A Path-Complete Approach for Optimal Control of Switched Systems

TL;DR

This paper introduces a path-complete graph framework to compute tractable upper bounds on the value function for discrete-time switched systems with exogenous switching, enabling better control synthesis.

Contribution

It develops a novel path-complete graph approach to approximate the value function, providing convex formulations and approximation guarantees for switched systems.

Findings

Provides LMI-based formulations for switched linear systems

Establishes approximation guarantees and asymptotic non-conservativeness

Demonstrates effectiveness through numerical examples

Abstract

We study the problem of estimating the value function of discrete-time switched systems under arbitrary switching. Unlike the switched LQR problem, where both inputs and mode sequences are optimized, we consider the case where switching is exogenous. For such systems, the number of possible mode sequences grows exponentially with time, making the exact computation of the value function intractable. This motivates the development of tractable bounds that approximate it. We propose a novel framework, based on path-complete graphs, for constructing computable upper bounds on the value function. In this framework, multiple quadratic functions are combined through a directed graph that encodes dynamic programming inequalities, yielding convex and sound formulations. For example, for switched linear systems with quadratic cost, we derive tractable LMI-based formulations and provide computational complexity bounds. We further establish approximation guarantees for the upper bounds and show asymptotic non-conservativeness using concepts from graph theory. Finally, we extend the approach to controller synthesis for systems with affine control inputs and demonstrate its effectiveness on numerical examples.