Distributionally Robust Model Predictive Control for Safety-Critical Systems: with Applications in Autonomous Driving

The availability of reliable methods for data-driven, optimal decision making under uncertainty is of central importance in several fields of study, including operations research, control, machine learning, and many more specialized application domains. Current (black-box) machine learning techniques, despite their many successes, still provide very few guarantees regarding correctness and safety, impeding their adoption for many high-stakes and safety-critical applications. This drawback can be partially mitigated by further increases in computational power and dataset sizes, but nevertheless, such a remedy is insufficient for applications akin to robotics, where computational hardware or availability of informative data may be limited. This thesis aims to address this shortcoming by developing novel methodologies for data-driven decision-making and control, which are applicable in online settings, and which provide guarantees regarding performance and constraint satisfaction for finite sample sizes. To this end, distributionally robust optimization (DRO) serves as our main tool, as it provides many of these desirable guarantees by design. Furthermore, its conceptual simplicity makes it a versatile approach for many specific problems. In particular, we highlight the following contributions: (i) We present distributionally robust techniques for synthesis of stabilizing controllers for Markov jump linear systems, where the distribution of the switching process is unknown; (ii) We develop a theoretical framework for distributionally robust model predictive control of general (nonlinear) Markov jump systems (also referred to as Markov switching systems). This gives rise to multi-stage risk-averse optimization problems, including nested risk measures in both the costs and the constraints; (iii) In order to efficiently solve these problems, we propose tractable reformulations and a tailored, massively parallellisable solver; (iv) We propose a novel DRO method, called Cost-Aware Distributionally Robust Optimization (Cadro), which, by exploiting the structure of the cost function in the design of its ambiguity set, results in less conservative solutions to data-driven DRO problems, while retaining  the same guarantees; and (v) We apply our developments to case studies within automated driving, and demonstrate empirically the effects of relevant design trade-offs in these examples.