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International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 03 Issue: 08 | Aug-2016 www.irjet.net p-ISSN: 2395-0072

A comprehensive study on different approximation methods of Fractional order system Asif Iqbal and Rakesh Roshon Shekh P.G Scholar, Dept. of Electrical Engineering, NITTTR, Kolkata, West Bengal, India P.G Scholar, Dept. of Electrical Engineering, NITTTR, Kolkata, West Bengal, India ------***------Abstract - Many natural phenomena can be more infinite dimensional (i.e. infinite memory). So for analysis, accurately modeled using non-integer or fractional order controller design, signal processing etc. these infinite order calculus. As the fractional order differentiator is systems are approximated by finite order integer order mathematically equivalent to infinite dimensional filter, it’s system in a proper range of frequencies of practical interest proper integer order approximation is very much important. [1], [2]. In this paper four different approximation methods are given and these four approximation methods are applied on two different examples and the results are compared both in time 1.1 FRACTIONAL CALCULAS: INRODUCTION domain and in frequency domain. Continued Fraction Expansion method is applied on a fractional order plant model Fractional Calculus [3], is a generalization of integer order and the approximated model is converted to its equivalent  calculus to non-integer order fundamental operator D t , delta domain model. Then step response of both delta domain l and continuous domain model is shown. where 1 and t is the lower and upper limit of integration and Key Words: Fractional Order Calculus, Delta operator the order of operation is . Oustaloup approximation method, Modified Oustaloup  d  approximation method, Continued Fraction Expansion (CFE)  R()  0 dx method, Matsuda approximation method, Illustrative   examples. l Dt   1 R()  0 (1)  t  (dx) ( ) R()  0  l 1. INTRODUCTION Where R  is the real part of . In this paper we consider Fractional calculus is the generalization of traditional only those cases where  is a real number, i.e.   R . calculus and it deals with the derivatives and integrals of There are some commonly used definitions are Grunwald- arbitrary order (i.e. non-integer).It is a 300 years old topic Letnikov definition, Riemann-Liouville definition and Caputo and the idea started when L’Hospital asked Leibniz about ½ definition. order derivative in 1695.Leibniz in a letter dated September The Riemann-Liouville definition of the fractional Integral is 30, 1695 (The exact birthday of fractional calculus) replied given below: that It will lead to an apparent paradox, from which one day m t useful consequences will be drawn [11]. In 1695 Leibniz  1  d  f ( ) l D t f (x)    d (2) mentioned derivatives of “general order” in his (m )  dx   (x  )1(m ) correspondence with Bernoulli. After that many scientist l worked on it but the actual progress started in June 1974 For m 1    m when B.Ross organized the first conference on application of Where,  p is the Euler’s gamma function. fractional calculus at the University of New Haven.

Most of the natural phenomena are generally fractional but The Grunwald-Letnikov definition, based on successive their fractionality is very small so that it can be neglected. differentiation, is given below: But for accurate modelling fractional calculus is more suited (x ) than traditional calculus. Due to the lack of solution methods h  1 (  k) of fractional differential equations scientists and l D t f (x)  lim f (x  kh) (3) h0   mathematicians had used the integer order model before () h k0 (k 1) late nineties’ but now there are some methods for So it can be seen from the above definition that by using the approximation of fraction derivative and integrals. So it can  fractional order operator D f (x) the fractional be easily used in areas of science (e.g. control system, circuit l t theory etc.). It has been found that fractional controller is integrator and fractional differentiator can be unified. more effective than integer order controller and it gives Another popular definition is the Caputo definition: better performance and flexibility. Fractional order system is

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 03 Issue: 08 | Aug-2016 www.irjet.net p-ISSN: 2395-0072

t (m) C  1 f d Bq D f (x)  (4) B  B  (14) l t  (m1 )  c T (  m) l (x  ) C  C  C (15) For m 1    m  c q

Let us consider the following fractional order differential D  Dc  Dq (16) equation which represents the dynamics of a system: Where A , B ,C & D are the state space matrix of n n1 0 c c c c an D y (t)  an1 D y (t)      a0 D y (t)  continuous time model. At very high sampling frequency m m1 0 bm D u (t)  bm1 D u (t)      b0 D u (t) delta domain model converges to its continuous counterpart. (5) Thus the system theory is unified by using delta operator based approach. Where are real coefficients and are positive ak ,bk k , k real numbers. Taking Laplace transform of the above equation and assuming zero initial condition, the fractional 1.3 OUSTALOUP APPROXIMATION METHOD order model takes the following form: m m1 0 Oustaloup approximation [5] is widely used in fractional Y(s) bms  bm1s      b0 s G(s)   (6) order control. This approximation method is based on the n n1 0 U(s) ans  an1s      a0s recursive distribution of poles and zero’s. It approximates Where, Ly(t) Y(s), Lu(t)U(s) the basic fractional order operator s (0    1) to a integer order rational form within a chosen frequency band. Let  ,   the desired frequency band is. Even within 1.2 DELTA OPERATOR l h that frequency band both gain and phase have ripples, which

decreases as the order of approximation increases. The The shift operator model of a system has the form: method is given below: qxk  Aq xk  Bquk N ' (7) s   k s  K (17) yk  Cq xk  Dquk  kN s  k At very high frequency T  0 this model fails because Where, N  Order of approximation Aq  I & Bq  O , to overcome this problem Middleton & Goodwin proposed a new operator called Delta operator. It  K  Gain  h is represented as: 1 f k 1T  f kT kN  1    f kT  (8) 2 2N 1 T '  h  k   q 1   Zeros  l     (9)  l  T 1 And the corresponding complex variable  is defined as k N  1  2 Z 1    2N 1     Poles    h  T k l   (10)  l  esT 1  T The delta operator model of a MIMO system is given below 1.4 MODIFIED OUSTALOUP APPROXIMATION

xk A xk  B uk (11) In Oustaloup approximation method the quality of yk C xk  D uk approximation is poor near the edges of desired frequency And it relates with the continuous model and shift operator band. To overcome this problem the authors modify this model and continuous time model as: approximation method [6], and present a new modified method, known as modified oustaloup approximation 1 T   e A d (12) method. This method gives very good fitting in the whole  frequency range of interest. This approximation is only valid T 0 for 0    1. For higher order derivative, s 2.3 e.g. , the Aq  I A  A  (13) 0.3  c T approximation method should be applied to s and the

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 03 Issue: 08 | Aug-2016 www.irjet.net p-ISSN: 2395-0072

2 0.3 original derivative is approximated by s s , for fractional s  p0 s  p1 s  p2 G(s)  a0     (17) 0.6 0.4 integrals, s  s . The algorithm is given below: a1  a2  a3  s Where, 2 N ' ai  i ( pi )   ds  bh s  s  k s  K   (14)  (s)  G(s) d1 s 2  b s  d  s   0  h  k N k s  p n i N  i1(s)   dl  k Where,  i (s)  ai K     '  b  kNk In this approximation method the sum of total number of n2k zero’s and total no of poles is N (order of '  d  2N 1 approximation).This method has a drawback, if N is odd the    l  k b approximated transfer function will be improper i.e., there   will be one more zero’s than poles. n2k 2N 1  bh  k     d  2. ILLUSTRATIVE EXAMPLE After extensive calculation it has been found that for the best fitting the values of b and d is 10 and 9. In this section two examples are shown and the results of different approximation methods are presented and compared. In the first example, fractional derivative s0.4 is 1.5 CFE APPROXIMATION approximated and the results are compared. In the second example different approximation methods are applied on a The CFE approximation [7-9], is given below: fractional order transfer function and the results are studied  both in time domain and in frequency domain Then CFE s  Gm s,  method is applied on second example to obtain an p ()s m  p ()s m1      p () (15)  m0 m1 mm approximated transfer function and the results are shown q ()s m  q ()s m1      q () and model reduction is done on the approximated transfer m0 m1 mm function and its equivalent model is obtained in delta domain. Where m is the order of approximation and Then its delta domain response is shown. pmi (n), qmi (n), i  0,1, , m are the coefficients. These Example 1: Here pure fractional order differentiator is coefficients can be calculated from the following formula. approximated by different approximation methods within the frequency of 0.01 to 100 rad/sec. The order of pmi (n)  qm,mi (n) (16) approximation is 3.The results of different approximation i methods are below:  1 Bm,in  i 1mi n  mi Oustaloup approximation: Where, B(m,i) = Binomial Coefficients 6.31s3  77.14s2  41.74s 1 g1 (18)  m! 3 2 i! (m i)! s  41.74s  77.14s  6.31 Modified Oustaloup approximation: uk = Pochammer symbol 10.08s5 1243s4 137660s3  7413s2 177.6s  u(u 1)(u  2)    (u  k 1) g2  s5  226.9s4  7809s3 14320s2 1220s  4.207 u0 1 (19)

CFE approximation:

11.42s3  63.65s2  42.43s  2.5 1.6 MATSUDA APPROXIMATION g3  (20) 2.5s3  42.43s2  63.65s 11.42 Matsuda approximation method [10], approximates Matsuda approximation: fractional differential operator into rational model in two 10.01s3 163.6s2  83.01s 1 steps. First a rational model of the irrational function is g4  (21) obtained by CFE and then it fits the original function in a set s3  83.01s2 163.6s 10.01 of some points which are logarithmically spaced. Assume The bode diagram of the approximated transfer functions that the points taken are pk , k  0,1, 2,  then the Are given below: approximation method has the following form:

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 03 Issue: 08 | Aug-2016 www.irjet.net p-ISSN: 2395-0072

Figure 1. Bode plot comparison Figure 3. Step response comparison Example 2: In this example approximated model of a Using CFE approximation method and taking the fractional order transfer function is obtained by different approximation methods and the results are compared. The approximation order as three the equivalent integer order fractional order transfer function given below: transfer function within the frequency band 0.01 rad/sec to 100 rad/sec is obtained as below: s  1 G(s)  (21) 10s3.2  185s2.5  288s0.7  1 6.781s10  481.5s9 11320s8 108900s7  448700s6  935700s5 1058000s4  Approximation methods are applied on fractional powered 3 2 terms s0.2 , s0.5 & s0.7 and the overall transfer function is 650800s  208000s  32120s 1883 Ga (s)  12 11 10 calculated. Here the order of approximation is 3 and the 142.1s 18020s  659000s  8266000 frequency range is 0.01 to 100. s9  43350000s8 114100000s7 185700000 The exact bode diagram of fractional order transfer function s6  217100000s5 172500000s4  76820000 and the approximated models are obtained. The responses 3 2 are presented in figure 2. s 1.6400000s 1.434000s  3.0530

(22)

The approximated model has the order of 10. On increasing the order of approximation more accurate approximation will be obtained but the order of the model will be very large. So model reduction is very important.

Using the algorithm presented in [10], the reduced model is obtained as:

0.0007923s2  0.00926s  0.003066 G s  (23) r s3  0.4406s2 1.636s  0.0497

The step response of the original model, approximated model and reduced model is given below: Figure 2. Bode plot comparison

The step response of the original model and the different approximation models are obtained. The responses are given in figure 3.

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more accurate fitting will be obtained. The equivalent delta domain model and simulation results are also given. From the results it is hard to tell which one of the approximation methods is the best. Even though some of them are better than others for some characteristics. The relative merits of different approximation methods depend on whether one is more interested in accurate frequency behavior or in accurate time responses.

REFERENCES

[1] M. Khanra, J. Pal, and K. Biswas, "Rational approximation and analog realization of Fractional order transfer function with multiple Fractional powered terms," Asian Journel of Control, Vol. 15 , No. 3, pp. 1-13, May 2013. [2] M. Khanra, J. Pal, and K. Biswas, "Reduced order Figure 4. Step response comparison approximation of MIMO Fractional order system," IEEE journel on emerging and selected topics in Circuits and The equivalent delta domain model for the sampling period Systems, Vol. 3, No. 3, September 2013. 0.01 is obtained using the Delta Transform. The resulting [3] M.D. Ortigueira," Fractional Calculas for Scientists and models for sampling period 0.01 is given below: Engineers,"Volume 84, London, New York : Springer, 2011. [4] R. Matusu," Application of Fractional order calculas to 0.0008 2  0.0093  0.0031 G    (24) control theory," International Journel of mathematical 1  3  0.456 2 1.6327  0.0496 models and methods in applied science, Issue 7, volume 5, 2011. The step response of the two delta domain models are [5] A. Oustaloup, F. Levron, B. Mathieu, and F.M. Nanot," Frequency-band complex noninteger differentiator: obtained. The responses are shown in figure 5. characterization and synthesis," IEEE Trans. Circuits Syst. 1, vol. 47, pp. 25-39,2000. [6] D. Xue, C. Zhao, and Y.Q. Chen," A Modified approximation method of Fractional order system," in proc. of the IEEE International conference on Mechatronics and Automation, Luoyang, China, June 25-28, 2006. [7] G. Maione," High-Speed digital realization of Fractional operators in Delta domain," IEEE Trans. on Automatic control, Vol. 56, No. 3, March 2011. [8] G. Maione," Continued fractions approximation of the impulse response of fractional-order dynamic systems," IET Control Theory Appl., Vol. 2, No. 7, pp. 564-572, 2008. [9] G.Maione,"Concerning Continued fractions representation of noninteger order digirtal differentiator," IEEE signal proc. Letters, Vol. 13, No. 12, December 2006.

[10] D.Y. Xue and Y.Q. Chen," Sub-optimal H 2 rational approximations to fractional order linear systems," In proc. of the ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Figure 5. Step response in Delta domain Engineering Conference, Long Beach, California, USA, 2005. [11] Y. Q. Chen, I. Petras and D. Xue, “Fractional Order Control 3. CONCLUSIONS – A Tutorial,” American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA, pp. 1397-1411, In this paper a comprehensive study on four different June 10-12, 2009. approximation methods of fractional order operator is presented with two different illustrative examples. The simulation results are also given so that one can easily study the quality of approximation of different approximation methods. If the order of approximation is increased the approximated model will converge to original model and © 2016, IRJET | Impact Factor value: 4.45 | ISO 9001:2008 Certified Journal | Page 1852

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BIOGRAPHIES

Asif Iqbal received the B. Tech degree in electronics and communication engineering from Techno India College of Technology, Kolkata, India, in 2013. Since august 2014, he is pursuing M. Tech in Mechatronics engineering in National Institute of Technical Teachers’ Training and Research, Kolkata, and working in the area of fractional order systems approximations and delta domain realization of fractional order systems.

Rakesh Roshon Shekh received the B. Tech degree in electronics and communication engineering from Asansol Engineering college, Asansol, India, in 2013. Since august 2014, he is pursuing M. Tech in Mechatronics engineering in National Institute of Technical Teachers’ Training and Research, Kolkata, and working in the area of fractional order systems approximations and delta domain realization of fractional order systems.