Approximate Dynamic Programming for Constrained Piecewise Affine Systems with Stability and Safety Guarantees

TL;DR

This paper introduces an efficient approximate dynamic programming approach for constrained piecewise affine systems, providing stability, safety guarantees, and reduced computational complexity compared to traditional hybrid MPC methods.

Contribution

It develops a novel ADP method with PWA function approximation for constrained PWA systems, offering online efficiency and formal stability and safety guarantees.

Findings

Achieves smaller-scale optimization problems than hybrid MPC

Provides stability and safety guarantees for closed-loop systems

Demonstrates effectiveness through simulations on pendulum and cruise control

Abstract

Infinite-horizon optimal control of constrained piecewise affine (PWA) systems has been approximately addressed by hybrid model predictive control (MPC), which, however, has computational limitations, both in offline design and online implementation. In this paper, we consider an alternative approach based on approximate dynamic programming (ADP), an important class of methods in reinforcement learning. We accommodate non-convex union-of-polyhedra state constraints and linear input constraints into ADP by designing PWA penalty functions. PWA function approximation is used, which allows for a mixed-integer encoding to implement ADP. The main advantage of the proposed ADP method is its online computational efficiency. Particularly, we propose two control policies, which lead to solving a smaller-scale mixed-integer linear program than conventional hybrid MPC, or a single convex quadratic program, depending on whether the policy is implicitly determined online or explicitly computed offline. We characterize the stability and safety properties of the closed-loop systems, as well as the sub-optimality of the proposed policies, by quantifying the approximation errors of value functions and policies. We also develop an offline mixed-integer linear programming-based method to certify the reliability of the proposed method. Simulation results on an inverted pendulum with elastic walls and on an adaptive cruise control problem validate the control performance in terms of constraint satisfaction and CPU time.