Learning Soft Constrained MPC Value Functions: Efficient MPC Design and Implementation providing Stability and Safety Guarantees

TL;DR

This paper introduces a framework combining soft constrained MPC with supervised learning to efficiently approximate the MPC value function, ensuring stability and safety in embedded control systems.

Contribution

It proposes a novel approach that integrates soft constrained MPC with supervised learning, providing stability and safety guarantees while enabling efficient implementation on embedded hardware.

Findings

The method guarantees stability and constraint satisfaction for nonlinear systems.

The value function approximation is Lipschitz continuous, facilitating learning.

Effective in a nonlinear numerical example.

Abstract

Model Predictive Control (MPC) can be applied to safety-critical control problems, providing closed-loop safety and performance guarantees. Implementation of MPC controllers requires solving an optimization problem at every sampling instant, which is challenging to execute on embedded hardware. To address this challenge, we propose a framework that combines a tightened soft constrained MPC formulation with supervised learning to approximate the MPC value function. This combination enables us to obtain a corresponding optimal control law, which can be implemented efficiently on embedded platforms. The framework ensures stability and constraint satisfaction for various nonlinear systems. While the design effort is similar to that of nominal MPC, the proposed formulation provides input-to-state stability (ISS) with respect to the approximation error of the value function. Furthermore, we prove that the value function corresponding to the soft constrained MPC problem is Lipschitz continuous for Lipschitz continuous systems, even if the optimal control law may be discontinuous. This serves two purposes: First, it allows to relate approximation errors to a sufficiently large constraint tightening to obtain constraint satisfaction guarantees. Second, it paves the way for an efficient supervised learning procedure to obtain a continuous value function approximation. We demonstrate the effectiveness of the method using a nonlinear numerical example.