Approximate Dynamic Programming for Constrained Linear Systems: A Piecewise Quadratic Approximation Approach

TL;DR

This paper presents a novel approximate dynamic programming method for constrained linear systems that combines model predictive control insights with a piecewise quadratic neural network to improve online computation and stability.

Contribution

It introduces a new ADP approach using a convex piecewise quadratic neural network to approximate the value function for constrained linear quadratic regulation problems.

Findings

The method achieves faster online computation.

It maintains stability with good value function approximation.

Comparative simulations show improved performance.

Abstract

Approximate dynamic programming (ADP) faces challenges in dealing with constraints in control problems. Model predictive control (MPC) is, in comparison, well-known for its accommodation of constraints and stability guarantees, although its computation is sometimes prohibitive. This paper introduces an approach combining the two methodologies to overcome their individual limitations. The predictive control law for constrained linear quadratic regulation (CLQR) problems has been proven to be piecewise affine (PWA) while the value function is piecewise quadratic. We exploit these formal results from MPC to design an ADP method for CLQR problems. A novel convex and piecewise quadratic neural network with a local-global architecture is proposed to provide an accurate approximation of the value function, which is used as the cost-to-go function in the online dynamic programming problem. An efficient decomposition algorithm is developed to speed up the online computation. Rigorous stability analysis of the closed-loop system is conducted for the proposed control scheme under the condition that a good approximation of the value function is achieved. Comparative simulations are carried out to demonstrate the potential of the proposed method in terms of online computation and optimality.