Prescribed-Time Optimal Control for a Class of Switched Nonlinear Systems

Abstract

This paper proposes a comprehensive framework for prescribed-time optimal switching and control (PTOSC) in switched systems. First, a new performance index function is defined, which considers the system state, specified time and accuracy, and control costs. This effectively incorporates the prescribed time control into the optimal control framework. Following this, a switched Hamilton-Jacobi-Bellman equation is derived. An event-triggered (ET) PTOSC algorithm, via reinforcement learning, is subsequently presented to solve this equation, and then the optimal control policies are derived. At each event-triggering instant, the switched controller determines which subsystem to activate, and the input controller updates the system inputs. The proposed PTOSC algorithm guarantees the stability of the switched systems and ensures the system states converge to a specified range within a specified time, all while minimizing energy consumption. Furthermore, the devised ET mechanism significantly reduces the communication burden and effectively avoids Zeno behavior. Finally, a simulation is performed to validate the proposed PTOSC algorithms’ effectiveness. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper addresses the critical problem in the field of switched systems of achieving prescribed-time optimal switching and control. Current practices might not adequately balance convergence accuracy, settling time, and cost-saving, which is the motivation for this paper. The presented control method is of great significance for many practical processes, such as power systems, robot control, and water-air amphibious vehicles. The feasibility of this new approach is confirmed through simulation. Future research can further refine this approach and explore how to achieve the prescribed-time optimal control for multi-agent systems or multi-player non-zero-sum games.