José Luis Mancilla-Aguilar

A condition for the stability of switched nonlinear systems

In this paper, we present a sufficient condition for the global asymptotic stability of a switched nonlinear system composed of a finite family of subsystems. We show that the global asymptotic stability of each subsystem and the pairwise commutation of the vector fields that define the subsystems (i.e., the Lie bracket of any pair of them is zero) are sufficient for the global asymptotic stability of the switched system. We also show that these conditions are sufficient for the existence of a common Lyapunov function.

Some results on the stabilization of switched systems

This paper deals with the stabilization of switched systems with respect to (w.r.t.) compact sets. We show that the switched system is stabilizable w.r.t. a compact set by means of a family of switched signals if and only if a certain control affine system whose admissible controls take values in a polytope is asymptotically controllable to that set. In addition we present a control algorithm that based on a family of open-loop controls which stabilizes the aforementioned control system, a model of the system and the states of the switched system, generates switching signals which stabilize the switched system in a practical sense. We also give results about the convergence and the robustness of the algorithm.

Input / output stability of systems with switched dynamics and outputs☆

This paper provides the extension of results on input-to-output stability properties of switched systems to switched systems whose dynamics are described by forced differential inclusions and whose outputs are obtained via switched set-valued maps. Lyapunov characterizations of these input/output stability properties, obtained in terms of certain conceptual output functions, are also presented.

Global Stability Results for Switched Systems Based on Weak Lyapunov Functions

In this paper we study the stability of nonlinear and time-varying switched systems under restricted switching. We approach the problem by decomposing the system dynamics into a nominal-like part and a perturbation-like one. Most stability results for perturbed systems are based on the use of strong Lyapunov functions, i.e. functions of time and state whose total time derivative along the nominal system trajectories is bounded by a negative definite function of the state. However, switched systems under restricted switching may not admit strong Lyapunov functions, even when asymptotic stability is uniform over the set of switching signals considered. The main contribution of the current paper consists in providing stability results that are based on the stability of the nominal-like part of the system and require only a weak Lyapunov function. These results may have wider applicability than results based on strong Lyapunov functions. The results provided follow two lines. First, we give very general global uniform asymptotic stability results under reasonable boundedness conditions on the functions that define the dynamics of the nominal-like and the perturbation-like parts of the system. Second, we provide input-to-state stability (ISS) results for the case when the nominal-like part is switched linear-time-varying. We provide two types of ISS results: standard ISS that involves the essential supremum norm of the input and a modified ISS that involves a power-type norm.

Invariance results for constrained switched systems

In this paper we address invariance principles for nonlinear switched systems with otherwise arbitrary compact index set and with constrained switchings. We present an extension of LaSalles invariance principle for these systems and derive by using detectability notions some convergence and asymptotic stability criteria. These results enable to take into account in the analysis of stability not only state-dependent constraints but also to treat the case in which the switching logic has memory, i.e., the active subsystem only can switch to a prescribed subset of subsystems.

A Characterization of Integral ISS for Switched and Time-Varying Systems

Most of the existing characterizations of the integral input-to-state stability (iISS) property are not valid for time-varying or switched systems in cases where converse Lyapunov theorems for stability are not available. This paper provides a characterization that is valid for switched and time-varying systems, and shows that natural extensions of some of the existing characterizations result in only sufficient but not necessary conditions. The results provided also pinpoint suitable iISS gains and relate these to supply functions and bounds on the function defining the system dynamics.

Invariance principles for switched systems with restrictions

In this paper we consider switched nonlinear systems with otherwise arbitrary compact index sets and subjected to different switching constraints. We present invariance principles for these systems and derive, by using observability-like notions, some convergence criteria which enable us to analyze the convergence of solutions of switched systems with restrictions originating from the timing of the switchings, state-dependent constrained switching and switching whose logic has memory, i.e., the active subsystem can only switch to a prescribed subset of subsystems.

Diseño de Observadores en Modos Cuasi-Deslizantes vía LMIs

En este trabajo se presenta un observador robusto por modos cuasi-deslizantes para plantas con modelo nominal lineal e incertidumbres/perturbaciones de cierta clase particular. Las senales utilizadas para la estimacion del vector de estados se suponen contaminadas con ruido del que solo se conocen cotas. El diseno del observador se plantea como un problema de factibilidad LMI (Linear Matrix Inequalities) y se encuentran cotas para el error de estimacion que pueden ser calculadas a priori. Posteriormente el diseno del observador se reformula como un problema de optimizacion GEVP (Generalized Eigen Value Problem) con el objeto de minimizar las cotas del error de estimacion. El trabajo incluye un ejemplo numerico y simulaciones de un brazo robotico con un eje manejado por un motor de corriente continua.

On zero-input stability inheritance for time-varying systems with decaying-to-zero input power

Abstract Stability results for time-varying systems with inputs are relatively scarce, as opposed to the abundant literature available for time-invariant systems. This paper extends to time-varying systems existing results that ensure that if the input converges to zero in some specific sense, then the state trajectory will inherit stability properties from the corresponding zero-input system. This extension is non-trivial, in the sense that the proof technique is completely novel, and allows to recover the existing results under weaker assumptions in a unifying way.

Robustness properties of an algorithm for the stabilisation of switched systems with unbounded perturbations

ABSTRACT In this paper, it is shown that an algorithm for the stabilisation of switched systems introduced by the authors is robust with respect to perturbations which are unbounded in the supremum norm, but bounded in a power-like sense. The obtained stability results comprise, among others, both the exponential input-to-state stability and the exponential integral input-to-state stability properties of the closed-loop system and give a better description of the behaviour of the closed-loop system.