Order reduction of linear systems with an improved pole clustering

Article

Order reduction of linear systems with an improved pole clustering

Journal of Vibration and Control 18(12) 1876–1885 ! The Author(s) 2011 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1077546311426592 jvc.sagepub.com

Ramesh Komarasamy1, Nirmalkumar Albhonso2 and Gurusamy Gurusamy2

Abstract A new model order reduction technique for the reduction of the linear time invariant system has been proposed in this paper. An improved clustering algorithm is employed in the proposed method to obtain the reduced order denominator polynomial. The reduced order numerator polynomial is obtained with a simple mathematical calculation as mentioned in the proposed scenario. The validity of the proposed method has been illustrated through some numerical examples. The improved clustering algorithm guaranteed the stability in the reduced model and also preserves the characteristics of the original system in the approximated one. The method is extended to the higher order multivariable system described by its matrix transfer function.

Keywords Dominant pole, ISE, ITAE, and IAE, order reduction, pole clustering, stability Received: 12 March 2010; accepted: 13 June 2011

1. Introduction Model reduction has attracted attention in system modeling and design for the last four decades. This continued interest and the huge number of methods available in the literature reflect the importance of producing a reliable reduced order model for the system analysis and design. Some of the papers were proposed, based on matching of Markov parameters and initial time moments of the original and reduced order systems such as Mittal et al. (2002), Prasad et al. (2003) and Wan Bai-Wu (1981). The concept of retaining the dominant dynamical characteristics of the original system in the reduced model is intuitive and has two appealing advantages: the reduced-order model retains the basic physical features (such as time constants) of the original system; and the stability of the simplified model is guaranteed. These characteristics confer upon the reduced-order models a greater physical meaning. The mode retention methods produce the reduced model such that it matches a certain number of coefficients computed from the original system. Lucas (1983) proposed a method which is an alternative approach for linear system reduction by Pade approximation presented in Lal and Mitra (1974) and Lucas (1978) to allow

retention of dominant modes. It avoids calculation of system time moments and the solution of Pade equations by simply dividing out the unwanted pole factors. This method adjusts the numerator polynomial coefficients based on the original systems pole values. The Routh approximation and the stability equation method are used to guarantee the stability of the reduced model. Research works proposed by Chen et al. (1980a,b), Gupta et al. (2002), Pal et al. (1995) and Shamash (1975) proves the quality of the reduction process. Most of the model order reduction techniques are concerned with preserving stability and matching initial time moments between the full and reduced systems. The stability of the system is preserved by obtaining the reduced order denominator polynomial based on selecting stable poles or using the properties of the Routh table. To preserve the steady state characteristics it is usual either to solve the Pade equations or invert 1 2

Velalar College of Engineering and Technology, Erode, India Bannari Amman Institute of Technology, India

Corresponding author: K Ramesh, Velalar College of Engineering and Technology, Thindal Post, Erode 638012, India Email: [email protected]

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a continued fraction, which yields the reduced numerator. Instead of using a single method to derive the reduced model, nowadays researchers prepare some mixed methods of model simplification for continuous time systems. In Bhagat et al. (2004) method stability preserving methods namely, (g-d) canonical expansion (1968), Gutman’s differentiation method (Gutman et al., 1982) and stability equation method (Chen et al., 1979) have been used to obtain the denominator of the original system and the Lucas factor division method (Lucas, 1983) is used to yield the numerator. The Mihailov criterion has been combined with the Pade and factor division method to obtain the better approximation. In Prasad and Vishwakarma (2009) the Mihailov criterion is combined with the Cauer second form for reducing the order of the large scale SISO systems. Recently evolutionary techniques such as the genetic algorithm and particle swarm optimization are applied to obtain the better approximation. A few methods proposed in Howitt and Luus (1990), Hwang (1984), Lamba et al. (1988), Mittal et al. (2004), Mukherjee and Mishra (1987, 1988), Puri and Lan (1988) and Vilbe and Calvez (1990 use the integral of the square error (ISE) as a performance parameter which produces a reduced order closer to a given higher order system behavior. In these methods, the denominator polynomial is obtaining by using any of the stability preserving criteria like the stability equation method, Mihailov stability criterion and Routh approximation etc. In this proposed method, the reduced order denominator polynomial has been obtained using an improved clustering approach and its corresponding reduced order model was obtained through a simple mathematical process. The clustering method proposed in this paper differs from the existing pole clustering technique by considering the distance of system poles from the first pole in the group clustering process. This process yields a better approximation in the reduction process. The results obtained from this method are compared with the papers which use the evolutionary techniques such as the genetic algorithm and particle swarm optimization. The results, which are highlighted in the following sections, show the validity of the proposed method.

2. Statement of problem Let the higher order transfer function of SISO linear time invariant system be in the form of GðSÞ ¼

a0 þ a1 s þ a2 s2    þ am1 sm1 þ am sm b0 þ b1 s þ b2 s2    þ bn1 sn1 þ bn sn

Where mn.

ð1Þ

m P

¼

j¼0 n P

aj s j ¼ bj s j

NðsÞ DðsÞ

ð2Þ

j¼0

The corresponding reduced order (‘r’) model should be in the form of Gr ðSÞ ¼

d0 þ d1 s þ d2 s2    þ dq1 sq1 þ dq sq e0 þ e1 s þ e2 s2    þ er1 sr1 þ er sr

ð3Þ

Where qr q P

¼

i¼0 r P

di s i ¼ ei s i

Nr ðsÞ Dr ðsÞ

ð4Þ

i¼0

The reduced model retains the important characteristics of the original system and approximates its response as closely as possible for the same type of inputs.

3. Proposed method The proposed model order reduction method consists of two steps: Step 1: Obtain the reduced order denominator polynomial with an improved pole clustering technique. Calculate the ‘n’ number of poles from the given higher order system denominator polynomial. The number of cluster centers to be calculated is equal to the order of the reduced system. The poles are distributed into the cluster center for the calculation such that none of the repeated poles present in the same cluster center. The minimum number of poles distributed per each cluster center is at least one. There is no limitation for the maximum number poles per cluster center. Let k number of poles be available in a cluster center: p1, p2, p3. . .pk. The  poles  are  arranged in a manner such that p1  5 p2  . . . 5 pk . The cluster center for the reduced order model can be obtained by using the following procedure. The procedure described in step 1 is similar to the case of the method proposed by Vishwakarma and Prasad (2009) but the pole cluster calculated in the proposed scenario is based on the dominant pole in that particular cluster center. 1. Let  k  number of poles available   bep1  5 p2  . . . 5 pk , 2. Set L ¼ 1, 3. Find pole as, k  the  P   cluster CL ¼ ½ð1=p1  þ 1=pi  p1 Þ  k1 , i¼2

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Journal of Vibration and Control 18(12)

4. Check for L ¼ k. If yes, then the final cluster center isCC ¼ CL and terminates the process. Otherwise proceed to next step. 5. Set L ¼ L þ 1, 6. The improved cluster center from   CL ¼ ½ð1=p1  þ 1=jCL jÞ  21 . 7. Check for L ¼ k. If no, then go to the step (5). Otherwise go to the next step. 8. Final cluster center isCC ¼ CL . If the system to be reduced has a pole at the origin, then that pole (s ¼ 0) is put into a cluster center as a single pole and the remaining poles are clustered in other groups based on the order to which the system has to be reduced. This process will retain the same pole(s ¼ 0) in the reduced model and remaining cluster centers are obtained through the procedure as described in step-2. The way in which clustering centers are obtained in this paper is similar to the method proposed by Vishwakarma and Prasad (2009) except that the pole cluster obtained in step 3 concentrates on the dominant pole in that particular cluster center. The inverse distance, measured from the dominant pole alone, is considered here. This leads to a good approximation in the reduced model and it is illustrated through various examples for SISO and MIMO systems in the following proceedings. The pole centers obtained through the proposed method are more dominant than the pole centers as obtained from the method proposed by Vishwakarma and Prasad (2009). While calculating the cluster center values, we have the following three cases as in Vishwakarma and Prasad (2009). Case (i). All the denominator poles are real: The corresponding reduced order denominator polynomial can be obtained as, Dr ðsÞ ¼ ðs þ CC1 Þðs þ CC2 Þ . . . ðs þ CCr Þ

ð5Þ

Where CC1 ,CC2 . . .CCr are the improved cluster values required to obtain the reduced order denominator polynomial of order ‘r’, Case (ii). All the poles are complex: Let‘t’ ( ¼ k=2) pairs of complex conjugate poles in a Lth cluster be,

  Dr ðsÞ ¼ ðs þ j1 jÞðs þ j2 jÞ . . . ðs þ j Þ

Where, j1 j 5 j2 j 5 . . . jL j. Apply the proposed algorithm individually for real and imaginary parts to obtain the respective improved cluster centers. The improved cluster center is in the form of ð6Þ

ð7Þ

Where, j ¼ r. The value of ‘r’ is to be of an odd number when odd ordered transfer function is required in the reduced model i.e., odd number of cluster centers were required to obtain the odd ordered reduced model. Case (iii). If some poles are real and some poles are complex in nature, then applying an improved clustering algorithm separately for real and complex terms. Finally obtained improved cluster centers are combined together to get the reduced order denominator polynomial. Step 2: Obtain the numerator polynomial of a reduced system. Equate the given higher order system transfer function with the general form of a reduced system transfer function. The reduced order denominator polynomial obtained from step 1 is utilized here to obtain the unknown values of reduced order system coefficients. a0 þ a1 s þ a2 s2    þ am1 sm1 þ am sm b0 þ b1 s þ b2 s2    þ bn1 sn1 þ bn sn d0 þ d1 s þ d2 s2    þ dq1 sq1 þ dq sq ¼ e0 þ e1 s þ e2 s2    þ er1 sr1 þ er sr

ð8Þ

On cross multiplying the above equation and comparing the same powers of ‘s’ on both sides, we get following (n þ 2) equations. a0 e0 ¼ b0 d0 a0 e1 þ a1 e0 ¼ b0 d1 þ b1 d0 a0 e2 þ a1 e1 þ a2 e0 ¼ b0 d2 þ b1 d1 þ b2 d0 .. . am e r ¼ bn dq On solving the above equations, we can find the unknown coefficients d0, d1. . .dq. The reduced order numerator polynomial in the form of Nr ðsÞ ¼ e0 þ e1 s þ e2 s2    þ er1 sr1 þ er sr

½ð1  j!1 Þ, ð2  j!2 Þ, ð3  j!3 Þ . . . ðt  j!t Þ

j ¼ Aj  jBj :

Where, Aj and Bj is the improved pole cluster values obtained for real and imaginary parts respectively. The corresponding reduced order denominator polynomial can be obtained as,

ð9Þ

While calculating the numerator polynomial coefficients using step 2, it may lead to negative coefficient values when given the higher order system is in minimum phase transfer function form. Unless otherwise, it will give the positive value for numerator coefficients in the reduced model.

Komarasamy et al.

1879 The reduced order model transfer function is,

3.1. Example 1 Consider a fourth order system in Shamash (1975) with a transfer function of GðsÞ ¼

28s3 þ 496s2 þ 1800s þ 2400 2s4 þ 36s3 þ 204s2 þ 360s þ 240

ð10Þ

The poles of the given system are s ¼ 1.1967  j0.6934 and 7.8033  j1.3576. By applying the improved pole clustering method, the improved cluster poles are obtained as m1 ¼ 1.50476 þ j0.6859 and m2 ¼ 1.50476  j0.6859. The corresponding reduced order denominator polynomial is, Dr ðsÞ ¼ s2 þ 3:3074s þ 2:73472

ð11Þ

By following step 2 in the proposed method, the reduced order numerator polynomial can be obtained as, Nr ðsÞ ¼ 11:4748s þ 27:3472

ð12Þ

Gr ðsÞ ¼

11:4748s þ 27:3472 s2 þ 3:3074s þ 2:73472

The reduced order model is compared with some of the existing methods and an integral square error calculated between the original and reduced order system is calculated as ISE ¼

n X

½Yðti Þ  Yr ðti Þ2

Prasad et al. (2003)

Gutman et al. (1982)

Prasad et al. (2003)

Prasad and Vishwakarma (2009)

Krishnamurthy and Seshadri (1978)

Table 1 gives the comparison of the proposed scenario with some of the existing methods. The proposed method concentrates on the distance between the first pole and the remaining poles available in the pole clustering group. It is used to produce a more dominant pole cluster value which will retain the important properties of the higher order system. The step responses of original and reduced order models were shown in Figure 1. The validity of the proposed method is evaluated by calculating an integral square error between

Reduced model 14s þ 11:903620 s2 þ 3:145997s þ 1:190362 17:64706s þ 70:58824 s2 þ 5:2491s þ 7:05582 22:532255s þ 11:903620 s2 þ 3:145997s þ 1:190362 6:80039s þ 11:9031 s2 þ 1:5730s þ 1:19031 9:046283s þ 13:043478 s2 þ 1:701323s þ 1:304348

Shamash (1975)

Manigandan et al. (2005)

8:83s þ 11:76 s2 þ 1:765s þ 1:176 14s þ 410:256 s2 þ 29:5897s þ 41:0256

Lucas (1983)

30s þ 40 3s2 þ 6s þ 4

Proposed method s2 ISE: Integral of the square error.

ð14Þ

i¼0

Table 1. Comparison of proposed method with existing methods for example 1 Method of reduction

ð13Þ

11:4748s þ 27:3472 þ 3:3074s þ 2:73472

ISE 81.3603

34.7607

27.4459

17.8716

12.0784

5.7634

4.3794

2.0610

0.0468

1880

Journal of Vibration and Control 18(12) The step responses of original and reduced order models were shown in Figure 2. Table 2 shows the validity of the proposed method along with some of the existing methods. The eigen values for the denominator polynomial in higher order system are 1, 2, 3 and 4. The dominant roots obtained through the proposed scenario are 1 and 2.0004. Figure 2 shows that the reduced model is closely matched with the higher order system. The undamped natural frequency (!n ) and damping ratio () for the second order system are 1.4144 and 1.0607 respectively.

Step Response

12

10

Amplitude

8

6

4

2

0

Higher order system Proposed method

0

1

2

3

4

5 6 Time (sec)

7

8

9

10

Figure 1. Step responses of original and reduced order models.

3.3. Example 3 Consider an eighth order system in Shamash (1975) with a transfer function of 

 18s7 þ 514s6 þ 5982s5 þ 36380s4 þ 122664s3 þ 222088s2 þ 185760s þ 40320  GðsÞ ¼  8 s þ 36s7 þ 546s6 þ 4536s5 þ 22449s4 þ 67284s3 þ 118124s2 þ 109584s þ 40320

Step Response

1.2

1

ð17Þ

Amplitude

0.8

By using the proposed method of model order reduction, the reduced order model is obtained as, 0.6

Gr ðsÞ ¼

0.4

s2

13:4491s þ 4:3505 þ 5:2298s þ 4:3505

ð18Þ

Higher order system Proposed method

0.2

0 0

1

2

3

4

5 6 Time (sec)

7

8

9

10

Figure 2. Step responses of original and reduced order models.

the original reduced order models. Table 1 gives the comparison of the proposed scenario with some of the existing methods.

3.2. Example 2 Consider a fourth order system in Mukherjee and Mishra (1987) with a transfer function of GðsÞ ¼

s3 þ 7s2 þ 24s þ 24 s4 þ 10s3 þ 35s2 þ 50s þ 24

ð15Þ

By using the proposed method of model order reduction, the reduced order model is obtained as, 0:8s þ 2:0004 Gr ðsÞ ¼ 2 s þ 3:0004s þ 2:0004

ð16Þ

The step responses of original and reduced order models were shown in Figure 3. Table 3 shows the validity of the proposed method along with some of the existing methods.

4. Multivariable system Consider the nth order transfer matrix of the higher order original system having k inputs and l outputs as, 2

a11 ðsÞ 6 a 21 ðsÞ   GðsÞ ¼ 1 6 6 : Dn ðsÞ 4 : al1 ðsÞ

3 a12 ðsÞ . . . a1k ðsÞ a22 ðsÞ . . . a2k ðsÞ 7 7 : : : 7 5 : : : al2 ðsÞ . . . alk ðsÞ

ð19Þ

  The general from of GðsÞcan be taken as 2 m1   þ am sm GðsÞ ¼ aij ðsÞ ¼ a0 þ a1 s þ a2 s    þ am1 s 2 n1 Dn ðsÞ b0 þ b1 s þ b2 s    þ bn1 s þ bn sn ð20Þ

Where, i ¼ 1, 2. . .l and j ¼ 1, 2. . .k. Let the transfer function matrix of rth order having k inputs and l outputs to be synthesized as,

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Table 2. Comparison of proposed method with existing methods for example 2 Method of reduction

Reduced model

Pal (1983)

ISE

s þ 34:2465 s2 þ 239:8082s þ 34:2465

Shieh and Wei (1975) s2

Parthasarathy and Jayasimha (1982)

s þ 2:3014 þ 5:7946s þ 2:3014

s þ 0:6997 s2 þ 1:45771:s þ 0:6997

Pal (1979)

16:0008s þ 24 30s2 þ 42s þ 24

Vishwakarma and Prasad (2008) method

0:189762s þ 4:5713 s2 þ 4:76187:s þ 4:5713

Parmar et al. (2007)

0:7442575s þ 0:6991576 s2 þ 1:45771s þ 0:6997

Manigandan et al. (2005)

0:0417s þ 1 0:0417s2 þ 1:125:s þ 1

Lucas (1983)

1.4763

0.3420

0.1188

0.0801

0.0176

0.0111

0.0033

0:833s þ 2 s2 þ 3:s þ 2

Proposed method

21.5719

0:8s þ 2:0004 s2 þ 3:0004s þ 2:0004

0.000216

ISE: Integral of the square error.

2

d11 ðsÞ 6 d21 ðsÞ   Gr ðsÞ ¼ 1 6 6 : Dr ðsÞ 4 : dl1 ðsÞ

Step Response

2.5

3 d12 ðsÞ . . . d1k ðsÞ d22 ðsÞ . . . d2k ðsÞ 7 7 : : : 7 5 : : : dl2 ðsÞ . . . dlk ðsÞ

ð21Þ

Higher order system Proposed method

The general form of the reduced order MIMO system is obtained as,

2

2 q1   þ dq s q Gr ðsÞ ¼ dij ðsÞ ¼ d0 þ d1 s þ d2 s    þ dq1 s 2 r1 Dr ðsÞ e0 þ e1 s þ e2 s    þ er1 s þ er sr ð22Þ

Amplitude

1.5

The proposed model order reduction method has been applied to the given higher order system model so that it will retains important characteristics of an original system in reduced model.

1

0.5

0 0

4.1. Example 4 1

2

3

4

5 6 Time (sec)

7

8

9

10

Figure 3. Step responses of original and reduced order models.

Consider a sixth order two input two output system described by the transfer function matrix in Bistritz and Shaked (1984)

1882

Journal of Vibration and Control 18(12) Table 3. Comparison of proposed method with existing methods for example 3 Method of reduction

Reduced model

Tomar and Prasad (2009)

ISE 23.9230

1:5725s þ 0:34134 s2 þ 0:9277s þ 0:3414

Shamash (1975)

2.7825

6:7786s þ 2 s2 þ 3s þ 2

Lucas (1983)

2.7818

6:78s þ 2 s2 þ 3s þ 2

Parmar et al. (2007)

2.6901

7:086314s þ 1:993259 s2 þ 3s þ 2

Proposed method

0.0081

13:4491s þ 4:3505 s2 þ 5:2298s þ 4:3505

ISE: Integral of the square error.

Table 4. Comparison of proposed method with existing methods for example 4 Reduced model ISE Method of reduction r11 Parmar et al. (2007) Vishwakarma and Prasad (2009) Proposed method

r12

ITAE

r21

r22

0.1454 0.0877 0.0255 0.0151 0.000785 0.0030 0.0040 0.0014

r11

r12

r21

IAE r22

r11

r12

r21

r22

0.1580 14.4436 9.5763 6.5988 15.7232 3.1058 2.2673 1.3547 3.3060 0.0469 1.4742 0.3234 0.7206 2.2062 0.7382 0.1676 0.3389 1.2485

0.000429 0.0869

0.2854 0.3244 0.0839

2.3461 0.2507 0.1968 0.0761 1.5012

ISE: Integral of the square error, ITAE: Integral of time multiplied by absolute error; IAE: Integral of absolute magnitude of the error.

" ½GðsÞ ¼

2ðsþ5Þ ðsþ1Þðsþ10Þ ðsþ10Þ ðsþ1Þðsþ20Þ



1 a11 ðsÞ DðsÞ a21 ðsÞ

ðsþ4Þ ðsþ2Þðsþ5Þ ðsþ6Þ ðsþ2Þðsþ3Þ

a12 ðsÞ a22 ðsÞ

# ð23Þ

a12 ðsÞ ¼ s5 þ38s4 þ459s3 þ2182s2 þ4160s þ 2400 ð27Þ a21 ðsÞ ¼ s5 þ30s4 þ331s3 þ1650s2 þ3700s þ 3000 ð28Þ

 ð24Þ

a22 ðsÞ ¼ s5 þ42s4 þ601s3 þ3660s2 þ9100s þ 6000 ð29Þ

Where, D(s) is the common denominator of given MIMO system and is given by,

By using the proposed method the cluster centers for the reduced model of the second order are obtained as S ¼ 1, 3.0847. The corresponding reduced order denominator polynomial is obtained as,

¼

DðsÞ ¼ ðs þ 1Þðs þ 2Þðs þ 3Þðs þ 5Þðs þ 10Þðs þ 20Þ

Dr ðsÞ ¼ D2 ðsÞ ¼ s2 þ 4:0847s þ 3:0847

DðsÞ ¼ s6 þ 41s5 þ 571s4 þ 3491s3 þ 10060s2 þ 13100s þ 6000 ð25Þ

The general form of the reduced order model is given by,   G2 ðsÞ ¼

and a11 ðsÞ ¼ 2s5 þ70s4 þ762s3 þ3610s2 þ7700s þ 6000 ð26Þ

ð30Þ

and

  1 d11 ðsÞ d12 ðsÞ D2 ðsÞ d21 ðsÞ d22 ðsÞ

ð31Þ

Komarasamy et al.

1883

Step Response

Step Response (b)

1

0.5

0.8

0.4 Amplitude

Amplitude

(a) 1.2

0.6

0.4

0.2 Higher order system Proposed method

0.2

0

0.3

0

1

2

3

4

5

6

7

8

9

Higher order system

0.1

Proposed method

0

10

0

1

2

3

4

5

6

7

8

9

10

Time (sec)

Time (sec)

Step Response

Step Response (d)

(c) 0.6

1

0.5 Amplitude

Amplitude

0.8 0.4 0.3

0.6 0.4

0.2 Higher order system Proposd method

0.1 0

0

1

2

3

4

5

6

7

8

9

Higher order system Proposed method

0.2

10

0

0

1

2

3

4

5

6

7

8

9

10

Time (sec)

Time (sec)

Figure 4. (a) Comparison of step responses: u1 ¼ 1, u2 ¼ 0. (b) Comparison of step responses: u1 ¼ 0, u2 ¼ 1. (c) Comparison of step responses: u1 ¼ 1. u2 ¼ 0. (d) Comparison of step responses: u1 ¼ 0, u2 ¼ 1.

d11 ðsÞ ¼ 1:30487s þ 3:0847

ð32Þ

d12 ðsÞ ¼ 1:0786s þ 1:23388

ð33Þ

d21 ðsÞ ¼ 0:5771s þ 1:54235

ð34Þ

d22 ðsÞ ¼ 2:0282s þ 3:0847

ð35Þ

The closeness between the original and reduced order models is analyzed by calculating the values of ISE, integral of time multiplied by absolute error (ITAE) and integral of absolute magnitude of the error (IAE) and is shown in Table 4. The proposed

method gives better result as compared with existing methods using the genetic algorithm in their order reduction scenario. The step responses of original and reduced order model were shown in Figures 4(a) to (d).

5. Conclusion In this method, an improved pole clustering method along with a simple mathematical procedure is proposed to obtain the reduced order system. Order reduction of SISO and MIMO systems were explained via some numerical illustrations in this paper. The closeness between the original and approximated system is calculated by using ISE, ITAE and IAE as quality

1884 parameters for the given step input (may also be calculated for other test inputs like ramp and impulse).Stability of the reduced order model is assured if the given higher order system is stable. The proposed scenario comparatively produced a better result as compared with methods which employed the genetic algorithm. The proposed method can be further improved by adjusting the obtained reduced order numerator polynomial by using the genetic algorithm, particle swarm optimization (PSO) and fuzzy logic on the basis of the low values of ISE, ITAE and IAE. The model order reduction method proposed in this paper gives the stable system in the approximated model provided that the higher order system is in the minimum phase transfer function form. It is not applicable for the higher order systems that are in non-minimum transfer function form. This scenario can be extended to the design of sub-optimal controller, State feedback controller, compensator and/or controller design. Funding This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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