An Adaptive Rational Block Lanczos-Type Algorithm for Model Reduction of Large Scale Dynamical Systems

Abstract

Mathematical modeling of complex electrical systems, has led us to linear mathematical models of higher order. Consequently, it is difficult to analyse and to design a control strategy of these systems. The order reduction is an important and effective tool to facilitate the handling and designing of a control strategy. In this paper we present, firstly, a reduction method which is based on the Krylov subspace and Lyapunov techniques, that we call Lyapunov-Global-Lanczos. This method minimizes the H∞ norm error, absolute error and preserves the stability of the reduced system. It also provides a better reduced system of order 1, with closer behaviour to the original system. This first order system is used to design PI (Proportional-Integral) controller. Secondly, we implement an adaptive digital PI controller in a microcontroller. It calculates the PI parameters in real time, referring to the error between the desired and measured outputs and the initial values of PI controller, that were determined from the first order system. Two simulation examples and a real time experimentation are presented to show the effectiveness of the proposed algorithms.

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]):…”

A matrix rational Lanczos method for model reduction in large‐scale first‐ and second‐order dynamical systems

“…Moreover, the output should be close to the output y of the original system, which means that the error should be small for an appropriate norm. The associated low‐order transfer function is denoted by There exist various model reduction approaches for MIMO state space systems such as balanced truncation, moment matching approximations, and the optimal Hankel norm approximation . A popular model reduction technique for large‐scale systems is the moment matching method considered first in Gallivan et al The main idea of this approach is to project the original system onto Krylov subspaces computed by Arnoldi or Lanczos processes for SISO and MIMO systems; see other works and the references therein.…”

“…Unfortunately, model reduction methods using a single frequency such as partial realization and Padé approximation tend to create reduced‐order models that poorly approximate low‐frequency dynamics. To overcome this problem, multipoint Padé approximants (denoted also multipoint rational interpolants) are proposed for approximating Equation using rational (or block rational) Krylov methods for SISO and MIMO systems; see other works . By multi‐point Padé approximation, we mean that the reduced system matches the moments of the original system at multiple interpolation points, these shifts have to be appropriately chosen to guarantee a good convergence of the process; see or works …”

“…There exist various model reduction approaches for MIMO state space systems such as balanced truncation, 1-3 moment matching approximations, [4][5][6][7] and the optimal Hankel norm approximation. 8 A popular model reduction technique for large-scale systems is the moment matching method considered first in Gallivan et al 9 The main idea of this approach is to project the original system onto Krylov subspaces computed by Arnoldi or Lanczos processes for SISO and MIMO systems; see other works [10][11][12][13][14][15] and the references therein.…”

“…To overcome this problem, multipoint Padé approximants (denoted also multipoint rational interpolants) are proposed for approximating Equation 1 using rational (or block rational) Krylov methods for SISO and MIMO systems; see other works. [5][6][7][16][17][18][19][20][21] By multi-point Padé approximation, we mean that the reduced system matches the moments of the original system at multiple interpolation points, these shifts have to be appropriately chosen to guarantee a good convergence of the process; see or works. 6,7,[22][23][24][25] The paper is organized as follows.…”