Regret and Risk Optimal Design in Control

Abstract: Engineering sciences deal with the problem of optimal design in the face of uncertainty. In particular, control engineering is concerned about designing policies/laws/algorithms that sequentially take decisions given unreliable data.This thesis addresses two particular instances of optimal sequential decision making for two different problems. The first problem is known as the H∞-norm (or l2-gain, for LTI systems) estimation problem, which is a fundamental quantity in control design through the small gain theorem. Given an unknown system, the goal is to find the maximum l2-gain which, in a model-free approach, involves solving a sequential input design problem. The H∞-norm estimation problem (or simply "gain estimation problem") is cast as the composition of a multi-armed bandit problem generating data, and an optimal estimation problem given that data. It is shown that the separation of the gain estimation problem into these two sub-problems is optimal in a mean-square sense, as the expected estimation error asymptotically matches the Cramér-Rao lower bound.In the second part of the thesis, we address the problem of risk-coherent optimal control design for disturbance rejection under uncertainty, where optimality is studied from an H2 and an H∞ sense. We consider a parametric model for the plant and the noise spectrum, where the modeling error between the model and the real system is uncertain. This uncertainty is condensed in a probability density function over the different realizations of the parameters defining the model. We use this information to design a controller that minimizes the risk of falling into poor closed-loop performance within a financial theory of risk framework. A systematic approach for the design of H2- and H∞-optimal controllers is proposed in terms of a quadratically-constrained linear program and a semi-definite program, respectively. An interesting application to H2-optimal design under covert attacks is also developed.