Stabilizing Switching Signals Research Articles

On the Design of Stabilizing Cycles for Switched Linear Systems

Given a family of systems, identifying stabilizing switching signals in terms of infinite walks constructed by concatenating cycles on the underlying directed graph of a switched system that satisfy certain conditions, is a well-known technique in the literature. This paper deals with a new {method to design} these cycles for stability of switched linear systems. We employ properties of the subsystem matrices and mild assumption on the admissible switches between the subsystems {for this purpose}. In contrast to prior works, {our construction of} stabilizing cycles does not involve design of Lyapunov-like functions and storage of sets of scalars in memory prior to the application of a cycle detection algorithm. As a result, {the} techniques {proposed in this paper} offer improved numerical tractability.

Robust matrix commutator conditions for stability of switched linear systems under restricted switching

This article treats global uniform exponential stability (GUES) of discrete-time switched linear systems under restricted switching. Given admissible minimum and maximum dwell times, we provide sufficient conditions on the subsystems under which they admit a set of switching signals that obeys the given restrictions on dwell times and preserves stability of the resulting switched system. Our analysis relies on combinatorial arguments applied to matrix commutators. The proposed set of stabilizing switching signals is characterized in terms of duration of activation of Schur stable subsystems and non-consecutive activation of distinct unstable subsystems.

Stabilization of discrete-time switched singular systems with state, output and switching delays

This paper is concerned with state feedback stabilization of discrete-time switched singular systems with time-varying delays existing simultaneously in the state, the output and the switching signal of the switched controller. On the basis of equivalent dynamics decomposition and Lyapunov–Krasovskii method, exponential estimates for the response of slow states of the closed-loop subsystems running in asynchronous and synchronous periods are first given. Exponential estimates for the response of fast states are also provided by establishing an analytic equation to solve the fast states and using some algebraic techniques. Then, by employing the obtained exponential estimates and the piecewise Lyapunov function approach with average dwell time (ADT) switching, sufficient conditions for the existence of a class of stabilizing switching signals and state feedback gains are derived, which explicitly depend on upper bounds on the delays and a lower bound on the ADT. Finally, two numerical examples are provided to illustrate the effectiveness of the obtained theoretical results.

Randomized algorithms for stabilizing switching signals

Qualitative behaviour of switched systems has attracted considerable research attention in the recent past. In this article we study 'how likely' is it for a family of systems to admit stabilizing switching signals. A weighted digraph is associated to a switched system in a natural fashion, and the switching signal is expressed as an infinite walk on this digraph. We provide a linear time probabilistic algorithm to find cycles on this digraph that have a desirable property (we call it 'contractivity'), and under mild statistical hypotheses on the connectivity and weights of the digraph, demonstrate that there exist uncountably many stabilizing switching signals derived from such cycles. Our algorithm does not require the vertex and edge weights to be stored in memory prior to its application, has a learning/exploratory character, and naturally fits very large scale systems.

Stabilizing switching signals: A transition from point-wise to asymptotic conditions

Characterization of classes of switching signals that ensure stability of switched systems occupies a significant portion of the switched systems literature. This article collects a multitude of stabilizing switching signals under an umbrella framework. We achieve this in two steps: Firstly, given a family of systems, possibly containing unstable dynamics, we propose a new and general class of stabilizing switching signals. Secondly, we demonstrate that prior results based on both point-wise and asymptotic characterizations follow our result. This is the first attempt in the switched systems literature where these switching signals are unified under one banner.

A graph theoretic approach to input-to-state stability of switched systems

This article deals with input-to-state stability (ISS) of discrete-time switched systems. Given a family of nonlinear systems with exogenous inputs, we present a class of switching signals under which the resulting switched system is ISS. We allow non-ISS systems in the family and our analysis involves graph-theoretic arguments. A weighted digraph is associated to the switched system, and a switching signal is expressed as an infinite walk on this digraph, both in a natural way. Our class of stabilizing switching signals (infinite walks) is periodic in nature and affords simple algorithmic construction.

Generalized switching signals for input-to-state stability of switched systems

This article deals with input-to-state stability (ISS) of continuous-time switched nonlinear systems. Given a family of systems with exogenous inputs such that not all systems in the family are ISS, we characterize a new and general class of switching signals under which the resulting switched system is ISS. Our stabilizing switching signals allow the number of switches to grow faster than an affine function of the length of a time interval, unlike in the case of average dwell time switching. We also recast a subclass of average dwell time switching signals in our setting and establish analogs of two representative prior results.

Stabilizing Switching Signals for Switched Systems

This article deals with stability of continuous-time switched systems under constrained switching. Given a family of systems, possibly containing unstable dynamics, we characterize a new class of switching signals under which the switched system generated by it and the family of systems is globally asymptotically stable. Our characterization of such stabilizing switching signals involves the asymptotic frequency of switching, the asymptotic fraction of activation of the constituent systems, and the asymptotic densities of admissible transitions among them. Our techniques employ multiple Lyapunov-like functions, and extend preceding results both in scope and applicability.

Constrained switching stabilization of linear uncertain switched systems using piecewise linear Lyapunov functions

The problem of switching stabilizability of linear uncertain switched systems with unstable modes has recently attracted increased interest. For a given collection of unstable linear uncertain subsystems, the problem is to determine asymptotically stabilizing switching signals or to ascertain the absence of such laws. In this paper, we deal with the switching stabilizability problem for a class of autonomous continuous-time switched linear systems with time-varying parametric uncertainties in the presence of additional state constraints. With a special emphasis on two-dimensional systems we extend previous relevant results in the literature, and we propose a computational technique that generates piecewise linear Lyapunov functions of a convex or non-convex nature. The algorithm is graph-based and consists of determining controlled invariant polytopes induced by systematic conic decompositions of the state-space and corresponding state-dependent switching control actions. Further extension of this computational tool to higher dimensional systems is also discussed.

On the [formula omitted]-stabilization of switched nonlinear systems via state-dependent switching rule

This paper considers switching stabilization of some general nonlinear systems. Assuming certain properties of a convex linear combination of the nonlinear vector fields, two ways of generating stabilizing switching signals are proposed, i.e., the minimal rule and the generalized rule, both based on a partition of the time-state space. The main theorems show that the resulting switched system is globally uniformly asymptotically stable and globally uniformly exponentially stable, respectively. It is shown that the stabilizing switching signals do not exhibit chattering, i.e., two consecutive switching times are separated by a positive amount of time. In addition, under the generalized rule, the switching signal does not exhibit Zeno behavior (accumulation of switching times in a finite time). Stability analysis is performed in terms of two measures so that the results can unify many different stability criteria, such as Lyapunov stability, partial stability, orbital stability, and stability of an invariant set. Applications of the main results are shown by several examples, and numerical simulations are performed to both illustrate and verify the stability analysis.