Robustness in Dynamic Factor Analysis Identification and Stochastic Optimal Control

This dissertation is divided into two main parts, the common thread being the prominent role of entropy-based metrics in the robust identification and control of stochastic models. The first part is concerned with the dynamic factor analysis problem. Factor analysis models boast a long tradition and find natural application in many engineering and scientific disciplines, including, for example, psychology, econometrics, system engineering, machine learning and statistics. In general, the attention for this kind of models is motivated by their effectiveness in complex-data representation. In this part of the thesis, inspired by the previous contributions on robust estimation and robust static factor analysis, we propose a novel approach to deal with the dynamic factor analysis problem for the case of zero-mean Gaussian stochastic processes. To robustly estimate the number of factors, we construct a confidence region centered in a finite sample estimate of the underlying model which contains the true model with a prescribed probability. In this confidence region, we seek for the most parsimonious factor model by solving a convex optimization problem. This paradigm is applied to the identification of moving-average factor models. The obtained result is then generalize to the larger class of autoregressive moving-average factor models by resorting to an iterative technique in which the autoregressive and the moving-average part of the model are identified separately. The second part of the thesis deals with a finite-horizon distributionally robust optimal control problem for linear stochastic uncertain systems. The linear quadratic Gaussian optimal control, which is one of the most fundamental ideas in control theory, suffers from a major disadvantage in that it does not provide systematic means for addressing the issue of robustness. On the other hand, in designing any feedback control law, a fundamental requirement is that of robustness, that is the ability to maintain satisfactory performances even in presence of misspecifications and perturbances in the plant model. In this thesis, a new paradigm is proposed for the robustification of the linear quadratic Gaussian controller against distributional uncertainties on the noise process. Our controller optimizes the closed-loop performances in the worst possible scenario under the constraint that the noise distributional deviation does not exceed a certain threshold limiting the relative entropy pseudo-distance between the actual noise distribution and the nominal one. The main novelty is that the bounds on the distributional deviation can be arbitrarily distributed along the whole disturbance trajectory.