A method for modelling and simulation of fractional systems

Transcription

1 Signal Processing 83 (23) wwwelseviercom/locate/sigpro A method for modelling and simulation of fractional systems T Poinot, J-C Trigeassou Laboratoire d Automatique et d Informatique Industrielle, Ecole Superieure d Ingenieurs de Poitiers, 4 Avenue du Recteur Pineau Poitiers Cedex, France Received 19 December 22 Abstract An original method for modelling and simulation of fractional systems is presented in this article The basic idea is to model the fractional system by a state-space representation, where conventional integration is replaced by fractional one with the help ofnon-integer integrator This operator is itselfapproximated by a N dimensional system composed ofan integrator and ofa phase-lead lter This method is compared to other techniques like direct discretization ofthe fractional derivator and diusive representation Numerical simulations exhibit the general applicability and exibility ofthis new approach to dierent types offractional models and to non-conventional non-integer derivation with limited spectral range? 23 Elsevier BV All rights reserved Keywords: Fractional systems; Fractional operator; Modelling; Simulation; State-space representation 1 Introduction Non-integer order systems, also known as fractional lters, have been introduced long ago in various elds ofscience such as electrochemistry [7], thermal engineering [1], acoustics [15], electromagnetism [22], etc where they are fundamentally used for the modelling ofdiusion processes These systems are characterized by long memory transients and innite dimensional structure Their dynamics depend on the well-known Diusion Equation and on the geometry ofthe considered problem Ifthe attention is focussed on the relation between variables at the boundary region, a theoretical modelling leads to an integrator Corresponding author Tel: ; fax: addresses: thierrypoinot@esipuniv-poitiersfr (T Poinot), jean-claudetrigeassou@esipuniv-poitiersfr (J-C Trigeassou) with order equal to 5 Generalization ofthis modelling to more complex situations needs the help ofa fractional model, characterized by its non-integer order, whose value can vary from and 1 Consequently, the modelling offractional systems turns out to be necessary for simulation, identication and control [4,8 14,18 2] This modelling is fundamentally based on noninteger derivation Then, the numerical simulation ofthese systems is highly linked to the modelling ofthe non-integer derivator, or equivalently ofthe non-integer integrator Two main approaches are commonly used A direct solution [19] is based on the discretization ofthe derivator: then, the fractional model is replaced by a dierence equation model, with long memory behaviour An indirect solution uses Diusive Representation [6,17], with nite discretization ofthe continuous fractional model into a N-dimensional state-space representation with conventional integer derivation /$ - see front matter? 23 Elsevier BV All rights reserved doi:1116/s (3)185-3

2 232 T Poinot, J-C Trigeassou / Signal Processing 83 (23) In this article, we propose a third approach based on a fractional integration operator in order to simulate a fractional system with a conventional state-space representation approach Fundamentally, simulation ofa state-space model needs integration operators; thus simulation ofa fractional model needs also a non-integer integration operator This new operator is dened in the frequency domain, with reference to the ideal integrator oforder n ( n 1) An approximation is necessary, because it is impossible to use an innite spectral range Thus the corresponding fractional operator acts with order n on a limited frequency domain and with order one outside this domain Moreover, this approximation is performed by a nite dimensional system or equivalently by its state-space representation Then, this fractional integration operator is used to simulate the corresponding system with an appropriate macro state-space representation Analysis ofthis operator permits a comparison with the other simulation techniques We demonstrate that it can be interpreted as a diusive representation system with special interesting features, permitting a general approach ofthe simulation offractional systems, without static errors This article is divided in four parts The rst one presents the main methods used in order to simulate fractional systems The second part is devoted to the denition and the modelling ofthe fractional integrator This new operator is used to dene the state-space representation ofa non-integer system in the third part Modelling ofdierent types offractional systems are illustrated by numerical simulations in the fourth part 2 Simulation of fractional systems Many methods are used in order to simulate non-integer or fractional systems Two types of methods can be considered The rst ones, also called direct methods, are based on an numerical approximation ofthe non-integer derivator operator The second ones, called indirect methods, are based on the simulation ofthe continuous fractional model, with the help ofa specic operator or representation 21 Direct methods In these methods the fractional derivator operator is replaced by a numerical approximation, in order to obtain a recurrent equation directly used for simulation Dierent types ofapproximations can be used The more usual approximation is the one directly related to the denition given by Grunwald [16]: d n dt n f(kh) ( ) 1 n =lim h h n ( 1) k f((k k)h); (1) k= k where h is the sampling period and ( ) n n(n 1)(n 2) (n k +1) = : (2) k k! In order to illustrate this approach, consider for instance the fractional system dened by d n y(t) dt n + a y(t)=b u(t) n 1: (3) This system will be used as a benchmark model for the dierent techniques ofsimulation Using previous numerical approximation, the following equation is obtained [19]: ( ) K ( 1) k n h n y((k k)h)+a y(kh) k= k =b u(kh); (4) where K is the number ofdata such as t = Kh The system output is given by y(kh)= b u(kh) K ( 1) ( k n ) k=1 h n k y((k k)h) a +1=h n : (5) This method is very simple to use On the other hand, the simulation requires, for each step, the computation ofsums ofincreasing dimension with time Other approximations can be used, refer for instance to [3,23] 22 Indirect method based on diusive representation Indirect methods refer to the simulation of the continuous fractional model with the help of a specic operator or representation Diusive Representation (DR) approach [6,17] is typically one ofthese methods

3 T Poinot, J-C Trigeassou / Signal Processing 83 (23) Consider a fractional system, whose impulse response is h(t): it is composed ofa weighted sum of an innity ofmodes e kt, where k varies from very small to very large values Thus K h(t)= ( k )e kt k : (6) k=1 Considering k and K, we get a continuous limit: h(t)= ()e t d; (7) where () is the weight density function of e t modes, it is called Diusive Representation (DR) Note that in Eq (7) h(t) can be interpreted as the transform (Laplace transform L) of (), where t s; t Thus, () can be obtained from h(t) using inverse transform (L 1 ): h(t)=l{()}; (8) ()=L 1 {h(t)}: (9) The main dierence with Laplace transform is that the two variables and t are only real ones An innite dimension state vector representation is attached to each fractional system Let u(t) be the input and y(t) the output; then the input/output representation ofthe system is given by y(t) =h(t) u(t) (1) or equivalently by Y (s)=h(s)u(s) (11) with H(s)=L{h(t)}: (12) Consider Eq (6) and dene c k = ( k ) k (13) then { K } K H(s)=L c k e c kt k = : (14) s + k k=1 k=1 Input/output representation (1) is equivalent to the state vector model derived from Eq (14): dx k (t) = k x k (t)+u(t) (k =1toK); (15) dt y(t)= K c k x k (t)= k=1 K ( k )x k (t) k : (16) k=1 Considering k and K, the continuous limit t) = x(; t)+u(t); y(t)= ()x(; t)d: (18) Numerical simulation ofa fractional system is based on discretization ofthe variable in Eqs (17) and (18), that is to say simulation is performed using Eqs (15) and (16) Consider previous benchmark model (3): H(s)= b a + s n n 1: (19) It is necessary to use the DR () corresponding to this model G Montseny has demonstrated [17] that: ()= b sin(n) n : (2) 2n cos(n) +2a n + a 2 The next step is the choice ofthe discretization for k A geometric progression for k and k is certainly a better choice than the arithmetic one because it reduces the number ofcomputations Thus we obtain the system (15) and (16) approximating the fractional model Finally, numerical integration ofthis system is the solution ofthe simulation problem Remark: the same approach can be used to simulate a non-integer integrator [6], whose impulse response is h i (t)= 1 (n) tn 1 t ; n 1 (21) and transfer function is I(s)= 1 s n : (22)

4 2322 T Poinot, J-C Trigeassou / Signal Processing 83 (23) This fractional integrator is characterized by the following DR [17]: ()= sin(n) n : (23) By discretization, a state space vector representation is attached to I(s), identical by its form to (15) and (16), diering essentially by the choice of c k weight Fundamentally, the fractional integrator is approximated by: I(s)= K k=1 c k s + k ; (24) where the weights c k are related to the DR (23) ρ db θ ω b ωb ω h ωh ω 9 n 9 ω Fig 1 Bode diagram ofthe fractional derivator 3 A fractional Integrator operator 31 Fractional derivator There are many ways to dene the ideal fractional derivator [19] Using the transform technique, let D(s)=s n be the Laplace transform of (d n =dt n ), where n 1 Then D(j!) is the Fourier transform de- ned by D(j!)=e j =! n ; = n 2 : (25) With this denition, the fractional domain where D(j!) acts as a non-integer derivator ranges from to A Oustaloup [18,19] has shown that the spectral range has to be necessarily limited to [! b ;! h ] Moreover, experiments show that this spectral range can be very limited, for instance over one decade when behaviour is caused by an articial diusion process [19] On the contrary, when the domain [! b ;! h ]is very large, there is not a great dierence with an ideal fractional derivator Thus, for narrow band derivators, it is necessary to dene in a realistic way the behaviour ofthe derivator outside the limited domain So, let us consider the Bode plot ofthe proposed practical derivator in Fig 1 It is composed ofthree parts: The intermediate part corresponds to non-integer action, characterized by the order n ( n 1) In the other two parts, the derivator has conventional action, characterized by its order 1 In this way, we dene a new operator D n (j!)(d n (s)) which is a conventional derivator, except on a limited band [! b ;! h ] where it acts like a n non-integer derivator Thus, this operator D n (s) is characterized by three parameters:! b,! h and n 32 Fractional integrator Practically, simulation ofsystems is not performed with derivators but with integrators It is easy to dene the practical non-integer integrator as the inverse of D n (s), thus I n (s)= 1 D n (s) : (26) So, our interest will be focussed on the modelling and simulation ofthe operator I n (s), whose Bode plot is dual ofthat ofd j! (s) This Bode diagram can be obtained using a transfer function like: I n (s)= G ( ) n n 1+(s=!b ) : (27) s 1+(s=! h ) The synthesis ofthis operator is performed by the association ofan integrator 1=s and ofthe conventional phase lead lter used by Oustaloup [19]: A v (j!)= N 1+j(!=! i) 1+j(!=! i ) (28) composed of N cells and characterized by four design parameters! i,! i,, (n is derived from and

5 T Poinot, J-C Trigeassou / Signal Processing 83 (23) ρ db θ ω b ωb ω h ωh log ω log ω n 9 9 Fig 2 Bode diagram ofthe fractional integrator according to (29)) Let us dene:! 1 : the lower pulsation,! N : the higher one with, for the ith cell:! i =! i with 1! i+1 =! i with 1: If N is suciently high, the Bode plot of A v (j!), inside [! 1 ;! N ], is characterized by a positive slope equal to n 2 db=dec and a constant positive phase equal to n 9, where n = log log : (29) Thus, combining A v (j!) with conventional integrator 1=j!, we obtain the Bode plot offig 2 This means that the operator I n (s) can be approximated by I n (s)= G n s N 1+s=! i 1+s=! i : (3) This operator is characterized by six parameters:! 1 and! N dene the frequency range (equivalently to! b and! h ), N is the number ofcells (it is directly related to the quality ofthe desired approximation), and are recursive parameters related to non-integer order n and G n is dened in order to have the same gain for 1=s n and I n (s) at the pulsation! u =1rd=s This operator is completely dened by the following relations:! i =! i;! i+1 =! i ; n=1 log log : (31) 33 State-space representation of the operator It is necessary to associate a state-space representation to I n (s) in order to simulate more complex systems An innity ofthese representations can be associated to I n (s) Because one ofour objectives is to estimate the parameters {! 1,! N, and }, we have privileged parsimonious models in order to facilitate the identication procedure [1,12 14,2] Because I n (s) is composed ofa product ofcells, we have dened the state variables as the output ofeach cell, according to Fig 3 With this natural decomposition, each state variable x n is only related to the preceding x n 1 by! n 1! n 1 ẋ n 1 +ẋ n =! n 1 (x n 1 x n ); (32) where!n 1! = n 1 Considering x n+1, we obtain ẋ n +ẋ n+1 =! n (x n x n+1 ): (33) The connection between these two rows (n and n 1) is realized by! n =! n 1 : Thus, this particular state-space representation uses only and, plus! 1 or! N So, we can write: 1 ẋ 1 1 ẋ ẋ N +1

6 2324 T Poinot, J-C Trigeassou / Signal Processing 83 (23) G n u x1 1+ s ω1 x2 1+ s ω2 x3 x N 1+ s ω N x N + 1 s 1+ s ω 1 1+ s ω Fig 3 I n(s) block diagram 2 1+ s ω N! 1! 1 =! 2! 2! N! N G n + u: x 1 x 2 x N +1 Then, dening M I, A I, B I and x I, we can write (where M I, A I and B I are parsimonious matrixes): M I ẋ I = A I x I + B I u (34) or equivalently: ẋ I = A I x I + B I u; (35) where A I = M 1 I A I ; B I = M 1 I B I are full matrixes necessary for the numerical simulation ofthe operator Refer to Fig 4 for a symbolic representation of this state-space model 4 State-space representation of a non-integer system 41 Principle Using the fractional integrator, one can construct the state-space representation ofgeneral fractional u = x 1 B * I u = x 1 In x y I I = x N + 1 () s A * I x I x N + 1 = yi Fig 4 State-space representation ofthe operator systems We will again consider the benchmark model (3): H(s)= Y (s) U(s) = b a + s n : (36) This system has an aperiodic response if n 1 and an oscillatoring one if1 n 2 This transmittance is equivalent to the dierential equation form: d n y(t) dt n + a y(t)=b u(t): (37) Let us dene x(t) such as X (s)= 1 s n U(s): + a (38) Thus, we obtain a macro state-space representation ofthis system (with macro parameters a and b ) d n x(t) dt n = a x(t)+u(t); y(t)=b x(t); (39) or equivalently using the operator ofpart 3: { ẋ1 = G n ( a x N +i + u) with y = b x N +i { i = 1 if n 1 i = 2 if1 n 2: (4)

7 T Poinot, J-C Trigeassou / Signal Processing 83 (23) u x 1 1 () x N + 2 I n s s a Fig 5 State-space representation ofthe system Fig 5 represents the block diagram ofthe system for 1 n 2 (in the case where n 1, it is necessary to remove the rst block integrator 1=s) Remark: This system is stable for n 2, unstable for n 2 42 Analysis of the state-space model This macro model (4) is only convenient for compact writing Practically, there are two imbricated state-space representations, one for the macro model, the other for the integrator operator, as represented in Fig 5 Because ẋ I = a x N +1 +u, we obtain for the global state-space model: Mẋ = Ax + Bu; y = C T x (41) with if n 1 G n a A = A I + ; B= B I 1 1 M = ; 1 C T =[ b ]; b y if1 n 2 G n a A = ; B= A I 1 1 M = 1 ; 1 C T =[ b ]: Remarks: [ BI N has to be very large in order to perform an appropriate approximation ofthe fractional integrator Then, x I is a large dimension vector, equivalent to long memory [2], the main feature of fractional systems The dierential equation is characterized by fractional order n This parameter is not explicitly used in the model, because it has been converted into four equivalent parameters! 1,! N, and This transformation enables us to simulate the fractional system with a conventional equivalent state-space model and so to estimate its parameters in identication applications [1,12 14,2] With and estimates it is possible to derive n, using n = 1 log =log Dierent state-space representations have been derived to simulate fractional systems A Oustaloup [19] has proposed an innite dimensional model based on the numerical approximation ofthe non-integer derivator Recently, new models have been introduced; refer for instance to [5,21] 43 Comparison with diusive representation approach Direct comparison ofoperator I n (s) with DR approach (22) does not exhibit any particular relation: ]

8 2326 T Poinot, J-C Trigeassou / Signal Processing 83 (23) we propose to show that these approaches are very close in their principle, but dierent in their use and potential applications Operator I n (s) can be expanded using! i pulsations: N I n (s)= G n s where c i = G n! i! i! i 1+s=! i 1+s=! i = G n s + N N j=1 j i c i s +! i ; (42) 1! i =! j 1! i =! j : (43) Inverting relation (42) using Laplace transform, we get the impulse response ofthe operator: N h n (t)=g n + c i e!it (44) which corresponds to the approximation offractional integrator (24) using its DR: N h(t)= ( i )e it i ; (45) where ( i ) i = c i ; i =! i : (46) The only dierence between h n (t) and h(t) concerns the supplementary coecient G n : is this dierence signicative? First, let us interpret the meaning ofthis coecient Consider a ctitious time constant and the corresponding mode G n e t= Then G n = lim G n e t= (47) that is to say G n represents the contribution ofan in- nite memory mode: G n avoids a hard truncation of i series when This was also interpreted initially as the need to link the fractional action to an integer one for!! 1 Secondly, let us show that G n is essential to the respect ofthe static gain ofthe simulated system In the previous example, H(s) =b =(a + s n )is approximated by H n (s)= b I n (s) 1+a I n (s) : (48) Static gain of H(s) is: G st = H() = b : (49) a Using H n (s), we get G nst = H n () = b Gn s N 1+a G n s N 1+(s=! i ) 1+(s=! i) 1+(s=! i ) 1+(s=! i) s= = b : (5) a On the other hand, ifwe had used the DR ofthe fractional integrator to perform the simulation in the same way as in our approach, we would have obtained: G DRst = b K k=1 c k= k K 1+a k=1 c b K: (51) k= k a The conclusion ofthis comparison is that the link between fractional action n and integer one (with the help of G n =s) is essential to the success ofthis fractional integrator operator On the other hand, because it is impossible to use directly the DR approximation ofthe fractional operator, it is necessary to compute, prior to the simulation, the DR () ofthe desired fractional system, and this is certainly a major constraint for the DR approach On the other hand, the only constraint with the operator I n (s) is the choice of {! 1 ;! N } and N appropriate to the simulation ofthe fractional system Then, knowledge of n directly refers to and parameters, which are fundamentally equivalent to the DR () Finally, this operator is included in a conventional macroscopic state-vector representation: thus, this approach has large exibility and it needs no additional knowledge on the fractional system model Moreover, we will exhibit in the last part that this approach is able to face fractional systems with limited fractional derivation 5 Numerical simulations 51 Benchmark fractional system 511 Simulations The modelling ofthe operator I n (s) permits to approach the idealfractional integrator when the

9 T Poinot, J-C Trigeassou / Signal Processing 83 (23) frequency domain is very large For two values of the fractional order n (5 and 15), step responses of the benchmark fractional system (3) have been simulated using the operator I n (s) and the direct method described in Section 21 Fig 6 (n =:5) and Fig 7 (n =1:5) show step responses obtained by these two methods with: a =1, b =1,! 1 =1 5 rd=s,! N =1 5 rd=s a number ofcells N = 3 One can notice that in the two cases, responses are superposed Decreasing the number ofcells N and observing the step responses, one can notice that with N = 1, small dierences between the two responses appear (see Fig 8) This result is ofcourse evident because the spectral domain is very large (1 decades) and the number ofcells used to approximate the fractional integrator is too small; nevertheless, the number ofcells necessary to perform a good approximation is quite reasonable ( 1) The frequency responses of the theoretical fractional system and its approximation using I n (s) are plotted in Fig 9 for n =:5 and Fig 1 for n =1:5 One can notice that in the frequency domain dened by! 1 and! N, the two responses coincide Beyond this domain, the system is equivalent to a rst order system in the rst case and to a second order system in the second one In the case where N =1, the frequency response is plotted in Fig 11 One can notice the phase oscillations directly linked to the insucient number ofcells 512 Conclusion The direct method gives satisfactory results for any value of n We obtain the same result with I n (s) when the number ofcells is suciently large We have not presented simulation results for the Diusive Representation approach but as expected, they are also satisfactory but only for n 1 We have already exhibited the limitations ofthe DR approach On the contrary, the direct one applies to any kind offractional systems: the essential drawback is the increasing dimension ofthe sums directly related to the numerical approximation (1) This is a real Time (s) Fig 6 Step response for n =:5 (, simulation with I n(s); +, simulation with the direct method)

10 2328 T Poinot, J-C Trigeassou / Signal Processing 83 (23) Time (s) Fig 7 Step response for n =1:5 (, simulation with I n(s); +, simulation with the direct method) Time (s) Fig 8 Step response for n =:5 and N = 1 (, simulation with I n(s); +, simulation with the direct method)

11 T Poinot, J-C Trigeassou / Signal Processing 83 (23) Magnitude (db) Phase ( ) w (rd/s) Fig 9 Bode plot for n =:5 and N = 3 (, approximated system;, theoretical system) 5 Magnitude (db) Phase ( ) w (rd/s) Fig 1 Bode plot for n =1:5 and N = 3 (, approximated system;, theoretical system)

12 233 T Poinot, J-C Trigeassou / Signal Processing 83 (23) Magnitude (db) Phase ( ) w (rd/s) Fig 11 Bode plot for n =:5 and N = 1 (, approximated system;, theoretical system) constraint when this approximation is used to simulate the model in identication and parameter estimation [4] u 1 s I n s b 1 b y 52 Simulation of a fractional system with bounded spectral range a 1 When the frequency domain where acts the non-integer order is not innite but limited to some decades, the operator dened in this article permits nevertheless to give corresponding time and frequency responses Consider the system H(s): H(s)= Y (s) U(s) = b + b 1 s n a + a 1 s n ; (52) + sn+1 where the fractional order n is limited to one decade Its state-space model is represented in Fig 12: For this example, the following parameters have been chosen: a =1,a 1 =1,b =1,b 1 =1, n =:4 a Fig 12 State-space representation ofthe system (52) Two simulations are performed The rst one (solid line) is done with a fractional integrator limited to one decade with N =1 cells,! b =:1 rd=s and! h =1 rd=s The second simulation (dotted line) is performed with an ideal fractional operator with the parameters N =3 cells,! b =1 5 rd=s and! h =1 5 rd=s Figs 13 and 14 exhibit respectively the step responses of H(s) and its bode plots in the two cases This last example exhibits clearly that the proposed fractional integrator operator is able to face conventional fractional modelling and simulation as well as non-conventional ones, like fractional derivation with

13 T Poinot, J-C Trigeassou / Signal Processing 83 (23) Time (s) Fig 13 Step responses ofthe system 1 Magnitude (db) Phase ( ) w (rd/s) Fig 14 Bode plots ofthe system

14 2332 T Poinot, J-C Trigeassou / Signal Processing 83 (23) limited spectral range It is also evident that neither direct numerical approximation nor diusive representation can take into account this non-conventional problem 6 Conclusion An original method for modelling and simulation offractional systems has been presented in this article This modelling is based on a new fractional integrator operator, associated to a N dimensional state-space representation A few parameters used for the design of the non-integer action and its spectral range are necessary to characterize this operator Theoretical and numerical comparisons with other techniques commonly used for the simulation of fractional systems have exhibited the performances ofthis original approach Its main interest is to propose a general framework for the modelling of fractional systems based on a macro state-space representation, where conventional integration is replaced by fractional one with the help of the integrator operator Another important feature of this new approach is its exibility because it applies to dierent types offractional models and to non-conventional non-integer derivation with limited spectral range This fractional operator has been already applied to identication problems with output error technique Various experiments [1,12 14,2] have conrmed the interest and the validity ofthis new approach for modelling and simulation of fractional systems in diffusive applications References [1] JL Battaglia, Methodes d identication de modeles a derivees d ordres non entiers et de reduction modale Application a la resolution de problemes thermiques inverses dans des systemes industriels, Habilitation a Diriger des Recherches, Universite de Bordeaux I, 22 [2] J Beran, Statistical methods for data with long-range dependance, Stat Sci 7 (1992) [3] YQ Chen, KL Moore, Discretization schemes for fractional dierentiators and integrators, IEEE Trans Circuits System 1: Fundamental Theory Appl 49 (3) (22) [4] O Cois, Systemes lineaires non entiers et identication par modele non entier: application en thermique, These de Doctorat, Universite de Bordeaux I, 22 [5] M Fliess, R Hotzel, Linear systems with a derivation of non integer order: introductory theory with an example, Proceedings CESA 96 Symposium on Models, Analysis and Simulation, Vol 1, France, 1996, pp [6] D Heleschewitz, D Matignon, Diusive realisations of fractional integrodierential operators: structural analysis under approximation, Proceedings IFAC Conference System, Structure and Control, Vol 2, Nantes, France, 1998, pp [7] M Ichise, Y Nagayanagi, T Kojima, An analog simulation of non integer order transfer functions for analysis of electrode processes, J Electroanal Chem Interfacial Electrochem 33 (1971) 253 [8] L Le Lay, A Oustaloup, J-C Trigeassou, F Levron, Frequency identication by non integer model, Proceedings IFAC Conference System, Structure and Control, Nantes, France, 1998, pp [9] L Le Lay, A Oustaloup, J-C Trigeassou, F Levron, Frequency identication by implicit dierentiation models, Proceedings AVCS 98 International Conference on Advance in Vehicle Control and Safety, Amiens, France, 1998, [1] J Lin, Modelisation et identication de systemes d ordre non entier, These de Doctorat, Universite de Poitiers, 21 [11] J Lin, T Poinot, Modelisation de systemes d ordre non entier, JDA 99 Journees Doctorales d Automatique, Nancy, France, 1999, pp [12] J Lin, T Poinot, JC Trigeassou, P Coirault, Parameter estimation of fractional systems Application to heat transfer, ECC 21, European Control Conference, Porto, Portugal, 21, pp [13] J Lin, T Poinot, JC Trigeassou, H Kabbaj, J Faucher, Modelisation et identication d ordre non entier d une machine asynchrone, CIFA 2, Conference Internationale Francophone d Automatique, Lille, France, 2 [14] J Lin, T Poinot, JC Trigeassou, R Ouvrard, Parameter estimation offractional systems: application to the modeling ofa lead-acid battery, SYSID 2, 12th IFAC Symposium on System Identication, USA, 2 [15] D Matignon, B d Andrea-Novel, P Depalle, A Oustaloup, Viscothermal losses in wind instrument: a non integer model, in: System and Networks: Mathematical Theory and Applications, Vol 2, Akademie Verlag, Berlin, 1994 [16] KS Miller, B Ross, An Introduction to the Fractional Calculus and Fractional Dierential Equations, Wiley-Interscience Publication, New York, 1993 [17] G Montseny, Diusive representation ofpseudo-dierential time-operators, Proceedings Fractional Dierential Systems: Models, Methods and Applications, Paris, 1998 [18] A Oustaloup, Systemes asservis lineaires d ordre fractionnaire, Masson, Paris, 1983 [19] A Oustaloup, La derivation non entiere: theorie, synthese et applications, Hermes, 1995

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