An extension of LaSalle's invariance principle for switched systems

In this paper we address invariance principles for a certain class of switched nonlinear systems. We provide an extension of LaSalle's invariance principle for these systems and state asymptotic stability criteria. We also present some related results on the compactness of the trajectories of these switched systems.     

Stabilization of continuous-time switched nonlinear systems

The paper considers three problems for continuous-time nonlinear switched systems. The first result of this paper is a open-loop stabilization strategy based on dwell time computation. The second considers a state switching strategy for global stabilization. The strategy is of closed loop nature (trajectory dependent) and is designed from the solution of what we call nonlinear Lyapunov–Metzler inequalities from which the stability condition is expressed. Finally, results on the stabilization of nonlinear time varying polytopic systems are provided.     

Model matching with strong stability in switched linear systems

This work deals with model matching by output dynamic feedback in switched linear systems. The plant and the model are assumed to be subject to different switching signals. A necessary and sufficient condition for achieving model matching with asymptotic stability of both the closed-loop compensated system and the compensator, for a sufficiently large dwell time, is proven. Such condition is obtained by specializing the structural condition with a suitable redefinition of the robust controlled invariant subspace involved, capable of capturing not only the structural aspect of the problem but also the stability aspects. The effect of the combined action of nonzero initial states and nonzero inputs is dealt with. The solution to a more demanding problem formulation, where the dwell time is assumed to be fixed and given is also discussed.     

On the asymptotic stability of switched homogeneous systems

The stability of switched systems generated by the family of autonomous subsystems with homogeneous right-hand sides is investigated. It is assumed that for each subsystem the proper homogeneous Lyapunov function is constructed. The sufficient conditions of the existence of the common Lyapunov function providing global asymptotic stability of the zero solution for any admissible switching law are obtained. In the case where we can not guarantee the existence of a common Lyapunov function, the classes of switching signals are determined under which the zero solution is locally or globally asymptotically stable. It is proved that, for any given neighborhood of the origin, one can choose a number L>0 (dwell time) such that if intervals between consecutive switching times are not smaller than L then any solution of the considered system enters this neighborhood in finite time and remains within it thereafter.     

一类含扰动的非线性切换系统稳定性分析

首先在非零扰动情况下,利用平均滞留时间的方法给出保证切换系统一致有界或一致终极有界的条件以及最终边界.当切换系统中含有不稳定子系统时,证明了当切换规则满足一定条件时,仍可保证切换系统是有界的.然后在零扰动情况下,给出了扰动非线性切换系统稳定的充分条件.最后通过仿真算例说明了所得结果的有效性.     

Stabilization of switched continuous-time systems with all modes unstable via dwell time switching

Stabilization of switched systems composed fully of unstable subsystems is one of the most challenging problems in the field of switched systems. In this brief paper, a sufficient condition ensuring the asymptotic stability of switched continuous-time systems with all modes unstable is proposed. The main idea is to exploit the stabilization property of switching behaviors to compensate the state divergence made by unstable modes. Then, by using a discretized Lyapunov function approach, a computable sufficient condition for switched linear systems is proposed in the framework of dwell time; it is shown that the time intervals between two successive switching instants are required to be confined by a pair of upper and lower bounds to guarantee the asymptotic stability. Based on derived results, an algorithm is proposed to compute the stability region of admissible dwell time. A numerical example is proposed to illustrate our approach.     

A converse Lyapunov theorem for nonlinear switched systems

In this paper we present a converse Lyapunov theorem for uniform asymptotic stability of switched nonlinear systems. Its proof is a simple consequence of some results on converse Lyapunov theorems for systems with bounded disturbances obtained by Lin et al. (SIAM J. Control Optim. 34 (1996) 124–160), once an association of the switched system with a nonlinear system with disturbances is established.     

A notion of passivity for switched systems with state-dependent switching

A passivity concept for switched systems with state-dependent switching is presented. Each subsystem has a storage function to describe the “energy” stored in the subsystem. The passivity property of a switched system is given in terms of multiple storage functions. Each storage function is allowed to grow on the “switched on” time sequence but the total growth is bounded by a certain function. Stability is inferred from passivity and asymptotic stability is achieved under further assumptions of a detectivity property of a local form and boundedness of the total change of some storage function on its inactive intervals. A state-dependent switching law that renders the system passive is also designed.     

On the algebraic characterization of invariant sets of switched linear systems

In this paper, a suitable LaSalle principle for continuous-time linear switched systems is used to characterize invariant sets and their associated switching laws. An algorithm to determine algebraically these invariants is proposed. The main novelty of our approach is that we require no dwell time conditions on the switching laws. By not focusing on restricted control classes we are able to describe the asymptotic properties of the considered switched systems. Observability analysis of a flying capacitor converter is proposed as an illustration.     

Passivity and output feedback passification of switched continuous-time systems with a dwell time constraint

This paper is concerned with the passivity analysis and output feedback passification for a class of switched continuous-time systems. A switching law and control units are designed to guarantee the passivity of closed-loop switched systems. Different from the exist results in state-dependent switching framework, we construct a novel switching law such that the switched system obeys a minimal dwell time between any two consecutive switchings. This avoids the possible arbitrarily fast switching caused by state-dependent switching law. Moreover, the switching law uses only the lower bound of the dwell time and partial measurable states of the closed-loop system, which is applicable in output feedback framework. Using multiple discretized storage functions, the controllers working on different time intervals are obtained, which ensure the closed-loop system is passive under the switching law. Finally, two examples are provided to illustrate the effectiveness of the developed results.     

Necessary and sufficient condition for stabilizability of discrete-time linear switched systems: A set-theory approach

In this paper, the stabilizability of discrete-time linear switched systems is considered. Several sufficient conditions for stabilizability are proposed in the literature, but no necessary and sufficient. The main contributions are the necessary and sufficient conditions for stabilizability based on the set-theory and the characterization of a universal class of Lyapunov functions. An algorithm for computing the Lyapunov functions and a procedure to design the stabilizing switching control law are provided, based on such conditions. Moreover, a sufficient condition for non-stabilizability for switched system is presented. Several academic examples are given to illustrate the efficiency of the proposed results. In particular, a Lyapunov function is obtained for a system for which the Lyapunov–Metzler condition for stabilizability does not hold.     

A control design method for a class of switched linear systems

In this note we consider the stability properties of a system class that arises in the control design problem of switched linear systems. The control design we are studying is based on a classical pole-placement approach. We analyse the stability of the resulting switched system and develop analytic conditions which reduce the complexity of the stability problem. We further consider two special cases for which strongly simplified conditions are obtained that support the analytic controller design.     

Design of switched controllers for discrete singular bilinear systems

In this paper, switched controllers are designed for a class of nonlinear discrete singular systems and a class of discrete singular bilinear systems. An invariant principle is presented for such switched nonlinear singular systems.The invariant principle and the switched controllers are used to globally stabilize a class of singular bilinear systems that are not open-loop stable.     

Design of switched controllers for discrete singular bilinear systems

In this paper, switched controllers are designed for a class of nonlinear discrete singular systems and a class of discrete singular bilinear systems. An invariant principle is presented for such switched nonlinear singular systems.The invariant principle and the switched controllers are used to globally stabilize a class of singular bilinear systems that are not open-loop stable.     

Dwell time and average dwell time methods based on the cycle ratio of the switching graph

A new method to find an upper bound on dwell time and average dwell time for switched linear systems is proposed. The method is based on computing the maximum cycle ratio and the maximum cycle mean of the directed graph that governs switchings. For planar switched systems, an upper bound for dwell time and average dwell time can be estimated by considering only the cycles of length two.     

Global stabilisation for a class of switched high-order stochastic nonlinear systems

This paper is devoted to the problem of global stabili sation for a class of switched high-order stochastic nonlinear systems under arbitrary switchings. By extending the adding -a -power -integrator technique and choosing an appropriate common Lyapunov function, a common state-feedback controller is explicitly constructed to render the closed-loop system globally asymptotically stable in probability. The effectiveness of the proposed approach is demonstrated by a simulation example.     

Convergence rate of nonlinear switched systems

This paper is concerned with the convergence rate of the solutions of nonlinear switched systems.We first consider a switched system which is asymptotically stable for a class of switching signals but not for all switching signals. We show that solutions corresponding to that class of switching signals converge arbitrarily slowly to the origin.Then we consider analytic switched systems for which a common weak quadratic Lyapunov function exists. Under two different sets of assumptions we provide explicit exponential convergence rates for switching signals with a fixed dwell-time.     

Absolute exponential stability and stabilization of switched nonlinear systems

This paper is concerned with the problems of absolute exponential stability and stabilization for a class of switched nonlinear systems whose system matrices are Metzler. Nonlinearity of the systems is constrained in a sector field, which is bounded by two odd symmetric piecewise linear functions. Multiple Lyapunov functions are introduced to deal with the stability of such nonlinear systems. Compared with some existing results obtained by the common Lyapunov function approach in the literature, the conservatism of our results is reduced. All present conditions can be solved by linear programming. Furthermore, the absolute exponential stabilization for the considered systems is designed by the state-feedback and average dwell time switching strategy. Two examples are also given to illustrate the validity of the theoretical findings.