Stability and performance of stochastic economic MPC - Stochastic characterization of the closed-loop asymptotics

TL;DR

This work develops a stochastic economic MPC framework that yields $P$-practical asymptotic stability for the closed loop without terminal constraints by leveraging stochastic dissipativity and turnpike properties, and it provides tight near-optimal performance bounds for both averaged and non-averaged criteria. It introduces an abstract MPC formulation on the space of random variables and proves that a practically implementable, pathwise MPC algorithm achieves the same closed-loop properties, including stability under several notions of convergence (in distribution, in $p$-th mean, etc.). The analysis rests on a stochastic Lyapunov function built from a modified cost and a storage function, and it extends classical deterministic results to the stochastic domain with explicit residual terms that vanish as the horizon grows. Numerical simulations on a nonlinear, one-dimensional example illustrate pathwise and distributional turnpikes and confirm the theoretical stability and performance guarantees. The results enhance the practical relevance of stochastic MPC by showing that sampling-based implementations can inherit rigorous stochastic stability and near-optimality properties, with clear avenues for future work on chance constraints and risk-sensitive criteria.

Abstract

Model Predictive Control (MPC) is well understood in the deterministic setting, yet rigorous stability and performance guarantees for stochastic MPC remain limited to the consideration of terminal constraints and penalties. In contrast, this work analyzes stochastic economic MPC with an expected cost criterion and establishes closed-loop guarantees without terminal conditions. Relying on stochastic dissipativity and turnpike properties, we construct closed-loop Lyapunov functions that ensure $P$-practical asymptotic stability of a particular optimal stationary process under different notions of stochastic convergence, such as in distribution or in the $p$-th mean. In addition, we derive tight near-optimal bounds for both averaged and non-averaged performance, thereby extending classical deterministic results to the stochastic domain. Finally, we show that the abstract stochastic MPC scheme requiring distributional knowledge shares the same closed-loop properties as a practically implementable algorithm based only on sampled state information, ensuring applicability of our findings. Our findings are illustrated by a numerical example.

Figures (5)

• Figure 1: Evolution of all possible realization paths for the optimal trajectories (left) and controls (right) for $N=3,5,7,\ldots,15$.
• Figure 2: 15 sample paths of the closed-loop trajectories $X_{\mu_N}$ for $N=3$ (left), $N=4$ (middle), and $N=5$ (right) together with the upper and lower bound of the realizations of the stationary process (red dashed).
• Figure 3: Approximation of the probability density function (left) and cumulative distribution function (right) of $X_{\mu_N}(150)$ for $N=3$ (orange), $N=4$ (green) and $N=5$ (blue) as well as the approximate stationary values (red).
• Figure 4: Approximation of the mean (left) and variance (right) of $X_{\mu_N}(150)$ for $N=3$ (orange), $N=4$ (green) and $N=5$ (blue) as well as the approximate stationary values (red).
• Figure 5: Cumulative costs (left) and stage cost of the stationary process (red) and approximation of the averaged costs (right) for $X_{\mu_N}$ with $N=3$ (orange), $N=4$ (green) and $N=5$ (blue).

Theorems & Definitions (57)

• Remark 2.1: Link to MDPs
• Theorem 3.1: Finite horizon DPP, cf. Gruene2017b
• Theorem 3.2: Infinite horizon DPP, cf.Gruene2017b
• Theorem 3.3: cf. Bertsekas1996b
• Lemma 3.4
• proof
• Corollary 3.5
• proof
• Remark 3.6
• Definition 4.1: Stationary stochastic processes