Distributionally Robust Model Predictive Control: Closed-loop Guarantees and Scalable Algorithms

Abstract

We establish a collection of closed-loop guarantees and propose a scalable optimization algorithm for distributionally robust model predictive control (DRMPC) applied to linear systems, convex constraints, and quadratic costs. Via standard assumptions for the terminal cost and constraint, we establish distribtionally robust long-term and stage-wise performance guarantees for the closed-loop system. We further demonstrate that a common choice of the terminal cost, i.e., via the discrete-algebraic Riccati equation, renders the origin input-to-state stable for the closed-loop system. This choice also ensures that the exact long-term performance of the closed-loop system is independent of the choice of ambiguity set for the DRMPC formulation. Thus, we establish conditions under which DRMPC does not provide a long-term performance benefit relative to stochastic MPC. To solve the DRMPC optimization problem, we propose a Newton-type algorithm that empirically achieves superlinear convergence and guarantees the feasibility of each iterate. We demonstrate the implications of the closed-loop guarantees and the scalability of the proposed algorithm via two examples. To facilitate the reproducibility of the results, we also provide open-source code to implement the proposed algorithm and generate the figures.

Figures (6)

• Figure 1: Convergence of FW and NT algorithms with adaptive (A) and fully adaptive (FA) stepsize calculations for a DRMPC problem ($N=10$) in terms of suboptimality gap as a function of iteration (left) and computation time (right).
• Figure 2: Comparison of computation times for the NT algorithm (fully adaptive stepsize) and LMI formulation solved by MOSEK for the horizon $N$.
• Figure 3: Closed-loop trajectories ($N=10$) with zero disturbance, i.e., $x(k)=\phi_d(k;x,\mathbf{0})$, for the first element of the state, denoted $x_1(k)$.
• Figure 4: Sample averages of $\mathbb{E}_{\mathds{P}}\left[|\phi(k;x,\mathbf{w}_{\infty})|^2\right]$ and $\mathcal{J}_k(x,d,\mathds{P})$, denoted $\tilde{\mathbb{E}}_{\mathds{P}}\left[|x(k)|^2\right]$ and $\tilde{\mathcal{J}}_k$, for $S=100$ realizations of the closed-loop trajectory. Shaded regions show plus/minus one standard deviation.
• Figure 5: Sample average of the performance metric $\mathcal{J}_T(x,d,\mathds{P})$, denoted $\tilde{\mathcal{J}}_T$, for $T=500$ as a function of the ambiguity radius $\varepsilon$ for $S=30$ realizations of the disturbance trajectory. The shaded region shows plus/minus one standard deviation and the dashed lines indicate the min/max values of $\tilde{\mathcal{J}}_T^s$ for all $s$.

Theorems & Definitions (33)

• Lemma 2.1: Policy constraints
• Definition 3.1: Robust positive invariance
• Theorem 3.1: DR long-term performance
• Theorem 3.2: DR stage-wise performance
• Corollary 3.3: DR, mean-squared ISS
• Theorem 3.4: Pathwise ISS
• Theorem 3.5: Exact long-term performance
• Corollary 3.6: DRMPC versus SMPC
• Remark 3.1: Detectable stage cost
• Lemma 4.1: DR cost decrease