Wasserstein Distributionally Robust Regret-Optimal Control in the Infinite-Horizon
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| Format: | Preprint |
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2023
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| author | Kargin, Taylan Hajar, Joudi Malik, Vikrant Hassibi, Babak |
| author_facet | Kargin, Taylan Hajar, Joudi Malik, Vikrant Hassibi, Babak |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_17376 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Wasserstein Distributionally Robust Regret-Optimal Control in the Infinite-Horizon Kargin, Taylan Hajar, Joudi Malik, Vikrant Hassibi, Babak Systems and Control Optimization and Control 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. |
| title | Wasserstein Distributionally Robust Regret-Optimal Control in the Infinite-Horizon |
| topic | Systems and Control Optimization and Control |
| url | https://arxiv.org/abs/2312.17376 |