[Paper Review] On convex problems in chance-constrained stochastic model predictive control

Eugenio Cinquemani, Mayank Agarwal+2 more|arXiv (Cornell University)|May 21, 2009

Advanced Control Systems Optimization21 references47 citations

TL;DR

This paper proposes a convex optimization framework for chance-constrained stochastic model predictive control in discrete-time linear systems with expected cost minimization. By parametrizing the control policy appropriately, it shows that probabilistic constraints lead to convex problems or tight convex approximations, enabling efficient and reliable solution of stochastic optimal control with hard constraints reformulated probabilistically.

ABSTRACT

We investigate constrained optimal control problems for linear stochastic dynamical systems evolving in discrete time. We consider minimization of an expected value cost over a finite horizon. Hard constraints are introduced first, and then reformulated in terms of probabilistic constraints. It is shown that, for a suitable parametrization of the control policy, a wide class of the resulting optimization problems are convex, or admit reasonable convex approximations.

Motivation & Objective

• To address the challenge of incorporating hard constraints into stochastic optimal control for linear systems with uncertainty.
• To reformulate deterministic constraints as probabilistic constraints to handle process and measurement noise.
• To identify conditions under which the resulting stochastic optimal control problem becomes convex or admits convex approximations.
• To enable efficient and tractable solution of finite-horizon stochastic control problems using convex optimization techniques.
• To provide a systematic framework for designing control policies that balance performance and constraint satisfaction under uncertainty.

Proposed method

• Formulate a finite-horizon optimal control problem with expected value cost minimization for linear stochastic systems.
• Introduce hard constraints on state and control variables, then reformulate them as probabilistic (chance) constraints.
• Apply a parameterized control policy, specifically affine policies depending on past disturbances, to ensure convexity.
• Use convex relaxation techniques to handle non-convex probabilistic constraints, enabling tractable solution via convex optimization.
• Demonstrate that under this policy parametrization, the overall problem structure preserves convexity or allows tight convex approximations.
• Leverage known results from convex optimization and chance-constrained programming to ensure computational tractability and constraint satisfaction.

Experimental results

Research questions

• RQ1 Under what conditions does a chance-constrained stochastic optimal control problem for linear systems become convex?
• RQ2 How can hard constraints in stochastic systems be effectively reformulated as probabilistic constraints without losing tractability?
• RQ3 What parametrization of the control policy ensures convexity or convex approximations in the resulting optimization problem?
• RQ4 Can a convex approximation of the chance-constrained problem be constructed that preserves performance and constraint satisfaction?
• RQ5 What is the relationship between the choice of policy parametrization and the convexity of the overall stochastic control problem?

Key findings

• The use of an affine control policy parametrization in the presence of stochastic disturbances leads to a convex optimization problem.
• Probabilistic constraints derived from hard constraints can be reformulated such that the overall problem remains convex or admits tight convex approximations.
• The resulting optimization problem is computationally tractable and suitable for real-time implementation in model predictive control.
• The framework enables reliable constraint satisfaction with a specified probability, even under process and measurement uncertainty.
• The approach provides a systematic way to balance expected cost minimization with probabilistic constraint enforcement in linear stochastic systems.
• The convexity of the problem is preserved under mild assumptions on the noise distribution and system dynamics.