Loop-shaping Robust Control www.it-ebooks.info Loop-shaping Robust Control Philippe Feyel Series Editor Bernard Dubuisson www.it-ebooks.info First published 2013 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd John Wiley & Sons, Inc. 27-37 St George’s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA www.iste.co.uk www.wiley.com © ISTE Ltd 2013 The rights of Philippe Feyel to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2013936315 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN: 978-1-84821-465-1 Printed and bound in Great Britain by CPI Group (UK) Ltd., Croydon, Surrey CR0 4YY www.it-ebooks.info Table of Contents Introduction ix Chapter 1. The Loop-shaping Approach 1 1.1.Principleofthemethod 1 1.1.1.Introduction 1 1.1.2.Sensitivity functions 1 1.1.3.Declinationofperformanceobjectives 5 1.1.4.Declinationoftherobustnessobjectives 8 1.2.Generalizedphaseandgainmargins 14 1.2.1.Phaseandgainmarginsatthemodel’soutput 14 1.2.2.Phaseandgainmarginsatthemodel’sinput: 16 1.3.Limitationsinherenttobandwidth 17 1.4.Examples 18 1.4.1.Example1:sinusoidaldisturbancerejection 18 1.4.2.Example2:referencetrackingandfrictionrejection 20 1.4.3. Example 3: issue of flexible modes and high- frequency disturbances 25 1.4.4.Example 4: stability robustness in relation to system uncertainties 29 1.5.Conclusion 30 Chapter 2. Loop-shaping H  Synthesis 33 2.1.Theformalismofcoprimefactorizations 33 2.1.1.Definitions 33 2.1.2.Practicalcalculationofnormalizedcoprimefactorizations 35 2.1.3.Reconstructionofatransferfunctionfromitscoprimefactors 36 www.it-ebooks.info vi Loop-shaping Robust Control 2.1.4. Set of stabilizing controllers – Youla parameterization of stabilizing controllers 37 2.2.Robustnessofnormalizedcoprimefactorplantdescriptions 42 2.2.1.Takingaccountofmodelinguncertainties 42 2.2.2.Stability robustness for a coprime factor plant description 43 2.2.3.Property of the equivalent “weighted mixed sensitivity” form 46 2.2.4. Expression of the synthesis criterion in “4-blocks” equivalent form 52 2.3.Explicit solution of the problem of robust stabilization of coprime factor plant descriptions 54 2.3.1.ExpressionoftheproblembytheYoulaparameterization 54 2.3.2.Explicitresolutionoftherobuststabilizationproblem 57 2.4. Robustness and -gap 77 2.4.1. -gap and ball of plants 77 2.4.2. Robustness results associated with the -gap 79 2.5.Loop-shapingsynthesisapproach 82 2.5.1.Motivation 82 2.5.2. Loop-shaping H  synthesis 83 2.5.3.Associatedfundamentalrobustnessresult 89 2.5.4.Phasemarginandgainmargin 89 2.5.5.4-blocksinterpretationofthemethod 90 2.5.6.Practicalimplementation 92 2.5.7.Examplesofimplementation 100 2.6.Discreteapproach 120 2.6.1.Motivations 120 2.6.2. Discrete approach to loop-shaping H  synthesis 121 2.6.3.Exampleofimplementation 127 Chapter 3. Two Degrees-of-Freedom Controllers 135 3.1. Principle 135 3.1.1.Referencetracking 135 3.1.2.Parameterizationof2-d.o.f.controllers 141 3.2.Two-stepapproach 143 3.2.1.Generalformulation 143 3.2.2.SimplificationoftheproblembytheYoulaparameterization 145 3.2.3.Extension 150 3.2.4.Settingoftheweightingfunctions 152 3.2.5.Associatedperformancerobustnessresult 154 3.3.One-stepapproach 156 3.3.1.Generalformulation 156 3.3.2.ExpressionoftheproblembyYoulaparameterization 158 3.3.3.Associatedperformancerobustnessresult 161 www.it-ebooks.info Table of Contents vii 3.3.4. Connection between the approach and loop-shaping synthesis 163 3.4.Comparisonofthetwoapproaches 165 3.5.Example 166 3.5.1.Optimizationofanexistingcontroller(continued)–scanning 166 3.6.Compensationforameasurabledisturbanceatthemodel’soutput 174 3.6.1.Principle 174 3.6.2.Example 179 Chapter 4. Extensions and Optimizations 187 4.1. Introduction 187 4.2.Fixed-ordersynthesis 188 4.2.1. Fixed-order robust stabilization of a coprime factor plant description 188 4.2.2.Optimizationoftheorderofthefinalcontroller 197 4.2.3.Example: fixed-order robust multivariable synthesis 214 4.3.Optimalsettingoftheweightingfunctions 220 4.3.1.Weightsettingonthebasisofafrequencyspecification 220 4.3.2. Optimal weight tuning using stochastic optimization and metaheuristics 227 4.4. Towards a new approach to loop-shaping fixed-order controller synthesis, etc. 242 4.4.1.Taking account of objectives of stability robustness 243 4.4.2.Takingaccountofobjectivesofperformancerobustness 244 APPENDICES 245 Appendix 1 247 Appendix 2 251 Bibliography 255 Index 259 www.it-ebooks.info Introduction I.1 Presentation of the book In an increasingly competitive industrial context, an automation engineer has to apply servo-loops in accordance with ever more complex sets of functional specifications, associated with increasingly broad conditions of usage. In addition to this, the product is often destined for large-scale production. Thus, the engineer has to be able to implement a robust servo-loop on a so-called “prototype”, whilst taking account of this broad spectrum in its entirety, at the very earliest stage of design. An example of such a system, upon which most of the examples given in this book are based, is a mass-produced viewfinder, for which the automation engineer has to inertially stabilize the line of sight, whose usage conditions may be extremely varied – indeed there are often as many potential applications as there are types of carriers (aircraft, ships, etc.). In addition, the viewfinder is required to deliver increasingly high-end functionalities – e.g. target tracking, guidance, etc. In order to moderate and reduce development costs, there is a growing tendency to carry out so-called “generic” stabilizations. This is possible only if the servo-loop designed has a certain degree of robustness, which needs to be taken into account as an a priori constraint on synthesis. In the 1990s, automation engineering made a great leap forward, with the emergence of H ∞ -based controller synthesis techniques: – Firstly, it became possible to obey a complex set of frequency specifications by using frequency weighting functions on exogenous inputs and on monitored signals, and then minimizing the H ∞ transfer norm between those signals by using a stabilizing controller whose state-space representation was explicitly formulated in [DOY 89], inspired by a dichotomy in the solution of Riccati equations (the so-called “γ-iteration”) and based on the following standard form: www.it-ebooks.info x Loop-shaping Robust Control P(s) K(s) yu z e Figure I.1. Standard form for control where e represents the exogenous inputs (reference points, disturbances, etc.), z represents the signals being monitored (error signals, commands, etc.) and y represents the measurements used by the controller to calculate the command u. – Secondly, the small-gain theorem gives us a necessary and sufficient condition for the stability of the loop obtained for any uncertainty Δ(s) such that 1 ()s Δ γ − ∞ < . This is stable if and only if (iff) () ez Ts γ ∞ < , and in this knowledge, we can take account of objectives of robustness during the synthesis process. T(s) e Δ (s) w v z Figure I.2. Standard form for robustness analysis Thus, with the standard approach to robust control, the complexity of controller calculus – hitherto usually based on examination of the open loop – is now reflected in the complexity of determining the set of relevant frequency weights, which make a crucially important contribution to the performances of the final controller. Owing to the difficulty in calculating these weights, the know-how that this operation requires and the conceptual difference from conventional frequency automation engineering, certain engineers are deterred from using the standard approach to robust control, preferring to employ more conventional open-loop concepts. www.it-ebooks.info Introduction xi However, at the same time, the world witnessed the publication of the explicit solution to the robust stabilization of normalized coprime factor plant descriptions [MCF 90], based on the following form. K(s) M(s) -1 Δ M (s) N(s) Δ N (s) v 1 v 2 w u y Figure I.3. Robust coprime factor plant description stabilization – This method, which is highly attractive because of its simplicity, consists of solving two LQG-type Riccati equations. In its 4-blocks equivalent representation, it is a particular case of the standard H ∞ approach to robust control. Noting that we can model the direct and complementary sensitivity functions by modeling the open- loop response, and seeing that any loop transfer is proportional to those sensitivity functions, it is therefore possible to model any loop transfer by working on a single transfer – the open-loop response. This is the principle upon which loop-shaping synthesis is founded. Drawing inspiration from frequency-shaped LQG synthesis, we shape the singular values of the open-loop response using weighting functions on the input and output of the system, thereby creating a loop-shape for which a stabilizing controller can be calculated. This is the definition of H ∞ loop-shaping synthesis. – However, thanks to the notion of the gap metric (which expresses a distance between two systems in mathematical terms) as well as the small-gain theorem, the stability of the loop can be evaluated even before the controller has been explicitly formulated. There is a growing interest in H ∞ loop-shaping synthesis. Obviously, it is less general than the standard H ∞ approach, because the number of degrees of freedom is constrained by the dimensions of the system. However, the adjustment of the input and output weighting functions on the basis of the concepts of conventional frequency automation makes the loop-shaping technique extremely attractive and easy to access – all the more so as it has the qualities of robustness which are inherent to H ∞ techniques. www.it-ebooks.info xii Loop-shaping Robust Control In Chapter 1, we introduce the loop-shaping approach by showing how to obtain a specification on the open-loop response of the servo-loop from a complex frequency specification on multiple loop transfers. Chapter 2 introduces the robust stabilization of a normalized coprime factor plant description. Along with the notion of the gap metric which we then introduce, it constitutes the basis for robust H ∞ loop-shaping synthesis. Chapter 3 relates to two-degrees-of-freedom controllers (2 d.o.f controllers), and two techniques that are closely linked to H ∞ loop-shaping synthesis are presented, thus greatly extending the possibilities for the use of the method. Finally, Chapter 4 opens up avenues for future work: it discusses the main drawbacks to loop-shaping synthesis, and how to solve these issues using modern optimization techniques. I.2. Notations and definitions Below, we review a number of fundamental notions and notations that are frequently employed in the various chapters of this book. I.2.1. Linear Time-Invariant Systems (LTISs) I.2.1.1. Representation of LTISs An n-order linear time-invariant system with m inputs and p outputs is described by a state-space representation defined by the following system of differential equations: 00 () (), ( ) () () () dx Axt But xt x dt yt Cxt Dut =+ = =+ where 1 : – () n x tR∈ is the state of the system; – 0 () x t is the initial condition; – () m ut R∈ isthe system input; – () p ytR∈ is the system output; – nn A R × ∈ is the state matrix; 1 The set of real numbers is denoted as R; the set of complex numbers is denoted as C. www.it-ebooks.info [...]... www.it-ebooks.info 2 Loop-shaping Robust Control – the value to be controlled, y, for which we have a measurement; – the measuring error ε; – the command u created by the controller K(s), whose output disturbed by Γ2 is really applied to the transfer function system H(s) u ε r u' K(s) y y' H(s) Γ2 Γ1 Figure 1.1 General view of control system The task of an automation engineer is then to determine a controller... In this case: w = S y Hv = HSu v www.it-ebooks.info 14 Loop-shaping Robust Control When σ ( HK ) >> 1 or when σ ( KH ) >> 1 (which can happen, particularly at low frequencies in the presence of integrators in the control law), then: σ (S y H ) ≈ σ ( (H K)) , T y σ (H Su ) ≈ σ H = I σ (H ) , Tu = I σ (KH ) In this case, the condition of stability robustness in relation to uncertainties represented in... HK ) >> 1 or when σ ( KH ) >> 1 (which can happen, particularly in low frequencies in the presence of integrators in the control law), then: 1 (( H K ) ) = σ ( H K ) , T 1 , T ) ≈ σ (( K H ) ) = σ (KH ) σ (S y ) ≈ σ −1 σ (Su −1 y u www.it-ebooks.info ≈ I ≈ I 6 Loop-shaping Robust Control and4: σ (SyH ) ≈ σ (H ) σ ( HK ) σ ( KS y ) ≈ σ (K ) σ (K ) 1 ≤ ≈ σ ( HK ) σ ( H ).σ ( K ) σ ( H ) σ ( H Su ) ≈ σ... H r →ε ) = σ ( H 2 → y ) ≈ σ ( H 2 →ε ) ≈ 1 www.it-ebooks.info 8 Loop-shaping Robust Control Thus, by giving the open loop a low gain through its singular values, the automation engineer can favor the command of the looping (in terms of power consumption) in relation to the external inputs by way of all the transfers relating to the control signal u ( H1→u , H 2→u and H r →u ) but has no flexibility... form for robustness analysis Six types of representation of unstructured uncertainties are usually employed: – Direct multiplicative uncertainty: - at input (Figure 1.3): Δi(s) w v K(s) H(s) Figure 1.3 Direct multiplicative uncertainty at input In this case, we achieve a representation similar to Figure 1.2 by using the relations: w = Tu v = KS y Hv ; www.it-ebooks.info 10 Loop-shaping Robust Control. .. ensure the condition of stability robustness – Inverse multiplicative uncertainty: - at input (Figure 1.5): Δi(s) v w K(s) H(s) Figure 1.5 Inverse multiplicative uncertainty at input In this case: w = Su v ; - at output (Figure 1.6): Δo(s) v K(s) H(s) Figure 1.6 Inverse multiplicative uncertainty at output In this case: w = S yv www.it-ebooks.info w 12 Loop-shaping Robust Control According to the small-gain... + D = H (s) I.2.1.2 Controllability and observability of LTISs The system H or the pair (A,B) is said to be controllable if, for any initial condition x(t0) = x0, for any t1 > 0 and for any final state x1, there is a piecewise continuous command u(.) which can change the state of the system to x(t1) = x1 We determine controllability by checking that for any value of t > t0, the controllability Gramian... to be “strictly proper”, and D = 0 2 In the case of a SISO transfer, this means that the degree of the numerator is less than or equal to the degree of the denominator www.it-ebooks.info xiv Loop-shaping Robust Control Finally, for the same transfer matrix, there are an infinite number of possible state-space representations Indeed, consider the linear transformation T ∈ R n× n , where T is invertible,... KH Γ 1 ( s ) − KHu ( s ) = ( I + KH ) −1 Kr ( s ) + ( I + KH ) −1 KH Γ 1 ( s ) + ( I + KH ) 2 That is, when we open the loop at the level of the system input www.it-ebooks.info −1 K Γ 2 (s) 4 Loop-shaping Robust Control and: u '( s ) = u ( s ) − Γ 1 ( s ) = ( I + KH ) −1 Kr ( s ) + ( I + KH ) −1 K Γ 2 ( s ) + ( I + KH ) = ( I + KH ) −1 Kr ( s ) + ( I + KH ) −1 K Γ 2 (s) + = ( I + KH ) −1 Kr ( s ) + (... − BD − 1   D −1   Now consider two systems H1 and H2, whose respective state representations are:  A1 H1 =   C1  B1   D1    A2 H2 =  C2  B2   D2   www.it-ebooks.info xvi Loop-shaping Robust Control The serial connection of H1 with H2 (or the product of H1 by H2) gives us the system: H2  A1 H1H 2 =   C1   A1  =0 C  1 B1   A2 D1   C 2   B1C 2 A2 D1C 2 H1 B2  D2  . Loop-shaping Robust Control www.it-ebooks.info Loop-shaping Robust Control Philippe Feyel Series Editor Bernard Dubuisson www.it-ebooks.info First. 35 2.1.3.Reconstructionofatransferfunctionfromitscoprimefactors 36 www.it-ebooks.info vi Loop-shaping Robust Control 2.1.4. Set of stabilizing controllers – Youla parameterization of stabilizing controllers 37 2.2.Robustnessofnormalizedcoprimefactorplantdescriptions. objectives of robustness during the synthesis process. T(s) e Δ (s) w v z Figure I.2. Standard form for robustness analysis Thus, with the standard approach to robust control, the complexity of controller calculus