A relaxation technique to ensure feasibility in stochastic control with input and state constraints

TL;DR

This paper introduces a relaxation method for stochastic control problems with probabilistic constraints, ensuring feasibility when disturbances have unbounded support, using a cascade of scenario-based optimization problems.

Contribution

It proposes a novel relaxation technique that guarantees feasibility in stochastic control with probabilistic constraints, applicable even with unbounded disturbances, via a computationally tractable cascade approach.

Findings

The relaxation method effectively recovers feasibility in unfeasible cases.

The scenario-based scheme provides high-confidence approximate solutions.

Simulation confirms the approach's effectiveness in practical scenarios.

Abstract

We consider a stochastic linear system and address the design of a finite horizon control policy that is optimal according to some average cost criterion and accounts also for probabilistic constraints on both the input and state variables. This finite horizon control problem formulation is quite common in the literature and has potential for being implemented in a receding horizon fashion according to the model predictive control strategy. Such a possibility, however, is hampered by the fact that, if the disturbance has unbounded support, a feasibility issue may arise. In this paper, we address this issue by introducing a constraint relaxation that is effective only when the original problem turns out to be unfeasible and, in that case, recovers feasibility as quickly as possible. This is obtained via a cascade of two probabilistically-constrained optimization problems, which are solved here through a computationally tractable scenario-based scheme, providing an approximate solution that satisfies the original probabilistic constraints of the cascade, with high confidence. A simulation example showing the effectiveness of the proposed approach concludes the paper.