[Paper Review] Receding-horizon Stochastic Model Predictive Control with Hard Input Constraints and Joint State Chance Constraints

Joel A. Paulson, Edward A. Buehler+2 more|arXiv (Cornell University)|Jun 28, 2015

Advanced Control Systems Optimization48 references21 citations

TL;DR

This paper proposes a receding-horizon stochastic model predictive control (SMPC) framework for discrete-time linear systems with unbounded stochastic disturbances, hard input constraints, and joint state chance constraints. By using a saturated affine disturbance feedback control law and conservative approximation via the Cantelli-Chebyshev inequality, the method converts the chance constraints into a convex second-order cone program (SOCP), ensuring recursive feasibility and closed-loop stability with bounded state variance.

ABSTRACT

This article considers the stochastic optimal control of discrete-time linear systems subject to (possibly) unbounded stochastic disturbances, hard constraints on the manipulated variables, and joint chance constraints on the states. A tractable convex second-order cone program (SOCP) is derived for calculating the receding-horizon control law at each time step. Feedback is incorporated during prediction by parametrizing the control law as an affine function of the disturbances. Hard input constraints are guaranteed by saturating the disturbances that appear in the control law parametrization. The joint state chance constraints are conservatively approximated as a collection of individual chance constraints that are subsequently relaxed via the Cantelli-Chebyshev inequality. Feasibility of the SOCP is guaranteed by softening the approximated chance constraints using the exact penalty function method. Closed-loop stability in a stochastic sense is established by establishing that the states satisfy a geometric drift condition outside of a compact set such that their variance is bounded at all times. The SMPC approach is demonstrated using a continuous acetone-butanol-ethanol fermentation process, which is used for production of high-value-added drop-in biofuels.

Motivation & Objective

• To develop a computationally tractable SMPC approach for linear systems with unbounded stochastic disturbances and hard input constraints.
• To address the intractability of joint chance constraints in stochastic MPC by conservative approximation using the Cantelli-Chebyshev inequality.
• To guarantee recursive feasibility and closed-loop stability in a stochastic sense despite unbounded disturbances.
• To enable direct handling of hard input constraints without relaxation into soft chance constraints, reducing conservatism.
• To demonstrate the method on a continuous acetone-butanol-ethanol fermentation process as a real-world application.

Proposed method

• A saturated affine disturbance feedback control law is used to incorporate feedback during prediction and enforce hard input constraints by saturating disturbances in the control policy.
• Joint state chance constraints are conservatively approximated as individual chance constraints using the Cantelli-Chebyshev inequality to enable convex reformulation.
• The resulting optimal control problem is formulated as a convex second-order cone program (SOCP) with soft constraints via the exact penalty function method to ensure feasibility.
• Recursive feasibility is guaranteed by softening the approximated chance constraints using an exact penalty function, avoiding infeasibility due to conservative approximations.
• Closed-loop stability is established by proving a geometric drift condition outside a compact set, ensuring bounded state variance over time.
• The method avoids worst-case conservatism of robust MPC and does not require bounded uncertainty bounds, making it suitable for systems with Gaussian or arbitrary unbounded disturbances.

Experimental results

Research questions

• RQ1 Can a convex SMPC formulation be derived for systems with unbounded stochastic disturbances and hard input constraints?
• RQ2 How can joint state chance constraints be conservatively approximated to enable convex optimization while preserving feasibility?
• RQ3 What control policy structure ensures both recursive feasibility and closed-loop stability under unbounded disturbances?
• RQ4 Can hard input constraints be directly enforced without relaxing them into soft chance constraints?
• RQ5 How can the trade-off between constraint satisfaction and performance be systematically managed in the presence of unbounded uncertainties?

Key findings

• The proposed SMPC approach transforms the original nonconvex, intractable problem into a convex second-order cone program (SOCP), enabling efficient online computation.
• Recursive feasibility is guaranteed through the exact penalty function method applied to softened chance constraints, ensuring the optimization problem remains solvable at each time step.
• Closed-loop stability is established via a geometric drift condition, proving that the expected state cost decays outside a compact set and that state variance remains bounded.
• The method achieves reduced conservatism compared to deterministic robust MPC by allowing probabilistic constraint violations within a specified tolerance.
• The approach successfully controls a continuous acetone-butanol-ethanol fermentation process, demonstrating practical applicability in a bioproduction context with unbounded disturbances.
• The use of saturated affine feedback control laws enables direct handling of hard input constraints even under unbounded disturbances, a limitation in prior stochastic tube and sample-based approaches.