The stabilization of linear continuous-time systems with time delay in the state and subject to saturating controls is addressed. Sufficient conditions obtained via a linear matrix inequality (LMI) formulation are stated to guarantee both the local stabilization and the satisfaction of some performance requirements. The method of synthesis consists in determining simultaneously a state feedback control law and an associated domain of safe admissible states for which the stability of the closed-loop system is guaranteed when control saturations effectively occur. Index Terms-Control saturation, linear matrix inequalities (LMI's), local stability, time-delay systems.placement: the concept of -stability is used (see [14], [15], and references therein). The method used is based on the Lyapunov-Krasovskii

V. CONCLUSIONThis note has developed a sliding-mode controller which requires only output information for a class of uncertain linear systems. The controller comprises both linear and nonlinear components and is static in nature, i.e., no compensation/observation is included. The novelty of the approach is in the rationale and method used to synthesize the linear control component. The reachability condition is not required to be satisfied globally. Instead, sliding is only expected to take place within a subset of the state-space containing the origin referred to as the sliding patch. This region is shown to be rendered invariant by the control law. The linear static output feedback control component is synthesized using an LMI optimization. The resulting LMI formulation can be solved easily by standard commercially available software. The efficacy of the approach has been demonstrated on a numerical example taken from the sliding-modeliterature. Control, vol. 3, pp. 115-132, 1993. [11] C. M. Kwan, "On variable structure output feedback controllers," IEEE Trans. Automat. Contr., vol. 41, pp. 1691-1693, Nov. 1996 [12] J. J. E. Slotine, J. K. Hedrick, and E. A. Misawa, "On sliding observers for nonlinear systems," ASME: J. Dyn. Syst., Meas. Control, vol. 109, pp. 245-252, Sept. 1987. [13] S. K. Spurgeon and R. Davies, "A nonlinear control strategy for robust sliding-mode performance in the presence of unmatched uncertainty," Int. J. Control, vol. 57, no. 5, pp. 1107Control, vol. 57, no. 5, pp. -1123Control, vol. 57, no. 5, pp. , 1993 Abstract-This note deals with the problem of local stabilization of linear discrete-time systems subject to control saturation. A linear matrix inequalitie-based framework is proposed in order to compute a saturating state feedback that stabilizes the system with respect to a given set of admissible initial states and, in addition, guarantees some dynamical performances when the system operates in the zone of linear behavior (i.e., when the controls are not saturated). REFERENCES

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This note addresses the problem of local stabilization of linear systems subject to control amplitude and rate saturation. Considering the actuator represented by a first-order system subject to input and state saturation, a condition for the stabilization of an a priori given set of admissible initial states is formulated from certain saturation nonlinearities representation and quadratic stability results. From this condition, an algorithm based on the iterative solution of linear matrix inequalities-based problems is proposed in order to compute the control law.Abstract-In this note, we present an algorithm to compute all solutions of the scalar algebraic Riccati equations that play an important role in finding feedback Nash equilibria of the scalar -player linear-quadratic differential game. We show that all appropriate solutions can be obtained by analyzing the eigenstructure of a related matrix.Index Terms-Eigenvalue problem, feedback Nash equilibria, nonzero-sum linear quadratic differential games, Riccati equations.

The purpose of this paper is to study the use of anti-windup gains for obtaining larger regions of stability for linear systems with saturating inputs. Considering that a linear dynamic output feedback has been designed to stabilize the linear system (without saturation), a method is proposed for designing an anti-windup gain that maximizes the estimation of the basin of attraction of the closed-loop system. It is shown that the closed-loop system obtained from the controller plus the anti-windup gain can be locally modeled by a linear system with a deadzone nonlinearity. From this model, stability conditionsbased on quadratic and Lure type Lyapunov functions are stated. Algorithms based on LMI schemes are proposed for computing both the anti-windup gain and an associated region of stability. h~u c t i o nStudies on the analysis and control design problems for linear systems with saturating actuators have followed two main approaches in the literature. Either a design is carried out directly taking into account the effect of the saturation or the effect of saturation is dealt with in a second step after a previous design performed disregarding the Saturation tenns.The anti-windup fits this second approach as it consists in introducing control modifications in order to recover, as much as possible, the performance induced by a previous design carried out on the basis of the unsaturated system. In particular, anti-windup schemes have been successfully applied in order to avoid or minimize the windup of the integral action in PID controllers, largely applied in the industry. In this case, most of the related literature focuses on the performance improvement in the sense of avoiding large and oscillatory transient responses (see, among others, [2], VI. W1).More recently, a special attention has been paid to the influence of the anti-windup schemes in the stability and the performances of the closed-loop system (see, for example, 131, [SI, [91, 1121, 1131, [141, [17]). Several results on the anti-windup problem are concemed with achieving global stability properties. Since global results cannot be achieved for open-loop unstable linear system in the presence of actuator saturation, local results have to be develaped. In this context, a key issue is the determination of domains of stability for the closed-loop system. With very few exceptions, most of the local results available in the literature of antiwindup do not provide explicit characterization of the domain of stability. It is worth to notice that the basin of attraction is modified by the anti-windup loop. In particular, if the resulting basin of attraction is not sufficiently large, the system can present a divergent behavior depending on its initialization and the action of disturbances.In this paper we focus on the structure of the obmer-based anti-windup [2,1]. F o d stability conditions for this antiwindup structure have been given in [9] on the basis of passivity arguments. A design algorithm is provided which, under certain conditions, achieves global as...