This paper first shows that this geometric distribution is only a particular distribution case and that many other distributions (an infinity) are in fact possible. From the networks obtained, a class of partial differential equations (heat equation with a spatially variable coefficient) is then deduced. This class of equations is thus another tool for power law type long memory behaviour modelling, that solves the drawback inherent in fractional heat equations that was proposed to model anomalous diffusion phenomena.
Journal of Advanced Research 25 (2020) 243–255 Contents lists available at ScienceDirect Journal of Advanced Research journal homepage: www.elsevier.com/locate/jare Beyond the particular case of circuits with geometrically distributed components for approximation of fractional order models: Application to a new class of model for power law type long memory behaviour modelling Jocelyn Sabatier IMS Laboratory, Bordeaux University, UMR CNRS 5218, 351 Cours de la liberation, 33400 Talence, France g r a p h i c a l a b s t r a c t Power law type behaviour of a r t i c l e T ð0;sÞ uð0;sÞ i n f o Article history: Received 20 February 2020 Revised April 2020 Accepted April 2020 Available online 23 April 2020 Keywords: Power law type long memory behaviours Fractional models Cauer networks Foster networks Heat equation Poles and zeros geometric distributions a b s t r a c t In the literature, fractional models are commonly approximated by transfer functions with a geometric distribution of poles and zeros, or equivalently, using electrical Foster or Cauer type networks with components whose values also meet geometric distributions This paper first shows that this geometric distribution is only a particular distribution case and that many other distributions (an infinity) are in fact possible From the networks obtained, a class of partial differential equations (heat equation with a spatially variable coefficient) is then deduced This class of equations is thus another tool for power law type long memory behaviour modelling, that solves the drawback inherent in fractional heat equations that was proposed to model anomalous diffusion phenomena Ó 2020 The Authors Published by Elsevier B.V on behalf of Cairo University This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Introduction It is well known that the diffusion equation of the form Peer review under responsibility of Cairo University E-mail address: Jocelyn.sabatier@u-bordeaux.fr @/x; tị @ /x; t ị ẳ Df @t @x2 https://doi.org/10.1016/j.jare.2020.04.004 2090-1232/Ó 2020 The Authors Published by Elsevier B.V on behalf of Cairo University This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) ð1Þ 244 J Sabatier / Journal of Advanced Research 25 (2020) 243–255 produces power law type long memory behaviours of order 0.5 (Df is a diffusion coefficient) That is why the Warburg impedance, À1=2 , defined in the frequency domain (variable x) by Z jxị ẳ jxị was introduced to model numerous diffusion-controlled processes in many domains such as electrochemistry [44,23,42,36,3], solid-state electronics and ionics [15,41,2] However, it is also well-known that there are processes whose behaviour cannot be modelled by the Warburg impedance as they exhibit a power law type behaviour of the form Z jxị ẳ jxị m 0 s : ; xl ỵ m Z xl B sinmpị ẳ C0 @dtị þ p xh As a first try, the following change of variable is used in relation (14): x ¼ az ¼ ezlnðaÞ xh xb m ðxh À xÞ C dxA x xl ịm s ỵ xị 22ị m ILb sị % a0 ỵ N X s kẳ0 xk ak ỵ1 23ị dx ẳ lnaịe thus Hsị can be rewritten as: Z H sị ẳ where dtị is the Dirac impulse function According to the previous comments, with a Rỵ l ezlnaị zlnaị s ỵ ezlnaị lnaịe Z dz ẳ zlnaị lnðaÞ l ezlnðaÞ s ezlnðaÞ À1 dz: ð27Þ dz: ð28Þ ỵ1 This transfer function can be approximated by: Ha sị ẳ N X lnaị kẳ0 l ezk lnaị s ezk lnaị ỵ1 Dz 29ị with with m a0 ẳ C xxl ị ; h m sinmpị ðxh Àxk Þm p ðxk Àxl Þm xk Dx; h xh xb ak ẳ C xxl ị xk ẳ xh ỵ kDx; Dx ẳ N : 24ị m Fraction expansion of IN ðsÞ in relation (4) can also be written as ImLb sị ẳ a0 ỵ N X kẳ0 a0 k s x ỵ1 sinmpị p 25ị lnxmax Þ lnðaÞ Dx ¼ Þ À lnlnðxðmin aÞ N zk ẳ lnxmin ị ỵ kDz: lnaị 30ị mzk lnaị lnaịe Dz C k ẳ sinmpị p lnaịe mỵ1ịzk lnaị Dz 31ị and k C kỵ1 xkm1 ẳ m1 Ck xkỵ1 lnxmin ị lnaị Such a discretisation permits the realization of Fig with: Rk ¼ A comparison of coefficients ak and a0 k is given by Fig It reveals that, for a large value of N, ak % a0 k and that the two approximations are very close If the transfer function HðsÞ of relation (12) is considered again (for simplicity but a similar analysis can be done with the transfer functions of Table A1), relation (20) highlights that the capacitors and resistors are linked by the recurrence relations: m1 Rkỵ1 xkỵ1 ẳ m1 Rk xk z0 ẳ ð26Þ It can be noticed, that unlike relation (10), the ratios linking two resistors or two capacitors are not constant and depend on k This discretization can be viewed as an alternative solution to algorithm 1, but as parameter N must be very large to have an accurate approximation on a large frequency band, it requires a very large number of components in the network of Fig Such a defect is due to the fixed step discretization of integral (11) To overcome this defect, it is possible to search for a change of variable that contracts the frequency domain, thus making the fixed step discretization more efficient xk ¼ ẳ ezk lnaị : Rk C k 32ị If ¼ 0:4; a ¼ 10, N ¼ 10, xl ¼ 0:001 rd=s, xh ¼ 1000 rd=s, the Bode diagrams of the approximation Ha ðsÞ in relation (29) with change of variable (27) are shown in Fig They are very similar to those of Fig obtained with relation (18) and N ¼ 106 , thus showing the interest of the change of variable (27) in reducing the size of the approximation Remark Whatever the value of a, and as: zkỵ1 zk ẳ lnxmin ị lnxmin ị ỵ k ỵ ị Dz k Dz ¼ Dz lnðaÞ lnðaÞ ð33Þ It can be noticed that Rkỵ1 emzkỵ1 lnaị ẳ mz lnaị ẳ emDzlnaị Rk e k C kỵ1 1emỵ1ịzkỵ1 lnaị ẳ mỵ1ịz lnaị ẳ emỵ1ịlnaị k Ck e 34ị And xkỵ1 ẳ ẳ eDlnaị : Rk C k xk ð35Þ The previous relation highlights a geometric distribution of the values of resistors, capacitors and corner frequencies, defined by the following ratios: a ẳ emDzlnaị g ẳ emỵ1ịlnaị : 36ị This geometric distribution generalises the one introduced by Oustaloup [19,20] The latter is indeed a particular case obtained with a ¼ 10, among the infinite number of distributions obtained for all the other values of a, and for other changes of variable that can be proposed instead of relation (27) Among this infinity, the following one is interesting as it also makes it possible to contract the frequency domain Using the following change of variable zn ¼ x or z ¼ x1=n with n N thus dx ẳ nzn1 dz 37ị relation (14) can be rewritten as: Fig Comparison of coefficients ak and a0 k , with N ¼2000 and xl ¼ rd=s, xh ¼ 106 rd=s (zoom inside the figure) m ¼ 0:3, H ð sÞ ¼ sinðmpÞ p Z zmn sinmpị nzn1 dz ẳ s ỵ zn p Z n zmn1 dz s ỵ1 zn 38ị 248 J Sabatier / Journal of Advanced Research 25 (2020) 243–255 Fig Bode diagram of Ha ðsÞ with change of variable (27) and comparison with the Bode diagrams of approximation (18) and permits the network of Fig with: Rk ẳ sinmpị p nzkmn1 Dz; C k ẳ sinmpị p nzkmn1ỵn Dz Extension to Cauer type networks ; 39ị and xk ẳ ẳ znk : Rk C k 40ị If m ẳ 0:4; n ẳ 60, N ¼ 10, xl ¼ 0:001 rd=s, xh ¼ 1000 rd=s is the Bode diagrams of the approximation Ha ðsÞ obtained by discretisation of integral (38) and change of variable (37) are shown in Fig They are compared with the Bode diagrams obtained with change of variable (27) The comparison reveals that the two changes of variable are of equivalent quality with the same complexity (N ¼ 10) As an infinity of changes of variable can be proposed, an infinity of Foster type networks can be used to generate a power law behaviour The following section shows that Cauer networks can also generate this type of behaviour with an infinity of different distributions The Cauer network of Fig is considered For the geometric distribution, such as the one defined by relations (5) and (6), an analytical result can be obtained to show that a Cauer type network generates a power law behaviour Considering the network in Fig 7, the following relations hold Ik1 sị Ik sịị ẳ U k sị sC k 41ị and U kỵ1 sị U k sị ẳ Rk U k sị: 42ị From relations (41) and (42) respectively, it can be written that U k sị ẳ U k1 sị ỵ sC k sC k Ik ðsÞ U k ðsÞ and Fig Bode diagram of Ha ðsÞ with the change of variable (37) and comparison with change of variable (27) ð43Þ 249 J Sabatier / Journal of Advanced Research 25 (2020) 243–255 If the network results from an infinitesimal slicing of a continuous medium of abscissa z, the ratio of two consecutive components (capacitor or resistor) denoted Fis given by: F kỵ1 F kdz ỵ dzịdz ẳ F ðkdzÞdz Fk where dz denotes the thickness of the considered slices, with dz ! Given that Fig Cauer type RC network Ik sị ẳ U k sị þ Rk U iþ1 ðsÞ I i ðsÞ Rk : 44ị I sị ẳ U sị R1 1ỵ F kdz ỵ dzị F kdzị ẳ F kdzị dz ! dz sC R1 sC R 1ỵ 12 sC R2 1ỵ 1ỵ : 45ị F kdz ỵ dzị ẳ F kdzị ỵ F kdzịdz F kỵ1 F kdzị dz: ẳ1ỵ F kdzị Fk Rkỵ1 ẳ r and Rk C kỵ1 ẳ q; Ck with and if Z sị ẳ relation (45) becomes I sị ẳ U sị ỵ 46ị Z sị 1ỵ : 47ị 1ỵ Introducing the function 1ỵ Z sị=r Z sị=rq 1ỵ Z sị=r2 q 1ỵ Z sị=r2 q2 1ỵ Z sị=rN qN 1ỵZ sị=rNỵ1 qN 1ỵ After resolution of the differential equation (57), function F ðkdzÞ is given by z ẵ0; 1ẵ: 58ị This shows that the lineic characteristics of the discretized medium that produces the network of Fig are defined by: 1ỵZ sị=rNỵ1 qN Z sị 57ị F kdzị ẳ F ekf kdz Z sị=r2 q2 Z sị=rN qN 1ỵ g Z sị; r; qị ẳ 56ị F kdzị ẳ kf F kdzị Z sị=r Z sị=rq Z sị=r2 q 1ỵ F kdzị dz ẳ ỵ kf dz F kdzị R1 1ỵ 55ị If this ratio, only a function of dz, is assumed constant 8k as in relation (46) and equal to K, using relation (55), Kẳ1ỵ Hsị ẳ ð54Þ the ratio of relation (52) becomes Suppose that in Fig 7, the resistors and capacitors are geometrically distributed and linked by the following ratios (as in [19]): sC R1 ð53Þ where F ðzÞ denotes the derivative of F ðzÞ and thus Combining relations (43) and (44), it can be shown that the input admittance of the network in Fig is defined by the continued fraction: Hsị ẳ 52ị Rzị ẳ R0 ekR z 48ị and C zị ẳ C ekC z z ẵ0; 1ẵ: 59ị The ratio of two consecutive resistors and capacitors is thus defined by: Rkỵ1 Rk ỵ 1ịdxị ẳ ẳ ekR Dz Rkdxị Rk C kỵ1 C k ỵ 1ịdxị ¼ ¼ e kC D z : C ðkdxÞ Ck and relation (48) becomes Hsị ẳ R1 I sị ẳ : U sị ỵ g ðZ ðsÞ; r; qÞ ð60Þ ð49Þ If N tends towards infinity, function g ðZ ðsÞ; r; qÞ meets the following property: Property Function g ðZ ðsÞ; r; qÞ meets the following relation g Z sị; r; qị ẳ 1ỵgZZssịị;q;rị R1 K r; qịZ sịm x ẵ1; 1ẵ k and C k ẳ C kdxị ẳ C ðkdxÞ C dx ð62Þ The ratio of two consecutive resistors and capacitors is thus defined by: k ð50Þ ð51Þ ð61Þ With an infinitesimal slicing of the continuous medium, the system can be characterised by the network of Fig with: kR Rkỵ1 R0 k ỵ 1ịdxị R dx k ỵ 1ị ẳ ẳ k k Rk k ị R R0 ðkdxÞ R dx Using theorem 1, demonstrated in Appendix A.2, for jZ ðsÞj ) 1, r > 1), admittance HðsÞ meets the relation Hsị % and C xị ẳ C xkC k Theorem With N ! (as Rxị ẳ R0 xkR Rk ẳ Rkdxị ẳ R0 kdxị R dx Thanks to property 1, the function g ðZ ðsÞ; r; qÞ can be written under the form of a rational function with descending powers g ðZ ðsÞ; r; qị ẳ Kr; qịZ sịm mkỵ1 k P1 Z sị ỵ C 2k r; qị Zrsị ỵ kẳ1 C 2k1 r; qị r P mkỵ1 k 1ỵ C q ; r ị Z s ị ị ỵ C ð q ; r Þ ð Z ð s Þ Þ 2k k¼1 2kÀ1 Now consider the change of variable z ẳ log xị, x ẵ1; 1ẵ, then relation (59) becomes k C kỵ1 C k ỵ 1ịdxị C dx k ỵ 1ị ẳ ẳ k k Ck ðkÞ C C ðkdxÞ C dx and kC ð63Þ These ratios are similar to those given by relation (26) for the Foster circuit of Fig The following change of variable z ẳ log xn ị, x ẵ1; 1ẵ, n Nỵ is now considered Relation (62) thus becomes 250 J Sabatier / Journal of Advanced Research 25 (2020) 243255 Rxị ẳ R0 xnkR and C xị ẳ C xnkC x ẵ1; 1ẵ: 64ị With an infinitesimal slicing of the continuous medium, the system can be characterised by the network of Fig with: Rk ¼ RðkdxÞ ¼ R0 ðkdxÞ nkR dx nkC and C k ẳ C kdxị ẳ C kdxị dx: Rkỵ1 ẳ ekR Dz Rk Rkỵ1 k ỵ 1ị ẳ k Rk ðkÞ R ð65Þ The ratio of two consecutive resistors and capacitors is thus defined by: nk nk and nkC C kỵ1 C k ỵ 1ịdxị C dx k ỵ 1ị ẳ ẳ nk nk Ck k ị C C ðkdxÞ C dx : ð66Þ These ratios are similar to those given by relation (39) for the Foster circuit of Fig These networks and the associated distributions are used in the next section to introduce a class of heat equation that exhibits a power law type long memory behaviour Heat equation with spatially variable coefficients for power law type long memory behaviour modelling The following heat equation with spatially dependent parameters is now considered @T z; t ị @ @T z; tị ẳ czị bzị @t @z @z 67ị with z Rỵ This equation is a simplified form of the equation studied in [13] Let uz; tị ẳ bzị @T z; tị : @z ð68Þ Discretisation of equation (68) with a discretisation stepDzleads to: 69ị and thus: nkR czị ẳ Dz uz; tÞ: bðzÞ ð70Þ Using relation (69), relation (67) can be rewritten as: @T z; t ị @ uz; tị ẳ cðzÞ : @t @z Spatial discretisation of Eq (71) with a discretisation step Dz leads to: czị ẳ Dz ckDzị ẳ C kDzịDz and Rk ẳ 75ị nkC and C kỵ1 k ỵ 1ị ẳ nk Ck k Þ C ðrelation ð72ÞÞ C e kC z and bzị ẳ R0 ekR z 76ị z C zkC bzị ẳ and 77ị R0 zkR z ẵ1; 1ẵ relation 60ịị 78ị 1 and bzị ẳ C znkC R0 znkR ẵ1; 1ẵ relation 63ịị czị ẳ z 79ị Of course, as previously explained, many other spatially varying coefficients can be obtained using other changes of variable than those proposed at the end of Section ‘Extension to Cauer type networks’ Discussions around some other distributions for further Now, among the infinity of distributions that can be obtained using changes of variable as shown in Section ‘Beyond geometric distribution’, the following is studied: or x ¼ zÀ1=m 1 thus dx ẳ zm1 dz m 80ị Using this change of variable, relation (14) becomes: sinðmpÞ Z z 1m1 dz z p m ỵ1 s ỵ z m Z 1m sinmpị z ẳ dz p m s ỵ z1m H sị ẳ sinmpị 81ị mp Z s z m dz ỵ1 82ị and permits the realization of Fig with: Ha sị ẳ PN Rk kẳ0 Rk C k sỵ1 Rk ẳ sinmpmpị Dz ẳ Cte mpzm C k ẳ sinmpịk 72ị For z ẳ kDz and if the following notations are introduced Ck ¼ À ðrelation ð62ÞÞ or after simplification ð71Þ @T ðz; t ị uz ỵ dz; tị uz; tị ẳ czị @t Dz czị ẳ uz ỵ dz; t ị uz; t ịị: Dz 74ị kC ẵ0; 1ẵ: relation 59ịị H sị ẳ T z; t ị T z ỵ dz; tị ẳ C kỵ1 k ỵ 1ị ẳ k Ck kị C and Rkỵ1 k ỵ 1ị ẳ nk Rk kị R z ẳ x m ; T z ỵ dz; t ị T z; t ị uz; tị ẳ bzị Dz kR relation ð46ÞÞ and according to the relations (59), (60) and (63), the heat equation (67) exhibits a power law type long memory behaviour if (as czị ẳ 1=C zị and bzị ẳ 1=Rzị according to relation (73)) nkR Rkỵ1 R0 k ỵ 1ịdxị R dx k ỵ 1ị ẳ ẳ nk nk Rk ðk Þ R R0 ðkdxÞ R dx C kỵ1 ẳ e kC D z Ck and Dz ẳ RðkDzÞDz ð73Þ bðkDzÞ discretisation of Eq (67) thus leads to the Cauer network of Fig As C k ¼ C kDzịDz and Rk ẳ RkDzịDz, according to relations (46), (62) and (72), the transfer function uð0; sÞ=T ð0; sÞ of the Cauer network of Fig exhibits a power law type long memory behaviour if p Dz À1 and xk ẳ R 1C ẳ zk m : 83ị k k If ¼ 0:4; N ¼ 10; 000, xl ¼ 0:001 rd=s, xh ¼ 1000 rd=s the Bode diagrams of the approximation Ha ðsÞ are shown in Fig They are compared with the Bode diagrams of approximation (18) and the one obtained with change of variable (37) As for approximation (18), parameterNmust be very large to have an accurate approximation of sÀm on a large frequency band, but the interest of this change of variable is not there The distribution of resistors and capacitors of relation (83) is now used to build the Cauer network of Fig 7, with m ¼ 0:4, N = 1000, Dz ¼ and 251 J Sabatier / Journal of Advanced Research 25 (2020) 243–255 (0,t) T(0,t) R1 (z,t) C1 (k z,t) Rk ((k+1) z,t) Ck T((z,t) Rk+1 T(k z,t) Ck+1 Fig Cauer type RC network resulting from the discretization of relation (67) Fig Bode diagram of Ha ðsÞ of relation (83) with change of variable (80) and comparisons with the Bode diagrams of approximation (18) and the one obtained with change of variable (37) C0 ẳ mp : sinmpị 84ị The resulting Bode diagram of the transfer function I0 ðsÞ=U ðsÞ is represented by Fig 10 This diagram shows yet again that a power law behaviour can be obtained without a geometric distribution of resistors and capacitors In this circuit, all the resistors have the same values and the capacitors are linked by the following relation 1 C kỵ1 ððN À k À 1ÞDzÞm ðN À k À 1Þm ¼ ¼ : 1 Ck ððN À kÞDzÞm ðN À kÞm ð85Þ This class of components distribution, that cannot be deduced using a change of variable in relation (59), and the resulting class of spatially varying coefficients in relation (67) will be studied by the author in future work Fig 10 Bode diagram of transfer function I0 ðsÞ=U ðsÞ of the Cauer type RC network with distribution of relation (83) 252 J Sabatier / Journal of Advanced Research 25 (2020) 243255 Conclusion h1 t ị ẳ This paper shows that an infinity of – pole and zero distributions (frequency modes) in classical integer transfer functions, – passive component value distributions (such as capacitors or resistors) in Foster type networks, can generate power law type long memory behaviours Hence, the geometric distributions [19,20] often encountered in the literature are a particular case among an infinity of distributions For the Foster type network the proof is easy to establish using several changes of variables, as this network results directly from the discretisation of a filter transfer function that exhibits a power law behaviour The proof for the Cauer type network is more tedious and is developed in the paper Due to the close link between Cauer type networks and heat equations (through discretisation), this paper also shows the ability of heat equations with a spatially variable coefficient to have a power law type long memory behaviour This class of equation is thus another tool for power law type long memory behaviour modelling that solves the drawback inherent in fractional heat equations This class of equation will be more deeply studied by the author Finally, this paper shows, without proof, that other distributions and thus heat equations with spatially variable coefficients also exhibit power law type long memory behaviours Moreover, by increasing the number of components in each branch of the Cauer network, it is possible to keep a power law behaviour, which suggests that there are a very large number of partial differential equations, other than the heat equations, which can produce a power law type long memory behaviour, some were already proposed in [27] With reference to other papers recently published by the author [33,35], this work is a new contribution to the dissemination of models not based on fractional differentiation but which exhibit power law type long memory behaviours Z 2pj cỵj1 H1 sịets ds with c > Àxl : For the computation of integral (A1.2), path C ¼ c0 [ ::: [ c7 of Fig A11 is considered with c > Àxl This path bypasses the negative axis around the branching point z ¼ Àxl and z ¼ Àxh for t > It thus avoids the complex plane domain for which the transfer function H1 sị is not defined, i.e the segment ẵxh ; Àxl On path C, the radii of sub-path c1 and c7 tend towards infinity, and the radius of sub-path c4 tends towards Using Cauchy’s theorem with c > xl : h1 t ị ẳ ỵ 2pj X Z cỵj1 H1 sịets ds ẳ cj1 2pj Z H1 ðsÞets ds CÀc0 ðA1:3Þ Â Ã Res H1 sịets : poles in C Since ResẵH1 sịets ẳ 0; ðA1:4Þ operator H1 ðsÞ being strictly proper, by Jordan’s lemma integrals on the large circular arcs of radius R, R ? can be neglected: Z c1 ỵc7 H1 sịets ds ẳ A1:5ị Let s ẳ xejp , x 21; xh on c2 and thus ds ¼ ejp dx Let also s ¼ xeÀjp , x ½xh ; 1½ on c6 and thus ds ¼ eÀjp dx Then Z H1 sịets ds ẳ c2ỵ c6 xl m ỵ m1 xh Z xh xl m xh m1 Z xh jp m1 xe ỵ xh m xejp ỵ xl ị ext ejp dx m1 xejp ỵ xh ị xt jp m e e dx ẳ J c2 ỵc6 tị xejp ỵ xl ị Compliance with ethics requirements This article does not contain any studies with human or animal subjects Declaration of Competing Interest The author has declared no conflict of interest Appendix A.1 Impulse response of some transfer functions that exhibit power law type long memory behaviours The approximations given in Section ‘Beyond geometric distribution’ are made on the integral form of the impulse response of the transfer function Hsị ẳ s1m The methodology used to derive the approximations and the change of variable used in Sections ‘Beyond geometric distribution’ and ‘Extension to Cauer type networks’ can be extended to other transfer functions The following one is now considered: s xh H1 ðsÞ ẳ C ỵ1 s xl m1 ỵ1 m xl with C ¼ 2 xh ỵ1 ỵ1 The impulse response of H1 ðsÞ is defined by 2m : mÀ1 ðA1:2Þ cÀj1 ðA1:1Þ Fig A1.1 Integration path considered ðA1:6Þ 253 J Sabatier / Journal of Advanced Research 25 (2020) 243–255 or J c2 ỵc6 tị ẳ Z xl m xh m1 þ mÀ1 ejðmÀ1Þp ðx À xh Þ ejmp ðx À xl Þm xh xl m xh mÀ1 Z To make the demonstration easier to read, the following notam,m0 ,K,ak ,bk to denote respectively tions are used: mðr; qÞ,mðq; rÞ,K ðr; qÞ,ak ðr; qÞ,bk ðr; qÞ Relation (A2.1) can thus be rewritten: ejp eÀxt dx " eÀjðmÀ1Þp ðx À xh ÞmÀ1 Àjp Àxt e e dx eÀjmp ðx À xl Þm xh ðA1:7Þ m g ðZ ðsÞ; r; qÞ ẳ KZ sị J c2 ỵc6 tị ẳ 0: A1:8ị Let s ẳ xl ỵ qejh , h À p; p and thus ds ¼ jqejh dh with q ! 0, then c4 Z c4 mkỵ1 ỵ a2k Z sịk kẳ1 a2k1 Z sị P1 m kỵ1 ỵ b2k Z sịk kẳ1 b2k1 Z sị # : Also, let K ,a k ,b k denote the functions K ðq; rÞ,ak ðq; rÞ,bk ðq; rÞ The function g ðZ ðsÞ; q; rÞ is thus defined by: g Z sị; q; rị ẳ K Z sị m0 " # P m0 kỵ1 1ỵ ỵ a0 2k Z sịk kẳ1 a 2k1 Z sị : P mkỵ1 1ỵ ỵ b0 2k Z sịk kẳ1 b 2kÀ1 Z ðsÞ ðA2:3Þ Now using Z ¼ ZrðsÞ, the function g ZrðsÞ ; q; r is defined by: p mÀ1 Â Z Àp xl m xh m1 H1 sịets ds ẳ 1ỵ P1 A2:2ị and thus Z 1ỵ xl ỵ xh ỵ qejh ị m qejh ị jh jqejh etxl ỵqe ị dh: A1:9ị ỵ g Z; q; rị ẳ As q ! I1m sịets ds ẳ 0: ỵ K Zm ỵ P1 mkỵ1 ỵ K a0 2k1 Z kỵ1 ỵ b0 2k Z k kẳ1 b 2k1 Z P1 ỵ kẳ1 b 2k1 Z mkỵ1 þ b0 2k Z Àk þ K a0 2k Z m k A2:4ị A1:10ị or Let s ẳ xejp , x ½xh ; xl on c3 and thus ds ¼ ejp dx Also let s ¼ xeÀjp , x ½xl ; xh on c5 and thus ds ẳ ejp dx Then Z c3 ỵc5 H1 sịets ds ẳ xl m xh m1 ỵ Z xl xejp ỵ xh xejp ỵ xl ị xh Z xb m xh mÀ1 ÁmÀ1 xh xl m eÀxt ejp dx A2:5ị xl m H1 sịets ds ẳ xh m1 c3 ỵc5 Z xb xh xịm1 jmp e À eÀjmp ÞeÀxt dx ðx À xl Þm xl m xh m1 Z xh xb ẳ xh xịm1 xt e dx: ðx À xl Þm h i P mkỵ1 rm0 Zsị1r ỵ ỵ b0 2k Z sịk kẳ1 b 2k1 Z sị h i m0 P b0 2k1 0 mm0 kỵ1 K ỵ ZsKị0 ỵ ỵ a0 2k1 Z sịm kỵ1 ỵ bK2k0 Z sịm k ỵ a0 2k Z sịk kẳ1 K Z sị Term-to-term identification of relations (A2.1) and (A2.7) leads to the following equations: m ỵ m ẳ or mr; qị ỵ mq; rị ¼ Z H1 ðsÞe ds ts CÀc0 ðA1:13Þ using relations (A1.3), (A1.4), (A1.10) and (A1.12), the impulse response of H1 ðsÞ is given by, for t > 0: sin ðmpÞ xl m p ðA2:6Þ ðA2:7Þ As h1 ðt Þ ẳ Z sị ỵ g Z sị; q; rị g Z sị; r; qị A1:12ị h1 tị ẳ 2pj g Z sị; r; qị ẳ and using relation (A2.5), the function g ðZ ðsÞ; r; qÞis given by: xh 2jsinmpị ẳ Taking into account property 1, m1 xejp ỵ xh ị xt jp m e e dx; A1:11ị xejp ỵ xl ị and then Z ỵ g ðZ; q; rÞ h i P b0 2kÀ1 ÀmÀm0 kỵ1 0 m0 ỵ ZK ỵ ỵ a0 2k1 Z m kỵ1 ỵ bK2k0 Z m k ỵ a0 2k Z k kẳ1 K Z m0 ẳK Z P mkỵ1 ỵ b0 2k Z k 1ỵ kẳ1 b 2k1 Z xh mÀ1 Z xh xb ðxh À xÞmÀ1 Àxt e dx: x xl ịm A1:14ị and Kẳ rm0 K0 or K r; qịK q; rị ẳ rmq;rị : A2:9ị Eq (A2.9) is symmetric in relation to q and ble to write that r It is thus possi- rmq;rị ẳ K r; qịK q; rị ẳ K q; rịK r; qị ¼ qmðr;qÞ The same calculation can be done, with other transfer functions, giving Table A1.1 Appendix A Demonstration of theorem A2:10ị and thus rmq;rị ẳ qmr;qị A2:11ị or taking the logarithm of relation (A2.10) Suppose that the function g ðZ ðsÞ; r; qÞ is given by: mðq; rÞlog ¼ mðr; qÞlog ðqÞ: mðr;qÞ g ðZ ðsÞ; r; qÞ ẳ K r; qịZ sị " # P mr;qịkỵ1 1ỵ ỵ a2k r; qịZ sịịk kẳ1 a2k1 r; qịZ sịị P1 mq;rịkỵ1 k ỵ kẳ1 b2k1 r; qịZ sịị ỵ b2k r; qịZ sịị A2:1ị where < mðr; qÞ < and < mðq; rÞ < ðA2:8Þ ðA2:12Þ Using relation (A2.8), relation (A2.12) permits the following equations mr; qị ẳ log rị log rị ỵ log qị and mq; rị ẳ log qị : log rị ỵ log qị A2:13ị 254 J Sabatier / Journal of Advanced Research 25 (2020) 243–255 Table A1.1 Table of inverse Laplace transform of some transfer functions with power law type long memory behaviour Hi ðsÞ PL b s bl PKlẳ0 l a hi tị Pr Pni r Y t ịesi t jẳ1 ij j iẳ1 a s k kẳ0 k ỵ p1 R ỵ1 PK PL a b sinak bl ịpịxak ỵbl lẳ0 k l P PK kẳ0 2a a2 x k ỵ kẳ0 k a pịxak þal a a cosðð k À l Þ k l K k l a eÀxt dx From [17] with demonstration in [26] si are the poles sinmpị R ỵ1 xt dx p xm e sinmpị R ỵ1 xl m eÀxt dx p ðxÀxl Þm sm m xl ỵ1 s xl m1 s xh þ1 sinðmpÞ p m xl m R xh ðxh ÀxÞmÀ1 Àxt e dx xh mÀ1 ðxÀxl Þm xl xl þ1 s m xh þ1 m s xl x ị h m xl ỵ1 dtị ỵ sinpmpị R xh s s xl s 1m ỵ1 1m x h 1m xl xh ỵ1 xb mịpị He tị ỵ sin p s p z asỵb ep asỵb a > 0; b > 0; z>0 p b=a b2kÀ1 k NÃ with b0 2k1 m0 ỵk1 ẳ a0 2k1 ỵ r K and b0 ẳ A2:15ị a2k ẳ b 2k r b2k with k N k b0 2k1 k ẳ a0 2k ỵ r K0 with k NÃ : Eq (A2.14) being symmetric in relation to q and rewritten as m0 ỵk1 b2k1 ẳ b2k1 qrị b0 2k2 m0 ỵk1 ỵ r K0 He t ị: Heaviside step function b2k ẳ K0 b rk 2kÀ1 À ðqrÞk k NÃ with and b0 ¼ K0 1Àr or given relation (A2.21) b2kÀ2 rk qkÀ1 b2k ¼ À qrịk qrịmỵk1 with k N and b0 ¼ ðA2:25Þ ðA2:17Þ b0 2kÀ2 qk rkÀ1 b0 2k ẳ qrịk qrịm ỵk1 with k N Kb 2k2 r b2k1 ẳ qrịm ỵk1 A2:18ị The set of relations (A2.13), (A2.21) and (A2.25) thus prove that the function g ðZ ðsÞ; r; qÞmeets the relation m with k N Ã and b ¼1 g Z sị; r; qị ẳ Kr; qịZ sị mkỵ1 k P1 Z sị ỵ C 2k q; rị Zrsị ỵ kẳ1 C 2k1 q; rịKq; rị r P m0 kỵ1 k 1ỵ C r ; q ịK r ; q Þ ð Z ð s Þ Þ þ C ð r ; q Þ ð Z ð s ị ị 2k1 2k kẳ1 A2:27ị with k N and b ẳ 1: A2:20ị K b2k2 qk1 b 2k1 ẳ qrịmỵk1 with k N and b0 ẳ 1: A2:21ị Also, Eq (A2.16) being symmetric in relation toqandr, it can be rewritten as a0 2k ¼ b2k qk with k NÃ and thus b0 2kÀ2 k r K0 log rị log rị ỵ log qị c2k r; qị ẳ k NÃ and with k NÃ qrị k k1 r q qrịmỵk1 1Àq ðA2:28Þ c2kÀ2 ðr; qÞ c0 ðr; qÞ ¼ rkÀ1 c1 ðr; qÞ ¼ ðA2:23Þ k < mr; qị < c2k1 r; qị ẳ qrịm ỵk1 A2:22ị enabling Eq (A2.17) to be rewritten as b2k ẳ b2k1 qrịk ỵ with mr; qị ẳ Eq (A2.20) being symmetric in relation toqandr, it can be rewritten as 0 and b ¼ 1: or using relation (A2.9): kÀ1 r ðA2:26Þ ðA2:19Þ ðA2:24Þ The symmetric form of relation (A2.25) in relation to q and permits to write enabling Eq (A2.15) to be rewritten as m0 ỵk1 ext dx A2:16ị r, it can be with k NÃ a0 2kÀ1 ¼ b2kÀ1 r xl ! ðxÀxl Þ1Àm xðxh ÀxÞ1Àm axÀb ðA2:14Þ with k N R xh dtị: Dirac impulse R ỵ1 cosðzpffiffiffiffiffiffiffiffi axÀbÞ Àxt pffiffiffiffiffiffiffiffi e dx Term-to-term identification of relations (A2.1) and (A2.7) also permits the following equations: a2kÀ1 ¼ b0 2k1 rmỵk1 ! xh xịm dx xxl ịm sỵxị ðA2:29Þ with k NÃ À f1g and ðA2:30Þ In relation (A2.27), only coefficient Kðr; qÞ remains to be computed It is possible to give an expression of Kðr; qÞ in the form of a ratio of two series as Eq (A2.27) can be rewritten as: J Sabatier / Journal of Advanced Research 25 (2020) 243–255 m4 g ðZ sị; r; qị ẳ Kr; qịZ sị 3 ỵ Kq; rịh1 Zrsị ; q; r ỵ h2 Zrsị ; q; r ỵ Kr; qịh1 Z sị; r; qị ỵ h2 Z sị; r; qị ðA2:31Þ or using relation (A2.9) m ỵ Krr;qị h1 Zrsị ; q; r ỵ h2 Zrsị ; q; r 5; ỵ Kr; qịh1 Z sị; r; qị ỵ h2 Z sị; r; qị g Z sị; r; qị ẳ Kr; qịZ sị m4 A2:32ị with h1 Z sị; r; qị ẳ X C 2k1 r; qịZ sịm kỵ1 ; A2:33ị C 2k r; qịZ sịk : A2:34ị kẳ1 and h2 Z sị; r; qị ẳ X kẳ1 Now usingZ sị ẳ 1in relation (A2.32), coefficient Kr; qị is given by: Kr; qị ẳ ! g 1; r; qị ỵ h2 ð1; r; qÞ À rm h1 ð1; q; rÞ : h1 1; r; qị ỵ h2 1; q; rÞ ðA2:35Þ References [1] Bingi K, Ibrahim R, Karsiti MN, Hassam SM, Rajah V Frequency Response Based Curve Fitting Approximation of Fractional-Order PID Controllers Int J Appl Math Comput Sci 2019;29(2):311–26 [2] Bisquert J, Garcia-Belmonte G Complex plane analysis of pn junction forwardvoltage impedance Electron Lett 1997;33(10):900–1 [3] Bisquert J, Garcia-Belmonte G, Fabregat-Santiago F, Bueno PR Theoretical models for ac impedance of finite diffusion layers exhibiting low frequency dispersion J Electroanal Chem 1999;475(2):152–63 [4] Bisquert J, Compte A Theory of the electrochemical impedance of anomalous diffusion J Electroanal Chem 2001;499:112–20 [5] Bourouba B, Ladaci S, Chaabi A Reduced-order model approximation of fractional-order systems using differential evolution algorithm J Control Autom Electr Syst 2018;29(1):32–43 [6] Caponetto R Fractional order systems: modeling and control applications Singapore: World Scientific; 2010 [7] Carlson GE, Halijak CA Approximation of fractional capacitors (1/s)1/n by a regular Newton process IEEE Trans on Circuit Theory 1964;11(2):210–3 [8] Charef A Analogue realisation of fractional-order integrator, differentiator and fractional PI/ˆ lambda/D/ˆ mu/ controller IEE Proc Control Theory Appl 2006;153:714–20 [9] De Oliveira E C., Tenreiro Machado J A (2014), A review of definitions for fractional derivatives and integral, Math Probl Eng 2014, Article ID 238459 [10] Evangelista LR, Lenzi EK Fractional diffusion equations and anomalous diffusion Cambridge University Press; 2018 [11] Ichise M, Nagayanagi Y, Kojima T An analog simulation of non-integer order transfer functions for analysis of electrode processes J Electroanal Chem Interfacial Electrochem 1971;33:253–65 [12] Khanra M, Pal J, Biswas K Rational approximation and analog realization of fractional order transfer function with multiple fractional powered terms Asian J Control 2013;15(3):723–35 [13] Levron F, Sabatier J, Oustaloup A, Habsieger L From partial differential equations of propagative recursive systems to non integer differentiation Fracti Calculus Appl Anal (FCAA): Int J Theory Appl 1999;2(3):246–64 [14] Li Z, Liu L, Dehghan S, Chen Y, Xue D A review and evaluation of numerical tools for fractional calculus and fractional order controls Int J Control 2017;90 (6) [15] Lindmayer J, Wrigley CY Fundamentals of semiconductor devices New York: Van Nostrand; 1965 255 [16] Manabe S The non-integer Integral and its Application to control systems ETJ Jpn 1961;6:83–7 [17] Matignon D Stability properties for generalized fractional differential systems ESAIM: Proc 1998;5:145–58 [18] Montseny G Diffusive representation of pseudo-differential time-operators ESAIM: Proc 1998;5:159–75 [19] Oustaloup A Systèmes Asservis Linéaires d’ordre Fractionnaire Paris (France): Masson; 1983 [20] Oustaloup A, Levron F, Mathieu B, Nanot F Frequency-band complex non integer differentiator: characterization and synthesis IEEE Trans Circ Syst I: Fundam Theory Appl 2000;47(1):25–39 [21] Petráš I Fractional derivatives, fractional integrals, and fractional differential equations in Matlab In: Assi A, editor Engineering education and research using MATLAB London: InTech; 2011 p 239–64 [22] Podlubny I Fractional differential equations Mathematics in sciences and engineering Academic Press; 1999 [23] Randles JEB Kinetics of rapid electrode reactions Discuss Faraday Soc 1947;1:11–9 [24] Raynaud HF, Zergaïnoh A State-space representation for fractional order controllers Automatica 2000;36:1017–21 [25] Sabatier J, Merveillaut M, Malti R, Oustaloup A How to impose physically coherent initial conditions to a fractional system? Commun Nonlinear Sci Numer Simul 2010;15(5) [26] Sabatier J, Farges C, Merveillaut M, Feneteau L On observability and pseudo state estimation of fractional order systems Eur J Control 2012;3:1–12 [27] Sabatier J, Nguyen HC, Moreau X, Oustaloup A Fractional behaviour of partial differential equations whose coefficients are exponential functions of the space variable Math Comput Modell Dyn Syst 2013;19(5):434–50 doi: https://doi.org/10.1080/13873954.2013.766805 [28] Sabatier J, Farges C, Trigeassou JC Fractional systems state space description: some wrong ideas and proposed solutions J Vib Control 2014;20(7):1076–84 [29] Sabatier J, Farges C, Fadiga L Approximation of a fractional order model by an integer order model: a new approach taking into account approximation error as an uncertainty J Vib Control 2016;22(8):2069–82 [30] Sabatier J, Farges C Comments on the description and initialization of fractional partial differential equations using Riemann-Liouville’s and Caputo’s definitions J Comput Appl Math 2018;339:30–9 [31] Sabatier J Solutions to the sub-optimality and stability issues of recursive pole and zero distribution algorithms for the approximation of fractional order models Algorithms 2018;11(7) p103 [32] Sabatier J, Rodriguez Cadavid S, Farges C Advantages of limited frequency band fractional integration operator in fractional models definition Conference on control, decision and information technologies, CoDIT 2019, Paris, France, April 23–26, 2019 [33] Sabatier J Power law type long memory behaviors modeled with distributed time delay systems Fract Fract 2020;4(1) [34] Sabatier J Non-singular kernels for modelling power law type long memory behaviours and beyond Cybernetics 2020 (accepted for publication) [35] Sabatier J, Farges C, Tartaglione V Some alternative solutions to fractional models for modelling long memory behaviors Mathematics 2020;8:196 doi: https://doi.org/10.3390/math8020196 [36] Yeager E, J.O.’M Bockris, B.E Conway, S Sarangapani,, editors Comprehensive treatise of electrochemistry, vol New York: Plenum Press; 1984 [37] Tartaglione V, Farges C, Sabatier J Dynamical modelling of random sequential adsorption Paper proposed to the European Control Conference ECC 2020, StPetersburg, Russia, 2020 [38] Tenreiro Machado JA, Jesus IS A suggestion from the past? Fract Calculus Appl Anal 2004;7(4):403–7 [39] Valerio D, Sá da Costa J Time-domain implementation of fractional order controllers IEE Proc – Control Theory Appl 2005;152(5):539 [40] Valério D, Trujillo JJ, Rivero M, Machado JT, Baleanu D Fractional calculus: A survey of useful formulas Eur Phys J Special Top 2013;222(8):1827–46 [41] Van Herle J, McEvoy AJ Oxygen diffusion through silver cathodes for solid oxide fuel cells J Phys Chem Solids 1994;55(4):339–47 [42] Vetter KJ Electrochemical kinetics New York: Academic Press; 1967 [43] Vinagre B, Podlubny I, Hernandez A, Feliu V Some approximations of fractional order operators used in control theory and applications Fract Calculus Appl Anal 2000;3(3):231–48 [44] Warburg E Ueber die Polarisationscapacität des Platins Ann Phys 1901;311 (9):125–35 [45] Wei Y, Wang J, Liu T, Wang Y Fixed pole based modeling and simulation schemes for fractional order systems ISA Trans 2019;84:43–54 ... diagrams of the approximation Ha ðsÞ are shown in Fig They are compared with the Bode diagrams of approximation (18) and the one obtained with change of variable (37) As for approximation (18), parameterNmust... yet again that a power law behaviour can be obtained without a geometric distribution of resistors and capacitors In this circuit, all the resistors have the same values and the capacitors are... discretisation of a heat equation, Section ‘Heat equation with spatially variable coefficients for power law type long memory behaviour modelling’ proposes a class of heat equation with spatially varying