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Mean-Square Stability of Continuous-Time Stochastic Model Predictive Control

Qi Lü, Bowen Ma, Enrique Zuazua

TL;DR

This work introduces a delayed stochastic model predictive control framework for unconstrained continuous-time SDEs, where each prediction step linearizes the nonlinear system at the origin to form a finite-horizon stochastic LQ problem. The resulting control, computed from the SLQ solution, is applied to the true nonlinear dynamics between sampling instants, yielding a time-delay closed-loop system. The authors prove global mean-square exponential stability for linear and mildly nonlinear SDEs via exponential convergence of the Riccati equation to the ARE, and establish local mean-square exponential stability for strongly nonlinear SDEs using stopping times and Grönwall-type estimates; polynomial growth is allowed, but exponential growth is ruled out. These results provide the first rigorous mean-square stability guarantees for SMPC with delayed state information in stochastic settings, enriching the theoretical foundation and guiding practical SMPC design for uncertain dynamical systems.

Abstract

We propose a stochastic model predictive control (SMPC) framework for a broad class of unconstrained controlled stochastic differential equations (SDEs) and establish its mean-square exponential stability in the infinite-horizon limit. At each prediction step of the MPC iteration, the nonlinear controlled SDE is approximated by its linearization at the origin, with the sampled state of the nonlinear system as initial condition, yielding a finite-horizon stochastic linear-quadratic (SLQ) optimal control problem. The resulting optimal control is then applied to the original nonlinear stochastic dynamics until the next sampling instant. This construction leads to a delayed SMPC scheme whose closed-loop behavior is governed by a coupled time-delay SDE system, a setting that has not been analyzed before. We prove global mean-square exponential stability for linear and mildly nonlinear SDEs by exploiting the exponential convergence of the Riccati equation to the algebraic Riccati equation (ARE). For strongly nonlinear SDEs, we establish local mean-square exponential stability by combining exponential Riccati convergence with stopping-time techniques and Grönwall-type estimates. It is observed that, to ensure the desired local stability properties, the nonlinearities of the SDE are allowed to have polynomial growth but not exponential growth, distinguishing SMPC from its deterministic counterpart. These results provide the first rigorous mean-square stability guarantees for SMPC of SDE systems with delayed state information, thereby advancing the theoretical foundations of stochastic predictive control.

Mean-Square Stability of Continuous-Time Stochastic Model Predictive Control

TL;DR

This work introduces a delayed stochastic model predictive control framework for unconstrained continuous-time SDEs, where each prediction step linearizes the nonlinear system at the origin to form a finite-horizon stochastic LQ problem. The resulting control, computed from the SLQ solution, is applied to the true nonlinear dynamics between sampling instants, yielding a time-delay closed-loop system. The authors prove global mean-square exponential stability for linear and mildly nonlinear SDEs via exponential convergence of the Riccati equation to the ARE, and establish local mean-square exponential stability for strongly nonlinear SDEs using stopping times and Grönwall-type estimates; polynomial growth is allowed, but exponential growth is ruled out. These results provide the first rigorous mean-square stability guarantees for SMPC with delayed state information in stochastic settings, enriching the theoretical foundation and guiding practical SMPC design for uncertain dynamical systems.

Abstract

We propose a stochastic model predictive control (SMPC) framework for a broad class of unconstrained controlled stochastic differential equations (SDEs) and establish its mean-square exponential stability in the infinite-horizon limit. At each prediction step of the MPC iteration, the nonlinear controlled SDE is approximated by its linearization at the origin, with the sampled state of the nonlinear system as initial condition, yielding a finite-horizon stochastic linear-quadratic (SLQ) optimal control problem. The resulting optimal control is then applied to the original nonlinear stochastic dynamics until the next sampling instant. This construction leads to a delayed SMPC scheme whose closed-loop behavior is governed by a coupled time-delay SDE system, a setting that has not been analyzed before. We prove global mean-square exponential stability for linear and mildly nonlinear SDEs by exploiting the exponential convergence of the Riccati equation to the algebraic Riccati equation (ARE). For strongly nonlinear SDEs, we establish local mean-square exponential stability by combining exponential Riccati convergence with stopping-time techniques and Grönwall-type estimates. It is observed that, to ensure the desired local stability properties, the nonlinearities of the SDE are allowed to have polynomial growth but not exponential growth, distinguishing SMPC from its deterministic counterpart. These results provide the first rigorous mean-square stability guarantees for SMPC of SDE systems with delayed state information, thereby advancing the theoretical foundations of stochastic predictive control.

Paper Structure

This paper contains 23 sections, 10 theorems, 164 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Formulation of the problem and main results
  3. Stochastic Model Predictive Control
  4. SMPC algorithm
  5. Statements of the main results
  6. Receding Horizon Control: Perfect Linear Plant Model
  7. Stochastic Model Predictive Control: Linear Growth Case
  8. Stochastic Model Predictive Control: Nonlinear Growth Case
  9. Main ideas for the proofs of Theorem \ref{['Th2.1']}--\ref{['Theorem 2.3']}
  10. Preliminary results
  11. Infinite horizon SLQ and the related ARE
  12. Exponential convergence of solutions of the Riccati equation \ref{['Riccati']} toward solutions of the ARE \ref{['ARE']}
  13. A Grönwall-type inequality involving time delay
  14. Conclusions, perspectives and open problems
  15. Appendix
  16. ...and 8 more sections

Key Result

Theorem 2.1

Let Assumption H1 hold. Then there exists a constant $K>0$, independent of $t$, $x$, $T$, $\tau$, and $G$, such that under either of the following conditions: it holds that Moreover, in this case, we have where $u_\infty^*(t)=\Theta_\infty Y^*_\infty (t)$ and $Y_\infty^*(\cdot)$ is the solution of with ${\cal A}_\infty,{\cal C}_\infty, \Theta_\infty$ given in nota.

Figures (1)

  • Figure 1: The SMPC scheme over time-horizon $t\in [k\tau,(k+1)\tau] \,(k\in {\mathbb{N}})$ in Algorithm 1. The closed-loop optimal strategy $\Theta(t-k\tau)$ for the auxiliary SLQ problem is computed offline. The SMPC control $u_{\rm M}(t)=\Theta(t-k\tau)\bar{X}_T(t)$ is applied in feedback form to regulate \ref{['OriCon']}, regardless of the realization of the sample path of $\bar{X}_T(t)$.

Theorems & Definitions (34)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.1
  • Remark 2.5
  • Theorem 2.1: Stability of RHC
  • Remark 2.6
  • Remark 2.7
  • Theorem 2.2: Stability of SMPC: Linear growth case
  • ...and 24 more