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Data-Driven Distributionally Robust System Level Synthesis

Francesco Micheli, Anastasios Tsiamis, John Lygeros

TL;DR

This paper tackles robust control of uncertain discrete-time LTI systems with unknown dynamics and disturbances by marrying System Level Synthesis with distributionally robust optimization. It introduces a doubly robust, data-driven state-feedback controller whose design is cast as a distributionally robust finite-horizon problem with Wasserstein ambiguity sets centered on an empirical predictive distribution, and whose radius can depend on the decision variables to ensure containment of the true distribution with high confidence. The authors derive a tractable LP-based reformulation for piecewise affine costs/constraints, provide finite-sample performance guarantees, and quantify the distribution shift between predictive and actual closed-loop distributions via a small-gain analysis. They demonstrate, through a numerical example, that the proposed approach safely controls the system under model mismatch and limited data while avoiding the conservatism of purely robust or purely stochastic methods. The framework is extensible to arbitrary initial conditions and affine SLS, with future work directed at episodic learning and online data updates.

Abstract

We present a novel approach for the control of uncertain, linear time-invariant systems, which are perturbed by potentially unbounded, additive disturbances. We propose a \emph{doubly robust} data-driven state-feedback controller to ensure reliable performance against both model mismatch and disturbance distribution uncertainty. Our controller, which leverages the System Level Synthesis parameterization, is designed as the solution to a distributionally robust finite-horizon optimal control problem. The goal is to minimize a cost function while satisfying constraints against the worst-case realization of the uncertainty, which is quantified using distributional ambiguity sets. The latter are defined as balls in the Wasserstein metric centered on the predictive empirical distribution computed from a set of collected trajectory data. By harnessing techniques from robust control and distributionally robust optimization, we characterize the distributional shift between the predictive and the actual closed-loop distributions, and highlight its dependency on the model mismatch and the uncertainty about the disturbance distribution. We also provide bounds on the number of samples required to achieve a desired confidence level and propose a tractable approximate formulation for the doubly robust data-driven controller. To demonstrate the effectiveness of our approach, we present a numerical example showcasing the performance of the proposed algorithm.

Data-Driven Distributionally Robust System Level Synthesis

TL;DR

This paper tackles robust control of uncertain discrete-time LTI systems with unknown dynamics and disturbances by marrying System Level Synthesis with distributionally robust optimization. It introduces a doubly robust, data-driven state-feedback controller whose design is cast as a distributionally robust finite-horizon problem with Wasserstein ambiguity sets centered on an empirical predictive distribution, and whose radius can depend on the decision variables to ensure containment of the true distribution with high confidence. The authors derive a tractable LP-based reformulation for piecewise affine costs/constraints, provide finite-sample performance guarantees, and quantify the distribution shift between predictive and actual closed-loop distributions via a small-gain analysis. They demonstrate, through a numerical example, that the proposed approach safely controls the system under model mismatch and limited data while avoiding the conservatism of purely robust or purely stochastic methods. The framework is extensible to arbitrary initial conditions and affine SLS, with future work directed at episodic learning and online data updates.

Abstract

We present a novel approach for the control of uncertain, linear time-invariant systems, which are perturbed by potentially unbounded, additive disturbances. We propose a \emph{doubly robust} data-driven state-feedback controller to ensure reliable performance against both model mismatch and disturbance distribution uncertainty. Our controller, which leverages the System Level Synthesis parameterization, is designed as the solution to a distributionally robust finite-horizon optimal control problem. The goal is to minimize a cost function while satisfying constraints against the worst-case realization of the uncertainty, which is quantified using distributional ambiguity sets. The latter are defined as balls in the Wasserstein metric centered on the predictive empirical distribution computed from a set of collected trajectory data. By harnessing techniques from robust control and distributionally robust optimization, we characterize the distributional shift between the predictive and the actual closed-loop distributions, and highlight its dependency on the model mismatch and the uncertainty about the disturbance distribution. We also provide bounds on the number of samples required to achieve a desired confidence level and propose a tractable approximate formulation for the doubly robust data-driven controller. To demonstrate the effectiveness of our approach, we present a numerical example showcasing the performance of the proposed algorithm.
Paper Structure (18 sections, 73 equations, 3 figures)

This paper contains 18 sections, 73 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Further related work
  3. Notation and preliminaries
  4. Problem formulation
  5. System Level Synthesis
  6. Sample Average Approximation
  7. Distributionally robust formulation
  8. Characterization of the distribution shift
  9. Tractable reformulation
  10. Extensions
  11. Arbitrary initial conditions
  12. Affine SLS formulation
  13. Numerical example
  14. Conclusions
  15. Proof of Lemma \ref{['Lem:1']}
  16. ...and 3 more sections

Figures (3)

  • Figure 1: Comparison of optimal predicted trajectories for the RR (blue) and SAA (green) approaches. The vertical black line represents the constraint on the first coordinate of the state.
  • Figure 2: Comparison of validation trajectories for the RR (red) and SAA (purple) approaches. The SAA optimal controller is not robust against the model mismatch and the disturbance realizations, resulting in higher cost and large constraint violations.
  • Figure 3: Comparison of optimization cost, validation cost and validation CVaR value, for the RR and SAA approaches.

Theorems & Definitions (2)

  • proof
  • proof