# Multiresolutional approach to optimal control

Control design is based on an approximate model of a real life system. In this thesis, a complex dynamical system is modelled using not one but several models of the same system. Based on interest for robust implementation of reduced order controllers in higher order models, we study the regulator problem for a dynamical system expressed as a framework of several models where the reduction process between them is made using singular perturbation theory. This multiresolutional representation portrays the model and cost at different levels of resolution, allowing us to study the correlation between the analytical solutions for the minimization problem at each level and determine cases in which the cost can be minimized without necessarily requiring the exact optimal control. Our main goal is to show that sometimes a small improvement in the reduced order optimal control may improve the minimum cost value for the high order model in a significant way. As result, we find that in a singularly perturbed linear quadratic regulator problem the solution for the high resolution model tends to a solution given for a reduced order model. In addition, for the same class of dynamical systems but accounting for actuator dynamics expressed as the discontinuous nonlinearity sign in their structure, we obtain an improvement over the original minimum cost value when a modified reduced order optimal control law is used as control scheme for the extended model