GUARANTEED PERFORMANCES DESIGN VIA MODEL SETS IDENTIFICATION S. Malan ∗ M. Milanese ∗ D. Regruto ∗ M. Taragna ∗ ∗ Dip. di Automatica e Informatica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy. stefano.malan@polito.it, mario.milanese@polito.it, diego.regruto@polito.it, michele.taragna@polito.it Abstract: In this paper an integrated robust identiﬁcation and control design procedure is proposed. It is supposed that the plant to be controlled is linear, time invariant, stable, possibly inﬁnite dimensional and that input-output noise-corrupted measurements are available. The emphasis is placed on the design of controllers guaranteeing robust stability and robust performances, and on the trade off between controller complexity and achievable robust performances. First, uncertainty models are identiﬁed, consisting of parametric models of different order and tight frequency bounds on the magnitude of the unmodeled dynamics. Second, Internal Model Controllers, guaranteeing robust closed loop stability and best approximating the “perfect control” ideal target, are designed using μ-synthesis techniques. Then, the robust performances of the designed controllers are computed, allowing to determine the level of model/controller complexity needed to guarantee desired closed loop performances. Copyright c  2003 IFAC. Keywords: robust control, identiﬁcation for control, uncertain systems. 1. INTRODUCTION The typical problem a control designer has to face in most practical situations can be brieﬂy described as follows: a physical plant is given and a control law has to be designed, able to drive the plant to reach, if possible, given performance speciﬁcations. The classical approach consists in building a mathe- matical model of the plant, on the basis of available in- formation on it (physical laws, time invariance, linear- ity, etc.) and of input-output measurements, and then designing a control that meets the desired performance speciﬁcations for the identiﬁed model. However, this way it is not taken into account that any identiﬁed model is only an approximation of the actual plant. Indeed, the performances that can be actually achieved on the plant may be very poor, according to the size of the modeling error, and even the closed loop sta- bility may be missed. These problems motivated the large interest devoted to robustness issues in the last decade. Robust control methodologies aim to design controllers guaranteeing to meet the speciﬁcations not for a single nominal model, but for all models ob- tained by given perturbations of the nominal model. However, the size of such perturbations has to account for the modeling error, which is not known and has to be estimated using available information and actual measurements on the plant. Moreover, the nominal model and the perturbation, indicated here as uncer- tainty model, have to be identiﬁed in a form suitable for the robust design, thus requiring strict interaction between identiﬁcation and the goal of control design. In this paper an integrated identiﬁcation and control design procedure is proposed. It is supposed that the plant P o to be controlled is linear, time invariant, stable, possibly inﬁnite dimensional and that input- output noise-corrupted measurements are available, together with some general information on the im- pulse response decay rate of the plant and on the char- acteristics of the noise. The emphasis is placed on the design of controllers guaranteeing robust stability and This research was supported in part by funds of Ministero dell’Universit` a e della Ricerca Scientiﬁca e Tecnologica under the Project “Robustness techniques for control of uncertain systems”. robust performances when used on the actual plant, and on the trade off between controller complexity and achievable robust performances. The proposed design procedure is based on the follow- ing main steps. First, uncertainty models are identiﬁed, consisting of models M of different order and frequency bounds on the magnitude of the modeling error Δ= P o - M . The models are selected within given classes of para- metric models M (p), estimating the parameter vector ˆ p minimizing the H ∞ norm of the modeling error. The nonparametric part of the uncertainty models accounts for the unmodeled dynamics, by evaluating tight fre- quency bounds W Δ (ω) on their frequency response, assuring that, under the considered assumptions, the plant P o is within the uncertainty models. Second, Internal Model Controllers (IMC), guaran- teeing robust closed loop stability and approximat- ing frequency domain “perfect control” ideal target, are designed for each identiﬁed model set using µ- synthesis techniques. Then, the robust performances of the designed controllers, i.e., the performances that can be guaranteed for all systems belonging to the un- certainty models, are derived. The derived controllers can be further reduced by means of standard approxi- mation techniques until the corresponding guaranteed performances are considered acceptable. A key point in the proposed procedure is the identiﬁ- cation of the uncertainty model. In recent years this problem has been widely studied. For an extensive list of references, see, e.g., the surveys Milanese and Vicino (1993), M¨ akil¨ a et al. (1995) and Ninness and Goodwin (1995). However, most of the papers in the literature use a nonparametric approach, leading to identiﬁed models of high order, greater or equal to the number of data. As a consequence, if the control is designed using H ∞ robust methods, the controller complexity is quite “high”. More importantly, only rough bounds are usually derived on the magnitude of the identiﬁcation error, whose degree of conser- vativeness is unknown or quite high. Consequently, the guaranteed performances may result very poor and conservative.