Bayesian Risk-averse Model Predictive Control with Consistency and Stability Guarantees

TL;DR

This work addresses uncertainty in online-learned MPC for stochastic nonlinear systems by formulating a Bayesian risk-averse MPC that guarantees consistency and stability via RAAS. It separates epistemic and aleatoric uncertainty using dynamically shrinking credible-interval ambiguity sets updated with a particle filter, and provides both optimal and sub-optimal receding-horizon policies with theoretical RAAS guarantees as learning becomes consistent. The nested DP framework and RA Lyapunov stability theory underpin stability proofs, while practical algorithmic choices enable real-time implementation. Validation on a nonlinear cart-pole demonstrates improved performance-safety-learning trade-offs compared to traditional risk-neutral and robust baselines.

Abstract

Model Predictive Control (MPC) is a powerful framework for constrained control, but its performance and safety can be severely degraded when the prediction model is learned online and thus remains uncertain. In this work, we develop a Bayesian risk-averse MPC framework for stochastic, discrete-time, nonlinear systems that provides theoretical guarantees on the consistency of Bayesian learning and closed-loop stability. First, we study Bayesian learning under the conditionally independent state transitions induced by feedback control and establish explicit conditions for Bayesian consistency on an infinitely countable parameter space. Second, we introduce a general notion of risk-averse asymptotic stability (RAAS), defined via comparison function classes and independent of any specific coherent risk measure or convergence rate, and we derive a risk-averse Lyapunov stability theorem together with MPC-specific stability conditions. Third, building on these foundations, we design a practical Bayesian risk-averse MPC scheme that separates epistemic (parametric) and aleatoric (disturbance) uncertainty: additive disturbances are treated in a risk-neutral fashion, while parametric uncertainty is managed via dynamically shrinking ambiguity sets constructed from Bayesian credible intervals, approximated online using particle filtering. To enable real-time implementation, we propose both an optimal and a sub-optimal receding-horizon control policy, the latter obtained by warm-starting from the previous solution, and prove that asymptotic RAAS is recovered as the Bayesian estimator becomes consistent.

Figures (2)

• Figure 1: Bayesian risk-averse MPC framework. We use $k$ to represent the sampling time instant, and $N$ to represent the prediction horizon of MPC problems at each sampling time instant.
• Figure 2: Performance comparison across Nominal, Tube, Stochastic, and the proposed Risk-verse MPC over 50 Monte Carlo trials on the cart-pole system with an unknown pole mass and length. Metrics include total cost, tracking error, estimation accuracy, angle constraint violations. Error bars denote one standard deviation. Our method consistently balances performance, robustness, and learning efficiency under epistemic uncertainty.

Theorems & Definitions (42)

• Definition 4.1: Bayesian Consistency Vaart1998AsymptoticStatistics
• Definition 4.2: Observational Distinguishability
• Proposition 4.1: Observational Distinguishability with Extended Contexts
• Definition 4.3: Blind Zone
• Proposition 4.2: Combination of Blind Zones
• proof
• Definition 4.4: Blind Region
• Proposition 4.3: Combination of Blind Regions
• proof
• Theorem 4.1: Bayesian Consistency Conditions