Optimization Algorithms for System Analysis and Identification

Abstract: Optimization is a powerful and frequently used tool in many fields of research. In this thesis two relevant and important problems from robust control system analysis and system identification are solved using optimization algorithms.Many of the most important examples of optimization in control and signal processing applications involve semidefinite programming with linear matrix inequality constraints derived from the Kalman-Yakubovich-Popov lemma. For realistic examples these semidefinite programs have a huge number of variables making them intractable for general purpose solvers. Three customized algorithms for this class of optimization problems are presented and compared to each other. Preprocessing of the semidefinite program that may improve numerical issues are discussed. This preprocessing also makes it possible to relax some assumptions usually made on the semidefinite program. Moreover, it is shown how to use the algorithms for other stability regions than the left half plane.Even though missing data is quite common in many control and signal processing applications, most system identification algorithms do not address this phenomenon in a good way. This often results in parameter estimates with a large bias. In this thesis the maximum likelihood criterion for identication of Autoregressive models with an exogenous signal subject to missing data is investigated. Two algorithms for identifying the models are presented and are compared to the expectation maximization algorithm. From optimality conditions is computed estimates of the asymptotic variance of the parameter estimates. In addition, it is discussed how a criterion equivalent to the maximum likelihood criterion opens up the possibility to apply a wide range of other optimization algorithms to the estimation problem. It is also shown what property of the data it is that determines why one model is more likely to have produced the data than another. Finally, the multiple optima problem is addressed.