Optimal Control of Switched Dynamical Systems Under Dwell Time Constraints—Theory and Computation

This article addresses the problem of optimal mode scheduling subject to dwell time constraints, which is the minimum amount of time a system has to spend in one mode before it can transition to another. The constraint is important since most physical systems cannot switch rapidly between different modes and its presence also eliminates the problem of chattering solutions by construction. We investigate the topology of the optimization space and show that it lacks structure to define local minima. A framework is developed for defining optimal solutions as stationary points of optimality functions and an optimality function is proposed for characterizing the necessary conditions for optimality. The challenges posed by dwell time constraints to algorithmic implementation are addressed by exploring the geometric properties of the so-called mode insertion gradient, and a technique is developed for rapidly updating of the mode sequence. The algorithm's convergence to an optimal solution is proved and simulation results are provided to demonstrate the algorithm's efficacy.

I. Introduction

Most real-world systems and processes are too complex to be represented using smooth linear or nonlinear dynamics, and are often modeled as switched dynamical systems made up of many subsystems or modes [1]. A switching law then acts as a control variable to determine the active subsystem from a given a set of modes. The resulting mode sequence, combined with the mode durations, is referred to as a mode schedule [2]. Switched systems arise in many application areas, such as humanoid and mobile robotics, cyber-physical systems, power converters, air traffic management, chemical processes, autopilots etc., see e.g., [3] and [4], and as such, their optimal control has recently been an active area of research.