H2 optimal model reduction for linear dynamic systems and the solution of multiparameter matrix pencil problems.

Abstract

This thesis mainly discusses three research questions. Firstly, the optimal H2 model reduction problem for SISO linear dynamic systems is considered. Then, the decoupled multiparameter matrix pencil problem is studied. Finally, a class of structured multiparameter matrix pencil problem is explored. The H2 optimal model reduction has received considerable attention due to its importance in dynamic system simulation and control. Recent results have combined the various solution approaches into a unified interpolation based framework that provided first order necessary conditions for local optimality of the H2 norm and resulted in some iterative numerical algorithms based on Newton updates that converge to local solutions. In this thesis, a novel reformulation of the first order necessary conditions for local optimality of the H2 norm is given in terms of finding all fixed points transformation of the reduced order model’s system matrix. The fixed points of the transformation are related to the optimal interpolation points of the original transfer function. This produces a set of nonlinear equations which are then reformulated as a structured multiparameter matrix pencil problem where the parameters are associated with the required fixed points. A simple procedure is presented to find the associated interpolation points. The multiparameter matrix pencil problem produced by the first order necessary conditions for local optimality of the optimal H2 model reduction problem has not been studied before in the literature. Hence, the second part of this thesis is dedicated to study of the unstructured multiparameter matrix pencil problem. Although a limited work has been done towards coupled multiparameter eigenvalue problems, a thorough research is needed to study the area of decoupled multiparameter matrix pencil problems. An important result pertaining to the structure of the one{parameter matrix pencil problem is highlighted. Furthermore, a new result for the simultaneous and decomposable features of the multiparameter matrix pencil is stated. Moreover, a novel solver for the decoupled multiparameter matrix pencil problem is developed. The structured multiparameter matrix pencil typically results in a singular multiparameter matrix pencil where the matrices are rank-deficient. Unfortunately, the H2 optimal model reduction problem involves the solution of a structured multiparameter matrix pencil problem where the coefficient matrices exhibit an almost block diagonal structure. The third part of this thesis is therefore devoted to study this class of structured multiparameter matrix pencil problems. We rely on the developed tools to construct the optimal H2 reduced order model. We 6 show that the H2 multiparameter problem is in the form of the structured multiparameter matrix pencil problem. A further development to the unstructured problem solver is required to deflate the singular and infinite regular parts associated with the pencil. Once the solutions are obtained, a simple procedure is implemented to identify the optimal solution. Numerical results are given to compare this procedure with the existing methods.