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Model- Based Predictive Control A Practical Approach CONTROL SERIES Robert H. Bishop Series Editor University of Texas at Austin Austin, Texas Published Titles Linear Systems Properties: A Quick Reference Venkatarama Krishnan Robust Control Systems and Genetic Algorithms Mo Jamshidi, Renato A Krohli...

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ModelBased Predictive Control A Practical Approach

CONTROL SERIES Robert H. Bishop Series Editor University of Texas at Austin Austin, Texas

Published Titles Linear Systems Properties: A Quick Reference Venkatarama Krishnan Robust Control Systems and Genetic Algorithms Mo Jamshidi, Renato A Krohling, Leandro dos Santos Coelho, and Peter J. Fleming Sensitivity of Automatic Control Systems Efim Rozenwasser and Rafael Yusupov Model-Based Predictive Control: A Practical Approach J.A. Rossiter

Forthcoming Titles Material and Device Characterization Measurements Lev I. Berger

ModelBased Predictive Control A Practical Approach J. A . RO SSI TER

CRC PR E S S Boca Raton London New York Washington, D.C.

1291 disclaimer Page 1 Monday, May 19, 2003 3:29 PM

This edition published in the Taylor & Francis e-Library, 2005. “To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.”

Library of Congress Cataloging-in-Publication Data Rossiter, J.A. Model-based predictive control : a practical approach / by J.A. Rossiter. p. cm. — (Control series) Includes bibliographical references and index. ISBN 0-8493-1291-4 (alk. paper) 1. Predictive control. 2. Control theory. I. Title. II. CRC Press control series TJ217.6.R67 2003 629.8—dc21

2003048996

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe.

Visit the CRC Press Web site at www.crcpress.com © 2004 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 0-8493-1291-4 Library of Congress Card Number 2003048996 ISBN 0-203-50396-1 Master e-book ISBN

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Overview The main aim of this book is to make the presentation less mathematically formal and hence more palatable for the less mathematically inclined. Insight is given in a non-theoretical way and there are a number of summary boxes to give a quick picture of the key results without the need to read through the detailed explanation. The book can serve a twofold purpose: first as a textbook for graduate students and industrialists covering a detailed introduction to predictive control with a strong focus on the philosophy answering the questions, ‘why?’ and ‘does it help me?’ The basic concepts are introduced and then these are developed to fit different purposes: for instance, how to model, to give robustness, to handle constraints, to ensure feasibility, to guarantee stability and to consider what options there are with regard to models, algorithms, complexity versus performance, etc. The second role of the book is to target researchers in predictive control. In places the book goes into more depth, particularly in those areas where Dr. Rossiter has expertise. In his research Dr. Rossiter has adopted a different style of presentation to that adopted by many authors and this style gives different insights to model-based predictive control. Dr. Rossiter uses this personal style and his own insight, hence forming a contrast to and complementing the other books available. Novel areas either not much discussed in other books or having recent developments are: (i) connections to optimal control and stability; (ii) the closed-loop paradigm; (iii) robust design in MPC; (iv) implementations of MPC using only small on-line computational burdens and (v) implicit modelling for predictive control. Dr. Rossiter would like to apologise for any obvious references or topics that have been missed. He found writing a book a far more demanding task than anticipated and it was necessary to draw a line, at some point, on the continual improvement. Nevertheless, he does believe that this book complements the existing literature. By all means let him know of the large gaps you find and he will bear them in mind for a second edition. Some MATLAB files have been written for readers of Model-Based Predictive Control: A Practical Approach. The files enable the user to design and simulate simple MPC controllers and moreover are easy to modify. They are available on the publisher’s Web site at www.crcpress.com.

Acknowledgments The main person I should thank is my friend and colleague, Prof. Kouvaritakis, who has had and continues to have a major influence on my work. Many of the insights in this book arose through working with Basil. Other collaborators from whom I have learned much that is in this book are Mark Cannon, Jesse Gossner and Luigi Chisci. I would also like to thank Prof. Shah, who encouraged me to write a book, CRC Press for their patience while waiting for it, my family for their constant support and, of course, God for giving me the opportunity and skills to be who I am.

About the author Dr. Rossiter has been researching predictive control since the late 1980s and he has published over 100 articles in journals and conferences on the topic. His particular contributions have focused on stability, feasibility and computational simplicity. He has a Bachelor’s degree (1st class, 1987) and a doctorate (1990) from the University of Oxford. He spent 9 years as a lecturer at Loughborough University and is currently a reader at: University of Sheffield Department of Automatic Control and Systems Engineering Mappin Street Sheffield, S1 3JD UK email: [email protected]

Contents

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Introduction 1.1 Overview of model-based predictive control . . . . . . . . 1.2 The main components of MPC . . . . . . . . . . . . . . . 1.2.1 Dependence of actions on predictions . . . . . . . 1.2.2 Predictions are based on a model . . . . . . . . . . 1.2.3 Selecting the current input . . . . . . . . . . . . . 1.2.4 Receding horizon . . . . . . . . . . . . . . . . . 1.2.5 Optimal or safe performance . . . . . . . . . . . . 1.2.6 Tuning . . . . . . . . . . . . . . . . . . . . . . . 1.2.7 Constraint handling . . . . . . . . . . . . . . . . 1.2.8 Systematic use of future demands . . . . . . . . . 1.2.9 Systematic control design for multivariable systems 1.3 Overview of the book . . . . . . . . . . . . . . . . . . . . 1.4 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Table of notation . . . . . . . . . . . . . . . . . . 1.4.2 Vectors of past and future values . . . . . . . . . 1.4.3 Toeplitz and Hankel matrices . . . . . . . . . . . 1.4.4 Common acronyms/abbreviations . . . . . . . . .

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Common linear models used in model predictive control 2.1 Modelling uncertainty . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Integral action and disturbance models . . . . . . . . . . . 2.1.2 Modelling measurement noise . . . . . . . . . . . . . . . 2.2 Typical models . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 State-space models . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Nominal state-space model . . . . . . . . . . . . . . . . . 2.3.2 Nonsquare systems . . . . . . . . . . . . . . . . . . . . . 2.3.3 Including a disturbance model . . . . . . . . . . . . . . . 2.3.4 Systematic inclusion of integral action with state-space models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Transfer function models (single-input/single-ouput) . . . . . . . 2.4.1 Disturbance modelling . . . . . . . . . . . . . . . . . . . 2.4.2 Consistent steady-state estimates with CARIMA models .

1 1 2 2 2 3 4 4 5 5 6 6 7 9 9 9 9 14 17 18 18 18 19 20 20 21 21 22 24 24 25

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25 26 26 26 27 27 28 29

Prediction in model predictive control 3.1 General format of prediction modelling . . . . . . . . . . . . . . 3.2 Prediction with state-space models . . . . . . . . . . . . . . . . . 3.3 Prediction with transfer function models – matrix methods . . . . 3.3.1 Prediction for a CARIMA model with T (z) = 1 – SISO case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Prediction with CARIMA model and T = 1 – MIMO case 3.3.3 Prediction equations with T (z)
= 1 . . . . . . . . . . . . 3.4 Using recursion to find matrices H, P, Q . . . . . . . . . . . . . 3.5 Prediction with FIR models . . . . . . . . . . . . . . . . . . . . 3.5.1 Impulse response models . . . . . . . . . . . . . . . . . . 3.5.2 Step response models . . . . . . . . . . . . . . . . . . . 3.6 Prediction with independent models . . . . . . . . . . . . . . . . 3.6.1 Structure and prediction set up with internal models . . . . 3.6.2 Prediction with unstable open-loop plant . . . . . . . . . 3.7 Numerically robust prediction with open-loop unstable plant . . . 3.7.1 Why is open-loop prediction unsatisfactory? . . . . . . . . 3.7.2 Prestabilisation and pseudo closed-loop prediction . . . . 3.8 Pseudo closed-loop prediction . . . . . . . . . . . . . . . . . . .

31 31 32 33

Predictive control – the basic algorithm 4.1 Summary of main results . . . . . . . . . . . . . . . . . . . . . 4.2 GPC algorithm – the main components . . . . . . . . . . . . . 4.2.1 Performance index and optimisation . . . . . . . . . . . 4.2.2 Restrictions on the predicted future control trajectory . . 4.2.3 The receding horizon concept . . . . . . . . . . . . . . 4.2.4 Constraints . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Multivariable systems . . . . . . . . . . . . . . . . . . 4.2.6 The use of input increments and obtaining integral action 4.2.7 Eliminating tracking offset while weighting the inputs . 4.2.8 Links to optimal control . . . . . . . . . . . . . . . . . 4.3 GPC algorithm formulation for transfer function models . . . .

53 53 54 54 56 57 57 58 59 60 61 61

2.5

2.6 2.7 2.8 3

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2.4.3 Achieving integral action with CARIMA models 2.4.4 Selection of T (z) for MPC . . . . . . . . . . . . FIR models . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Impulse response models . . . . . . . . . . . . 2.5.2 Step response models . . . . . . . . . . . . . . Independent models . . . . . . . . . . . . . . . . . . . Matrix-fraction descriptions . . . . . . . . . . . . . . . Modelling the dead times in a process . . . . . . . . . .

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33 35 36 39 41 41 42 43 43 44 48 48 49 52

4.3.1 Steps to form a GPC control law . . . . . . . . . . . . . 4.3.2 Transfer function representation of the control law . . . 4.3.3 Closed-loop transfer functions . . . . . . . . . . . . . . 4.3.4 GPC based on MFD models with a T-filter (GPCT) . . . 4.4 Predictive control with a state-space model . . . . . . . . . . . . 4.4.1 Simple state augmentation . . . . . . . . . . . . . . . . 4.4.2 State-space models without state augmentation . . . . . 4.5 Formulation for finite impulse response models . . . . . . . . . 4.6 Formulation for independent models . . . . . . . . . . . . . . . 4.6.1 IM is a transfer function or MFD . . . . . . . . . . . . . 4.6.2 Closed-loop poles in the IM case with an MFD model . 4.6.3 IM is a state-space model . . . . . . . . . . . . . . . . . 4.7 General comments on stability analysis of GPC . . . . . . . . . 4.8 Constraint handling . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 The constraint equations . . . . . . . . . . . . . . . . . 4.8.2 Solving the constrained optimisation . . . . . . . . . . . 4.8.3 Hard and soft constraints . . . . . . . . . . . . . . . . . 4.8.4 Stability with constraints . . . . . . . . . . . . . . . . . 4.9 Simple variations on the basic algorithm . . . . . . . . . . . . . 4.9.1 Alternatives to the 2-norm . . . . . . . . . . . . . . . . 4.9.2 Alternative parameterisations of the degrees of freedom . 4.9.3 Improving response to measurable disturbances . . . . . 4.10 Predictive functional control (PFC) . . . . . . . . . . . . . . . . 4.10.1 Predictive functional control with one coincidence point 4.10.2 PFC tuning parameters . . . . . . . . . . . . . . . . . . 4.10.3 PFC with two coincidence points . . . . . . . . . . . . . 4.10.4 Limitations and summary . . . . . . . . . . . . . . . . . 4.11 Other performance indices . . . . . . . . . . . . . . . . . . . . 5

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61 62 63 64 65 66 68 71 72 72 73 73 74 75 75 78 78 79 80 80 80 80 81 81 82 82 82 83

Examples – tuning predictive control and numerical conditioning 5.1 Matching closed-loop and open-loop behaviour . . . . . . . . . . 5.2 Single-input/single-output examples . . . . . . . . . . . . . . . . 5.2.1 Effect of varying the output horizon . . . . . . . . . . . . 5.2.2 Effect of varying the input horizon . . . . . . . . . . . . . 5.2.3 Effect of varying the control weighting . . . . . . . . . . 5.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The benefits of systematic constraint handling . . . . . . . . . . . 5.3.1 Simple example of weakness in a saturation policy . . . . 5.3.2 Numerical illustration of the weaknesses in saturation policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Unstable examples . . . . . . . . . . . . . . . . . . . . . . . . .

85 85 86 86 86 87 87 92 93 93 95

5.4.1

5.5 5.6 5.7 6

Effects of tuning parameters on unstable systems – a counter intuitive result . . . . . . . . . . . . . . . . . . . . . . . . 95 Numerical ill-conditioning with open-loop unstable systems . . . . 97 5.5.1 How does ill-conditioning arise? . . . . . . . . . . . . . . 98 MIMO examples . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Summary of guidelines . . . . . . . . . . . . . . . . . . . . . . . 101

Stability guarantees and optimising performance 6.1 Prediction mismatch in MPC . . . . . . . . . . . . . . . . . . . . 6.1.1 Illustration of ill-posed objective . . . . . . . . . . . . . . 6.1.2 Example where prediction mismatch causes instability . . 6.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Feedforward design in MPC . . . . . . . . . . . . . . . . . . . . 6.2.1 Structure of the set point prefilter . . . . . . . . . . . . . 6.2.2 Mismatch between predictions and actual behaviour . . . 6.2.3 Making better use of advance information . . . . . . . . . 6.3 Infinite horizons imply stability . . . . . . . . . . . . . . . . . . . 6.3.1 Definition of the tail . . . . . . . . . . . . . . . . . . . . 6.3.2 Infinite horizons and the tail . . . . . . . . . . . . . . . . 6.3.3 Only the output horizon needs to be infinite . . . . . . . . 6.4 Stability proofs with constraints . . . . . . . . . . . . . . . . . . 6.4.1 Are infinite horizons impractical? . . . . . . . . . . . . . 6.4.2 Alternatives to optimal control . . . . . . . . . . . . . . . 6.5 Dual mode control – an overview . . . . . . . . . . . . . . . . . . 6.5.1 What is dual mode control? . . . . . . . . . . . . . . . . 6.5.2 The structure of dual mode predictions . . . . . . . . . . . 6.5.3 Overview of MPC dual mode algorithms . . . . . . . . . 6.5.4 Two possible dual mode algorithms . . . . . . . . . . . . 6.5.5 Is a dual mode strategy guaranteed stabilising? . . . . . . 6.5.6 How do dual mode predictions make infinite horizon MPC more tractable ? . . . . . . . . . . . . . . . . . . . . . . . 6.6 Implementation of dual mode MPC . . . . . . . . . . . . . . . . . 6.6.1 The cost function for linear predictions over infinite horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Forming the cost function for dual mode predictions . . . 6.6.3 Constraint handling with dual mode predictions . . . . . . 6.6.4 Computing the dual mode MPC control law . . . . . . . . 6.6.5 Remarks on stability and performance of dual mode control . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103 104 104 106 107 107 108 108 109 112 113 114 115 116 116 117 117 117 118 118 119 119 120 121 121 122 123 123 124

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Closed-loop paradigm 7.1 Introduction to the closed-loop paradigm . . . . . . . . . . . . . 7.1.1 Overview of the CLP concept . . . . . . . . . . . . . . . 7.1.2 CLP predictions . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 CLP structure . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Setting up an MPC problem with the closed-loop paradigm . . . . 7.2.1 Setting up the cost function and computing the control law for state-space models . . . . . . . . . . . . . . . . . . . 7.2.2 Including the constraints for state-space models . . . . . . 7.2.3 The constrained optimisation . . . . . . . . . . . . . . . . 7.3 Different choices for mode 2 of dual mode control . . . . . . . . . 7.3.1 Dead beat terminal conditions (SGPC) . . . . . . . . . . . 7.3.2 No terminal control (NTC) . . . . . . . . . . . . . . . . . 7.3.3 Terminal mode by elimination of unstable modes (EUM) . 7.3.4 Terminal mode is optimal (LQMPC) . . . . . . . . . . . . 7.3.5 Summary of dual mode algorithms and key points . . . . . 7.4 Are dual mode-based algorithms used in industry? . . . . . . . . . 7.4.1 Efficacy of typical industrial algorithm . . . . . . . . . . . 7.4.2 The potential role of dual mode algorithms . . . . . . . . 7.5 Advantages and disadvantages of the CLP over the open loop predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Cost function for optimal stable predictive control . . . . . 7.5.2 Improved numerical conditioning with the CLP . . . . . . 7.5.3 Improved robust design with CLP . . . . . . . . . . . . . Constraint handling and feasibility issues in MPC 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Description of feasibility . . . . . . . . . . . . . . . . . . 8.1.2 Feasibility in MPC . . . . . . . . . . . . . . . . . . . . . 8.1.3 Overview of chapter . . . . . . . . . . . . . . . . . . . . 8.2 Constraints in MPC . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Hard constraints . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Soft constraints . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Terminal constraints . . . . . . . . . . . . . . . . . . . . 8.3 Why is feasibility important? . . . . . . . . . . . . . . . . . . . . 8.3.1 Consequences of infeasibility . . . . . . . . . . . . . . . 8.3.2 Recursive feasibility . . . . . . . . . . . . . . . . . . . . 8.4 What causes infeasibility in predictive control? . . . . . . . . . . 8.4.1 Incompatible constraints due to overambitious performance requirements . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Conflicts with terminal mode control laws . . . . . . . . .

127 127 128 130 132 133 134 135 136 136 137 140 140 143 144 145 145 146 146 147 149 151 153 153 153 154 154 154 155 155 155 156 156 156 158 158 158

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159 159 160 160 161 162 162 163 164 164 165

Improving robustness – the constraint free case 9.1 Key concept used in robust design for MPC . . . . . . . . . . . . ...