202 Asian Journal of Control, Vol. 7, No. 2, pp. 202-208, June 2005 Manuscript received May 30, 2003; revised March 11, 2004; accepted May 12, 2004. The authors are with the Control & Intelligent Processing, Center of Excellence, Department of Electrical & Computer Engineering, University of Tehran, P.O. Box: 14395/515, Te- hran, Iran (e-mails: hrkarimi@ut.ac.ir, yazdan@ut.ac.ir). －Brief Paper－ ROBUST CONTROL FOR A CLASS OF UNCERTAIN STATE-DELAYED SINGULARLY PERTURBED SYSTEMS H.R. Karimi and M.J. Yazdanpanah ABSTRACT This paper considers the problem of robust control for a class of uncer- tain state-delayed singularly perturbed systems with norm-bounded nonlin- ear uncertainties. The system under consideration involves state time-delay and norm-bounded nonlinear uncertainties in the slow state variable. It is shown that the state feedback gain matrices can be determined to guarantee the stability of the closed-loop system for all ε ∈ (0, ∞) and independently of the time-delay. Based on this key result and some standard Riccati ine- quality approaches for robust control of singularly perturbed systems, a constructive design procedure is developed. We present an illustrative ex- ample to demonstrate the applicability of the proposed design approach. KeyWords: Robust stability, disturbance attenuation, singularly per- turbed systems, time-delay. I. INTRODUCTION Singularly perturbed systems often occur naturally because of the presence of small parasitic parameters mul- tiplying the time derivatives of some of the system states. Singularly perturbed control systems have been intensively studied for the past three decades; see, for example, [1]. A popular approach adopted to handle these systems is based on the so-called reduced technique [2]. The composite de- sign based on separate designs for slow and fast subsys- tems has been systematically reviewed by Saksena, et al. [3]. Recently, the robust stabilization of singularly per- turbed systems based on a new modeling approach has been investigated in [4]. The stability problem (ε-bound problem) in singularly perturbed systems differs from conventional linear systems, which can be designed as: characterizing an upper bound ε 0 of the positive perturbing scalar ε such that the stability of a reduced-order system would guarantee the stability of the original full-order system for all ε ∈ (0, ε 0 ) [5]. It is known, by the lemma of Klimushchev and Krasovskii [1,2], that if the reduced-order system is an asymptotically stable, then this upper bound ε 0 always exists. Researchers have tried various ways to find either the stability bound ε 0 or a less conservative lower bound for ε 0 , see [1,5,6]. Also, Shao and Rowland in [7] considered a linear time-invariant sin- gularly perturbed system with single time-delay in the slow states. Then, the research on time-scale modeling was ex- tended to include singularly perturbed systems with multi- ple time-delays in both slow and fast states [8]. Recently, the problem of robust stabilization and disturbance attenua- tion for a class of uncertain singularly perturbed systems with norm-bounded nonlinear uncertainties has been con- sidered in [9]. Also, the robust stability analysis and stabil- ity bound improvement of perturbed parameter (ε) in the singularly perturbed systems by using linear fractional transformations and structured singular values approach (μ) has been investigated by Karimi and Yazdanpanah in [10]. In the sequel of [9], this paper presents new results on control synthesis for robust stabilization and robust distur- bance attenuation for linear state-delayed singularly per- turbed systems with norm-bounded nonlinear uncertainties.