Evolutionary Algorithm and Fuzzy C -Means Clustering Based

Evolutionary Algorithm and

Fuzzy C-Means Clustering Based

Order Reduction of Discrete

Time Interval Systems Ch. Naga Sandhya1

, *M. Siva Kumar2

1 Department of Electrical and Electronics

Engineering, Gudlavalleru Engineering college,

Gudlavalleru, India. 2Department of Electrical and Electronics

Engineering, Gudlavalleru Engineering college,

Gudlavalleru, India.

E-mail:

[email protected],shivamangipudi

[email protected]

Abstract

This paper deals with a mixed method for model

order reduction of linear discrete time interval

systems. This method is based on Particle Swarm

Optimization; Fuzzy C means Clustering and

Kharitonov’s theorem. The reduced order interval

denominator coefficients are determined by using

Fuzzy C means Clustering technique while

Particle Swarm Optimization algorithm is used for

obtaining numerator coefficients by minimizing

ISE. This method is always generated to a stable

reduced order interval system if the original higher

order system is stable system. The proposed

method has to be applied to a typical numerical

example available in the literature. The results are

comparing with the results are determined by

using some other familiar methods.

Keywords: Interval system, Model order reduction,

Particle Swarm Optimization, Fuzzy C means Clustering,

Kharitonov’s theorem, Integral Square Error.

International Journal of Pure and Applied MathematicsVolume 114 No. 8 2017, 309-319ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu

309

1. Introduction Higher order model systems are complicate to handle due

to the computational complexities and they are also very

difficult to be used in real time problems. So it is

desirable that the higher model is to be replaced with

lower order model. Many number of methods for model

order reductions are available in literature [1]. Even

though several methods available, no approach gives the

best result for all systems. In literature, a very powerful

method which involves some simple algebraic calculations

i.e. moments matching continued fraction, and Pade

approximation [2], [3] are available. In some cases, Pade

approximant may turn out to be unstable even though the

original system is stable.

Routh approximation based methods are available

[2],[3],[6],[8] for continuous-time systems and for discrete-

time systems [4],[5],[7].There are more methods are

available for the order reduction of continuous time

systems, but very few methods extended to the discrete-

time systems. The methods for discrete-time systems are

classified into two types. The first one it consists of already

existed in continuous-time algorithm, G(z) into another

one G1(w), by using bilinear transformation z= (1+w)/(1-

w)[1]. Then one of known techniques for continuous-time

systems is applied to determine a reduced approximant

R1(w) for G1(w). Finally the inverse transformation w=

φ(z) converts R1(w) to the required approximant R(z).

Many systems the coefficients are constants but

uncertain within a finite range. Such systems are

classified as interval systems. Some methods like [9-15]

are proposed for order reduction of interval systems in

both continuous and discrete time systems. In [10],a

method for interval systems is γ − δ Routh Approximation

is proposed. A modified Routh approximation method is

presented for obtaining stable reduced models of

continuous-time systems. The reduced models give good

approximates of the initial transient response and the

steady-state response of the original system. In [11-16]

Kharitonov’s theorem is used for reduction technique of

linear interval systems Routh Approximation is used to be

generates stable reduced interval model.

In this paper, a proposed method for model order reduction

of Discrete time interval systems based up on Particle

Swarm Optimization (PSO) and Fuzzy C means Clustering

technique. The reduced interval model denominator

polynomials obtained by Fuzzy C means clustering method

International Journal of Pure and Applied Mathematics Special Issue

310

while the numerator coefficients are obtained by using

minimization of ISE by PSO. In this method produces a

stable reduced model if the original higher order system is

to be stable.

The proposed paper is to be organized as, section 2 consists

of a problem formulation, section 3 consists of proposed

method, section 4 consists of numerical example and

simulation results and section 5 consists of conclusion and

references.

3. Determination of Reduced Order Denominator polynomials:

For first Kharitonov's transfer function, Fuzzy C-Means

technique is applied to determine reduced order

polynomials.Fuzzy clustering method is a powerful

unsupervised method for analysis of data and construction

models. In many situations, fuzzy c means clustering is

natural compared to hard clustering. The boundaries

between many classes are not forced to fully belong to one

of the classes, but rather then the membership degrees

between 0 and 1 indicating the partial membership. Fuzzy

c-means algorithm is most widely used. The sum of

degrees of belongingness of the data point to all clusters is

always equal to unity: uij

ci=1 = 1 (1)

The cost function for FCM is generalization is

J= Jici=1 = um

ijnj=1

ci=1 d2

ij (2)

Where uij is between 0 and 1, ci is the cluster center of

fuzzy group i; dij = 𝑐𝑖 − 𝑥𝑗 is the Euclidean distance

between the 𝑖𝑡ℎ cluster center and the 𝑗𝑡ℎ data point; and

m Є [1,∞]. The necessary conditions for (9) to reach to its

minimum are

𝑐𝑖= 𝑢𝑖𝑗

𝑚𝑋𝑗𝑛𝑗=1

𝑢𝑖𝑗𝑚𝑛

𝑗=1

(3)

and 𝑢𝑖𝑗 =1

[ 𝑐𝑖−𝑋𝑗

𝑐𝑘−𝑋𝑗

2(𝑚−1)

𝑐𝑘=1

(4)

This algorithm works by assuming the membership to

each data point according to each cluster centers. The

distance between the cluster center and data point should

be more then the data nearer to the cluster center more is

its membership towards the particular cluster center.

Clearly, combination of membership of each data point

should be equal to one.


311

Fig 1 FCM algorithm

4 Determination of Reduced Order numerator polynomial:

PSO is an evolutionary algorithm that can be used

for solving the nonlinear equations. It is a kind of swarm

intelligence that is based on social-psychological principles

and provides insights into social behaviour, as well as

contributing to engineering applications. In PSO, the

velocity and position are randomly chosen for a set of

particles. During the start, the initial position is taken as

the best position and the velocity is updated.

The main purpose of this optimization method is

a) A global optimum for the nonlinear system may be

found,

b) It can produce a different number of substitute

solutions,

c) There are no mathematical limitations on the

formulation of the problem,

d) Comparatively very simple in execution and

e) Numerically strong.

In Table 1, the typical parameters for PSO optimization

routines, used in the present study are given.

TABLE1

Typical parameters used by PSO

Name Value(type)

Number of generations 100

Population size 60

Maximum Particle velocity 2

Epoch 100

Termination method Maximum Generation


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5. Numerical Example: Consider a higher order discrete time interval system

G(z) =

2.6 , 3 + 3.4 , 3.55 𝑧+ 3.15 , 3.45 𝑧2+ 2.05 , 2.15 𝑧3

+ 1.1 , 1.25 𝑧4+[0.85 , 0.95]𝑧5

2.2 , 2.25 + 1.3 , 1.6 𝑧+ 1.1 , 1.2 𝑧2+ 5.3 , 5.6 𝑧3+

6.2 , 6.65 𝑧4+ 8.1 , 8.45 𝑧5+[10.15 , 10.35]𝑧6

With the help of Kharitonov's theorem, the higher order interval

system can be represented as four Kharitonov's higher order transfer

functions are given as

𝐺61 𝑧 =

0.85𝑧5+1.1𝑧4+2.15𝑧3+3.45𝑧2+3.4𝑧+2.6

10.35𝑧6+8.1𝑧5+6.2𝑧4+5.6𝑧3+1.2𝑧2+1.3𝑧+2.2

𝐺62 𝑧 =

0.95𝑧5+1.1𝑧4+2.05𝑧3+3.45𝑧2+3.55𝑧+2.6

10.35𝑧6+8.45𝑧5+6.2𝑧4+5.3𝑧3+1.2𝑧2+1.6𝑧+2.2

𝐺63 𝑧 =

0.85𝑧5+1.25𝑧4+2.15𝑧3+3.15𝑧2+3.4𝑧+3

10.15𝑧6+8.1𝑧5+6.65𝑧4+5.6𝑧3+1.1𝑧2+1.3𝑧+2.25

𝐺64 𝑧 =

0.95𝑧5+1.25𝑧4+2.05𝑧3+3.15𝑧2+3.55𝑧+3

10.15𝑧6+8.45𝑧5+6.65𝑧4+5.3𝑧3+1.1𝑧2+1.6𝑧+2.25

Step 1:

Consider the first Kharitonov's transfer function

𝐺61 𝑧 =

0.85𝑧5+1.1𝑧4+2.15𝑧3+3.45𝑧2+3.4𝑧+2.6

10.35𝑧6+8.1𝑧5+6.2𝑧4+5.6𝑧3+1.2𝑧2+1.3𝑧+2.2

(4.6)

The poles of 𝐺61 𝑧 are

𝜆1 = -0.0773+0.8618i 𝜆4 = -0.7546-0.2622i

𝜆2 = -0.0773-0.8618i 𝜆5 = 0.4406+0.5008i

𝜆3 = -0.7546+0.2622i 𝜆6 = 0.4406-0.5008i

The second order reduced model is desired to be approximated for

the system as

𝐺𝑟1 𝑧 =

𝑏2𝑧2+𝑏1𝑧+𝑏0

𝑎2𝑧2+𝑎1𝑧+𝑎0

By using FCM algorithm the poles of the reduced order system are

found to be

-0.241567± 0.5475i and the denominator of the ROM is obtained

as

D2(s) = 𝑧2 + 0.4831𝑧 + 0.3581

The reduced order model is:𝐺21 𝑧 =

𝑏12𝑧2+𝑏11𝑧+𝑏10

𝑧2+0.4831𝑧+0.3581

Repeat the FCM algorithm for remaining Kharitonov’s transfer

function the reduced order models are:

𝐺22 𝑧 =

𝑏22𝑧2+𝑏21𝑧+𝑏20

𝑧2+0.5412𝑧+0.3692 ;𝐺2

3 𝑧 =𝑏32𝑧

2+𝑏31𝑧+𝑏30

𝑧2+0.4936𝑧+0.3689;𝐺2

4 𝑧 =

𝑏42𝑧2+𝑏41𝑧+𝑏40

𝑧2+0.5484𝑧+0.3793


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Step2 :

The reduced order numerator coefficients are determined by the

minimization of ISE using PSO algorithm for 4 Kharitonov’s

transfer functions and there given as

𝑁21 𝑧 = 0.05534 𝑧2 +0.3731z+0.2846; 𝑁2

2 𝑧 =

0.05348 𝑧2 +0.4874z+0.2012

𝑁23 𝑧 = 0.0212 𝑍2 +0.4095z+0.3021 ;𝑁2

4 𝑧 =

0.0211𝑍2+0.4157z+0.3191

The four reduced order Kharitonov’s transfer functions are:

𝐺21 𝑧 =

0.05534𝑧2+0.3731𝑧+0.2846

𝑧2+0.4831𝑧+0.3581; 𝐺2

2 𝑧 =

0.05348𝑧2+0.4874𝑧+0.2012

𝑧2+0.5412𝑧+0.3692

𝐺23 𝑧 =

0.0212𝑧2+0.4095𝑧+0.302

𝑧2+0.4936𝑧+0.3689 ;𝐺2

4 𝑧 =

0.0211𝑧2+0.4157𝑧+0.3191

𝑧2+0.5484𝑧+0.3793

Therefore, the reduced order interval model is obtained as

G2(z) =

0.0211 , 0.05534 𝑧2+ 0.3731 , 0.4913 𝑧

+[0.2017 0.3191]

1,1 𝑧2+ 0.4831,0.5484 𝑧+[0.3581, 0.3793]

Comparing the proposed model with reduced order model obtained

by the method given in [9]

G2(z) =

0.02285 , 0.0791 𝑧2+ 0.3665, 0.4956 𝑧

+[0.2067 ,0.2999]

1,1 𝑧2+ 0.4286,0.712 𝑧+[0.2478, 0.399]

Fig. 3: Step Response comparison for 1st Kharitonov’s TF


314

Fig. 4: Step Response comparison for 2nd

Kharitonov’s TF

Fig. 5: Step Response comparison for 3rd

Kharitonov’s TF

Fig. 6: Step Response comparison for 3rd

Kharitonov’s TF

The step responses of the four reduced order Kharitonov's models

are compared with the method given in [9]. It has been observed


315

from figure 3 to figure 6 that the step response of four Kharitonov's

transfer functions of higher order time interval system and the

respective reduced models obtained by the present method are

closely matching than the method given in [9].This method also

retains the stability of the reduced order interval model since the

original system is stable. It is observed that the reduced order model

obtained by the present method has better matching of transient and

steady state response than the method in [9]. From Table 1 it is

observed that the ISEs of lower and upper bounds of reduced

interval model determined by the present method are less than the

method given in [9].

Table 1

Comparison of ISEs for reduced interval system model

MOR ISE for lower bound ISE for upper bound

Proposed Method 0.292 0.3061

Pade & Pole clustering

Method[11] 0.4052 0.3545

5. Conclusion In this proposed paper, the model order reduction

for a linear discrete time interval system is done by

combining the pole clustering technique and an

evolutionary algorithm based soft computing technique i.e.

PSO algorithm. The reduced model is determined by using

the Kharitonov’s theorem and Fuzzy C-means Clustering

algorithm for denominator coefficients, while numerator

coefficients are determined by using PSO algorithm. The

main objective to find the numerator is minimization of

ISE. The use of interval arithmetic has the possibility of

producing an unstable reduced model. To avoid this

problem, Kharitonov’s theorem is used to make reduced

interval models robustly stable. It is observed that, since

the original discrete time interval system is stable the

reduced order polynomials are also stable, has better

matching response. Therefore, the original higher order

interval system is reduced to second order interval model

by minimizing the ISE.

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Evolutionary Algorithm and Fuzzy C -Means Clustering Based