Transition-time optimization for switched-mode dynamical systems

This note considers the problem of determining optimal switching times at which mode transitions should occur in multimodal, hybrid systems. It derives a simple formula for the gradient of the cost functional with respect to the switching times, and uses it in a gradient-descent algorithm. Much of the analysis is carried out in the setting of optimization problems involving fixed switching-mode sequences, but a possible extension is pointed out for the case where the switching-mode sequence is a part of the variable. Numerical examples testify to the viability of the proposed approach.

I. Introduction

Switched dynamical systems are often described by differential inclusions of the form

$${\mathdot {x}}(t)\in \{g_{\alpha}(x(t),u(t))\}_{\alpha \in A}\eqno{\hbox{(1)}}$$ where , , and is a collection of continuously differentiable functions, parameterized by belonging to some given set . The time is confined to a given finite-length interval . Such systems arise in a variety of applications, including situations where a control module has to switch its attention among a number of subsystems [12], [15], [19], or collect data sequentially from a number of sensory sources [4], [6], [11]. A supervisory controller is normally engaged for dictating the switching law, i.e., the rule for switching among the functions in the right-hand side of (1).