Balanced truncation-rational Krylov methods for model reduction in large scale dynamical systems

Abstract

In this paper, we consider the balanced truncation method for model reductions in large-scale linear and time-independent dynamical systems with multi-inputs and multioutputs. The method is based on the solutions of two large coupled Lyapunov matrix equations when the system is stable or on the computation of stabilizing positive and semi-definite solutions of some continuous-time algebraic Riccati equations when the dynamical system is not stable. Using the rational block Arnoldi, we show how to compute approximate solutions to these large Lyapunov or algebraic Riccati equations. The obtained approximate solutions are given in a factored form and used to build the reduced order model. We give some theoretical results and present numerical examples with some benchmark models.

Computational methods for model reduction in large-scale MIMO dynamical systems

“…To remedy this drawback we can apply implicit restarts techniques [8,70] • The obtained reduced-order model tends to approximate high frequencies. This problem can be solved by using Rational Krylov approaches [1,15,17,50,51,52,69]. • For some problems we have a loss of orthogonality and one need to use re-orthogonalisations.…”

“…, V m } of the block Krylov subspace K b m (A, V ), see [137] while an extended block Arnoldi algorithm was developed in [86,144] to obtain such a basis for the extended block Krylov subspace K eb m (A, V ). For the rational block Krylov subspace K rb m (A, V ), a block version of the classical rational Arnoldi was recently developed in [1]. Next we show how to apply these block methods for solving large-scale Lyapunov matrix equations.…”

“…. , s m , we refer to [1,50,51,69,156,157]. We notice that we can also define a rational version of the block Lanczos method [17,18,70].…”

“…For small-to-medium scale problems, Lyapunov balancing can be implemented efficiently. However, for large-scale settings, exact balancing is expensive to implement because it requires dense matrix factorizations and results in a computational complexity of O(n 3 ) and a storage requirement of O(n 2 ), see [1,8,21,72]. For large problems, direct methods could not be applied and then Krylovbased [2,19,56,88,89,94,98,104,25] or ADI-based methods [21,23,130] are required to compute these Gramians that are given in factored forms which allows to save memory.…”

Lyapunov-Global-Lanczos Algorithm for Model Order Reduction & Adaptive PI Controller of Large Scale Electrical Systems

“…Where m and n are the order of reduced and original system, respectively. In table 2, we report the computational complexity of the proposed algorithm Lyapunov-Global-Lanczos (Lyap-GL) compared to selected state-of-art algorithms (Global Lanczos (GL) [18], [21], Lanczos [9], [15], [22], Rational Lanczos (RL) [21], [23], Arnodli (Ar) [9], [24], Rational Arnoldi (RA) [9], [14], [19], Balanced Truncation (BTR) [9], [25]):…”