CONTROL SYSTEMS, ROBOTICS, AND AUTOMATION Vol. IX Uncertainty Models For Robustness Analysis - A. Garulli, A. Tesi and A. Vicino

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1 UNCERTAINTY MODELS FOR ROBUSTNESS ANALYSIS A. Garulli Dipartiento di Ingegneria dell Inforazione, Università di Siena, Italy A. Tesi Dipartiento di Sistei e Inforatica, Università di Firenze, Italy A. Vicino Dipartiento di Ingegneria dell Inforazione, Università di Siena, Italy Keywords: Uncertainty odel, Model set, Robustness, Unstructured uncertainty, Structured uncertainty, Paraetric uncertainty, Robust stability, Robust perforance, Sall gain theore, Robust stability argin, Standardized canonical for, H uncertainty, Interval plant, Kharitonov systes, Polytopic plant, Interval atrix, Polytope of atrices, Matrix stability radius. Contents. Introduction. Notation and definitions 3. Uncertainty representation and robustness probles 4. Unstructured uncertainty odels 5. Structured uncertainty odels 6. Highly structured (paraetric) uncertainty odels 7. State space uncertainty odels 7.. Unstructured State Space Uncertainty 7.. Paraetric State Space Uncertainty 8. Conclusions Acknowledgeents Glossary Bibliography Biographical Sketches Suary In any engineering context, it is coon practice to represent physical systes by atheatical odels. Clearly, this representation is not exact. The discrepancy between the physical syste and the atheatical odel is due to two ain sources: i) the lack of inforation on the structure or the behavior of the physical syste; ii) the need for a siple odel in order to apply available analysis and design techniques. Such a discrepancy, although unknown, ust be odeled in a ore or less accurate way in order to account for it in the design of the control law. In this chapter an overview of the ost used approaches to uncertainty odeling is provided. Several unstructured and structured uncertainty odels for both input-output and state-space settings are presented. The properties of these odels in control design probles involving robustness issues are also discussed.

2 . Introduction When odeling physical systes for control purposes, it is necessary to provide odel descriptions that capture the ain features of the syste behavior and are atheatically tractable at the sae tie. An extreely accurate odel of a physical process ay turn out to be unsuitable for application of the available analysis and design techniques. By contrast, an over-siplified odel, which isses significant inforation on the syste structure, ay lead to unacceptable design perforance. A careful balance between capturing the true behavior of the physical syste and generating atheatically tractable odels requires a great effort fro the control designer. A standard way is to assue a siplified odel as the noinal syste. The discrepancy between the syste and the adopted noinal odel is usually represented as a perturbation on the noinal odel. The resulting odel, which is therefore coposed of the noinal one and the perturbation, is usually referred to as the uncertain odel or odel set. For exaple, an infinite-diensional syste is usually represented by a finite-diensional approxiation as the noinal odel, and a perturbation accounting for the neglected dynaics. In order to obtain a satisfactory control design, it is andatory that the control syste perfors well, not only on the adopted noinal odel, but also on the actual physical process. This leads directly to the requireent of control design robustness, which deands that satisfactory perforance is achieved for the uncertain odel, i.e., the noinal odel and the class of possible perturbations. Two approaches are coonly used to describe the uncertainty involved in the physical syste description: i) unstructured; ii) structured. Roughly speaking, the unstructured uncertainty representation is used to describe unodeled or difficult to odel dynaics and it is usually given as a bound on soe easure of the error signal between syste and noinal odel outputs for a chosen class of input signals. The structured uncertainty is represented by an eleent (e.g. a finite diensional vector or an operator) in soe pre-specified uncertainty set of a suitable space. For exaple, in highly structured (or paraetric) uncertainty description, the uncertain eleents ay be the coefficients of a transfer function, or the coponents of syste atrices in a state-space realization. After introducing the eployed definitions and notation in Section, the ain features of the two approaches to describe uncertainty are outlined in Section 3, by focusing on an input-output setting. Section 4 suarizes the ost popular unstructured uncertainty odels eployed for odeling feedback control systes, also illustrating the related sources of syste uncertainty. More structured uncertainty odels are introduced in Section 5, where a standard odel for uncertain control systes is presented. Section 6 is devoted to highly structured, paraetric uncertainty odels. Finally, state-space odels are discussed in Section 7.. Notation and Definitions In this section, the basic aterial required for the representation of uncertainty odels is

3 introduced. Vector and atrix nors are defined in the usual way. In particular, the -nor of a vector n v R is equal to n v i= i axiu singular value of A, denoted by σ [ A]., while the -nor of atrix A is given by the Linear tie-invariant dynaic systes are addressed via the associated real rational transfer function atrices. Figure shows a generic ulti-input ulti-output (MIMO) p syste, with input signal ut () R and output signal yt () R. Figure : Syste G with input u and output y. The corresponding p transfer function atrix is denoted by Gs () = NsD () () s, where N and D are polynoial atrices in the coplex variable s, and Gs () is the p Laplace transfor of the syste ipulse response gt () R satisfying y() t = g( t τ ) u( τ) dτ. When = p=, the syste is said single-input single-output 0 Ns () (SISO) and its transfer function Gs () = is a rational function of s. Ds () In order to evaluate the perforance of a control syste, it is custoary to quantify the size of the involved signals. This is usually done by eans of suitable signal nors. For a signal ut () R, nors that frequently arise in control systes are: - the L nor (or energy nor) u u ( ) 0 i t = dt ; i= - the L nor

4 u = sup ax u ( t). t i =,, i For a scalar signal, the L nor represents the aount of energy associated with the signal, while the L nor is the axiu agnitude attained by the signal. A standard way to assess the perforance of a control syste is to look at the size of the output signal, once the size of one or ore input signals (coands and/or disturbances) is fixed. If the syste G is considered as an operator fro the input space to the output space, the achievable perforance of the syste can be easured according to the induced nor of G, defined as Gu G = sup u. () u 0 Depending on the nors used in () for signals u and y = Gu, different syste induced nors are obtained. Let Gs ( ) be a atrix transfer function with poles in the open left half plane, and gij () t be the entry ( i, j) of the corresponding ipulse response atrix gt ( ). The ost popular syste nors when dealing with uncertainty odels for robust control are: - the H nor [ G jω ] G = sup σ ( ) ; ω R - the H nor G trace ( ) ( ) G = j ω G j ω d ω π ; 0 - the L nor i p 0 =,, j= G = ax g ( t ) dt. ij The interpretation of the above syste nors in ters of input-output gain () are reported in Table for SISO systes. An uncertain polynoial faily of order n is defined as { ( sp ) a n n ( ) n ps a n ( ps δ ) a ( ps ) a 0( p ) p B } ; = ,, ()

5 where p = ( p p, q ) is the paraeter vector, B R is the paraeter set, and ai : B R, i =,, n are given functions. It is usually assued that B is arcwise connected and ai ( ) are continuous functions. Also, an( p) 0, p B, to guarantee an invariant degree polynoial faily. q Bibliography Table : Syste induced nors for SISO systes. TO ACCESS ALL THE 5 PAGES OF THIS CHAPTER, Click here Barish B.R. (994). New Tools for Robustness of Linear Systes. New York, NY: MacMillan Publishing. [A book on robust control of systes with structured paraetric uncertainties] Bhattacharyya S.P., Chapellat H., Keel L.H. (995). Robust Control: The Paraetric Approach. Upper Saddle River, NJ: Prentice Hall. [A coprehensive text dealing with analysis and design issues of control systes with paraetric uncertainties] Dahleh M.A., Diaz-Bobillo I. (995). Control of Uncertain Systes: A Linear Prograing Approach. Englewood Cliffs, NJ: Prentice Hall. [A book on robust control for unstructured uncertainty odels, ainly focused on L robust control] Dahleh M., Tesi A., Vicino A. (993). An overview of extreal properties for robust control of interval plants. Autoatica 9(3), [A survey of the ost significant robustness results concerning highly structured uncertainty odels]. Doyle J.C. (98). Analysis of feedback systes with structured uncertainties. IEE Proceedings, Part D 33, [The paper that introduced the structured singular value µ, for robustness analysis of structured uncertainty odels] Doyle J.C., Francis B.A., Tannenbau A.R. (99). Feedback Control Theory. New York, NY: MacMillan Publishing. [An essential and siple treatent of H robust control, for input-output odels with unstructured uncertainty] Dullerud G.E., Paganini F. (000). A Course in Robust Control Theory: A Convex Approach, New York: Springer-Verlag. [A graduate-level course on robust control theory, focused on convex optiization techniques] Horowitz I. (963). Synthesis of Feedback Control Systes. New York, NY: Acadeic Press. [One of the

6 first books in which uncertainty in control systes is explicitely considered] Šiljak D.D. (969). Nonlinear Systes: Paraetric Analysis and Design. New York, NY: John Wiley & Sons. [An early approach to paraetric analysis of robust nonlinear control systes] Zaes G. (98). Feedback and optial sensitivity: odel reference transforations, ultiplicative seinors, and approxiate inverses. IEEE Transactions on Autoatic Control 6(), [The paper that introduced the H approach to control design and disturbance rejection] Zhou K., Doyle J.C., Glover K. (996) Robust and Optial Control, Upper Saddle River, NJ: Prentice Hall. [A state space approach to optial H and H control] Biographical Sketches Andrea Garulli was born in Bologna, Italy, in 968. He received the Laurea in Electronic Engineering fro the Universit`a di Firenze in 993, and the Ph.D. in Syste Engineering fro the Universit`a di Bologna in 997. In 996 he joined the Dipartiento di Ingegneria dell'inforazione of the Universit`a di Siena, where he is currently Associate Professor. He serves as Associate Editor for the IEEE Transactions on Autoatic Control and he is eber of the Conference Editorial Board of the IEEE Control Systes Society. He is author of ore than 60 technical publications and co-editor of the book "Robustness in Identification and Control", Springer, 999. His present research interests include robust identification and estiation, robust control, LMI-based optiization, obile robotics and autonoous navigation. Alberto Tesi received the Laurea in Electronic Engineering fro the Universit`a di Firenze, Italy, in 984. In 989 he obtained the Ph.D. degree fro the Universit`a di Bologna, Italy. In 990 he joined the Dipartiento di Sistei e Inforatica of the Universit`a di Firenze, where he is currently a Professor of Autoatic Control. He has served as Associate Editor for the IEEE Transactions on Circuits and Systes fro 994 to 995 and the IEEE Transactions on Autoatic Control fro 995 to 998. Presently he serves as Associate Editor for Systes and Control Letters. His research interests are ainly in robust control of linear systes, analysis of nonlinear dynaics of coplex systes, and optiization. Antonio Vicino was born in 954. He received the Laurea in Electrical Engineering fro the Politecnico di Torino, Torino, Italy, in 978. Fro 979 to 98 he held several Fellowships at the Dipartiento di Autoatica e Inforatica of the Politecnico di Torino. He was assistant professor of Autoatic Control fro 983 to 987 at the sae Departent. Fro 987 to 990 he was Associate Professor of Control Systes at the Universit`a di Firenze. In 990 he joined the Dipartiento di Ingegneria Elettrica, Universit`a di L'Aquila, as Professor of Control Systes. Since 993 he is with the Universit`a di Siena, where he founded the Dipartiento di Ingegneria dell'inforazione and covered the position of Head of the Departent fro 996 to 999. Fro 999 he is Dean of the Engineering Faculty. In 000 he founded the Center for Coplex Systes Studies (CSC) of the University of Siena, where he presently covers the position of Director. He has served as Associate Editor for the IEEE Transactions on Autoatic Control fro 99 to 996. Presently he serves as Associate Editor for Autoatica and Associate Editor at Large for the IEEE Transactions on Autoatic Control. He is Fellow of the IEEE. He is author of 70 technical publications, co-editor of books on `Robustness in Identification and Control', Guest Editor of the Special Issue `Robustness in Identification and Control' of the Int. Journal of Robust and Nonlinear Control. He has worked on stability analysis of nonlinear systes and tie series analysis and prediction. Presently, his ain research interests include robust control of uncertain systes, robust identification and filtering, obile robotics and applied syste odeling.