Robust and Stochastic Control of Uncertain Systems - From Scenario Optimization to Adjustable Uncertainty Sets

The main theme of this thesis is the development of methods for synthesizing controllers for constrained uncertain systems. Generally speaking, two control paradigms exist for addressing uncertain systems: robust control and stochastic control. Strategies based on robust control treat all elements within a given uncertainty set equally, and aim at satisfying a given performance criterion (e.g., stability and constraint satisfaction) for all possible uncertainty realizations. In contrast, stochastic control strategies assume a probabilistic description of the uncertainty that allows us to optimize the performance of a controller towards events with a high probability of occurrence, while neglecting unlikely events. Synthesizing robust and stochastic controllers for uncertain systems is often more challenging than for deterministic systems; in many cases, approximations must be made to achieve computational tractability. The challenge to efficiently design robust and stochastic controllers motivates the work of this thesis and defines its two parts: the first part is concerned with methods to synthesize stochastic controllers, while the second part of this thesis addresses a novel class of robust control problems. In the first part, we study randomized algorithms as a tool for synthesizing stochastic controllers. Randomized algorithms have initially been introduced to the control community in the 1990s for addressing NP-hard problems associated to the analysis and control synthesis of uncertain stochastic systems, but have soon developed into a research area of their own right. The key challenge in randomized algorithms is to establish the appropriate sample size such that the controller, which is designed based on these “seen” samples only, also behaves well for “unseen” samples (“generalization property”). Determining the appropriate sample size is important in practice since sample sizes that are too large unnecessarily complicate the control synthesis problem, while sample sizes that are too small exhibit bad generalization properties and lead to poor control performance. Unfortunately, the sample sizes required by existing methods can, at times, be prohibitively large for control synthesis tasks. As the first contribution of this thesis, we address the sample size issue by leveraging recent breakthroughs in scenario-based optimization. Specifically, we show that by exploiting structure present in common control synthesis problems, the sample sizes can be significantly reduced. We embody our results in a general framework that allows us to recognize and exploit such problem structure. The efficacy of the framework is demonstrated on stochastic Model Predictive Control problems, for which we derive sample sizes that are orders of magnitude smaller than those provided by existing methods. This not only dramatically reduces computational