Thuật toán giảm bậc bảo toàn điểm cực dựa trên phân tích Schur

Kien Ngoc Vu et al Journal of SCIENCE and TECHNOLOGY 127(13): 101 - 106

101

MODEL REDUCTION IN SCHUR BASIS WITH POLE RETENTION

Kien Ngoc Vu1,*, Du Huy Dao1

, Cong Huu Nguyen2

1University of Technology – TNU; 2Thai Nguyen University

ABSTRACT

Model order reduction is a research direction which is more interested scientists in recent years.

There have been many order reduction algorithm introduced to many different approaches in

which retaining the important poles of the original system in the reduced root system is the right

approach and has many advantages.

This paper presents a new model order reduction algorithm, the order reduction algorithm based on

Schur analysis, based on the idea of keeping the important poles of the original system in the order

reduction process. This algorithm transforms matrix A of the higher-order original system to upper

- triangle matrix on which the poles are arranged in descending important properties on the main

diagonal of the upper – triangle matrix. The illustrative examples show the correctness of the

model order algorithm.

Keywordss: Model order reduction, Schur analysis, important poles.

INTRODUCTION*

In the previous paper [1] the authors

introduce balanced truncation algorithms. The

truncation [1] for the system is based on

Hankel singular value (It removes the state

corresponding to Hankel small singular

values) leading to important climax points of

the root system without conserved reduced

order system. However, important climax

poles (dominant poles) are invariant in the

real system, so it should be preserved in

process of order reduction. Therefore, this

paper, we introduce a new algorithm, the

order reduction algorithm based on Schur

analysis, based on the idea of keeping the

important poles of the original system in the

order reduction process. The illustrative

examples shows the correctness of the model

order algorithm.

MODEL ORDER REDUCTION ALGORITHM

Problem of order reduction model

A linear system is given with continuous-time

constant parameters available multiple-inputs

multiple-outputs described in state space by

the following equations:

x=Ax+Bu

y=Cx

(1)

* Tel: 0965869293; Email: [email protected]

In which, x  Rn

, u  Rp, y  Rq

, A  Rnxn, B

 Rnxp, C  Rqxn.

The goal of the order reduction problem with

model described by (1) is to find models

described by systems of equations:

r r r r

r r r

x =A x +B u

y =C x

(2)

In which, xr  Rr

, u  Rp, yrRq

, Ar  Rrxr, Br

 Rrxp, Cr  Rqxr, với r  n;

So that the model described by (2) can be

replaced by the model described in (1) to

apply in analysis, design and control system.

Model reduction in Schur basis with pole

retention

Model reduction in Schur basis with pole

retention was developed by the researched

team based on truncation technique and

analysis of Schur.

Truncation technique is a method of simple

order reduction. In which, the main idea of it

can be divided into 2 steps: the step 1 is to

convert the high original system to the

equivalent system with a non-singular

transformation in state space. The step 2 is to

delete rows and columns of similar systems to

generate the reduced order system. The two

most typical algorithms for truncation