The Distributionally Robust Infinite-Horizon LQR
We explore the infinite-horizon Distributionally Robust (DR) linear-quadratic control. While the probability distribution of disturbances is unknown and potentially correlated over time, it is confined within a Wasserstein-2 ball of a radius $r$ around a known nominal distribution. Our goal is to de...
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Main Authors | , , , |
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Format | Journal Article |
Language | English |
Published |
12.08.2024
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Subjects | |
Online Access | Get full text |
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Summary: | We explore the infinite-horizon Distributionally Robust (DR) linear-quadratic
control. While the probability distribution of disturbances is unknown and
potentially correlated over time, it is confined within a Wasserstein-2 ball of
a radius $r$ around a known nominal distribution. Our goal is to devise a
control policy that minimizes the worst-case expected Linear-Quadratic
Regulator (LQR) cost among all probability distributions of disturbances lying
in the Wasserstein ambiguity set. We obtain the optimality conditions for the
optimal DR controller and show that it is non-rational. Despite lacking a
finite-order state-space representation, we introduce a computationally
tractable fixed-point iteration algorithm. Our proposed method computes the
optimal controller in the frequency domain to any desired fidelity. Moreover,
for any given finite order, we use a convex numerical method to compute the
best rational approximation (in $H_\infty$-norm) to the optimal non-rational DR
controller. This enables efficient time-domain implementation by finite-order
state-space controllers and addresses the computational hurdles associated with
the finite-horizon approaches to DR-LQR problems, which typically necessitate
solving a Semi-Definite Program (SDP) with a dimension scaling with the time
horizon. We provide numerical simulations to showcase the effectiveness of our
approach. |
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DOI: | 10.48550/arxiv.2408.06230 |