Error minimization of reduced models of large-scale systems by solving time and frequency restricted lyapunov equations

Abstract

The usage of mathematical models of physical systems are increasing day by day in various disciplines of science and engineering for simulation, optimization, or control. Descriptor systems is a special kind of formation of physical systems arisen in many practical oriented fields whose dynamics maintain Differential- Algebraic Equations. Such type of systems are originated by finite elements or difference methods which becomes huge and complex to analyze along with the increment of the fineness of the grid resolution. As a result, the necessity of model order reduction comes up in order to minimize the complexity of the models during controlling by preserving the input-output relation of the original large-scale models. However, although reducing the dimension of large-scale models on infinite time and frequency domains has a great theoretical significance, the reduced order models of original large-scale models on restricted time and frequency intervals are more demandable to the analyzers and engineers for practical investigation. This dissertation elaborately discusses the model generations for data extraction to form the large-scale state-space systems and the projection based techniques to calculate the approximate low-rank solutions of the original state-space systems on definite time and frequency intervals. We impose relevant governing equations to create physical as well as data models. Balanced truncation is one of the most notable methods for the reduction of the model dimensions of linear time-invariant systems which requires computing the numerical solutions of two Lyapunov equations, commonly known as Gramians. Among some widely used approaches, Rational Krylov Subspace Method is one of the most effective procedures for finding the Gramians of the Lyapunov equations of the large-scale sparse dynamical systems which has already been developed to compute the low-rank time and frequency indefinite solutions. Besides, it has been also reformed to compute the low-rank approximation of the standard Lyapunov equations on limited time and frequency intervals for small-dense state-space systems. However, in this thesis, we establish algorithms intending to obtain the low-rank solutions of the Lyapunov equations on restricted time and frequency intervals constructed centering around large-scale sparse index-I and index-II descriptor systems by creating no explicit projection. Moreover, we develop algorithm to compute the matrix exponential through power series expansion avoiding typical Schur decomposition in order to retain the sparsity for less memory consumption and analyze the existing algorithms of matrix logarithm computation. For gaining the fastest convergent solutions, we do a comparative analysis of the existing shift parameters essential for solving the linear time-invariant systems. We also develop algorithms for getting low-rank solutions on finite time interval with non-homogeneous initial condition, i.e., non-zero initial value. Although balanced truncation gives the guarantee to preserve the stability of the reduced order models deduced from the stable full models on infinite time and frequency domains, it fails to give the stable time and frequency restricted reduced order models of stable original systems.

Hence, an algorithm is proposed here as a remedy to this problem. At the final stage, the numerical outcomes by applying our proposed algorithms on various types of existing data models including our generated models are exhibited to demonstrate the efficiency and exactness by minimizing the errors on the restricted time and frequency intervals. in addition, the comparative analysis between the domain restricted and unrestricted reduced order models are performed to show that on limited time and frequency intervals, our proposed algorithms give better approximations of the original large-scale sparse systems