Numerical study on continuous algebraic riccati equation arising from large-scale sparse descriptor systems

Abstract

The mathematical models derived from the physical models are the pivot ingredient in science and engineering, especially, in the control theory. Most of the physical models have a large number of components with critical combinations and sophisticated designs. These models are infeasible for the computing tools ac-cording to the time dealing and memory allocation. To nd the remedy of current adversity and attain desired execution results, the models are to be approximated as structure-preserving Reduced-Order Models (ROM) and machine-executable designs. Memory allocation and time management are the most eye-catching factors in the simulations of the large-scale sparse Linear Time-Invariant (LTI) systems, especially, the descriptor systems. Numerical techniques can be applied practically to control, stabilize and optimize the physical models.

In this thesis, rstly, the projection-based Rational Krylov Subspace Method (RKSM) has been proposed to compute the solution of Continuous Algebraic Riccati Equations (CARE) governed from large sparse index-1 descriptor systems. Iterative RKSM is not only time saving but also computationally feasible for mem-ory allocation in nding the solution of the CAREs utilizing the Reduced-Order Models (ROM). The novelties of RKSM are sparsity preserving techniques and the implementation of time convenient recursive adaptive shift parameters. Secondly, the machine-independent Alternating Direction Implicit (ADI) technique based nested iterative Kleinman-Newton (KN) method has been modi ed and adjusted to solve the CAREs governed from large sparse index-1 descriptor systems. Then compare results achieved by the Kleinman-Newton method with that of using the RKSM.

The objective has been mainly focused on nding optimal feedback matrix for Riccati based feedback stabilization for the unstable index-1 descriptor systems applying the proposed methods. The applicability and adaptability of the pro-posed methods have been justified through the power system models and their transient behaviors have been analyzed.

Finally, numerical results have been shown both in tabular and graphical form to verify the robustness and accuracy of the proposed methods In addition, their comparative analysis for the target models has been illustrated in detail.