Iterative methods for solving large sparse Lyapunov equations and application to model reduction of index 1 differential-algebraic-equations

Abstract

To implement the balancing based model reduction of large-scale dynamical systems we need to compute the low-rank (controllability and observability) Gramian factors by solving Lyapunov equations. In recent time, Rational Krylov Subspace Method (RKSM) is considered as one of the efficient methods for solving the Lyapunov equations of large-scale sparse dynamical systems. The method is well established for solving the Lyapunov equations of the standard or generalized state space systems. In this paper, we develop algorithms for solving the Lyapunov equations for large-sparse structured descriptor system of index-1. The resulting algorithm is applied for the balancing based model reduction of large sparse power system model. Numerical results are presented to show the efficiency and capability of the proposed algorithm.

Computationally Efficient Optimal Control for Unstable Power System Models

“…, μ (i) j 􏽮 􏽯 from the eigenpair defined in (54). ( 5) while ‖W (i) j ‖ 2 > ‖τW (i) 0 ‖ 2 do (6) j � j + 1 (7) Solve the linear system (56) for V (i) j . (8) if Im(μ i ) � 0 then…”

“…e optimal feedback matrix K o � B T XE can be achieved by the feasible solution X of Riccati equation (5). en, applying A s � A − BK o , optimally stabilized LTI continuous-time system can be written as (7). To preserve the structure of the system, it needs to back to the original Input: E, A, C, τ (tolerance), i max (number of iterations), and shift parameters μ j 􏽮 􏽯…”

“…In most of the state-space representations, the direct transmission remains absent and because of that D � 0. Because E is singular (i.e., det(E) � 0), system (2) is called the descriptor system [7,8].…”

SVD-Krylov based Sparsity-preserving Techniques for Riccati-based Feedback Stabilization of Unstable Power System Models

“…The vectors 1 ( ) ∈ ℝ n 1 , 2 ( ) ∈ ℝ n 2 are for state vectors, whereas ( ) ∈ ℝ and ( ) ∈ ℝ represent the input (control) and output vectors, respectively. The sub-matrices 11 and 22 have the full rank [1]- [2]. ( ) needs to be eliminated from the algebraic (second) part of the Equation (1).…”

“…The sub-matrices 11 and 22 have the full rank [1]- [2]. ( ) needs to be eliminated from the algebraic (second) part of the Equation (1). Then the Schur complements of the system (1) can be formed as…”