Numerical methods for large scale matrix equations with applications in LTI system model reduction

Abstract

LTI (Linear Time Invariant) systems arise frequently in different branches of engineering. This thesis mainly concerns the properties and numerical methods for large scale matrix equations related to LTI systems, the final goal is model reduction. Due to the importance of small to medium scale matrix equations, we firstly made formulation improvements to the two standard direct methods (the Bartels-Stewart's method and the Hammarling's method) for Sylvester and Lyapunov equations. Numerical evidence and flop counts show the better performance of our modified formulations. The low rank solution property of large scale Lyapunov equations is the basis for any algorithm that computes low rank approximate solutions. We study the eigendecay rate of the solution since eigen-decay rate is closely related to the low rank solution property. New eigen-decay rate bounds and estimated rates are established for general nonsymmetric Lyapunov equation. Connections between the solution of Lyapunov equation and some special matrices are discussed, which further reveal different properties of the solution. We also present new bounds on the conditioning of Lyapunov operator. Finally, we develop an AISIAD (Approximate Implicit Subspace Iteration with Alternating Directions) framework for model reduction. Two new approaches within this framework are constructed. The efficiency of these approaches are demonstrated by numerical results.