Identification of fractional order systems using modulating functions method
Da-Yan Liu, Taous-Meriem Laleg-Kirati, Olivier Gibaru, Wilfrid Perruquetti
aa r X i v : . [ m a t h . NA ] M a r Identification of fractional order systems using modulating functionsmethod
Da-Yan Liu, Taous-Meriem Laleg-Kirati, Olivier Gibaru and Wilfrid Perruquetti
Abstract —The modulating functions method has been usedfor the identification of linear and nonlinear systems. In thispaper, we generalize this method to the on-line identificationof fractional order systems based on the Riemann-Liouvillefractional derivatives. First, a new fractional integration byparts formula involving the fractional derivative of a modulatingfunction is given. Then, we apply this formula to a fractionalorder system, for which the fractional derivatives of the inputand the output can be transferred into the ones of the modulatingfunctions. By choosing a set of modulating functions, a linearsystem of algebraic equations is obtained. Hence, the unknownparameters of a fractional order system can be estimated bysolving a linear system. Using this method, we do not need anyinitial values which are usually unknown and not equal to zero.Also we do not need to estimate the fractional derivatives of noisyoutput. Moreover, it is shown that the proposed estimators arerobust against high frequency sinusoidal noises and the ones dueto a class of stochastic processes. Finally, the efficiency and thestability of the proposed method is confirmed by some numericalsimulations.
I. I
NTRODUCTION
Fractional differential equations and fractional integrals aregaining importance in research community because of theircapacity to accurately describe real world processes. The flowof fluid in a porous media, the conduction of heat in a semi-infinite slab, the voltage-current relation in a semi infinitetransmission line are such examples of processes naturallymodeled by fractional differential equations or fractional inte-grals.This paper is dealing with the identification of fractionalorder dynamical systems. The identification of such systemshas been used for instance, for the estimation of the state ofcharge of lead acid batteries [1], and for the identification inthermal systems [2], [3]. The goal of system identification is toestimate the parameters of a model from system input/outputmeasurements. Different methods have been proposed for theidentification of fractional order systems. Most of them consistin the generalization to fractional order systems of standardmethods that were used in the identification of systems withinteger order derivatives. We can classify these methods into
D.Y. Liu, and T.M. Laleg-Kirati are with Computer, Electrical andMathematical Sciences and Engineering Division, King Abdullah univer-sity of science and technology, KSA
[email protected] ; [email protected] O. Gibaru is with LSIS (CNRS, UMR 7296), Arts et M´etiersParisTech, 8 Boulevard Louis XIV, 59046 Lille Cedex, France [email protected]
W. Perruquetti is with LAGIS (CNRS, UMR 8146), ´Ecole Centralede Lille, BP 48, Cit´e Scientifique, 59650 Villeneuve d’Ascq, France [email protected]
O. Gibaru and W. Perruquetti are with L’ ´Equipe Projet Non-A, INRIALille-Nord Europe, 40, Avenue Halley, 59650 Villeneuve d’Ascq, France. time domain methods and frequency domain methods. Time-domain methods have been introduced for example in [4], [5],where a method based on the discretization of a fractionaldifferential equation using Grunwald definition has been in-troduced, and the parameters have been estimated using leastsquare approach. In [6], a method based on the approxima-tion of a fractional integrator by a rational model has beenproposed. In [7], the use of methods based on fractionalorthogonal bases has been introduced. Other techniques canbe also found for example in [8], [9], [10] and the referencestherein.In this paper, we are interested in the identification offractional order systems using modulating functions methodin case of noisy measurements. Modulating functions methodhas been developed by Shinbrot in [11], [12] to estimate theparameters of a state space representation. Thanks to the prop-erties of modulating functions, the fractional differential equa-tion defining a fractional order system is transformed into alinear system of algebraic equations. Hence, instead of solvinga fractional differential equation where the initial values areoften unknown, the problem of identification is transformedinto solving a linear system where the initial conditions arenot required. Generalization of modulating functions methodto fractional systems has been already proposed, for examplein [13]. However, the authors of this paper proposed only toreduce the orders of the derivatives in a fractional differentialequation. Moreover, the noisy case has not been considered.In the next section, basic definitions of fractional derivativesand modulating functions are recalled. Then in Section III,modulating functions method is applied to the identificationof fractional order linear systems. Error analysis in both thecontinuous and discrete cases are presented in Section IV.Numerical results are presented in Section V, followed byconclusions summarizing the main results obtained.II. P
RELIMINARY
In this section, first we recall the definitions and some usefulproperties of the Riemann-Liouville fractional derivative andmodulating functions. Then, a new fractional integration byparts formula is given.
A. Riemann-Liouville fractional derivative
Let f be a continuous function defined on R , then theRiemann-Liouville fractional derivative of f is defined by (see[14] p. 62): ∀ t ∈ R ∗ + , D αt f ( t ) := 1Γ( l − α ) d l dt l Z t ( t − τ ) l − α − f ( τ ) dτ, (1)here l − ≤ α < l with l ∈ N ∗ , and Γ( · ) is the Gammafunction (see [15] p. 255). As an example, using (1) thefractional derivative of an n th ( n ∈ N ) order polynomial isgiven by (see [14] p. 72): ∀ t ∈ R ∗ + , D αt t n = Γ( n + 1)Γ( n + 1 − α ) t n − α . (2)We assume that the fractional derivative and the Laplacetransform of f both exist, then the Laplace transform of thefractional derivative of f is given by (see [16] p. 284): ∀ s ∈ C , L { D αt f ( t ) } ( s ) = s α ˆ f ( s ) − l − X i =0 s i (cid:2) D α − i − t f ( t ) (cid:3) t =0 , (3)where ˆ f denotes the Laplace transform of f , and s denotesthe variable in the frequency domain.Finally, we recall some results on the existence and theinitial values of the fractional derivative in the followingproposition. Proposition 1: (see [14] pp. 75-77) If f ∈ C l − ([0 , T ]) and f ( l ) ∈ L ([0 , T ]) where T ∈ R ∗ + , then the Riemann-Liouvillefractional derivative D αt f ( t ) exists, where l − ≤ α < l with l ∈ N ∗ . Moreover, the condition f ( i ) (0) = 0 , for i = 0 , . . . , l − , is equivalent to the following condition: [D αt f ( t )] t =0 = 0 . B. Modulating functions
Let l ∈ N ∗ , T ∈ R ∗ + , and g be a function satisfying thefollowing properties: ( P ) : g ∈ C l ([0 , T ]) ; ( P ) : g ( j ) (0) = g ( j ) ( T ) = 0 , ∀ j = 0 , , . . . , l − ,then g is called modulating function of order l [17]. Hence,according to Proposition 1, the modulating function g has alsothe following properties: ( P ) : ∀ ≤ β < l , D βt g ( t ) exists; ( P ) : ∀ ≤ β < l , h D βt g ( t ) i t =0 = 0 . C. Fractional integration by parts
Another useful property of the modulating functions is givenin the following theorem.
Theorem 1:
Let y be a function such that the α th orderfractional derivative exits and g be a modulating function oforder l with l − ≤ α < l with l ∈ N ∗ . Then, we have: Z T g ( T − t ) D αt y ( t ) dt = Z T D αt g ( t ) y ( T − t ) dt, (4)where T ∈ R ∗ + . Proof:
By applying the convolution theorem of theLaplace transform (see [15], p. 1020), we get: L (Z T g ( T − t ) D αt y ( t ) dt ) ( s ) = ˆ g ( s ) L { D αt y ( t ) } ( s ) . (5)Then, using (3) we obtain: ˆ g ( s ) L { D αt y ( t ) } ( s ) =ˆ g ( s ) s α ˆ y ( s ) − l − X i =0 s i ˆ g ( s ) (cid:2) D α − i − t y ( t ) (cid:3) t =0 . (6) Moreover, using (3) and ( P ) we get: L { D αt g ( t ) } ( s ) = s α ˆ g ( s ) , (7) L n g ( i ) ( t ) o ( s ) = s i ˆ g ( s ) , (8)for i = 0 , . . . , l − . Consequently, by applying (7), (8) andthe inverse of the Laplace transform to (6), we obtain: L − { ˆ g ( s ) L { D αt y ( t ) } ( s ) } ( T ) = L − {L { D αt g ( t ) } ( s ) ˆ y ( s ) } ( T ) − l − X i =0 g ( i ) ( T ) (cid:2) D α − i − t y ( t ) (cid:3) t =0 . (9)Using ( P ) , the initial conditions (cid:2) D α − i − t y ( t ) (cid:3) t =0 , for i =0 , , . . . , l − , can be eliminated.Finally, this proof can be completed by applying the con-volution theorem of the Laplace transform to (9).According to the previous theorem, we can see that byworking in the frequency domain: on the one hand, we canobtain an integral formula which can be considered as thegeneralization of the classical integration by parts formula.Another fractional integration by parts formula has also beengiven in [18], however the Caputo fractional derivative wasinvolved in the formula; on the other hand, the initial con-ditions of the fractional derivatives of y can be eliminatedusing a modulating function. In fact, the idea of obtainingthis theorem is inspired by the recent algebraic parametricestimation technique [19], [20], [21], [22], [23], [24], [25],[26], which eliminates the unknown initial conditions byapplying algebraic manipulations in the frequency domain.III. I DENTIFICATION OF FRACTIONAL ORDER SYSTEMS
A. Fractional order linear systems
In this section, we consider a class of fractional orderlinear systems which are defined by the following fractionaldifferential equation: ∀ t ∈ I = [0 , T ] ⊂ R ∗ + , L X i =0 a i D α i t y ( t ) = M X j =0 b j D β j t u ( t ) , (10)where y is the output, u is the input, a i , b j ∈ R ∗ are unknownparameters to be identified, and α i , β j ∈ R + are assumed ≤ α < α < · · · < α L , ≤ β < β < · · · < β M with L, M ∈ N .Let y ̟ be a noisy observation of y on the interval I : ∀ t ∈ I, y ̟ ( t ) = y ( t ) + ̟ ( t ) , (11)where ̟ is an integrable noise . We are going to estimatethe unknown parameters in (10) using the input u and theobservation y ̟ .One of the standard methods for the identification ofsystems with integer order derivatives is to use the least-squares method [28]. This method was generalized to frac-tional order systems in [4], [5]. In this method, we need More generally, the noise is a stochastic process, which is integrable inthe sense of convergence in mean square [22]. o estimate the fractional derivatives of y and u using afractional order differentiator [29], [30], [31]. However, afractional order differentiator often contains a truncated termerror. An alternative method consists in solving the problemin the frequency domain by applying the Laplace transform to(10). However, according to (3), this application can produceunknown initial conditions. In the next subsection, we aregoing to apply modulating functions method to eliminate theseunknown initial conditions. B. Application of modulating functions method
We denote W = L + M + 1 , α = max ( α L , β M ) , and l = ⌈ α ⌉ , where ⌈ α ⌉ denotes the smallest integer greater thanor equal to α . Then, we take a set of modulating functions { g n } Nn =1 with W ≤ N ∈ N . Using Theorem 1, we can givethe following proposition. Proposition 2:
Let { g n } Nn =1 be a set of modulating func-tions of order l . If we assume that b = 1 in the fractional orderlinear system defined by (10), then the unknown parameters inthis fractional order linear system can be estimated by solvingthe following linear system: (cid:0) U N Y ̟N (cid:1) (cid:18) ˜ B ˜ A (cid:19) = I N , (12)where ˜ B = (cid:16) ˜ b , · · · , ˜ b M (cid:17) T , ˜ A = (˜ a , · · · , ˜ a L ) T are estima-tors of the unknown parameters, and U N ( n, j ) = − Z T D β j t g n ( t ) u ( T − t ) dt, (13) Y ̟N ( n, i + 1) = Z T D α i t g n ( t ) y ̟ ( T − t ) dt, (14) I N ( n ) = Z T D β t g n ( t ) u ( T − t ) dt, (15)for n = 1 , . . . , N , j = 1 , . . . , M and i = 0 , . . . , L . Proof:
By multiplying the modulating functions g n to theequation (10) and by integrating between and T , we get: Z T L X i =0 a i g n ( T − t ) D α i t y ( t ) dt = Z T M X j =0 b j g n ( T − t ) D β j t u ( t ) dt, (16)for n = 1 , . . . , N . Then, using Theorem 1, we get M X j =1 b j U N ( n, j )+ L X i =0 a i Y N ( n, i +1) = Z T D β t g n ( t ) u ( T − t ) dt, (17)where U N ( n, j ) = − R T D β j t g n ( t ) u ( T − t ) dt , and Y N ( n, i + 1) = R T D α i t g n ( t ) y ( T − t ) dt . Finally, this proofcan be completed by substituting y by y ̟ in (17).Consequently, thanks to Theorem 1, instead of estimating(resp. calculating) the fractional derivatives of y (resp. u ), wecalculate the fractional derivatives of the modulating functions.On the one hand, comparing to y , the modulating functions are known and without noise. On the other hand, if thefractional derivatives of u cannot be analytically calculatedor are difficult to calculate, we can solve the problem bycalculating the ones of the modulating functions.Finally, let us mention that since the proposed estimators aregiven in causal case, if we take the value of T to be equal to thetime where we estimate the parameters, then these estimatorscan be used for on-line identification applications.IV. E RROR ANALYSIS
In this section, we are going to study the noise effect in theintegrals obtained in Proposition 2. For this purpose, we studythe noise error contributions due to a high frequency sinusoidalnoise and the ones due to a class of stochastic processes incontinuous case and in discrete case, respectively.
A. Error analysis in continuous case
There are many applications where the output signal iscorrupted by a sinusoidal noise of higher frequency [26].Hence, we assume that the noise ̟ is a high frequencysinusoidal noise in this subsection.By writing y ̟ = y + ̟ , the integral Y ̟N ( n, i + 1) given in(14), for n = 1 , . . . , N and i = 0 , . . . , L , can be divided into: Y ̟N ( n, i + 1) = Y N ( n, i + 1) + e ̟N ( n, i + 1) , (18)where Y N ( n, i + 1) is given in (17), and the associated noiseerror contribution is given by: e ̟N ( n, i + 1) = Z T D α i t g n ( t ) ̟ ( T − t ) dt. (19)Consequently, the estimation errors for the estimators given inProposition 2 only come from these noise error contributions.In the following proposition, we give error bounds for thesenoise error contributions. Proposition 3:
We assume that ∀ t ∈ I , ̟ ( t ) = c sin( ωt + φ ) with c, ω ∈ R ∗ + and φ ∈ [0 , π [ . Moreover, we assumethat D α i +1 t g n ( t ) exists and is continuous on [0 , T ] , for n =1 , . . . , N and i = 0 , . . . , L . Then, we have: | e ̟N ( n, i + 1) | ≤ cω T C α i +1 + cω | [D α i t g n ( t )] t = T | , (20)where C α i +1 = sup t ∈ [0 ,T ] (cid:12)(cid:12) D α i +1 t g n ( t ) (cid:12)(cid:12) , and e ̟N ( n, i +1) is givenby (19). Proof:
Since ddt { D α i t g n ( t ) } = D α i +1 t g n ( t ) exists, thenby applying integration by parts and ( P ) , we get: e ̟N ( n, i + 1) = Z T D α i t g n ( t ) c sin( ω ( T − t ) + φ ) dt = − cω Z T D α i +1 t g n ( t ) cos( ω ( T − t ) + φ ) dt + cω cos( φ ) [D α i t g n ( t )] t = T . (21)If D α i +1 t g n ( t ) ∈ C ([0 , T ]) , then this proof can be completedusing (21).ccording to the previous proposition, if the frequency ofthe sinusoidal noise is high, then the associated noise errorcontributions can be negligible. Consequently, the estimatorsgiven in Proposition 2 can cope with this kind of noises. B. Error analysis in discrete case
From now on, we assume that the noisy observation y ̟ defined in (11) is given in a discrete case. Let y ̟ ( t j ) = y ( t j ) + ̟ ( t j ) be a noisy discrete observation of y given withan equidistant sampling period T s , where T s = Tm , m ∈ N ∗ ,and t j = jT s , for j = 0 , · · · , m .Since y ̟ is a discrete measurement, we apply a numericalintegration method to approximate the integrals in (12). Let w ≥ , w m ≥ and w j > for j = 1 , . . . , m − bethe weights for a given numerical integration method, whereweight a (resp. a m ) is set to zero when there is an infinitevalue at t = 0 (resp. t m = T ). Then, the integral Y ̟N ( n, i +1) given in (14), for n = 1 , . . . , N and i = 0 , . . . , L , can beapproximated by: Y ̟N,m ( n, i + 1) := T s m X j =0 w j g ( α i ) n ( t j ) y ̟ ( t m − j ) , (22)where g ( α i ) n ( t j ) := [D α i t g n ( t )] t = t j . The integrals given in (13)and (15) can be approximated in a similar way.By writing y ̟ ( t j ) = y ( t j ) + ̟ ( t j ) , we get: Y ̟N,m ( n, i + 1) = Y N,m ( n, i + 1) + e ̟N,m ( n, i + 1) , (23)where Y N,m ( n, i + 1) = T s m X j =0 w j g ( α i ) n ( t j ) y ( t m − j ) , (24) e ̟N,m ( n, i + 1) = T s m X j =0 w j g ( α i ) n ( t j ) ̟ ( t m − j ) . (25)Thus the integral Y ̟N ( n, i + 1) is corrupted by two sources oferrors: • the numerical error which comes from a numerical inte-gration method, • the noise error contribution e ̟N,m ( n, i + 1) .Consequently, the estimation errors for the estimators givenin Proposition 2 come from both the numerical errors and thenoise error contributions in the discrete noisy case.It is well known that if the value of T is set, then when T s tends to , i.e. m → + ∞ , the numerical errors tend to . Inthe next subsection, we are going to study the effect of thesampling period on the noise error contributions. C. Influence of sampling period on noise error contributions
In this subsection, we consider a family of noises which arestochastic processes satisfying the following conditions: ( C ) : for any t, s ∈ I , t = s , ̟ ( t ) and ̟ ( s ) areindependent; ( C ) : the mean value function of ̟ ( · ) denoted by E[ · ] belongs to L ( I ) ; ( C ) : the variance function of ̟ ( · ) denoted by Var[ · ] isbounded on I .Note that the white Gaussian noise and the Poisson noisesatisfy these conditions. Then, we can give the followingproposition. Proposition 4:
Let ̟ ( · ) be a stochastic process satisfyingconditions ( C ) − ( C ) , and ̟ ( t j ) , for j = 0 , · · · , m , be asequence of { ̟ ( · ) } with an equidistant sampling period T s .If D α i t g n ( t ) ∈ L ( I ) , then we have the following convergencein mean square of the noise error contribution in the integral Y ̟N ( n, i +1) given in (14), for n = 1 , . . . , N and i = 0 , . . . , L : e ̟N,m ( n, i + 1) L ( I ) ==== ⇒ T s → Z T D α i t g n ( t ) E [ ̟ ( T − t )] dt, (26)where e ̟N,m ( n, i + 1) is given in (25). Moreover, if ∀ t ∈ I , E [ ̟ ( t )] = 0 , then we have: e ̟N,m ( n, i + 1) L ( I ) ==== ⇒ T s → . (27)The proof of the previous proposition can be obtained ina similar way to the one given in [27]. Moreover, a similarresult was studied using the non-standard analysis in [32].Consequently, according to the previous proposition, thenoise error contributions can be increasing with respect to thesampling period.Finally, let us mention that solving the linear system givenin Proposition 2 in noisy case is related to the matrix pertur-bation theory. The accuracy and the stability of the proposedestimators not only depend on the noise error contributions,but also depend on the condition number of the associatedmatrix. This condition number depends both on the inputand the output of the fractional order system and on theused modulating functions. In general, we should choose themodulating functions that can give a small condition number.This study is out of the scope of this paper.V. S IMULATION RESULTS
In order to illustrate the accuracy and robustness withrespect to corrupting noises of the proposed estimators, wepresent some numerical results in this section.Let us consider a fractional order system defined by thefollowing fractional differential equation: ∀ t ∈ [0 , , a D α t y ( t ) + a D α t y ( t ) + a D α t y ( t ) = u ( t ) , (28)where a = 3 , a = 2 , a = 1 , α = 0 , α = 0 . and α = 1 . . We assume that the output is y ( t ) = sin(3 t ) + 1 .Hence, the initial conditions of y are not equal to . Moreover,the expression of the input can be obtained using (2) and thefollowing formula (see [33] p. 83): D α i t sin(3 t ) =3 t − α i Γ(2 − α i ) F (cid:18)
1; 12 (2 − α i ) ,
12 (3 − α i ); −
14 3 t (cid:19) , (29)where p F q ( c , . . . , c p ; d , . . . , d q ; · ) is the generalized hyper-geometric function given in [33] p. 303.n our identification procedure, we use the following mod-ulating functions, the fractional derivative of which are simpleto calculate: g n ( t ) = ( T − t ) n t N +1 − n , (30)where n = 1 , , · · · , N with N = 13 . The value of T is takento be equal to the time where we estimate the parameters. Letus recall that this kind of functions has been obtained whenthe algebraic parametric estimation technique was applied tothe parameter estimation for signals described by differentialequations [21].In the two following examples, we estimate the parameters a , a and a using the noisy observation of y where thenoise is a high frequency sinusoidal noise and a gaussiannoise, respectively. Moreover, we apply the trapezoidal ruleto numerically approximate the integrals in our estimators. Example 1.
In this example, we assume that y ̟ ( t j ) = y ( t j )+0 . t j ) with T s = 0 . . We can see this discretenoisy signal in Figure 1. In our identification procedure, wetake T = t i for t i ∈ [1 . , . The obtained estimations andthe associated relative estimation errors are shown in Figure 2and Figure 3. Hence, we can see that the proposed estimatorscan cope with a high frequency sinusoidal noise. Example 2.
In this example, we assume that y ̟ ( t j ) = y ( t j ) + σ̟ ( t j ) , where T s = 0 . , ̟ ( t j ) is simulated froma zero-mean white Gaussian iid sequence, and σ ∈ R ∗ + isadjusted in such a way that the signal-to-noise ratio is equalto SN R = 22 dB. This noisy observation is shown in Figure 4.The obtained estimations and the associated relative estimationerrors are given in Figure 5 and Figure 6. We can see that theproposed estimators are robust against a gaussian noise. ϖ y Fig. 1. The noise-free output and its noisy observation.
VI. C
ONCLUSIONS
In this paper, the modulating functions method has beengeneralized to the on-line identification problem of fractionalorder systems. Using this method, the unknown parametershave been estimated by solving a linear system of algebraicequations involving the input and the noisy output. Thanksto the properties of modulating functions, we do not need toestimate the fractional derivatives of the output, to calculatethe ones of the input. We do not need to know the initial estimated0 a a estimated1 a a estimated2 a Fig. 2. The exact parameters and their estimations with a sinusoidal noise. estimated0 a estimated1 a estimated2 Fig. 3. Relative estimation errors in the sinusoidal noise case. ϖ y Fig. 4. The noise-free output and its noisy observation. conditions either. Moreover, the integral given in the estimatorscan reduce the noise effects due to a high frequency sinusoidalnoise or a class of stochastic processes. The efficiency and therobustness against corrupting noises have been confirmed bynumerical examples. In order to improve the robustness againstnoises, some methods such as the instrumental variable methodwill be applied [34]. It was mentioned that the proposedestimators also depend on the choice of modulating functions.This problem will be studied in the future work. Moreover, theestimation of the fractional derivative orders in a time-delayedfractional order system will be considered. estimated0 a a estimated1 a a estimated2 a Fig. 5. The exact parameters and their estimations with a gaussian noise. estimated0 a estimated1 a estimated2 Fig. 6. Relative estimation errors in the gaussian noise case. R EFERENCES[1] J. Sabatier, M. Aoun, A. Oustaloup, G. Gregoire, F. Ragot and P. Roy,Fractional system identification for lead acid battery state of chargeestimation, Signal Processing, 86, pp. 2645-2657, 2006.[2] J.D. Gabano and T. 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