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Wasserstein Distributionally Robust Regret-Optimal Control in the Infinite-Horizon

Taylan Kargin, Joudi Hajar, Vikrant Malik, Babak Hassibi

TL;DR

The paper develops distributionally robust regret-optimal control for discrete-time LTI systems over an infinite horizon, using a Wasserstein-2 ambiguity set to hedge against distributional uncertainty in time-correlated disturbances. By applying a strong duality framework, it reduces the problem to a suboptimal, gamma-parameterized formulation and derives a frequency-domain fixed-point method to synthesize the controller, despite the optimal policy being non-rational. The gamma-suboptimal controller is characterized by a finite-dimensional parameter and a spectral-factor based relation, enabling efficient computation via a fixed-point iterations and spectral factorization, while guaranteeing stabilization for $eta>eta_{ m RO}$. The approach interpolates between $H_2$ and regret-optimal (RO) performance as the ambiguity radius changes, and numerical experiments on aircraft-like systems demonstrate superior worst-case regret performance with favorable computation times compared to horizon-dependent SDPs. This work advances robust control by accommodating arbitrarily correlated disturbances and providing scalable infinite-horizon synthesis methods with practical applicability.

Abstract

We investigate the Distributionally Robust Regret-Optimal (DR-RO) control of discrete-time linear dynamical systems with quadratic cost over an infinite horizon. Regret is the difference in cost obtained by a causal controller and a clairvoyant controller with access to future disturbances. We focus on the infinite-horizon framework, which results in stability guarantees. In this DR setting, the probability distribution of the disturbances resides within a Wasserstein-2 ambiguity set centered at a specified nominal distribution. Our objective is to identify a control policy that minimizes the worst-case expected regret over an infinite horizon, considering all potential disturbance distributions within the ambiguity set. In contrast to prior works, which assume time-independent disturbances, we relax this constraint to allow for time-correlated disturbances, thus actual distributional robustness. While we show that the resulting optimal controller is non-rational and lacks a finite-dimensional state-space realization, we demonstrate that it can still be uniquely characterized by a finite dimensional parameter. Exploiting this fact, we introduce an efficient numerical method to compute the controller in the frequency domain using fixed-point iterations. This method circumvents the computational bottleneck associated with the finite-horizon problem, where the semi-definite programming (SDP) solution dimension scales with the time horizon. Numerical experiments demonstrate the effectiveness and performance of our framework.

Wasserstein Distributionally Robust Regret-Optimal Control in the Infinite-Horizon

TL;DR

The paper develops distributionally robust regret-optimal control for discrete-time LTI systems over an infinite horizon, using a Wasserstein-2 ambiguity set to hedge against distributional uncertainty in time-correlated disturbances. By applying a strong duality framework, it reduces the problem to a suboptimal, gamma-parameterized formulation and derives a frequency-domain fixed-point method to synthesize the controller, despite the optimal policy being non-rational. The gamma-suboptimal controller is characterized by a finite-dimensional parameter and a spectral-factor based relation, enabling efficient computation via a fixed-point iterations and spectral factorization, while guaranteeing stabilization for . The approach interpolates between and regret-optimal (RO) performance as the ambiguity radius changes, and numerical experiments on aircraft-like systems demonstrate superior worst-case regret performance with favorable computation times compared to horizon-dependent SDPs. This work advances robust control by accommodating arbitrarily correlated disturbances and providing scalable infinite-horizon synthesis methods with practical applicability.

Abstract

We investigate the Distributionally Robust Regret-Optimal (DR-RO) control of discrete-time linear dynamical systems with quadratic cost over an infinite horizon. Regret is the difference in cost obtained by a causal controller and a clairvoyant controller with access to future disturbances. We focus on the infinite-horizon framework, which results in stability guarantees. In this DR setting, the probability distribution of the disturbances resides within a Wasserstein-2 ambiguity set centered at a specified nominal distribution. Our objective is to identify a control policy that minimizes the worst-case expected regret over an infinite horizon, considering all potential disturbance distributions within the ambiguity set. In contrast to prior works, which assume time-independent disturbances, we relax this constraint to allow for time-correlated disturbances, thus actual distributional robustness. While we show that the resulting optimal controller is non-rational and lacks a finite-dimensional state-space realization, we demonstrate that it can still be uniquely characterized by a finite dimensional parameter. Exploiting this fact, we introduce an efficient numerical method to compute the controller in the frequency domain using fixed-point iterations. This method circumvents the computational bottleneck associated with the finite-horizon problem, where the semi-definite programming (SDP) solution dimension scales with the time horizon. Numerical experiments demonstrate the effectiveness and performance of our framework.
Paper Structure (39 sections, 11 theorems, 61 equations, 2 figures, 1 table, 2 algorithms)

This paper contains 39 sections, 11 theorems, 61 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Regret-Optimal (RO) Control.
  3. Distributionally Robust (DR) Control.
  4. Contributions
  5. Stabilizing Infinite-Horizon Controller.
  6. Robustness to Arbitrarily Correlated Disturbances.
  7. Computationally Efficient Controller Synthesis.
  8. Preliminaries and Problem Setup
  9. Notations
  10. A Linear Dynamical System
  11. Controller.
  12. Cost.
  13. The Regret Measure
  14. Distributionally Robust Regret-Optimal Control
  15. Main Theoretical Results
  16. ...and 24 more sections

Key Result

theorem 4

Let $\mathcal{C}_{\mathcal{K}} \coloneqq \mathcal{T}_{\mathcal{K}}^\ast \mathcal{T}_{\mathcal{K}} - \mathcal{T}_{\mathcal{K}_\circ}^\ast \mathcal{T}_{\mathcal{K}_\circ}$ and $\mathcal{K} \in \mathfrak{H}_2^{+}$ be a a causal and time-invariant policy. Under assumption asmp:nominal, the infinite-hori Furthermore, the worst-case disturbance, $\mathsf{w}_\star$, can be identified from the nominal dis

Figures (2)

  • Figure 1: (a) The worst-case expected regret cost of each controller for different values of $r$, for system [REA4]. (b) The percentage difference in the worst case regret relative to the DR-RO controller. (a) and (b) show that DR-RO minimizes the cost at all $r's$, and for small (large) $r$, the cost of DR-RO controller matches that of ${\cal H}_2$ (RO). The cost of the DR controller is less than that of $H_2$ and RO by $14.5\%$, and of $H_\infty$ by $89.7\%$ for $r=0.639$.
  • Figure 2: The operator norm, $\|T_K^\ast(\mathrm{e}^{{j}\omega})T_K(\mathrm{e}^{{j}\omega})\|$, of each controller at different frequency values, for system [REA4]. The cost of the DR-RO controller interpolates between $H_2$ and RO according to the value of $r$, across all frequencies. For a small (large) $r$, DR matches $H_2$ (RO) across all frequencies.

Theorems & Definitions (14)

  • definition 1: Worst-Case Expected Regret
  • theorem 4: Strong Duality for \ref{['eq:worst_case_regret']}
  • remark 1
  • lemma 1: Duality for the Suboptimal Problem \ref{['prob:suboptimal_DR_RO']}
  • theorem 6: suboptimal DR-RO Controller
  • corollary 1
  • lemma 2
  • corollary 2
  • theorem 7: Fixed-Point Solution
  • theorem 8: Strong Duality in the Finite-Horizon [Theorem 2 in DRORO]
  • ...and 4 more