Design of Fractional Order Differentiators and Integrators using Indirect Discretization Approach

Richa Yadav Advanced Electronics Lab, Division of Electronics and Communication Engineering, Netaji Subhas Institute of

Technology, Sector-3, Dwarka, New Delhi 110075, India Email: yadav.richa1@gmail.com

Maneesha Gupta Advanced Electronics Lab, Division of Electronics and Communication Engineering, Netaji Subhas Institute of

Technology, Sector-3, Dwarka, New Delhi 110075, India Email: maneeshapub@gmail.com

Abstract—This paper deals with continued fraction expansion

(CFE) based indirect discretization scheme for finding the rational approximation of fractional order differentiators and integrators and their discretized transfer functions. Indirect discretization approach is used for Al-Alaoui’s Optimized four segment rule and Simpson 1/3 rule based differentiators and integrators of order ½ and ¼[15]. These rational transfer functions are first stabilized for minimum phase then the resultant rational approximation is discretized by using s to z transforms. The curves for their magnitude responses, absolute magnitude errors and the phase responses are drawn with the help of MATLAB. These curves are then compared with ideal characteristics of differentiators and integrators.

Keywords—fractional order integrator, fractional order differentiator, continued fraction expansion ,Al-Alaou’s Optimized four segment rule,Simpson 1/3 rule.

I. INTRODUCTION Fractional calculus is known for describing a real object

more accurately than the classical “integer order” methods as most of these objects are fractional in nature [2]. The main point for implementing fractional order controllers (FOC) digitally is by discretization of the fractional order differentiator sr and integrator 1/sr. There are two commonly used discretization techniques (1) direct discretization and (2) indirect discretization [3]. In direct discretization direct power series expansion [5, 10] or continued fraction of s to z transform is used [4]. We can replace any power series by a sequence of rational functions because these converge more rapidly. In indirect discretization first the transfer function is fitted in continuous time domain and then this fit s-transfer function is discretized. In this paper new fractional order differentiators and integrators are suggested using indirect discretization technique.

This paper is organized as follows: in section 2, indirect discretization technique is briefly introduced. Section 3 deals with magnitude and phase responses of different differentiators and integrators of orders ½ and ¼ after stabilization. Finally, section 4 concludes this paper with some additional remarks.

II. INDIRECT DISCRETIZATION

A. Rational Approximation

Two definitions that are commonly used for fractional order differentiation and integration operators are the Grunwald- Letnikov (GL) definition and the Rieman – Liovuville (RL) definition. Here we have used simple method of indirect discretization by employing a continued fraction expansion using GL definition.

Grunwald – Letnikov (GL) definition is:

( )∑∞

=→ −⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

00 )1(1lim)(

k

k ktfk

tfD τα

ττα

(1)

Where

)1()1()1(

+−++=⎟⎟

⎠

⎞⎜⎜⎝

⎛kkk αττ

ατα

(2)

Rational approximations are used mainly in continuous time domain to restrict the fractional order systems to a finite order difference equation because they converge more rapidly and have a wider domain of convergence in the complex plane [5]. Our main target is to first find rational approximation for H(s) in the Laplace domain by limiting its order and then digitize this limited rationalized fractional integrodifferential function.

A.N. Khovanskii [9] has obtained the continued fraction expansion for

(1 ) (1 ) (2 )(1 ) 11 2 3 2 .......

xx α α α α α− + −+ = ++ + + +

(3)

The above continued fraction expansion is convergent and we only use the first ten terms of equation (1) and we replace x by ( s-1) in this equation. The resultant rational operator in s-domain is limited to the fifth order [1] and is given by:

542

33

24

15

0

542

33

24

15

0

QsQsQsQsQsQPsPsPsPsPsPs

++++++++++=α (4)

��Numerator and denominator polynomials in the rational approximation form are given in table 1.

2010 International Conference on Advances in Recent Technologies in Communication and Computing

978-0-7695-4201-0/10 $26.00 © 2010 IEEE

DOI 10.1109/ARTCom.2010.67

126

TABLE I. NUMERATOR AND DENOMINATOR POLYNOMIALS IN THE RATIONAL APPROXIMATION

Coefficients with ∝ Parameter

5 4 3 20 5 1 5 8 5 2 2 5 2 7 4 1 2 0P Q α α α α α= = − − − − − −

5 4 3 21 4 5 45 5 1005 3250 3000P Q α α α α α= = + + − − −

5 4 3 22 3 1 0 3 0 4 1 0 1 2 3 0 4 0 0 0 1 2 0 0 0P Q α α α α α= = − − + + − −

5 4 3 23 2 1 0 3 0 4 1 0 1 2 3 0 4 0 0 0 1 2 0 0 0P Q α α α α α= = − − + + −

5 4 3 24 1 5 4 5 5 1 0 0 5 3 2 5 0 3 0 0 0P Q α α α α α= = − + − − + −

5 4 3 25 0 1 5 8 5 2 2 5 2 7 4 1 2 0P Q α α α α α= = − + − + −

B. Operators used for rational approximation Rational approximation is discretized by using s to z transforms. We have employed two rules for these transformations. These are:-

(1) Al- Alaoui,s optimized 4-segment rule[15]

H1(z) = 5549.04438.0

0499.06635.08416.02

2

−−++

zzz (5)

(2)Simpson 1/3 integration rule H2(z) =

)1(314

2

2

−++

zzz (6)

C. Considerations for stability of operators used Before expanding these two rules into a rational function we will concentrate on their stability criterion first i.e., both operators should have all poles and zeros inside the unit circle. Al-Alaoui,s Optimized Four segment rule fulfill this condition so this is stable. But Simpson 1/3 digital differentiator has one of the poles (3.7321) outside the unit circle. So we have to reflect this pole inside the circle by using method suggested by Steigglitz. K. in [13] for stability and minimum phase. Stabilized Simpson 1/3 digital differentiator is given by equation (7).

2 2

2 2

3( 1) 3 ( 1)( 1) 1( )13.73214 1 0.8038 0.4307 0.0577( 0.2679)

3.7321

z z z zh zz z z zz z

− − + −= = =+ + + +⎛ ⎞+ +⎜ ⎟

⎝ ⎠

(7)

III. DISCRETIZATION BY RATIONALIZED FUNCTION

A. Employing Fifth order rational s-domain operator When we substitute both Al-Alaoui’s Optimized Four

Segment rule and Stabilized Simpson 1/3 rule i.e., equation (5) and (7) in rationalized function i.e., equation (4) in place of s , we obtain two discretized transfer functions in the z-domain given by (8) and (9) respectively.

H_optfour =

109

28

37

46

55

64

73

82

91

100

1092

83

74

65

56

47

38

29

110

0

hzhzhzhzhzhzhzhzhzhzhgzgzgzgzgzgzgzgzgzgzg

++++++++++++++++++++ (8)

H_simp =

1092

83

74

65

56

47

38

29

110

0

1092

83

74

65

56

47

38

29

110

0

jzjzjzjzjzjzjzjzjzjzjizizizizizizizizizizi++++++++++

++++++++++ (9)

TABLE II. EXPRESSIONS FOR THE DIGITAL FILTER COEFFICIENTS FOR AL-ALAOUI’S OPTIMIZED FOUR ESGMENT RULE

Coefficients of Al- Alaoui’s Optimized Four Segment Discretized Transfer function

G0 = 0.4222p5 + 0.5017p4 + 0.5961p3 + 0.7083p2 +0.8416p1 + 1p0;

G1 = 1.6643p5 + 1.3593p4 + 0.8808p3 +0.1738 p2 - 0.8305p1 - 2.2190p0;

G2 = 2.7494p5 +1.0094p4 - 0.5780p3 - 1.7233p2 - 2.0014p1 - 0.8049p0;

G3 = 2.4636p5 - 0.4963p4 - 1.61478p3 - 0.8463p2 + 1.4156p1 + 4.0512p0;

G4 = 1.2972p5 - 1.2393p4 - 0.5689p3 + 1.3261p2 + 2.1607p1 -

0.0057p0;

G5 = 0.4090p5- 0.8159p4 + 0.5554p3 +1.0294p2 - 0.7054p1 - 3.1467p0;

G6 = 0.0769p5 - 0.2628p4 + 0.5308p3 - 0.1772p2 - 1.1996p1 + 0.0031p0;

G7 = 0.0087p5 - 0.0444p4 + 0.1723p3 - 0.3629p2 - 0.0289p1 + 1.2474p0;

G8 = 0.0006p5 - 0.0040p4 + 0.0246p3 -0.1152p2 + 0.2651p1 + 0.1375p0;

G9 = -0.0002p4 + 0.0016p3 - 0.0123p2 + 0.0780p1 - 0.2104p0;

G10 = -0.0004p2+ 0.0047p1 -0.0526p0;

H0 = 0.4222q5 + 0.5017q4 + 0.5961q3 + 0.7083q2 +0.8416q1 + 1q0;

H1 = 1.6643q0 + 1.3593q1 + 0.8808q2 +0.1738 q3 - 0.8305q4 - 2.2190q5;

H2 = 2.7494q5 +1.0094q4 - 0.5780q3 - 1.7233q2 - 2.0014q1 - 0.8049q0;

H3 = 2.4636q5 - 0.4963q4 - 1.6147q3 - 0.8463q2 + 1.4156q1 + 4.0512q0;

H4 = 1.2972q5 - 1.2393q4 - 0.5689q3 + 1.3261q2 + 2.1607q1 - 0.0057q0;

H5 = 0.4090q5- 0.8159q4 + 0.5554q3 +1.0294q2 - 0.7054q1 - 3.1467q0;

H6 = 0.0769q5 - 0.2628q4 + 0.5308q3 - 0.1772q2 - 1.1996q1 + 0.0031q0;

H7 = 0.0087q5 - 0.0444q4 + 0.1723q3 - 0.3629q2 - 0.0289q1 + 1.2474q0;

H8 = 0.0006q5 - 0.0040q4 + 0.0246q3 -0.1152q2 + 0.2651q1 + 0.1375q0;

H9 = -0.0002q4+ 0.0016q3 - 0.0123q2 + 0.0780q1 – 0.2104q0;

H10 = -0.0004q2+ 0.0047q1 -0.0526q0;

Expressions for the digital filter coefficients for Al-Alaoui’s Optimized Four segment Rule discretized transfer function H_optfour and Stabalized Simpson discretized transfer function H_simp are given in Table 2 and Table 3 respectively.

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TABLE III. EXPRESSIONS FOR THE DIGITAL FILTER COEFFICIENTS OF STABALIZED SIMPSON 1/3 RULE

Coefficients for Stabalized Simpson 1/3 Discretized Transfer Function

i0 = 0.3355p0 + 0.4174p1 +0.5193p2 + 0.6461p3 +0.8038p4 +1p5;

i1 =0.8990p0 + 0.8947p1 + 0.8348p2 + 0.6924p3 + 0.4307p4;

i2 = 1.0838*p0 + 0.4215*p1 -0.4795*p2 - 1.6600*p3 - 3.1575*p4 - 5*p5;

i3 = 0.7743p0- 0.4451p1 - 1.4699p2 – 2.0275p3 - 1.7228p4;

i4 = 0.3631p0- 0.6884p1 - 0.5588p2 + 1.1068p3 + 4.5920p4 +10p5;

i5 = 0.1167p0 - 0.4173p1 + 0.4396p2 + 1.9281p3 + 2.5842p4;

i6 = 0.0261p0 - 0.1462p1 + 0.4791p2 + 0.1787p3 - 2.8690p4 - 10p5;

i7 = 0.0040p0 - 0.0319p1 + 0.1911p2 - 0.5433p3 - 1.7228p4;

i8 =0.0004p0 - 0.0043p1 + 0.0398p2- 0.2683p3 + 0.5730p4 + 5p5;

i9 = 0p0 - 0.0003p1 + 0.0043p2 - 0.0497p3 + 0.4307p4;

i10 = 0p0 - 0p1 + 0.0002p2 - 0.0033p3+ 0.0577p4 -1p5;

j0 = 0.3355q0 + 0.4174q1 +0.5193q2 + 0.6461q3 +0.8038q4 +1q5;

j1 =0.8990q0 + 0.8947q1 + 0.8348q2 + 0.6924q3 + 0.4307q4;

j2 = 1.0838q0 + 0.4215q1 -0.4795q2 - 1.6600q3 - 3.1575q4 - 5q5;

j3 = 0.7743q0- 0.4451q1 - 1.4699q2 - 2.0275q3 - 1.7228q4;

j4 = 0.3631q0- 0.6884q1 - 0.5588q2 + 1.1068q3 + 4.5920q4 +10q5;

j5 = 0.1167q0 - 0.4173q1 + 0.4396q2 + 1.9281q3 + 2.5842q4;

j6 = 0.0261q0 - 0.1462q1 + 0.4791q2 + 0.1787q3 - 2.8690q4 - 10q5;

j7 = 0.0040q0 - 0.0319q1 + 0.1911q2 - 0.5433q3 - 1.7228q4;

j8 =0.0004q0 - 0.0043q1 + 0.0398q2- 0.2683q3 + 0.5730q4 + 5q5;

j9 = 0q0 - 0.0003q1 + 0.0043q2 - 0.0497q3 + 0.4307q4;

j10 = 0q0 - 0q1 + 0.0002q2 - 0.0033q3+ 0.0577q4 -1q5;

B. Two desirable properties of discretization approximation • Discrete time rational transfer function should be stable

and minimum phase.

• Poles and zeros of discrete approximation should be interlaced along the line z Є (-1,1) for a better fit to continuous frequency response [5].

Figure 1 and 2 are the pole-zero diagrams of Al-Alaoui’s Optimized digital differentiators and integrators of order ½ and ¼.Figure 3 and 4 are the pole-zero plot of stabilised Simpson digital differentiators and integrators of order ½ and ¼. Both the discretized transfer functions almost fulfill these two properties as they also have some pair of complex poles and

zeros which are slightly above and below the z line if not exactly interlaced on real z-axis.

Figure 1. Pole-zero plot of optimized four segment differentiators and integrators of order 1/2

Figure 2. Pole-zero plot of optimized four segment differentiators and integrators of order 1/4

Figure 3. Pole-zero plot of Simpson differentiators and integrators of order 1/2

Figure 4. Pole-zero plot of Simpson differentiators and integrators of order 1/4

Main advantage is that all the poles and zeros are inside the unit circle for differentiators and integrators of the orders ½ and ¼ for both Al-Alaoui’s Optimized as well as Stabilized Simpson discretized transfer function . So these two transfer functions are stable.

IV. SIMULATION RESULT The magnitude , phase responses and magnitude error (dB) of the integrators and differentiators evaluated at T=1 sec. as shown in figures 5 to 17 by simulating discretized transfer

128

functions H_optfour and H_simp with the help of MATLAB R2006a .

Figure 5. Magnitude response of differentiators of order ½

Figure 6. Phase response of differentiators of order 1/2

Figure 7. Magnitude error response of differentiators of order 1/2

Figure 8. Magnitude response of integrators for order 1/2

Figure 9. Phase response of integrators of order ½

Figure 10. Magnitude error response of integrators of order 1/2

Figure 11. Magnitude response of differentiators of order 1/4

Figure 12. Phase response of differentiators of order ¼

129

Figure 13. Magnitude error response of differentiators of order 1/4

Figure 14. Magnitude response of integrators for order 1/4

Figure 15. Phase response of integrators of order 1/4

Figure 16. Magnitude error response of integrators of order 1/4

In this paper we have presented designs for fractional order digital differentiators and integrators using indirect discretization technique. It can be observed that magnitude response of stabilized Simpson differentiators as well as integrators of orders ½ and ¼ outperform the corresponding Al-Alaoui’s optimized four segment differentiators and integrators. As stabilised Simpson curves starts following the curve of ideal differentiator and integrator earlier than curves of Optimized discretized function.Magnitude error (dB) for Simpson rule reaches zero value as it exactly copies ideal at frequency 0.8Hz. for all differentiators and integrators of order

½ and ¼. Finally magnitude error (dB) gets stabilize at approximately -15 db for both differentiators and integrators of order 1/2 and at -20 dB for differentiators and integrators of order ¼. Phase responses of stabilized Simpson rule perform better than that of optimized four segment differentiator and integrator.

V. CONCLUSION From the magnitude response plots, phase plots and magnitude error (dB) plots it is to be noted that stabilised Simpson discretized function has superior performance compared to Optimized four segment discretized function for digital differentiators and integrators of order ½ and ¼.

ACKNOWLEDGMENT The authors would like to thank Mrs. Pragya Varshney,

Lecturer, NSIT, Dwarka, Delhi for providing constructive comments and help in improving the contents of this paper.

REFERENCES [1] B.T. Krishna, K.V.V.S. Reddy, “Design of fractional Order Digital

Differentiators and Integrators using Indirect Discretization,” An International Journal for Theory and Applications, vol. 11, pp.143-151, Number 2, 2008.

[2] Y.Q. Chen,I. Petras and Dingyu Xue,”Fractional Order Control – A Tutorial”, Proceedings of the 2009 conference on American Control Conference, p.1397-1411, June10-12 June, 2009, St. Louis, Missouri, USA.

[3] Y.Q. Chen, B.M. Vinagre and I.Podlubny, “Continued Fraction Expansion Approaches to discretizing fractional order Derivatives – an Expository Review,” Nonlinear Dynamics 38,2004 Kluwer Academic publishers, pp155-170,March 2004

[4] Y. Q. Chen and K. L. Moore,” Discretization schemes for fractional- order differentiators and integrators”. IEEE transactions on Circuits and Systems-I: vol. 49, no. 3, pp 363- 367, Mar.2002.

[5] B. M. Vinagre, Y. Q. Chen, and I. Petras. “Two direct tustin discretizati- -on methods for fractional-order differentiator/ integrator”. Journal of the Franklin Institute, 340:349–362, 2003.

[6] M. A. Al-Alaoui, “Novel digital integrator and differentiator”, Electronics Letters, 29(4):376–378, 1993.

[7] K. S. Miller and B. Ross.,”An Introduction to the Fractional Calculus and Fractional Differential Equations”, Wiley and Sons, New York, 1993.

[8] C. C. Tseng,” Design of fractional order digital FIR differentiators”, IEEE Signal Processing Letters, 8(3):77–79, 2001.

[9] A.N. Khovanskii, The Application of Continued Fractions and Their Generalizations to Problems in Approximation Theory (Transl. by Peter Wynn). P. Noordhoff Ltd. (1963).

[10] B.M. Vinagre, I. Podlubny, A. Hernandez and V. Feliu“Some Approximations of Fractional Order Operators used in Control Theory”, Fractional Calculus and Applied Analysis,vol. 3, n0. 3, pp 231- 248, 2000.

[11] Guido Maione, “ A Rational Discrete Approximation to the Operator s0.5 “, IEEE Signal Processing Letters, Vol.13, No..3, March 2006.

[12] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).

[13] Steigglitz. K., “Computer – aided design of recursive digital filters”, IEEE Trans., AU-18, pp 123-129, 1970.

[14] M.A. Al-Alaoui, “Al-Alaoui’s Operator and the alpha approximation for Discretization of Analog Systems”, Elect. Energ., Vol. 19, No. 1, pp 143-146, April. 2006.

[15] M.A.Al-alaoui, “Novel class of Digital Integrators and Differentiators”,IEEE Transactions on Signal Processing, Vol.-pp, issue :99, 19th Aug.,2008.

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