Bounds and Invariant Sets for a Class of Switching Systems with Delayed-state-dependent Perturbations

TL;DR

This paper introduces a componentwise method for computing bounds and invariant regions in switching linear systems with delayed-state-dependent perturbations, improving accuracy and reducing conservativeness over norm-based approaches.

Contribution

It extends previous results to include switching systems with delayed-state-dependent perturbations, clarifies the relationship with common quadratic Lyapunov functions, and provides a technique for their computation.

Findings

Componentwise bounds can be computed without using norms.

The method applies to systems with affine delayed-state-dependent perturbations.

Conditions for local, semi-global, and global bounds are established.

Abstract

We present a novel method to compute componentwise transient bounds, ultimate bounds, and invariant regions for a class of switching continuous-time linear systems with perturbation bounds that may depend nonlinearly on a delayed state. The main advantage of the method is its componentwise nature, i.e. the fact that it allows each component of the perturbation vector to have an independent bound and that the bounds and sets obtained are also given componentwise. This componentwise method does not employ a norm for bounding either the perturbation or state vectors, avoids the need for scaling the different state vector components in order to obtain useful results, and may also reduce conservativeness in some cases. We give conditions for the derived bounds to be of local or semi-global nature. In addition, we deal with the case of perturbation bounds whose dependence on a delayed state is of affine form as a particular case of nonlinear dependence for which the bounds derived are shown to be globally valid. A sufficient condition for practical stability is also provided. The present paper builds upon and extends to switching systems with delayed-state-dependent perturbations previous results by the authors. In this sense, the contribution is three-fold: the derivation of the aforementioned extension; the elucidation of the precise relationship between the class of switching linear systems to which the proposed method can be applied and those that admit a common quadratic Lyapunov function (a question that was left open in our previous work); and the derivation of a technique to compute a common quadratic Lyapunov function for switching linear systems with perturbations bounded componentwise by affine functions of the absolute value of the state vector components.