Guisheng Zhai

Stability analysis of switched systems with stable and unstable subsystems: An average dwell time approach

We study the stability properties of switched systems consisting of both Hurwitz stable and unstable linear time-invariant subsystems using an average dwell time approach. We propose a class of switching laws so that the entire switched system is exponentially stable with a desired stability margin. In the switching laws, the average dwell time is required to be sufficiently large, and the total activation time ratio between Hurwitz stable subsystems and unstable subsystems is required to be no less than a specified constant. We also apply the result to perturbed switched systems where nonlinear vanishing or non-vanishing norm-bounded perturbations exist in the subsystems, and we show quantitatively that, when norms of the perturbations are small, the solutions of the switched systems converge to the origin exponentially under the same switching laws.

Disturbance Attenuation Properties of Time-Controlled Switched Systems

In this paper, we investigate the disturbance attenuation properties of time-controlled switched systems consisting of several linear time-invariant subsystems by using an average dwell time approach incorporated with a piecewise Lyapunov function. First, we show that when all subsystems are Hurwitz stable and achieve a disturbance attenuation level smaller than a positive scalar γ0, the switched system under an average dwell time scheme achieves a weighted disturbance attenuation level γ0, and the weighted disturbance attenuation approaches normal disturbance attenuation if the average dwell time is chosen sufficiently large. We extend this result to the case where not all subsystems are Hurwitz stable, by showing that in addition to the average dwell time scheme, if the total activation time of unstable subsystems is relatively small compared with that of the Hurwitz stable subsystems, then a reasonable weighted disturbance attenuation level is guaranteed. Finally, a discussion is made on the case for which nonlinear norm-bounded perturbations exist in the subsystems.

Quadratic stabilizability of switched linear systems with polytopic uncertainties

In this paper, we consider quadratic stabilizability via state feedback for both continuous-time and discrete-time switched linear systems that are composed of polytopic uncertain subsystems. By state feedback, we mean that the switchings among subsystems are dependent on system states. For continuous-time switched linear systems, we show that if there exists a common positive definite matrix for stability of all convex combinations of the extreme points which belong to different subsystem matrices, then the switched system is quadratically stabilizable via state feedback. For discrete-time switched linear systems, we derive a quadratic stabilizability condition expressed as matrix inequalities with respect to a family of non-negative scalars and a common positive definite matrix. For both continuous-time and discrete-time switched systems, we propose the switching rules by using the obtained common positive definite matrix.

Qualitative analysis of discrete-time switched systems

We investigate some qualitative properties for time-controlled switched systems consisting of several linear discrete-time subsystems. First, we study exponential stability of the switched system with commutation property, stable combination and average dwell time. When all subsystem matrices are commutative pairwise and there exists a stable combination of unstable subsystem matrices, we propose a class of stabilizing switching laws where Schur stable subsystems are activated arbitrarily while unstable ones are activated in sequence with their duration time periods satisfying a specified ratio. For more general switched system whose subsystem matrices are not commutative pairwise, we show that the switched system is exponentially stable if the average dwell time is chosen sufficiently large and the total, activation time ratio between Schur stable and unstable subsystems is not smaller than a specified constant. Secondly, we use an average dwell time approach incorporated with a piecewise Lyapunov function to study the /spl Lscr//sub 2/ gain of the switched system.

Stability analysis of switched systems with stable and unstable subsystems: an average dwell time approach

We study the stability properties of linear switched systems consisting of both Hurwitz stable and unstable subsystems using an average dwell time approach. We show that if the average dwell time is chosen sufficiently large and the total activation time of unstable subsystems is relatively small compared with that of Hurwitz stable subsystems, then exponential stability of a desired degree is guaranteed. We also derive a piecewise Lyapunov function for the switched system subjected to the switching law and the average dwell time scheme under consideration, and we extend these results to the case for which nonlinear norm-bounded perturbations exist in the subsystems. We show that when the norms of the perturbations are small, we can modify the switching law appropriately to guarantee that the solutions of the switched system converge to the origin exponentially with large average dwell time.

Practical stability and stabilization of hybrid and switched systems

In this note, practical stability and stabilization problems for hybrid and switched systems are studied. The main results of this note include a direct method for the /spl epsi/-practical stability analysis of hybrid systems and sufficient conditions for the practical stabilizability of switched systems. We construct an /spl epsi/-practically stabilizing switching law in the proof of the practical stabilizability result and apply it to a tracking problem to show its effectiveness.

Quadratic stabilizability of discrete-time switched systems via state and output feedback

We study quadratic stabilizability via state and output feedback for switched systems composed of several discrete-time linear time-invariant subsystems, under the assumption that all subsystem matrices are unstable. We derive a sufficient condition expressed as a matrix inequality under which the switched system is quadratically stabilizable via a state-based switching strategy, and we show that the sufficient condition is also necessary if the number of subsystems is two. When a robust detectability condition is satisfied in addition to the sufficient condition, we construct a quadratically stabilizing switching strategy based on the measurement output.

Stability and /spl Lscr//sub 2/ gain analysis for switched symmetric systems with time delay

In this paper, we study the stability and /spl Lscr//sub 2/ gain properties for a class of switched systems which are composed of a finite number of linear time-invariant symmetric systems with time delay. We show that when all subsystems are asymptotically stable in the sense of satisfying an LMI, the switched system is asymptotically stable under arbitrary switching. Furthermore, we show that when all subsystems are asymptotically stable and have the /spl Lscr//sub 2/ gains /spl gamma/ in the sense of satisfying an LMI, the switched system is asymptotically stable and has the same /spl Lscr//sub 2/ gain /spl gamma/ under arbitrary switching. The key idea for both stability and /spl Lscr//sub 2/ gain analysis in this paper is to establish a common Lyapunov function for all subsystems in the switched system.

Stability analysis of switched linear singular systems

This paper addresses the stability analysis problem for switched linear continuous-time singular systems. First, based on the equivalent dynamics decomposition form, a refined description for state jumps of the switched singular system is presented, which indicates that overall state jumps are resulted by two sequential state jumps. Second, sufficient conditions for exponential stability of the switched singular system with stable subsystems are presented. It is shown that the stability property of the system is completely determined by the switched reduced-order dynamic subsystem and the switching law induced state jumps. Then, a sufficient stability condition for the switched singular system with both stable and unstable subsystems is obtained. Finally, numerical examples are presented to illustrate the effectiveness of the proposed approach.

Hybrid output feedback stabilization of two-dimensional linear control systems

We solve for two-dimensional SISO linear control systems the open problem whether there exists finite-state hybrid output feedback to stabilize the systems. We show that under reasonable assumptions, such as stabilizability and detectability conditions, the answer to this question is affirmative for the case of 2-state output feedback. We also address a simulation procedure to locate the stabilizable regions for switched systems consisting of two subsystems. We demonstrate the applicability of our results by considering several specific examples.