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IRJET-V3I8330.Pdf 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) h0 mathematicians had used the integer order model before () h k0 (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 © 2016, IRJET | Impact Factor value: 4.45 | ISO 9001:2008 Certified Journal | Page 1848 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 (m1 ) 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 n1 0 c c c c an D y (t) an1 D y (t) a0 D y (t) continuous time model. At very high sampling frequency m m1 0 bm D u (t) bm1 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 m1 0 Oustaloup approximation [5] is widely used in fractional Y(s) bms bm1s b0 s G(s) (6) order control. This approximation method is based on the n n1 0 U(s) ans an1s a0s recursive distribution of poles and zero’s. It approximates Where, Ly(t) Y(s), Lu(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: qxk Aq xk Bquk N ' (7) s k s K (17) yk Cq xk Dquk kN 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 1T f kT kN 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 xk A xk B uk (11) In Oustaloup approximation method the quality of yk C xk D uk 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 © 2016, IRJET | Impact Factor value: 4.45 | ISO 9001:2008 Certified Journal | Page 1849 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 bh s s k s K (14) (s) G(s) d1 s 2 b s d s 0 h k N k s p n i N i1(s) dl k Where, i (s) ai K ' b kNk In this approximation method the sum of total number of n2k 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. n2k b 2N 1 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 m1 p () (15) m0 m1 mm approximated transfer function and the results are shown q ()s m q ()s m1 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.
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