PII: sooo5-1o!M(%)ooo!M-2 zyxwvutsrqponmlkjihgfedcbaZYXWVUTS Aummarica, Vol. 32, No. I I, pp. 1575-1579, 19% Copyright 0 19% Elsevier Science Ltd Printed in Great Britain. All rights reserved tm5-10981% Sl5.00 + 0.00 Brief Paper zyxwvutsrqpon K-based Synthesis for a Robust Controller of Interval Plants* KANT1 B. DAlTA_F and VIJAY V. PATELT Key Words-Robust controller; HZ-based synthesis; interval plants; controller design with structured perturbations; design via Kharitonov’s polynomials. Abstract-The synthesis of a robust controller for a SISO interval plant is carried out by converting the numerator and denominator parametric uncertainty to an uncertainty band around the numerator and denominator polynomials of the nominal plant model and then applying the standard H, method for the mixed sensitivity problem in which the selection of weighing functions is made using Kharitonov’s polynomials. Copyright 0 1996 Elsevier Science Ltd. 1. Introduction In most practical systems at least two types of uncertainties are present: first, unmodeled dynamics, which represent high-frequency uncertainties, and, second, parametric uncer- tainty, representing lack of precise knowledge of the actual parameters. They cause the deterioration of system performance, and may even drive the system unstable. Analysis of parametric robust stability is carried out via the edge theorem in Bartlett et al. (1988), the [p-norm approach in Tsypkin and Polyak (1991), and a generalization of Kharitonov’s theorem in Chapellat and Bhattachryya (1989) and Chapellat et al. (1990). However, the synthesis procedure for a robust controller that can stabilize the plant having parametric perturbations has not been fully developed. A few synthesis methods are now available in the literature to design a robust controller for plants where the uncertainty is modelled as an unstructured perturbation (Kimura, 1984; Kwakemaak. 1985: Francis. 1987: Dovle et al.. 1989). In Kimura (1984) a necessary and sufhcieni condition for robust stabilizability is derived based on Nevanlinna-Pick theory for the class of plants characterized by a frequency-dependent uncertainty band function around the transfer function of a nominal model. For the existence of a robust controller, the values of the uncertainty band function should be restricted within a certain range at the unstable poles of the nominal model. A synthesis procedure for a robust controller is provided along with the parametrization of the set of all robust controllers. To design a (robust) controller using standard H, techniques, a weighted sum of sensitivity and control sensitivity functions is minimized, for which a state-space approach (Doyle et al., 1989) and a frequency-domain polynomial approach (Kwakernaak, 1985, 1991) are avail- able. The controller designed employing the above methods provides robustness to the plant for a designed amount of unstructured perturbations. Recently, in Bhattacharya et al. (1993) an attempt was made to synthesize a robust controller for the interval plant by converting the parametric uncertainty to an uncertainty band around the transfer function of the nominal model. This *Received 12 July 1994, revised 13 March 1995; received in final form 21 January 19%. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor F. Delebecque under the direction of Editor Ruth F. Curtain. Corresponding author Professor K. B. Datta. Tel. +91 03222 55224 7612; E-mail kbd@ee.iitkgp.emet.in. t Department of Electrical Engineering, Indian Institute of Technology, Kharagpur 721 302, India. is virtually equivalent to transforming the synthesis problem for an SISO interval plant to the synthesis of an H, controller for unstructured additive perturbations. Specifi- cally, the transformed problem is to synthesize an H,-robust controller when the plant uncertainty can be represented by a norm-bounded additive unstructured perturbation rep- resented by P(s) = PO(s) + 6P(s), ISP(jo)( 5 r(jw), where the uncertainty band r(s) is a stable, proper and rational function. Given the nominal plant e,(s) and the uncertainty r(s), H, synthesis methods can be used to check if the given class of plants is robust-stabilizable and, if so, to find a family of robust stabilizers for the given family of plants. In this paper, instead of additive unstructured perturba- tions, numerator-denominator perturbations are considered by representing the plant as P(s) = N(s)&‘(s), with P,,(s) = H,,(s)&‘(r) as the nominal plant, where, N(s) = N,,(s) + &V(s), D(s) = Q,(s) + 6D(s); &V(s) and SD(s) are polynomials. In place of additive perturbation structure, if this numerator-denominator perturbation structure is used, the following merits are its direct outcome. (a) Stability and performance robustness are simultaneously possible, while in the previous case they are not. (b) Unlike in Bhattacharva et al. (1993), where P;,(s) and f(s) must have the same number of unstable poles, in the present case Q,(s) and D(s) may have different numbers of unstable poles. The iterative procedure in Bhattacharya et al. (1993) to maximize E is not required in the present method. More explicitly, in their method the value of E is first fixed for which r,(jo) is determined by computing the maximum value of IP,(jo) - P&O)], when w varies from 0 to 2, where P,(s) is one of the 32 segments of the plant family for the fixed F. Then r(jw) is constructed as a stable proper rational function bounding the magnitude of r,(jw) for all frequencies. The controller is found by minimizing the H, norm of the function r(s)C(s)[l + C(s)P(s)]-‘, and if it is less than one then the controller stabilizes the family for the E which was fixed. If it is not less than one then the value of F is decreased and the whole procedure is repeated. These iterations are not necessary in our case. Also, we need to consider only two Kharitonov polynomials-one for the numerator and the other for the denominator of the interval plant-instead of 32 plant segments, which decreases the calculational burden significantly. I. Problem and main result The family of interval plants under consideration arc represented without loss of generality by (Chapellat et al., 1990) I+,&):= N(s E) n (&)S’ + + n,,(e) P(s):P(s)=----= ‘I D(s, E,) d,,(e)sQ +. +d,,(e) ’ 1575