Efficient solution of lyapunov equation for descriptor system and application to model order reduction

Abstract

In this thesis under the title “Efficient Solution of Lyapunov Equation for Descriptor System and Application to Model Order Reduction ", an efficient solution of Lyapunov equation will be derived and the application of model order reduction to reduce the large system to lower dimensional system will be shown.

Mathematical models of physical systems are extending in engineering fields which can be used for simulation, optimization or control. Structured descriptor systems play important roles in many applications. Such systems are used to analyze properties of the system or simulate the system. However, many of these models are too complex, large and sometime impossible to handle for standard analysis or control system design. Hence, there is a need to reduce the complexity of models preserving the input-output behavior of the original complex model as far as possible. The existing techniques produce complexity for large dimensional state space systems.

In this thesis, the projection based techniques are considered to compute the low rank solutions of the state space systems. Recently, balanced truncation is being considered as a prominent technique for model reduction of linear time invariant systems. The most expensive part of the technique is the numerical solution of two Lyapunov matrix equations. Rational Krylov subspace method is one of the efficient methods for solving the Lyapunov equations of large-scale sparse dynamical systems. The method is well established to compute the low rank solution of the Lyapunov equations for standard state space systems. The main advantage of this solution technique is that the projected Lyapunov equations can be handled easily. We develop algorithms to solve the Lyapunov equations for large sparse structured index-1 descriptor system. The accuracy and suitability of the proposed method is demonstrated through different examples of different orders and the results are compared and discussed. Then resulting Lyapunov solutions have been applied for the balancing based model reduction.

Finally, numerical results are shown to illustrate the efficiency and accuracy of the proposed methods.