Optimal Control of Path-Constrained Switched Systems With Guaranteed Feasibility

This article considers the optimal control problem for path-constrained switched systems with guaranteed feasibility. First, a method is proposed to reduce continuous path constraints of the switched systems into discrete abstract constraints in terms of the switch time vector and the input. Second, differentiability of the optimal value function, i.e., the cost function optimized over the input, is analyzed by using perturbation analysis theory. For a given switch time vector, the derivative of the optimal value function is proved equal to the derivative of the cost function evaluated at the optimal input provided that there is a unique optimal input in the optimal set of the cost function for continuous inputs. Third, a bilevel algorithm is proposed to find optimal switch times and the optimal input with guaranteed satisfaction of path constraints over the entire time horizon. At the inner level, the input is optimized by iteratively solving dynamic optimization problems of path-constrained switched systems with fixed switch time vector, and at the outer level, the fixed switch time vector is optimized by sequentially solving quadratic programs related to the optimal value function of switch time vectors. Fourth, it is rigorously proven that the bilevel algorithm terminates within a finite number of iterations. Finally, a numerical example is provided to illustrate the effectiveness of the proposed algorithm.

I. Introduction

Switched systems are hybrid dynamic systems consisting of several subsystems and a switching law specifying the active subsystems at each time instant. Such systems have been used in a wide variety of fields such as automotive industry, traffic control, and networked control systems. This topic has been extensively researched and a lot of progress has been made both theoretically and computationally. A wealth of important literature is available for the modeling and dynamic analyzing of switched systems including stability [1]–[3], controllability, and observability [4]–[6].