Wasserstein Distributionally Robust Control and Optimization for Autonomous Systems

Abstract

Distributionally robust control (DRC) and optimization (DRO) have recently become popular approaches for handling uncertain distributional information in stochastic systems with accuracy. In this work, we develop novel control methods for autonomous systems in situations where only limited information is available about the uncertainties in system or environment models. To achieve this, we estimate the uncertainty distribution using disturbance samples or state-of-the-art learning techniques and construct an ambiguity set around the nominal distribution. Our ambiguity set contains all distributions whose Wasserstein distance from the nominal one is less than the given radius. We then solve the optimal control problem with respect to the worst-case distribution within the ambiguity set. However, the resulting problem is infinite-dimensional and intractable. Therefore, we apply modern tools from DRO to develop several methods for solving the Wasserstein DRC (WDRC) problem in various settings with different theoretical properties and applications. Our first method proposes a novel safety specification tool, the distributionally robust risk map (DR-risk map), for motion planning and control of a mobile robot in a learning-enabled environment. The DR-risk map reliably assesses the conditional value-at-risk of collision with obstacles whose movements are inferred by Gaussian process regression. Our tool measures the risk under the worst-case distribution within the ambiguity set to account for errors in the inferred distribution. To resolve the intractability, we develop a semidefinite programming (SDP) formulation that provides an upper bound of the risk. We apply the DR-risk map to perform motion planning and control of autonomous systems in learning-enabled environments. Our second method introduces a novel learning-based motion control tool that uses an uncertainty propagation scheme based on an unscented transform to achieve better prediction accuracy and computational efficiency. In addition, this approach replaces the DR-risk constraint for any arbitrary safety loss function with a novel simpler upper bound. The WDRC framework can be applied not only to fully observable systems but also to partially observable systems, which are more realistic. In our next stage, we focus on the WDRC problem for partially observable linear stochastic systems and present a new approximation scheme. This method leverages the Gelbrich bound of the Wasserstein distance to penalize deviations from the nominal distribution. We derive a closed-form expression for the optimal control policy and a tractable SDP problem for the worst-case distribution policy in both finite-horizon and infinite-horizon average-cost settings. Our proposed method features several salient theoretical properties, such as a guaranteed cost property and a probabilistic out-of-sample performance guarantee, demonstrating the distributional robustness of our controller. Furthermore, the resulting controller ensures the closed-loop stability of the mean-state system. Finally, we present a novel distributionally robust differential dynamic programming algorithm for approximately solving the general nonlinear WDRC problem in a tractable and scalable way. It provides a closed-form control policy for nonlinear stochastic systems and therefore is applicable to learning-enabled environments. Our approach features a novel decomposition of the value function and its iterative local-quadratic approximations, making our method tractable and scalable without the need for numerically solving any minimax optimization problems. We analyze and demonstrate the effectiveness of our methods through simulation studies on various systems, ranging from oscillator synchronization to autonomous driving problems. Our contributions enable controllers that can handle distributional uncertainties in both system and environment dynamics, as well as learning outcomes.