936 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 36, NO. 5, SEPTEMBER 2006 A Linear Matrix Inequality Approach for Guaranteed Cost Control of Systems With State and Input Delays Olga I. Kosmidou, Member, IEEE, and Yiannis S. Boutalis, Member, IEEE Abstract—The robust control problem for linear systems with parameter uncertainties and time-varying delays is examined. By using an appropriate uncertainty description, a linear state- feedback control law is found, ensuring the closed-loop system’s stability and a performance measure, in terms of the guaranteed cost. A linear matrix inequality objective minimization approach allows to determine the “optimal” choice of free parameters in the uncertainty description, leading to the minimal guaranteed cost. Index Terms—Guaranteed-cost control, linear matrix inequali- ties (LMIs), optimal control, time delays, uncertain systems. I. I NTRODUCTION M ANY physical systems are characterized by model uncertainties and time delays that have to be taken into account in the design of control laws in order to avoid poor performances and even instability of the control systems [20], [21], [25]. Stability analysis and robust control problems of uncertain dynamic systems with time delays have been widely studied in recent control-systems literature; for details and references, see, e.g., [14], [26], and [28]. The design of robust controllers for uncertain time-delay systems leads to complex problems lacking of analytical solutions; hence, linear matrix inequality (LMI) techniques are often used to provide computational solutions for continuous-time [6], [9], [11], [13], [16]–[18], [22], [24], [30], [31] and discrete-time systems [12], [29]. All of the above techniques consider uncertain dynamic systems with delayed states. However, since control input delays are often imposed by process-design demands, as in the case of transmission lines in hydraulic or electric networks, it is necessary to consider uncertain systems with both state and input time delays. Moreover, when the delays are imperfectly known, one has to consider uncertainty on the delay terms as well. To the authors’ best knowledge, only a few methods deal with the class of systems with both state and input uncertain delays [7], [19], [23]. Besides, it is known that robust control approaches for qua- dratic stability and guaranteed cost are often characterized by conservatism, since the corresponding control laws are derived from sufficient conditions. In order to reduce conservatism, one has to solve some auxiliary parameter optimization problem Manuscript received March 23, 2004; revised December 2, 2004 and March 30, 2005. This paper was recommended by Associate Editor G. C. Calafiore. The authors are with the Department of Electrical and Computer Engi- neering, Democritus University of Thrace, 67100 Xanthi, Greece (e-mail: kosmidou@ee.duth.gr; ybout@ee.duth.gr). Digital Object Identifier 10.1109/TSMCA.2005.855788 [19]. As shown by Dorato et al. [8], this is an analytically intractable problem; in effect, the guaranteed-cost bound can be minimized only in a limited number of particular cases. On the other hand, the approaches in [7] and [23] make use of an LMI optimization technique in order to minimize the guaranteed- cost bound. Their drawback consists of using arbitrarily chosen scalars to cope with a nonconvex LMI objective minimization problem. Moreover, the existence of a suboptimal solution in [7] requires some strong conditions to be satisfied. In this paper, a different approach is proposed in order to remove the above drawbacks. Control laws for systems with uncertain parameters and uncertain time delays affecting the state and the control input are designed by extending the notions of quadratic stability and guaranteed-cost control. The uncer- tain parameters affecting the state, input, and delay matrices are allowed to vary into prespecified ranges. They enter into the system description in terms of the so-called uncertainty matrices that have a given structure. Different unity-rank de- compositions of the uncertainty matrices are possible by means of appropriate scaling. This description is convenient for many physical-system representations [3]. An LMI optimization so- lution [4] is then sought in order to determine the appropri- ate uncertainty decomposition. Since a minimal value of the guaranteed cost is desired, one has to solve an LMI objective minimization problem. Due to the presence of uncertain time delays to the input, the resulting objective function to be min- imized is nonconvex. In the proposed approach, this function is approximated by a convex one. The closed-loop system’s quadratic stability follows as a direct consequence. The paper is organized as follows: The problem formulation and basic notions are given in Section II. Computation of a solution in the LMI framework is presented in Section III. Section IV presents a numerical example. Finally, conclusions are given in Section V. II. PROBLEM STATEMENT AND DEFINITIONS Consider the uncertain time-delay system described in state-space form ˙ x(t)=[A 1 +ΔA 1 (t)] x(t)+[A 2 +ΔA 2 (t)] x (t - d 1 (t)) +[B 1 +ΔB 1 (t)] u(t)+[B 2 +ΔB 2 (t)] u (t - d 2 (t)) (1) for t ∈ [0, ∞) and with x(t)= φ(t) for t< 0. 1083-4427/$20.00 © 2006 IEEE