Proceedings zyxwvutsrqponm of zyxwvutsrqp the 39'b JEEE Conference on Decision and Control Sydney, Australia December, 2OOO Controller validation for stability and performance based on a frequency domain uncertainty region obtained by stochastic embedding+ zyx Xavier Bombois(') , Michel Geverdl) , Gkrard Scorletti(2) CESAME, Universitk Catholique de Louvain B6timent EULER, 4 av. Georges Lemaitre, B-1348 Louvain-la-Neuve, Belgium Tel: +32 10 472596, Email: zyxwv {Bombois , Gevers}Ocsam. ucl . ac .be ('1 LAP ISMRA 6 boulevard du Markha1 Juin, F-14050 Caen Cedex, F'rance Tel: +33 2 31452712, Email: gerard.scorlettiOgreyc.ismra.fr Abstract This paper presents a robustness analysis for an un- certainty set deduced from stochastic embedding tech- niques and made up of ellipsoids at each frequency in the Nyquist plane. Our robustness analysis focuses on the validation of a controller both for robust sta- bility and for robust performance, over all systems in such frequency domain uncertainty region. Our valida- tion procedure for stability ensures that the controller stabilizes all systems in this nonstandard uncertainty set. Our validation procedure for performance com- putes the worst case performance over all closed loop systems made up of the controller and all plants in the frequency domain uncertainty region. 1 Introduction This paper is part of our continuing effort to connect time-domain prediction error identification and robust- ness theory [3, 21. In our previous papers, we have analysed the robustness properties of an uncertainty set zyxwvu D delivered by classical prediction error identification methods and to which the true system zyxwvutsr GO was known to belong with some prescribed probability. This un- certainty set zyxwvutsrq 2) was defined as a set of parametrized rational transfer functions whose parameter vector lies in an ellipsoidal confidence region. Such uncertainty region results naturally from a prediction error identi- fication experiment when the system is assumed to be in the model set. The uncertainty set is thus entirely defined by covariance errors on the parameters. This restriction can be relaxed using the stochastic embed- ding approach of [5] to construct uncertainty regions that then take into account both bias and variance er- t The authors acknowledge the Belgian Programme on Inter- university Poles of Attraction, initiated by the Belgian State, Prime Minister's Office for Science, Technology and Culture. The scientific responsibility rests with its authors. rors in the estimated transfer functions. In the present paper, we develop robust analysis tools for such uncer- tainty regions obtained using a stochastic embedding technique. Uncertainty region C. Stochastic embedding tech- niques developed in e.g. [5] allow one to design a fre- quency domain uncertainty region around a possibly biased identified model with k e d denominator (such as FIR or Laguerre models). This uncertainty region t contains the stable unknown true system Go at a certain probability level and is made up of ellipsoids at each frequency in the Nyquist plane around the fre- quency response of the identified model. These uncer- tainty ellipsoidsare designed using the assumption that the unmodelled dynamics of the true system can be considered as a stochastic process and that the param- eters that describe the second-order properties of this stochastic process can be estimated from the data. This parameter estimation step is achieved using a maxi- mum likelihood technique. One of the contributions of the present paper is to extend the stochastic embedding technique to closed-loop identification. Controller validation for stability. Our validation procedure for stability ensures that a given controller C stabilizes all systems in an uncertainty region C ob tained by stochastic embedding. Robust stability the- ory developed in e.g. [9, 7, 61 provides some necessary and sufficient conditions for the stabilization, by some given controller C, of all plants in an uncertainty re- gion that is defined in a general LFT (linear fractional transformation) framework. Our contribution with the proposed stability validation procedure is to show that one can rewrite the closed-loop connection of the con- troller C and all plants in such uncertainty region C as a particular LFT where the uncertainty part is a transfer vector whose frequency response is real. In that particular LFT, the (real) stability radius can be 0-7803-663&7/00$10.00 0 2000 IEEE 689