Invariance Control Synthesis for Switched Systems: An Interval Analysis Approach

TL;DR

This paper introduces an interval analysis-based method for invariance control in discrete-time switched nonlinear systems, enabling precise computation of controlled invariant sets without requiring common equilibrium points.

Contribution

It develops a novel interval analysis approach for computing controlled invariant sets and invariance controllers, avoiding Lyapunov-based assumptions and providing convergence guarantees.

Findings

Computes outer and inner approximations of invariant sets.

Provides algorithms for finite-step invariant set computation.

Demonstrates effectiveness through three application examples.

Abstract

This paper focuses on the invariance control problem for discrete-time switched nonlinear systems. The proposed approach computes controlled invariant sets in a finite number of iterations and directly yields a partition-based invariance controller using the information recorded during the computation. In contrast with Lyapunov-based control methods, this method does not require the subsystems to have common equilibrium points. Algorithms are developed for computing both outer and inner approximations of the maximal controlled invariant sets, which are represented as finite unions of intervals. The general convergence results of interval methods allow us to obtain arbitrarily precise approximations without any stability assumptions. In addition, invariant inner approximations can be computed provided that the switched system satisfies a robustly controlled invariance condition. Under the same condition, we also prove the existence of an invariance controller based on partitions of the state space. Our method is illustrated with three examples drawn from different applications and compared with existing work in the literature.