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Paper machine cross-directional control near spatial domain boundaries 2004

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PAPER MACHINE CROSS-DIRECTIONAL CONTROL NEAR SPATIAL DOMAIN BOUNDARIES B y Stevo Mijanovic Dipl . Ing. (Electrical Engineering) University of Montenegro, Serbia and Montenegro A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F E L E C T R I C A L A N D C O M P U T E R E N G I N E E R I N G We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A March 2004 © Stevo Mijanovic, 2004 Abstract This work is concerned with the modification of an existing industrial paper machine cross- directional (CD) control law near spatial domain boundaries (paper sheet edges), taking into account relevant control engineering criteria: closed-loop stability, performance, and robustness. Paper machine C D control systems belong to a set of large, multivariable, spatially- distributed control systems, having 30-300 control inputs and 200-2000 process outputs. The objective of C D control is to reduce the variations of a particular paper sheet property (basis weight - weight per unit area, moisture content, or thickness) in the cross-direction (the direction perpendicular to the sheet travel direction) as much as possible. C D control systems can properly be described as two-dimensional systems, with one time dimension and one spatial dimension (cross-direction). The state-of-the-art industrial C D controllers of interest in this work are designed assuming spatially-invariant C D processes. Indeed, a lot of recently developed techniques for the design of spatially-distributed control laws make use of the spatial-invariance assumption. However, very many of the real-life systems (including paper machine C D processes) are not spatially-invariant. Paper machine edges represent a clear disruption of the assumed spatial-invariance. As a result, initially designed spatially-invariant control laws must be modified before implementation on the real (spatially-variant) paper machines. The current industrial techniques for modifying C D control laws near spatial domain boundaries are based on techniques for extending finite-width signals, borrowed from the field of signal processing. As these techniques do not take into account relevant control engineering criteria, they can lead to very poor control near the edges, and potentially even destabilize the overall C D control system. The main contributions of this work are the three novel approaches to modifying the existing industrial C D control law that directly take into account important control engineering criteria. In addition, the newly developed closed-loop approach has also been successfully tested on a paper machine in a working paper mill . A developed closed-loop stability transfer approach is a straightforward perturbation technique for the spatially-invariant C D controller, that is guaranteed to stabilize a closed- loop system with the actual (spatially-variant) C D plant. Next, the similarities between effects observed near spatial domain boundaries of the industrial C D control systems and the well-known Gibbs effect are illustrated. Subse- quently, based on the techniques for mitigating the Gibbs effect, the so-called open-loop approach to modifying the existing C D control law is developed and illustrated with a closed-loop simulation example. ii Finally, in a closed-loop approach to modifying the existing industrial C D controller, the objective is restated in terms of a block-decentralized static output feedback design problem. Static modifications of the existing controller's two constant matrix components are then sequentially computed by the use of a novel low-bandwidth static output feed- back controller design algorithm. The relevant control engineering criteria (closed-loop stability, performance, and robustness) are all systematically taken into account with this approach. Since the resulting closed-loop system robustness margins near the sheet edges are directly considered, the possibility of C D control instability originating from the edges and 'creeping' into the rest of the system is eliminated with the new approach. The new approach has a clear economic benefit for the papermakers, since with a stable, robust, and performance improving control law near the sheet edges, the quality of the paper sheet near the edges can be significantly improved, thus resulting in less paper being trimmed off and more on-spec paper being produced from which the papermaker can extract his orders. The newly developed closed-loop approach to modifying the existing industrial C D control law near spatial domain boundaries is tested and verified on a paper machine in a working paper mill. The obtained closed-loop control results are presented. i i i Table of Contents Abstract i i List of Tables v i i List of Figures v i i i Acknowledgments x i i Chapter 1 Introduction 1 1.1 Industrial Paper Making - The Paper Machine 1 1.2 Paper Machine Cross-Directional (CD) Control Systems 5 1.2.1 Basis Weight Control 5 1.2.2 Moisture Control 6 1.2.3 Caliper Control 7 1.3 C D Process Attributes 8 1.3.1 Approaches to C D Control 10 1.4 Spatially-Distributed Control Systems 12 1.5 Industrial Paper Machine C D Control Systems 14 1.5.1 Problems Near the Sheet Edges (Spatial Domain Bound- aries) 16 1.6 Aims and Contributions of the Work 18 1.7 Thesis Overview 19 Chapter 2 Problem Statement 21 2.1 Explicit and Implicit Boundary Conditions 21 2.2 Paper Machine C D Control System 25 2.2.1 Industrial C D Controller Tuning Technique: Two-Dimensional Loop Shaping 28 2.3 Objective of the Work 29 Chapter 3 A Closed-Loop Stability Transfer Between Systems 32 3.1 Control Systems with Known Plant Deviations 32 3.2 Relationships Between Plant Models 33 3.3 Augmentation of Feedback Controllers 35 3.4 Application of Theorem 1 to C D Control 37 3.5 Other Applications of Theorem 1 39 3.5.1 Actuator and Sensor Failures 39 iv 3.5.2 Smith Predictor 41 3.5.3 Recycle Compensator 42 3.6 Summary . 43 Chapter 4 Open-Loop Approach to C D Controller Modifications 45 4.1 Gibbs Phenomenon and Spatial Filtering 45 4.2 C D Control Modifications Near the Boundaries 47 4.3 Simulation Example 49 4.3.1 Edge Filter Design 49 4.3.2 Closed-Loop Simulations 51 4.4 Summary 56 Chapter 5 Closed-Loop Approach to C D Controller Modifications 57 5.1 Static Output Feedback (SOF) Controller Synthesis 58 5.1.1 Synthesis Algorithm 62 5.2 Computation of C D Controller Modifications 62 5.2.1 Modifications Near One Sheet Edge ( C e and De) . . . . 63 5.2.2 Computation of Ce and De 66 5.3 Hardware-In-The-Loop Simulator Example 67 5.3.1 Process and Controller Parameters 68 5.3.2 Controller Modifications and Closed-Loop Simulations . 69 5.4 Stabilization Procedure (Rarely Required) 76 5.4.1 Example 77 5.5 Summary 80 Chapter 6 Industrial Trial 81 6.1 C D Control Setup in the M i l l 81 6.2 Trial Setup and Procedure 83 6.3 Trial Results 87 6.3.1 Process and Controller Parameters 87 6.3.2 Computed Controller Modifications 6C and SD . . . . . 87 6.3.3 Closed-Loop Control Results: Data Set 1 91 6.3.4 Closed-Loop Control Results: Data Set 2 95 6.4 Summary 99 Chapter 7 Concluding Remarks 100 7.1 Summary of the Thesis 100 7.2 Future Work 102 v Bibliography 104 Appendices 111 Appendix A Proof of Theorem 1 111 A . l Process Additive Perturbation (Case a) I l l A.2 Process Inverse Additive Perturbation (Case b) 112 A.3 Process Multiplicative Input Perturbation (Casec) 112 A.4 Process Inverse Multiplicative Input Perturbation (Case d) . . . 113 A.5 Process Multiplicative Output Perturbation (Casee) 113 A.6 Process Multiplicative Output Perturbation (Casef) 114 Appendix B Matrix Optimization 115 Appendix C Proofs of Theorems 4—5 118 C . l Supporting Relationships 118 C.2 Proof of Theorem 4 119 C. 3 Proof of Theorem 5 119 Appendix D Closed-loop transfer functions used for defining LFTs 120 D. l Closed-loop transfer functions that make up Pe(z) in Figure 5.2 . 120 vi List o f Tables 2.1 Matrix coefficients in (2.6) resulting from the representation of spatial filters (order lh = 1) with various boundary conditions 24 2.2 Stability of the system in (2.5) with lh = 1, n = 20, and filter coefficients h0 = 0.8, hi = 0.1, in case of various boundary conditions in Table 2.1. . . 25 3.1 Closed-loop transfer functions in Figure 3.1b for the various configurations of Gp and Kp in Theorem 1 37 4.1 Boundary layer coefficients of the controller matrix D in case of reflective boundary conditions and d = [do, • • • , c^], d-j = dj for j = 1, 2, 3 52 4.2 2-norms of the steady-state process output and control signal profiles shown in Figures 4.4 - 4.6 55 5.1 2-norm of the process output and control signal steady-state profiles shown in Figures 5.8 - 5.11 75 6.1 2-norms of the process output and control signal profiles shown in Figures 6.6 - 6.9 (Data Set 1) 95 6.2 2-norms of the process output and control signal profiles shown in Figures 6.10-6.13 (Data Set 2). . .,. 98 vii List of Figures 1.1 Schematic view of the paper machine showing typical positions of the scan- ners) and various C D actuator arrays, as well as illustrating machine and cross directions (Figure courtesy of Honeywell Process Solutions - North Vancouver) 2 1.2 Illustration of trim squirts, used for trimming-off of narrow paper sheet strips, in the sheet-forming section 3 1.3 Illustration of the scanning sensor's measuring path (Figure courtesy of J . Fan [20]) 4 1.4 Slice lip basis weight control: measured (thin line) and modelled (thick line) steady-state bump response shapes (lower figure) in the case of deflection of 3 out of 36 slice lip actuators (upper figure). Data obtained during the industrial trial described in Chapter 6 6 1.5 Illustration of the two-dimensional characteristics of the C D processes: C D process model basis weight response to a slice lip actuator 9 1.6 Steady-state singular values of a typical C D process model 10 1.7 Simplified diagram of an industrial C D control system 15 1.8 Data flow in an industrial C D control system; HR: High-Resolution (Scanner spatial resolution) L R : Low-Resolution (Actuator spatial resolution). . . . 16 1.9 Illustration of the problems ('actuator picketing' in the lower portion of the screenshot) that often occur when implementing current C D control tech- niques near the sheet edges 17 2.1 The template structure and explicit boundary layer 50, (denoted by o for i = 0 and i = n + 1) of a spatiotemporal filter with lh = 1 in (2.1)-(2.3). (The row of o at k == -1 indicates the initial conditions of the causal filter and are not important for the case being considered.) 22 2.2 The non-zero elements of the matrix Ha in (2.7) (a); matrix Hc in (2.8) (b); and the difference AH — Hc - Hd (c). 24 2.3 The industrial C D control system 26 2.4 The industrial C D controller structure. (Compare with Linear time-invariant CD controller in Figure 1.8) 27 2.5 Idealized cross-directional control system with periodic boundary conditions. . 28 2.6 C D control system with control law modifications, 5C and 5D, near spatial domain boundaries 30 2.7 Problem reformulated in terms of linear fractional transformation (LFT) . . . . 30 vii i 3.1 (a) Original and (b) modified closed-loop control systems 33 3.2 Block diagrams for various Gp in terms of Go and A G - (a) Additive per- turbation, (b) Inverse additive perturbation, (c) Multiplicative input per- turbation, (d) Inverse multiplicative input perturbation, (e) Multiplicative output perturbation, (f) Inverse multiplicative output perturbation (com- pare Figure 8.5 in [58]) 34 3.3 Block diagrams indicating the various configurations oi Gp and Kp described in Theorem 1 . 36 3.4 Position of the non-zero elements of: (a) the Toeplitz matrix Go = Gd(z); (b) the circulant symmetric matrix Gp = Gc(z); and (c) the difference between the two: A G = -AG{z) = Gd(z) - Gc{z) 38 3.5 Paper machine cross-directional control system, initially computed with the two-dimensional loop shaping technique resulting in a spatially-invariant process and controller models G c (z ) , G c , Dc, stabilized by the use of Theo- rem 1 39 3.6 Location of the non-zero elements of (a) the nominal plant model Go, (b) the additive perturbation due to failure of the 7th sensor, and (c) the corresponding transfer matrix model Gp 40 3.7 Smith-predictor design for plants with pure time delay (compare with Figure 3.3a) 42 3.8 Recycle compensator for plants with recycle dynamics (compare with Figure 3.3b) 43 4.1 Traditional illustration of Gibbs effect in Fourier analysis (a) and its reduction achieved by using Lanczos filter (b) 46 4.2 Desired (full line) and achieved (dotted line) frequency responses 50 4.3 Process output disturbance (at zero temporal frequency LU = 0) 53 4.4 Steady state process output and actuator array in case of the reflective bound- ary conditions in Table 4.1 • 54 4.5 Steady state process output and actuator array when the controller with matrix Df, given with (4.2) and (4.7), is used 54 4.6 Steady state process output and actuator array when the approach presented in Chapter 3 (controller with the structure illustrated in Figure 3.5) is used. 55 5.1 Diagram of the lower linear fractional transformation J-~i(N, K) 58 5.2 Isolating system inputs/outputs near one edge 64 5.3 Transforming a sub-block into a full-block design problem. 65 ix 5.4 Illustration of the rapid decrease of the Hankel singular values of the closed- loop transfer functions that define a generalized plant Pe(z) in Figure 5.2: Hankel singular values of Pi : de —> ye for the C D control system presented in Section 5.3 66 5.5 Linear fractional transformations for computing (a) Ce and (b) De modifications. 66 5.6 Schematic of the simulator trial setup 67 5.7 Process output disturbance d (at zero temporal frequency u = 0) 69 5.8 Steady-state process output (a) and control signal (b), using the current in- dustrial technique - reflection padding 73 5.9 Steady-state process output (a) and control signal (b), using the new technique - conservative tuning (kp = 300 and kp = 0.2 in (5.24)) 74 5.10 Steady-state process output (a) and control signal (b), using the new technique - balanced tuning (kP = 1600 and kR = 0.2 in (5.24)) 74 5.11 Steady-state process output (a) and control signal (b), using the new technique - aggressive tuning (kp = 2400 and kit = 0.5 in (5.24)) 75 5.12 A gradual elimination of the process and controller circulant symmetric matri- ces' 'ears' with the parameter A £ [0,1] 77 5.13 Actuator array shape (upper figure) and the corresponding process steady-state response (lower figure) for the process model given with (5.37) 78 6.1 Stevo Mijanovic near machine on which the industrial trial was carried out. In the background: machine's forming section (left photo), and the press and dryer sections (right photo) 82 6.2 A simplified schematic of the mill's C D control setup 83 6.3 A simplified schematic of the industrial trial setup 84 6.4 The dataflow diagram between the Matlab prototype software and the indus- trial software packages 85 6.5 Model identification: The upper plot illustrates the shape of the actuator array used for process excitation. The middle plot shows the measured process output profile. The bottom plot illustrates the identified model, as given by the parameters bj, j = 0,1, 2 , 8 in (6.1) 88 6.6 Data set 1 process output (a) and control signal (b), using the current industrial technique - average padding 93 6.7 Data Set 1 process output (a) and control signal (b), using the new technique - conservative tuning (kP = 180 and kp = 0.2 in (6.3)) 93 6.8 Data Set 1 process output (a) and control signal (b), using the new technique - balanced tuning (kP = 480 and kR = 0.2 in (6.3)) 94 x 6.9 Data Set 1 process output (a) and control signal (b), using the new technique - aggressive tuning (kp = 720 and kp = 0.4 in (6.3)) 94 6.10 Data Set 2 process output (a) and control signal (b), using the current indus- trial technique - average padding 96 6.11 Data Set 2 process output (a) and control signal (b), using the new technique - conservative tuning (kp = 180 and ka = 0.2 in (6.3)) 97 6.12 Data Set 2 process output (a) and control signal (b), using the new technique - balanced tuning (kp = 480 and kp, = 0.2 in (6.3)) 97 6.13 Data Set 2 process output (a) and control signal (b), using the new technique - aggressive tuning (kp = 720 and kp = 0.4 in (6.3)) 98 A . l Diagram used to analyze internal stability for the configuration given in The- orem 1 - case (a) I l l A.2 Diagram used to analyze internal stability for the configuration given in The- orem 1 - case (b) 112 A.3 Diagram used to analyze internal stability for the configuration given in The- orem 1 - case (c) 112 A.4 Diagram used to analyze internal stability for the configuration given in The- orem 1 - case (d) 113 A.5 Diagram used to analyze internal stability for the configuration given in The- orem 1 - case (e) 114 A.6 Diagram used to analyze internal stability for the configuration given in The- orem 1 - case (f) 114 xi Acknowledgments This project has been carried out as a collaboration between the University of British Columbia and the industrial partner Honeywell Process Solutions - North Vancouver. I would like to thank my research supervisors Dr. Greg Stewart from Honeywell Process Solutions, and Profs. Guy Dumont and Michael Davies from the University of British Columbia (UBC) for their advice and guidance throughout the course of my graduate studies. Prof. Michael Davies admitted me into U B C graduate school and Prof. Guy Dumont encouraged me to do a PhD thesis.. Over the past few years, I have greatly benefited from the regular (weekly) meetings with my industrial supervisor Dr. Greg Stewart. Valuable discussions during these meetings have left an indelible impact on this work. This project would not have been possible without the technical and financial assistance of the industrial partner Honeywell Process Solutions - North Vancouver. I consider this industrial collaboration a particularly useful experience for myself. Numerous people from Honeywell have in or the other way contributed to this project. In particular, I would like to acknowledge the help of the following people: Cristian Gheorghe, Joyce Choi, Bijan Nazem, Paul Baker, Stephen Chu, Chuck Chung, Roger Chen, Amor Lahouaoula, Max Kean, Pengling He, Dan Stevens, Scott Morgan, Johan Backstrom, Rhonda Kieper, and Bob Vyse. The financial support of my research work, over the past few years, by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Science Council of British Columbia is also gratefully acknowledged. During my graduate studies at U B C , I have benefited from the technical discus- sions/talks with a large number of people. Particularly useful have been the discussions with the past and present members of the U B C Pulp and Paper Centre: Junqiang Fan, Shiro Ogawa, Michael Chong Ping, Kayvan Najarian, Leonardo Kammer, Zoran Nesic, Mihai Huzmezan, Ahmed Ismail, Manpreet Sidhu, Stephan Bibian, Tatjana Zikov, Setareh Aslani, and David Yang. Also I would like to gratefully acknowledge the help of the Pulp and Paper Centre staff (past and present), in particular: Brenda Dutka, Brian MacMillan, Lisa Brandly, and Ken Wong. I would have never reached graduate school if there had not been a steady and con- tinuous support of my parents, Savo and Slavojka Mijanovic, during my earlier education years. I owe a great deal to their love and support. In addition, the encouragement of my father and other immediate family from my native land, over the course of the last few years, has made my graduate school years much easier. Finally, I would like to specially acknowledge the help and support of my wife Mi la xi i Mijanovic. Her love and patience have tremendously helped me during my graduate studies over the last five years, as well as with all my endeavours since 1996. She provides a constant inspiration and feedback in my life, and helps me lead a good life. (The good life is one inspired by love and guided by knowledge - Bertrand Russell.) xiii Dedicated to the everlasting memory of my mother Chapter 1 Introduction 1.1 Industrial Paper Making - The Paper Machine Although the ancient Egyptians produced the world's first writing material, real paper making began in China about 1900 years ago. In the following centuries, it spread to Europe and North America through the Middle East. Paper continued to be made by hand until the beginning of the 19th century when the first paper machine was built in England. In the past two centuries, paper machines have been developed, making it possible to increase production, and to establish rigorous standards of quality. Today, paper making is a multi-billion dollar industry employing hundreds of thousands of people worldwide. The fibrous raw material for paper making is called pulp. Pulp fibers are usually of vegetable origin, but other types (mineral, animal, or synthetic fibres) have been used in some special applications as well [61]. The task of the paper machine is to transform a slurry of water and pulp fibres into a sheet of paper conforming to the required standards of quality. A typical fourdrinier-type paper machine is illustrated in Figure 1.1. After numerous stages of pulp pre-processing (not shown in Figure 1.1), a very dilute mixture of water and pulp fibres (approximately 99.5% water and 0.5% fibres) enters the headbox at the left of the figure. The dry paper is wound up on the reel (100-200 metres downstream from the headbox) at the right of the figure by which stage the composition has become about 95% fibre and 5% moisture. Today's paper machines are typically between 3-10m wide. The following is a very brief description of the modern paper making process; detailed descriptions can be found in [25, 61]. The overall machine is divided into four sections (Figure 1.1) [61]: • Sheet- forming section in the Wet end of the paper machine. The mixture of water and pulp fibres (with only ~0.5% fibres concentration) is delivered onto the moving forming fabric (in today's modern machines, almost exclusively, made of plastic web materials). The forming fabric is moving at speeds that could possibly be in excess of 100km/h, in the so-called machine direction (MD). By the use of various drainage elements (forming board, hydrofoil assemblies, table rolls, vacuum boxes, etc.), a significant amount of water is removed, resulting in formation of a paper sheet with ~20% fibres concentration leaving the machine's Wet end. 1 Chapter 1. Introduction 2 W E T END slice lip dilution actuators Machine Direction (MD) infrared steam boxes heaters r e W e t Cross Direction showers (CD) Sheet-forming section J L Press section — r ~ Dryer section DRY END scanning sensor induction heating J L Post-drying operations section Figure 1.1: Schematic view of the paper machine showing typical positions of the scan- ner^) and various C D actuator arrays, as well as illustrating machine and cross directions (Figure courtesy of Honeywell Process Solutions - North Vancouver). • Press section of the paper machine. In this section, the sheet is heated by steam boxes, before being pressed and further dewatered by the counter-rotating rolls, resulting in a sheet with approximately 40% fiber concentration just before the Dryer section. • In the D r y e r section, the sheet's water content is further reduced through evapo- ration, by the use of a series of large-diameter, rotating, steam filled cans. The sheet leaving the Dryer section contains only 5-9% water content by weight (91-95% fiber concentration). • P o s t - d r y i n g operat ions section is the final section of the paper machine, where the paper sheet thickness (caliper) and surface properties (e.g. gloss) are being controlled. At the end of the paper machine, which is, as pointed out earlier, located 100-200m away from the headbox, the sheet is finally wound up onto the reel. The direction of sheet travel is known in the paper making industry as the machine- direction (MD). The direction perpendicular to machine-direction is called cross-direction (CD). The machine and cross directions are illustrated in Figure 1.1. It is also important to note that paper machines usually have high-pressure water jets, Chapter 1. Introduction 3 located very near the end of the Sheet-forming section (in the machine-direction) and not far from the machine edges (in the cross-direction). These jets are called 'trim squirts' as they trim off narrow strips of the paper sheet, on both sides of the machine, before it enters Press section [61], as illustrated in Figure 1.2. WET END CD < Trim DRY END Figure 1.2: Illustration of trim squirts, used for trimming-off of narrow paper sheet strips, in the sheet-forming section. The most important paper sheet properties are: (1) the sheet weight per unit area (usually given in terms of grams per square meter - gsm), (2) the sheet moisture content (given in terms of a percentage of the sheet weight - %), and (3) the thickness (caliper) of the finished sheet (given in pm). These properties are measured by a scanning sensor, located near the end of the machine (in the Dry end), that traverses the moving sheet back and forth in the cross-direction (CD), measuring these properties at 200-2000 points (see Figure 1.3). The quality of the paper sheet is defined in terms of weight, moisture content, and caliper variations [1]. The smaller the variations are about the target values, the better the paper sheet. The scan time (the time required for the scanner to traverse the sheet once in cross- direction) depends on the width of the sheet and the scanner's speed, and is usually Chapter 1. Introduction 4 between 15 — ASseconds. If a roll of paper consists of m scans and the sheet properties are measured at n locations across the sheet, the sheet measurements are given by y(i,j), where i = 1,2,n and j = 1, 2 , m , and the average value is: = E E y(hj) = 1 j=l m • n (1.1) An important sheet quality factor is the variance of the measured property, known as 'two sigma' within the industry, defined by [1], 2a \ E E i=i j=i (y(i,j) - y)2 m • n — 1 ' (1.2) where y is given by (1.1). Since the scanning sensor traverses the moving sheet, it traces a diagonal path along the sheet (as illustrated in Figure 1.3), resulting in the measurement sheet profiles containing both MD and CD sheet variations. As a result of process design and the nature of the actuators, the industrial approach to paper machine control is to consider the MD and CD control problems separately. M D sheet travel direction C D Paper sheet Scanner head Measuring path Figure 1.3: Illustration of the scanning sensor's measuring path (Figure courtesy of J. Fan [20]). Machine-direction (MD) control deals with controlling the average value of each mea- surement scan, and the resulting MD control loops are relatively simple Single-Input- Single-Output (SISO) loops. Paper machine MD basis weight control isvusually realized by controlling the concentration of pulp fibres being delivered to the machine headbox. MD moisture control, on the other hand, is typically achieved by controlling the overall Chapter 1. Introduction 5 steam flow into the machine's dryer section. Machine-direction (MD) control is outside the scope of this work, and no M D actuators are illustrated in Figure 1.1. 1.2 Paper Machine Cross-Directional (CD) Control Systems Cross-direction (CD) control is concerned with controlling dynamically varying, but'zero- mean measurement profiles, and is performed by an array of actuators distributed across the machine (in cross-direction). In most cases, paper machines have at least one actuator array for controlling each of the important sheet properties (weight, moisture, and caliper). Except for some of the most recent approaches to C D control [6, 7, 20], different sheet properties are controlled by separate and independent control systems. The most common ways of realizing industrial paper machine C D control are detailed in Sections 1.2.1-1.2.3 below. 1.2.1 Basis Weight Control Basis weight (weight per unit area), expressed in grams per square meter (gsm) or pounds per ream (lbs/ream)1, is a fundamental property of the paper sheet, and its variations cause variations in most other sheet properties [12]. The desired target values for basis weight vary from about 35gsm for a light weight 'telephone directory grade' paper sheet, 45gsm - newsprint, 300gsm - book cover, to 450gsm for a cardboard sheet [65]. The role of C D basis weight actuators is to distribute pulp fibers evenly across the machine width (in the cross-direction) so that variability of the basis weight C D profile is minimized. Weight actuators are located at the headbox (at the far left in Figure 1.1), furthest upstream from the scanning sensor. As a-result, a.significant delay (dead time) is a dominant characteristic of industrial C D weight control systems. Two main types of actuators are used for controlling C D weight profiles: slice lip actuators and dilution actuators. The traditional way of achieving C D weight control is by the use of slice lip actuators. A mixture of water and pulp fibres exits the headbox through a gap (slice), which is between l -6cm tall depending on the type of paper being produced, and as wide as the paper machine. The slice has an adjustable top lip and a fixed apron or bottom lip. The upper lip, in addition to being adjustable up or down as a unit, can also be locally bent (moved up or down) by the use of C D actuators. The larger the actuator opening, the more fibres are delivered, thus resulting in a heavier sheet in the localized area around that actuator. As an illustration, a possible steady-state response 11 [ream] = 3000 [square feet] = > 1 [lbs/ream] = 1.6289 [gr/m2] Chapter 1. Introduction 6 of the basis weight profile to slice lip actuators is shown in Figure 1.4. The actuator 15 20 25 •ACTUATOR NUMBER 50 100 150 CD LOCATION [measurement point] Figure 1.4: Slice lip basis weight control: measured (thin line) and modelled (thick line) steady-state bump response shapes (lower figure) in the case of deflection of 3 out of 36 slice lip actuators (upper figure). Data obtained during the industrial trial described in Chapter 6. spacing (xa) and a total number of actuators (n) can vary significantly, depending on the installation, and can be anywhere between xa =70-200mm and n =30-118 or more actuators [65]. A more recent approach to C D basis weight control is by the use of dilution actuators [74], located across the back of the paper machine headbox (see Figure 1.1). These actu- ators locally alter the concentration of pulp fibres in the headbox by injecting a stream of low consistency white water. Obviously, an increase in the low consistency stream locally reduces the concentration of fibres, thus resulting in a local decrease of basis weight. The most important advantage of dilution actuators, in comparison to slice lip actuators, is a significantly smaller actuator spacing (xa = 35mm) and a narrower spatial response, which translates into better C D controllability characteristics. 1.2.2 Moisture Control The three main types of C D moisture actuators are: steam boxes, rewet showers, and infrared heaters [12]. Steam box arrays, in two possible locations, and one array of rewet Chapter 1. Introduction 7 showers as well as an array of infrared heaters, are illustrated in Figure 1.1. Steam showers improve water removal (through the principle of hot pressing) by locally increasing the sheet temperature [12, 61]. In the hot pressing procedure, steam is applied to the sheet structure, where it condenses, giving up its heat, and thus resulting in an increase of the sheet temperature. The higher sheet temperature increases water fluidity, thus making the water removal by the presses easier and more efficient. An important economic benefit resulting from the use of steam showers before or in the press section (see Figure 1.1), is a subsequent dryer load reduction. It has been found that improved water removal before the dryer section leads to overall net energy (steam) savings [12]. The location of steam shower actuators mainly depends on the type of paper being produced. In case of heavier grade papers, steam showers located on the fourdrinier forming table tend to work better, while steam showers located in the press section give better results in the case of light grade papers [12]. The number of steam box actuators can be as high as n — 171, and the actuator spacing is usually between xa =75-150mm. Steam boxes are generally slow actuators with a time constant of approximately 200-250sec [62]. In the case of the appearance of (over)dry streaks in the paper sheet, these are re- moisturized by the use of rewet showers (water sprays). Rewet shower actuators apply an atomized water spray directly to the sheet, thus increasing the moisture content in the localized area. The spacing between these actuators is usually in the range xa =70- 150mm, and the number can be as high as n = 120 in some wider machine applications. In contrast to steam boxes, rewet showers are very fast actuators with the time constant comparable to scan time (Ts = 15 — 45sec) and the dynamics is mainly dominated by the transport delay (dead time) [62]. 1.2.3 Caliper Control In the final section of the paper machine (Dry end), the paper sheet is fed to a vertical stack of rotating rollers, known as a calender stack. The rollers exert pressure onto the sheet with the objective of smoothing and evening the sheet thickness (caliper). Cross- directional caliper variations are modified by the use of CD caliper actuators which locally change the pressure exerted onto the paper sheet. While early CD caliper control was done mainly by the use of hot and cold air showers, modern CD caliper control systems almost exclusively use induction heaters. High fre- quency electric current induces eddy currents in a calender roll made of ferrous materials. Such induced eddy currents cause a local heat build up, resulting in a temperature in- crease which also causes the roll diameter to increase. This, in turn increases the pressure locally exerted onto the sheet causing the caliper (thickness) to be reduced. A decrease of the roll's diameter, achieved through temperature reduction, has clearly the opposite Chapter 1. Introduction 8 effect, i.e. the sheet's caliper will increase. The number of induction heaters, on a typical paper machine can be as high as n = 150, and the usual actuator spacing is xa — 75mm [62]. Of all the actuators considered in this chapter, induction heaters are by far the slowest. The response time generally varies from very slow to almost an integrating processes. Considering the location of inductions heaters (in the Dry end, near the scanning sensor), the delay of these systems, in contrast to weight and moisture control systems, is usually quite negligible. It is interesting to note, from a historical point of view, that the first computer controlled paper machine C D control systems were actually C D caliper control systems, bought and installed in 1973 (see graphs in Figure 612 in [12]). 1 . 3 C D P r o c e s s A t t r i b u t e s As demonstrated in Sections 1.1-1.2, paper machine C D processes are large, multivariable, spatially-distributed processes with 30-300 inputs (actuators) and 200-2000 outputs (mea- surement points). In the C D control literature, C D processes are most often considered to have separable dynamic and spatial responses [18, 20, 23, 45, 62, 65], y(z) = G(z).u(z), G(z)=g(z)-G0 (1.3) where y(z) € Cm, u(z) 6 Cn are the Z-transforms of the output (measurement) profile and the input (actuator) profile respectively (200 < m < 2000 and 30 < n < 300), g(z) is a -Z-transform of the process time response, and Go € 7Zmxn is a constant, process spatial response matrix. The temporal response g(z) is usually modelled as a stable, first-order-plus-deadtime (FOPDT) scalar transfer function, 9(*) = - i (1-4) 1 — CIQZ 1 where d is the process delay in samples and ao is a process pole (determined by the process time constant and control system's sampling time). Depending on the property being controlled and the type of actuators being used, process time constant and delay can vary significantly. As pointed out in Section 1.2, some actuators have a very fast, almost instantaneous, response (e.g. rewet showers modifying moisture profiles), while some others are extremely slow, almost integrating processes (e.g. induction heaters affect on caliper profiles). On the other hand, the process delay (dead time) d is mainly determined by the distance between the actuators and a scanning sensor and machine speed. As a result, it can vary from less than one scan (caliper control by the use of induction heating Chapter 1. Introduction 9 actuators) to up to 9 scans (basis weight control by the means of slice lip or dilution actuators). In terms of their spatial responses, modelled by the matrix Go in (1.3), C D processes can also vary significantly. Spatial response to a single actuator can be as narrow as a few centimetres (basis weight control by the use of dilution actuators) or as wide as a few metres (slice lip basis weight control on a heavy grade paper) [62]. As an illustration of the spatiotemporal nature of C D processes, a typical C D process model response, in the case of a spatial impulse and temporal step input signal (i.e. one actuator 'bumped' to a predetermined value and kept at that value), is illustrated in Figure 1.5. It is a model of basis weight response to a slice lip actuator. CD [SPACE] CD [SPACE] Figure 1.5: Illustration of the two-dimensional characteristics of the C D processes: C D process model basis weight response to a slice lip actuator. Another very important characteristic of C D process models is their inherent severe ill-conditioning [23, 33, 45, 62]. In other words, the singular values of the matrix Go in (1.3) vary significantly. The ratio between the largest and the smallest singular value (condition number 7): l(Go) = ^ » 1 , (1-5) QL(C0) and can reach into thousands. As an example, the singular values of a basis weight control process model with 36 slice lip actuators, at steady-state (cu = 0), are given in Figure 1.6. The maximum singular value, for this particular model, is 1.674 • 10~ 2, and the minimum 5.101 • 10~6, resulting in a condition number y{G(z)z=i) = 3281.71. As a result of the process model separability (1.3)—(1.4), ill-conditioning is present across all dynamic frequencies since the condition number remains unchanged (and very large) with frequency u: j(G(eju>)) = 7 (G 0 ) , 0 < u < n. Significant process model uncertainty is another important attribute of industrial C D control systems. C D process models are usually identified from noisy input/output data, and there are numerous sources of model uncertainty. Uncertainty can arise, for example, Chapter 1. Introduction 10 0.018 0.006 p 0.004 h 0.002h 0' 5 10 15 20 25 30 35 SINGULAR VALUE NUMBER Figure 1.6: Steady-state singular values of a typical CD process model. from (1) wandering of the paper web, (2) paper shrinkage characteristics, (3) flow pat- tern of extruded liquid paper stock in the initial stage of the paper making process (on the forming wire) [32]. Al l of these factors generally change with time, depend on the paper grade being made, machine speed, etc. As a result, it has become recognized, that industrial CD control systems must deal systematically with the inherent process model uncertainty [9, 10, 18, 19, 45, 62, 65, 73]. To summarize, some of the most important characteristics of the CD control systems are: large-scale (30-300 inputs and 200-2000 outputs), ill-conditioning (condition number potentially in the order of thousands across all dynamical frequencies), and a significant process model uncertainty. Some of the approaches developed for addressing these char- acteristics are outlined in Section 1.3.1 below. 1.3.1 Approaches to CD Control Major advances in the cross-directional control of web forming processes were achieved in [18]. Therein, controllability of the cross-directional processes, in terms of the spatial Fourier components of the process output y(t), was analyzed, and it was shown that, for control purposes, it is enough to identify only those spatial frequency components of the CD profile that are controllable. After this, the parametrization of CD process received a lot of attention [20, 21, 22, 38, 42, 45, 62]. Parametrization describes the compact repre- sentation of the system input, output, and the resulting interaction matrix with a certain set of basis functions. The Fourier transform [18, 20, 62], Gram polynomials [38, 42], Chapter 1. Introduction 11 and singular value decomposition (SVD) [18] represent different ways of parameterizing the CD control system. The main objective of parametrization is to identify compactly those components that can be controlled. System parametrization leads to problem size reduction, which in turn leads to a substantial computational load reduction [21, 22, 42]. In [18] for the first time, the robustness of CD control systems was thoroughly an- alyzed. In that thesis, the author gave some very important insight about the robust stability of CD control systems for web forming processes with the controller C(z) = c(z)[GlG0\-1Gl = c(z)C, where G0 € K m x n is the plant model from (1.3), the scalar transfer function c(z) contains dynamical response of the controller, and the constant ma- trix C G !f t n x m contains the spatial response of the controller. Results were compared using analysis in an artificial case of infinite width web with infinite actuator array. In both cases robustness was analyzed in cases of additive uncertainty in the process plant. In the analysis of the actual control system with controller C(z) — c(z)C, robustness in case of the process spatial response (C70) additive uncertainty A , with limited maxi- mum singular value (<r(A) < Q), was investigated. The analysis considered closed-loop singular values, in which case the analysis of the MIMO system with a given stabilizing controller can be carried out as the analysis of a set of n (n - is the number of actuators) SISO loops. It was found that robust stability can be guaranteed through detuning either the dynamic response or the spatial response of the controller. However, these methods have the drawback that all controllable modes will be detuned and not only the poorly controllable ones. Further advances in the analysis and design of robust controllers were made in [19, 45]. In [45], the paper machine CD control problem was analyzed from the robust performance point of view. The analysis was done for the case of a square interaction matrix. A continuous time model was used, separability of dynamic and spatial responses assumed, and model uncertainty Gn was expressed in terms of ranges on the elements of interaction matrix G and the parameters of the common dynamic part g(s). Two design techniques were presented, one with decentralized controller structure with C(s) = c(s)S, where S is a diagonal matrix and the other with model-inverse-based controller C(s) = c(s)G~1. The former technique can be applied only in the case of a positive definite spatial response process interaction matrix G. If G is not positive definite then the model-inverse-based controller has to be used. In case of decentralized controller design, circulant symmetric matrix theory was used to obtain bounds on the plant within which robust performance is guaranteed. Closed-loop robustness was achieved through detuning of the controller dynamics c(s). This would usually lead to a conservative control design since all the controllable modes are detuned. Main drawback of these techniques is that neither the decentralized nor model-inverse control is suitable for ill-conditioned plants. Chapter 1. Introduction 12 In [19], a robust design procedure was developed for separable controllers, given with C(s) = c(s)C. Singular value decomposition of the system was performed and the set of n SISO systems was analyzed. Analysis was done both for the case of simultaneous uncertainties in the spatial response part of the interaction matrix (limited maximum singular value uncertainty) and for the dynamical part (parameter uncertainty). Detuning of the controller in order to achieve robustness can be done either by detuning its dynamics c(s) or by detuning the pre-compensator matrix C = [GTG + Q,I\~lGT. In both cases there is a degradation of the performance of well-controllable modes of the system but this occurs to a lesser extent in the case of detuning the pre-compensator matrix C. The problems can also arise if, due to uncertainty, the sign of the process gain at high spatial frequencies changes. In that case, the closed-loop system might be destabilized. In [9] a larger class of uncertainty structures was allowed which included: additive, mul- tiplicative input, multiplicative output, inverse multiplicative input and inverse multiplica- tive output uncertainties. The analysis in case of controller structure C(s) = VT,(s)cUT, with process matrix G(s) = g(s)UY,VT = UT,(s)VT where £(s) is a matrix with pseudo singular values <Tj(s) was performed. A modification of the DK-iteration (p synthesis) algorithm was presented where K-step of the design procedure was reduced to the design of n independent robust SISO controllers, where n is the number of actuators. The major advancement of this approach is its ability to handle a larger set of model uncertainties, but the controller design technique remains fairly complex. However, further refinements of this approach were presented in [73]. Robustness of CD control systems with respect to different basis functions used for system representation was analyzed in [10]. The comparison was made between mini- mum variance controllers designed using representation based on orthogonal polynomials, Fourier methods, singular value decomposition, splines and wavelets. The comparison was performed by analyzing the condition number (ratio of maximum and minimum singular value) of the system models obtained using different methods. It was found that in all of these cases robustness is practically the same except in the case of representation with splines. For spline representation, the system was found to be significantly less robust with respect to uncertainties in actuator response shape. Therefore, the choice of system representation (except in case of representation with splines) can be left to considerations other than robustness. 1 . 4 S p a t i a l l y - D i s t r i b u t e d C o n t r o l S y s t e m s As illustrated in previous sections, paper machine CD control systems clearly belong to a broader class of spatially-distributed control systems that, in turn, form an important Chapter 1. Introduction 13 subset of large, multivariable, coupled industrial control systems. In addition to flat sheet forming industries (including paper, steel, and plastics making processes), spatially- distributed control systems arise in various other applications. For example: flow control, control of vehicular platoons, microelectromechanical systems (MEMS), space telescopes, systems described by partial differential equations with constant coefficients and spatially- distributed actuation and measurements, all belong to a set of spatially-distributed control systems. As a result of an increasing interest of the control engineering community in these systems, there have been many new tools developed recently for the analysis and controller synthesis of spatially distributed control systems [8, 15, 31, 34, 35, 46, 62, 66], some of which are discussed below. First, it should be noted that an important assumption, usually necessary for the application of the above mentioned techniques, is spatial invariance. Spatial invariance means that process dynamics are (assumed) invariant with respect to translation in some spatial coordinates(s). As a result of this assumption, the subsequent controller synthesis procedure can be significantly simplified [8]. The systems analyzed in [8] are infinite- dimensional. The analysis and controller synthesis for such systems (see for example [11]), in general, is significantly more complex than for finite-dimensional systems. How- ever, with the spatial invariance assumption, it was shown [8] that quadratically optimal controllers can be synthesized by solving a parameterized (over spatial frequency) family of finite-dimensional problems. It was also shown in [8] that controllers computed via quadratic optimal techniques, including LQR, Ti.2, and TLoo optimization, preserve the spatially invariant characteristics of the process. In addition, it was demonstrated that such optimal controllers have a degree of localization similar to that of the plant, justifying implementation of localized controllers in the case of localized processes (plants). Distributed control of spatially-invariant systems, by the use of linear matrix inequal- ities (LMIs), was investigated in [13, 15]. Therein, a state-space approach is used on continuous-time and spatially-discrete systems with spatial coordinates Si E Di, where D, is a set of integers Z or some finite set {1, 2 , N i } . As a performance criterion, the I2- induced norm is used, with the space I2 being a set of functions mapping D\ x • • • x DL to R*, where L is a number of spatial coordinates and M* is a set of real valued finite vectors. Applications of the proposed approach, based on LMIs, were presented in [14, 24]. It was demonstrated in [24], that a distributed control approach yields far superior results in comparison to the decentralized control techniques, and results comparable to centralized control techniques at a fraction of required computational time. However, very often, practical control systems (including CD control systems) are not spatially-invariant, and the above techniques can not be implemented without further modifications. Even more, because of the idealized boundary conditions (spatial invari- Chapter 1. Introduction 14 ance) assumed in the design process, there is no guarantee of performance (not even stabil- ity) around the boundaries. In other words, there exist destabilizing boundary conditions. A class of bounded, spatially distributed systems with associated boundary conditions, for which stability and performance are guaranteed after implementing a controller designed under process spatial invariance assumption is presented in [44]. The importance of boundary conditions is also very well known in the theory of partial differential equations (PDE) [68]. The class of PDEs in which boundary conditions are specifically taken into account are the so called boundary value problems (BVP). In PDE theory, there are three types of boundary conditions that are most often implemented: Dirichlet, Neumann, and Robin boundary conditions. In the case of Dirichlet boundary conditions (BC), the finite-width signals are extended with a constant (predefined) value. In the case of Neumann BC, the signal extension is defined such that its first derivative (with respect to the spatial coordinate in question) remains constant, and in the case of Robin boundary conditions such that a linear combination of the signal extension and its first derivative has a predefined value. In addition to the theory of PDEs, the importance of boundaries has also long been recognized in the field of signal processing, where the boundaries of a signal typically require modification to the filtering. Temporal (causal) filters require initial conditions to be specified, and noncausal filters (e.g. image processing) may require both initial and final conditions to be specified [16]. As illustrated in Section 1.3 above, paper machine CD processes belong to an important class of filters that are causal in one direction (time), but noncausal in space. It is illustrated in Section 2.2 below, that industrial CD controllers, of particular interest in this work, belong to the same class of spatio-temporal filters. These filters (systems) carry the additional risk of instability due to incorrect design of boundary conditions. These boundaries may be interpreted as points of discontinuity in the signal. The effect of the boundaries on the filtered signal is clearly influenced by the way the filtering is modified to handle these. 1 . 5 I n d u s t r i a l P a p e r M a c h i n e C D C o n t r o l S y s t e m s This work is concerned with modifying a particular realization of the industrial paper machine CD control law - a Honeywell CD controller - around spatial domain bound- aries. The controller structure, tuning technique, and implementation are presented in [62, 64, 65, 66]. Industrial CD control systems with the specific controller structure un- der consideration in this work are currently installed on more than 4200 paper machines worldwide. Chapter 1. Introduction 15 A simplified diagram of the industrial CD control system relevant to this work is illus- trated in Figure 1.7. Industrial CD controller software resides on a PC that is connected by network to the paper machine's actuator array and scanning sensor (the scanner is sometimes connected to the PC via serial connection). Based on the measurements y(t) obtained from the scanning system and the desired target value r(i), the controller al- gorithm generates a control signal u(t), that is subsequently sent to the actuator array. INPUT SIGNAL, u(t) LAN connection | LAN (or Serial) connection | OUTPUT SIGNAL, y(t) Figure 1.7: Simplified diagram of an industrial CD control system. A more detailed block diagram, indicating data flow in this industrial CD control system, is given in Figure 1.8. The scanning sensor measures the sheet properties at 200- 2000 points, thus generating the so called high-resolution (scanner's spatial resolution) vector profile yHR.{t)- After the each scan, this vector of measurements is delivered to the industrial controller's measurement processing section. The measured signal is then dynamically filtered in order to separate machine-direction (MD) and cross-direction (CD) components of the profile. As pointed out earlier, the MD component is controlled by a separate control loop, not illustrated in Figures 1.7-1.8. Since the M D / C D separation involves the use of a dynamic filter, it clearly introduces additional dynamics into the CD control loop. As the number of CD actuators (30-300) is usually ~3-10 times smaller than the number of measurement points, the high-resolution CD profile is spatially low-pass filtered (spatial anti-alias filtering), before being downsampled to the actuator resolution (spatial low-resolution) yz,fi(0- Finally, the error between the measurement profile and the desired profile is passed to the linear time-invariant CD controller algorithm. The industrial linear time-invariant CD controllers are essentially 2D (spatio-temporal) Chapter 1. Introduction 16 eLR(t) Linear time-invariant CD controller Decoupling Dahlin _ w Setpoint Actuator Actuator window controller smoothing setpoint maintenance array 7t6 Paper machine Spatial Spatial MD/CD Scanning downsampling antialias filter separation i YHRW sensor Measurement processing Figure 1.8: Data flow in an industrial CD control system; HR: High-Resolution (Scanner spatial resolution) LR: Low-Resolution (Actuator spatial resolution). filters, causal in time and noncausal in the spatial domain [31, 50, 66]. They consist of three blocks (1) a spatial decoupling algorithm, (2) a Dahlin controller, and (3) a set- point smoothing algorithm, connected in cascade as illustrated in Figure 1.8. In current industrial practice, these blocks (i.e. CD controller) are most often tuned using a two- dimensional loop shaping technique [62, 65, 66]. A spatial invariance approximation is central to this technique, which is an extension of traditional loop shaping and addresses the closed-loop performance and robust stability criteria in terms of the spatial and tem- poral frequency domains. The output of the linear time-invariant CD controller passes through the Actuator setpoint maintenance section, a nonlinear block that does constant checking/verification to ensure the setpoints do not violate physical constraints, before the setpoints are finally delivered to the actuator array. 1.5.1 P r o b l e m s N e a r the Sheet Edges (Spat ia l D o m a i n B o u n d - aries) The two-dimensional loop shaping technique has been successfully implemented on more than 100 paper machines worldwide to date. However, as the process characteristics near the sheet edges are clearly different from those in the centre of the sheet, the spatially- invariant CD controllers (spatio-temporal filters) are modified at the edges on a real paper machine [50]. The current industry practice uses methods (that extend the signal beyond Chapter 1. Introduction 17 the sheet edges) based on the techniques in the field of signal processing. These will be described in detail in Section 2.1 and include: zero padding, signal average padding, and reflection padding [67]. Unfortunately, often these approximations then lead to unsatis- factory control at the edges, and control systems can even be destabilized [50, 63]. One such example is illustrated in Figure 1.9. a a - i | g i i * Iffm |B|m i|ini,.. arm | •»». | »c&..| j j t i . l >» Igw. IB-* I H ^ l !!«• I >.»C.B6<SP MBW Figure 1.9: Illustration of the problems ('actuator picketing' in the lower portion of the screenshot) that often occur when implementing current CD control techniques near the sheet edges. Figure 1.9 is a screenshot of the Honeywell hard ware-in-the-loop paper machine simu- lator. It illustrates the behavior of a simulated CD moisture control system, with process output (CD moisture) and actuator array profiles shown. It can be seen that the process output profile is very good (fiat). However, the actuator array, while being very-well be- haved (smooth) away from the edges, shows clear signs of instability near the edges. The edge actuators are moving in the opposite directions, developing a well-known mode of instability referred to as 'actuator picketing' by paper makers, because of the picket fence appearance of the actuator profile. Such a control signal is most likely not reducing paper sheet profile variations as CD processes are essentially spatial low-pass filters [17, 18], but it is also indicative of a low stability margin on a(Tuci) - the maximum singular value of the closed-loop transfer matrix gain from the process output disturbance d to the control signal u. Chapter 1. Introduction 18 'Actuator picketing', originating from the edges, often 'creeps' into the rest of the system, finally destabilizing the whole control system. In order to prevent instability, in practice the edge actuators of the paper machine CD control systems very often are placed in open-loop [63]. 1 . 6 A i m s a n d C o n t r i b u t i o n s o f t h e W o r k The spatial invariance assumption is central to the two-dimensional loop shaping tech- nique, used for tuning the industrial paper machine cross-directional control systems. As a result of the assumed periodic (idealized) boundary conditions, the control laws are modified near the spatial domain boundaries (paper sheet edges) before implementation on a real paper machine. The current industrial techniques for extending the finite-width signals are borrowed from the field of signal processing, and do not take into account rele- vant control engineering criteria or physical reality. As a result CD controller performance near the edges can be very poor. The objective of this work is to modify the existing industrial CD control law, ini- tially designed with the process spatial-invariance assumption, so that important control engineering criteria, such as: closed-loop stability, performance, and robustness are taken into account. At the same time, the structure and complexity of the existing control law are to be unchanged. The main contributions of this work are: 1. The introduction of a straightforward technique for achieving a stabilizing controller for a known plant by modifying a controller that is known to stabilize a second, related plant. The technique can be applied to a broad class of systems with mul- tivariable linear time-invariant transfer matrix models and controllers, including industrial paper machine cross-directional control systems. Although the main ob- jective of this technique is closed-loop stability transfer, in certain circumstances, a successful controller design in terms of closed-loop performance can also be achieved. 2. The development of a novel stability-preserving method for modifying boundary conditions (BCs) of a spatially-distributed controller, initially computed assuming idealized spatially-invariant BCs. An analogy between the effects observed when implementing such controllers, with various boundary conditions, on the actual (i.e. finite, non-periodic) processes and the well-known Gibbs phenomenon is observed. Stability-preserving modifications, based on a method for reducing the Gibbs effect, are developed. The technique is demonstrated in the case of the industrial paper machine CD controller. Chapter 1. Introduction 19 3. A new approach to the redesign of industrial paper machine cross-directional (CD) controllers near spatial domain boundaries is presented. The method directly takes into account relevant control engineering criteria, such as closed-loop stability, per- formance, and robustness. The new approach modifies the CD control law near the paper sheet edges (spatial domain boundaries) by sequentially applying a novel low-bandwidth static output feedback design algorithm on two matrix components of the existing industrial controller. The existing industrial CD controller structure or complexity are not changed with the new approach. 4. A successful implementation of the above approach has been tested on an industrial paper machine. It is demonstrated that with the new approach a trade-off between the final product quality (paper sheet smoothness) and controller aggressiveness can be achieved. When compared against the existing industrial practice, the imple- mentation of the new technique resulted in improved paper sheet quality and lower actuator usage (i.e. smaller control signal magnitudes). 1 . 7 T h e s i s O v e r v i e w The remainder of the thesis is organized as follows. The concept of explicit and implicit boundary conditions (BCs), in terms of spatio-temporal filters of interest in this work, is introduced in Chapter 2. The spatio-temporal filters under consideration are the key element in defining paper machine cross-directional (CD) process and controller models. The main characteristics of industrial paper machine CD control systems and the two- dimensional loop shaping technique (tuning tool for the industrial CD controllers) are also presented in Chapter 2. The objective of this work is specified at the end of this chapter. In Chapter 3, a novel and straightforward closed-loop stability preserving perturba- tion technique, for CD controllers initially designed under the idealized process spatial- invariance assumption, is presented. The technique is simple and requires no additional computation on the part of the designer, and, as illustrated in Chapter 3, can be imple- mented on a broad class of linear time-invariant systems with known plant perturbations. However, in terms of the objectives of this work, performance requirements other than closed-loop stability are not addressed with this approach. The similarities between the effects observed in the industrial paper machine CD con- trol systems and the well-known Gibbs phenomenon in Fourier analysis are summarized in Chapter 4. Next, CD controller modification technique, inspired by a method for reduc- ing the Gibbs effect is presented. The proposed technique guarantees the stability of the resulting CD controller. A closed-loop simulation example of this controller modification technique is also presented in Chapter 4. Chapter 1. Introduction 2 0 In Chapter 5, the objective of modifying paper machine CD control law is stated in terms of a block-decentralized static output feedback (SOF) design problem via appro- priately defined linear fractional transformation (LFT). The subsequent design approach is based on a novel low-bandwidth static output feedback design algorithm, sequentially implemented on two constant matrix components of the existing industrial CD controller. This approach systematically takes into account all objectives of this work, and important control engineering criteria: closed-loop stability, performance, and robustness. At the end of this chapter, a closed-loop simulation example with the Honeywell hardware-in-the-loop paper machine simulator is presented. The new approach to CD control near the paper sheet edges, presented in Chapter 5, has been tested on a paper machine in a working paper mill. The industrial trial procedure and results are given in Chapter 6. Three different sets of the CD control law modifications (conservative, balanced, and aggressive) were computed and implemented. The results are then compared with the results achieved by current industrial practice. Finally, conclusions and suggestions for future research are given in Chapter 7. Chapter 2 Problem Statement Considering the two-dimensional nature of CD control systems, CD controllers (of interest in this work) and CD processes can be viewed as 2-D (spatiotemporal) filters. Further tak- ing into account the main focus of the thesis (CD control near spatial domain boundaries), boundary conditions (BC) of such filters are clearly of particular importance here. Section 2.1 introduces the concept of explicit and implicit boundary conditions in terms of the class of spatiotemporal filters that are the basic building block for the paper machine cross-directional process and controller models of interest in this work. Industrial CD process and controller models under consideration are detailed in Section 2.2. The main characteristics and tasks of the industrial cross-directional control systems, including an overview of the CD controller tuning method (two-dimensional loop shaping technique [65, 66]), are also presented in Section 2.2. Finally, the objective of this work is specified in Section 2.3. 2.1 E x p l i c i t a n d I m p l i c i t B o u n d a r y C o n d i t i o n s A well known problem in engineering is the filtering of signals with discontinuities. We can find examples of this problem related to temporal as well as spatial filters. In the temporal domain, it is well known that it is of particular importance to handle sudden changes of the variables that define the state of such systems. The transients that occur as a consequence of those abrupt changes (i.e. discontinuities) can pose significant problems, and engineering systems have to be able to withstand such transients. An analogous effect occurs with spatial filters. The unwanted consequences of such filtering are spatial ripples (equivalent to temporal transients), for example edge blurring in image processing [28]. The importance of spatial filters' ability to handle input signal discontinuities is even greater in cases where spatial filters are placed in a feedback loop (see Figure 2.4). In such cases, the unwanted effects of signals' discontinuities can be magnified as the filtered discontinuous signal is being brought back to the filter's input. In order to place our problem in a familiar context, we will consider the issue of spatial boundary conditions as appearing in partial differential equations and image processing. 21 Chapter 2. Problem Statement 22 Consider the update equation of a two-dimensional signal, ih y(i,k + 1) = ^2 hi • y(i + j, k),i E ft = {i : 1 < i < n} (2.1) j=-ih where y 6 7Z (7Z - set of real numbers), and (2.1) represents the implementation of a non-causal FIR filter in the first index (spatial dimension), and a causal, first-order IIR filter in the second index (temporal dimension), with scalar filter coefficients hj G 7Z, and j = —lh,. •. ,lh- We will consider filtering of a signal y(i,k) that is defined on a finite spatial domain consisting of n discrete locations i — 1,..., n. Immediately it can be seen that the relation (2.1) is incomplete as it requires information from y(i, k) on a boundary layer 50. = {i : 1 - lh < i < 0 U n + 1 < i < n + lh) (2.2) as illustrated in Figure 2.1. 3 0 I o o o o o _L_ • X X X • • • • • • • • o o o o o o J I I I I L_ • • O • • O • • o • • o o o o _! I l_ 1 2 n Figure 2.1: The template structure and explicit boundary layer 8Q (denoted by o for i = 0 and i = n + 1) of a spatiotemporal filter with lh = 1 in (2.1)-(2.3). (The row of o at k = -1 indicates the initial conditions of the causal filter and are not important for the case being considered.) The need for such auxiliary conditions arises in numerical solutions to partial dif- ferential equations [68] and image processing [16]. The three most important kinds of boundary conditions in PDEs are Dirichlet, Neumann, and Robin conditions [68]. In the case of Dirichlet boundary conditions (BC), the finite-width signals are extended with a constant (predefined) value, while in the case of Neumann BC, the signal extension is defined such that its first derivative (with respect to the spatial coordinate in question) remains constant. Finally, in the case of Robin boundary conditions a linear combination of the signal extension and its first derivative has a predefined value. Image processing [67] provides another set of filter modifications for boundary conditions, often referred Chapter 2. Problem Statement 23 to as edge padding. Common padding techniques include reflection, zero, average, and periodic padding. Each of the above boundary conditions may be represented by creating an explicit boundary layer on 50 by executing: n y(i,k + l) = ^2gij-yU,k+l) + w(i,k-rl), ie50 (2.3) 3 = 1 with corresponding scalar constants g^, exogenous signal w(i,k), and 50 as defined in (2.2), following the update in (2.1). However, as we are concerned only with the values of y(i,k) within the spatial domain for i 6 0 we can eliminate the explicit boundary layer by solving (2.1) and (2.3) obtaining1 the implicit form n y(i,k + l) = J2hv-yti,k), i£0 (2.4) 3 = 1 As a result, the relation (2.1) along with any one of the boundary conditions is imple- mentable with the matrix equation, Y(k + 1) = H-Y(k), Y(k) = [y(l,k),...,y(n,k)}T (2.5) where H £ fZnxn [s a constant matrix whose elements are given by { hj-i, \j-i\<lh and lh + l < i < n - l h 0, \j — i\>lh and lh + l < i < n — lh (2-6) 5hij, 1 < i < lh and n — lh + 1 < i < n where j ' = 1, • • • ,n, and the coefficients 5hij determine the implicit boundary conditions. Examples are presented in Table 2.1 obtained by solving in (2.4) for the common BCs (assuming a spatially symmetric filter in (2.1) with lh = 1, i.e., hj = h-\).2 It is illustrated in Section 2.2 that both CD process and industrial CD controller represent a slight generalization of the system given by (2.5). Since two cases of BCs will be used often throughout the thesis, we introduce a short- hand notation. The matrix H in (2.5) defining a spatial filter with Dirichlet BCs, corre- 1 We are considering the homogeneous form of each, as it is general enough to include all the cases of interest in CD control. 2 The parameter b that appears in Table 2.1 in case of Robin boundary conditions is a predefined function of spatial and temporal domains. Chapter 2. Problem Statement 24 Boundary Cond. Coefficients bh^ in (2.6) 1. Dirichlet 5hu = ho 5h\2 = h\ 2. Neumann 5h\\ = ho + hi 5hi2 = hi 3. Robin 5hn = ho + b • hi 6hi2 = hi 4. Reflection Shn = h0 5. Zero 5h\\ — ho Shi2 = hi 6. Average Shn=ho + ^ 5h12 = hi + ^ 5hlj = ^ 7. Periodic 8h\\ — ho 5hi2 = hi 5hin = hi Table 2.1: Matrix coefficients in (2.6) resulting from the representation of spatial filters (order lh = 1) with various boundary conditions. sponding to a Toeplitz matrix Ht, given by T(h, n) = Hd := •j-i, -h < j - i < h otherwise (2.7) with h = [h-ih, • • • ,hih]. The matrix arising from the imposition of periodic BCs on a spatial filter, corresponding to a circulant matrix Hc, given by C(h,n) = Hc:= I hj—i, hj—i—ni hj—i+m 0, -k< j -i < k n-lh< j - i j - i < -(n - lh) otherwise (2.8) The non-zero elements of the Toeplitz and circulant matrices Ht and Hc, as well as the difference between the two AH Figure 2.2. Hc — Ht, in case n = 36 and lh — 4 are illustrated in Figure 2.2: The non-zero elements of the matrix Hd in (2.7) (a); matrix Hc in (2.8) (b); and the difference AH = Hc — Hd (c). Chapter 2. Problem Statement 25 Boundary Conditions Stability 1. Dirichlet Stable 2. Neumann Marginally stable 3. Robin (with 6 = 1.1) Unstable 4. Reflection Stable 5. Zero Stable 6. Average Marginally stable 7. Periodic Marginally stable Table 2.2: Stability of the system in (2.5) with lh = 1, n = 20, and filter coefficients ho = 0.8, hi — 0.1, in case of various boundary conditions in Table 2.1. The stability of the system given in (2.5) is completely determined by the eigenvalues of the matrix H. The system in (2.5) is stable (marginally stable) if and only if all the eigenvalues of H are in the open (closed) unit circle. The implementation of the various boundary conditions in (2.6) requires modification of the first and last lh rows of H in (2.5). As such, it affects the locations of the eigenvalues and possibly the stability of the system. An illustrative example, in case of filter in (2.1) with lh = 1, n = 20, and filter coefficients ho = 0.8, hi = 0.1, is presented in Table 2.2, demonstrating that the BCs of the filter can influence the stability of the system given in (2.5). 2.2 P a p e r M a c h i n e C D C o n t r o l S y s t e m Paper machine CD control systems usually have a significantly greater number of mea- surements (200-2000) than the number of actuators (30-300). In industrial systems the measurement array is typically low-pass filtered and downsampled to the number of ac- tuators (see Figure 1.8 in Section 1.5) thus permitting the use of square transfer matrix models. The standard model of a paper machine CD control system shown in Figure 2.3, subject to process output disturbances, is given by: y(z) = G(z)-u(z)+d(z), u(z) = K(z)-y(z) (2.9) where y(z),u(z) G Cn (C - set of complex numbers) are the 2-transforms of the output (measurement) profile and the input (actuator) profile respectively, and d(z) € Cn is the 2-transform of the process output disturbance. The objective of the CD controller K(z) G Cnxn is rejection of disturbances d(z). The transfer matrix G(z) G Cnxn can be written as: G(z) = [I- Az~lYxBz-d (2.10) C h a p t e r 2. P r o b l e m S t a t e m e n t 26 O(z) >6 d(z) y(z) Figure 2.3: The industrial CD control system. where the constant matrices A and B G 1Znxn represent spatial filters with Dirichlet BCs (2.7): A d = T ( a , n ) B d = T ( b , n ) a = a 0 b= [b-tb,--- A l (2-H) where the coefficients modelling the spatial response [ b ~ i b , • • • , bib], the discrete time pole a 0 , and the process model delay d are identified from input-output data, e.g. using com- mercial software described in [32]. Typically the paper sheet response to an actuator is symmetric with = bj in (2.11) and much narrower than the paper sheet so that lb <C n . This structure and the use of a band-diagonal Toeplitz matrix B d in (2.10) is standard in the modelling of CD processes [23]. The structure of the industrial CD controller of interest in this work (illustrated in Fig- ure 2.4) has been obtained through the years of theoretical analysis and industrial testing. Presently, there are more than 4200 installations of this particular controller perform- ing CD control on paper machines worldwide. The controller structure, detailed below, represents a specific realization of the L i n e a r t i m e - i n v a r i a n t C D c o n t r o l l e r , illustrated in Figure 1.8 in Section 1.5, with its three distinctive sections (Decoupling window, Dahlin controller, and the Setpoint smoothing section) connected in cascade. Decoupling window is represented with a static matrix, Dahlin controller with a scalar transfer function, and a Setpoint smoothing section with a static matrix in a local dynamic feedback loop. The industrial controller K ( z ) in (2.9) of interest in this work (and illustrated in Figure 2.4) is given by [49, 50, 52, 53, 62, 63, 64, 65]: K ( z ) = [ I - D z - 1 } - l D - C - c ( z )  1 (2.12) Chapter 2. Problem Statement 27 Decoupling window Dahlin controller Setpoint smoothing Figure 2.4: The industrial CD controller structure. (Compare with Linear time-invariant CD controller in Figure 1.8) where the matrices (spatial filters) C and D G TZnxn are defined as Cij Cj-i, \ j - i\<lc and lc + l<i<n — nci 0, \ j ~i\> lc and ncj + 1 < i < n — nci Scij, 1 < i < nci and n — nc\ + 1 < i < n \ j — i \< Id and rapi + 1 < i < n — nm \ j - i\> Id and no\ + 1 < i < n — npj 1 < i < n£,\ and n — UD\ + 1 < i < n and j = 1, • • • , n. The coefficients collected in: (2.13) (2.14) c = c. (2.15) with c_/t = Cfc for fc = 1, • • • , / c and d_j = dj for j = 1, • • • , l& are determined by controller tuning. Typically, the elements dj are greater than zero, i.e. d > 0. The size of the implicit boundary layer must be at least as large as the corresponding filter order (nci > lc and n>D\ > Id)- The scalar transfer function c(z) in (2.12) is a standard deadtime compensator known as the Dahlin controller in the process industries [62], given by: c(z) = (1 - a)(l - aoz-1) ^d-l ( l -ao)[l + ( l - a ) E L i z ~ (2.16) where a 0 and d are process dynamic parameters, defined in (2.10)—(2.11), and a € [0,1] is a closed-loop control system tuning variable. Chapter 2. Problem Statement 28 2.2.1 Industrial CD Controller Tuning Technique: Two-Dimensional Loop Shaping State-of-the-art method for tuning the above industrial CD controller (i.e. determining the values for c, d, and a in (2.15)—(2.16)) is the two-dimensional loop shaping technique [62, 65, 66]. Spatial invariance assumption is central to this technique, as detailed below. The two-dimensional loop shaping approach to CD controller tuning is as follows. First, the boundary conditions of the process model are changed from Dirichlet to periodic by substituting B = Bc in (2.10), where • • Bc = C(b, n) (2.17) is a symmetric circulant matrix in (2.8). This imposes spatial shift invariance on the plant model G(z) in (2.9). A CD controller is then synthesized based on the spatially invariant plant model using the two-dimensional loop shaping technique described in [62, 66]. This generates the coefficients c and d in (2.15) for the matrices C and D in (2.12) with periodic BCs, Cc = C(c,n), Dc = C(d,n) (2.18) This design technique results in a stable controller, stable closed-loop, and desired closed- loop performance assuming all boundary conditions are periodic. Such an idealized cross- directional system is illustrated in Figure 2.5. c(z) u(z). 1-%Z~ d(z) y(z) Figure 2.5: Idealized cross-directional control system with periodic boundary conditions. However, the controller must be implemented on the system modeled with Dirichlet BCs given by B = Bd in (2.11). From (2.10)-(2.11) and (2.17), it can easily be seen that the difference between the true (Toeplitz symmetric) and idealized (circulant symmetric) CD process models is given by: AG(z) = T[C(6,n) - T(b,n)} = Gc(z) - Gd(z), (2.19) Chapter 2. Problem Statement 29 and contains the 'ears' of the circulant symmetric transfer matrix Gc(z). (Figure 2.2 illustrates the location of non-zero entries of the corresponding Toeplitz and circulant matrices as well as the difference between the two.) Similarly, the difference between the controller resulting from periodic and Toeplitz symmetric matrices is given by, AC = C{c,n)-T(c,n), AD = C(d, n) - T(d, n), (2.20) and contains the 'ears' of the controller circulant symmetric matrices Cc and Dc. Current industrial practice for modifying idealized CD control law, given with (2.12) and (2.18), before implementation on a real paper machine, consists of replacing con- troller's idealized (periodic) BCs with the corresponding Dirichlet, Average, or Reflection boundary conditions. As these techniques do not take into account relevant closed-loop control engineering criteria, they can lead to poor control (even instability) near spatial domain boundaries (paper sheet edges). One such example was illustrated in Figure 1.9 in Section 1.5.1. 2.3 Objective of the Work Briefly stated, the objective of this work is to modify the existing CD control law - initially computed assuming process spatial invariance using a two-dimensional loop shaping tool - near spatial domain boundaries in a way that satisfies controller performance requirements. Let us define the modifications to existing controller matrices in (2.12) in terms of additive matrix perturbations SC,SD G TZnxn in Figure 2.6. The elements of 5C and 5D are given by, nc\ + 1 < i < n nC2 + 1 < j < n (2.21) ! 5dij, 1 < i < noi and n — noi + 1 <i <n 1 < j < no2 and n — nm + 1 < j < n (2.22) 0, otherwise with lc < nci < n/2, Id < noi < n/2, and 1 < nc2,nD2 < n. However, normally nci: nC2, n,D\, n£>2 <C n, resulting in only upper-left and lower-right corners of the matrices 5C and 5D being different from zero. The parameters lc and ld are the respective widths of the matrix bands of C and D in (2.13)-(2.14). Also, nci and n^i represent the length of the implicit boundary layers of the spatial filters C and D respectively. It should be noted fi<kj > 1 < i < ^ c i and n • [5C]ij = | 1 < j < nc2 and n • 0, otherwise Chapter 2. Problem Statement 30 yc sc yD SD -*cA c(z) l+Q D r>Ch G(z) P(z) Figure 2.6: CD control system with control law modifications, SC and SD, near spatial domain boundaries. that the existing industrial controller modifications near paper sheet edges (based on zero, average, and reflection padding) can be represented with the perturbation matrices SC and SD in (2.21)-(2.22), with nci = lc, r̂>i = 1 < nc2,nr)2 < n. By factoring out controller perturbations SC and SD as shown in Figure 2.6, a lower linear fractional transformation (LFT) as illustrated in Figure 2.7, can be defined. The generalized plant P(z) in Figure 2.7 consists of the closed-loop transfer functions that can be obtained, after some straightforward algebra, from the system shown in Figure 2.6. Figure 2.7: Problem reformulated in terms of linear fractional transformation (LFT). As pointed out in Section 2.2, the modifications SC and SD currently used in industrial CD control systems do not take into account relevant control engineering criteria and can lead to very poor control near spatial domain boundaries. The objective of this work is to find a compensator E in Figure 2.7, such that: 1. 5D,SC € 1Znxn are static matrices with nci , nc2, nrn, ^ D 2 < f i i n (2.21) and (2.22). Chapter 2. Problem Statement 31 2. The resulting closed-loop system in Figure 2.7 is stable. 3. The closed-loop performance of the system, as measured by the 2-norm of the process output vector at low frequencies, is improved: \\y(e^,E)\\2 < ||y(e^,0)|| 2, V \u\ < oob, (2.23) for some Uh > 0. 4. The gain M(e J W , £") : d —> u is limited across all the frequencies, i.e weight W{e^): a(M(ej",E)) < W(ejuJ), Vw, where CT(-) denotes the maximum singular value. The first requirement above is a consequence of the main objective of this work: designing a localized modification of the existing industrial control law near the paper sheet edges without changing the controller structure. The need for the second requirement is obvious. The third requirement is in accordance with the main objective of CD control: the reduc- tion of process output variations as measured by their 2-norm [1]. The fourth requirement is a result of the desired limit on control action, so that robustness characteristics of the system (as measured by \\K[I — GK]-1^) are preserved. In the two-dimensional loop shaping procedure, the process model uncertainty is modelled as additive uncertainty, and consequently \\K[I — G i ^ ] - 1 ^ is a measure of system robustness. for a given (2.24) Chapter 3 A Closed-Loop Stability Transfer Between Systems In this chapter, a novel and straightforward perturbation technique is developed for mod- ifying the spatially-invariant CD controller, in a manner that is guaranteed to stabilize a closed-loop control system with the true (spatially-variant) CD process model. The tech- nique is based on the known difference between the idealized (spatially-invariant) and the true (spatially-variant) paper machine cross-directional process models and requires no further computation on the part of the designer. Closed-loop stability (requirement 2 in Section 2.3) is maintained in an efficient and straightforward manner with this technique, however, the other requirements from Section 2.3 are not considered at this stage. It was demonstrated in [51] that this technique can also be implemented on a broad class of linear control systems with known plant perturbations. In Section 3.2, the set of transfer matrix models under consideration in this chapter is presented. A straightfor- ward controller perturbation technique, guaranteeing closed-loop stability for the class of systems in question, is given in Section 3.3. The application of the technique to industrial paper machine cross-directional control systems is illustrated in Section 3.4. Another three examples of the use of the presented technique, taken from very different applications, are demonstrated in Section 3.5. In that regard, Section 3.5 strays from the main object of this work - paper machine CD control near the sheet edges - but examines some familiar controller design techniques in terms of the controller perturbation technique from Section 3.3. 3 . 1 C o n t r o l S y s t e m s w i t h K n o w n P l a n t D e v i a t i o n s A control engineer is often faced with implementing a controller for a plant Gp that is different from the plant Go for which a feedback controller KQ has been designed. The deviation of the true plant from its mathematical model can be separated into two categories - unknown and known. While a practical control system will often contain both types of deviations, this chapter focuses on the known deviations. It is not uncommon in control engineering for the plant model used for controller synthesis to differ from the true plant by a known amount. In many cases, the use of all knowledge of a plant model may overly complicate the synthesis of the feedback controller. 32 Chapter 3. A Closed-Loop Stability Transfer Between Systems 33 Figure 3.1: (a) Original and (b) modified closed-loop control systems. Deliberate model simplifications, prior to controller design occur, for example: • to replace spatially-distributed plant models with more convenient spatially-invariant models [8, 15, 17, 34, 45, 49, 65, 66], • to eliminate time delay from plant models [58, 59, 60], • to eliminate the recycle dynamics in certain chemical processes [57, 70], • to remove complicated high-order dynamics [58], • to replace a multivariable process model with a diagonal process model in order to facilitate the design of a decentralized controller [54, 58, 75]. Deliberate model approximation is not the only source of known faults in a plant model. Factors whose influence is potentially known include: • failure of actuators or sensors [37, 40, 58, 77], • multiple models for various operating points [29, 56], • model changes resulting from re-identification of all or part of the transfer matrix. While there already exist various control strategies for many of the examples listed above (see, for example, references in [51]), the contribution of this work is a straightfor- ward perturbation technique for the modification of a controller KQ, originally designed for G 0 , such that closed-loop stability is guaranteed for the known plant Gp. 3 . 2 R e l a t i o n s h i p s B e t w e e n P l a n t M o d e l s In this section we present the class of transfer matrix models under consideration. We consider relationships between transfer matrix models in six standard configurations taken from [58]. Let the original or nominal plant model be the linear, time-invariant transfer matrix Go 6 cmxn. Consider a linear, time-invariant transfer matrix KQ 6 Qnxm r ep r esenting Chapter 3. A Closed-Loop Stability Transfer Between Systems 34 (a) (b) -H Af -> G0 (d) O- (e) Figure 3.2: Block diagrams for various Gp in terms of Go and A G . (a) Additive perturba- tion, (b) Inverse additive perturbation, (c) Multiplicative input perturbation, (d) Inverse multiplicative input perturbation, (e) Multiplicative output perturbation, (f) Inverse mul- tiplicative output perturbation (compare Figure 8.5 in [58]). a feedback controller such that the closed-loop system in the configuration illustrated in Figure 3.1a is internally stable. Now denote a second linear, time-in variant transfer matrix plant model by Gp 6 Cmxn where Gp ^ Go- In general, there are many ways to represent the difference between two matrices, but here we will restrict the study to six standard configurations, illustrated in Figure 3.2: (a). GP = G 0 + A G (b) GP = ( I - G „ A G ) - (c) GP = G 0 ( / + A G ) (d) GP = G 0 ( I - A G ) - (e) GP = (/ + A G ) G 0 (/) GP = ( J - A G ) " 1 G L G n (3.1) Although the presentation of Figure 3.2 is reminiscent of the representation of model uncertainty (see for example [58, 76]), for the purposes of this work we will assume full knowledge of the perturbation A G . The problem at hand is to replace the original feedback controller KQ with a feedback Chapter 3. A Closed-Loop Stability Transfer Between Systems 35 controller Kp, resulting in the closed-loop control system illustrated in Figure 3.1b, such that: 1. Internal stability is guaranteed for the closed-loop system defined by Kp and Gp. 2. Modifications to the existing controller KQ are straightforward and require less work than would a full redesign of Kp for Gp. Section 3.3 presents results that may be used for modifying the original feedback controller K0 to achieve a new controller Kp such that the closed-loop system in Figure 3.1b is internally stable. 3.3 A u g m e n t a t i o n o f F e e d b a c k C o n t r o l l e r s In this section we present a controller perturbation technique that allows the straight- forward generation of a stabilizing controller Kp for Gp in Figure 3.1b, given a known stabilizing controller K0 for the plant Go in Figure 3.1a. Theorem 1 (Stability Transfer) If the system in Figure 3.1a with (Go, K0) is internally stable, then the system in Figure 3.1b with (Gp,Kp) is internally stable if is stable, AK = —AQ, and (a) KP = (I - KOAK)-1^; GP = G 0 + A G (b) KP = K0 + AK; GP = (/ - G 0 A G ) - 1 G 0 (c) KP = {I-AK)-LKQ- GP = G 0 ( / + A G ) with stable (I + A G ) _ 1 (d) KP = (I + AK)K0; GP = G 0 ( / - A G ) - 1 with stable (I — A G ) _ 1 (e) KP — K0(I - AK)*1; GP = (/ + A G ) G 0 with stable (I + A G ) _ 1 (/) KP = K0(I + AK); GP = (I - A G ) - J G 0 with stable (I — A G ) _ 1 Proof. Given in Appendix A. Remark 1: The only restriction placed on the design of the controller KQ is that it stabilizes Go in Figure 3.1a. Theorem 1 then transfers only the stability to the per- turbed system in Figure 3.1b. The issue of closed-loop performance in either system is not addressed by Theorem 1. Remark 2: In [71, 72], the nominal plant-controller pair has been written in terms of stable coprime factors as Go = ND~l and KQ = XY~l. Subsequently, it has been shown that the family of controllers Kp = [X + DQ]\Y — NQ\~l stabilizes the set of perturbed plants Gp = [N + YS][D — XS}-1 if and only if the perturbations (S, Q) form a closed-loop stable pair. In light of this result, Theorem 1 is equivalent to providing - Chapter 3. A Closed-Loop Stability Transfer Between Systems 36 (a). Kg h~CH~ (b) X A G « G0 ~ r K0 « AK « G0 ! Ir (e). >C^rn K„ k n n f ! (f). i rL=Gj r^\~Go~U<)— KAf<-C>4 Figure 3.3: Block diagrams indicating the various configurations of Gp and Kp described in Theorem 1. without any additional computation - a controller perturbation Q that stabilizes the given plant perturbation S for each of the cases (a)-(f). So closed-loop stability is recovered by providing a stabilizing design for the problem in [71, 72] and without the use of the restrictive assumption that (Gp,K0) is a stable pair as in the double-Youla formulation [3]. On the other hand [3] and [71, 72] provide an additional degree of freedom to allow design for performance, while our proposed technique does not. Remark 3: The additional requirement on stability of (I + A G ) - 1 or (I — A Q ) - 1 in the cases (c)-(f) in Theorem 1 is necessary to avoid cancellation of unstable modes between the plant and controller. Remark 4: Theorem 1 does not necessarily require stability of the transfer matrices Go, Gp, KQ, or Kp. However the various restrictions on the stability of A G , (I + A c ) - 1 , Chapter 3. A Closed-Loop Stability Transfer Between Systems 37 and/or (I — A G ) - 1 may indirectly place stability restrictions on Kp or Gp. Remark 5: The closed-loop systems corresponding to each of the cases in Theorem 1 are illustrated in Figure 3.3. There is an easily-recognized pattern relating the configu- ration of Kp (expressed in terms of KQ and A K ) to the configuration oi Gp (expressed in terms of Go and A G ) . This duality arises from the fact that the blocks are configured so that for Ax = — A G , the controller perturbation will partially 'cancel' the plant pertur- bation. As a result, some of the closed-loop transfer functions of the perturbed systems will be identical to the corresponding closed-loop transfer functions of the original systems (see Table 3.1). As pointed out in Remark 1 above, the issue of closed-loop performance is not addressed by Theorem 1. However, in certain circumstances this technique may also achieve a successful controller design in terms of closed-loop performance. For example, the Smith predictor [54, 59, 60] and the recycle compensator [57, 70] are shown to be special cases of this method that have been applied in industrial situations without a need for further modification. More details about various applications of Theorem 1 can be found in [51] and its references. Table 3.1: Closed-loop transfer functions in Figure 3.1b for the various configurations of Gp and Kp in Theorem 1. wo to u WJ to y wo to y W] to u case M = (I — KG)~1K M = (I — GK)~1G M = (I - GK)-1 M = (I - K G ) - 1 (a) MP = M0 [M0 + AG(I - K0Go)-L}- (I + AGK0)Mo Mo(I + K0AG) (I + K0AG) (b) MP = [M0-AG(I-G0K0)-1]- M0 Mo'I - GoAG) (I-AGGo)M0 (I - G0AG) (c) MP = (7 + A G ) - 1 M 0 M0(I + AG) M0 (7 + A G ) - 1 M 0 ( / + A G ) (d) MP = ( / - A G ) M 0 Moil-Ac)-" M0 (I-AG)M0(I- A G ) - 1 (e) MP = Moil + Ao)'1 (I + AG)Mo (I + AG)M0(I + A G ) - 1 M0 (f) MP = M0 (I - AG) {I- A G ) - l M 0 (I-AG)-lMo(I-Ac) M0 3 . 4 A p p l i c a t i o n o f T h e o r e m 1 t o C D C o n t r o l Let the nominal plant Go G CNXN be the idealized CD process model with periodic bound- ary conditions, Go = " d ^(b,n) = Z d XBC (3.2) 1 — aQz 1 1 — a0z 1 and KQ G CNXN a corresponding spatially-invariant controller obtained by the two-dimensional loop shaping technique and defined with (2.12) and (2.18). Also, let the actual (spatially- variant) CD process model be denoted with Gp G C n x n , lT(b,n) = - (3.3) 1 — CIQZ 1 1 — CIQZ 1 Chapter 3. A Closed-Loop Stability Transfer Between Systems 38 As pointed out earlier, in Section 2.2, the difference between the two plants AG(z) = Gc(z) — Gd(z) = Go(z) — Gp(z) is the 'ears' of the circulant symmetric matrix Go(z), given with (2.19). Based on Theorem 1, a stabilizing controller for the actual CD process model with Dirichlet boundary conditions Gp is given by Kp = (I — KQAK)~1KQ configured as in Figure 3.3a with AK = — A G = AG(z) in (2.19). Such a modified controller may be represented as, u = K0 (y + Atf • u) = K0 (y + AG • u) (3.4) Note the necessary structure of a stabilizing controller for Gp. In general, the controller Kp is an n x n transfer matrix with n2 elements. However, a consequence of Theorem 1 is that, if one begins with a stabilizing spatially-invariant controller KQ = Kc(z) for a spatially-invariant process model Go = Gc(z), then the closed-loop with the actual (spatially-variant) CD process model may be stabilized with an n x n transfer matrix AK in (3.4) with only It, • (lb + 1) nonzero elements, (where lb « n is the process spatial response parameter in (2.11)) - independent of the large size of the original problem. Figure 3.4 illustrates the location of non-zero entries in Gp, Go, and Ac = AG(z) for a typical industrial example with n = 54 actuators and the process spatial response parameter lb = 7. The brute-force design of a multivariable controller would require the synthesis of all n 2 = 2916 transfer matrix elements in Kp. However if one begins with the controller KQ, then one may achieve a stabilizing controller via AK in (3.4) by designing only I • (I + 1) = 56 transfer matrix elements of - fewer than 2% of the full design. Figure 3.4: Position of the non-zero elements of: (a) the Toeplitz matrix Go = Gd{z); (b) the circulant symmetric matrix Gp =-Gc(z); and (c) the difference between the two: AG = -AG{z) = Gd{z)-Gc{z). Paper machine cross-directional control system with the actual (spatially-variant) pro- Chapter 3. A Closed-Loop Stability Transfer Between Systems 39 cess model Gd(z) in (2.10)—(2.11) and the initially designed spatially-invariant controller KQ = [I — DcZ'^Dc • Cc • c(z) modified based on the results of Theorem 1, is illustrated in Figure 3.5. KP(z) = [I-K0(z)AG(z)]1K0(z) GP=GJz) = Gc(z)-AG(z) Figure 3.5: Paper machine cross-directional control system, initially computed with the two-dimensional loop shaping technique resulting in a spatially-invariant process and con- troller models Gc(z), Cc, Dc, stabilized by the use of Theorem 1. Even as the required spatially-invariant controller perturbation is straightforward and the resulting closed-loop system stability guaranteed by Theorem 1, the other three re- quirements from Section 2.3 are not directly taken into account with this approach. How- ever, the issue of performance of C D control systems based on this approach will be briefly revisited in Section 4.3.2 in Chapter 4. 3.5 Other Applications of Theorem 1 This section will stray from the main topic of this work (paper machine C D control), as three additional examples of the use of Theorem 1, drawn from very different applications, are presented. A n application in the case of actuator and/or sensor failure in a multivari- able control system is presented in Section 3.5.1. Finally, two familiar industrial controller design techniques - the Smith predictor and the recycle compensator - are examined in terms of Theorem 1 in Sections 3.5.2 and 3.5.3. 3.5.1 Actuator and Sensor Failures A n application in which the deviation of the true plant from the nominal plant model may potentially be known arises in the study of closed-loop control systems with actuator and/or sensor failures. If the failure is diagnosed and the effect is known, then Theorem Chapter 3. A Closed-Loop Stability Transfer Between Systems 40 1 provides a simple technique that may be used to reconfigure the controller to guarantee closed-loop stability. For example, the so called fault-tolerant sensor configuration (see Figure 7b in [40]) is equivalent to the modifications illustrated in Theorem 1 (case (a)). Consider a linear open-loop stable transfer matrix plant model Go and a feedback controller KQ forming an internally stable system in Figure 3.1a. Now consider the same control system but with failed sensors and/or actuators. Let J denote the set of indices of failed sensors. Let J denote the set of indices of failed actuators. The plant with failed sensors and actuators may be modelled using an additive matrix perturbation to the original model Go: GP A G (z , j ) Figure 3.6 illustrates this for a plant with 20 sensors and 20 actuators where the 7th sensor has failed. In other words, 1 = {7} and J = 0 (the empty set) in (3.5). A failed actuator would result in a column of non-zero entries in A G . = G 0 + A G , | -G0(i,j), ielorjej 1 0, otherwise (a) 10 20 a a • « • • • • • • • • « » • • • • « • e o a a a a a a a a a a a a a a a a a a • e « e » * a « a « a a e a e a o « « « 10 G„ 20 (C) 10 20 «««•«•«•••••*•««»«• a e a a a a a a a a s a o a a a v a a O B « « o a o « » « » f « « a « » a o a a a a a a a a a a a a a a a a o a a a 10 20 Figure 3.6: Location of the non-zero elements of (a) the nominal plant model Go, (b) the additive perturbation A G due to failure of the 7th sensor, and (c) the corresponding transfer matrix model Gp. Closed-loop stability is not guaranteed in general for the plant with actuator and/or sensor failures Gp in (3.5) and the original controller KQ. However, modification of the controller as in Figure 3.3a KP = (I- KQAK^KQ, (3.6) and setting A# = — A G in (3.5) leads to an internally stable closed-loop (Gp,KP) if A G is stable according to Theorem 1. In the case of sensor failure, effectively replacing the Chapter 3. A Closed-Loop Stability Transfer Between Systems 41 failed sensor signals with their modelled response, so that the control law is modified to u = K0{y - A G • u). Note that while no guarantees are provided for closed-loop performance, Theorem 1 allows the recovery of closed-loop stability without the need to perform any new controller synthesis. A fact that makes it attractive in practical fault-tolerant schemes such as in 3.5.2 Smith Predictor One of the earliest examples of the intentional simplification of the plant model for con- troller design is the well-known Smith predictor controller. The controller structure was proposed in 1950's by Otto Smith [59, 60] in order to improve control of the plants with dead-time dynamics, where g/(s) is a finite-dimensional transfer function. The basic idea of the Smith predictor scheme is initially to design a controller ko(s) for the plant g/(s) in (3.7) with no delay, and afterwards modify the controller to account for the delay in gp(s) in (3.7). A detailed analysis of the Smith predictor controller characteristics, as well as a modern interpretation of this design technique (via the internal model control principle), can be found in [29, 54]. The results in Section 3.3 can be interpreted from an IMC standpoint, and used with gp(s) in (3.7) present an alternative derivation of the Smith predictor. First, the transfer function gp(s) in (3.7) may be factored into an additive perturbation as g P ( s ) = g0(s) + Ag(s) with where go(s) is a finite-dimensional transfer function and A g ( s ) contains the deadtime of the original transfer function gp(s) in (3.7). Then using a typically low-order stabilizing controller ko(s) for the stable transfer function go(s) in (3.8), applying Theorem 1 for the configuration in Figure 3.3a yields the controller, [40]. 9P(S) = g f ( s ) e (3.7) 9o(s) = g f ( s ) , and A g ( s ) = g f ( s ) { e - 9 a - 1) (3.8) k P ( s ) = k Q ( s ) Afc(s) = g0(s)(e 1) (3.9) 1 - k 0 ( s ) A k { s ) and is equivalent to the Smith predictor illustrated in Figure 3.7. (The difference in sign where AK(S) = A G in (3.9) is due to negative feedback in Figure 3.7, but AK = — A G in Theorem 1, due to the positive feedback.) Chapter 3. A Closed-Loop Stability Transfer Between Systems 42 kP(s) r(s) : gP(s) K>—H kn(s) g0(s)(e-9 s-D go(s) - o go(s)(e-es-D Y(s) Figure 3.7: Smith-predictor design for plants with pure time delay (compare with Figure 3.3a). A word regarding closed-loop performance is needed here. While Theorem 1 was not necessarily developed for use as a controller design technique, it is well known that the Smith predictor can be designed to provide acceptable closed-loop performance as a dead- time compensator. In general, the modification of k0('s) to kp(s) with the configuration in Figure 3.3a or Figure 3.7 will move the poles of the controller and potentially alter the closed-loop performance. For example, if ko(s) is designed with integral action, then the perturbation configuration kp(s) in Figure 3.3a could potentially eliminate it. However, the Smith predictor in Figure 3.7 is a special case since the perturbation |AG(JW)| —* 0 as LU —> 0 in (3.8) meaning that gp(jto) —> go(jto) and kp(juj) —> k0(jco) as to —> 0. Thus the behaviour of the perturbed loop approaches that of the nominal loop at low frequencies important for performance. As a final comment, it is noted in [54] that - internal stability and steady-state per- formance notwithstanding - the design of the nominal controller ko(s) is important for closed-loop performance for u > 0 where |AG(JW)| ^ 0 in (3.8). Incautious designs of ko(s) that disregard the delay in gp(s) in (3.7) will not lead to good performance of the Smith predictor system in Figure 3.7. 3.5.3 Recyc le Compensa to r Recycle streams in chemical industries are used to feed back some of the process output for further processing [43]. For economical and environmental reasons (e.g. saving energy and materials), plants with recycle streams are becoming more common [57]. It has been shown in [55] that the overall dynamics of such plants can be very different from those of plants having no recycle streams and should be taken into account in the controller design procedure. A chemical process with a recycle stream can be described in terms of a plant model with an inverse additive perturbation given in Figure 3.2b, where go(s) and A s (s ) are the Chapter 3. A Closed-Loop Stability Transfer Between Systems 43 r(s): kP(s) gP(s) * 0 w k0(s) +2 gr(s) e" gf(s) e -es gr(s) eH y(s) Figure 3.8: Recycle compensator for plants with recycle dynamics (compare with Figure 3.3b). models of the process forward path and recycle path respectively, g0(s) = gf{s)e-ti\ and Ag(s) = gr(s)e-*s (3.10) and <7/(s) where gr(s) are stable (often first-order) transfer functions. One of the controller design procedures for the processes with recycle streams, explained in [43, 55, 57], can be presented in terms of the results given in Section 3.3. First one synthesizes a stabilizing controller fcn(s) for the forward path go(s) in (3.10) (possibly using the Smith predictor of Section 3.5.2 above). Since Ag(s), as defined in (3.10), is a stable transfer function, the controller modification procedure outlined in Theorem 1 (case (b)) and illustrated in Figure 3.3b is used in the second step. A block diagram of the closed-loop control system with a recycle compensator is illustrated in Figure 3.8. As in the case of Smith predictor, the controller modification is A*, = A f l (as opposed to Afc = — Ag in Theorem 1) due to the negative feedback being used in the recycle compensation in Figure 3.8. As with the Smith predictor, the recycle compensator will typically be used as illus- trated in Figure 3.8 without further modification due to the fact that for this particular configuration the complementary sensitivity function is unchanged by the perturbations, [1 + gpkp]~lgPkp = [1 + gako}"1 gQkQ (3.11) and thus the nominal performance is recovered. 3.6 Summary A novel and straightforward modification technique with which a controller that stabilizes one plant may be modified so that it stabilizes a second (related) plant has been presented in this chapter. In addition to paper machine CD control, the presented technique can also be implemented on a broad class of multivariable linear time-invariant control systems. Chapter 3. A Closed-Loop Stability Transfer Between Systems 44 Some of the well-known industrial controller design methods have been shown to be special cases of the presented technique. Although the technique guarantees only the resulting system closed-loop stability, in certain industrial applications (e.g. Smith predictor and recycle compensator), it does achieve a successful design in terms of closed-loop performance as well. In terms of CD controllers initially designed under the (idealized) spatial-invariance assumption, a developed technique is guaranteed to stabilize a true (spatially-variant) process models in a straightforward and efficient manner. However, the other require- ments (closed-loop performance and robustness), specified in Chapter 2, are not directly considered with this approach. Chapter 4 Open-Loop Approach to CD Controller Modifications The effects that arise in paper machine CD control systems near spatial domain boundaries (see Figure 1.9 in Introduction) have qualitative similarities with the Gibbs phenomenon encountered in Fourier analysis. The Gibbs effect is a well-known consequence of ap- proximating a discontinuous function with a truncated Fourier series. Unavoidably, this reconstruction exhibits overshoots and undershoots around the point (s) of discontinuity. Even though a truncated Fourier series is equivalent to filtering a signal with an ideal low-pass filter, overshoot and undershoot near discontinuities are evident in all forms of signal filtering. As described in Chapter 2, the paper machine CD controllers are essen- tially spatial and temporal low-pass filters. Also, the paper sheet edges are process spatial domain discontinuities. Considering the above similarities, the methods used for reducing the Gibbs phe- nomenon [41] are used as an inspiration for the CD control modifications proposed in this chapter. An overview of Gibbs phenomenon and its relation to paper machine cross- directional control near the sheet edges is presented in Section 4.1. Proposed CD controller modifications are given in Section 4.2, and a closed-loop simulation example in Section 4.3. The proposed modifications meet the first requirement in Section 2.3 (the requirement for static modifications of the existing industrial CD control law). The second requirement (closed-loop stability) is subsequently checked and confirmed. The last two requirements from Section 2.3 are not addressed with the approach presented in this chapter. 4.1 Gibbs Phenomenon and Spatial Filtering The Gibbs phenomenon is an effect of a special case of discontinuous signal filtering that has been very well studied. It occurs when a discontinuous input signal is filtered with the ideal low-pass filter, which has an impulse response in the original domain (e.g. 1-D spatial domain x) as: sinc{x) = . The ideal low-pass filters cannot be implemented in practice as they are infinite order filters. The infinite order constraint would be impossible to satisfy for either temporal or spatial filters. To illustrate the Gibbs phenomenon, consider the original signal given as a function 45 Chapter 4. Open-Loop Approach to CD Controller Modifications 46 f(x), which is defined as: /(*) = 1 for 0.25 < x < 0.75 0 otherwise (4.1) The function f(x) (shown in Figures 4.1a and 4.1b with dotted lines) obviously has two points of discontinuity, at x = 0.25 and x = 0.75. After modifying the Fourier coefficients of the original signal f(x) by setting high frequency coefficients to zero while keeping the low frequency coefficients unchanged (which is equivalent to filtering f(x) with the ideal low-pass filter) and doing inverse Fourier transform with such modified Fourier coefficients, the modified signal fi(x) is obtained. The signal fi(x) is shown in Figure 4.1a with full line. The oscillating error effects of the original signal discontinuities, at points x = 0.25 and x = 0.75, are clearly seen in Figure 4.1a. This error is called Gibbs phenomenon. 1 0.8 0.6 0.4 0.2 o I 1 1 1 1 1 1 1 1 (a)- - ^ 1 1 1 |Vw^—-~- 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 4.1: Traditional illustration of Gibbs effect in Fourier analysis (a) and its reduction achieved by using Lanczos filter (b). Various methods for reducing and eliminating the Gibbs phenomenon are presented in [41] and its references. One of the proposed methods is based on the Lanczos filtering. In this method, the modified signal fi(x) (a signal obtained by setting high frequency Fourier coefficients of the original discontinuous signal f(x) to zero) is further filtered with a Lanczos filter. The Lanczos filter is a low-pass filter whose frequency response is Chapter 4. Open-Loop Approach to CD Controller Modifications 47 denned by the sinc(u) function in frequency domain. More details on Lanczos filter, shape of its magnitude response, and its use in this work are given in Section 4.3. The original signal f{x) and modified signal filtered with Lanczos filter fi(x) are given in Figure 4.1b. By comparing the signals f\(x) and fi(x) given in Figures 4.1a and 4.1b respectively, it can clearly be seen that Lanczos filter has substantially reduced overshoots and undershoots around points of discontinuity x = 0.25 and x = 0.75. Paper machine CD controllers act as spatial and temporal low-pass filters and the paper sheet edges can be understood as the points of discontinuity. Taking this into account, the analogy between the Gibbs phenomenon (briefly illustrated above) and spatial filtering as realized by CD controllers around the edges can be observed. As mentioned earlier, paper machine CD control systems exhibit overshoots and undershoots when implemented near the sheet edges, which is also in accordance with the Gibbs phenomenon. 4 . 2 C D C o n t r o l M o d i f i c a t i o n s N e a r t h e B o u n d a r i e s As pointed out in Section 2.2, two-dimensional loop shaping technique [62, 66] typically produces a controller K(z) in (2.12) with the periodic boundary conditions (2.18) and coefficients d > 0. Theorem 2 states that, in such a case, replacing the periodic boundary conditions with the corresponding Dirichlet boundary conditions, will not destabilize the controller. Theorem 2 The controller K(z) given with (2.12)-(2.15) is stable with Cd = T(c, n) and Dd = T(d,n) if K(z) is stable with Cc — C(c,n) and Dc = C(d,n) and d > 0 with d„j = dj for j = 1, • • • ,ld- Proof. A necessary and sufficient condition for controller stability (marginal stability) is that all the eigenvalues of D in (2.12) are in the open (closed) unit circle. The spectral ra- dius of the matrix corresponding to periodic BCs is given as [36]: p(Dc) = <io + X3 p =i — 1. The eigenvalues of the equivalent Toeplitz symmetric matrix Dd (corresponding to Dirichlet BCs) are bounded [36] by a function: / ( A ) = d0 + J22dPcos(Px)> AG [0 ,2TT) P=I Finally, p(Dd) <d0 + £'d=1 2dp = p(Dc) < 1.0 Chapter 4. Open-Loop Approach to CD Controller Modifications 48 Consider further modifying the boundary conditions of Dd in Theorem 2 by multiplying it by a real nx n matrix, Df = F • Dd F = (4.2) where f\ and f2 are each n/ x nf real constant matrices and I is the identity matrix of order n — 2-nf. Note that each of the matrices Df, Dd, and Dc has elements defined by (2.14), the only difference being those elements 6dij in the first and last no\ = nf rows that define the implicit boundary layer. The proposed design of the matrices fi and f2 in (4.2) is inspired by research on the Gibbs phenomenon. To mitigate the effect of the Gibbs ripples, a straightforward technique is to locally filter the jump discontinuities for smoothness. Approximating a function by a truncated Fourier series is equivalent to performing a convolution of the function with an ideal low pass filter (usually referred to as a 'sine' function). However, overshoot and undershoot are not limited to sine functions and may occur in other types of non-ideal filtering. Thus we propose to design the matrices f\ and f2 in (4.2) to provide local low-pass filtering that mitigates the effect of the transition induced by the implicit boundary con- ditions. The following Theorem provides a conservative result for directing the design of the matrices f\ and f2 in (4.2) such that controller stability is preserved. Theorem 3 If the controller K(z) given with (2.12)-(2.15) is stable with Cd = T(c,n) and Dd = T(d,n) (with d_j = dj for j = 1, • • • ,ld) then it is stable with Df = F • Dd, if &(F) < 1; where a(F) is the maximum singular value of F. Proof. For the symmetric matrix Dd we have p(FDd) < o(FDd) < a(F)a{Dd) = a(F)p(Dd) < 1 where the final inequality is implied by stability of K(z) with D = Dd.() Note that for the block-diagonal matrix F in (4.2), we need only to ensure that <T(/I) < 1 and ^(fi) < 1 for the nf x nf matrices to satisfy Theorem 3. This can lead to a much easier calculation if nf <C n. This is fortunate since the number of actuators is potentially as large as n = 300 for some cross-directional control systems. It is not difficult to see from (4.2) and (2.20) that the above proposed modifications can be represented with the controller matrices' additive perturbations 8C and 6D in Chapter 4. Open-Loop Approach to CD Controller Modifications 49 (2.21)-(2.22) denned as, 6C = -AC, 5D = -AD + (F - I) • Dd, (4.3) where AC and A D are the 'ears' of the controller circulant symmetric matrices, defined in (2.20), matrix F € 5f n x n as given in (4.2), and Dd a controller's matrix (spatial filter) D in (2.12), corresponding to Dirichlet boundary conditions. The next section contains an example where a standard finite impulse response (FIR) filter is used in the design of fc and fc in (4.2) to modify the spatial boundary conditions of a cross-directional controller K(z) in (2.12). 4 . 3 Simulation Example 4.3.1 Edge Filter Design Based on the techniques used for the reduction and elimination of the Gibbs phenomenon presented in [41], and the theorems given in Section 4.2, a modification of CD controller boundary conditions is proposed. Modifying the control law includes smoothing the con- trol signal around paper machine edges with a low-pass filter computed based on the techniques presented in [41]. As will be shown below, the proposed control law modifica- tions preserve controller stability while also maintaining closed-loop stability. Controller stability is proved using Theorem 3 in Section 4.2, and closed loop stability is checked and confirmed. It will also be shown that in contrast to the original (unmodified) control law (computed implementing reflection edge padding, as defined in Section 2.1), the actuator array obtained with a modified control law does not contain high spatial frequency con- tent. The gradual development and buildup of the high spatial frequency content in the actuator array often leads to control system instability in CD control applications. The smoothing filter, proposed here, is based on the Lanczos filters [41], used for the reduction of Gibbs phenomenon exhibited in truncated Fourier series. The original Fourier series of a signal f(x), with discontinuities, is given as: n (4.4) fc=i where ak and bk are the Fourier coefficients of the signal, and is modified as: n sin(/c7r/n) fn(x) = a0 + ̂ 2 kir/n [ak cos(kx) + bk sin(fcx)] (4.5) fc=i Chapter 4. Open-Loop Approach to CD Controller Modifications 50 The factor Cfc = s ' ° | * " ^ in (4.5) has a smoothing influence on the Fourier series repre- sentation [41] and reduces the Gibbs effect exhibited at signal discontinuities. This was illustrated in Section 4.1 with the example shown in Figures 4.1a and 4.1b. It is also inter- esting to notice that the filter (k can be cascaded for even better convergence of the series (i.e. smaller overshoots and undershoots around discontinuities). However, in that case, the transition of such a representation from one level to the other around discontinuity is less sharp [41]. Using this factor m times simply means using £™ instead of £fc in (4.5). In the example shown in Figures 4.1a and 4.1b, in Section 4.1, Q was used. A finite impulse response (FIR) digital filter that is equivalent to the above Lanc- zos filter with a smoothing factor has been designed using M A T L A B function firls [47]. Based on the Lanczos filter used for the Fourier series in [41] and shown in (4.5), the target and achieved (using the above mentioned M A T L A B function) frequency re- sponses are shown in Figure 4.2. The frequency response, shown in Figure 4.2 with ,1 1 !_ , , , C 0 0.5 1 1.5 2 2.5 3 Normal ized f requency Figure 4.2: Desired (full line) and achieved (dotted line) frequency responses dotted line, is the least square approximation to the Lanczos filter and has been ob- tained with a symmetric fourth order FIR filter whose impulse response coefficients are [h0,h1,h2,h3,h4\ = [ho,huh2,hitho] = [-0.0306,0.2149,0.6148,0.2149,-0.0306]. Based Chapter 4. Open-Loop Approach to CD Controller Modifications 51 on this, the edge filtering matrix H is defined as follows: h2 h h0 0 0 0 0 . . 0 0 h h2 h0 0 0 0 . . 0 0 h0 h2 h h0 0 0 . . 0 0 0 h0 hi h2 h h0 0 . . 0 0 0 0 0 0 1 0 0 . . 0 0 0 0 0 0 0 1 0 . . 0 0 0 0 0 0 0 0 1 . . 0 0 0 0 . . 1 0 0 0 0 0 0 0 0 . . 0 1 0 0 0 0 0 0 0 . . 0 0 1 0 0 0 0 0 0 . . 0 h0 h h2 hi h0 0 0 0 . . 0 0 h0 hr h2 hi h0 0 0 . . 0 0 0 h0 hi h2 hi 0 0 . . 0 0 0 0 h0 hi h2 Hi Ho (4.6) which can be written in a block diagonal form as H = diag(Hi,I, H 2 ) . It is important to notice that in order to obtain a block-diagonal structure, in addition to the matrix rows on which edge filtering is being implemented, two additional rows (because there are two non-zero off-diagonal elements in H i and H 2 ) are being included in blocks H i and H 2 . Also, the block H 2 is obtained just by flipping the block H i twice, first its rows and then its columns. 4.3.2 Closed-Loop Simulations The process model and feedback controller, used in this chapter, were obtained from an industrial paper machine. This particular system was described in detail in [64]. An array of n = 54 slice lip actuators is used to control the basis weight profile of a sheet of light weight 'telephone directory grade' paper. The parameters of the process model in (2.10)—(2.11) were identified using software described in [30, 32] as It — 7 and {bQ, h , b 7 } = {0.0713, 0.0337, -0.0167/-0.0200, -0.0050,0.0006,0.0005,0.0001} a0 = 0.8311 , d = 3 The feedback controller K(z) in (2.12)—(2.15) is designed using the two-dimensional loop shaping technique [62, 66]. We first replace the Dirichlet boundary conditions of the Chapter 4. Open-Loop Approach to CD Controller Modifications 52 Coeffs. Sdij in (2.14) with nd = 3 and reflective BCs 5du = d0 + 2(da + d2 + d3) 8d12 = 0 Sd13 = 0 8d2i = d1 + 2(d2 + d3) Sd22 = do - d2 8d23 = di - d3 8d3X = d2 + 2d3 8d32 — di — dz 8d33 = 0 Table 4.1: Boundary layer coefficients of the controller matrix D in case of reflective boundary conditions and d = [do, • • •. , d3], d_j = dj for j = 1,2,3. process Bd = T(b, 54), with a model using periodic boundary conditions Bc = C(b, 54), then the controller matrices Cc — C(c, 54) and Dc = C(d, 54) with lc — 5 and ld = 3 with {c 0 ,c a , ...,c5} {d0,di,d2,d3} {-10.4708, -3.5297,1.4841, -0.0042,0.0017,0.0006} {0.9860, 0.0046,0.0020,0.0004} The parameters of the Dahlin controller c(z) in (2.12) are also produced by the design, but are not central to the spatial boundary condition issue and may be found in [64]. The current industrial practice for modifying the controller coefficients is to replace the periodic boundary conditions with reflection conditions as given in Table 4.1. The proposed design technique involves first replacing the periodic boundary conditions with Dirichlet conditions D = T(d, 54), according to Theorem 2. Next, a matrix F in (4.2) is designed for further modification of the control near the sheet edges. As discussed earlier, the submatrices fi and f2 in (4.2) are synthesized based on a finite impulse response (FIR) digital filter, computed as an approximation to the Lanczos filter. The resulting symmetric fourth order FIR filter has impulse response coefficients [8f2,8fi,8f0,8fi,8f2] — [-0.0306,0.2149,0.6148,0.2149,-0.0306]. The matrices fi and f2 with nf = 5, in (4.2), are then given by: fi = 0.9595 • Sfo 5fi 8f2 0 0 Sfi 5fo Sfi 8f2 0 Sf2 Sfi 8f0 8fi 8f2 0 0 0 1 0 0 0 0 0 1 f2 = 0.9595 • 1 0 0 0 0 0 1 0 0 0 5f2 8fi 5fQ 5fi 5f2 0 8f2 8fi Sf0 8fi . 0 0 8f2 5fi 5f0_ (4.7) The coefficient 0.9595 in (4.7) has been introduced in order to satisfy the requirement of Theorem 3 (a(F) < 1). As a result, the stability of the controller with the matrix Df as defined in (4.2)-(4.7) is guaranteed by Theorem 3. Although the proposed design is mainly concerned with stability of the controller, we will present simulation results of the closed-loop behaviour. In order to compare the Chapter 4. Open-Loop Approach to CD Controller Modifications 53 1.5 0 5 10 15 20 25 30 35 40 45 50 Cross-direction Figure 4.3: Process output disturbance (at zero temporal frequency to = 0). standard industrial approach with the proposed design, closed-loop simulations have been performed with the steady state process output disturbance, d(z) in (2.9), as shown in Figure 4.3. The disturbance has been constructed to allow comparison of the closed-loop performance at the sheet edges and also away from the spatial boundaries. The same localized disturbance is introduced at the edge and also near the middle of the sheet, as illustrated in Figure 4.3. The closed loop simulation results are shown in Figures 4.4 and 4.5. Figure 4.4 i l - lustrates the closed-loop steady state process output and actuator profiles obtained with controller K(z) using the reflection boundary conditions given in Table 4.1. It can be seen that control signal at the edges has significant high frequency content with maximum and minimum values varying between -4.85 and 4.4. At the same time, the control signal away from the edges has significantly smaller high frequency component with the maximum and minimum values varying between -2.1 and 2.1. Such actuator profile at the edges would be unacceptable in real life paper machine CD control systems. In Figure 4.5, the steady state values of the process output and control signal are shown in case when the controller using Df matrix, in (4.2)-(4.7), is used. Compared with the results achieved using reflective boundary conditions, the control is significantly less active at the sheet edges while, as expected, the performance away from the edges is unchanged. From Figure 4.5, it can be seen that the proposed control modifications at the edges result in the less active (more conservative) control than the control away from the edges. Figure 4.4 illustrates that the control designed with reflective boundary conditions is more active at the sheet edges, indicating a reduced robust stability margin compared to the original design. However, the obtained control signal at the edges is significantly more acceptable than the signal obtained using the current industrial practice (see Table 4.2 below). At the end of this chapter, the approach to modifying CD control law near spatial domain boundaries developed in Chapter 3 is briefly revisited. As pointed out earlier, the approach to modifying CD control developed in Chapter 3 (detailed in Section 3.4) guarantees only the resulting system closed-loop stability. However, for comparison Chapter 4. Open-Loop Approach to CD Controller Modifications 54 10 15 20 25 30 35 40 45 50 o 3 o < 10 15 20 25 30 35 40 45 50 Cross-direction Figure 4.4: Steady state process output and actuator array in case of the reflective bound- ary conditions in Table 4.1. CO CO o to 0 o . < 1 1 1 . i i i . 1 i i i in i • i | 11 i . t 10 15 20 25 30 35 40 45 50 Cross-direction Figure 4.5: Steady state process output and actuator array when the controller with matrix Df, given with (4.2) and (4.7), is used. Chapter 4. Open-Loop Approach to CD Controller Modifications 55 Q. 1.5 1 0.5 0 -0.5 -1 -1.5 1 I 1 I [ 1 i i i i i - H A ~ 4 u ^ • i / V i i i i i i i i i i 10 15 20 25 30 35 40 45 50 co o to o < 20 25 30 35 Cross-direction Figure 4.6: Steady state process output and actuator array when the approach presented in Chapter 3 (controller with the structure illustrated in Figure 3.5) is used. reasons, closed-loop simulations with the process and controller parameters given above and the controller structure illustrated in Figure 3.5 (based on Theorem 1) are shown in Figure 4.6. As expected, given controller's circulant symmetric matrices Cc and D c , a non-zero disturbance near only one edge (Figure 4.3), results in non-zero control signal at the both edges. This would, of course, be unacceptable in the industrial setting. Finally, the results obtained with all three approaches (current industrial practice and the approaches from Chapters 3 and this chapter) are summarized in Table 4.2. Current technique (Reflection padding) S T T approach (Chapter 3) Open-loop approach (Chapter 4) Process output \\y\\2 2.5297 2.5772 2.6928 Actuator array \\u\\2 8.9678 6.0388 4.47 Output y and control signal u near left edge l | y ( i : io) | | 2 1.7629 1.8244 1.9899 | K 1 : 10)||2 7.8876 3.9402 1.3327 Table 4.2: 2-norms of the steady-state process output and control signal profiles shown in Figures 4.4 - 4.6. Chapter 4. Open-Loop Approach to CD Controller Modifications 56 4.4 Summary In this chapter, a novel technique for modifying CD controllers near spatial domain bound- aries, based on the method for mitigating the well-known Gibbs effect, has been developed. A closed-loop simulation example has also been presented at the end of the chapter, where the new approaches (developed in Chapters 3 and 4) to modifying CD control law near the sheet edges have been compared against the existing industrial practice. Considering that CD controllers are spatial (and temporal) low-pass filters and that paper sheet edges represent clear spatial domain discontinuities, the observed similarities between the effects occurring in CD control systems near the edges and the well-studied Gibbs phenomenon are no surprise. Based on a Lanczos filter, used for mitigating the Gibbs effect, a CD controller stability-guaranteeing modification technique has been de- veloped. While, the proposed technique does not alter the structure/complexity of the industrial CD controller (the first requirement in Chapter 2), the closed-loop performance and robustness requirements are not directly considered with this approach. Chapter 5 Closed-Loop Approach to CD Controller Modifications In this chapter, a method for modifying the existing industrial paper machine CD control law is presented directly taking into account all of the requirements from Section 2.3. The proposed modifications to the existing controller's static matrices (requirement 1 in Section 2.3), and the resulting closed-loop system stability, performance, and robustness (requirements 2-4 in Section 2.3) are directly and systematically considered with this approach. There are a few important observations that should be made about the problem defined in Section 2.3 and illustrated in Figure 2.7. Since the desired compensator E is a (block- diagonal) static matrix, the problem is a static output feedback (SOF) design problem. While such a problem is very easy to state, a wide variety of SOF problems are still unsolved and represent a significant design challenge (see [69] and references therein). In this work the design of 5D and SC in Figure 2.7 will be performed sequentially as it is very difficult to design a static output feedback compensator E with the additional block- diagonal structure constraint. A low-bandwidth static output feedback controller design algorithm, used for computing 5D and 5C, is outlined in Section 5.1. The generalized plant P(z), in Figure 2.7, consists of the CD control system closed- loop transfer functions. Considering the CD control system size (the number of actuators 30 < n < 300), and high-order dynamics introduced by the process delay d in (2.10) and Dahlin compensator c(z) in (2.12), the number of states included in these closed- loop functions can easily be of the order ~10 3. This renders implementation of most of the existing controller design algorithms, including the efficient one presented in Section 5.1, practically intractable. However, knowing that the desired controller modifications (2.21)-(2.22) are localized to a small fraction of the total matrices, the order of the re- sulting generalized plants used for computing modifications 5C and 5D can be reduced significantly, as detailed in Section 5.2. Implementation of the low-bandwidth static output feedback (SOF) compensator de- sign algorithm, to be presented in Section 5.1, presumes stable systems. Based on numer- ous simulation and industrial data, replacing the process and controller circulant symmet- ric matrices with the corresponding Toeplitz symmetric matrices (which is the first step of the computation of CD controller modifications procedure, presented in Section 5.2) 57 Chapter 5. Closed-Loop Approach to CD Controller Modifications 58 typically results in closed-loop stable systems. This is not surprising considering that the original (spatially-invariant) controllers are obtained by the two-dimensional loop shaping technique, in which system robustness, with respect to the unstructured process additive perturbations, is a requirement [65, 66]. However, if the closed-loop system, with pro- cess and controller Toeplitz symmetric matrices, happens to be unstable (very rarely), a stabilization algorithm, outlined in Section 5.4, has to be implemented. In preparation for an industrial trial, the simulation studies, presented in Section 5.3, were carried out using the Honeywell hardware-in-the-loop paper machine simulator and the industrial identification and controller tuning software detailed in [32, 65]. Newly developed Matlab prototype software, based on the approach presented in this chapter, was used for computing CD control law modifications 5D and SC in Figure 2.7. The results obtained with the new approach are subsequently compared against the existing industrial practice. A novel static output feedback compensator design algorithm, used for computing CD controllers modifications in Sections 5.2-5.4, is detailed in Section 5.1. The algorithm is presented in full generality, independent of CD control problem. However, in order to address performance and robustness conditions (requirements (3) and (4) in Section 2.3), a generalized plant N(z) with two exogenous outputs, wa(z) for performance and Wb(z) for robustness, is considered in Section 5.1, as illustrated in Figure 5.1. Transfer functions that make up the generalized plant N(z) will be made explicit in Sections 5.2-5.4. Figure 5.1: Diagram of the lower linear fractional transformation J-~i(N, K). 5 . 1 Static Output Feedback (SOF) Controller Syn- thesis The static output feedback problem considers the linear, time-invariant plant x(k + 1) = A0x(k) + B0u(k), y(k) = C0x(k) + DQu{k) (5.1) Chapter 5. Closed-Loop Approach to CD Controller Modifications 59 with the controller u(k) = K0y(k) (5.2) where KQ is a constant matrix of appropriate dimensions. In general, the SOF problem is concerned with designing KQ such that various closed-loop properties of the control system given by (5.1)—(5.2) are satisfied [69]. In this work, we are concerned with a stable, finite-dimensional generalized plant N(z): N(z) = 'Nn(z) N12(z) N21(z) N22(z) N31(z) N32{z) (5.3) illustrated in Figure 5.1. The signal d(z) represents the exogenous input into the control system, wa(z) and Wb(z) represent the exogenous outputs, y(z) represents the feedback signal, and u(z) represents the control signal. Let the generalized plants Na(z) and Nb(z) be defined as, Na(z) = Nu(z) N12(z) N3l(z) N32(z) Nb(z) = N21(z) . N22(z) N31{z) N32(z) (5.4) then the input-output transfer functions, d(z) —> wa(z) and d(z) —> wb(z), are given by lower linear fractional transformations (LFTs): Tt(Na(z),K(z)) = 7V11(z) + 7V 1 2(z)^(z)(/-iV3 2(z)^(z))- 17V3i(z) ^(Nb{z),K(z)) = N ^ + N^Kiz^I-N^Kiz))-1^^) (5.5) Our objective is to design a compensator such that: (a) the controller K(z) = KQ is a static matrix, (b) the feedback system in Figure 5.1, with K(z) = KQ is stable, (c) the compensator improves the closed-loop performance as measured by the Frobenius norm, \mNa{en,Ko)\\F < \\Tt(Na(en,0)\\F, V \u\ < ujb, (5.6) for some tob > 0, (d) the performance at higher frequencies is not overly degraded. In other words, a constraint, \\HNb(z), KQ) < 1, (5.7) is satisfied. Chapter 5. Closed-Loop Approach to CD Controller Modifications 60 It can be seen that the above requirements (a)-(b) completely correspond to the require- ments (l)-(2) in Section 2.3. Also, the requirement (c) above is closely related to the requirement (3), as the Frobenius norm is equal to the sum of all singular values [5, 58]: WH\\F = /El^l 2 = JY,°l(H), (5-8) where hij indicates the element in the ith row and jth column, and Ofc(-) denotes the kth singular value. Since N(z) in (5.3) is stable, the internal stability of the closed-loop system in Figure 5.1 is equivalent to the input-output stability of K(z)(I — N32(z)K(z))~x in (5.5). We can then write down the familiar parametrization of stabilizing controllers K(z) for the feedback system in Figure 5.1, K(z) = Q(z)(I + N32(z)Q(z))-1 (5.9) for stable Q(z) (see for example [76]), leading to the convenient form of the LFTs in (5.5), Ti(Na(z),K(z)) = Nu(z) + N12(z)Q(z)N31{z) (5.10) ^(Nb(z),K(z)) = N21(z) + N22(z)Q(z)N31(z) (5.11) Consider the low-frequency requirement on the Frobenius norm in (5.6). Using (5.10) we can write the LFT at steady-state (LU = 0), HNatf0), K{ei°)) = Nn(e>°) + N12(e>°) • Q(e>°) • N ^ 0 ) (5.12) Now consider the following optimization problem motivated by (5.12), Q0 = argmmJ(iV a(e i 0),^,Q) J(Na(e>°),PiQ) = ||A^n(eJ'°) + iV 1 2 (^ 0 ) .g -7V3 1 (e^) | | ; + p | |Q | | F (5.13) A closed-form solution to this optimization problem for a real static matrix Q0 is given in Appendix B. Subsequently, the resulting Qo from (5.13) is used to form the static controller KQ (requirement (a) above), K0 = Q0(I + Nviet^Qo)-1 (5.14) The first term in optimization (5.13) is intended to address the above specified perfor- mance requirement (5.6), while the second term is intended to limit the magnitude of the Chapter 5. Closed-Loop Approach to CD Controller Modifications 61 synthesized matrix Q0. Allowing the optimization weight p —> 0 in (5.13) would produce the static matrix K0 in (5.14) such that the matrix norm \\Fi(Na(ej0), i^o)||f is globally minimized. The conditions on p and generalized plant N(z), such that the stability condition (b) and the dynamical condition (d) above are satisfied, are determined by Theorem 4 below. Theorem 4 (Stability and Full Bandwidth Performance Limit) If N(z) in (5.3) is stable, a(Fi(Nb(ejuJ), 0)) < 1 for all LO, and p > (5 in (5.13) where fi=Vw^ • 5(A'i2(eJ 0))<>(Af3,(eJ 0))5(A'1 1(e*0))^ { | | ^ M - W a ( ^ ) | | . + " ^ » - J ^ W » - } (5.15) with the integers r\2 and r 3 1 denoting the number of nonzero singular values of iV^e-?0) and iV 3 1 (e j 0 ) ; respectively. Then KQ synthesized from (5.13)-(5.14) stabilizes the feedback system in Figure 5.1 and a{Fi'Nb{e?u\KQ)) < 1, for all co (5.16) where a(-) denotes the maximum singular value. Proof. Given in Appendix C. Note that since 1 > \\Fi(Nb(z), 0)11̂  = | | A ^ i ( - 2 ) | | 0 0 in Theorem 4, the denominator in (5.15) is always greater than 0. The closed-loop performance improvement, as defined in (5.6), with the CD controller designed using the above outlined algorithm is guaranteed by Theorem 5. Theorem 5 (Low Frequency Performance Improvement) If N(z) in (5.3) is stable, then for any KQ 7̂  0 constructed from (5.13)-(5.14) that stabilizes the system in Figure 5.1, there exists a frequency Ub > 0 such that • \\n'Na(e>u),K0)\\F< I I ^ A ^ e ^ O ) ^ (5.17) for all \ui\ < u>b- Proof. Given in Appendix C. The value for the weight p in (5.13) is determined through bisection on p to produce a KQ in (5.14) such that the requirements (b)-(d) are successfully traded off, and is initialized with the value computed based on Theorem 4. Subsequently, the bisection continues as long as the stability and full bandwidth requirements are satisfied, and until the difference Chapter 5. Closed-Loop Approach to CD Controller Modifications 62 between the two consecutive values of the weight p in (5.13) is smaller than some specified value of the tolerance e > 0. A more detailed outline of the static compensator synthesis algorithm is given in Section 5.1.1 below. 5.1.1 Synthesis A l g o r i t h m The overall algorithm for determining the weight p in the optimization (5.13) and computing the corresponding static compensator K0, satisfying all the requirements (a)-(d) above, is given as follows: 1. INITIALIZATION Set pi = 0 and specify tolerance e > 0. Based on Theorem 4, find ph > 0 that is guaranteed to satisfy the stability condition (b) and the performance condition (d). 2. Set p = ph and compute the corresponding Qo and K0 based on (5.13) and (5.14) re- spectively. (Theorems 4 and 5 guarantee that so computed compensator KQ satisfies all the requirements (a)-(d) above.) 3. IF ph - pi < e GOTO STEP 7. 4. Find p = £ h ^ £ i and compute the corresponding Q* and K* based on (5.13) and (5.14) respectively. 5. (Verifying conditions (b) and (d) for the system with the above computed compen- sator K*). IF either one of the requirements (b) and (d) is not satisfied THEN set pi = p and GOTO STEP 3. 6. Set KQ = K* and ph = p. GOTO STEP 3. 7. END The above algorithm converges to a non-zero K0 satisfying all the requirements (a)-(d) above, for every tolerance e > 0. 5.2 Computation of CD Controller Modifications The process spatial invariance assumption, given with (2.17), significantly facilitates CD controller design and is central to the two-dimensional loop shaping technique [66]. How- ever, the implementation of the controller with circulant symmetric matrices in (2.18) would mean a computation of the control signal near one paper machine edge based on Chapter 5. Closed-Loop Approach to CD Controller Modifications 63 the measurements and previous control signal near the other edge (for a typical paper ma- chine that is between 3 and 11 metres away). This is clearly unwarranted considering that the actual process model (2.10) is characterized by actuators with a localized response on the paper sheet. As a result, the first step in the proposed CD controller modification is a replacement of the initially computed spatially-invariant (circulant) controller matrices Cc and Dc in (2.18) with their corresponding Toeplitz symmetric matrices. Next, the final controller modifications are computed using the SOF algorithm, presented in Section 5.1, in turn on the SD and SC matrices in Figures 2.6 and 2.7. Since the SOF algorithm presumes stable systems, the closed-loop stability of the system with Toeplitz symmetric process and controller models, is assumed. Based on numerous industrial data, as well as simulation studies, this is not a restrictive assumption and is only violated in certain pathological examples. However, if the system with Toeplitz symmetric process and con- troller models is not stable, then a stabilization procedure is required. Such a stabilization procedure is presented in Section 5.4. Based on the above, the overall algorithm for computing CD controller modi- fications SC and SD is given as follows: • •  . 1. Replace the controller circulant symmetric matrices Cc and Dc in (2.18) with their corresponding Toeplitz symmetric matrices Cd and Dd, respectively, and check sta- bility of the resulting closed-loop system. 2. IF Toeplitz system is stable GOTO 4. 3. (Rarely required) Use the stabilization procedure given in Section 5.4. 4. Compute SC and SD modifications (for performance) as detailed in Sections 5.2.1- 5.2.2. 5. END. Since the two edges of the paper machine are modelled to be identical, it is enough to retune the controller at one edge only, e.g. upper left corners of the matrices SC and SD. Subsequently, the corresponding modifications at the other edge can easily be found by symmetry arguments. 5.2.1 Modi f i ca t ions N e a r One Sheet Edge ( C e and De) Factoring out of the control system inputs and outputs near one edge, based on the closed- loop system in Figure 2.7, is performed with rectangular weights Wi, i = 1,2,... ,7, as illustrated in Figure 5.2. Chapter 5. Closed-Loop Approach to CD Controller Modifications 64 PJz) Figure 5.2: Isolating system inputs/outputs near one edge. The rectangular weighting matrices i = 2, 3, 4, 5 are defined as: W2 = [ I n o i O n D i x ( n - n D i ) ] i ^ 3 = [If ici ®nC\ X ( n - n c i ) ] ) W4 = [I„y O n ! / X ( n - n y ) ] , n c i x n ^ 5 [In„ 0 n u X ( n — n u ) ] , (5.18) and the rectangular weighting matrices Wi, i = 1,6, 7: wx = P(n-nd)xnd_ , w6 = I n D 2 P(n-nD2)xnD2_ , w7 = InC2 P(n-nC2)xnc2_ nxn<i nxn.02 nxnc2 (5.19) The matrices W2, W3, W7 are used to convert the matrix sub-block design into a full-block design problem, as illustrated in Figure 5.3. From (5.18)—(5.19) and Figure 5.3, it can be seen that the elements are given by: [De]ij = Sdij, 1 < i < n m and 1 < j < nD2, [Celtj = Scij, 1 < i < nCi and 1 < j < nC2, (5.20) where Scij and Sd^ are the same as those elements in (2.21)-(2.22). In other words, Ce € TZnciXnc2 and De £ fcnmXn°2 are the non-zero, upper-left elements of the matrices 5C and SD in (2.21)-(2.22), respectively. The matrices W i , W 4 , W 5 , on the other hand, are used to isolate the sheet edges for Chapter 5. Closed-Loop Approach to CD Controller Modifications 65 w6> 5D -+ W2T 4 M SC W3T Figure 5.3: Transforming a sub-block into a full-block design problem. consideration in the performance design. Al l the closed-loop transfer functions that define Pe(z) in Figure 5.2 are given in Appendix D. However, elimination of inputs/outputs with the matrices Wi,i = 1,2,... ,7 does not necessarily reduce the number of states of the corresponding closed-loop transfer functions. As a result, the order of the transfer matrix Pe(z) in Figure 5.2, in typical CD control systems, can easily reach into the thousands. Fortunately, most of these states have very little impact on input/output behavior of the corresponding closed-loop transfer functions. This is the case because many of the states are mainly related to the inputs/outputs located in the middle of the sheet and other machine edge, i.e. those inputs/outputs that were eliminated with the rectangular weights Wj. The (in)significance of the states, in terms of the corresponding transfer function in- put/output behavior, can be quantified using Hankel singular values [2, 5, 26, 58, 76]. For example, in the case of the system to be presented in Section 5.3, out of 144 Hankel singu- lar values of the closed-loop transfer function Px : de —> ye, 80 are smaller than 2.2 -10~ 1 6, 105 smaller than 10 - 9 , and 125 smaller than 10 - 3 (as illustrated in Figure 5.4). Such a rapid decrease of the Hankel singular values is quite representative of these systems and not surprising, considering the localized nature of the CD processes and controllers. The order reduction procedure, based on Hankel singular values, has two important characteristics [4, 5]: 1. Stability preservation (stability of Pe(z) implies stability of Pr(z)), 2. An apriori computable upper bound of the approximation error Ji^ norm: ||Pe(,z) — P r (z)| |oo < 2 * y^^discarded Hankel singular values) (5-21) Chapter 5. Closed-Loop Approach to CD Controller Modifications 66 0.5 0.4! 0.3 r 0.2 [ < 1(T3 <10" 9 , - \ < 2.2*1 cr1,6 \ 1 _1_ _ I J.....I __l _ _ l  1 1 20 40 60 80 100 120 140 State number Figure 5.4: Illustration of the rapid decrease of the Hankel singular values of the closed- loop transfer functions that define a generalized plant Pe(z) in Figure 5.2: Hankel singular values of Pi : de —> ye for the CD control system presented in Section 5.3. 5.2.2 Computation of Ce and De Based on the diagram given in Figure 5.2, the linear fractional transformations (LFT's) for computing modifications Ce and De in Figure 5.3 can be defined. They are presented in Figures 5.5a and 5.5b. The design of Ce and De is performed by alternately synthesizing one matrix component while holding the other fixed, as detailed below. The synthesis procedure is the low-bandwidth procedure of Section 5.1. (a) (b) Pc(z) PJz) Figure 5.5: Linear fractional transformations for computing (a) Ce and (b) De modifica- tions. The algorithm for computing Ce and De is given as follows: Chapter 5. Closed-Loop Approach to CD Controller Modifications 67 1. Compute De by the use of the static compensator synthesis procedure in Section 5.1.1, based on the linear fractional transformation given in Figure 5.5b. 2. Based on (5.20), find the corresponding controller modification 5D in (2.22) and update the controller matrix D. 3. Compute Ce by the use of the static compensator synthesis procedure in Section 5.1.1, based on the linear fractional transformation given in Figure 5.5a. 4. Based on (5.20), find the corresponding controller modification 5C in (2.21) and update the controller matrix C. It should be noticed here that this sequential static output feedback (SOF) controller design procedure does not increase the order of the resulting system Pe(z), unlike conven- tional sequential decentralized control (see discussion and references in [39]). This is, of course, a consequence of the designed compensator being a static matrix at each iteration of the design and so does not contribute to the number of states. 5.3 Hardware-In-The-Loop Simulator Example In preparation for testing the new technique on a real paper machine, the simulation studies presented in this section were carried out using the industrial identification and controller tuning software detailed in [32, 65] (residing on one computer), and a hardware- in-the-loop simulator (residing on another computer), as illustrated in Figure 5.6. 5. END. INDUSTRIAL IDENTIFICATION AND TUNING and Honeywell L A N t Honeywell L A N Figure 5.6: Schematic of the simulator trial setup. The industrial controller algorithm, implemented in the simulator, was modified to accommodate the control law changes according to Figure 2.6. The process simulator was set up with parameters to correspond to those of the actual paper machine on which the Chapter 5. Closed-Loop Approach to CD Controller Modifications 68 real testing was afterwards carried out (Chapter 6). The simulator testing procedure was as follows. The open-loop 'bump test', with three actuators stepped up (down) for 150 microns, was carried out for identification purposes. Process identification and controller tuning were performed using the industrial software presented in [32, 65]. Next, the design for various controller modifications De and Ce in Figure 5.5 was completed, and these parameters were transferred over the L A N into the correct database location for use by the industrial controller (the information flow between the Matlab prototype software and the industrial software packages is given in Figure 6.4 in Chapter 6). Finally, the performance of the resulting closed-loop systems was observed, recorded, and is presented below. This is the same procedure that was followed in a working paper mill (Chapter 6). 5.3.1 Process and Con t ro l l e r Paramete rs The CD control system, presented below, describes an array of n = 36 slice lip actuators being used to control the paper sheet basis weight profile. More details about basis weight control using slice lip actuators were given in the Introduction. The parameters of the process model in (2.10)—(2.11) were identified using software described in [32] with the size of the matrix B band lb = 6 in (2.11) and {60, & i , . . . , b6} = 10~3 • {0.1652,0.2044,0.0789, -0.0382, -0.0169, -0.0009,0.0001} a 0 = 0.855, d = 2 (5.22) The feedback controller K(z) in (2.12) was designed using the standard two-dimensional loop shaping technique [65, 66]. First, the process Toeplitz symmetric matrix Bd in (2.11) is replaced with the corresponding circulant symmetric matrix Bc in (2.17), resulting in a (spatially-invariant) controller with circulant symmetric matrices (2.18). The controller parameters obtained had matrix band sizes lc = 4,Z<j = 1, with: {c0,Cl,C2,c3,c4} = {-0.2089,-0.2129,-0.0487,0.0856,0.0481} {d 0,di} = {0.9878,0.0061} (5.23) The tuning parameter a of the Dahlin controller c(z) in (2.12) was also produced by the two-dimensional loop shaping design, a = 0.8506. Subsequently, the initially computed circulant-symmetric controller matrices are re- placed with the corresponding Toeplitz symmetric matrices. After confirming stability of the system with the process and controller Toeplitz symmetric matrices, the procedure for modifying CD control near the edges, presented in Section 5.2, can be implemented. Chapter 5. Closed-Loop Approach to CD Controller Modifications 69 5.3.2 Controller Modifications and Closed-Loop Simulations The closed-loop simulations have been performed with the steady-state process output disturbance, d(z) in (2.9), as shown in Figure 5.7. Near one edge (left side), the disturbance has a significant high spatial frequency content, and near the other (right side), has a smoother appearance. The first type of disturbance very often leads to system instabilities in real life CD systems as the actuator array is trying to remove the uncontrollable (high frequency) modes of the process output disturbance. The second type of disturbance (introduced near the right edge) is usually successfully attenuated by the control systems. Cross -d i rec t i on Figure 5.7: Process output disturbance d (at zero temporal frequency u = 0). The closed-loop simulation results are shown in Figures 5.8-5.11, and summarized in Table 5.1. Figure 5.8 illustrates the closed-loop steady-state process output and actuator profile using the reflection padding (one of the techniques currently used in industry). It can be seen that, in spite of excessive control action near left edge, the process output profile is not particularly good, with maximum and minimum varying between 1.176 and -1.235. Simulation results, with the controller tuned in turn conservatively, balanced, and aggressively using the new approach presented in Sections 5.1 and 5.2, are given in Figures 5.9 - 5.11. In all three cases, the sizes of the rectangular weights Wj, i = 1,..., 7 in (5.18)—(5.19) were chosen as, nci — 5 , n c 2 = 8, nm = 3,nr>2 = 8, nu = 8,n<2 = 8,ny = 8. The output vectors wa(z) and Wb(z) in Figure 5.1 were chosen as: wa(z) = [kP • y(z) u(z)]T, wb(z) = \ • u(z), (5.24) 1 + kR where y(z) and u(z) are the process output and control signal respectively, and coefficients kp and k^ are the tuning variables. Chapter 5. Closed-Loop Approach to CD Controller Modifications 70 Parameter kp is a closed-loop performance tuning variable, as it affects the gener- alized plant iVQ in (5.6). The point of reference for kp is the inverse of the process maximum singular value at steady-state. For the CD process model given with (5.22), <T(G(eJ' 0)) = 00043  = 232.56, where cf(-) denotes the maximum singular value. Variable kR, on the other hand, is a closed-loop robustness tuning variable, since it affects the generalized plant iVj, in (5.7). Parameter kp is determined considering a maximum al- lowed performance degradation at higher frequencies (requirement (d) in Section 5.1). For example, a maximum allowed degradation of 10% corresponds to kp = 0.1. In the case of conservative tuning (Figure 5.9), the tuning variables kp and kp in (5.24) were chosen as kp = 300, = 0.2. The computed controller modifications 5C and SD are given with (5.25)-(5.26) respectively1. SC = 0.01284 -0.02690 -0.05871 -0.07497 -0.04865 0.01139 0.02531 0.02930 0.06390 -0.07756 -0.02907 -0.03857 0.06664 0.02755 0.11821 0.22949 0.04778 0.05045 0.20296 0.19367 0.01151 -0.00872 -0.03373 0.11884 0.19066 0.03860 -0.05851 -0.11619 0.07592 0.06971 -0.07427 -0.02425 -0.15091 -0.05849 -0.05596 -0.17865 -0.10772 0.17263 0.03482 -0.07492 (5.25) SD = -0.08934 -0.00653 0.01573 0.00946 -0.02752 -0.05200 -0.04448 0.00710 0.02074 -0.01803 0.00926 -0.01185 -0.06767 0.02585 0.01692 -0.00943 0.01168 -0.01664 0.00547 -0.01097 -0.01788 0.00885 0.00188 -0.00230J (5.26) The resulting upper left sections of the resulting control law matrices Cd+5C and Dd+SD, in the case of conservative tuning, are given in (5.27)-(5.28). Cd(l : nciA • nC2) + 5C = -0.19606 -0.20151 -0.07777 0.04703 0.05961 -0.23980 -0.18359 -0.14626 -0.02115 0.07688 -0.10741 -0.18360 -0.09069 -0.16245 -0.08243 0.01063 0.01520 0.01659 -0.00594 -0.09406 -0.00055 0.00804 -0.00092 -0.01923 -0.01824 0.03860 0.07592 0.06971 -0.01041 -0.07427 -0.02425 -0.03059 -0.10281 -0.05849 -0.10466 -0.09305 -0.05962 -0.04027 -0.01388 0.01068 (5.27) 1 A l l the computed matrix modifications in this Chapter and Chapter 6 are given with the precision 10 - 5, as that level of accuracy has been achieved in the communication link between the Matlab prototype software and the industrial software packages. Chapter 5. Closed-Loop Approach to CD Controller Modifications 71 Dd(l : ri£)i, 1 : nm) + SD = ' 0.89846 -0.00043 0.01573 0.00946 -0.04590 0.94332 0.01320 0.02074 0.00926 -0.00575 0.92013 0.03195 -0.02752 -0.00943 0.01168 0.00885 -0.01803 -0.01664 0.00547 0.00188 0.01692 -0.01097 -0.01788 -0.00230. (5.28) In the case of balanced tuning (Figure 5.10), the tuning variables kp and kp, in (5.24) were chosen as kp = 1600, k^ = 0.2, and the computed controller modifications are given with (5.29)-(5.30). 5C = 0.01067 -0.01959 0.01019 0.02468 -0.01064 -0.02047 0.03168 -0.02143 -0.00089 0.01550 0.01566 -0.02330 0.02107 -0.05615 0.01087 0.00062 0.00434 -0.00808 0.05064 -0.03997 -0.00071 -0.00675 0.00353 -0.02132 0.04554 -0.00375 0.00885 -0.00198 -0.00315 -0.02272 -0.00422 0.00482 0.00041 0.00791 -0.00035 0.01422 -0.01655 0.00169 -0.00904 0.00431 (5.29) SD = 0.00605 -0.00448 0.00050 -0.00394 -0.00041 0.00139 -0.00233 0.00171 0.00080 0.00721 -0.00421 -0.00348 0.00467 0.00016 -0.00390 -0.00497 0.00906 -0.00781 0.00025 0.00459 -0.00274 -0.00154 -0.00085 -0.00299. (5.30) The upper left sections of the resulting control law matrices Cd + SC and Dd + SD, in the case of balanced tuning, are given in (5.31)-(5.32). Cd(l: nc-i.1 :nC2)+5C = -0.19823 -0.23337 -0.03304 0.08622 0.04739 -0.00375 -0.00422 -0.23249 -0.17722 -0.23620 -0.04436 0.07885 0.05695 0.00482 -0.03851 -0.23433 -0.18783 -0.22098 -0.04517 0.08362 0.04851 0.11028 -0.04959 -0.26905 -0.15826 -0.23422 -0.05185 0.09351 0.03746 0.10110 -0.03783 -0.25287 -0.16336 -0.23562 -0.04905 0.01422 -0.01655 0.00169 0.03906 0.08991 (5.31) Chapter 5. Closed-Loop Approach to CD Controller Modifications 72 Dd{l •• nDi, 1 : nm) + SD = "0.99385 0.00216 -0.00041 0.00139 0.00162 0.99501 0.00189 -0.00348 0.00050 0.00113 0.99686 -0.00171 -0.00233 0.00171 0.00080 -0.00154 0.00467 0.00016 -0.00390 -0.00085 0.00025 0.00459 -0.00274 -0.00299. (5.32) Finally in the case of aggressive tuning, the tuning parameters kp and kp in (5.24) were chosen as kp = 2400, k^ = 0.5, and the computed controller modifications SC and SD are given with (5.33)-(5.34). SC-. 0.00139 0.09924 0.14139 0.36969 -0.22865 -0.09556 -0.06312 -0.08458 -0.09298 0.36187 0.12427 -0.26003 -0.14097 -0.65551 0.15454 -0.04591 0.32070 0.12526 0.56045 -0.50466 0.01286 -0.21818 -0.09832 -0.37441 0.44304 0.02910 0.07590 0.02563 0.00775 -0.25590 0.00224 0.07775 0.10160 0.29926 -0.10725 0.07400 -0.21568 -0.06081 -0.18947 0.28026 (5.33) 6D = 0.00854 -0.00493 0.00013 -0.00446 0.00907 -0.00532 -0.00089 0.00142 -0.00418 -0.00462 -0.00233 0.00238 0.00518 0.00051 0.01133 -0.00892 -0.00022 0.00504 0.00097 -0.00436 -0.00290 -0.00067 -0.00275 -0.00288J (5.34) The upper left sections of the resulting control law matrices Cd + SC and Dd + SD, in the case of aggressive tuning, are given in (5.35)-(5.36). Cd(l : nciA : nC2) + SC = -0.20751 -0.30846 0.07557 0.03969 -0.11366 -0.27202 -0.47293 0.27200 0.09269 -0.29748 -0.34987 -0.08764 0.45529 -0.14168 -0.86841 0.35155 -0.18055 0.44747 0.10584 -0.71756 0.06096 -0.13258 -0.14702 0.02910 0.12400 0.11123 -0.58731 -0.04095 0,00224 0.07775 0.14970 0.38486 0.23414 -0.46880 -0.15595 0.07400 -0.21568 -0.06081 -0.14137 0.36586 (5.35) Chapter 5. Closed-Loop Approach to CD Controller Modifications 73 Figure 5.8: Steady-state process output (a) and control signal (b), using the current industrial technique - reflection padding. Dd{l •• nDi, 1 : nD2) + 5D = "0.99634 0.00164 -0.00089 0.00142 0.00117 0.99687 0.00192 -0.00462 0.00013 0.00078 0.99913 -0.00282 -0.00233 0.00238 0.00097 -0.00290 0.00518 0.00051 -0.00436 -0.00067 -0.00022 0.00504 -0.00275 -0.00288. (5.36) It can be seen from Figures 5.9 - 5.11 and Table 5.1 that, by using the approach presented in Sections 5.1 and 5.2, a successful trade-off between performance (a flat CD profile) and the corresponding control signal magnitude, can be achieved. From Table 5.1, it can be noticed that in case of all three tunings based on the new approach, the output profile has been improved in comparison to the result achieved with the current industrial technique. Also, in the cases of conservative and balanced tunings, such a result has been achieved with less actuator usage. Chapter 5. Closed-Loop Approach to CD Controller Modifications 74 21 1 1 1 , 1 1 r (a) 5 10 15 20 25 30 35 cn c o o 'E 15 20 25 Cross -d i r ec t i on Figure 5.9: Steady-state process output (a) and control signal (b), using the new technique - conservative tuning (kp = 300 and kp = 0.2 in (5.24)). 10 15 20 25 30 35 15 20 C ross -d i r ec t i on Figure 5.10: Steady-state process output (a) and control signal (b), using the new tech- nique - balanced tuning (kp = 1600 and kp = 0.2 in (5.24)). Chapter 5. Closed-Loop Approach to CD Controller Modifications 75 Figure 5.11: Steady-state process output (a) and control signal (b), using the new tech- nique - aggressive tuning (kp = 2400 and kR — 0.5 in (5.24)). Current industrial New technique technique Reflection Conservative Balanced Aggressive padding tuning: tuning: tuning: kP = 300 kP = 1600 kP = 2400 kR = 0.2 kR = 0.2 kR = 0.5 Process output [gsm] 1.9913 1.8305 1.4467 1.0577 Control signal [microns] 3307 2376 2974 4436 Table 5.1: 2-norm of the process output and control signal steady-state profiles shown in Figures 5.8 - 5.11. Chapter 5. Closed-Loop Approach to CD Controller Modifications 76 5.4 Stabilization Procedure (Rarely Required) Modification of process and controller matrices from circulant into Toeplitz symmetric, as proposed in the overall algorithm presented in Section 5.2, clearly represents a one- step elimination of the corresponding circulant matrices' 'ears' AC and AD in (2.20). As pointed out earlier, it has been established (based on numerous industrial and simulation data analyzed) that, except in some pathological cases, the resulting closed-loop systems with Toeplitz symmetric process and controller matrices are nominally stable. However, if that is not the gradual elimination of the circulant symmetric matrices' 'ears' AGc(z),AC, and A D in (2.19)-(2.20) is proposed. This is illustrated in Figure 5.12 with the parameter A 6 [0,1]: A = 0 corresponds to periodic boundary conditions (circulant symmetric process and controller models), and A = 1 corresponds to Dirichlet boundary conditions (Toeplitz symmetric process and controller models). Further controller mod- ifications SC and SD in Figure 5.12, to be computed in accordance with the algorithm below, are defined with (2.21)-(2.22). The overall stabilization algorithm is given as follows: 1. Find the maximum.value of A € [0,1] for which the system is closed-loop stable. IF A = 1 GOTO STEP 6. 2. Based on the diagram in Figure 5.2, with Wi: i = 1,2,3,5,6,7 given with (5.18)— (5.19) and W4 = 0nyXn, define linear fractional transformations (LFT's) for comput- ing modifications Ce and De in Figure 5.3 (Note: W 4 = 0, i.e. process output y is not a part of the LFT's exogenous output vector, as in the stabilization procedure we are exclusively interested in reducing the gain d —> u). 3. Using the algorithm from Section 5.2.2, compute modifications De and Ce and find the corresponding controller modifications SC and SD in (2.21)-(2.22). 4. Update the controller matrices C and D with the above obtained SC and SD - as illustrated in Figure 5.12. 5. GOTO STEP 1. 6. END. At the end of the above algorithm, the process and controller 'ears' AG(z), AC and A D in (2.19)-(2.20) are eliminated, and further controller modifications, that optimize the gain from the process output disturbance d to process output y and control signal u, as detailed in Section 5.2, can be carried out. Chapter 5. Closed-Loop Approach to CD Controller Modifications 77 -A-AC c(z) -AAD -A-AG(zt \d(z) u(z) sc yd SD Figure 5.12: A gradual elimination of the process and controller circulant symmetric matrices' 'ears' with the parameter A € [0,1]. 5.4.1 E x a m p l e As pointed out above, the stabilization procedure, when going from periodic to Dirichlet boundary conditions, is very rarely (practically never) required for the actual CD control systems designed with the two-dimensional loop shaping technique. However, after exten- sive simulation studies, one CD control system, destabilized by the change of the boundary conditions from periodic to Dirichlet (as defined in Section 2.1), was found. The simulated system had 36 actuators, with the fabricated but realistic process parameters resulting in a very wide bi-modal spatial response, illustrated in Figure 5.13. The specific parameters of the process model in (2.10)—(2.11) are given as, {60, & i , . . . , b12} = 10~3 • {0.56966,0.85783,0.8600,0.010764, -0.63285, -0.47079, -0.091566, 0.047077,0.029923,0.0053444, -0.00039524, -0.00027924, -0.00001078} a 0 = 0.916, d = 3 (5.37) Next, the corresponding feedback controller K(z) in (2.12) was intentionally designed with (unacceptably) small stability margins, {c0, ci, ca, c3, c4, c5} = {-0.14168, -0.15141, -0.11392, 0.0046317,0.090511,0.068683} {do.di} = {0.9998,0.00010119}, (5.38) and the tuning parameter a of the Dahlin controller c(z) in (2.12), a = 0.88959. The resulting circulant (spatially-invariant) closed-loop system (corresponding to A = 0, SC — Chapter 5. Closed-Loop Approach to CD Controller Modifications 78 0.015 5 10 15 20 25 30 35 I _j i i i i i 10 15 20 25 30 35 ACTUATOR NUMBER Figure 5.13: Actuator array shape (upper figure) and the corresponding process steady- state response (lower figure) for the process model given with (5.37). 0,6D = 0 in Figure 5.12) is (barely) stable1 with the maximum closed-loop pole magnitude equal to 0.99997. However, the system with the same parameters (5.37)-(5.38), and Dirichlet boundary conditions (i.e. process and controller Toeplitz symmetric matrices, corresponding to A = 1,8C = 0,5D = 0 in Figure 5.12) is unstable - the maximum pole magnitude equal to 1.0004. Using the stabilization algorithm in Section 5.4, it was possible to stabilize the above system. First (following Step 1 of the stabilization algorithm), the maximum value of A 6 [0,1] for which the system remains stable was found to be A = 0.46 (resulting in a maximum closed-loop pole magnitude: 0.9999). Next, following Steps 2—4, the sizes of the rectan- gular weights Wj, z = 1,2,3,5,6, 7 in (5.18)—(5.19) were chosen as, ric\ = 5, n G 2 = 8, nD1 = 3,nx)2 = 8, n u = 8, rid = 8, and the output vector ujb(z) in Figure 5.2 as uib{z) = 0.5u(z). to rr O o < 2 1.5 1 0.5 0 -0.5 -1 •"The controller (5.38) should certainly never be implemented on a real paper machine having CD process parameters (5.37) as the stability margins are unacceptably small for industrial implementation. Chapter 5. Closed-Loop Approach to CD Controller Modifications 79 Subsequently, the controller modifications SC and SD were computed1, 6C(1: nCi, 1 : nC2) = 0.03229 -0.04907 0.00116 -0.01859 0.24680 0.10920 -0.00029 0.02756 -0.05612 -0.03441 0.01572 0.00011 -0.00887 -0.06934 -0.01601 -0.00013 0.00905 -0.01706 -0.04335 -0.02383 -0.00609 -0.00088 0.00379 -0.01454 -0.01608 0.02133 0.00062 -0.01221 0.02257 0.02055 -0.00397 0.00072 0.00192 0.09182 0.06249 -0.01116 0.00018 0.00615 0.09304 0.05152 (5.39) 5D(1 : n m , 1 : nD2) = -1.00632 0.11875 0.00024 -0.81129 0.00020 0.10728 -0.26057 1.07935 -0.49785 1.90570 -1.28327 1.75901 -1.12241 0.35009 0.11752 -0.89432 -1.58477 0.22826 0.94664 -1.75554 -0.94426 -0.25715 0.98224 -1.00001. (5.40) The resulting upper left sections of the controller matrices CC+5C and DC + SD, in Figure 5.12, are given with (5.41)-(5.42). Cc(l:nCi,l:nc2) + SC = -0.10939 -0.20048 -0.09820 -0.01138 0.08442 0.09001 -0.00397 -0.15025 -0.14197 -0.15130 -0.11405 0.00375 0.09113 0.06940 -0.13251 -0.12385 -0.15055 -0.14236 -0.11013 -0.00758 0.09243 0.25143 -0.17004 -0.22075 -0.15874 -0.16595 -0.09135 "0.09645 0.19971 -0.02978 -0.15727 -0.17524 -0.15776 -0.13086 -0.05143 -0.01116 0.00018 0.07483 0.18355 0.05616 (5.41) £> c (l : UDi, 1 : n D 2 ) + 5D = -0.00652 0.11885 -0.26057 1.07935 -1.12241 0.00034 0.18851 -0.49775 1.90570 -1.58477 0.00020 0.10738 -0.28347 1.75911 -0.94426 0.35009 0.11752 0.22826 0.94664 -0.25715 0.98224 -0.89432 -1.75554 -1.00001. (5.42) Finally, following Step 5 of the stabilization algorithm, it can be found that a complete elimination of the 'ears' of the matrices C c , Dc, and Gc(z) can be done (i.e. A set to 1), without destabilizing the system, as the resulting maximum closed-loop pole magnitude 1Note the magnitude of SD being significantly larger in this case of modification for stabilization than in the cases of modifications for performance illustrated in Section 5.3.2. Chapter 5. Closed-Loop Approach to CD Controller Modifications 80 is 0.99978. In other words, with the above stabilization algorithm, the maximum closed- loop pole magnitude has been reduced from 1.0004 (unstable system) to 0.99978 (stable system). However, it should be stressed again that the above stabilization procedure is practically never needed for the practical CD control systems, designed with the reasonable (practically acceptable) stability margin. 5.5 Summary A novel method for modifying a CD control law near spatial domain boundaries that sys- tematically takes into account all the requirements from Chapter 2 has been developed in this chapter. A simulation example, with the use of Honeywell hardware-in-the-loop paper machine simulator, comparing the newly proposed technique with the current industrial practice has also been presented. The objective of modifying the industrial CD controller near spatial domain bound- aries has been posed, in this chapter, in terms of a block-diagonal static output feedback (SOF) compensator design problem. The problem has been solved and the control law modifications computed by implementing a novel low-bandwidth SOF controller design al- gorithm. The algorithm is implemented sequentially on the existing industrial controller's two constant matrix components. The newly proposed approach systematically takes into account all the closed-loop requirements (stability, performance, and robustness) specified in Chapter 2. Chapter 6 Industrial Trial The method for designing CD edge controllers presented in Chapter 5 has been evaluated in an industrial setting. Three different control law modifications (conservative, balanced, and aggressive), 5C and 5D in Figure 2.7 were computed and implemented during the trial. The mill was producing the paper grade on which the tests were carried out for about 15 hours on the day of the trial. During this period, the trial setup was arranged and the controller testing carried out. The newly developed approach provides a systematic way of modifying the existing, industrial, CD control law near the sheet edges, guaranteeing the resulting closed-loop system stability, robustness margins, and performance improvement. Considering that the resulting closed-loop system robustness margins near the sheet edges are directly taken into account, the possibility of CD control instability originating from the edges (as illustrated in Figure 1.9) and 'creeping' into the rest of the system is eliminated with the new approach. The new technique has a clear economic benefit for the papermakers, since with a stable, robust, and performance improving control law, the quality of the paper sheet near the edges can be significantly improved. In some papermaking situations, the 'trim squirts' (Figure 1.2) can be moved outward, thus resulting in less paper being trimmed off and more on-spec paper is being produced from which the papermaker can extract his orders. Two photos of the author taken in the mill where the trial was completed are given in Figure 6.1. Details regarding the paper machine used for testing, as well as the mill's existing CD control setup, are given in Section 6.1. The overall trial procedure and setup are detailed in Section 6.2. In Section 6.3, the results obtained during the industrial trial are presented. 6.1 CD Control Setup in the Mill The inaugural test site for the Matlab prototype tuning tool was selected to be a Canadian mill producing a wide variety of paper grades. During the site visit, the mill was producing a 66[lbs/ream] grade paper1. The particular paper machine, on which the testing was done, uses slice lip actuators to reduce the variations of the paper sheet cross-directional t l [lbs/ream] = 1.6289 [gr/m2] 81 Chapter 6. Industrial Trial 82 Figure 6.1: Stevo Mijanovic near machine on which the industrial trial was carried out. In the background: machine's forming section (left photo), and the press and dryer sections (right photo). dry weight1 profile, as discussed in the Introduction. The machine has 36 actuators spaced on xa = 95.6mm centres and distributed across the machine width of 3.44m. The total number of scanner measurement points was 216. High-pressure water jets ('trim squirts' - illustrated in Figure 1.2 in Section 1.1), in the sheet-forming section of the paper machine, were located approximately two actuator zones (or 12 measurement points or 191.2mm) from the machine edges. As a result of this trimming in the sheet-forming section, of 191.2mm wide sheet strips, on both sides of the machine, only the measurement points 13-205 were actual paper sheet measurements (so called 'on-sheet' measurements). Accordingly the actuators 3-34, corresponding to the 'on-sheet' measurements, are called 'on-sheet' actuators. The 'off-sheet' actuators (actuators 1-2 and 35-36) were in open- loop (at constant values) throughout the whole trial. The exact setpoint values for the 'off-sheet' actuators are given in Section 6.3 below. (An illustration of the 'on-sheet' and P a p e r sheet d r y we igh t is de f ined as t he sheet bas is we igh t r e d u c e d b y t he wa te r (mo is tu re ) con ten t . Chapter 6. Industrial Trial 83 SLICE LIP ACTUATOR A R R A Y SETPOINTS, u(t) L A N connection RocketPort Interface T_ RS-485 connection J DRY WEIGHT MEASUREMENT A R R A Y , y(t) Figure 6.2: A simplified schematic of the mill's CD control setup. 'off-sheet' actuators locations is also given in Figure 1.2.) The period over which the scanner was making one set of measurements (the control system sampling time) was Ts = 30s. A simplified schematic of the mill's CD control system is shown in Figure 6.2 (compare with Figure 1.7 given in Introduction). Slice lip actuators are connected via local L A N connection to a PC with the industrial CD controller software, while the scanning sensor is serially connected via RS-485 connection through a RocketPort interface to a RS-232 connection to computer. 6.2 Trial Setup and Procedure A simplified schematic of the industrial trial setup is given in Figure 6.3. The mill's ex- isting PC with the industrial CD controller software had to be replaced with the laptop on which the modified industrial controller software had been loaded (Laptop 1 in Figure 6.3). The industrial controller software was modified in order to accommodate the control law changes according to Figure 2.6. The existing control law modifications were com- puted, based on the approach presented in Chapter 5, using a newly developed Matlab prototype tuning tool. Proper dataflow between the Matlab prototype tuning tool and the industrial software packages (outlined in Figure 6.4), as well as correct implementa- tion of the computed modifications by the industrial control software had previously been thoroughly tested and verified [48]. Chapter 6. Industrial Trial 84 LAPTOP 2: MODIFIED INDUSTRIAL IDENTIFICATION AND TUNING SOFTWARE and M A T L A B P R O T O T Y P E SOFTWARE L A N connection LAPTOP 1: MODIFIED INDUSTRIAL PAPER MACHINE \ \ RS485-RS232 adapter i RS-485 connection Figure 6.3: A simplified schematic of the industrial trial setup. The overall procedure followed for the field trial is outlined as follows: 1. Al l the system parameters, for the particular paper grade being tested, were copied from the mill's PC with the industrial CD controller algorithm (PC in Figure 6.2) onto the laptop computer (Laptop 1 in Figure 6.3) with the specially modified con- troller algorithm. 2. The control system actuators were placed in open-loop and left at the positions assumed just before taking them out of the closed-loop. 3. The scanner was disconnected from the mill's PC and connected (through a RS485- RS232 adapter) to Laptop 1. It was subsequently confirmed that the communication link between the paper machine scanner and Laptop 1 was working properly and that controller on Laptop 1 was receiving the scanner measurements. 4. The actuator array of the controller on Laptop 1 was verified to be in Manual mode so that a smooth transition from the mill's existing PC to Laptop 1 control software could be ensured. The mill's PC was disconnected from the mill's L A N , thus detaching the paper machine's actuator array from the existing control software Chapter 6. Industrial Trial 85 M A T L A B P R O T O T Y P E S O F T W A R E INPUT DATA: Identified Process Model & Spatially-invariant controller parameters (obtained by 2-D loop shaping technique) MODIFED INDUST. IDENT. & TUNING SOFTWARE INPUT DATA OUTPUT DATA MATLAB PROTOTYPE SOFTWARE U S E R TUNING w„ and wh MODIFED INDUST. CD CONTROL SOFTWARE M A T L A B P R O T O T Y P E S O F T W A R E OUTPUT DATA: Computed C D controller modifications 8C and 6D in Figure 2.7 SPECIF IED PARAMETERS: in Figure 5.1 Figure 6.4: The dataflow diagram between the Matlab prototype software and the indus- trial software packages. residing on the PC. Afterwards, Laptop 1 (previously having been assigned the same TCP-IP address as the above mentioned PC) was connected onto the local L A N . 5. The communication link between the paper machine actuator array and Laptop 1 control software was checked and confirmed to be working properly. The modi- fied industrial controller software on Laptop 1 was receiving the actuators' current position information. 6. The actuator array was put on control (i.e. in closed-loop) with Laptop 1 controller settings the same as on the mill's industrial controller PC (the setting had been previously copied as outlined in Step 1 above). The control system was left in closed- loop for about an hour, to ensure it operated correctly with Laptop 1 controller software. 7. A local TCP-IP address was assigned to the second laptop computer (Laptop 2 in Figure 6.3) with the industrial CD process identification and controller tuning software [32, 65], as well as the Matlab prototype software for computing controller modifications near the paper sheet edges, as detailed in Chapter 5. Laptop 2 was successfully connected onto the local L A N . 8. The control system actuator array was put in open-loop (off control), and the stan- dard process identification procedure [32] was initiated from Laptop 2. The open- loop 'bump test', with three actuators stepped up (down) for 200 microns, was Chapter 6. Industrial Trial 86 carried out and process identification and controller tuning completed using the industrial software presented in [32, 65] (see Figure 6.5). 9. The design for various modifications De and Ce in Figure 5.5 was completed accord- ing to the algorithm outlined in Section 5.2.2, and these parameters were transferred by the modified identification and tuning software on Laptop 2 over the L A N into the correct database location, for use by the modified industrial controller software on Laptop 1. (The dataflow between the Matlab prototype software and the industrial software packages is outlined in Figure 6.4.) 10. The trial control system was put on closed-loop with one of the existing indus- trial techniques for CD control near the paper sheet edges - average padding. The resulting closed-loop data were logged over the course of about an hour. 11. The three sets of controller modifications (referred to as conservative, balanced, and aggressive), computed in Step 9 above, were implemented in turn (each one over the course of about an hour). The data logged in Step 10, with the current industrial technique, and in this step with the three sets of tuning numbers obtained with the new technique are called Data Set 1 in Section 6.3 below. 12. The steps 10-11 above were repeated once more. This logged data are called Data Set 2 (Section 6.3). 13. The control system actuators were put back in open-loop so that a smooth transition back to the mill's existing industrial controller system on the PC could be carried out. 14. The scanner was disconnected from Laptop 1 and connected back to the mill's PC with the industrial CD controller software. Subsequently, Laptop 1 was also dis- connected from the local L A N , and the mill's PC connected back thus enabling the communication between the paper machine actuators and the PC with the industrial controller software. 15. After it was confirmed that the communication link between the paper machine and PC was working properly, the control system was put back in closed-loop with the original settings found upon arrival at the mill. The closed-loop data obtained during the trial procedure, outlined above, are presented and analyzed in Section 6.3. Chapter 6. Industrial Trial 87 6.3 Trial Results In Section 6.3.1, the process and controller parameters, as obtained by the industrial identification and tuning software [32, 65], are presented. The subsequently computed controller modifications 8C and 5D in (2.21)-(2.22), based on the approach outlined in Chapter 5, are given in Section 6.3.2. Finally, the closed-loop control data obtained during the trial are presented and analyzed in Sections 6.3.3 and 6.3.4. 6.3.1 Process and Controller Parameters As detailed in Step 8 of the field test trial procedure in Section 6.2, an open-loop 'bump test' with three actuators stepped up (down) for 200 microns was carried out. The system scan time (sampling period), as pointed out in Section 6.2, was Ts = 30s. The parameters of the process model in (2.10)—(2.11) were identified using software described in [32] with the size of the matrix B in (2.11) band lb = 8 and { r j 0 , 6 8 } = 10~2 • {0.1362,0.1033,0.0216,-0.0364,-0.0302, -0.0073,0.0005,0.0005,0.0001} a 0 = 0.759, d = 2 (6.1) The process model spatial response identification results are illustrated in Figure 6.5. Subsequently, the feedback controller K(z) in (2.12) was designed using the standard two-dimensional loop shaping technique [65, 66]. The controller parameters obtained had matrix band sizes lc = 4, Id = 1, with: {c0, ci, c2, c3, c4} = {-0.338689, -0.232651, -0.024331,0.070768, -0.002905} {d 0,di} = {0.991399,0.004301} (6.2) The tuning parameter a of the Dahlin controller c(z) in (2.16) was also produced by the two-dimensional loop shaping design: a = 0.8506. • 6.3.2 Computed Controller Modifications 5C and 5D Next, the design for various controller modifications (conservative, balanced, and aggres- sive tuning) De and Ce in Figure 5.5 and described by the algorithm in Section 5.2.2 was completed and these were transferred over the L A N into the correct database location for use by the industrial controller software on Laptop 1 (see Figures 6.3 and 6.4). In all three cases (conservative, balanced, and aggressive tuning), the sizes of the rectangular weights Wi,i — 1,...,7 in (5.18)-(5.19) were chosen as, nci = 5,nc 2 = Chapter 6. Industrial Trial 88 i r 1000 2000 3000 CD POSITION [mm] Figure 6.5: Model identification: The upper plot illustrates the shape of the actuator array used for process excitation. The middle plot shows the measured process output profile. The bottom plot illustrates the identified model, as given by the parameters bj, 3 = 0,1, 2,...,8 in (6.1). 8, nm = 3, nm = 8, nu = 8,na = 8, ny = 8. Also, the output vectors wa(z) and wb(z) in Figure 5.1 were chosen as: wa(z) = [kP • y(z) u(z)}T, wb(z) = 1 • u(z), (6.3) 1 + kp where y(z) and u(z) are the process output and control signal respectively, and coefficients k p and k^, are the t u n i n g v a r i a b l e s . As explained in more detail in Section 5.3.2, kp is a closed-loop performance tuning variable, and kp is a a closed-loop robustness tuning variable. For the CD process model given with (6.1), inverse of the process' maximum I singular value at steady-state (i.e. the point of reference for choosing kp) is =(G(eJ°)) = 59.74. 0.01674 In the case of c o n s e r v a t i v e t u n i n g , the tuning variables kp and kp in (6.3) were cho- sen as k p = 180, kR, = 0.2. Subsequently, using Matlab prototype software, the industrial Chapter 6. Industrial Trial 89 controller modifications SC and SD given with (6.4)-(6.5) respectively were obtained. SC = 0.03849 -0.15373 0.01922 -0.03055 0.03328 0.03452 -0.00357 -0.03630 -0.02833 -0.08007 0.00158 0.10040 -0.00537 0.11562 -0.03483 -0.00733 0.01442 0.02771 0.15004 0.08597 0.00579 -0.08664 -0.00198 0.00179 0.08674 0.00955 -0.03759 -0.03342 -0.11406 -0.00287 0.08151 0.00068 -0.04143 -0.03174 -0.10293 -0.01349 0.11252 0.05602 0.09988 -0.05167 (6.4) 6D = -0.04205 0.00652 0.01487 0.03983 -0.03423 0.00459 -0.00329 0.00521 -0.00765 -0.05617 0.05445 -0.02583 0.00602 -0.01773 0.02016 -0.00503 -0.04141 0.03139 -0.00380 -0.00736 -0.01085 0.00119 -0.00059 0.01548 (6.5) The upper left sections of the resulting control law matrices Cd + SC and Dd + SD, in the case of conservative tuning, are given in (6.6)-(6.7). Cd(l:nci,l ••nC2)+5C = -0.30020 -0.19813 -0.02275 0.06343 0.00288 0.00955 -0.00287 -0.38639 -0.34226 -0.13226 -0.00992 -0.01588 -0.04049 0.08151 -0.00511 -0.26895 -0.34406 -0.20494 -0.02631 0.03735 -0.00223 0.04021 -0.05266 -0.11703 -0.18865 -0.23086 -0.13839 0.02934 0.03038 -0.00930 -0.05916 -0.14668 -0.25195 -0.26439 -0.12726 -0.01349 0.11252 0.05602 0.09697 0.01910 (6.6) Dd(l : nm, 1 : 7 i D 2 ) +5D = "0.94935 0.04413 -0.03423 0.00459 0.01083 0.93523 0.05875 -0.02583 0.01487 -0.00073 0.94999 0.03569 -0.00329 0.00521 -0.00765 0.00119 0.00602 -0.01773 0.02016 -0.00059 -0.00380 -0.00736 -0.01085 0.01548 . (6-7) In the case of balanced tuning, the tuning variables kp and kp in (6.3) were chosen as k p = 480, k R = 0.2. Subsequently computed modifications SC and SD are given in Chapter 6. Industrial Trial 90 (6.8)-(6.9), respectively. SC = -0.02680 0.04499 0.03625 0.10985 0.05526 0.01935 -0.09894 0.02275 0.02207 0.07139 0.00090 0.01944 -0.02307 -0.00882 -0.01473 -0.03639 0.13122 -0.00725 0.07469 -0.01478 0.02692 -0.12908 0.03103 0.01370 0.08206 0.01608 -0.04844 -0.00458 -0.02780 0.00500 -0.02549 0.11027 -0.01791 0.01941 -0.04921 -0.01109 0.04539 -0.00541 0.01187 -0.01843 (6.i 6D = 0.00116 -0.00272 0.00228 0.00256 -0.00109 -0.00449 -0.00724 0.00979 0.00189 0.00569 -0.02097 0.00140 -0.00653 0.02342 -0.00597 0.00641 -0.02121 0.00740 -0.00604 0.01315 -0.00971 0.00284 0.00105 0.00802_ (6.9) The upper left sections of the resulting control law matrices Cd + 8C and Dd + SD, in the case of balanced tuning, are given in (6.10)-(6.11). Cd(l:nci,l:nC2)+SC = -0.36549 -0.21330 -0.02343 0.03438 0.02402 0.01608 -0.02549 -0.18767 -0.43763 -0.21321 0.10689 -0.05831 -0.05134 0.11027 0.01192 -0.20990 -0.36176 -0.23990 0.00669 0.06619 -0.02081 0.18062 -0.00226 -0.24147 -0.26400 -0.21895 -0.05213 0.09018 0.05235 0.14216 -0.03906 -0.24743 -0.25663 -0.22765 -0.07355 -0.01109 0.04539 -0.00541 0.00897 0.05234 (6.10) Dd{l • nm, 1 : nD2) + SD - 0.99256 0.00687 -0.00724 0.00569 -0.00653 0.00158 0.99031 0.01409 -0.02097 0.02342 0.00228 -0.00019 0.99329 0.00570 -0.00597 0.00641 -0.00604 0.00284 -0.02121 0.01315 0.00105 0.00740 -0.00971 0.00802 (6.11) Finally, in the case of aggressive tuning, the tuning variables kp and kp in (6.3) were chosen as k P = 720, k R = 0.4, and the resulting controller modifications SC and Chapter 6. Industrial Trial 91 SD are given in (6.12)-(6.13), respectively. 6C = -0.09772 0.29758 0.01688 0.20798 -0.00883 0.05064 -0.24731 0.06060 0.02322 0.15726 0.05991 -0.16736 -0.02607 -0.13026 0.00132 -0.06116 0.27843 -0.05450 0.02426 -0.13232 0.02286 -0.08941 0.01129 -0.02241 0.02982 -0.00860 0.01597 0.00495 0.02265 0.00226 -0.04174 0.09766 0.03017 0.11865 0.02909 0.04844 -0.21222 0.03833 -0.02289 0.09746 (6.12) SD 0.00528 -0.00359 0.00102 -0.00122 -0.00434 0.00568 -0.00692 0.00428 0.00522 -0.02034 0.02512 -0.00439 0.00599 -0.00146 -0.00618 0.00658 -0.00586 0.00297 -0.02162 0.01246 0.00124 0.00881 -0.00960 0.00731. (6.13) The upper left sections of the resulting control law matrices Cd + SC and Dd + SD, in the case of aggressive tuning, are given in (6.14)-(6.15). Cd{\ : n Q i , l :nC2)+SC = -0.43641 -0.18201 0.03558 0.00961 0.01995 -0.00860 -0.04174 0.06493 -0.58600 -0.40001 0.25409 -0.01864 0.01307 0.09766 -0.00745 -0.17205 -0.36476 -0.28715 -0.01304 0.07572 0.02726 0.27875 -0.00111 -0.36291 -0.31443 -0.25507 -0.00169 0.18941 -0.01173 0.22803 -0.02301 -0.36497 -0.30886 -0.23039 0.00476 0.04844 -0.21222 0.03833 -0.02580 0.16822 (6.14) Dd(l : nD1,l : nD2) + 5D = 0.99668 0.00308 -0.00434 0.00568 -0.00692 0.00071 0.99567 0.00952 -0.02034 0.02512 0.00102 -0.00009 0.99738 0.00284 -0.00618 0.00658 -0.00586 0.00297 -0.02162 0.01246 0.00124 0.00881 -0.00960 0.00731. (6.15) 6.3.3 C losed -Loop C o n t r o l Resu l t s : D a t a Set 1 The Data Set 1 results (Steps 10-11 in the procedure outlined in Section 6.2) are illustrated in Figures 6.6-6.9 and Table 6.1. The closed-loop results obtained using the current industrial practice (with average padding) are shown in Figure 6.6, and the results obtained Chapter 6. Industrial Trial 92 with the new technique (conservative, balanced, and aggressive tuning in turn) in Figures 6.7-6.9. Al l the plots in Figures 6.6-6.9 were obtained using Honeywell software for paper machine cross-directional control systems data analysis. As detailed in Section 6.2, the closed-loop data was logged over the course of 40-60 min for each set of tuning numbers (the existing industrial technique - average padding and the three sets of tuning numbers obtained with the new technique). In other words, between 80-120 scans (as the system sampling time was Ts = 30s as specified in Section 6.3.1) of the closed-loop data in each case. In order to avoid transient effects in changing controller tuning, the data analysis software was used to average the last 20 scans in each case. In each figure, in addition to maximum, minimum, and average signal value, the 2-norm of the corresponding signal is given. Note that in the case of the actuator array profile, only the 2-norm of the on-sheet actuators is given as the off-sheet actuators (2 on each side u(l : 2) and u(35 : 36)) were in open-loop and at constant positions throughout the trial: [it(l); it(2); u(35); u(36)] = [-444.2; -296.4; -271.2; -370.8]. From Figures 6.6-6.9 and Table 6.1, it can be seen that in the cases of the balanced and aggressive tunings with the new technique, the paper sheet variations are smaller than with the existing industrial technique and this improvement was achieved with smaller control signals (i.e. actuator usage). For example, in the case of balanced tuning, the paper sheet dry weight variations are reduced by 5.8% with the overall control signal magnitude reduced by 9.4%. The difference between the control signal magnitudes is even more dramatic if only the first (and last) 5 on-sheet actuators are compared (u(3 : 7) and u(30 : 34) respectively in Table 6.1) since these are the actuators directly modified with the new technique1. In the case of conservative tuning, as expected, the sheet variations are larger than those obtained with the existing technique. This is not surprising since a significantly smaller actuator usage is invoked, particularly near the left edge (see u(3 : 7) in Table 6.1). It can also be noticed that, in the case of aggressive tuning, the sheet variations are the same as those obtained with the balanced tuning. Obviously for the particular process model and process output disturbance encountered during the trial, making the CD controller modifications near the sheet edges more aggressive than those obtained with the balanced tuning (given with (6.8)-(6.9)) is not warranted. 1 Partitioning the process output in the same way would not be appropriate since considering the identified process model given with (6.1), 5 edge actuators on each side directly influence the process output across 13 zones on the corresponding side. As a result, a total of 26 (out of 32) zones are directly affected by the modified actuator zones. Chapter 6. Industrial Trial 93 67.5 „ 67 E CO S 66.5 co " 66 65.5 Average = 66.3, Min = 65.8, Max = 66.89 llyll2= 2.706 (a) 200 h 100 0 -100 -200 0 500 1000 1500 2000 2500 3000 CD Position [mm] (b) . l l l l l l l l l l l l l l l . • ' Average = 43.1, Min = -205.5, Max = 164.7 I I  164.7 ||u||2= 556.10 10 15 20 25 Actuator Number 30 35 Figure 6.6: Data set 1 process output (a) and control signal (b), using the current indus- trial technique - average padding. 65.5 200 100 500 1000 1500 2000 2500 CD Position [mm] o o 0 -100 -200 II" .--I 3000 l l l l l l l l l l l l l l . . . Average = 43.13, Min = -110.60, Max = 158.0 ||U|L= 498.89 (b) l|l 10 15 20 25 Actuator Number 30 35 Figure 6.7: Data Set 1 process output (a) and control signal (b), using the new technique - conservative tuning (kp = 180 and kp_ = 0.2 in (6.3)). Chapter 6. Industrial Trial 94 E cc! CD i CO JO 67.5 67 h 66.5 66 65.5 Average = 66.22,' Min = 65.7^, Max = 66.8 (a) lly|l2= 2-548 500 1000 1500 2000 2500 CD Position [mm] 3000 200 100 0 -100 -200 r • i r i i i u r i l i i L . ^ . Average = 43.12, Min = -163.15, Max = 155.50 (b) 'II |u||2= 503.63 10 15 20 25 Actuator Number 30 35 Figure 6.8: Data Set 1 process output (a) and control signal (b), using the new technique - balanced tuning (kp = 480 and kp = 0.2 in (6.3)). 200 - „ 100 - 2 o- o " -100 - -200 - 1000 1500 2000 2500 CD Position [mm] 3000 (b) A llllllllllllllll._-a__l Average = 43.15, Min = -194.08, Max = 165.53 H ||u|| = 540.40 10 15 20 25 Actuator Number 30 35 Figure 6.9: Data Set 1 process output (a) and control signal (b), using the new technique - aggressive tuning (kp — 720 and kp — 0.4 in (6.3)). Chapter 6. Industrial Trial 95 Current technique New approach Average Conservative Balanced Aggressive padding tuning: tuning: tuning: kP = 180 kP = 480 kP = 720 kp = 0.2 kp = 0.2 kp = 0.4 ||y||2 [lbs/ream] 2.706 3.151 2.548 2.548 \\u(3 : 34)112 [microns] 556.10 498.89 503.63 540.40 On-sheet actuator array sections ||w(3 : 7)||2 [microns] 226.91 140.38 182.44 205.42 ||«(8 : 29)||2 [microns] 478.16 456.06 444.26 473.00 ||u(30 : 34)||2 [microns] 169.22 145.61 151.64 161.58 Table 6.1: 2-norms of the process output and control signal profiles shown in Figures 6.6 - 6.9 (Data Set 1). 6.3.4 Closed-Loop Control Results: Data Set 2 The Data Set 2 results (Step 12 in the procedure outlined in Section 6.2) are illustrated in Figures 6.10-6.13 and Table 6.2. The closed-loop results obtained using the current industrial practice (with average padding) are illustrated in Figure 6.10, and the results obtained with the new technique (conservative, balanced, and aggressive tuning in turn) in Figures 6.11-6.13. The data is analyzed in the same way (using the Honeywell cross-directional control data analysis software) as Data Set 1 (Section 6.3.3). The off-sheet actuators were again in open-loop and at the constant positions: [u(l);u(2); u(35); u(36)] = [-444.2; -296.4; -271.2; -370.8]. From Figures 6.10-6.13 and Table 6.2, it can be noticed that the results are similar to those from Data Set 1, presented in Section 6.3.3. The paper sheet dry weight variations are again reduced by using the new technique with balanced and aggressive tunings in comparison to the existing industrial practice. In this data set, the sheet variations are reduced by 8.7% with a reduction of the control signal magnitude by 5.5% by using the balanced tuning in comparison to the existing industrial practice (see Table 6.2). However, while smaller than the variations resulting from the use of the existing industrial technique, the variations obtained with the aggressive tuning are larger than those obtained with the balanced tuning. This confirms as has been stated in Section 6.3.3, that for the particular process and disturbance characteristics encountered on the day of trial, making the CD control law modifications near the sheet edges more aggressive than those obtained with the balanced tuning (given with (6.8)-(6.9)) is not warranted. Also, as expected, in the case of conservative tuning, the sheet dry weight variations are larger Chapter 6. Industrial Trial 96 E CB a> In 200 To 100 o o 0 -100 -200 h 1000 1500 2000 2500 3000 C D Position [mm] r . i r i i i t a l l b i . . . Average = 43.0, Min = -199.65, Max = 174.83 (b) ||u||2= 543.51 10 15 20 25 Actuator Number 30. 35 Figure 6.10: Data Set 2 process output (a) and control signal (b), using the current industrial technique - average padding. in comparison to the existing technique, although the actuator usage level is significantly lower (Table 6.2). Chapter 6. Industrial Trial 97 67.5 67 h nj 66.5 CD I 66 65.5 65 Average = 65.73, Min = 65.25, Max = 66.54 l l y l l 2 = 2.962 500 1000 1500 2000 CD Position [mm] (a) 2500 3000 o o E 200 100 0 -100 -200 ,. - . I l l l l l l l l l l l l l l . . (b) Average = 43.13, Min = -105.59, Max = 153.70 | |U|| 2 = 494.05 10 15 20 25 Actuator Number 30 35 Figure 6.11: Data Set 2 process output (a) and control signal (b), using the new technique - conservative tuning (kp = 180 and kp = 0.2 in (6.3)). E ns 65.5 200 100 o ! 0 -100 -200 500 1000 1500 2000 2500 CD Position [mm] 3000 r ,1 l l l l l l l l l l l l l l . . . . . . . . Average = 43.16, Min = -142.87, Max = 160.62 (b) H |u| | 2 = 513 .72 10 15 20 25 Actuator Number 30 35 Figure 6.12: Data Set 2 process output (a) and control signal (b), using the new technique - balanced tuning (kp = 480 and kR = 0.2 in (6.3)). Chapter 6. Industrial Trial 98 CO o o 'E 67.5 67 h £ 66.5 h 1/5 .O 66 65.5 I 200 100 0 -100 -200 Average = 66.16, Min = 65.61, Max = 66.85 IIVlL= 2.887 (a) 500 1000 1500 2000 2500 C D Position [mm] 3000 i l l l l l l l l l l l l l l . . . . . . Average = 43.24, Min = -176.64, Max = 156.74 ||u|L= 526.21 (b) 10 15 20 25 Actuator Number 30 35 Figure 6.13: Data Set 2 process output (a) and control signal (b), using the new technique - aggressive tuning (kp = 720 and kp = 0.4 in (6.3)). Current technique New approach Average Conservative Balanced Aggressive padding tuning: tuning: tuning: kP = 180 kP = 480 kP = 720 kp = 0.2 kp = 0.2 kp = 0.4 | | 7 / | | 2 [lbs/ream] 2.953 2.962 2.695 2.887 ||u(3 : 34)||2 [microns] 543.51 494.05 513.72 526.21 On-sheet actuator array sections ||ti(3 : 7)||2 [microns] 214.79 111.12 166.98 183.46 ||u(8 : 29)||2 [microns] 473.39 455.90 456.7 458.29 ||tf(30 : 34)||2 [microns] 158.65 155.34 165.67 182.24 Table 6.2: 2-norms of the process output and control signal profiles shown in Figures 6.10 - 6.13 (Data Set 2). Chapter 6. Industrial Trial 99 6.4 Summary A novel technique for modifying the industrial CD controllers near spatial domain bound- aries, developed in Chapter 5, has been successfully tested in a working paper mill and the results are presented in this chapter. The trial was carried out on a paper machine CD control system with slice lip actuators controlling the paper sheet dry weight profile. This trial demonstrates that with the new approach it is possible to achieve a successful trade-off between the CD control system closed-loop performance and the required control signal magnitude. In other words, the level of the resulting CD control law 'aggressiveness' near the sheet edges can be chosen with the new approach. The new technique was also compared with the existing industrial practice, and a better performance with smaller control signal magnitude was achieved by implementing the newly developed technique. Paper sheet quality improvement, achieved with a stable and robust control law near the sheet edges (as provided with the newly developed technique), has a clear economic benefit for the papermakers. In some papermaking situations, the 'trim squirts' (Figure 1.2) can be moved outward, thus resulting in less paper being trimmed off and more on-spec paper is being produced from which the papermaker can extract his orders. Chapter 7 Concluding Remarks This work has focused on modifying an existing industrial paper machine cross-directional control law near spatial domain boundaries (paper sheet edges). As the process spatial- invariance assumption (i.e. neglecting of paper sheet edges) is central to the current industrial controller tuning technique, thus computed CD control laws have to be modified near spatial domain boundaries before implementation on a paper machine. A brief review of the motivation, objectives, approaches, and results presented in this thesis is given below. 7.1 Summary of the Thesis It is illustrated in Chapter 1, that paper machine CD control systems belong to a set of large, multivariable, spatially-distributed control systems. The task of a particular cross-directional actuator array, on a paper machine, is to reduce the variations of the corresponding paper sheet property (weight, moisture content, or caliper) as much as possible in the cross-direction. Depending on the installation, the number of CD actuators (inputs) in an array is between 30-300 and the number of measurement points (outputs) 200-2000. The most frequently used types of CD actuators, various CD control techniques, and the problems encountered near the sheet edges are illustrated in Chapter 1. In Chapter 2, it is shown that the CD process and industrial controller models can be viewed as two dimensional (spatio-temporal) filters, causal in temporal and non-causal in spatial (cross-direction) domain. The two-dimensional loop shaping technique (briefly outlined in this Chapter 2), being used for tuning the industrial CD control systems, as- sumes idealized spatially-invariant CD process characteristics. As a result, the computed CD control law is also spatially-in variant. Indeed the spatial-invariance assumption is common to many of the recently developed techniques for the analysis and controller syn- thesis of spatially-distributed systems. The controllers obtained by the implementation of these techniques are generally spatially-invariant. In the case of paper machine CD con- trol, spatial invariance is equivalent to assuming that a paper machine produces a tube rather than a sheet of paper. Since the paper machine edges represent a clear disruption of the assumed spatial-invariance, the control laws obtained by the two-dimensional loop shaping technique must be modified near the sheet edges before being implemented on the paper machine. Current industrial practice addressing these issues is based on techniques 100 Chapter 7. Concluding Remarks 101 for extending finite-width signals, that have been borrowed from signal processing. Such methods very often result in poor control near the edges. The object of this work, dis- cussed in Chapter 2, is a modification of the existing industrial CD control law near spatial domain boundaries, considering relevant control engineering criteria (closed-loop stabil- ity, performance, and robustness), without a change in the controller structure and/or complexity. In Chapter 3, a straightforward perturbation technique with which a controller stabi- lizing one plant may be modified so that it stabilizes a second, related, plant is presented. The technique is based on the known difference between the two plants, and it is shown that various application examples (other than CD control) can be viewed in terms of this result. While the implementation of the proposed technique, in the case of the initially designed spatially-invariant CD control law, is very simple, and the resulting closed-loop stability guaranteed, it does not, however, address other important requirements - perfor- mance, robustness, and the desired preservation of the existing CD controller structure. Considering that CD controllers are essentially two dimensional low-pass filters and the paper sheet edges are clear spatial domain discontinuities, a similarity between the effects occurring near the sheet edges in the industrial CD control systems and the well- known Gibbs effect, is observed in Chapter 4. Subsequently, a CD control law modification technique, based on methods for reducing the Gibbs effect, is presented. The technique, which does not change the structure of the CD controller, guarantees the resulting CD controller stability. As illustrated in a simulation example in Chapter 4, the approach eliminates the 'actuator picketing' near the sheet edges. However, the resulting closed- loop performance and robustness are not systematically considered with this approach, and closed-loop stability has to be verified after the modifications have been carried out. Modification of the CD control law, while taking into account relevant control engi- neering criteria (closed-loop stability, performance, and robustness), and without changing the controller structure or complexity, is presented in Chapter 5. Al l the requirements from Chapter 2, have thus been taken into account with this approach. Modifications to the two constant controller matrices are computed by the sequential implementation of a novel low-bandwidth static output feedback design algorithm. It is demonstrated, using the Honeywell hardware-in-the-loop simulator example, that a successful trade-off between the performance and the corresponding actuator array signal magnitude can be achieved using the newly proposed approach. The technique presented in Chapter 5 was successfully tested in a paper mill on a paper machine. The trial was carried out at a Canadian mill, with the slice lip actuator array being used for controlling the paper sheet CD weight profile. Based on the algorithm developed in Chapter 5, Matlab prototype software was developed for computing the CD Chapter 7. Concluding Remarks 102 controller modifications near the sheet edges. The industrial controller algorithm was also modified so that the computed control law modifications could be implemented and tested. The field trial results are given in Chapter 6. As predicted by the hardware-in-the-loop simulator example, it was possible to achieve a trade-off between the performance and the corresponding control signal magnitude during the industrial trial. Also, the final prod- uct (paper sheet) quality was improved and the actuator usage reduced, with the new technique in comparison to the results obtained with the industrial state-of-the-art prac- tice. It is also particularly important to notice that, in contrast to the existing industrial techniques, the developed approach provides a systematic way of modifying the existing CD control law near the sheet edges, taking into account important control engineering criteria. The new approach guarantees the resulting closed-loop system stability, as well as robustness margins, as measured by the H.^ norm of the gain from the process output disturbance to control signal. The performance improvement, as measured by the Frobe- nius norm of the gain from the process output disturbance to the process output, at low frequencies, is also guaranteed with the new approach. 7 . 2 F u t u r e W o r k Some of the possible research directions directly related to the work presented in this thesis are outlined below. Modelling of CD processes near the sheet edges with the CD process parameters from the centre of the sheet and Dirichlet boundary conditions is currently accepted industrial practice. Obviously, a more accurate identification of CD processes near spatial domain boundaries would allow for even better CD control. In particular, the use of more accurate process models in conjunction with the closed-loop approach technique presented here should result in a better CD control. The transfer functions that define linear fractional transformation used in the closed- loop approach initially have a lot of insignificant states (related to the centre of sheet), subsequently eliminated by the use of Hankel singular values. Since state elimination (i.e. dynamical order reduction) could require a considerable computational effort (particularly in the case of very wide paper machines), a better way of eliminating (or even not including in the first place) insignificant states would be of clear benefit. As pointed out in Chapters 5 and 6, two tuning variables (kp and kp) were used in the newly proposed closed-loop approach to modifying industrial CD controller near spatial domain boundaries. This work provided guidelines for choosing the values of these variables. Further analysis of the influence kp and kp have on the resulting closed- loop system behavior in case of various industrial CD process and controller parameters Chapter 7. 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Appendix A Proof of Theorem 1 Internal stability of the perturbed systems is established by the stability of the transfer functions between the disturbance inputs (di, i = 1, 2, 3, 4) inserted in front of each block in the corresponding block diagram, and the outputs (?/,, i = 1,2,3,4) of each of the blocks. The corresponding closed-loop system is stable if and only if all 16 transfer functions from di to fji (i = 1,2,3,4) are stable. In all the cases below, closed-loop transfer functions S0 and Si are defined as S0 = [/ - GK]~\ Si = [I- KG]-1, and [yx y2 y 3 VA]t = T-[d1 d2 dz d4]T. It can easily be confirmed that, subject to the conditions given in Theorem 1, the systems in Figures A.1-A.6 are stable. A . l Process Additive Perturbation (Case a) Y4 d 2 r d 4 id, K u 3 l a Figure A . l : Diagram used to analyze internal stability for the configuration given in Theorem 1 - case (a). Tn = S0G S0GKAG S0GK AGSiKG AGS%KAG + AG AGSZK S,KG SiKAG SiK -AGSJ<G —AGSiKAG -AGSiK -S0GKAG -AGSiKAG —SiK AG -AG + AGStKAG (A.l) 111 A. Proof of Theorem 1 112 |d 2 1*0*- id4 <du—iAj£r- Mj*Cr> G 1 1 »C» K 13 Figure A.2: Diagram used to analyze internal stability for the configuration given in Theorem 1 - case (b). A . 2 Process Inverse Additive Perturbation (Case b) S0G S0GAG S0GK -S0GAG AGS0G AGS0GAG + A G AGSQGK - A G S 0 G A G KS0G KS0GAG KS0 -KS0GAG - A G S 0 G - A G S 0 G A G AGS0GK AGSQGAG - AG (A.2) All the closed-loop transfer functions in this case can easily be obtained from (A.l) by simultaneously swapping G and K, and A G and — A G . A.3 Process Multiplicative Input Perturbation (Case c) | d 3 c- y 2 iyi Figure A.3: Diagram used to analyze internal stability for the configuration given in Theorem 1 - case (c). A. Proof of Theorem 1 113 S 0 G [I + A G ) - L A G K S 0 G K S 0 G - [ I + A G } - L A G K S 0 G S 0 G A G [I + A G ] ~ 1 A G K S 0 G A G + A G K S 0 G A G — [I + A G \ ~ 1 A G K S 0 G A G S 0 G K [I + A G ^ A G K S O K S 0 - [ I + A G ] - L A G K S 0 — S 0 G A G - { I + A G ] - L A G { - K S 0 G A G + I ) - A G — K S 0 G A G - [ I + A G ] - L A G ( ~ K S 0 G A G +1) (A.3) A . 4 Process Inverse Multiplicative Input Perturba- tion (Case d) jd4 l d 3 ^64 K A ; y4 y2 Ac |d2 Mi yi Figure A.4: Diagram used to analyze internal stability for the configuration given in Theorem 1 - case (d). TR F = S 0 G S 0 G [ I — AG]~1G[I — A G ] _ 1 A G S 0 G K - S 0 G [ I - AG]-HG A G K S 0 G ( A G K S 0 G + I)[I-AG]AG AGKSa (-AGKS0G[I - AG]~l + I - [I - AG]-l)AG \ K S 0 G K S „ G [ I - Ac}-1 AG K S A - K S A G [ I - A G J - ' A G A G K S 0 G 6 g K S 0 G [ I - A G ] A G - A G K S 0 AGKS0G[I — AG]AG — AG (AA) A . 5 Process Multiplicative Output Perturbation (Case e) S 0 G S 0 G K [ I + A G } - 1 A G S 0 G K - S A G K [ I + A G ] ~ L A G T = A G S A G A G S 0 G K [ I + A G } - L A G + A G A G S 0 G K — A G S 0 G K [ I + A G ] ~ 1 A G K S 0 G K S A [ I + A G } - 1 A G K S 0 - K S 0 G K [ I + A G ] _ 1 A G — A G S 0 G — A G S 0 G K [ I + A G ] _ 1 A G — A G A G S A G K A G S A G K [ I + A G ] _ 1 A G A. Proof of Theorem 1 114 y4 •Ac • K G l u 4 l u 2 | d 3 Idi •64 Figure A.5: Diagram used to analyze internal stability for the configuration given in Theorem 1 - case (e). ' ' A.6 Process Multiplicative Output Perturbation (Case f) "3 y4 Ac • j d 4 i d 2 wo y2 K yi Figure A.6: Diagram used to analyze internal stability for the configuration given in Theorem 1 - case (f). SiK —[I — AG]~1AGGSiK GSiK [I - Aa^^oGSiK SiKAG [I-AG]~iAGGSlKAG-AG -GSlKAG -[I - Ac^AcGSiKAc SiKG -[I-Ac^AcGSi GSt [I — Ac]~1AaGSi SiKAG [I- A G ] - 1 A G ( G S i . f Y A G + /) + A G GSlKAG [I — AG]~1AG(GSiKAG + I) (A.6) Appendix B Matrix Optimization Given matrices A e Kmxn, B <E Hkxl, C G 1Zmxl, and scalar p > 0. Consider the static matrix optimization problem, Q* = argmin(||^Q5 - C\\2F + p\\Q\\2F) (B.l) with respect to matrix Q £ TZnxk. The Frobenius norm ||-| | F of a matrix is given in (5.8). Write the standard singular value decomposition, A = UAY,AVj, B = UBEBVi (B.2) where the m x m matrix UA and the n x n matrix VA are unitary and the rn x n matrix S A is given by, S 4 = 0 m> n (B.3) S A = [Ex 0], m < n (BA) where Si = diag{cri(^),... ,ap(A)}, p = min(m, n) (B.S) Similar expressions apply to the matrices and VB in (B.2). Now form the new matrices, X = VjQUB, Y = UTACVB (B.6) then the sum of matrix norms on the right hand side of (B.l) may be rewritten as the equivalent expression, \\AQB - C\\2F + p \\Q\\2F = \\XAXZB - Y\\2F + p \\X\\2F (B.7) by substituting (B.6) in (B.l) and due to the fact that multiplication by unitary matrices 115 B. Matrix Optimization 116 does not affect the Frobenius norm [27, 58]. We can expand the Frobenius norms in (B.7) as \ Z A X Z B - Y \ \ 2 F J^Y/(^(A)aJ(B)xlj-ylJ)2+ £ J>5 + E £ V% (B.8) i = l j = l I=TA+1 3=1 *=1 J=»"B+1 and n A: P P * = E E ' - 4 = 1=1 i=l M n k n k EX > 4 + E E^-4 + E E *>-4 (B-9) i = l 7 = 1 1 = ^ + 1 j=l i—1 j = r g - f - l Then the expression in (B.7) can be rewritten as the sum over terms, \\EAXEB-Y\\2F + P\\X\\F = E E (ai(A)ai(B)XiJ ~ Vijf + P • xij i=l j=l n k n k + E E^'4 + E E i = 7 \4+ l j = l i = l j—rB+l m l m l + E E 4 + E E 4 (B.IO) 1=^+1 j=l i=l j=rB+l We can minimize the right hand side of (B.IO) by noting that each term in the sum- mation (B.IO) contains at most one distinct element xi3- of the matrix X in (B.6). This allows the overall optimization to be decoupled in terms of the individual matrix elements. Optimizing term-by-term in (B.IO) is straightforward and leads to the solution, xh=i^mh-y^ l < ^ < r , and ! < , < r B ^ [ 0, otherwise so that the n x k matrix X* optimizing (B.7) has only rArB nonzero elements, where rA < min(m, ri) and rB < min (A;, I) are the number of nonzero singular values of matrices A and B respectively. Finally the nx k matrix Q* that minimizes (B.l) can be obtained from the expression B. Matrix Optimization 117 for X in (B.6), Q* = VAX*Ul (B.12) So the solution of the optimization problem (B.l) for the matrix Q involves taking the SVD of two matrices in (B.2), computing two matrix products in (B.6), an element-by- element construction in (B. l l ) , and computing a final matrix product in (B.12). Appendix C Proofs of Theorems 4-5 First a few supporting relationships, used in proving Theorems 4-5, are given in Section C . l . Next, the proofs are given in Sections C.2-C.3. C.l Supporting Relationships Given a static compensator K0, the feedback system in Figure 5.1 is internally stable if and only if the dynamic transfer matrix given by, R(z) = K0(I - N32(z)K0) - l (C.l) is stable. The stability of R(z) can be computed easily in state-space. First define the factors, R(z) = Dr N32(z) = A32 B32 C32 D32 then Ar can be written in terms of (C.l) and (C.2) as, Ar = A32 + B32K0(I - D32KQyYC, 32 (C.2) (C.3) and closed-loop stability is equivalent to the stability of all eigenvalues of the matrix Ar in (C.3). A conservative (sufficient) stability result can be obtained by substituting KQ from (5.14) into (C.l) to obtain R(z) = Q0[I- (N32(z) - N32(eJ°))Q0] 1 (C.4) Then since QQ is static and N32(z) is stable, small gain arguments lead to the result that R(z) in (C.4) is stable if, \\(N32(z)-N32(e>°))-Q0\\oQ<l (C.5) The following relationship indicates that the optimization weight p may be used to 118 C. Proofs of Theorems 4-5 119 govern the size of the resultant matrix QQ in (5.13), s m £ ] ] q 4 f £ y ^ - W ^ M ^ ) ) ( a 8 ) where a(-) denotes the maximum singular value, the integers r\2 and r 3 1 denote the number of nonzero singular values in N12(e:'0) and A r31(e j 0) respectively. The first inequality is standard for any matrix and may be found in, for example [27, 58]. The second inequality in (C.6) holds for Q0 in (5.13) and may be verified using (5.13) and (B.11)-(B.12). C.2 Proof of Theorem 4 Let, T = V^rTi • °{N12{e^)) a{N31{ei0)) a(Nn(ej0)) • \\N32(z) - N32(ej0)L (C.7) Substituting p > V in (C.7) into (C.6) results in Q0 satisfying (C.5), i.e. the system is stable. Since, by comparing (C.7) and (5.15), ft > T then p > ft in (5.13) will result in an internally stable system also. Using Ko in (5.14) we can write where A(e^) = N32(e?u) - N32{ej0). Then substitute (C.6) into (C.8) with p > (5 to obtain (5.16). 0 C.3 Proof of Theorem 5 First prove (5.17) for ui = 0. Note that if QQ optimizes J(iV a(e j 0),p,Q) in (5.13), then J(Na(e>°),p,Q0) < J(Na(e>°),p,0) = \\Ti(Na(eP°),0)||F and since Q0 ^ 0 then \\Q0\\F > 0 and using (5.14) we get | |^(iV a(e J ' 0), / Y O ) | | F < J(Na(ej0), p, Q0) leading to the steady-state result \\Ti{Na(eja),KQ)\\F < \\Jri(Na(ei°),0)||F. Then it follows there exists some e > 0 for which | | | ^ ( J V a ( e i O ) , 0 ) | | F _ \\^Na(ePQlKQ)\\F | > e , and since ^(N^z), K0) is a stable, finite-dimensional transfer matrix, there will exist u)b > 0 for which | \\Jri(Na(e:'0J), i \o) | | F — \\Fi(Na(ej0),K0)\\F | < e for -ub < u < cub and (5.17) follows. 0 Appendix D Closed-loop transfer functions used for denning L F T s D . l Closed-loop transfer functions that make up P e ( z ) in Figure 5.2 Closed-loop transfer functions of interest in case of optimization for performance: de -~* Ve Pi = WA[I - G(z)K(z)]-1Wi de * Vce Pla = W3[I - G(z)K(z)]~1Wi de - ue P 2 = W5K(z)[I - G^Kiz^Wi de ' -> Vde Ps = W2[I - Dz-1 - c(z)CG{z)D]-1c(z)CWi Ude -> ue PA = W5[I - Dz-1 - c(z)DCG(z)]- 1iy 6 Ude Ve P 5 = W4G(z)[I - Dz- 1 - c(z)L»CG(z)]- 1 W 6 Ude -> Vce Pha = W3G(z)[I - Dz-1 - c(z)DCG(2)]-1W6 Ude —> Ude P, = W2[I - Dz'1 - c{z)CG{z)D]-1- [c(z)CG{z) + z-1I]W6 Uce ~> Ve Pi = WA[I -G(z)[I - Dz-^-cMC]-1- G(z)[I - Dz-1)-1c{z)W7 Uce -> Vce P?a = W3[I-G(z)[I - Dz-1)-1^^)-1- G(z)[I - Dz-1]-1c(z)W7 Uce —> ue ps = W5[I -[I- Dz-^c^CGiz)}-1- [I - Dz-1}-1c(z)W7 Uce 2Afe p9 = W2[I - Dz'1 - c(z)CG(z)D}-1W7 (D.l) The rectangular matrices Wi} i = 1,2,... ,7 are defined in (5.18)—(5.19). 120

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