Lecture Notes in Control and Information Sciences 192 Editor: M. Thoma Roy S. Smith dna Mohammed Dahleh (Eds.) The Modeling of Uncertainty ni Control Systems Proceedings of the 1992 Santa Barbara Workshop galreV-regnirpS London Berlin Heidelberg weN York Paris Tokyo Hong Kong Barcelona Budapest Series Advisory Board A. Bensoussan • M.J. Grimble • P. Kokotovie • H. Kwakernaak • J.L. Massey Y. Z. Tsypkin EdRo~ Roy S. Smith, Phi) Electrical and Computer Engineering Department, University of California at Santa Barbara, Santa Barbara, California 93106-9560, USA Mohammed Dahleh, PhD Mechanical and Environmental Engineering Department, University of California at Santa Barbara, Santa Barbara, California 93106-9560, USA This book was processed using the I~TEX macro package with the LMAMULT style and additional modifications by Roy S. Smith ISBN 3-540-19870-9 Springer-Vedag Bedin Heidelberg New York ISBN 0-387-19870-9 Springer-Vedag New York Berlin Heidelberg British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designasn dP atentAsc t 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 of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. © Springer-Verlag London Limited 1994 Printed in Great Britain The publisher makes no representation, express or implied, with regard to the accuracy, of the informaticoonn tained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Typesetting: Camera ready by editors Printed and bound by Antony Rowe Ltd., Chippenham, Wiltshire 69/3830-543210 Printed on acid-free paper Preface This volume collects together a series of essays and papers arising from a workshop held at the University of California, Santa Barbara, 18-20th June, 1992. The official title was "The Modeling of Uncertainty in Control Sys- tems," and the sponsorship was provided by the National Science Foundation (NSF) and Air Force Office of Scientific Research (AFOSR). 1 The format of the workshop was somewhat novel and is worthy of com- ment here. For logistical reasons, it was held the week prior to the 1992 American Control Conference (ACC) in Chicago. As almost all workshop at- tendees would also be at the ACC, there was little point in having a program of formal presentations -- only to have it repeated the following week. The format chosen was suggested by members of the advisory committee with the emphasis being placed on noissucsid groups rather than formal .snoitatneserp of A tour lareves noitacifitnedi and lortnoc experiments ta the teJ Propulsion Laboratory was conducted rof the to prior attendees the workshop. Attendees yllacipyt had experience research ni rehtie traditionM -ifitnedi cation or robust .lortnoc In each area, some progress has been made towards the common goal of a developing deifinu noitacifitnedi and design paradigm. Re,archers ni the noitacifitnedi are area now considering models with un- ytniatrec Several interval. confidence as expressed groups of robust lortnoc srehcraeser have begun the considering noitacifitnedi ni worst a case gnittes which attempts obtain to models compatible with the gnitsixe robust lortnoc approaches. design Others era involved ni the studying seitreporp and -ilppa of cation evitareti ugised/noitacifitnedi schemes. Research ni these areas si only just beginning and of objective the siht workshop was to broaden the sevitcepsrep those of working ni these ,saera etaitini ,snoitaroballoc and out- enil common research .slaog Each academic tnapicitrap was a bring to asked graduate student detseretni ni area. the Most did and these often students played an active elor ni the ,snoissucsid contributing rieht own ideas sa llew as gnitifeneb from the esolc involvement with experienced .srehcraeser The stnapicitrap were four into divided groups rof deliated snoissucsid on I The workshop was funded by NSF and AFOSR under NSF number: grant ECS- 9203908 vi Preface a particular topic. The topics were essentially defined by the participants and are loosely categorized as: Probablistic uncertainty modeling. This group covered the probablistic bound- ing approaches to the problem. This is in the line of the more traditional identification theory and addressed such things as the effects of under- modeling. Worst case identification. The theory of worst case deterministic identifica- tion, in the presence of bounded noise, was discussed here. This area has seen a recent burst of activity. Iterative identification/control design. Several of these schemes have been proposed, both in probabilistic and Hoo settings. The discussion focused on whether a closed loop approach was necessary and what benefits might be gained from it. "What is it that eot really toant from control modular ~ A long term view was taken by the pea~ticipants in this discussion group in an effort to find some unifying ideas or approaches. Each discussion group was given the task of summarizing their area and identifying open problems and research directions. After a series of such dis- cussions each group presented the results of their discussion to the wider audience and lead a general discussion on the overall context of this work. This format proved to be very successful with most participants gaining an understanding of the issues involved in the other related areas. An effort has been made to maintain some of the informal discussion aspects in this proceedings volume. After the workshop, participants were invited to submit papers in either or both of two categories: speculative essays and formal papers. The essays are intended to provide an overview of the area and outline future research directions. This perspective is often difficult to obtain purely from research papers. This research area is in its infancy and the essays allow others to see the motivating issues and potential research directions. The papers, on the other hand, give a concrete indication of where the field currently stands. Both the essays and the papers were written after the workshop, giving the authors an opportunity to present their ideas and work in the context of the wider issues. Part I contains twelve essays. Several give general comments and opinions on the field and others address specific aspects in more detail. Pramod Khar- gonekar, in ~'Identification and Robust Control," provides a broad overview of the identification for robust control field. The extensive references will give a reader new to the axes an excellent starting point. ~An Essay on Identification of Feedback Systems, ~ by Pertti Miikil~ and Jonathon Partington outlines some of the issues that arise in deeidingibetween probabilistic or deterministic frameworks. In "Thoughts on Identification .for Control," Michael Safonov discusses the achievements of modern robust and robust adaptive control and~how Prefsce vii they are limited by the lack of appropriate identification for control tech- niques. The mathematical as well as the philosophical implications of a priori knowledge of the systems' structure, in addition to the separation between identification and control paradigm, are discussed. A similarly broad view of the problem is taken by Hidenori Kimura in ``An Essay on Roblst Control. ~ This essay analyzes the role of models in control systems design, the difficulty of arriving at such models, and contrasts control design based on models with control design based on mere insight and experience. The impact of modeling of uncertainty on control design is discussed. Robert Kosut, in ``On the Character of Uncertainty for System Identifica- tion and Robust Control Design, ~ points out that most traditional identifica- tion methods do not provide information for error estimates. Robust control calls for a nominal model and an error estimate. This essays contrasts the deterministic and probabilistic paradigms for uncertainty modeling. A parametric approach is discussed by John Norton in ``Ezfensions of Parametric Bounding Formulation of Identification for Robust Control De- sign." In this note the parametric bounding methods are discussed with the aim of motivating the incorporation of more a priori information, such as joint bounds on transfer function parameters. It also includes a discussion of matching of identification and control design methods, performance guaran- tees for modeling techniques, and performance guarantees for controllers. "Connecting Identification and Robust Control, ~ by Michel Gevers dis~ cusses the gap between the estimation of uncertain bounds and the current identification for robust control design. Iterative identification/control design schemes are presented as a means of bridging this gap. Iterative methods are also considered by Paul Van dHeonf , Ruud Schrama and Peter Bongers in ``On Nominal Models, Model Uncertainty and Iterative Methods in Identification and Control Design. ~ Their paper discusses the issues related to the role of nominal models and model uncertainty in the problem of joint identification and control design. General viewpoints on the construction of iterative schemes and their related criteria are discussed. Model validation takes a somewhat different approach to the problem. Roy Smith discusses this area in ``An Informal Review of Model Validation. ~ The problem is discussed in the context of identification for robust control. Future research prospects, as well as the current state of the art, are outlined for both the ooH and 11 cases. Munther Dahleh and John Doyle, in ``From Data to Control, ~ discuss an iterative validation approach. The idea involves characterizing and determin- ing the set of models that are unfalsified by the data, designing controllers for these models, and collecting further data to reduce the set ofunfalsified mod- els. The controller performance is improved as the set of unfalsified models shrinks. The applicability of worst case design approaches is called into question by Jeffrey Kantor and Billie Spencer, Jr. in their essay, "Is Robust Control viii Preface ~ f Reliable They suggest methods based on estimating structural reliability as a reliable alternative. A significant proportion of the researchers camef rom application specific backgrounds. Daniel River,, in "Modeling Uncertainty in Control Systems: A Process Control Perspective," discusses the requirements of any identi- fication/design theory from the chemical process control point of view, A distillation column is considered as an example of a benchmark problem. Nineteen technical papers are included in Part II. They can be considered as falling within several broad categories. The first of these is the problem of identification in a worst case setting. The probabilistic framework is con- sidered next ~nd then the iterative identification and design approaches are covered. Th, "olume closes with several papers discussing practical applica- tions and benchmark problems. "A Note on oeH System Identification with Probabilistie A Priori Infor- mation," by Clas Jacobson and Gilead Tadmor, deals with control-oriented identification in the ooH setting. They consider a probabilistic setting of a priori information and show how, under certain specifications of a priori in- formation, the probabilistic problem setting can be converted to a related worst case problems An algorithm to solve the related worst case problem is given. Jie Chert, Carl Nett and Michael Fan, present worikn the identificatifoonr Heo area in "A Worst Case Identification Method Using Time Series Data. ~ A discrete-time, SISO, linear time-invariant system is considered. Noise is assumed to be additive, at the system output, and oel bounded. It is also assumed that system has an impulse response bounded by an a priori specified decaying exponential. An algorithm for generating a nominal system, with a calculable Heo error bound, is presented. The exponentially bounded system and oel bounded noise are assumptions common to much of the work in this area. The concepts of algorithm convergence and optimality under these assumptions axe also discussed. The paper, "Identification in oeH using Time-Domain Measurement Data," by Guoxiang Gu, looks at a similar problem for systems under feedback. Sine- dwell experiments are proposed, leading to an assumed frequency domain noise bound. The identification error is quantified in terms of the directed gap metric. A time domain approach is taken by Pertti M~ilii and Jonathon Parting- ton in "Identification of Feedback Systems from Time Series." They discuss closed loop identification methods with a gap metric. The focus is on ARX models where the AR and X parts are coprime. A different approach to this problem is taken by Tong Zhou and Hidenori Kimura in "Input-Output Ertrapolation-Minimization Theorem and its Ap- plication to Model Validation and Robust Identification. ~ They characterize the smallest Heo norm transfer function matching input-output data in the time domain. Details are provided for the SISO noiseless case although the work can be extended to include noise and MIMO problems. This result is Preface ix applied to model validation for additive, multiplicative or co-prime pertur- bation descriptions. The results are also applied to the identification in Hoo problem in the case where the system poles are specified a priori. Each of these applications leads to a convex optimization problem. Richard Hakvoort and Paul Van den Hof consider a similar problem in "Identification of Model Error Bounds in il and Hoo-Norm." A discrete- time formulation is used and the system pulse response is assumed to satisfy an a priori exponential bound. The additive perturbation can be a priori frequency weighted and the noise is assumed to satisfy a time-domain ool bound. In the case where an 11 norm bound is applied to the perturbation, the approach results in a linear programming problem. In the Hoo bound case, the perturbation is overbounded and a linear programming approach is again used to find the minimum overbound. In "Asymptotic Worst-Case Identification with Bounded Noise, ", Munther A. Dahleh presents an overview of the work of the author and co-workers in the area of worst-case identification with bounded noise. The objective of the study is to address the fundamental limitations and capabilities of worst-case identification along with their impact on input design and algorithms. The approach is applied to a variety of model sets such as stable LTI systems and nonlinear systems with fading memory. An adaptive approach is taken in "Sequential Approzimation of Uncer- tainty Sets via Parallelotopes," by Antonio Vicino and Giovanni Zappa. This paper adopts a deterministic setting for disturbances through hard bounds. An adaptive algorithm is presented for constructing recursively an outer bounding parallelotopic approximation of the parameter uncertainty set. This can be employed in a purely parametric or in mixed parametric/non-parametric settings. Tung-Ching Tsao and Michael Safonov also consider an adaptive approach in A`` Robust Eilipsoidal-Bound Approach to Direct Adaptive ~ Control. Their direct robust adaptive control scheme overcomes some of the conservatism in ellipsoidal-bounding indirect schemes. A key idea is the introduction of a fic- titious reference signal which facilitates direct identification of the controller to meet performance specifications using measurements of plant input and output. No use is made of the separation of identification and control which is essential for indirect methods. Er-Wei Bai and Sundar Raman propose an on-line approach in nO" Line Model Uncertainty Quantification: Hard Upper Bounds and Conven3ence." They present a recursive, on-line, adaptive identification scheme to determine bounds in either an//2, Hoo or pointwise sense. The models are of restricted complexity and the order selection issues are discussed. Douwe de Vries and Paul Van den Hof consider a mix of both frame- works in Aa Mixed Deterministic-Probabilistic Approach for Quantifying -nU certainty in Transfer Function Estimation." They present a procedure to ob- tain an estimate of the transfer function of a linear system together with an upper bound on the error, using only limited a priori information on the x ecaferP data generating process. The model error consists of both probabilistic and deterministipca rts. ~Estimation for Robust Control," by Brett Ninness and Graham Good- win, considers the estimation otfh e frequency response of LTI systems from observed noise corrupted data. In contrast to related work it examines the extension of the stochastic paradigm to problems of undermodeliTnhgi.s ap- proach requires certain a priori information to be imposed on the class of undermodelings which in turn makes the problem amenable to analysis by stochastic estimation techniques. The consequences of time variation are considered by H~kan Hjaimarason in Vanishing "Non- Model Errors." This paper considers time-varying systems that are essentially time-invariant but with small, arbitrary fast, fluctuations of the dynamics. The approach aims at providing a method for detecting such time-varying terms, and to obtain a methodology to determine whether the assumption that with infinitely many data it is possible to get a perfect model is valid. A more classical identification approach is taken by Wallace Larimore in ycaruccA~' Confidence Bands Including the Bias of Model Under-Fitting. ~ A parametric model is obtained from an underlying linear system in the presence of stochastic noise and disturbances. Confidence bands are obtained as well as a nominal model. Model order selection issues are also discussed. The iterative identification and design approach is addressed in several papers. In the first of these, "Iterati~e Identification and Control Design: A Worked Out Example, ~ by guud Schrama and Paul Van den Hot, the ooH normo f a particular transfer matrix is applied as the performance criterion. A weightiedde ntification problem is used to minimize the error between this transfer matrix for the nominal model and the actual system. The tradeoff between nominal performance and robust stability is also adjusted, via a user specified parameter, during the iteration. A simulation example illustrates that a good open-loop fit is not necessary for the design of a good robust controls ystem. An iterative framework is outlined by David Bayard and Yeung Yam in ycneuqerF" Domain Identification for Robust Control Design. ~ The approach given here involves iterative reweighting of the identification scheme. This contrasts with the previous approach in that new input-output data is not obtained with the controller designed at each stage of the iteration. The underlying system is assumed to satisfy an a priori exponential decay on the impulse response. The noise and disturbances are assumed to be Gaussian and the approach gives a statistical confidence bound on the ooH error of the identified model. A simulation example illustrates the application of the approach. Humid Ajbar and Jeffrey Kantor present an integrated design and di- agnostic approach in ~Time Domain Approach to the Design of Integrated Control and Diagnosis Systems. ~ They consider the 11 design framework and develop a design framework which allows fault detection in the presence of Preface xi unknown bounded perturbations. One of the major motivations in this field is the difficulties that arise when applying standard robust control techniques to certain physical systems. It is therefore fitting to close the volume with some benchmark problem ap- plications. Elling Jacobsen and Signrd Skogestad give a detailed description of a heat exchanger in "Identification of Ill.Conditioned Plants ~ A Bench- mark Problem." A standard identification procedure is shown to give a very poor model in the multivariable sense. A nonlinear model, linearized model and Matlab based simulation are provided for this system to allow other researchers to investigate and apply their own techniques to this problem. Peter Bongers and Gregor van Baars describe a wind turbine system in "Control Design and Implementation Based on Experimental Wind Turbine Models." The physical aspects of this problem are used to illustrate some of the issues that typically arise in the practical design of robust controllers. Multiple input-output data is taken and multiple models are obtained. The gap metric is used to pick a suitable nominal with the criteria being the smallest gap between the nominala nd all other identified models. The design is done in order to obtain robustness with respect to this size gap. As these outlines illustrate, many viewpoints are represented in the fol- lowing pages. This research area is fundamental to the application of robust controlt echniques to practical problems. The work is only just beginning and we hope that volume serves as an interesting introduction of the field to the wider control community. Roy htimS Mohammed helhaD University of California, Santa Barbara January, 1993.