# MIMO System Reduction Using Modified Pole Clustering and Genetic Algorithm

Authors: C. B. Vishwakarma; R.  Prasad
Journal: Modelling and Simulation in Engineering
 (2009)

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## Abstract

A new mixed method for reducing the order of the large-scale linear dynamic multi-input-multi-output (MIMO) systems has been presented. In this method, the common denominator polynomial of the reduced-order transfer function matrix is synthesized by using modified pole clustering while the coefficients of the numerator elements are computed by minimizing the integral square error between the time responses of the original and reduced system element using Genetic Algorithm. The modified pole clustering generates more dominant cluster centres than cluster centres obtained by pole clustering technique already available in literature. The proposed algorithm is computer-oriented and comparable in quality. This method guarantees stability of the reduced model if the original high-order system is stable. The algorithm of the proposed method is illustrated with the help of an example and the results are compared with the other well-known reduction techniques.


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## Body

## 1. Introduction

Every physical
system can be translated into mathematical model. The mathematical procedure of
system modeling often leads to comprehensive description of a process in the
form of high-order differential equations which are difficult to use either for
analysis or controller synthesis. It is, therefore, useful, and sometimes
necessary, to find the possibility of finding some equations of the same type
but of lower order that may be considered to adequately reflect the dominant
characteristics of the system under consideration. Some of the reasons for
using reduced-order models of high-order linear systems could be as follows:(i)
to have a better understanding of the system,(ii)
to reduce computational complexity,(iii)
to reduce hardware complexity,(iv)
to make feasible controller design. Several reduction methods are available in literature for
reducing the order of large-scale linear MIMO systems in frequency domain [1–4].
Further, some mixed methods have been suggested by combining the algorithm of
two different reduction methods [5–7].
In spite of having several reduction methods, none always gives the
satisfactory results for all the systems.The optimization-based reduced-order modeling has already
been suggested in the work [8–10] in which the numerator coefficients are
computed by minimizing the integral square error (ISE) between the step
responses of the original and the reduced system while the denominator
polynomial is obtained by using existing order-reduction technique.The Genetic
Algorithm (GA) is a popular optimization technique in the science and
engineering nowadays. This is based on the process of Darwin
’s Theory of Evolution. By starting
with a set of potential solutions and changing them during several iterations,
the GA hopes to converge on the most “fit” solution. The process begins with a
set of potential solutions or chromosomes (usually in the form of bit strings)
that are randomly generated or selected. The entire set of these chromosomes
comprises a population. The chromosomes evolve during several iterations or
generations. New generations (offspring) are generated using the crossover and
mutation technique. Crossover involves splitting two chromosomes and then
combining one half of each chromosome with the other pair. Mutation involves
flipping a single bit of a chromosome. The chromosomes are then evaluated using
a certain fitness criteria and the best ones are kept while the others are
discarded. This process repeats until one chromosome has the best fitness and
thus is taken as the best solution of the problem [11].The authors
suggested a modified pole clustering based on the concept of the clustering
technique [12, 13] in which clusters of the poles of the original high-order
system are formed and then each cluster is replaced by its cluster centre by
using inverse distance measure criterion. The modified pole clustering
technique generates more dominant cluster centre than the cluster centre obtained
by authors [12, 13].In the present
study, the authors proposed a new mixed method for reducing the order of the
linear MIMO systems in which the common denominator polynomial of the reduced
system is obtained by using modified pole clustering while the coefficients of
the numerator element of the reduced model are computed by using GA.


## 2. Reduction Method

Let thenth-order transfer function matrix of the
high-order original system having “p” inputs and “q” outputs be (1)[G(s)]=1Dn(s)[a11(s)a12(s)⋯a1p(s)a21(s)a22(s)⋯a2p(s)⋮⋮⋮⋮aq1(s)aq2(s)⋯aqp(s)], or let  [G(s)]=[gij(s)], i=1,2,…,q;j=1,2,…,p be a q×p transfer matrix.The general form ofgij(s) can be taken as (2)gij(s)=aij(s)Dn(s)=a0+a1s+⋯+an−1sn−1b0+b1s+⋯+bnsn, where i=1,2,…,q;j=1,2,…,p.Let the reduced transfer function matrix of thekth-order having “p” inputs and “q” outputs to be synthesized as (3)[Rk(s)]=1Dk(s)[c11(s)c12(s)⋯c1p(s)c21(s)c22(s)⋯c2p(s)⋮⋮⋮⋮cq1(s)cq2(s)⋯cqp(s)], or let [Rk(s)]=[rij(s)],i=1,2,…,q;j=1,2,…,p  be a q×p transfer matrix.The general form ofrij(s) can be taken as (4)rij(s)=cij(s)Dk(s)=c0+c1s+⋯+ck−1sk−1d0+d1s+⋯+dksk, where i=1,2,…,q;j=1,2,…,p.The objective of this paper is to realize a reduced
transfer matrix of thekth-order in the form of (3) from the original
high-order transfer matrix (1) such that it retains the important features of
the original high-order system [G(s)].Further, the reduction procedure consists of the following
two steps.Step 1.
Determination
of the common denominator polynomialDk(s) for the reduced transfer matrix using modified
pole clustering technique.The brief
algorithm for realizing the denominator polynomial by using the modified pole
clustering is as follows.Let there ber real poles in the ith cluster: p1,p2,…,pr,
where |p1|<|p2|⋯<|pr|,
and then modified cluster centre pei can be obtained by using the algorithm of the modified
pole clustering suggested in this paper.Letm pairs of complex conjugate poles in a jth cluster be (5)[(α1±jβ1),(α2±jβ2),…,(αm±jβm)],  where |α1|<|α2|⋯<|αm|.Now using the
same algorithm separately for real and imaginary parts of the complex conjugate
poles, the modified cluster centre is obtained as(6)Φej=Aej±jBej, where Φ∗ej=Aej+jBej and Φ∘ej=Aej−jBej.An interactive
computer programme has been
developed that automatically finds the modified cluster centre and is given as follows:(1)
letr number of the real poles in a cluster be |p1|<|p2|⋯<|pr|,(2)
setj=1,(3)
find
pole cluster centre bycj=[∑i=1r(−1/|pi|)÷r]−1,(4)
setj=j+1,(5)
now
find a modified cluster centre fromcj=[(−1/|p1|+−1/|cj−1|)÷2]−1,(6)
isj=r?, if
No, then go to step (4) otherwise
go to step (7),(7)
take a modified cluster centre of thekth-cluster as pek=cj.For synthesizing the common denominator
polynomial, one of the following cases may occur.Case 1.
If all the modified cluster centres are real, then
denominator polynomial of thekth-order reduced model can be obtained as (7)Dk(s)=(s−pe1)(s−pe2)⋯(s−pek), where pe1,pe2,…,pek are the
first-, second-,…,kth-modified cluster centre, respectively.Case 2.
If all the modified
cluster centres are complex conjugate, then thekth-order denominator polynomial is taken as (8)Dk(s)=(s−Φ∗e1)(s−Φ∘e1)⋯(s−Φ∗ek/2)(s−Φ∘ek/2).Case 3.
If some cluster centres
are real and some are complex conjugate, then this case holds.Let(k−2) cluster centres are real and one pair of
cluster centre is complex conjugate, then the kth-order denominator can be obtained as (9)Dk(s)=(s−pe1)(s−pe2)⋯(s−pe(k−2))(s−Φ∗e1)(s−Φ∘e1).Step 2.
Determination of the
numeratorcij(s) of the reduced transfer matrix rij(s) using GA.In this paper,
GA has been applied to minimize the objective function “Fij,”  which is an integral square error ISE [9] between
the unit step responses of the elements of the original and the reduced system and
is given by (10)Fij=∫0∞[gij(t)−rij(t)]2dt,i=1,2,…,q;j=1,2,…,p,  where gij(t) and rij(t) are the
unit step responses of the elements of original high-order system [G(s)] and the
reduced-order model [Rk(s)],  respectively. The coefficients cj;1≤j≤k−1 of the numerator element cij(s) are determined with suitable GA parameters.To ensure the
zero steady-state error, the coefficientc0 is obtained as (11)c0=a0b0⋅d0. The
computational flowchart of the GA optimization is shown in Figure 1.Figure 1
Flowchart of genetic algorithm.


## 3. Numerical Example

To demonstrate the accuracy and usefulness of the proposed method,
one numerical example is taken from the literature and the second-order reduced
model is obtained using the suggested algorithm.Consider a sixth-order system [2] having two-input and two-output
described by the transfer matrix: (12)[G(s)]=[2(s+5)(s+1)(s+10)(s+4)(s+2)(s+5)(s+10)(s+1)(s+20)(s+6)(s+2)(s+3)],=1D6(s)[a11(s)a12(s)a21(s)a22(s)], where D(s) is the common denominator polynomial.The poles are −1, −2, −3,  −5, −10, −20. Let the first and
second clusters contain
the poles  −1, −2 and −3, −5, −10, −20, respectively for getting second-order
reduced model.Now using Step1, the modified cluster centres are obtained
as (13)pe1=−1.1428,pe2=−3.1946..Using Case 1, the denominator
polynomial for the second-order reduced model is obtained as (14)D2(s)=(s+1.1428)(s+3.1946)=3.65079+4.3374s+s2. Step 2 of the proposed algorithm is successively applied to
each element of the high-order original MIMO system and the reduced models rij(s) of the [R2(s)] and
then cij(s):1≤i≤2;1≤j≤2 are obtained as (15)c11(s)=3.65079+1.18156s,c12(s)=1.46031+1.04664s,c21(s)=1.82539+0.49819s,c22(s)=3.65079+1.6911s. Therefore, finally the second-order reduced model is obtained
as (16)[R2(s)]=13.65079+4.3374s+s2×[3.65079+1.18156s1.46031+1.04664s1.82539+0.49819s3.65079+1.6911s]. The step responses of the  reduced-order models rij(s) are compared with the original system gij(s) and are shown in Figure 2 and the proposed method is compared
with the other well-known order-reduction techniques available in literature
and are shown in the Table 1 by comparing the integral square error ISE in
between the step responses of the original gij(s) and reduced rij(s)-order  systems. The ISE is given as (17)ISE=∫0∞[gij(t)−rij(t)]2dt, where, i=1,2,j=1,2 and gij(t),rij(t) are the unit step responses of the original
and reduced systems respectively for ith-output with jth-input.Table 1
Comparison of reduction methods.
ISE forrij(i=1,2;j=1,2)Reduction methodr11r12r21r22Proposed method0.0015157.845×  10−50.0002990.004681Parmar et al. [9]0.0144980.0087440.0025380.015741Prasad and Pal [14]0.1364840.0024460.0402910.067902Safonov and Chiang [15]0.5906170.0371290.0073281.066123Prasad et al. [1]0.0306890.0002560.2619630.021683Comparison of the
step responses.
(a)(b)(c)(d)


## 4. Conclusions

A
new mixed method which combines the advantages of the modified pole clustering
and the integral square error minimization by using GA has been presented to
derive stable reduced-order models of the linear time-invariant MIMO
systems.In
this method, the common denominator polynomial of the reduced transfer function
matrix is synthesized using the modified pole clustering, and the coefficients
of the numerator of the reduced model are computed by minimizing the integral square error
between the step responses of the original and reduced system using GA. The
method is simple, rugged, and computer-oriented. The proposed algorithm has
been applied on the system having two inputs and two outputs to get second-order
reduced model. The comparison of the step responses of the original and reduced
system is shown in Figure2 and ISE is calculated between the original and
reduced system and compared in the tabular form as given in Table 1, from which
it is clear that the proposed method is comparable in quality with the other well-known
order-reduction techniques for MIMO systems. The method also preserves the
model stability and avoids any steady-state error between the time responses of
the original and reduced systems. The proposed method may be extended to
unstable systems and is reported elsewhere.



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*Source: 540895-2009-03-26.xml*