Publication
Recent years have witnessed a growing integration of automatic control into a wide range of cyber-physical systems, including electrical grids and autonomous transportation networks. These systems operate in uncertain and dynamic environments, exhibit complex behaviors arising from the intricate interactions between multiple decision-makers, and are subject to both safety and real-time computational constraints. Their automation must, therefore, address the challenges related to these three characteristics. In particular, this involves developing novel control and learning solutions that ensure reliable operation despite unmodeled uncertainty, leverage the availability of large volumes of data to improve closed-loop performance, and maintain computational efficiency.
This thesis contributes to addressing the aforementioned challenges in modern control systems. Departing from classical probabilistic or adversarial uncertainty models and moving towards designing adaptive controllers that adjust to the true disturbance realizations, in the first part of this thesis we study optimal control from the perspective of regret minimization. Focusing on finite-horizon control, we first present a convex optimization approach for synthesizing both the noncausal optimal policy in hindsight and a causal policy that tracks it as closely as possible. We then consider a model predictive control scheme based on the repeated computation of regret-optimal controllers, and establish performance, safety, and stability guarantees for the resulting closed-loop system.
To address the lack of accurate mathematical models describing the dynamics of modern control systems and the uncertainty they are affected by, in the second part of this thesis we focus on data-driven robust control methods. For the case where the dynamics are unknown, we derive suboptimality bounds for safely learning constrained linear quadratic Gaussian regulators from noisy data. Our analysis shows that the suboptimality of the proposed method converges to zero approximately as a linear function of the mismatch between the nominal and the true dynamics, provided that these are sufficiently close. For the case where the probability distribution of the uncertainty is unknown, we present new duality results to reformulate Wasserstein distributionally robust optimal control problems with empirical center distributions and possibly bounded uncertainty supports as semidefinite programs.
Shifting our attention to numerical methods for solving optimization problems, in the third part of this thesis we study iterative optimization algorithms from a dynamical system perspective. Our key contribution is a complete and unconstrained parametrization of algorithms that are convergent for smooth, possibly non-convex, objective functions. This is achieved by viewing convergent algorithms as the superposition of a gradient-descent step and a stable term that can be chosen freely. The resulting framework is directly