The Scenario Approach for Stochastic Model Predictive Control with Bounds on Closed-Loop Constraint Violations

TL;DR

The work addresses SMPC under uncertainty by reframing state-constraint satisfaction as a time-average property and leveraging the multi-stage horizon of MPC to dramatically reduce the number of required scenarios. It introduces SCMPC with a finite-horizon scenario program and a flexible a-posteriori scenario removal mechanism, enabling a controllable trade-off between performance and computational load. Theoretical results establish bounds on violation probabilities via a problem-specific support rank and provide conditions for maintaining the desired average violation level, both in finite time and asymptotically. Numerical results on a stochastic two-dimensional system demonstrate substantial reductions in sample complexity with modest impacts on average performance, highlighting practical scalability for large-scale problems.

Abstract

Many practical applications of control require that constraints on the inputs and states of the system be respected, while optimizing some performance criterion. In the presence of model uncertainties or disturbances, for many control applications it suffices to keep the state constraints at least for a prescribed share of the time, as e.g. in building climate control or load mitigation for wind turbines. For such systems, a new control method of Scenario-Based Model Predictive Control (SCMPC) is presented in this paper. It optimizes the control inputs over a finite horizon, subject to robust constraint satisfaction under a finite number of random scenarios of the uncertainty and/or disturbances. While previous approaches have shown to be conservative (i.e. to stay far below the specified rate of constraint violations), the new method is the first to account for the special structure of the MPC problem in order to significantly reduce the number of scenarios. In combination with a new framework for interpreting the probabilistic constraints as average-in-time, rather than pointwise-in-time, the conservatism is eliminated. The presented method retains the essential advantages of SCMPC, namely the reduced computational complexity and the handling of arbitrary probability distributions. It also allows for adopting sample-and-remove strategies, in order to trade performance against computational complexity.

Figures (4)

• Figure 1: Schematic overview of the SCMPC algorithm, for the case with scenario removal ($R>0$) and without scenario removal ($R=0$).
• Figure 2: Upper bound on the expected violation probability $\mathop{\mathrm{\mathbf{E}}}\nolimits^{KN}\bigl[V_{t}\,|\,x_{t} \bigr]$, as a function of the sample size $K$, for different scenario removals $R$ and support dimensions $\rho_{1}=2$ (solid lines) and $\rho_{1}=5$ (dashed lines).
• Figure 3: Phase plot of closed-loop system trajectory (red: violating states; black: other states). The axis lines mark the boundary of the feasible set $\mathbb{X}$.
• Figure 4: Phase plot of closed-loop system trajectory (blue, red, purple: violating states of $\mathbb{X}_1$, $\mathbb{X}_2$, $\mathbb{X}_1$ and $\mathbb{X}_2$; black: other states). The axis lines mark the boundaries of the feasible sets $\mathbb{X}_1$ and $\mathbb{X}_2$, respectively.

Theorems & Definitions (14)

• Remark 3: Alternative Formulations
• Remark 4: Terminal Cost
• Remark 6: Alternative Formulations
• Remark 7: Control Parameterization
• Definition 8: Removal Algorithm
• Definition 9: Support Rank
• Lemma 10: Pure Additive Disturbances
• Lemma 11: Additive and Multiplicative Disturbances
• Lemma 12: Upper Bound on Distribution
• Definition 13: Admissible Sample-Removal Pair