 (Bleed valve opening) 50 100 150 Input3 (Slice opening) 50 100 Time in seconds 150 200 200 Figure 4.17: Other Two Inputs. Output responses > 0.5 <D Q D 0 Q. O-0.5 100 10 0> CD b, 50 CD Q 1 ! r 1 = Level — = Air pressure - . - = Slice pressure 50 . 100 150 Inputl (stock valve opening) 50 100 150 Time in seconds 200 200 Figure 4.18: Response for a Major Disturbance in Level. Chapter 4. Multivariable Control Simulations 88 80 $60 c?40 20h 0.1 CM E c us "|0.05|-Q. O Input2 (Bleed valve opening) 50 100 150 Input3 (Slice opening) 50 100 Time in seconds 150 200 200 Figure 4.19: Other Inputs During the Disturbance. Figures 4.20 - 4.23 show the response of the system when a step change in level and total head set points occur in the system. This simulation makes it easier to see the tracking properties of the controller. This also shows the decoupling effect of the controller. Note that, in practice, paper machine set points are not changed in this way. The set point for the total head is changed only during the grade change process and then is not changed as a step, but as a ramp. Figure 4.20 clearly shows that the change in total head does not affect the level. Similarly Figure 4.22 shows that the set point change in level does not affect the total head. Now a grade change event will be simulated. The operating points for 45# and 54# papers are in Table 4.7.4-4.7.4. The model includes the slice opening as one of its inputs and so it is possible to achieve the grade change simply by selecting proper weighting matrices and reference trajectory. The Chapter 4. Multivariable Control Simulations 89 Output responses r t i i i i 1 1 1 • i 1 i i = Slice Pressure j — = Air Pressure > = Level \ i i Integrated Squared Error Is 226.99 0 20 40 60 80 100 120 140 160 180 200 m Inputl (stock valve opening) $ I 1 1 1 ! ! ! ! r o>80 - : : CD . Time In seconds Figure 4.20: Change in Total Head Set Point. Input2 (Bleed valve opening) $80 D) O •£60 ? E40H o & »20h 20 40 60 80 100 120 140 160 180 200 0.1 r ^0.08^ o>0.06h 8-0.04 \ S0.02H 20 40 Input3 (Slice opening) 60 80 100 120 140 Time in seconds 160 180 200 Figure 4.21: Change in Inputs During the Set Point Change. Chapter 4. Multivariable Control Simulations 90 0.1 3. 0.05 .9 > -0.05 Output responses -0.1 -. = Slice Pressure - = Air Pressure = Level \ •• Integrated Squared Error Is 563.96 0 20 40 60 80 100 120 140 160 180 200 inputl (stock valve opening) Figure 4.22: Change in Stock Level Set Point. |80 0 60 O) 1 40 « 8-120 > Input2 (Bleed valve opening) 1 1 1 1 1 1 1 i t i i i 20 40 60 80 100 120 140 160 180 200 0.1 £0.08 c g>0.06 c §•0.04 o j§0.02 20 40 Input3 (Slice opening) 60 80 100 120 140 160 180 200 Time In seconds Figure 4.23: Change in Inputs During the Set Point Change. Chapter 4. Multivariable Control Simulations 91 45# grade paper Stock level 0.365 m Airpad pressure 6.1 m Total head 5.53 m consistency 0.6 Stock flow 0.842m3/sec Slice opening 0.0804m2 Bleed valve Cg 484 Table 4.4: 45 lb Grade of Paper. 54# grade paper Stock level 0.365 m Airpad pressure 5.04 m Total head 4.5 m consistency 0.65 Stock flow 0.829m3/sec Slice opening 0.0879m2 Bleed valve Cg 526 Table 4.5: 54 lb Grade of Paper. Chapter 4. Multivariable Control Simulations 92 weighting matrices selected for the grade change are Qkaug — R-kaug — 5 0 0 0 0 0 0 0.01 0 0 0 0 0 0 300 0 0 0 5 0 0 30 0 0 0 0 0 0 500 0 0 0 0 0 0 500 0.01 0 0 0.01 0 0 0 0 0 0 0 150000 0 0 0 0.1 For the grade change event simulation, the constraint on the slice lip opening is relaxed. Now the slice lip is allowed to move within certain ranges (from 0.04 to 0.12 m2). The reference trajectory for the total head and the airpad pressure are increased slowly with a time constant of 100 seconds. The response of the system is shown in Figures 4.24-4.26. 4.7.5 Simulation Results with MPC The simulink implementation of MPC controller is shown in Figure 4.27: Regulation: The effect of sinusoidal variation in stock flow on the total head can be simulated by intro-ducing a sinusoidal disturbance as an unmeasured disturbance to the system. The simulation results are shown in Figures 4.28 and 4.29. Clearly the performance is not better than that of LQG controller. Response to small step disturbances in level and airpad pressures also indicated similar results. It thus appears that LQG performs well when there are small disturbances. But sometimes a major disturbance may occur. Sometimes senors may fail sending a major measurement error. It was observed Chapter 4. Multivariable Control Simulations 93 Figure 4.24: Grade-change Simulation. 90-80-70-Inpull and Input2 during the grade change ~i 1 1 1 Bleed valve opening(travel) ,ij.>v i^'iti' i i iii'M»'r^¥^ |M^-' f;.v'v.''. |!'w'.'.'y'. y"> ' 5 0 h SAO\-Stock valve openlng(travel) >30h 20i-10H 100 200 300 Time In seconds 400 500 600 Figure 4.25: Inputs During the Grade-change. Chapter 4. Multivariable Control Simulations 94 0.1 0.095 0.09 0.085 y CM E .£ 0.08 O) c I 0.075 CO 0.07 0.065 h 0.06 h 0.055 0.05 lnput3:(Slice opening) during the grade change 100 200 300 Time In seconds 400 500 600 Figure 4.26: Slice Opening During the Grade-change. • {stock Valve Nonlinearity Bleed Valve Nonlinearity L J L > Inputs Load DSUI nputsl bancs Scope Headbox Model Subsystem SI" Clock Sum Suml 1 1 - L Stock level - C Z J Airpad Pressure J L Total head Consistency Sn6 output 1 Figure 4.27: Simulink Implementation of MPC for Nonlinear Simulations. Chapter 4. Multivariable Control Simulations 95 0.05 Output responses to 1 % deviation in stock flow (MPC) 3 Q. B o > (D Q -0.05. • 1 A f\ 1 =Level —=Air pressur e --Slice p i ressure 50 100 150 Input!: Stockflow 200 50 100 150 200 Figure 4.28: Response to a Sinusoidal Disturbance (MPC). Controller output2: for bleed valve control 40 20 w <s 4> o> 0 -20 -40 0.02 I 0 -0.02 50 100 150 Controller output3: slice opening 200 50 100 Time in seconds 150 200 Figure 4.29: Controller Outputs During the Disturbance (MPC). Chapter 4. Multivariable Control Simulations 96 in Figure 4.18 that LQG could not cope with a large disturbance in level as LQG can not accommodate input/output constraints optimally. Figure 4.30 shows the response of the system to a large disturbance when constrained MPC is used. Note the improvement in performance. The strength of MPC becomes apparent when the constraints on inputs and outputs become active. 0.5r > o Q 3 a 3 O -0.5. 1 i i — = Air pressure ' 7 - . - = Slice pressure _ = Level 50 100 150 Inputl (stock valve opening) 200 100 Time in seconds Figure 4.30: A Major Disturbance in Level (MPC). Servo Performance: Figures4.32-4.33 show the response of the system when a step change in level and total head setpoints occur in the system using MPC as the controller. Figures4.32 - 4.34 clearly show the decoupling effect of the controller. The 0.5 m change in total head changed the level by less than 0.002 m. Similarly 0.03 m change in level caused negligible effect on the total head. The control signals are also smooth. Grade Change: Figures4.35-4.36 depict the grade change event with MPC control action. The new set points are tracked without any steady state errors. The inputs are also ramped smoothly to the final values with little oscillations. Chapter 4. Multivariable Control Simulations 97 Input2 (Bleed valve opening) 0 i 1 , , 1 0 50 100 150 200 Input3 (Slice opening) 0.11— 1 1 1 1 £ 0 . 4 Ql ' ' 1 1 0 50 100 150 200 Time in seconds Figure 4.31: Inputs During the Disturbance (MPC). 8 X 15 0 E -1 ( 0.1 8 1 •s 0 0.1 se 0 -0.1. 20 20 20 40 Set point change in Total head 1 1 1 1 1 I . I r* ! I I ! I ! I 1 1 1 ! | J j J Tot^ l head J 40 60 60 80 100 120 Time in seconds 20 40 60 80 100 120 140 160 180 200 40 60 80 100 120 140 160 180 200 I I 1 1 1 1 i i i i j j Stock level j i < i i 1 1 1 ! I i i i i 1 1 1 ! 1 1 1 1 1 1 1 1 1 1 1 I 100 120 140 160 180 200 I I I 1 1 1 1 1 nr. i . • i. J 1 1 1 1 1 1 1 1 _ 1 . , 1 1 1 | ] Consistency] j j . L ,. L, . ., . 1 i -, i 1 , R - 1. - l •v., . 1. ... . •i •• 1 1 -1- r V i - -! ! ! ! ! ! 140 160 180 200 Figure 4.32: Set point Change in Total Head(MPC). Chapter 4. Multivariable Control Simulations 98 Set point change in Steak level 1 1 1 1 1 , - 1 1 1 1 1 j | Consistency] , 1 I n 1 1 ... 1 1 j 1 J I 1 1 1 i ' ' i r • 1 1 1 1 1 1 1 1 1 0 20 40 60 80 100 120 140 160 180 200 Time in seconds Figure 4.33: Set point Change in Stock Level (MPC). 60 |50 0 •o 40' ( 80 s §,70 0 •o 60 ( 2 1 0 -2 ( 1 3*0.5 0 20 20 20 20 Inputs during the Set point change in Stock level 1 I I 1 1 l 1 ! ! T 1 i i • - i i 40 40 40 40 60 80 100 120 140 160 180 200 1 I I Ble#d valve Opening; 60 80 100 120 140 160 180 200 i 1 ! 1 j Slice lip opening 60 80 100 120 140 160 180 200 Consistency: 60 80 100 120 140 160 160 200 Time in seconds Figure 4.34: Inputs During the Set point Change in Level (MPC). Chapter 4. Multivariable Control Simulations 99 0.2 Output responses during the grade change using MPC ,-0.2 E --0.4 :-0.6 -0.8 -1.2 1 i ,\\ V \ \ \ = Stoc (level \ i s S — = AJrpad pressure Slice head \ s \ 100 200 300 400 Time in seconds 500 600 Figure 4.35: Grade-change Simulation (MPC). ,44.9995 44.999 Inputs during the grade change event! : Stock valve opening 100 200 300 400 500 600 80 860 Q o>40 •o 20 0 0.09 50.085 CO 0.08 100 _=Bleed.varve.openirig. = Slice Hp opening • 200 300 400 Time in seconds 100 200 300 400 500 600 500 600 Figure 4.36: Inputs During Grade-change (MPC). Chapter 4. Multivariable Control Simulations 100 Some simulation results with MPC are presented below to illustrate the capability of MPC to cope with different noise models and input output constraints. In Figure 4.37 the noise model is changed to filtered white noise. The variance of the noise is taken as 0.0004. With a Dynamic Matrix Control (DMC) estimator, the performance of the controller is not good. It is to be noted here that DMC controller assumes white measurement noise and process noise to be white noise passed through an integrator. Thus the performance of DMC estimator deteriorates as the spectrum of noise changes. In Figure 4.38 an estimator is used which accounts for the nature of the disturbance and the performance of the controller improves. The variance of total head is much lower in the second case. Output responses Inputl: Stock valve opening) °> 40 'c « 8" 20 I 0 .s I -20 > tS-40 0 SO 100 150 200 250 300 Constrained reponses. DMC estimator. Figure 4.37: Constrained MPC with DMC estimator. In Figures 4.39 and Figure 4.40 the servo performance of MPC controllers is compared. Figure 4.39 shows the reponse when the inputs are constrained by clipping while Figure 4.40 shows the responses when an optimal constrained MPC algorithm is used. It is clearly seen that the settling time is less for the later case. Figure 4.41 and Figure 4.42 show grade-change simulations with clipped constraining and the optimal constraining of the inputs respectively. Again it is clearly seen that the performance Chapter 4. Multivariable Control Simulations 101 Output responses 0.21 1 1 1— Constrained reponses. New estimator. Figure 4.38: Constrained MPC with a Different Estimator. is not satisfactory with the clipped version of MPC but improves with the optimal handling of constraints. 4.7.6 Discussion For brevity only a few simulation results are included. The strength of MPC to accommodate constraints is well illustrated when a large disturbance upsets the system. In Figure 4.18 the system became unstable as a large step disturbance occurred in the level under the LQG control. However the same situation was handled smoothly by MPC in Figure 4.30. It is to be noted here that the effect on total head is almost negligible showing the excellent decoupling effect. Of cource the control of stock level is a bit slow. Liquid level is not as important as the total head from basis weight point of view. Thus it can be concluded that the performance of LQG is satisfactory if there are no constraints in inputs and outputs but deteriorates with the presence of constraints. MPC compromises the performance but allows the hard constraints to be considered in optimization. Note, too, that robustness issues will also go against the LQG design. Chapter 4. Multivariable Control Simulations 102 Output responses • i i • i i i \ = Slice Pressure \v i' \\ — = Air Pressure i i t . — = Level i i i • i i i 1 1 0 10 20 30 40 50 60 70 80 90 100 Input!: Stock valve opening) constrained (by clipping) reponses (or ts=1 sec Figure 4.39: Constrained (Clipped) MPC with DMC Estimator. Output responses 1 1 1 i 1 1— i i i = Slice Pressure / / i i — = Air Pressure t = Level i i i i • i i 11 1 1 1 1 I I I I I I 0 10 20 30 40 50 60 70 80 90 100 Input!: Stock valve opening) 1 1 1 T7 1 1 1 J 1 1 : : : i i i i i i i i i 0 10 20 30 40 50 60 70 80 90 100 Constrained reponses forls=1 sec Figure 4.40: Constrained MPC with DMC estimator. Chapter 4. Multivariable Control Simulations 103 Output responses Q_ 1 t o 0.5 c a 0 .9 cs -0.5 > -1 -1.5 i r r — - i — r -= Slice Pressure ' — = AirPressure ^ ^ ^ ^ ^bevel • i i g> 40 c I 20 8 » o 2-20 1 D - 4 0 50 100 150 200 Inputl: Stock valve opening) 250 50 100 150 200 250 constrained (by clipping) reponses for ts=1 sec 300 300 Figure 4.41: Grade-change with a Constrained (clipped) MPC. Output responses I 1 o.5 y . = Slice Pressure = Air Pressure 1-0.5 _/= bevel -1.5L 50 100 150 200 250 300 Inputl: Stock valve opening) ? «>y C §• 2oy I •20 h .a. S O-40r 50 100 150 200 Constrained reponses for ts=1 sec 250 300 Figure 4.42: Grade-change with a Truly Constrained MPC. Chapter 5 Conclusions The thesis started with the development of a nonlinear dynamic model of a Sym-Flo headbox. The model and subsequent simulation results revealed that the dynamic characteristics of a Sym-Flo headbox are similar to those of a conventional airpad headbox when level control in the attenuator is also considered. It is, however, noteworthy that the presence of the turbulence generator has introduced an additional order higher dynamics to the system. The dynamics of level, airpad pressure and the total head at the slice-lip can be represented by a third order nonlinear system with three inputs and three outputs. The three inputs being considered are the thin stock flow, the bleed valve opening and the slice-lip opening. Three outputs are the stock level in the attenuator, the airpad pressure and the slice lip pressure. Including a limited consistency dynamics in the headbox, a fourth order system is resulted. The fourth input is the consistency of the stock inflow and the fourth output is the consistency of the stock leaving the headbox. The theoretical nonlinear model presented in this thesis accounts for the complex cross-section of the attenuator chamber. The model also accounts for the staggered configuration of turbulence generator tube array (honey-comb). The model, however, does not account for the intensity, scale and distribution of turbulence. Nevertheless the model is capable of showing the major dynamic characteristics of the headbox in response to external disturbances. The model is also very useful for studying the start-up and grade change behaviour as it includes the geometric nonlinearity of the attenuator cross section and valve nonlinearities. Having the slice lip opening as one of the inputs is beneficial for simulating the implementation of grade change event automatically. Some simulation results show that the grade change can 104 Chapter 5. Conclusions 105 be achieved within 5 minutes. As the event takes place under control, the product quality will not be inferior. The controller adjusts all four inputs in concert to bring all four outputs to the desired values. The problem is thus reduced to the selection of proper weighting matrices for the inputs and outputs (Q and R) in the case of LQG and proper weighting matrices and prediction and control horizons (P and M) for MPC controller. This is definitely an advantage from formulating the control system considering the headbox as a four by four system. The regulation properties of the controller are also satisfactory despite the fact that the third input, the slice opening, is not manipulated for regulation purposes. The thesis also includes the results of the wet end pressure pulsation analysis performed in a fine paper mill. The results show that there are some periodic disturbances related to the rotating machines of the wet-end components. To eliminate or suppress these disturbances, a good understanding of the headbox model and type of control to be used are essential. The thesis includes various simulation results to illustrate the regulation and servo properties of the system in the presence of periodic disturbances. It has also been shown in the thesis that LQG is able to solve most of the problems associated with the headbox control but suffers from drawbacks. The lack of inherent integral action poses a problem of steady state error. It has been shown that the controller may suffer from wind-up problem if the integrators are used to tackle the steady state problem when input constraints are imposed. Yet another drawback which is felt conspicuously is the lack of provision for accounting input/output constraints optimally. On the other hand, the MPC controller being a finite horizon predictive method, does not have these drawbacks. More tuning parameters are available in MPC than in LQG. Besides the weights on the input outputs, the horizons can also be tuned to establish a smoother control action. It is always easier to tune horizons than weights to influence the control actions. The thesis shows the strength of MPC to accommodate constraints optimally through some simulations. Consistency dynamics are not explored in depth. The results of the wet-end pressure pul-sation experiments clearly showed the presence of short term and medium term variability Chapter 5. Conclusions 106 indicating blending problems in the system. In order to study the consistency dynamics, the flow circuit from the basis weight valve up to the headbox should be considered. The recircu-lation loops, the consistency of the white water and time delay involved in the system are also to be considered. Future work is desirable in this direction. In order to account for the nonlinearity present in the system, an adaptive predictive control can be used which estimates and updates the parameters continuously. Here the model is presented in a state-space form. The model can be converted into an input/output form which makes the implementation of system identification easier. It is quite possible to identify the A, B, C, D matrices required for the state-space form directly too. Chapter 5. Conclusions 107 Nomenclature Symbol Description Unit Ai Area of attenuator-plate holes. m? A2 Area (turbulence generator, circ. end). vn? Ae Area (turbulence generator, rect. end). m 2 A Area of stock surface in attenuator. 2 m Ci Valve recovery index. Cds Coefficient of slice discharge. Valve sizing coefficient for gas flow. cd Consistency of dilution white water. % Cf Consistency of thin stock in fan pump. % Ci Consistency of thin stock entering headbox. - % Cs Consistency of thin stock at slice lip. % Cth Thick stock consistency. % Dcir Diameter of turbulence generator tubes. m NR Reynold's number. N R Number of rows (turbulence generator tube array). Po Atmospheric pressure. mH20 Qscfh Gas flow rate. scfh Hydraulic radius (turbulence generator, circ. end). m Rrect Hydraulic radius (turbulence generator, rect. end). m sb Area of bleed valve opening. rn2. Si Area of slice lip. rr?. So Area of overflow valve opening. 2 mr Vi Velocity of stock (at attenuator holes). m/sec v2 Velocity of stock (at turbulence generator inlet). m/sec Va Volume of air in attenuator. TO3 Chapter 5. Conclusions 108 Vs Volume of stock in attenuator m 3 7 Specific weight of stock. N/m3 P- Viscosity of stock. Pa- Density of air in attenuator. kg/m3 Pw Density of stock. kg/m3 hf Frictional head loss. m hi Turbulence generator inlet head. m hj Head at slice lip. m.H20 qi Stockflow into headbox. m3/sec. qe Airflow through bleed valve. Kg/sec qt Airflow into attenuator. Kg/sec q0 Stockflow through overflow valve. m3/sec q* Stockflow out of slice lip. m3/sec Qd White water flow rate. m3/sec V Total flow out of fan pump. m3/sec qth Volume flow rate of thick stock. m3/sec vc Velocity of stock (overflow tube). m/sec VaB Velocity of air (bleed valve). m/sec vt Velocity of stock (turbulence generator outlet). m/sec G Gas specific gravity. L Length of turbulence generator and slice channel. m P Absolute airpad pressure. m.H20 R Radius of upper portion of attenuator. m T Absolute temperature of gas at inlet. ° Rankine W Width of headbox. m f Frictional factor. g Acceleration due to gravity. m/sec2 Chapter 5. 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Ltd., Kawerau, New Zealand, Papermaker's conference 1986. Bibliography 115 [53] Patel, Rajnikant V.; Munro Neil, Multivariable system theory and design. Pergamon Press Ltd., Headington Hill Hall, Oxford 0X3 OBW, England. 1982. [54] Sanders, H.T. : Thesis, Lawrence University, Appleton, Wisconsin, USA, June 1970. [55] Scattolini,R. : Multi-rate self -tuning predictive controller for multi-variable systems. In-ternational Journal of Systems Science, vol. 23, no. 8, pp. 1347-1359, 1992. [56] Streeter, Victor Lyle; Wylie, E.Benjamin : Fluid mechanics. McGraw-Hill, Inc., 1979. [57] Hem, Svein : The Dynamic behaviour of a pressurized paper machine headbox as deter-mined by computers. M.Sc. thesis, University of Manchester, 1963. [58] Sbarbaro, D. G.; Jones, R.W. : Multivariable nonlinear control of a paper machine headbox. IEEE, New York, NY, USA, 24-26, Aug. 1994. [59] Soeterboek, Ronald : Predictive control: a unified approach. Pretice-Hall, Inc., Englewood Cliffs, NJ, 1992. [60] Tanttu, Juha T.; Cameron, Frank; Lisitzin, Hillel : Experimental comparison of some multivariable PI controller tuning methods. IECON'91, pp. 1818-1823, 1991. [61] Trufitt, Alton D. : Design Aspects of Manifold Type Flow-spreaders. Tappi Journal, vol. 60, no. 10, Oct. 1977. [62] Turunen, R. : Experiences of Sym-Headbox and automatic profile control. Tappi Paper-makers Conference, New Orleans, Notes:233-236, April 14-16, 1986. [63] Washlstrom, Borje. : Headbox design development, high speed and slow speed. Wet end operations, 1985. [64] Xia Qi-Jun; Sun You-Xian; Zhou Chun-Hui : An Optimal decoupling controller with application to the control of paper machine headboxes. Pergamon, oxford, UK, 23-25 Aug. 1988. Bibliography 116 [65] Xia, Q.J.; Rao, M.; Sun, Y. X.; Ying, Y. Q. :• New technique for decoupling control. International journal of systems science, vol. 24, no. 2, pp. 289-300, 1993. [66] Xia Qi-Jun; Sun You-Xian; Ying Y.; Rao, M. : Systematic modeling and decoupling control of a pressurized headbox. Proceedings of Canadian Conference on Electrical and Computer Engineering, 1993. Appendix A Equipment used in Wet end Pressure Pulsation Analysis. A . l Portable Validyne Pressure Transducers: The general operating instructions of the portable Validyne pressure transducers used to mea-sure the wet end pressure pulsations are described below: The transducer system consists of two parts: Sensor (transducer) and Carrier Demodulator. A.1.1 Sensor (Transducer): The transducer consists of a diaphragm of magnetically permeable stainless steel clamped be-tween two blocks of stainless steel. Embedded in each block is an inductance coil. This coil assembly, covered by an Inconel disc, has a corrosion resistant surface. In the undeflected position, the diaphragm is centered with equal gaps between it and the legs of the core to pro-vide equal reluctance for the magnetic flux paths of each coil. The pressure difference applied through the pressure ports deflects the diaphragm toward the cavity with the lower pressure so decreasing one gap and increasing the other. As the magnetic reluctance varies with the gap and determines the inductance value of each coil, the deflection causes the inductance of one coil to increase and that of the other to decrease. See FigureA.l. These two coils form half of the arm of an AC bridge and the transformer, T l , in the carrier demodulator form the other half arm as shown in FigureA.l. The change in inductance is linearly converted into the change in the bridge signal output. The carrier demodulator takes this signal, amplifies it, demodulates and filters it into a dc current of 4 to 20 milli amps. 117 Appendix A. Equipment used in Wet end Pressure Pulsation Analysis. 118 TRANSDUCER CARRIER DEMODULATOR General specifications: Accuracy Overpressure Excitation: Pressure media: Temperature: Frequency Response: Figure A . l : Transducer Bridge Circuit ±0.25%FS 200% FS with less than 0.5% zero shift. Rated: 5V rms, 3kHz to 5 kHz. Corrosive liquid and gases compatible with 410 Stainless Steel and inconel. -65 F to 250 F 0-50-100-1000Hz (depending on carrier demodulator) The following capabilities of the transducer make it suitable for a paper mill pressure mea-surement: 1. Good Dynamic Response : Low volumetric displacement (full scale diaphragm deflection is only about 0.0013 inch) results in good dynamic response. 2. High Overload capability: The internal cavity walls provide effective overload stops; this allows high overloads. 3. High Output Signal: The transducer provides a high output signal with low susceptibility to electrical noise, a recognized advantage of carrier systems. Appendix A. Equipment used in Wet end Pressure Pulsation Analysis. 119 DP15-42-N-1-S-4-A 20 PSI Tapered Header DP15-44-N-1-S-4-A 32 PSI Screen Inlet DP15-46-N-1-S-4-A 50 PSI Cleaner Accept DP15-48-N-1-S-4-A 80 PSI Fan Pump Table A . l : Pressure Transducers 4. Acceptance of corrosive liquids and gases: The Wet parts are made of corrosion resistant 410 Stainless Steel and Inconel. 5. Replaceable Diaphragm: The diaphragm of the sensor can be replaced if necessary. The model, max. pressure range and location of use are tabulated below: A . 1.2 Carrier Demodulator The CD379 is a portable, completely self contained digital transducer indicator and carrier demodulator. In addition to the front panel liquid crystal digital display, the unit also provides a 4 to 20 mA analog dc output for measurement or control purpose. The unit is for use with all variable reluctance sensors. The CD379 provides transducer excitation and signal conditioning zero and span controls. The 4-1/2 digit display provides up to ±1999 counts for full scale output from the transducer. The analog output is available from the rear panel. A . 1.3 Specifications: Transducer Excitation: 1.2V 5K Hz square wave output: Analog 4-20 mA Temperature: 0 to 160 F A . 1.4 Calibration: For the greatest accuracy, the transducer and its carrier demodulator with the actual cabling are to be calibrated as a system. A known source of pressure is required for the calibration. Appendix A. Equipment used in Wet end Pressure Pulsation Analysis. 120 The following steps are followed: 1. A dc power supply (0-24V) in series with a digital milliammeter (multimeter) is connected to the -I- and - terminals of TBI on the CD379 rear panel. 2. With 0% pressure input, the current should be 4.0 ± 0 . 1 mA. The screwdriver poten-tiometer labelled "ZERO" in the front panel is adjusted to bring the reading to the desired point.. 3. With 100% pressure input, the current should be 20.0 ± 0 . 1 mA. The screwdriver poten-tiometer "S" can be adjusted to set the desired current. The most accurate source of air in the Instrument Lab has got 50 PSI as the maximum so all the three lower ranged pressure sensors are calibrated with 100% input while the fourth one with a maximum 80 PSI is calibrated with 50% input. 4. Finally reducing the pressure back to zero, the current should also reduce down to 4.0±0.1 mA. If not, "ZERO" screw is adjusted again and step three is repeated again. A.2 Signal conditioning devices: The analog signal obtained from the pressure transducers is in the form of 4-20 mA current. Precision resistors of 220 ohms are used as shunts to convert the currents into voltages. The analog voltage signal is then filtered and sampled before interfacing with the computer. A set of low pass RC filters with a cut-off frequency of 280 Hz has been built. This filter avoids the problem of aliasing. Sampling and interfacing with the computer is done through a high speed, multichannel, portable data acquisition system. First two experiments were performed with a plug-in data acquisition card manufactured by National Instrument. The third experiment was performed with a more versatile instrument manufactured by IOtech called DaqBook200. Some relevant details of these systems are outlined below: Appendix A. Equipment used in Wet end Pressure Pulsation Analysis. 121 A.2.1 The plug-on A/D board: This board is a 12 bit, 100 KHz, multichannel data acquisition card. It supports 8 single-ended /4 differential analog channels. The input voltage range for differential mode is ± 5 V. Being a plug-in card there is no heed of external power supply. A.2.2 The DaqBook200: The DaqBook200 is a data acquisition system, with high speed, multi-function I/O capability connectable to notebook PCs for data gathering. The DaqBook attaches directly to the PC's parallel port and provides a second parallel connector for attaching a parallel printer. The unit can transfer data bidirectional at up to 170 Kilobytes/s for a conventional printer port and up to 1 Mbytes/sec for an EPP printer port. Real time acquired data can be stored in PC's memory and on its hard drive. The DaqBook provides 16 single-ended, 8 differential analog channels (expandable up to 256 ); single-ended /differential operation is software programmable. The DaqBook200 unit can be powered via several sources, including a standard 12V car battery, the included AC adapter or an optional rechargeable nickel cadmium battery mod-ule. This makes it ideal for portable and remote data acquisition applications such as process monitoring in paper mills. The DaqBook200 offers flexible analog input capabilities. The A / D sample rate is lOOKHz, with a 16 channel multiplexer and a programmable gain input amplifier. It has 16-bit A / D resolution. The DaqBook200 has broad-ranging software support. Two Windows program are included: DaqView3, a set-up and data acquisition package and PostView, a post-acquisition waveform-display package. DaqView3 is an easy to use software which allows us to select desired channels, modes, sampling rate, data collecting duration etc. with a click of mouse. Data can be saved in three formats: Raw Binary, Text, or Postview. Appendix A. Equipment used in Wet end Pressure Pulsation Analysis. 122 Specifications for Analog Inputs are given below: Number of Channels: Resolution: Ranges: Single ended: Differential: Maximum Overvoltage: Input Impedance: Channel scanning: Channel to channel Rate: 16 single-ended, 8 differential; 16 bits 0 to +.10V . 0 to+ / -5V 30VDC > lOOMohmm parallel with 100 PF (typical) 10 uS/channel, fixed A.3 Data Storing Device: A 386Dx33MHz laptop with 12 MB RAM was used to save the data in the first two experiments. A 486Dx66MHz notebook with 540 MB hard drive and 16 MB RAM was used to collect the data in the third experiment. The computer was connected to the DaqBook200 through the printer port. The computer could also be connected to the DaqBook200 through PCMCI card. This method increases the speed of data transfer by about 25% . A.4 Practical Hints Paying some attention before doing the experiment can save effort later. Some important points to be noted and implemented are given below: A.4.1 Sensor Locations Except for the fixed mill sensors, all the sensors should be located at the best practical site available. While taking the single point basis weight data, the Scanner should be positioned Appendix A. Equipment used in Wet end Pressure Pulsation Analysis. 123 approximately at the middle of the sheet to get best MD variation. The pressure sensors should be located preferably on a vertical upward flowing pipe run at least 8 pipe diameters down stream and 2 pipe diameters upstream from any obstructions in the pipe i.e. valves, elbows, bends, pumps, expansions and reductions[37]. The site should not be prone to vibrations and other disturbances. Further the sensors should be kept in such a way that the diaphragm is in vertical position. This avoids the errors introduced by the gravity effects. A.4.2 Configuration It is always preferable to go with differential configuration of data acquisition to reduce the effects of common mode noise and other disturbances. However, in the differential mode the range of input voltage reduces to half (of course ± ) . The Differential Configuration also creates a problem of reference values. There can be two possible differential configurations: 1. Floating Differential. 2. Referenced Differential. Floating differential measurements are generally made when low-level signals must be mea-sured in the presence of relatively high levels of common mode noise. The most common example would be a non-grounded pressure transducer. When the signal source has no inherent connection to the ground, a resistor of 10-100K is connected to the one of the two signal line and the analog common of the data acquisition system. If there are many such pressure transducers and they are all powered by a single power supply unit, (we need to convert the current signals from the sensors into voltages so we use resistors in shunts) then we can go with Referenced Differential configuration. In this configuration, all the shunts share a common supply terminal along a bus. This is equivalent to the single ended configuration however the difference lies on the fact that the sensor current does not flow through the common bus so does not produce undesirable effects. Appendix A. Equipment used in Wet end Pressure Pulsation Analysis. 124 A.4.3 Grounding The shields of the sensor cables should be grounded at one end only. All ground loops should be avoided. If possible a power isolation transformer should be used to separate the data acquisition system and sensors from the power supply source. A.4.4 File Format It was mentioned before that the Dakbook200 can save the acquired data in three file formats: raw binary, text, and postview. Saving the files as raw binary saves time and harddisk memory. The file can easily be read into MATLAB by writing a few lines of MATLAB code as shown below. Following M-File enables us to read the binary file directly into MATLAB and analyze the data. fexl=f open('expt2cur.bin', 'r') ; '/, expt2cur.bin i s the binary f i l e saved % during the experiment. N = 1000*60*3*8; % 3 minutes data each time, for j=i:5, % repeat 5 times f o r 15 min data, a = zeros(i,N); fseek(fexl,2*N*(j-l),-1); a=fread(fexl,N,'ushort'); f o r i=l:5, ch(:,i)=a(i:8:N); % extract a l l the channels. ahi=fix(ch(:,i)/256) •; f i n d the high byte. alow=rem(ch(:,i),256); % f i n d the low byte. a_new=alow*256+ahi; % swap the bytes only i f UNIX i s used. ch(:,i)=(a_new-2-15)/2~15*5 '/. convert into voltage s i g n a l . m=mean(ch(:,i)); °/o f i n d the mean. c=ch(:,i) ; Appendix A. Equipment used in Wet end Pressure Pulsation Analysis. 125 clear ch; '/, to save the memory. c=decimate(c,10) ; '/, to reduce the vectors and save memory. ch_new(:,i)=c; ch_new(:,i)=m/mean(c)*ch_new(:,i); % To avoid level change due to decimation, end ch_old=[ch_old;ch_new]; % add the values everytime. clear ahi alow a_new a; % so ch_old wi l l have a l l the '/, channels. end fclose(fexl); Appendix B M-Files for the Analysis and Simulations of the Headbox Model Following M-flie represents the nonlinear model of the headbox as a SIMULINK S-fuction. The function can be invoked as any other SIMULINK S-functions. %**********S-Function for nonlinear simulations*********** function [sys.xO] = nonmodKt ,x,u,f lag) This function represents nonlinear state-space model of a Sym-Flo headbox. i f nargin ~ 0 flag=0; end i f abs(flag) ==1, if x(l) < 0.475, 7. i f level is < 0.475 m A=4.35*( 0.610-2*(0.475-x(l))/1.732); V=(4.35*pi*0.355*0.355*0.5+4.35*(asin((0.655-0.475)/0.355)) *0.355*0.355+(0.655-0.475)*0.610/2*4.35 +(0.475-x(l)) *(0.610+A/4.35)/2*4.35); else % i f level is > 0.475 m A=(4.35*(0.71*cos(asin((0.655-x(l))/0.355)))); V=(4.35*pi*0.355*0.355*0.5+4.35*(asin((0.655-x(l))/0.355)) *0.355*0.355+(0.655-x(1))*A/4.35/2*4.35); end; 126 Appendix B. M-Files for the Analysis and Simulations of the Headbox Model Vt = 2.0705; % t o t a l volume of the attenuator chamber. T = 27; V, temperature of a i r i n the attebuator. Aa = 0.2670; % area of attenuator holes. At = 0.3155; % area of turbulence generator holes, c l = 30; % valve recovary c o e f f i c i e n t . ae = 0.7854/39.375/39.375/2.9; % max. bleed valve opening. a l = 1.31e-4; '/, max. area of a i r i n l e t valve. k = 1.4; % s p e c i f i c heat r a t i o . g = 9.81; % aceleration due to gravity. rhow = 999.1; '/, density of water. rhoa = 1.293; % density of a i r . q i = 0.1; % mass flow rate of a i r . atm = 10.34; '/, atmospheric pressure. P0=atm; '/, atmospheric pressure. friction=0.13; % f r i c t i o n a l factor of turbulence gen. Nr=4; % no. of rows i n turbulence gen. R = 287; °/« universal gas constant. angle=3417/cl*sqrt((x(2))/(x(2)+atm)); i f angle >=90, angle=90; end GT = 32+T*9/5 + 460; '/„ temperature i n Rankine scale, qe = (x(2)+atm)*14.7/atm*sqrt(520/GT) *u(2)*sin(pi*angle/180)/13.1/2.2/3600; Appendix B. M-Files for the Analysis and Simulations of the Headbox Model % mass rate of air bleed ql = u(l); % stock flow to the headbox as inputl % u(3) is the jet area at the vena-contracta. % slice l i p opening as the third input. qs = 0.98*u(3)*sqrt(2*9.81*x(3)); qo = 0.98*0.002*sqrt(2*9.81*(x(2))); if qo > 0.019, qo=0.019 % allow the overflow upto 300 gpm only, end vb = qs/Aa; 7, velocity in the attenuator plate, vt = qs/At; °/0 velocity in the turbulence generator, vj = qs/u(3) ; '/, slice jet velocity. sys(l) = l/A*(ql-qs-qo); sys(2) = P0/rhoa*k*((x(2)+atm)/P0)~((k-l)/k)/V *((qi- qe)+rhoa*((x(2)+atm)/P0)~(l/k)*A*sys(l)); hO = vb~2/2/g*(1-16*2/25-2)~2; % i f (qs < ql) °/„ hO = -hO; % end hi = (x(2)) + x(l) - hO; '/, head at the attenuator end. ht = x(3) - vt~2/2/g; hf = 1.07*4*friction*Nr*vt~2/2/g; % frictional headloss in the turbulence generator. °/.sys(3) = At/u(3)/0.44*sqrt(2*9.81*x(3))*(hi-x(3)-hf).; sys(3) = 1/1.14*sqrt(2*9.81*x(3))*(hi-x(3)-hf); Appendix B. M-Files for the Analysis and Simulations of the Headbox Model sys(4) = ql*(u(4) - x(4))/(Vt-V); elseif flag == 0; 7. Return i n i t i a l condition sys =[4 0 4 4 0 0]; xO = [0.365 6.1 5.53 0.6]; % xO = [0.1 0 0.1 0]; 7, to study the start-up behaviour, elseif flag ==3, sys = x; else sys = [] ; end 7.**********Steady state calculation********** x(l)=0.365; 7o stock level in the attenuator. cs=0.6; 7o s l i c e - l i p contraction factor, i f grade==72 x(3) =2.7937; 7. total head. slis=0.922*25.4/0.6; 7o s l i c e - l i p opening. x(4)=0.7; 7, output consistency. u(4)=0.7; 7» input consistency. end i f grade==54 x(3)=177.2/39.375; 7. total head. slis=0.82*25.4/0.6; 7c s l i c e - l i p opening. x(4)=0.65; 7o output Consistency. u(4)=0.65; 7o input Consistency. Appendix B. M-Files for the Analysis and Simulations of the Headbox Model end i f grade==451 % old 45# grade paper. x(3)=5.53 slis=0.75*25.4/0.6; x(4)=0.6; u(4)=0.6; end i f grade==452 °/, New 45# grade paper after wet end modification. x(3)=252.7/39.375; slis=0.901*25.4/0.6; x(4)=0.47; u(4)=0.47; end Aa= 0.2670; % area of holes i n the attenuator plate. At = 0.3155; % area of holes i n the turbulence generator, c l = 30; % Valve recovary c o e f f i c i e n t assumed constant, a l = 1.31e-4; % area of a i r i n l e t valve. ae = 0.7854/39.375/39.375/2.9; '/, max. Area of a i r bleed valve, k = 1.4; % s p e c i f i c heat c o e f f i c i e n t , g =9.81; °/0 acceleration due to gravity, rhow = 999.1; °/, density of stock, rhoa = 1.293; °/0 density of a i r . q i = 0.1; '/. mass rate flow of a i r into the headbox. %qi=0.25; y.atm = 1.013e5; atm = 10.34; °/, atmospheric constant. Appendix B. M-Files for the Analysis and Simulations of the Headbox Model 131 friction=0.13; °/0 frictional factor for the turbulence generator. Nr=4; '/, number of rows in the turbulence generator. T=27; 7. temperature of air in the headbox. GT=32+9*T/5+460; '/, temperature in Rankine scale. u(3)=cs*4.350*slis/1000; 7. Slice-lip area in m2. qs = 0.98*u(3)*sqrt(2*9.81*x(3)) '/, stock forced out through the s l i c e - l i p . vb = qs/Aa % velocity at the attenuator plate. vt=qs/At % velocity at the turbulence generator. vj = qs/u(3); % velocity at the sl i c e - l i p (jet velocity). ht = x(3); % head at the l i p . hi = ht+1.07*4*friction*Nr*vt~2/2/g; % 7% more for turbulence. % head at the s l i c e - l i p . hO = vb~2/2/g*(1-16*2/25-2)"2 % headloss in the attenuator plate. x(2) = (hi - x(l) + hO) % airpad pressure. qo = 0.98*0.002*sqrt(2*9.81*(x(2))) 1 overflow stock, i f qo > 0.019, qo=0.019 V, allow the overflow upto 300 gpm only, end angle=3417/cl*sqrt((x(2))/(x(2)+atm)); i f angle >=90, angle=90; end u(2)=qi*(13.1)*2.2*3600/sin(pi*angle/180)/((x(2)+atm)*14.7/atm)/sqrt(520/GT); u(l)=(qs+qo); Appendix B. M-Files for the Analysis and Simulations of the Headbox Model 132 %*****#****lineaxization********** steady; % c a l l the program to calculate the steady state values. [a,b,c,d]=linmod('newsimu',x,u); 7, linearize the model around that point. eig(a) % eigen-values step(a,b,c,d, 1) ; °/, step response to the stock flow. **********gjmujjnjj implementation********** SIMULINK diagram are used to simulate various regulatory, servo and grade change events with LQG controller. Figure B.2: Simulink Model of Headbox Control System Using LQG. 'i'H^^'l"l"f^'i*JJlotting t i l6 rCSTlltS^ "^ H^^ ^H^H^^ ^H^ Following program plots the responses obtained during the grade-change simulations. ul=u(:,1); u2=u(:,2); Appendix B. M-Files for the Analysis and Simulations of the Headbox Model u3=u(:,3); u4=u(:,4); loss=ul'*ul+u2'*u2+u3'*u3+u4'*u4+yl'*yl+y2'*y2+y3'*y3+y4'*y4; s=sprintf ('The loss function is °/08.2f' ,loss) ; subplot(211) plot(t,yl,'-',t,y2,'—',t , y 3 , ' - . ' ) title('Output responses'); ylabeK'Deviations in outputs'); text(25,0.75,'-.-. = Slice Pressure') text(25,-0.5,'— = Air Pressure') text(25,-0.25,'__ = Level') text(50,-1.25,s) text(15,0.25,'Grade change from 45# to 54#') subplot(212) plot(t,u(:,1)) title('Inputl (stock flow) during the grade change'); ylabeK'Deviation in stock flow'); grid xlabeK'Time in seconds'); axis([0 600 -10e-3 10e-3]); f igure subplot(211) plot(t,u(:,2)) title('Input2 (Bleed valve coef.) during the grade change'); ylabeK'Deviation in valve coefficient'); grid axis([0 600 -100 100]); Appendix B. M-Files for the Analysis and Simulations of the Headbox Model 134 subplot(212) plot(t,u(:,3)) title('Input3 (Slice opening) during the grade change'); ylabeK'Deviation in slice opening'); grid xlabeK'Time in seconds'); axis([0 600 -15e-3 15e-3]); Appendix C M-Files to Implement MPC Controllers. Following program illustrates the effect of noise and estimator on regulation. load abcdnew; 7, load a,b,c,d and ak.bk,ck.dk matrices. ts=l; 7o take the sampling time as 1 second. [phi,gm]=c2dmp(ak,bk,ts) ; '/, convert into descrete form. mod=ss2mod(phi,gm,c,d,ts) 7o convert to the mod format. imod=mod; ywt=100*[2 0 5 2]; 7o weigh the t o t a l head more. uwt=[0.1 0.01 15000 1] 7. penalize the s l i c e l i p heavily. p=20; 7o look 20 steps ahead. m=l; 7o control horizon as 1. t=300; % simulate f o r 300 seconds. gll=poly2tfd(l, [1 11,0,0); 7. introduce a f a u l t i n l e v e l sensor gl2=poly2tfd(0,1,0,0) gl3=poly2tfd(0,l,0,0) gl4=poly2tfd(0,l,0,0) g21=poly2tfd(0,1,0,0) g22=poly2tfd(l, [1 1],0,0); '/, a f a u l t i n airpad pressure sensor g23=poly2tfd(0,l,0,0); g24=poly2tfd(0,1,0,0); dmod=tfd2mod(ts,4,gl1,gl2,gl3,gl4,g21,g22,g23,g24); imod=addumd(mod,dmod) ; '/, add the disturbance model. 135 Appendix C. M-Files to Implement MPC Controllers. 136 Q=[l 0; 0 1]; R=diag([l 1 1 1]); kest=smpcest(imod,Q,R); % use general estimator. °/.kest=[]; •/. DMC estimator. ks=smpccon(imod,ywt,uwt,m,p); ulim=[] ; ylim=[-0.135 -4 -4 -0.2 0.435 2 2 0.2]; plant=imod; r=[0 0 0 0] ; randn('seed' ,8); nl=0.01*randn(t,4); % generate noise with std = 0.01 n2=zeros(t,4); n2(l, :) = [0 0 0 0]; for j=2:t '/, pass through a f i r s t order f i l t e r . n2(j,:)=0.8*n2(j-l,:)+nl(j,:); end z= [n2] ; v=[]; w=zeros(40/ts,2); % last row wi l l be repeated for rest 200 w(40,l)=0.5; 7, w(100,2)=0.5; [dy,du] =smpcs im(plant,imod,ks,t,r,ulim,kest,z,v,w); mpcregu; figure(l); xlabelCUnconstrained reponses. New estimator.'); ulim=[-45 -67 0 -0.2 45 23 0 0.2 20 27 0.001 0.01]; [dy,du]=smpcsim(plant,imod,ks,t,r,ulim,kest,z,v,w); Appendix C. M-Files to Implement MPC Controllers. figure mpcregu; * * * * * * * * * * F o r plotting the results********** This program is run just following the previous program to plot the responses. u l -du( : , l ) ; u2=du(:,2); % Input deviations u3=du(:,3); u4=du(:,4); yll=dy(:,l); y22=dy(:,2); y33=dy(:,3); % output deviations y44=dy(:,4); tl=l:length(yll); loss=ul'*ul+u2'*u2+u3'*u3+u4'*u4+yi1'*y1l+y22'*y22+y33'*y33+y44'*y44; s=sprintf('The variance of total head is %8.6f',cov(y33)); subplot(211) plot(tl,yll,'-',tl,y22,'—',tl,y33,'-.') title('Output responses'); ylabeK'Deviations in outputs'); axis([0 300 -0.2 0.2]) text(75,0.12,'-.-. = Slice Pressure') text(75,0.07,'— = Air Pressure') text(150,0.07,'__ = Level') text(75,-0.12,s) subplot(212) plot(tl,ul) title('Input1: Stock valve opening'); Appendix C. M-Files to Implement MPC Controllers. 138 ylabel('Deviation i n stock opening'); g r i d xlabel('Time i n seconds'); axis([0 300 -45 45]); figure subplot(211) plot(tl,u2) t i t l e ( ' I n p u t 2 : Bleed valve opening'); ylabeK'Deviation i n valve opening'); g r i d axis([0 300 -80 30]); subplot(212) plot(tl,u3) t i t l e ( ' I n p u t 3 : S l i c e opening'); ylabel('Deviation i n s l i c e opening'); g r i d xlabeK'Time i n seconds'); axis([0 length(yll) -35e-3 35e-3]); **********Constrained/Unconstrained MPC********** This program illustrates the effect of unmeasured disturbances. The responses are calculated for both the constrained, clipped and unconstrained cases hence it is possible to compare the results. load abcdnew; % load the system matrices ts=l; % sampling time [phi,gm]=c2dmp(ak,bk,ts); % change the model to descrete mod=ss2mod(phi,gm,c,d,ts) change the model to mod format imod=mod; Appendix C. M-Files to Implement MPC Controllers. 139 ywt=100*[2 0 5 2]; output weighting uwt= [0.1 0.01 15000 1] '/, input weighting p=20; % prediction horizon m=2; % control horizon t=300; den =[1.0000 18.078 89.42 78.36 0.166]; 7. the following transfer functions represent the effects of unmeasured disturbances. gll=poly2tfd([0.0006198 0.0112 0.05 0.003156]*0.09,conv(den,[1 0 0.09]),0); gl2=poly2tfd([0.01813 0.326 1.415 0.000227] *0.09,conv(den,[1 0 0.09]),0); gl3=poly2tfd([0.0 0.171323 1.252148 0.002893]*0.09,conv(den, [1 0 0.09]),0); gl4=poly2tfd(0,1,0); g21=poly2tfd([0.0000 0.000017 0.006862 0.04923],den,0); g22=poly2tfd([-0.02032 -0.36552 -1.592497 -0.048136],den,0); g23=poly2tfd([0.0000 -0.185662 -1.35639 0.0009337],den,0); g24=poly2tfd(0,1,0); g31=poly2tfd(le3*[-0.004858 -0.0802 -0.3295 0.0211],den,0); g32=poly2tfd(le3*[-0.14214 -2.33725 -9.4893 .00152],den,0); g33=poly2tfd(le3*[-0.2126 -2.9152 -9.9560 -0.0227],den,0); g34=poly2tfd(0,1,0); '/ the unmeasured disturbances are converted into mod format below dmod=tfd2mod(ts,4,gll,gl2,gl3,gl4,g21,g22,g23,g24,g31,g32,g33,g34); imod=addumd(mod,dmod) ; disturbance model is added to the system kest=[]; °/0 DMC estimator ks=smpccon(imod,ywt,uwt,m,p); ulim= [] ; Appendix C. M-Files to Implement MPC Controllers. ylim=[-0.135 -4 -4 -0.2 0.435 2 2 0.2]; plant=imod; %r= [0 0 0 0]; r=zeros(140,4); r(140,3)=0.5; randnCseed',8) ; nl=0.01*randn(t,4); 7, white measurement noise end z=[nl] ; v=[]; w=zeros(40/ts,3) ; 7, last row will be repeated. w(40,l)=3; °/, unmeasured disturbance after 40 second in slice % opening % w(80,2)=0.5; % disturbance in airflow after 80 seconds [dy,du]=smpcsim(plant,imod,ks,t,r,ulim,kest,z,v,w); mpcregu; figure(l); xlabelC Unconstrained reponses.'); ulim=[-45 -67 -le-12 -0.2 45 23 le-12 0.2 20 27 0 0.01]; [dy,du]=smpcsim(plant,imod.ks,t,r,ulim,kest,z,v,w); figure mpcregu; figure(3) xlabelCconstrained (by clipping) reponses.'); [dy,du]=scmpc(plant,imod,ywt,uwt,m,p,t,r,ulim,ylim,kest,z,v,w); f igure mpcregu; figure(5); xlabeK'Constrained reponses.'); Appendix C. M-Files to Implement MPC Controllers. 141 ******** Grade-change simulation******** The following program simulates grade-change event using MPC. The effect of optimal handling of constrains are illustrated here. load abcdnew; grade=451; % run ts=l; [phi,gm]=c2dmp(ak,bk,ts); mod=ss2mod(phi,gm,c,d,ts); imod=mod; ywt=10*[2 2 2 2] ; uwt=[0.1 0.01 150 1] p=20; m=l; t=300; kest=[] ; ks=smpccon(imod,ywt,uwt,m,p); ulim=[-45 -67 -0.0075 -0.2 45 23 0.0075 0.2 20 27 0.001 0.01]; ylim=[-0.135 -4 -4 -0.2 0.435 2 2 0.2]; r=zeros(100/ts,4); for j=10/ts:100/ts r(j ,2)=-1.03*j/100; '/, ramp the setpoints r(j,3)=-1.06*j/100; r(j,4)=0.2*j/100; end plant=imod; [dy,du]=smpcsim(plant,mod,ks,t,r); Appendix C. M-Files to Implement MPC Controllers. 142 mservo; % for p l o t t i n g the responses f i g u r e ( l ) ; xlabel('Unconstrained reponses.'); [dy,du]=smpcsim(plant,mod,ks,t,r,ulim); figure mservo; figure(3) xlabel('constrained (by clipping) reponses.'); r= [0 0 0 0] ; z=[]; v=[]; d=l; [dy,du]=scmpc(plant,imod.ywt,uwt,m,p,t,r,ulim,ylim,kest,z,v,d); figure plotalK [y3] , [u3] ,ts) ; mservo; figure(5) xlabel('Optimal Constrained reponses.'); **********Nonlinear s i m u i a t ions with MPC********** Following Simulink diagram shows the nonlinear implementation of MPC. The parameters for the nonlinear MPC controller are supplied by running another program listed just after this diagram. It is noteworthy here that the numerical method selected for the simulation is very important for the nonlinear simulations. Select Rk5 or Gear for better results. %********program supplying MPC parameters for nonlinear simulation********** The nonlinearity of the valves and the system can be included using the simulink blocks and functions. Regulation, servo and grade change behaviour can be studied by adjusting the weightings on inputs/output, Appendix C. M-Files to Implement MPC Controllers. 143 • •5-Stock Valve Nonlinearity Bleed Valve Nonlinearity Inputs Load DisuJI bance H3 Scope nputsi Headbox Model Subsystem S h Clock Sum HDr-tf Sum1 - L Stock level Airpad Pressure Total head l ] Consistency Sn6 output! Figure C.3: Simulink Implementation of MPC for Nonlinear Simulations. prediction horizon, and control horizon. Cl i c k i n g the "load data" button i n the SIMULINK program given above w i l l run the following program. clear load abcdnew; 7. grade=451; % grade 45# T=l; % sampling time tend=300; '/« simulate f o r 300 seconds °/, run; 7> t h i s program calculates a l l the a b e d matrices [phi,gm]=c2dmp(ak,bk,T) ; °/, convert to discrete time model imod=ss2mod(phi,gm,ck,dk,minfo) ; °/, convert to mod format minfo=[T,4,4,0,0,4,0] ; [model.dmodel]=mod2step(imod,30); Appendix C. M-Files to Implement MPC Controllers. 144 ywt=100* [5 0 5 1]; uwt=5*[0.1 0.01 15000 1]; p=20; m=2; 7. P=20; % M=[3 4 5] ; K=[] ; ylim=[0.21 0 0 0.4 0.5 8 8 0.7]; ulim=[0 0 0.0804-ie-12 0.4 90 90 0.0804+le-12 0.7 20 27 le-12 0.03]; 7. ulim=[] ; usat=ulim; 7. tfilter=[300 0 0 0;0 0 0 0] tf ilter=[] ; y0=[0.365 6.1 5.53 0.6]; u0=[45 67.6 0.0804 0.6]; r=[0.365 6.1 5.53 0.6]; Kmpc=mpccon(model,ywt,uwt,m,p); 0/ o**********Plotting after nonlinear simulations*********** Following program plots the responses after the nonlinear simulations using MPC. ul=du(:,1) u2=du(:,2) u3=du(:,3) u4=du(:,4) yll=yl-0.365*ones(size(yl)); 7. deviations in inputs Appendix C. M-Files to Implement MPC Controllers. y22=y2-6.1*ones(size(yl)); 7, deviations in outputs y33=y3-5.53*ones(size(yl)); y44=y4-0.6*ones(size(yl)); loss=ul'*ul+u2'*u2+u3'*u3+u4'*u4+y11'*y1l+y22'*y22+y33'*y33+y44'*y44; s=sprintf ('The loss function is 7,8.2f' .loss) ; subplot(211) plot(t,yll,'-',t,y22.'--',t.y33.'-.') title('Output responses'); ylabeK'Deviations in outputs'); text(25,0.75,'-.-. = Slice Pressure') text(25,-0.5,'— = Air Pressure') text(25,-0.25,'__ = Level') text(50,-1.25,s) subplot(212) plot(t.ul) title('Inputl (stockflow)'); ylabeK'Deviation in stock flow'); grid xlabeK'Time in seconds'); axis([0 200 -10e-3 10e-3]); f igure subplot(211) plot(t,u2) title('Input2 (Bleed valve coef.)'); ylabeK'Deviation in valve coefficient'); grid axis([0 200 -100 100]); Appendix C. M-Files to Implement MPC Controllers. subplot(212) plot(t,u3) title('Input3 (Slice opening)'); ylabel('Deviation in slice opening'); grid xlabelC Time in seconds'); axis([0 200 -15e-3 15e-3]); 

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