Contributions to Model Approximation

This dissertation focuses on the approximation problem of models in the form of linear operators and in the form of polynomial dynamical systems. For approximation of linear operators, Schmidt and Mirsky have shown the existence of an optimal approximant which minimizes the induced Euclidean norm distance between the original operator and all possible lower rank approximant. This result is regarded as an important step in the development of model approximation for dynamical systems. In this thesis, a possibility of extending the result of Schmidt and Mirsky to a general induced norm is discussed. For approximation of dynamical systems, three computational schemes are introduced for several classes of polynomial nonlinear systems. The main contribution of this thesis lies on these three schemes. The first computational scheme is heuristic in nature. The second one is derived based on a reachability approach. These two schemes are mainly to compute a reduced order model for a certain class of polynomial nonlinear systems such that the error model is finite gain L2 stable. The third scheme is an approach to generalize the balanced truncation method of linear systems to a class of polynomial nonlinear systems. The three schemes utilize the power of sum of squares programming which is amenable to computer solution.