Regret-Optimal Control under Partial Observability

This paper studies online solutions for regret-optimal control in partially observable systems over an infinite-horizon. Regret-optimal control aims to minimize the difference in LQR cost between causal and non-causal controllers while considering the worst-case regret across all $\ell_2$-norm-bounded disturbance and measurement sequences. Building on ideas from Sabag et al., 2023, on the the full-information setting, our work extends the framework to the scenario of partial observability (measurement-feedback). We derive an explicit state-space solution when the non-causal solution is the one that minimizes the $\mathcal H_2$ criterion, and demonstrate its practical utility on several practical examples. These results underscore the framework's significant relevance and applicability in real-world systems.Comment: Submitted to ACC 202

Wasserstein Distributionally Robust Regret-Optimal Control under Partial Observability

This paper presents a framework for Wasserstein distributionally robust (DR) regret-optimal (RO) control in the context of partially observable systems. DR-RO control considers the regret in LQR cost between a causal and non-causal controller and aims to minimize the worst-case regret over all disturbances whose probability distribution is within a certain Wasserstein-2 ball of a nominal distribution. Our work builds upon the full-information DR-RO problem that was introduced and solved in Yan et al., 2023, and extends it to handle partial observability and measurement-feedback (MF). We solve the finite horizon partially observable DR-RO and show that it reduces to a tractable semi-definite program whose size is proportional to the time horizon. Through simulations, the effectiveness and performance of the framework are demonstrated, showcasing its practical relevance to real-world control systems. The proposed approach enables robust control decisions, enhances system performance in uncertain and partially observable environments, and provides resilience against measurement noise and model discrepancies

Modeling of Political Systems using Wasserstein Gradient Flows

The study of complex political phenomena such as parties' polarization calls for mathematical models of political systems. In this paper, we aim at modeling the time evolution of a political system whereby various parties selfishly interact to maximize their political success (e.g., number of votes). More specifically, we identify the ideology of a party as a probability distribution over a one-dimensional real-valued ideology space, and we formulate a gradient flow in the probability space (also called a Wasserstein gradient flow) to study its temporal evolution. We characterize the equilibria of the arising dynamic system, and establish local convergence under mild assumptions. We calibrate and validate our model with real-world time-series data of the time evolution of the ideologies of the Republican and Democratic parties in the US Congress. Our framework allows to rigorously reason about various political effects such as parties' polarization and homogeneity. Among others, our mechanistic model can explain why political parties become more polarized and less inclusive with time (their distributions get "tighter"), until all candidates in a party converge asymptotically to the same ideological position

Modeling of Political Systems using Wasserstein Gradient Flows

The study of complex political phenomena such as parties' polarization calls for mathematical models of political systems. In this paper, we aim at modeling the time evolution of a political system whereby various parties selfishly interact to maximize their political success (e.g., number of votes). More specifically, we identify the ideology of a party as a probability distribution over a one-dimensional real-valued ideology space, and we formulate a gradient flow in the probability space (also called a Wasserstein gradient flow) to study its temporal evolution. We characterize the equilibria of the arising dynamic system, and establish local convergence under mild assumptions. We calibrate and validate our model with real-world time-series data of the time evolution of the ideologies of the Republican and Democratic parties in the US Congress. Our framework allows to rigorously reason about various political effects such as parties' polarization and homogeneity. Among others, our mechanistic model can explain why political parties become more polarized and less inclusive with time (their distributions get "tighter"), until all candidates in a party converge asymptotically to the same ideological position