8357 FACTA UNIVERSITATIS Series: Electronics and Energetics Vol. 35, No 3, September 2022, pp. 313-331 https://doi.org/10.2298/FUEE2203313D © 2022 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND Original scientific paper A NEW APPROACH FOR DIRECT DISCRETIZATION OF FRACTIONAL ORDER OPERATOR IN DELTA DOMAIN Sujay Kumar Dolai1, Arindam Mondal2, Prasanta Sarkar3 1Dream Institute of Technology, Faculty of Electrical Engineering, India 2Pailan College of Management & Technology, Faculty of Electrical and Electronics Engineering, India 3NITTTR Kolkata, Faculty of Electrical Engineering, India Abstract. The fractional order system (FOS) comprises fractional order operator. In order to obtain the discretized version of the fractional order system, the first step is to discretize the fractional order operator, commonly expressed as s, 0 <  < 1. The fractional order operator can be used as fractional order differentiator or integrator, depending upon the values of . In general, there are two approaches for discretization of fractional order operator, one is indirect method of discretization and another is direct method of discretization. The direct discretization method capitalizes the method of formation of generating function where fractional order operator s is expressed as a function of Z in the shift operator parameterization and continued fraction expansion (CFE) method is then utilized to get the corresponding discrete domain rational transfer function. There is an inherent problem with this discretization method using shift operator parameterization (discrete Z-domain). At fast sampling time, the discretized version of the continuous time operator or system should resemble that of the continuous time counterpart if the sampling theorem is satisfied. At very high sampling rate, the shift operator parameterized system fails to provide meaningful information due to its numerical ill conditioning. To overcome this problem, Delta operator parameterization for discretization is considered in this paper, where at fast sampling rate, the continuous time results can be obtained from the discrete time experiments and therefore a unified framework can be developed to get the discrete time results and continuous time results hand to hand. In this paper a new generating function is proposed to discretize the fractional order operator using the Gauss-Legendre 2- point quadrature rule. Additionally, the function has been expanded using the CFE in order to obtain rational approximation of the fractional order operator. The detailed mathematical formulations along with the simulation results in MATLAB, with different fractional order systems are considered, in order to prove the newness of this formulation for discretization of the FOS in complex Delta domain. Key words: continuous fraction expansion, direct discretization, delta operator, fractional order operator, fractional order system Received September 30, 2021; revised January 26, 2022; accepted July 4, 2022 Corresponding author: Arindam Mondal Pailan College of Management & Technology, Faculty of Electrical and Electronics Engineering, India E-mail: arininstru@gmail.com 314 S. K. DOLAI, A. MONDAL, P. SARKAR 1. INTRODUCTION Around 300 years ago the concept of fractional calculus [1-2] came into existence. It has been an untouched and undiscovered part of engineering until the conceptual furtherance of fractional calculus eventuated in the mid nineteenth century. With time this part attracted the researchers towards its diversified properties that can be implemented in various field of engineering, as well as various part of science [3-7]. The postulation of fractional order calculus has an immense perspective to change the technique we see, manipulate and design the nature that is around us. The fundamental unit of the non-integer order system is the operator (s), which can also be coined as fractional order differentiator or integrator [8-9] for variation of  by making it either positive or negative. The important part of digital realization of fractional order system is the discretization of this operator. In order to implement the FOS in real time, the rationalization is the only procedure, either in continuous time or in discrete time. There are various methods for continuous time approximation of fractional order operators [10-12]. Once it is converted to continuous time rational transfer function [13], there are methods of discretization to get the discretized version of the FOS [14-18]. This is known as indirect method of discretization of FOS. There is a second method known as direct discretization method, where the rational transfer function in Z-domain is directly obtained via different generating functions, as proposed by Euler, Tustin, Al-Alauoi. In the subsequent step, the generating function is expanded using methods such as continued fraction expansion (CFE) [19]. There has been an increased demand in digital system implementation. In order to implement the FOS digitally, the sampling rate must be increased to at least 10 times the original system bandwidth. The increased sampling rate makes the poles closer to each other in Z- domain transfer function and gets focused near the point (1,0) in the discrete Z- plane. This will result in an unstable system due to finite word length effect [15]. The conventional or shift operator representation of discrete time system fails to furnish the significant portrayal of the conventional continuous-time system at fast sampling rate. To circumvent this problem delta operator parameterization is introduced [20] where, at very high sampling frequency the continuous time results and discrete time results are obtained at the same time. The superiority of the delta operator parameterization along with its various applications are found in [21-29]. In this paper, a method is proposed by which the fractional order operator is directly discretized [30-31] in delta domain. Initially, a generating function is proposed in delta domain by using one of the useful numerical computational tools known as Gauss-Legendre 2-point quadrature rule [32]. The classical CFE method is adopted to expand this generating function to get the rational approximation of the fractional operator in discrete delta domain. The significant contributions are made in this paper as given below: earlier research work so far done on the discretization of the fractional order system through the discretization of the fractional order operator in shift operator parameterization. In this work the FOS has been directly discretized using delta operator parameterization so that at a very fast sampling frequency, the discrete time results resemble that of the continuous time counterpart. One more important contribution of this work is that here Gauss-Legendre 2-point quadrature rule is used for the close form approximation of the log(1 + x) function to minimize the approximation error. The comparison with the other standard methods are done to prove the efficacy of this proposed method. The paper has been well organized in the following sections as indicated: in Section 2, fractional order operator and systems are discussed; Section 3 enlightens the direct A New Approach for Direct Discretization of Fractional order Operator in Delta Domain 315 discretization method of FO operator in delta domain; simulation and result analysis are discussed with different examples in Section 4; and in Section 5 the conclusion is drawn. 2. FRACTIONAL ORDER SYSTEM AND ITS DISCRETIZATION (DIRECT METHOD) USING TRADITIONAL METHODS 2.1. Fractional order operator and fractional order system Fractional order system literally means the order of the system is no longer integer that is non-integer order. A system of fractional order is represented as fractional order differential equation and Laplace transform of the system can be performed to get the transfer function. A non-integer order system can be portrayed by the following equation [30]. )(....)()()(....)()( 0101 0101 tuDbtuDbtuDbtyDatyDatyDa mmnn mmnn  −  −  +++=+++ −− Where, ( ) ( 0) 1 ( 0) ( 0) m d d mD d                = =      (1) is known as integro-diffrerentiator operator. The Laplace transform of the Eq. (1) under consideration of zero initial condition, the transfer function that we get is: 1 0 1 0 m 1 0 n 1 0 .....( ) ( ) ( ) ..... m m n n m n b D b D b DY s G s U s a D a D a D       − − − − + + + = = + + + (2) where, [ ( )] ( ), [ ( )] ( )L y t Y s L u t U s= = . If the fractional differential equation as given in Eq. (2) may be coined as commensurate order which further gets reduced to the following form. 0 0 ( ) ( ) m n ka ka k kk k a D y t D u t = = =  (3) where, + = ,, k aa kk There are two popular definitions, such as Grünwald-Letnikov (GL) and Riemann- Liouville (RL) definitions, to express this operator   mD . Here, RL definition is considered. The RL definition is 1 1 ( ) ( ) ( ) ( ) k k k d p mD t dp k dt t p       − + =  − −  (4) Where m and  are the bounds of operation and  is used to represent the Euler's gamma function. 316 S. K. DOLAI, A. MONDAL, P. SARKAR For the analysis purpose, the fractional order differentiator is considered in this section. The fractional order system (differentiator) is realized in complex S-domain for the ease, which can be acquired by taking the Laplace transform of the Eq. (4), thus the Laplace transform of the equation is { ( )} ( ), 0 1L mD t s s for     =     (5) 3. DELTA DOMAIN DISCRETIZATION METHOD OF FRACTIONAL ORDER OPERATOR In contrast to get better finite-word-length effect under fast sampling, forward shift operator is going to be replaced by the delta operator [20]. The forward difference operator of delta operator is defined as 1q  − =  (6) Where q is the forward shift operator and  is termed as sampling time or internal. Employing a differentiable signal x(t), at high sampling time (→0) the delta (  ) operator gravitates with continuous-time derivative operator as shown in Eq. (7). dt tdxtxtx tx )()()( )( limlim 00 =  −+ = →→ (7) The variable corresponding to z in the shift operator parameterization is denoted by  in complex delta domain and relationship between the two complex variables are given in Eq. (8)[20]. 1 1 Δ Δ sΔ z e γ − − = = (8) At high sampling time limits (→0) the delta discrete-time frequency variable () coincides with the continuous-time frequency variable (s) as follows and it is the philosophy which is capitalized in this work. 2 2 0 00 1 .... 1 1 2! lim lim lim s s s e s  → →→  +  + + − − = = =   (9) To obtain the mapping between s and  , we need to replace sz e = in Eq. (8) as shown above. After taking logarithm on both sides the relationship between the two domains can be established by Eq. (10). 1 ln(1 )s = +   (10) Now, ln(1 )+ function is approximated in a closed form and the CFE expansion is made possible. Upon applying different Trapezoidal quadrature rule [32], the close form approximation of ln(1 )x+ is obtained through 2P-GILOG approximation as follows: 2 2 66 36 )1ln( xx xx x ++ + + (11) This Approximation is known as 2P-GILOG. A New Approach for Direct Discretization of Fractional order Operator in Delta Domain 317 Now replacing x by  in Eq. (11), the expression becomes, 2 2 6 3( ) ln(1 ) 6 6( ) ( )       +  +   +  +  (12) The Eq. (10) is re-established by using Eq. (12) and Eq. (13) and is obtained as follows: 2 2 2 1 6 3 ln(1 ) 6 6 s       +   = +       +  +     (13) At fast sampling limit ( 0 → ) the discrete-time frequency variable (  ) in delta domain coincides with the continuous-time frequency variable ( s ) as can be found out from Eq.(13) Therefore, at fast sampling limit, the complex variable in continuous domain is approximated as the complex variable in discrete delta domain. A FO differentiator is framed as: ( ) (0 1) r G s s r=   (14) CFE2P-GILOG method is used for discretization of r s directly in delta domain. The fractional order operator discretization is accomplished in two stages. Initially, the required generating function is selected and that is going to define the approximate mapping between delta discrete-time variable (γ) and continuous-time variable ( s ). In the next stage, to obtain the discrete time approximation of rs in the form of transfer function in delta domain, the selected generating function is expanded. In this work, Eq. (13) is chosen as the generating function and CFE method is used to expand it to get respective integer order approximation of sr in delta domain.                 ++ +   r r CFEGs 2 2 )()(66 )(36 )( (15) The mathematical expression for CFE approximation is as follow: .....2 )3( 5 )2( 2 )2( 3 )1( 2 )1( 1 1)1( + − + + + − + + + − + +=+ pr pr pr pr pr rp p r (16) 2 2 2 6 3 1 6 6  +  − +  +   is substituted in place of p in the Eq. (16) to get the equivalent form of Eq. (15). Now executing CFE approximation of 2 2 2 6 3 6 6 r   +      +  +      for third order, and fifth order in delta domain are obtained as given in Eq. (17) and Eq. (18) respectively. 3122030 3323130 22 2 3 66 36 )( bbbb aaaa CFEGs r r +++ +++ =                 ++ + = −−− −−−  (17) 318 S. K. DOLAI, A. MONDAL, P. SARKAR 5 1 4 2 3 3 2 4 1 5 0 5 1 4 2 3 3 2 4 1 5 0 22 2 5 66 36 )( bbbbbb aaaaaa CFEGs r r +++++ +++++ =                 ++ + = −−−−− −−−−−  (18) Table 1 Numerator coefficients for fifth order approximation in Delta Domain 15 14 13 12 5 11 10 9 8 7 6 (( 1)( 2)(1073741824 16911433728 13354663936 1002254106624 3869945888768 20886278111232 129327405203456 138817498447872 1829934470742016 712034173267968 12 num D r r r r r r r r r r r r = + + + + − − + + − − − + 5 4 3 2 608455533286400 13516106683236096 41479456532696640 59408759887249392 55098059015583104 92016444345172880))) r r r r r + − − + + Coefficients Numerator 0a 17 16 15 14 13 12 11 10 9 ( ((3 ) (1073741824 20132659200 66236448768 928367247360 6849998880768 7271932231680 184246347759616 290937273384960 1987732155678720 6479472582389760 8 681248407199 r / r r r r r r r r r r −  − + + − − + − + + 7 6 5 4 3 2 5 8464 49917404936559360 24285774583584448 156814916118867136 206087133711558336 138493101617423408 386245451066684864 184032888690345760)) num r r r r r r r / D + + − + − 1 a 15 16 14 13 12 11 10 9 8 ( ((3 ) (140928614400 8053063680 320612597760 7395229040640 42899897057280 102849152286720 1256995122708480 791103107235840 15398862690017280 30803 - / r r r r r r r r r r 7 6 5 4 3 2 5 473842032640 81700461363356160 272889395307594240 112347728802349920 1002215818766418432 425460690581907136 1423614499033773056 1274294568187684864)) num r r r r r r r / D 2a 2 15 2 14 2 13 2 12 2 11 2 10 2 9 2 8 2 7 ( ((3 / ) (28185722880 422785843200 65179484160 26212722278400 87432002273280 570237360537600 3060794314260480 4379676740812800 43583729456762880 7143762905395 r r r r r r r r r r −   −  +  +  −  −  +  +  −  + 2 6 2 5 2 4 2 3 2 2 2 2 5 200 300546723808853760 267563840484326400 992987073349200768 1262317112875803648 1327256882051046912 2019580911193378048 )) / num r r r r r r D  +  −  −  +  +  −  3a 3 13 3 14 3 12 3 11 3 10 3 9 3 8 3 7 3 6 (-((3 / ) (634178764800 - 56371445760 2247811399680 - 44392513536000 12672785448960 1194755633971200 - 1827599513026560 - 15451885751500800 32314828390440960 1020310420 r r r r r r r r r r      +  +    +  + + 3 5 3 4 3 3 3 2 3 3 5 11494400 - 243411076800568320 -330417309695155200 842112615992487808 436883676571452608 - 1161233166549210368 )) / num r r r r r D    +  +   4a 4 13 4 12 4 11 4 10 4 9 4 8 4 7 4 6 4 5 ( ((3 / ) (63417876480 396361728000 4510596464640 27834502348800 122752979435520 750819041280000 1602561749483520 9729145062604800 10683837557406720 643703179525 r r r r r r r r r r −   −  −  +  +  −  −  +  +  − 4 4 4 3 4 2 4 4 5 05600 34908135243891840 208843823902442400 46559875383003776 276641682514481024 )) / num r r r r D  −  +  +  −  5a 5 12 5 10 5 8 5 6 5 4 5 2 5 5 ((3 / ) (31708938240 2255298232320 61376489717760 801280874741760 5341918778703360 17454067621945920 23279937691501888 )) / r num r r r r r r D   −  +  −  +  −  +  A New Approach for Direct Discretization of Fractional order Operator in Delta Domain 319 Table 2 Denominator coefficients of fifth order approximation in Delta domain 2 3 5 4 5 6 7 8 6 (( 1)( 2)(55098059015583104 - 59408759887249392 - 41479456532696640 13516106683236096 12608455533286400 - 712034173267968 -1829934470742016 -138817498447872 - 712034173267968 -182993447074 Dn r r r r r r r r r r r = + + + + 7 8 9 10 11 12 13 14 15 2016 -138817498447872 129327405203456 20886278111232 - 3869945888768 -1002254106624 13354663936 16911433728 1073741824 92016444345172880)) r r r r r r r r r + + + + + + Coefficients Denominator 0b 2 3 4 5 6 7 8 9 10 ((386245451066684864 138493101617423408 206087133711558336 156814916118867136 24285774583584448 49917404936559360 6812484071998464 6479472582389760 1987732155678720 290937273384960 r r r r r r r r r r + − − + + + − − + 11 12 13 14 15 16 17 5 184246347759616 7271932231680 6849998880768 928367247360 66236448768 20132659200 1073741824 184032888690345760)) / r r r r r r r Dn + + − − + + + + 1b 2 3 4 5 6 7 8 ((1274294568187684864 1423614499033773056 -425460690581907136 -1002215818766418432 -112347728802349920 272889395307594240 81700461363356160 - 30803473842032640 -15398862690017280 791 r r r r r r r r  +     +  +    + 9 10 11 12 13 14 15 16 5 103107235840 1256995122708480 102849152286720 -42899897057280 - 7395229040640 320612597760 140928614400 8053063680 )) / r r r r r r r r Dn  +  +    +  +  +  2b 2 2 2 2 2 3 2 4 2 5 2 6 2 7 ((1327256882051046912 2019580911193378048 1262317112875803648 992987073349200768 267563840484326400 300546723808853760 7143762905395200 43583729456762880 4379676740812800 r r r r r r r  +  −  −  +  +  −  −  −  2 8 2 9 2 10 2 11 2 12 2 13 2 14 2 15 5 3060794314260480 570237360537600 87432002273280 26212722278400 65179484160 422785843200 28185722880 )) / r r r r r r r r Dn +  +  −  −  +  +  +  3b 3 3 3 2 3 3 3 4 3 5 3 6 3 7 3 ((436883676571452608 1161233166549210368 842112615992487808 330417309695155200 243411076800568320 102031042011494400 32314828390440960 15451885751500800 1827599513026560 r r r r r r r  +  −  −  +  +  −  −  +  8 3 9 3 10 3 11 3 12 3 13 3 14 5 1194755633971200 12672785448960 44392513536000 2247811399680 634178764800 56371445760 )) / r r r r r r r Dn +  −  −  −  +  +  320 S. K. DOLAI, A. MONDAL, P. SARKAR 4b 4 4 4 2 4 3 4 4 4 5 4 6 4 7 4 8 ((46559875383003776 276641682514481024 208843823902442400 34908135243891840 64370317952505600 10683837557406720 9729145062604800 1602561749483520 750819041280000 12275 r r r r r r r r  +  −  −  +  +  −  −  +  + 4 9 4 10 4 11 4 12 4 13 5 2979435520 27834502348800 4510596464640 396361728000 63417876480 )) / r r r r r Dn  −  −  +  +  5b 5 5 2 5 4 5 6 5 8 5 10 5 12 5 5 ((23279937691501888 17454067621945920 5341918778703360 801280874741760 61376489717760 2255298232320 31708938240 )) ) / r r r r r r Dn  −  +  −  +  −  +   4. SIMULATION AND RESULT ANALYSIS To prove the effectiveness of the portrayed approach, three examples are taken. Example 1: A 1/4th order differentiator is considered in this example [25] with transfer function as shown below: 0.25 ( ) r G s s s= = (19) The direct discretization of 1/4th order differentiator in delta domain is expressed as follows: 0.25 2 0.25 2 2 2 0.01 6 3 ( ) 6 6 P GILODEL s G CFE − =    +         +  +     (20) The third and fifth order approximation of 0.25s in delta domain after continued fraction expansion of                   ++ + 25.0 22 2 66 36   results in Eq. (21) and Eq. (22) respectively. The sampling time is considered to be 0.01s = 0.25 2 0.25 2 3 2 20.01 0.01 7 3 2 7 3 2 6 3 ( ) 6 6 5.48 0.0003722 0.06519 1.8 1.317 0.0001026 0.02198 1 P GILODEL s G CFE − = = − −    +     =    +  +      +  +  + =  +  +  + (21) 0.25 2 0.25 2 5 2 20.01 0.01 11 5 8 4 5 3 2 12 5 9 4 6 3 2 6 3 ( ) 6 6 3.238 3.685 1.466 0.002369 0.1322 1.439 7.781 9.632 4.252 0.0007911 0.05562 1 P GILODEL s G CFE                − = = − − − − − −   +    =    +  +    + + + + + = + + + + + (22) A New Approach for Direct Discretization of Fractional order Operator in Delta Domain 321 For G2P−GILOGDel5() the denominator and numerator coefficient are calculated using Table 1 and Table 2 taking r = 0.25 and  = 0.01. The frequency responses of delta domain transfer functions, G2P−GILOGDel3() and G2P−GILOGDel5 are shown in Fig. 1. The magnitude and phase error of the third order and fifth order approximate transfer function with respect to the original 1/4th order differentiator are demonstrated in Fig. 2. It can be seen through the graph that as the order of approximation goes higher, the precision of approximation gets better. Fig. 1 Fifth order and third order approximation of 0.25 s in delta domain using proposed method Fig. 2 Error comparison between fifth order and third order approximation of 0.25 s in delta domain using proposed method While taking the whole range of frequency into consideration, the magnitude is more accurate as compared to the phase response. The approximation is compared on the basis of the maximum absolute magnitude and phase error as shown in Table 3. As we can see 322 S. K. DOLAI, A. MONDAL, P. SARKAR that the approximation results for the fifth order are more prominent than those of the third order, therefore fifth order CFE approximation has been chosen to develop the frequency responses for the different systems considered in this paper. At a sampling time of 0.01s = , the fifth order discrete realization of 1/4th order differentiator is considered based upon the four methods described in this paper namely CFE of Al-Alaoui (CFEAL), CFE of Tustin (CFETO), CFEDO and CFE of 2P GILOG in Delta domain (CFE2P- GILOGDel) and following results are obtained. -1 -2 -3 -4 -5 5 -1 -2 -3 -4 -50.01 (1409-3221z +2435z -639.5z +6.82 z +5.449 z ) ( ) (430.9-861.9 z +533.6 z -90.88 z -7.06 z +z ) Al G z = = (23) -1 -2 -3 -4 -5 5 -1 -2 -3 -4 -50.01 (226.5 - 56.63 - 245.4 43.65 51.03 - 3.761 ) ( ) (60.24 15.06 - 65.25 -11.61 13.57 ) Tus z z z z z G z z z z z z= + + = + + + (24) 11 8 4 5 3 2 2 5 12 5 9 4 6 3 20 01 3.238 3.685 1.466 0.002369 0.1322 1.439 ( ) 7.781 9.632 4.252 0.000791 0.05562 1 P GILOGDel Δ . γ γ γ γ γ G γ γ γ γ γ γ − = + + + + + = + + + + + (25) 5 5 4 5 3 5 2 4 5 5 4 4 5 3 5 2 50.01 7157 1.282 4.186 3.512 7.025 2057 ( ) 2417 7.373 3.56 4.158 1.242 6526 CFEDO G =  +  +  +  +  +   +  +  +  +  + (26) Table 3 Absolute maximum phase error and magnitude error for discretization of 0.25th - order differentiator using CFE2P-GILOGDel Approximation order Maximum magnitude error (dB) Maximum phase error (degree) Fifth 0.92 7.7415 Third 1.27 30.5 Example 2: A fractional order system [25] is considered: 1 0.97 2.813 ( ) 0.191G s s = + (27) For the discretization of the above system, sampling time considered is s0001.0= . The discretization of this continuous time transfer function results in four rational approximation T.F. as given by Eq. (28), Eq. (29), Eq. (30) and Eq. (31), by using four methods CFEAL, CFETO, CFEDO and CFE2P-GILOGDel, respectively,. 7 7 1 7 2 7 3 6 4 4 5 5 7 8 1 8 2 7 30 0001 7 4 5 5 1 693 4 564 4 33 1 665 2 032 2 54 ( ) 8 85 2 387 2 266 8 719 1 064 1 33 - - - - - Al - - -Δ . - - . - . z + . z - . z + . z + . z G z . - . z + . z - . z + . z + . z = = (28) A New Approach for Direct Discretization of Fractional order Operator in Delta Domain 323 5 5 -1 5 -2 5 -3 4 -4 4 -5 5 -1 -2 -3 -4 -50.0001 (3.142 -3.048 z - 2.177 z + 2.052 z + 1.822 z -1.486 z ) ( ) (21.15+20.51z -14.65 z -13.81 z +1.226 z +z ) Tus G z = = (29) 18 4 13 4 9 3 5 2 5 18 5 13 4 8 3 20.0001 7157 1.076 3.584 4.511 0.1549 9.672 ( ) 6.698 5.628 1.874 0.0002356 0.8057 CFEDO G − − − − − −=  +  +  +  +  +  =  +  +  +  +  35.91+ (30) 5 4 4 5 3 6 2 5 2 5 4 5 5 4 5 3 5 20 0001 4238 3.512 7.721 1.341 6.34 6.204 ( ) 2.207 2.25 4 603 2.431 2861 24.51 P GILOGDel Δ . γ γ γ γ γ G γ γ γ . γ γ γ − = + + + + + = + + + + + (31) Example 3: The FO system [14] is chosen and the transfer function is as follows: 2 0.638 41.89 4 68) 28.(G s s = + (32) Here the sampling rate is made higher and that is considered as 0.00001s = . The discretization of this continuous time transfer function results in four rational approximation T.F., as given by Eq. (33), Eq. (34), Eq. (35) and Eq. (36), by using four methods CFEAL, CFETO, CFEDO and CFE2P-GILOGDel, respectively. 8 9 1 9 2 8 3 7 4 6 5 5 6 6 1 6 2 6 30 00001 5 4 5 8.257 2.07 1.782 5.872 4 794 3.057 ( ) 1.926 4 829 4.156 1.37 1.118 7132 Al Δ . z z z . z z G z . z z z z z − − − − − − − −= − − − + − + + = − + − + + (33) 7 7 1 7 2 7 3 6 4 6 5 5 4 4 1 4 2 4 30 00001 4 5 2 732 1 743 2 541 1 277 4 282 1 033 ( ) 6 372 4 065 5 927 2 978 9987 2410 Tus Δ . . . z . z .. z . z . z G z . . z . z . z z z − − − − − − − −= − − − − + + − = − + + + − (34) 6 5 7 4 8 3 8 2 7 4 5 4 5 5 4 5 3 5 20 00001 4 5 662 7.637 2 036 1.436 2.053 7 853 ( ) 1 315 1 751 4 964 2.913 3.077 546.5 CFEDO Δ . . γ γ . γ γ γ . G γ . γ . γ . γ γ γ = + + + + + = + + + + + (35) 21 5 15 4 9 3 2 4 2 5 23 5 17 4 12 30 00001 7 2 5 507 5 827 2 116 0 0003013 13 34 7 089 ( ) 1 285 1 359 4 936 7 029 0 03112 165 3 P GILOGDel Δ . . γ . γ . γ . γ . γ . G γ . γ . γ . γ . γ . γ . − − − − − − −= + + + + + = + + + + + (36) 324 S. K. DOLAI, A. MONDAL, P. SARKAR Fig. 3 Frequency response comparison after discretization of G(s) using four methods at 0.25r = and 01.0= Fig. 4 Frequency response comparison after discretization of G1(s) using four methods 0.97r = and 0001.0= Fig. 5 Frequency response comparison after discretization of G2(s) using four methods at 0.638r = and 0.00001= A New Approach for Direct Discretization of Fractional order Operator in Delta Domain 325 Four different discretization methods are utilized to discretize three fractional order systems as shown in three examples. The frequency responses of all the systems (fractional order) along with the frequency responses of their corresponding discrete-time approximated systems are shown in Fig. 3, Fig. 4, and Fig. 5, respectively. In all the discretization methods magnitude approximation turns out to be superior over the phase approximation. From the Fig. 3, Fig. 4 and Fig. 5, it is evident that the proposed method, CFE2P-GILOGDel produces excellent frequency responses in the frequency range of (0.001 rad/s to 1000rad/sec). Therefore, through experimental analysis, the proposed method is more promising than the other three approaches for discretization with respect to approximation of original fractional order system. Moreover, the comparison of the outcomes with another method developed in the delta domain been made and superiority of the proposed method is established. The CFE2P-GILOGDel method at high sampling time ( 0.00001 = ) provides frequency responses very much closer to the original fractional order system as can be seen from Fig. 5. This leads to a development of a unified approach towards the discretization of fractional order operator or system in complex delta domain means at high sampling rate the continuous time result and discrete time results can be obtained at the same time and is a sole reason for the development of discrete time systems’ in delta operator parameterization. Fig. 6 Magnitude and phase error after discretization of G(s) using four methods at 0.25r = and 01.0= Fig. 7 Magnitude and phase error after discretization of G1(s) using four methods at 0.97r = and 0.0001= 326 S. K. DOLAI, A. MONDAL, P. SARKAR Fig. 8 Magnitude and phase error after discretization of G2(s) using four methods at 0.638r = and 00001.0= Table 4 Absolute maximum magnitude error and phase error for four discretization methods for different systems FOS Max. magnitude error (dB) Max. phase error (degree) CFE2PG ILOGDel CFEDO Al- Alaoui Tustin CFE2P- GILOGDel CFEDO Al- Alaoui Tustin 0.25 ( )G s s= 0.72 1.06 1.11 1.2 7.74 18.3 44.79 44.88 1 0.97 2.813 ( ) 0.191G s s = + 1.66 2.12 5.83 24.27 44.46 45.1 79.83 88.02 2 .0.638 41.89 ( ) 428.68G s s = + 7.6 7.94 28.76 35.78 82.54 82.8 103.52 112.44 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 x 10 5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 111111 1 1 111111 1 1 1e+0052e+0053e+0054e+0055e+005 Pole-Zero Map Real Axis Im a g in a ry A x is CFE2P-GILOG 3rd order, r=0.97 Pole-Zero Map Real Axis Im a g in a ry A x is -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 x 10 5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 111111 1 1 111111 1 1 1e+0052e+0053e+0054e+0055e+005 CFE2P-GILOG 5th order, r=0.97 Fig. 9 Pole-zero plot for the third-order and fifth order approximation of 97.0 s using CFE2P-GILOGDel method A New Approach for Direct Discretization of Fractional order Operator in Delta Domain 327 -400 -350 -300 -250 -200 -150 -100 -50 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1 111111 1 1 50100150200250300350400 111111 1 Pole-Zero Map Real Axis Im a g in a ry A x is CFEDO, 3rd order, , r=0.97 Pole-Zero Map Real Axis Im a g in a ry A x is -10 0 10 20 30 40 50 60 -100 -80 -60 -40 -20 0 20 40 60 80 100 0.2 0.28 0.38 0.52 0.68 0.88 20 40 60 80 100 20 40 60 80 100 0.06 0.12 0.2 0.28 0.38 0.52 0.68 0.88 0.06 0.12 CFEDO, 5th Order, r=0.97 Fig. 10 Pole zero plot for the third-order and fifth order approximation of 97.0 s using CFE-DO method Pole-Zero Map Real Axis Im a g in a ry A x is -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0.2/T 0.3/T 0.4/T 0.5/T 0.6/T 0.7/T 0.8/T 0.9/T /T 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1/T 0.2/T 0.3/T 0.4/T 0.5/T 0.6/T 0.7/T 0.8/T 0.9/T /T 0.1/T Tustin 3rd order r=0.97 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0.2/T 0.3/T 0.4/T 0.5/T 0.6/T 0.7/T 0.8/T 0.9/T /T 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1/T 0.2/T 0.3/T 0.4/T 0.5/T 0.6/T 0.7/T 0.8/T 0.9/T /T 0.1/T Pole-Zero Map Real Axis Im a g in a ry A x is Tustini, 5th order, r=0.97 Fig. 11 Pole-zero plot for the third-order and fifth order approximation of 97.0 s using Tustin method -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0.2/T 0.3/T 0.4/T 0.5/T 0.6/T 0.7/T 0.8/T 0.9/T /T 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1/T 0.2/T 0.3/T 0.4/T 0.5/T 0.6/T 0.7/T 0.8/T 0.9/T /T 0.1/T Pole-Zero Map Real Axis Im a g in a ry A x is Al-alaoui, 3rd order r=0.97 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 /T 0.1/T 0.2/T 0.3/T 0.4/T 0.5/T 0.6/T 0.7/T 0.8/T 0.9/T /T 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1/T 0.2/T 0.3/T 0.4/T 0.5/T 0.6/T 0.7/T 0.8/T 0.9/T Pole-Zero Map Real Axis Im a g in a ry A x is Al-alaoui, 5th order, r=0.97 Fig. 12 Pole-zero plot for the third-order and fifth order approximation of 97.0 s using Al- Alaoui method 328 S. K. DOLAI, A. MONDAL, P. SARKAR From the Table 4, it is clearly observed that when the sampling time is increased to a very high limiting value 0.00001s = , the maximum absolute magnitude error and phase error is much higher in case of discretization using Tustin and Al-Alaoui method in Z- domain in comparison to the discretization using Delta operator parameterization. The graphical representation can also be viewed from Fig. 8. Also, it can be seen that the proposed method is superior to the other methods in the literature. At the same time, a comparison has been made for the fifth order approximation of s0.97 using another delta domain based approach, CFEDO method, where poles are in the right half of the plane Fig. 10, thus making the rational transfer function of the system unstable, whereas the method proposed in this paper shows that in both third order and fifth order the poles in the region itself are making the system stable. So, it is evident that the proposed method delivers preferable approximation amidst all four discretization methods and is a viable alternative in the literature of direct discretization of fractional order operator in delta domain. The following analysis has been done to prove the novelty of the direct discretization of fractional order operator (s, 0 <  < 1) over the indirect discretization of the fractional order operator in delta domain. For the illustration purpose, a 1/4th order differentiator is considered for the discretization purpose. This operator is discretized using indirect discretization using Oustaloup approximation [33] method as an intermediate step. Rational approximation of 25.0s is obtained using [33] as given in Eq. (37). 7 6 5 4 3 2 7 6 5 4 3 2 3 162 1899 2 411 05 7 763 06 6 586 07 1 472 08 8 343 07 1 07 834 3 1 472 05 6 586 06 7 763 07 2 411 08 1 899 08 3 162 07 . s + s + . e s + . e s + . e s + . e s + . e s + e s + . s + . e s + . e s + . e s + . e s + . e s + . e (37) Eq. (37) is discretized in delta domain to get the rational approximation of 25.0s . 7 6 5 4 3 2 7 6 5 4 3 2 3 162 1532 1 745 05 5 357 06 4 459 07 9 897 07 5 595 07 6 7 06 667 1 056 05 4 52 06 5 243 07 1 619 08 1 273 08 2 119 07 . γ + γ + . e γ + . e γ + . e γ + . e γ . e γ . e γ + γ + . e γ + . e γ + . e γ + . e γ + . e γ + . e + + (38) The rational approximation of 25.0s in delta domain using proposed direct discretization method is illustrated in Eq. (39) 5 4 3 2 5 4 3 2 2 55954 0 0235 0 000042 2 6066 ( 08) 6 5524 ( 12) 5 75821 ( 16) 0 00556 0 000007 4 25183 ( 9) 9 63178 ( 13) 7 7805 ( 17) . γ . γ . γ . e - γ . e - γ . e γ + . γ + . γ + . e - γ + . e + . e - + + + + − (39) A comparative analysis between the direct discretization and indirect discretization using delta operator based parameterization is graphically demonstrated in Fig. 13 and Fig. 14 respectively. A New Approach for Direct Discretization of Fractional order Operator in Delta Domain 329 Fig. 13 Frequency Response using Indirect Discretization of 25.0s at ∆=0.001s Fig. 14 Frequency response using direct discretization of 25.0s at ∆=0.001s From the above figure it is clear that using the direct discretization the magnitude and phase plot resembles that of the 1/4th order differentiator in continuous time domain, whereas there is a notable deviation of the magnitude and phase curve when indirect discretization is approached. Therefore, direct discretization of the fractional operator in delta domain is superior over indirect discretization. 330 S. K. DOLAI, A. MONDAL, P. SARKAR 5. CONCLUSION In this paper, a new direct discretization method for fractional order operator is proposed. The traditional discretization method for fractional order operator works in the discrete Z-domain and at a high sampling frequency, the resulting system fails to provide meaningful information. Instead, delta operator parameterized systems give continuous time results at high sampling frequency. In this work, an approximation mapping between the S-domain and delta domain is established through trapezoidal quadrature rule and traditional CFE, method is used to obtain rational transfer function corresponding to the fractional order operator in discrete delta domain. 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