Data-Driven Distributionally Robust Control for Interacting Agents under Logical Constraints

In this paper, we propose a distributionally robust control synthesis for an agent with stochastic dynamics that interacts with other agents under uncertainties and constraints expressed by signal temporal logic (STL). We formulate the control synthesis as a chance-constrained program (CCP) with STL specifications that must be satisfied with high probability under all uncertainty tubes induced by the other agents. To tackle the CCP, we propose two methods based on concentration of measure (CoM) theory and conditional value at risk (CVaR) and compare the required assumptions and resulting optimizations. These approaches convert the CCP into an expectation-constrained program (ECP), which is simpler to solve than the original CCP. To estimate the expectation using a finite set of observed data, we adopt a distributionally robust optimization (DRO) approach. The underlying DRO can be approximated as a robust data-driven optimization that provides a probabilistic under-approximation to the original ECP, where the probability depends on the number of samples. Therefore, under feasibility, the original STL constraints are satisfied with two layers of designed confidence: the confidence of the chance constraint and the confidence of the approximated data-driven optimization, which depends on the number of samples. We then provide details on solving the resulting robust data-driven optimization numerically. Finally, we compare the two proposed approaches through case studies.