Stochastic Model Predictive Control for Sub-Gaussian Noise

TL;DR

This work addresses safe control under non-Gaussian, time-varying noise by introducing stochastic MPC for sub-Gaussian disturbances. It replaces a scalar variance proxy with a matrix variance proxy $\Sigma$, derives linear propagation rules, and constructs probabilistic reachable sets to tighten constraints while preserving closed-loop chance guarantees. The authors prove recursive feasibility and provide an asymptotic performance bound, demonstrating that the method is less conservative than robust or distributionally robust approaches. Numerical experiments on mass-spring-damper and vision-based surgical planning tasks illustrate reliable constraint satisfaction (at a target $1-\delta=0.95$) and improved performance, with practical calibration using sample-based variance proxies and an open-source implementation.

Abstract

We propose a stochastic Model Predictive Control (MPC) framework that ensures closed-loop chance constraint satisfaction for linear systems with general sub-Gaussian process and measurement noise. By considering sub-Gaussian noise, we can provide guarantees for a large class of distributions, including time-varying distributions. Specifically, we first provide a new characterization of sub-Gaussian random vectors using matrix variance proxy, which can more accurately represent the predicted state distribution. We then derive tail bounds under linear propagation for the new characterization, enabling tractable computation of probabilistic reachable sets of linear systems. Lastly, we utilize these probabilistic reachable sets to formulate a stochastic MPC scheme that provides closed-loop guarantees for general sub-Gaussian noise. We further demonstrate our approach in simulations, including a challenging task of surgical planning from image observations.

Figures (3)

• Figure 1: Overview of the proposed stochastic MPC for sub-Gaussian noise at the example of the surgical planning (\ref{['sec:env']}). We first obtain (high-dimensional) measurements, estimate the vector state and compute a sub-Gaussian characterization of the state estimation error. Then, we provide a simple method to propagate uncertainty and compute probabilistic reachable sets (PRS, Section \ref{['sec:prob_guar', 'sec:prop']}). The resulting probabilistic reachable sets of states are utilized in the stochastic MPC to provide probabilistic safety guarantees (Section \ref{['sec:IF-SMPC-KF']}).
• Figure 2: Comparison of $95\%$ confidence bound sizes quantified by different methods in the mass-spring-damper environment with truncated Student-t noise. Blue lines represent quantiles from test samples. All approaches compute the global maximum confidence bound to address the heteroscedastic noise. The confidence bound from our sub-Gaussian approach is greater than the quantiles but less conservative than robust and distributionally robust approaches.
• Figure 3: Plans from sub-Gaussian and DR MPC approaches in 100 trials from (left) MSD and (right) SP environments, respectively. For MSD, the $x$ and $y$ axes are time and the first state, respectively. For SP, they correspond to the first 2 dimensions of the states. The confidence levels of displayed examples are set at $95\%$. The yellow lines represent the boundary constraints. In all problems, the proposed approach satisfies the safety-critical constraints with the chosen probability $95\%$.

Theorems & Definitions (20)

• Definition 1: $\sigma-$sub-Gaussian vershyninHighdimensionalProbabilityIntroduction2018
• Remark 1
• Definition 2: Sub-Gaussian with matrix (co-)variance proxy
• Lemma 1
• Theorem 1: Propagation of matrix variance proxy
• proof
• Lemma 2: Half-space bound
• proof
• Theorem 2: Elliptical bound
• Corollary 1