Bounds on set exit times of affine systems, using Linear Matrix Inequalities

TL;DR

This paper develops semi-definite programming methods to compute tight upper bounds on exit times of stable affine systems, improving efficiency in trajectory planning for hybrid systems.

Contribution

It introduces a novel SDP-based approach to bound exit times of affine systems within generalized ellipsoids, enhancing previous methods.

Findings

Bounds are tighter than previous results.

Method requires modest additional computation.

Numerical experiments validate effectiveness.

Abstract

Efficient computation of trajectories of switched affine systems becomes possible, if for any such hybrid system, we can manage to efficiently compute the sequence of switching times. Once the switching times have been computed, we can easily compute the trajectories between two successive switches as the solution of an affine ODE. Each switching time can be seen as a positive real root of an analytic function, thereby allowing for efficient computation by using root finding algorithms. These algorithms require a finite interval, within which to search for the switching time. In this paper, we study the problem of computing upper bounds on such switching times, and we restrict our attention to stable time-invariant affine systems. We provide semi-definite programming models to compute upper bounds on the time taken by the trajectories of an affine ODE to exit a set described as the intersection of a few generalized ellipsoids. Through numerical experiments, we show that the resulting bounds are tighter than bounds reported before, while requiring only a modest increase in computation time.