Da-Yan Y Liu
email: dayan.liu@kaust.edu.sa

Olivier Gibaru
email: olivier.gibaru@ensam.eu

Wilfrid Perruquetti
email: wilfrid.perruquetti@inria.fr

Taous-Meriem M Laleg-Kirati
email: taousmeriem.laleg@kaust.edu.sa

Fractional order differentiation by integration and error analysis in noisy environment: Part 1 continuous case

Keywords: Fractional order differentiator, Riemann-Liouville fractional derivative, Caputo fractional derivative, Jacobi orthogonal polynomial filter, Error analysis. EDICS: SSP-FILT; ASP-ANAL. I

. We are going to generalize this method from the integer order to the fractional order so as to estimate the fractional order derivatives of noisy signals. For sake of clarity, this work has been divided into two parts. The first part presented in this paper focuses on the continuous case while the second part that has been presented in another paper

[3] deals with the discrete case with on-line applications. In this paper, two kinds of fractional order differentiators are deduced from the Jacobi orthogonal polynomial filter, using the Riemann-Liouville and the Caputo fractional order derivative definitions respectively. Exact and simple formulae for these differentiators are given by integral expressions. Hence, they can be used both for continuous-time and discrete-time models in on-line or off line applications. Then, some error bounds are provided for the corresponding estimation errors in continuous case. These bounds will be used to study the design parameters' influence on the obtained fractional order differentiators in the second part. Finally, numerical simulations are given to show the accuracy and the robustness with respect to corrupting noises

Lanczos generalized derivative [START_REF] Lanczos | Applied Analysis[END_REF] (p. 324) in 1956 (see [START_REF] Diekema | Differentiation by integration using orthogonal polynomials, a survey[END_REF] for more details). The idea of the Lanczos generalized derivative is to use an integral of an unknown signal and the Legendre orthogonal polynomial to estimate the first order derivative of the signal. Recently, Mboup, Join and Fliess [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF], [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF] applied an algebraic setting to estimate high order derivatives by integration in noisy case, where the Jacobi orthogonal polynomials were introduced in the integral. Hence, we call the obtained differentiator integer order Jacobi differentiator. This differentiator greatly improves the convergence rate of the Lanczos generalized derivative [START_REF] Liu | Differentiation by integration with Jacobi polynomials[END_REF]. Moreover, it was shown in [START_REF] Liu | Convergence Rate of the Causal Jacobi Derivative Estimator, Curves and Surfaces[END_REF] that it can also be obtained by taking the integer order derivative of the Jacobi orthogonal polynomial filter considered as the extension of the Legendre polynomial filter [START_REF] Persson | Smoothing by Savitzky-Golay and Legendre filters[END_REF], [START_REF] Szegö | Orthogonal polynomials[END_REF]. Bearing this idea in mind, a fractional order Jacobi differentiator was introduced in [START_REF] Liu | Fractional order differentiation by integration with Jacobi polynomials[END_REF] by taking the Riemann-Liouville fractional order derivative of the Jacobi orthogonal polynomial filter. Similarly, the Jacobi orthogonal polynomial filter was used to estimate the Caputo fractional order derivatives so as to solve fractional order differential equations in [START_REF] Doha | A new Jacobi operational matrix: An application for solving fractional differential equations[END_REF]. In [START_REF] Liu | Fractional order differentiation by integration with Jacobi polynomials[END_REF], we showed that the fractional order Jacobi differentiator is more accurate than the one based on the classical Savitzky-Golay filter. Nevertheless, no error analysis was provided for this differentiator.

Let us mention that the algebraic differentiation method used in [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF], [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF] was introduced in [START_REF] Fliess | Compression différentielle de transitoires bruités[END_REF], where the idea was inspired from the algebraic parametric estimation methods [START_REF] Fliess | Questioning some paradigms of signal processing via concrete examples[END_REF], [START_REF] Mboup | Parameter estimation for signals described by differential equations[END_REF], [START_REF] Liu | An error analysis in the algebraic estimation of a noisy January 19, 2013 DRAFT sinusoidal signal[END_REF], [START_REF] Liu | Parameters estimation of a noisy sinusoidal signal with time-varying amplitude[END_REF]. The reader may find additional theoretical foundations in [START_REF] Fliess | Analyse non standard du bruit[END_REF], [START_REF] Fliess | An algebraic framework for linear identification[END_REF]. In particular, by using non standard analysis techniques, Fliess [START_REF] Fliess | Analyse non standard du bruit[END_REF], [START_REF] Fliess | Critique du rapport signal à bruit en communications numériques -Questioning the signal to noise ratio in digital communications[END_REF] showed that these methods exhibit good robustness properties with respect to corrupting noises without the need of knowing their statistical properties. Moreover, some accurate error analysis were given in [START_REF] Liu | Error analysis of Jacobi derivative estimators for noisy signals[END_REF], [START_REF] Liu | Error analysis for a class of numerical differentiator: application to state observation[END_REF] to show the robustness properties of the integer order Jacobi differentiator with respect to noises issued from stochastic process models. Consequently, the fractional order Jacobi differentiator proposed in [START_REF] Liu | Fractional order differentiation by integration with Jacobi polynomials[END_REF] can have the same robustness properties. In [START_REF] Liu | Non-asymptotic fractional order differentiators via an algebraic parametric method[END_REF], by applying these methods to a truncated fractional Taylor series expansion, we estimated the Jumarie's modified Riemann-Liouville derivative defined in [START_REF] Jumarie | Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results[END_REF] without any error analysis. This work has been divided into two parts presented in two different papers for sake of clarity. This first part presented in this paper focuses on the continuous case where we first extend the differentiation by integration method from the integer order to the fractional order so as to estimate both the Riemann-Liouville and the Caputo fractional order derivatives for noisy signals. Then, we provide some analysis for the corresponding estimation errors in continuous case. In Section II, we recall the definitions and some useful properties of the Riemann-Liouville and the Caputo fractional order derivatives. Then, two fractional order differentiators are given by exact and simple integral formulae in Section III. They are obtained by calculating the Riemann-Liouville and the Caputo fractional order derivatives of the January 19, 2013 DRAFT Jacobi orthogonal polynomial filter respectively. In Section IV, by providing some error bounds, we study different sources of errors for these differentiators in continuous case. These bounds will be used to study the design parameters' influence on the obtained fractional order differentiators in the second part [START_REF] Liu | Fractional order differentiation by integration and error analysis in noisy environment: Part 2 discrete case[END_REF]. In order to show the efficiency and the stability of these differentiators in off-line applications, numerical simulations results are given in Section V. Finally, we give some conclusions for this part and the perspectives for the second part in Section VI. To simplify the presentation, all the proofs are deferred to the appendix.

II. PRELIMINARY

In this section, we are going to recall the definitions and some useful properties of the Riemann-Liouville fractional derivative, the Caputo fractional derivative and the Jacobi orthogonal polynomials, respectively.

A. Riemann-Liouville and Caputo fractional order derivatives

Let l ∈ N * , a ∈ R, and f ∈ C l (R) where C l (R) refers to the set of functions being l-times continuously differentiable on R. Then, the Riemann-Liouville fractional order derivative (see [START_REF] Podlubny | Fractional Differential Equations[END_REF] p. 62) of f is defined

as follows: ∀ t > a, R D α a,t f (•) := 1 Γ(l -α) d l dt l t a (t -τ ) l-α-1 f (τ ) dτ, (1)
where l -1 ≤ α < l, and Γ(•) is the Gamma function (see [START_REF] Abramowitz | editeurs, Handbook of mathematical functions[END_REF] p. 255). Moreover, the Caputo fractional order derivative (see [START_REF] Podlubny | Fractional Differential Equations[END_REF] p. 79) of f is defined as follows: ∀ t > a,

C D α a,t f (•) := 1 Γ(l -α) t a (t -τ ) l-α-1 f (l) (τ ) dτ, (2)
where l -1 < α < l. If f ∈ C l+1 (R), then the Caputo fractional order derivative is also valid for α = l.

In this paper, we only consider the case where α ∈ R + \ N. By applying integration by parts to (1), we get (see [START_REF] Podlubny | Fractional Differential Equations[END_REF] pp. 75-76):

∀ t > a, R D α a,t f (•) = l-1 i=0 (t -a) i-α Γ(i + 1 -α) f (i) (a) + C D α a,t f (•), (3)
where l -1 < α < l with l ∈ N * . As an example, if we take f (t) = (ta) n with n ∈ N and a < t ∈ R, then by using (1) and (2), we obtain (see [START_REF] Podlubny | Fractional Differential Equations[END_REF] p. 72):

R D α a,t f (•) = Γ(n + 1) Γ(n + 1 -α) (t -a) n-α , (4)
C D α a,t f (•) =    0, if n < α, Γ(n+1) Γ(n+1-α) (t -a) n-α , if n > α, (5)
January 19, 2013 DRAFT where α ∈ R + \ N. As we can see, unlike the Riemann-Liouville fractional order derivative the Caputo fractional order derivative of the polynomial f can be defined at t = a, i.e. C D α a,t f (•) t=a = 0. From now on, we denote the α th (α ∈ R + \ N) order derivative of f by D α a,t f (•). This notation can refer both to the Riemann-Liouville and the Caputo fractional order derivatives when the context is clear. Otherwise, we use the specific notations introduced in (1) and [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF]. Some useful properties of these fractional order derivatives are recalled as follows:

• Linearity (see [START_REF] Podlubny | Fractional Differential Equations[END_REF] p. 91): ∀ t > a,

D α a,t {λ 1 f 1 (•) + λ 2 f 2 (•)} = λ 1 D α a,t f 1 (•) + λ 2 D α a,t f 2 (•), (6)
• Scale change (see [START_REF] Oldham | The Fractional Calculus[END_REF] p. 76):

∀ t > a, D α a,λ t f (•) = 1 λ α D α a λ ,t f (λ •), (7)
• Translation (see [START_REF] Oldham | The Fractional Calculus[END_REF] p. 89):

∀ t ∈ R * + , D α a,a+t f (•) = D α 0,t f (• + a), (8)
where l -

1 < α < l with l ∈ N * , λ ∈ R * + , λ 1 , λ 2 ∈ R and f, f 1 , f 2 ∈ C l (R).

B. Jacobi orthogonal polynomials

In this article, we use the n th (n ∈ N) order shifted Jacobi orthogonal polynomial defined on [0, 1] as follows (see [START_REF] Abramowitz | editeurs, Handbook of mathematical functions[END_REF] p. 775):

P (µ,κ) n (τ ) := n j=0 n + µ j n + κ n -j (τ -1) n-j τ j , (9)
where µ, κ ∈] -1, +∞[, and

n + µ j = Γ(n + µ + 1) Γ(n + µ -j + 1)Γ(j + 1) , n + κ n -j = Γ(n + κ + 1) Γ(κ + j + 1)Γ(n -j + 1)
.

Let g 1 and g 2 be two continuous functions defined on [0, 1], then we define the scalar product •, • µ,κ of these functions by (see [START_REF] Abramowitz | editeurs, Handbook of mathematical functions[END_REF] p. 773):

g 1 (•), g 2 (•) µ,κ = 1 0 w µ,κ (τ ) g 1 (τ ) g 2 (τ ) dτ, (10)
where w µ,κ (τ ) = (1τ ) µ τ κ is the associated weighted function. Thus, by denoting its associated norm by • µ,κ , we obtain:

P (µ,κ) n 2 µ,κ = Γ(µ + n + 1) Γ(κ + n + 1) Γ(µ + κ + n + 1) Γ(n + 1) (2n + µ + κ + 1) . ( 11
)
By applying the linearity of the fractional order derivative given in ( 6) to (9) and using ( 4) and ( 5), we obtain the fractional order derivatives of the Jacobi polynomial in the following lemma.

Lemma 1 The α th (α ∈ R + \ N) order derivatives of the n th order Jacobi orthogonal polynomial P (µ,κ) n defined in [START_REF] Oldham | The Fractional Calculus[END_REF] are given as follows:

∀ τ ∈]0, 1], R D α 0,τ P (µ,κ) n (•) = n j=0 n-j i=0 c µ,κ,n,j,i Γ(n -i + 1) Γ(n -i + 1 -α) τ n-i-α , (12)
C D α 0,τ P (µ,κ) n (•) =        0, if n < α, n j=0 min(n-j,n-l) i=0 c µ,κ,n,j,i Γ(n -i + 1) Γ(n -i + 1 -α) τ n-i-α , if n ≥ α, (13)
where c µ,κ,n,j,i = (-1) i n+µ j n+κ n-j n-j i , and l ∈ N * with l -1 < α < l.

III. FRACTIONAL ORDER DIFFERENTIATION BY INTEGRATION

In this section, we are going to extend the method of differentiation by integration involving some Jacobi polynomials, introduced in [1], [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF], from the integer order to the fractional order. Let y be a continuous function defined on R, and y ̟ be a noisy observation of y on an interval

I = [a, b] ⊂ R of length h = b -a: ∀ t ∈ I, y ̟ (t) = y(t) + ̟(t), (14)
where the noise ̟ is integrable and locally essentially bounded, i.e. locally bounded except on a set of measure zero. We are going to estimate the α th (α ∈ R + \ N) order derivative of y using the observation

y ̟ .
In this part, we ignore the noise. The Jacobi orthogonal series expansion of y(a

+ h•) ([48] p. 6) is
given by:

∀ ξ ∈ [0, 1], y(a + hξ) = +∞ i=0 P (µ,κ) i (•), y(a + h•) µ,κ P (µ,κ) i 2 µ,κ P (µ,κ) i (ξ). (15)
By taking the truncated Jacobi orthogonal series expansion of y(a + h•), we define an N th (N ∈ N)

order polynomial which approximates y on I:

∀ ξ ∈ [0, 1], D (0) h,µ,κ,N y(a + hξ) := N i=0 P (µ,κ) i (•), y(a + h•) µ,κ P (µ,κ) i 2 µ,κ P (µ,κ) i (ξ). (16)
If we take κ = µ = 0, then the Jacobi orthogonal polynomials become the Legendre orthogonal polynomials. Hence, this Jacobi polynomial filter can be considered as the extension of the Legendre polynomial filter (see [START_REF] Persson | Smoothing by Savitzky-Golay and Legendre filters[END_REF], [START_REF] Szegö | Orthogonal polynomials[END_REF] p. 29).

January 19, 2013 DRAFT

Let us recall that the integer order derivative of the previous approximation polynomial was used for the integer order differentiation by integration in [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF], [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF], [START_REF] Liu | Convergence Rate of the Causal Jacobi Derivative Estimator, Curves and Surfaces[END_REF], [START_REF] Liu | Differentiation by integration with Jacobi polynomials[END_REF]. Hence, by taking the fractional order derivative of this polynomial, we extend this method of differentiation by integration from the integer order to the fractional order in the following theorem.

Theorem 1 Let y ̟ be the noisy observation defined in [START_REF] Podlubny | Analogue realizations of fractional-order controllers[END_REF]. If we assume y ∈ C l (R) with l ∈ N * , then an estimate for the fractional order derivative D α 0,hξ y(a + •) (l -1 < α < l) of y is defined as follows:

∀ ξ ∈]0, 1], D (α) h,µ,κ,N y ̟ (a + hξ) := 1 h α 1 0 Q µ,κ,α,N (τ, ξ) y ̟ (a + hτ ) dτ, ( 17
)
where h ∈ R * + , N ∈ N, µ, κ ∈] -1, +∞[, and
Q µ,κ,α,N (τ, ξ) = w µ,κ (τ ) N i=0 P (µ,κ) i (τ ) P (µ,κ) i 2 µ,κ q µ,κ,i (ξ), ( 18
)
with q µ,κ,i (ξ) = D α 0,ξ P (µ,κ) i (•). This differentiator refers to R D (α) h,µ,κ,N y ̟ (a + hξ) (resp. C D (α) h,µ,κ,N y ̟ (a + hξ)) which estimates R D α 0,hξ y(a + •) (resp. C D α 0,hξ y(a + •)). Moreover, we have: ∀ ξ ∈]0, 1], D (α) h,µ,κ,N y ̟ (a + hξ) = 1 h α N i=0 P (µ,κ) i (•), y ̟ (a + h•) µ,κ P (µ,κ) i 2 µ,κ q µ,κ,i (ξ). (19)
On the one hand, according to [START_REF] Tseng | Design of fractional order digital differentiator using radial basis function[END_REF], this proposed differentiator

D (α)
h,µ,κ,N y ̟ (a + hξ) uses an integral involving the Jacobi polynomials to estimate the fractional order derivative of a noisy signal. We will call this differentiator fractional order Jacobi differentiator, and we will call this method fractional order differentiation by integration. It can be used both for continuous-time and discrete-time models.

In continuous case, we need to calculate the integral of the noisy signal y ̟ and a sum of the Jacobi polynomials. This integral corresponds to a convolution in discrete case.

On the other hand, it can be used both for off-line and on-line applications. For the off-line applications, we fix the length of the interval I = [a, b]. Hence, by setting the parameters κ, µ and N , we can calculate the scalar product and norm values in each term of the sum in [START_REF] Barbosa | Time domain design of fractional differintegrators using least-squares[END_REF]. Then, the values of D

IV. ERROR ANALYSIS IN CONTINUOUS CASE

The estimation error for the integer order Jacobi differentiator has been studied in [START_REF] Liu | Differentiation by integration with Jacobi polynomials[END_REF], [START_REF] Liu | Convergence Rate of the Causal Jacobi Derivative Estimator, Curves and Surfaces[END_REF], [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF] in the continuous case. We will use a similar procedure to determine some error analysis for the fractional order Jacobi differentiator in this section.

By writing the following equality:

∀ ξ ∈]0, 1], D (α) h,µ,κ,N y ̟ (a + hξ) -D α 0,hξ y(a + •) = D (α) h,µ,κ,N y ̟ (a + hξ) -D (α)
h,µ,κ,N y(a + hξ)

+ D (α) h,µ,κ,N y(a + hξ) -D α 0,hξ y(a + •) , (20)
the estimation error for the fractional order Jacobi differentiator in the noisy case can be divided into two sources:

• the noise error contribution:

e ̟ µ,κ,h,α,N (ξ) = D (α) h,µ,κ,N y ̟ (a + hξ) -D (α) h,µ,κ,N y(a + hξ),
which is due to the noise of signal y ̟ ;

• the N th order truncated term error:

e ∞ µ,κ,h,α,N (ξ) = D (α) h,µ,κ,N y(a + hξ) -D α 0,hξ y(a + •),
which comes from the truncated part in the Jacobi series expansion of y in [START_REF] Oustaloup | Frequency-band complex noninteger differentiator: Characterization and synthesis[END_REF].

Since the fractional order Jacobi differentiator is a linear operator, then the noise error contribution can be studied in the following proposition.

Proposition 1 Let y ̟ be the noisy observation defined in [START_REF] Podlubny | Analogue realizations of fractional-order controllers[END_REF], where y ∈ C l (R) with l ∈ N * and l -1 < α < l. Then, the noise error contribution in the fractional order Jacobi differentiator D

h,µ,κ,N y ̟ (a + h•) can be given as follows:

∀ ξ ∈]0, 1], e ̟ µ,κ,h,α,N (ξ) = 1 h α 1 0 Q µ,κ,α,N (τ, ξ) ̟(a + hτ ) dτ, ( 21
)
where h ∈ R * + , Q µ,κ,α,N (•, •
) is given by [START_REF] Tseng | Improved design of digital fractional-order differentiators using fractional sample delay[END_REF] with N ∈ N and µ, κ ∈] -1, +∞[. Moreover, if the essential supremum 1 of |̟| exists on I, i.e. δ = ess sup |̟| < ∞, then we have:

∀ ξ ∈]0, 1], e ̟ µ,κ,h,α,N (ξ) ≤ δ h α E µ,κ,α,N (ξ), ( 22
)
where E µ,κ,α,N (ξ) = Consequently, it is shown that the noise error contribution is decreasing with respect to h. By studying the influence of the parameters κ, µ and N on the value E µ,κ,α,N (ξ), we can also deduce their influence on the noise error contribution e ̟ µ,κ,h,α,N (ξ). In the following proposition, we study the truncated term error.

Proposition 2 Let y ̟ be the noisy observation defined by [START_REF] Podlubny | Analogue realizations of fractional-order controllers[END_REF], where we assume that y ∈ C N +1 (R) with N ∈ N * . If the Taylor series expansion of y at the point a converges on I, then the N th (N ∈ N * ) order truncated term error in the fractional order Jacobi differentiator D (α) h,µ,κ,N y ̟ (a + h•), where α < n = min(N, N ) with α ∈ R + \ N, can be given as follows:

∀ ξ ∈]0, 1], e ∞ µ,κ,h,α,N (ξ) = h n+1-α 1 0 Q µ,κ,α,N (τ, ξ) I y n,h (τ ) dτ + I y n,α,h (ξ) , ( 23
)
where h ∈ R * + , and Q µ,κ,α,N (•, •) is given by ( 18) with µ, κ ∈] -1, +∞[, and

I y n,h (τ ) = τ n+1 n! 1 0 (1 -u) n y (n+1) (a + uhτ ) du, I y n,α,h (ξ) = ξ n+1-α Γ(n + 1 -α) 1 0 (1 -τ ) n-α y (n+1) (a + ξhτ ) dτ. Moreover, if M n+1 = y (n+1)
∞ exists on I, then e ∞ µ,κ,h,α,N (•) can be bounded as follows:

∀ ξ ∈]0, 1], e ∞ µ,κ,h,α,N (ξ) ≤ h n+1-α M n+1 C µ,κ,α,n,N (ξ), (24)
where C µ,κ,α,n,N (ξ) = Iµ,κ,α,n,N (ξ)

(n+1)! + ξ n+1-α Γ(n+2-α) with I µ,κ,α,n,N (ξ) = 1 0 Q µ,κ,α,N (τ, ξ) τ n+1 dτ .
Consequently, the fractional order Jacobi differentiator can give the exact fractional order derivatives of an n th order polynomial in continuous noise-free case. Moreover, the convergence rate for the truncated term error can be improved by increasing n = min N, N . Let us recall that N refers to the order of the approximation polynomial defined in [START_REF] Charef | Fractal system as represented by singularity function[END_REF] which can be an arbitrary non-zero integer, and N depends on the smoothness of the function y. We can take n = N = N . Hence, the convergence rate is h N +1-α .

Consequently, by providing some additional smoothness hypotheses on y the convergence rate can be improved. In the case where y is analytic, it is sufficient to take n = N .

Unlike the noise error contribution, the truncated term error can be increasing with respect to h. An appropriate value of h is then proposed in the following corollary.

Corollary 1 Let y ̟ be the noisy observation defined by [START_REF] Podlubny | Analogue realizations of fractional-order controllers[END_REF]. We also assume that ̟ and y satisfy the hypotheses given in Proposition 1 and Proposition 2 respectively. Then, the total estimation error in the fractional order Jacobi differentiator D (α) h,µ,κ,N y ̟ (a + h•) can be bounded as follows:

D (α) h,µ,κ,N y ̟ (a + hξ) -D α 0,hξ y(a + •) ≤ h n+1-α M n+1 C µ,κ,α,n,N (ξ) + δ h α E µ,κ,α,N (ξ). ( 25
)
January 19, 2013 DRAFT

Moreover, if we choose

h * = α E µ,κ,α,N (ξ) δ (n + 1 -α)M n+1 C µ,κ,α,n,N (ξ) 1 n+1
, then we have:

D (α) h,µ,κ,N y ̟ (a + h * ξ) -D α 0,h * ξ y(a + •) = O(δ n+1-α n+1 ).
In the second part of this work [START_REF] Liu | Fractional order differentiation by integration and error analysis in noisy environment: Part 2 discrete case[END_REF], we will give some error bounds for the noise error contributions due to a large class of stochastic processes in discrete case. Then, these noise error bounds and the error bound for the truncated term error obtained in this section will be used to study the design parameters' influence on the obtained fractional order differentiators.

V. SIMULATION RESULTS

In order to show the accuracy and robustness with respect to corrupting noises of the fractional order Jacobi differentiator, we present some numerical results in this section. The reader may find some comparisons between the fractional order Jacobi differentiator and some exiting fractional order differentiators in [START_REF] Liu | Fractional order differentiation by integration with Jacobi polynomials[END_REF], [START_REF] Chen | Digital Fractional Order Savitzky-Golay Differentiator[END_REF]. Let y ̟ (t i ) = sin(5t i ) + σ̟(t i ) be a noisy signal with σ adjusted such that the signal-to-noise ratio SN R = 10 log 10 [START_REF] Haykin | Signals and Systems[END_REF] for this well known concept in signal processing). We also set h=4,µ=0,κ=0,N =15 y ̟ (ti) -R D 0.5 0,t i y(•) . classical Riemann-Liouville fractional order derivative of sin(5•) (see [START_REF] Miller | An Introduction to the Fractional Calculus and Fractional Differential Equations[END_REF] p. 83):

|y ̟ (ti)| 2 |σ̟(ti)| 2 is equal to SN R = 20dB (see
t i = iT s ∈ I = [0, h] with T s = I M , for i = 0, • • • , M ∈ N * . Let
R D α 0,t sin(5•) = 5t 1-α Γ(2 -α) 1 F 2 1; 1 2 (2 -α), 1 2 (3 -α); - 1 4 5 2 t 2 , (26)
where 1 F 2 is the generalized hypergeometric function (see [START_REF] Miller | An Introduction to the Fractional Calculus and Fractional Differential Equations[END_REF] p. 303). Then, the Caputo fractional order derivative of sin(5•) can be deduced by using (3).

In the following examples, we fix the value of M to M = 10 3 . Then, we use the fractional order Jacobi differentiator R D (α),θ h,µ,κ,N y ̟ (t i ) given in Theorem 1 to estimate the half order (α = 0.5) derivative January 19, 2013 DRAFT of y. In the estimation procedure, we use the trapezoidal rule to numerically calculate the scalar products in [START_REF] Barbosa | Time domain design of fractional differintegrators using least-squares[END_REF]. Then, the values of R D 0.5 0,ti y(•) can be estimated by taking different values of ξ i in q µ,κ,i (ξ i ), where ξ i = ti Ts , for i = 1, 2, • • • , M . Example 1. In this example, we take h = 1, σ = 0.077. The noise ̟(t i ) is simulated from a zeromean white Gaussian iid sequence by the Matlab function 'randn' with STATE reset to 0. We can see this discrete noisy signal in Figure 1. By taking µ = κ = 0 and N = 5 in R D (α),θ h,µ,κ,N y ̟ (t i ), the obtained estimation and the corresponding estimation error are shown in Figure 2. Hence, we can see that the proposed fractional order Jacobi differentiator is accurate and robust with respect to a zero-mean Gaussian noise. However, the estimation errors near the extremities of the interval are relatively larger. This phenomena will be explained in the second part [START_REF] Liu | Fractional order differentiation by integration and error analysis in noisy environment: Part 2 discrete case[END_REF].

Example 2. In this example, we increase the length of the interval I by taking h = 4. The noise σ̟(t i ) is still a zero-mean white Gaussian noise with σ = 0.074. As done in the previous example, we take

µ = κ = 0 in R D (α),θ h,µ,κ,N y ̟ (t i ).
In this case, we take N = 15. Then, we obtain the estimation and the corresponding estimation error in Figure 3. Hence, we can see that this value of N guarantees the validity of the fractional order Jacobi differentiator.

In these two examples, on the one hand, we assume that the noisy signals are given in the whole interval I, then the values of the fractional order derivatives are estimated on each point of I by taking different value of ξ. This corresponds to an off-line application of the fractional order Jacobi differentiator.

In the second part of this work [START_REF] Liu | Fractional order differentiation by integration and error analysis in noisy environment: Part 2 discrete case[END_REF], we will show how to use the fractional order Jacobi differentiator for on-line applications. On the other hand, we take µ = κ = 0 in R D (α),θ h,µ,κ,N y ̟ (t i ). The value of N is equal to 5 when h = 1, and it is increased to 15 when h = 4. The analysis on these choices of µ, κ and N will also be given in the second part [START_REF] Liu | Fractional order differentiation by integration and error analysis in noisy environment: Part 2 discrete case[END_REF].

VI. CONCLUSION

In this paper, firstly we have extended the differentiation by integration method from the integer order to the fractional order by proposing two kinds of fractional order Jacobi differentiators. These differentiators can accurately estimate the Riemann-Liouville and the Caputo fractional order derivatives by using exact integral formulae, without complex mathematical deduction. Secondly, accurate error analysis has been proposed by proving some error bounds for the corresponding estimation errors in continuous case.

Finally, numerical simulations have been given to show the accuracy and the robustness with respect to a corrupting Gaussian noise of the fractional order Jacobi differentiator in off-line applications. The discrete case is considered in a second paper [START_REF] Liu | Fractional order differentiation by integration and error analysis in noisy environment: Part 2 discrete case[END_REF] where we give a guideline on how to choose the design January 19, 2013 DRAFT parameters on which the fractional order Jacobi differentiators depend. Then, according to the knowledge of the design parameters' influence, an improved digital fractional order Jacobi differentiator is given by a recursive algorithm for on-line applications.

APPENDIX A

PROOFS

Proof of Theorem 1. We denote the α th order derivative of the polynomial D

h,µ,κ,N y(a + •) defined in [START_REF] Charef | Fractal system as represented by singularity function[END_REF] as follows:

∀ ξ ∈]0, 1], D (α) h,µ,κ,N y(a + hξ) := D α 0,hξ D (0) h,µ,κ,N y(a + •) . (27)
By using the scale change propriety of the fractional order derivative given in [START_REF] Pommier | CRONE control of a nonlinear hydraulic actuator[END_REF], we obtain:

∀ ξ ∈]0, 1], D (α)
h,µ,κ,N y(a

+ hξ) = 1 h α D α 0,ξ D (0) h,µ,κ,N y(a + h•) . (28)
Then, we apply the linearity of the fractional order derivative to [START_REF] Charef | Fractal system as represented by singularity function[END_REF]. It yields:

∀ ξ ∈]0, 1], D (α)
h,µ,κ,N y(a

+ hξ) = 1 h α N i=0 P (µ,κ) i (•), y(a + h•) µ,κ P (µ,κ) i 2 µ,κ D α 0,ξ P (µ,κ) i (•). (29)
By using Lemma 1 in (29), we obtain: ∀ ξ ∈]0, 1],

D (α) h,µ,κ,N y(a + hξ) = 1 h α 1 0 Q µ,κ,α,N (τ, ξ) y(a + hτ ) dτ. (30)
Finally, this proof can be completed by substituting y by y ̟ in [START_REF] Liu | Differentiation by integration with Jacobi polynomials[END_REF].

Proof of Proposition 1. By using y ̟ = y +̟ in [START_REF] Tseng | Design of fractional order digital differentiator using radial basis function[END_REF], the noise error contribution for D

(α) h,µ,κ,N y ̟ (a+h•) can be obtained: ∀ ξ ∈]0, 1], e ̟ µ,κ,h,α,N (ξ) = D (α) h,µ,κ,N y ̟ (a + hξ) -D (α) h,µ,κ,N y(a + hξ) = 1 h α 1 0 Q µ,κ,α,N (τ, ξ) ̟(a + hτ ) dτ.
Then, this proof can be completed by taking the absolute value of e ̟ µ,κ,h,α,N (ξ). Before giving the proof of Proposition 2, we need the two following lemmas.

Lemma 2 We assume that y n is an n th order polynomial defined on I. If n ≤ N with N ∈ N, then we have: Proof. Since y n is an n th order polynomial defined on I with n ≤ N , its Jacobi orthogonal series expansion can be given by ( 16), i.e. we have y n (a + h•) ≡ D

∀ ξ ∈]0, 1], D (α) h,µ,κ,N y n (a + hξ) = D α 0,hξ y n (a + •), (31)
h,µ,κ,N y n (a + h•). Then, this proof can be completed by taking the α th order derivative of y n (a + •) and using [START_REF] Diekema | Differentiation by integration using orthogonal polynomials, a survey[END_REF].

Lemma 3 We assume that y ∈ C n+1 (R) with n ∈ N * , and α < n with α ∈ R + \ N. Then we have:

∀ ξ ∈]0, 1], R D α 0,hξ y(a + •) = n i=0 (hξ) i-α Γ(i + 1 -α) y (i) (a) + h n+1-α Γ(n + 1 -α) ξ 0 (ξ -τ ) n-α y (n+1) (a + hτ ) dτ, (32) C D α 0,hξ y(a + •) = n i=l (hξ) i-α Γ(i + 1 -α) y (i) (a) + h n+1-α Γ(n + 1 -α) ξ 0 (ξ -τ ) n-α y (n+1) (a + hτ ) dτ, ( 33
)
where l -1 < α < l ≤ n with l ∈ N * .

Proof. By substituting f (•) in (3) by y(a + •), we obtain:

R D α 0,hξ y(a + •) = l-1 i=0 (hξ) i-α Γ(i + 1 -α) y (i) (a) + C D α 0,hξ y(a + •). (34)
By using (2), the following change of variables τ → hτ gives us:

C D α 0,hξ y(a + •) = h l-α Γ(l -α) ξ 0 (ξ -τ ) l-1-α y (l) (a + hτ ) dτ. (35)
Finally, this proof can be completed by applying nl + 1 times integration by parts in [START_REF] Fliess | Compression différentielle de transitoires bruités[END_REF].

Consequently, by using ( 4) and ( 5), the formulae given in ( 32) and ( 33) can be considered as the Riemann-Liouville and the Caputo fractional order derivative of the Taylor series expansion of y(a + h•)

with an integral remainder, respectively.

Proof of Proposition 2. If the Taylor series expansion of y at point a converges on I, then the Taylor's formula gives us (see [START_REF] Abramowitz | editeurs, Handbook of mathematical functions[END_REF] p.

Hence, the utilization of ( 36) in [START_REF] Fliess | An algebraic framework for linear identification[END_REF] gives us:

where I y n,α,h (ξ) = ξ n+1-α Γ(n+1-α) 1 0 (1τ ) n-α y (n+1) (a + ξhτ ) dτ. Finally, this proof can be completed by taking the absolute value of e ∞ µ,κ,h,α,N (•) and the following inequalities:

I y n,h (τ ) ≤ M n+1 τ n+1 n! 1 0 |(1 -u) n | du = M n+1 τ n+1 (n + 1)! , I y n,α,h (ξ) ≤ M n+1 ξ n+1-α Γ(n + 1 -α) 1 0 (1 -τ ) n-α dτ = M n+1 ξ n+1-α Γ(n + 2 -α) .
Proof of Corollary 1. Let us denote the error bound by: ψ(h) = h n+1-α M n+1 C µ,κ,α,n,N (ξ) + δ h α E µ,κ,α,N (ξ).

Consequently, we can calculate its minimum value. It is obtained for h * = α Eµ,κ,α,N (ξ) δ (n+1-α)Mn+1Cµ,κ,α,n,N (ξ)

The proof is thus complete.

  ,κ,N y ̟ (a+hξ) can be easily obtained for any ξ ∈]0, 1]. For the on-line applications, the length of the interval I varies with time (b = t). If we fix the value of ξ to 1, then in this case the fractional Jacobi differentiatorD (α)h,µ,κ,N y ̟ (a + hξ) only estimates the value of D α 0,t y(a + •) at point t = b = a + h. In order to give a guideline on how to choose the parameters κ, µ and N , we are going to study the estimation error for the fractional order Jacobi differentiator D (α) h,µ,κ,N y ̟ (a + hξ) in the next section. January 19, 2013 DRAFT

1 0

 1 |Q µ,κ,α,N (τ, ξ)| dτ . 1 The essential supremum of |̟|, denoted by ess sup |̟|, is defined by: ess sup |̟| := inf{c ∈ R : η({t ∈ I : |̟(t)| > c}) = 0}, where I is a set, and η is a measure on I. January 19, 2013 DRAFT

Fig. 1 .

 1 Fig. 1. Signal y and the noisy signal y ̟ (ti) = sin(5ti) + 0.077̟(ti).

  i y(•) and RD (α=0.5),θ=1 h=1,µ=0,κ=0,N =5 y ̟ (ti).

  ,κ=0,N =5 y ̟ (ti) -R D 0.5 0,t i y(•) .

Fig. 2 .

 2 Fig. 2. Example 1: I = [0, 1] and ̟(ti) is a zero-mean Gaussian noise.

Fig. 3 .

 3 Fig. 3. Example 2: I = [0, 4] and ̟(ti) is a zero-mean Gaussian noise.

( 1 -

 1 14): ∀ ξ ∈ [0, 1], y(a + hξ) u) n y (n+1) (a + uhξ) du.(36)Then, we define the following truncated Taylor series expansion:∀ ξ ∈ [0, 1],y n (a + hξ) α th order derivative of y n (a + •) can be defined and calculated as follows:∀ ξ ∈]0, 1], R D α 0,hξ y n (a + •) = n i=0 (hξ) i-α Γ(i + 1α) y (i) (a),(38)C D α 0,hξ y n (a + •) = n i=l (hξ) i-α Γ(i + 1α) y (i) (a). (39)Let us consider the following equality:∀ ξ ∈]0, 1], D(α)h,µ,κ,N y(a + hξ) -D α 0,hξ y(a + •) = D (α) h,µ,κ,N y(a + hξ) -D α 0,hξ y n (a + •) + D α 0,hξ y n (a + •) -D α 0,hξ y(a + •) .

( 40 )h α 1 0

 401 Then, by using Lemma 2 and (30) we get:D (α) h,µ,κ,N y(a + hξ) -D α 0,hξ y n (a + •) = 1 Q µ,κ,α,N (τ, ξ) (y(a + hτ )y n (a + hτ )) dτ.

D 1 0Q 1 0 ( 1 - 1 0Q

 1111 (α) h,µ,κ,N y(a + hξ) -D α 0,hξ y n (a + •) = h n+1-α µ,κ,α,N (τ, ξ) I y n,h (τ ) dτ,(42)whereI y n,h (τ ) = τ n+1 n! u) n y (n+1) (a + uhτ ) du.Consequently, by using (40), (38) (resp.[START_REF] Liu | Parameters estimation of a noisy sinusoidal signal with time-varying amplitude[END_REF]) and applying the following change of variables τ → ξτ in (32) (resp. (33)) we get:∀ ξ ∈]0, 1], e ∞ µ,κ,h,α,N (ξ) =D (α)h,µ,κ,N y(a + hξ) -D α 0,hξ y(a + •) =h n+1-α µ,κ,α,N (τ, ξ) I y n,h (τ ) dτ + h n+1-α I y n,α,h (ξ),

  C µ,κ,α,n,N (ξ)) α n+1 E µ,κ,α,N (ξ)

January 19, 2013 DRAFT


Da-Yan Y Liu
email: dayan.liu@kaust.edu.sa

Olivier Gibaru
email: olivier.gibaru@ensam.eu

Wilfrid Perruquetti
email: wilfrid.perruquetti@inria.fr

Taous-Meriem M Laleg-Kirati
email: taousmeriem.laleg@kaust.edu.sa

Fractional order differentiation by integration and error analysis in noisy environment: Part 2 discrete case

, the differentiation by integration method has been generalized from the integer order to the fractional order so as to estimate the fractional order derivatives of noisy signals. The estimation errors for the proposed fractional order Jacobi differentiators have been studied in continuous case. In this paper, the focus is on the study of these differentiators in discrete case. Firstly, the noise error contribution due to a large class of stochastic processes is studied in discrete case. In particular, it is shown that the differentiator based on the Caputo fractional order derivative can cope with a class of noises, the mean value and variance functions of which are time-variant. Secondly, by using the obtained noise error bound and the error bound for the bias term error obtained in the first part, we analyze the design parameters' influence on the obtained fractional order differentiators. Thirdly, according to the knowledge of the design parameters' influence, the fractional order Jacobi differentiators are significantly improved by admitting a time-delay. In order to reduce the calculation time for on-line applications, a recursive algorithm is proposed. Finally, numerical simulations show their accuracy and robustness with respect to corrupting noises.

Index Terms

Digital fractional order differentiator, Error analysis, Time-delay.

EDICS: SSP-FILT; ASP-ANAL.

I. INTRODUCTION

Fractional derivatives are gaining importance in research community because of their capacity to accurately describe real world processes (see, e.g., [START_REF] Podlubny | Fractional Differential Equations[END_REF], [START_REF] Oldham | The Fractional Calculus[END_REF]). The fractional order differentiator is concerned with estimating the fractional order derivatives of an unknown signal from its discrete noisy observation. It can be used to improve the performance and robustness properties in control theory (see, e.g., [START_REF] Oustaloup | From fractal robustness to the CRONE control[END_REF], [START_REF] Pommier | CRONE control of a nonlinear hydraulic actuator[END_REF]) and in signal processing applications which can be on-line or off-line (see, e.g., [START_REF] Mathieu | Fractional differentiation for edge detection[END_REF], [START_REF] Benmalek | Digital fractional order operators for R-wave detection in electrocardiogram signal[END_REF]). When designing a fractional order differentiator, the accuracy and the robustness to noise effect must be considered.

Moreover, unlike the classical integer order derivative which can be estimated using a sliding window, the fractional order derivative is an hereditary operator and needs a total memory of past states [START_REF] Podlubny | Fractional Differential Equations[END_REF].

Hence, the computation time must also be considered in a fractional order differentiator. In the first part of this work [START_REF] Liu | Fractional order differentiation by integration and error analysis in noisy environment: Part 1 continuous case[END_REF], we have proposed two kinds of fractional order Jacobi differentiators by generalizing the differentiation by integration method from the integer order to the fractional order. Accurate analysis for the corresponding estimation errors has been given in the continuous case where the noise was assumed to be integrable and locally essentially bounded. The aim of this paper is to study the fractional order Jacobi differentiators in discrete case in order to show their robustness to noises belonging to a large class of stochastic processes. We also show how they can be used for on-line applications. This paper is organized as follows: In Section II, we provide some error bounds for the noise error contributions due to a large class of stochastic processes in discrete case. These error bounds permit to study the influence of design parameters for the proposed differentiators in Section III. In Section IV, by using the knowledge of the design parameters' influence, an improved digital fractional order differentiator is introduced. Then, a recursive algorithm is given to reduce the computation time for on-line applications. In order to show the efficiency and the stability of the proposed differentiators, numerical simulations results are given in Section V. Finally, we give some conclusions and perspectives for our future work in Section VI. To simplify the presentation, all the proofs are deferred to the appendix. If we assume y ∈ C l (R) with l ∈ N * , then an estimate for the fractional order derivative D α 0,hξ y(a + •) (l -1 < α < l) of y, called fractional order Jacobi differentiator, was defined in Theorem 1 in the first part [START_REF] Liu | Fractional order differentiation by integration and error analysis in noisy environment: Part 1 continuous case[END_REF] as follows: ∀ ξ ∈]0, 1],

D (α) h,µ,κ,N y ̟ (a + hξ) := 1 h α 1 0 Q µ,κ,α,N (τ, ξ) y ̟ (a + hτ ) dτ, (1)
where

h ∈ R * + , N ∈ N, µ, κ ∈] -1, +∞[, Q µ,κ,α,N (τ, ξ) = w µ,κ (τ ) N i=0 P (µ,κ) i (τ ) P (µ,κ) i 2 µ,κ q µ,κ,i (ξ), (2)
P (µ,κ) i
is the i th order shifted Jacobi orthogonal polynomial defined on [0, 1], • µ,κ is the associated norm with respect to the weighted function w µ,κ (τ ) = (1τ ) µ τ κ , and

D α 0,ξ P (µ,κ) i (•) is denoted by q µ,κ,i (ξ). This differentiator refers to R D (α) h,µ,κ,N y ̟ (a + hξ) (resp. C D (α) h,µ,κ,N y ̟ (a + hξ)) which estimates R D α 0,hξ y(a + •) (resp. C D α 0,hξ y(a + •)). Moreover, we have: ∀ ξ ∈]0, 1], D (α) h,µ,κ,N y ̟ (a + hξ) = 1 h α N i=0 P (µ,κ) i (•), y ̟ (a + h•) µ,κ P (µ,κ) i 2 µ,κ q µ,κ,i (ξ). (3)
From now on, we assume y ̟ (t i ) = y(t i )+̟(t i ) be a noisy discrete observation of y with an equidistant sampling period T s , where T s = h M , M ∈ N * , and t i = a + hξ i with ξ i = i M , for i = 0, • • • , M . We also consider a family of noises which are stochastic processes satisfying the following conditions:

(C 1 ) : for any t, s ∈ I, t = s, ̟(t) and ̟(s) are independent;

(C 2 ) : the mean value function of ̟(•) denoted by E[•] belongs to L 1 (I); (C 3 ) : the variance function of ̟(•) denoted by Var[•] is bounded on I.
Note that some important classes of noises such as the white Gaussian noise and the Poisson noise satisfy these conditions. We are going to study the fractional order Jacobi differentiator in this discrete case.

A. Fractional order Jacobi differentiator in discrete case

Since y ̟ is a discrete observation, we need to use a numerical integration method to approximate the integral in the fractional order Jacobi differentiator given in [START_REF] Liu | Fractional order differentiation by integration and error analysis in noisy environment: Part 1 continuous case[END_REF].

Firstly, we consider the case where κ ≥ 0 and µ ≥ 0. It is clear that the smaller the sampling period is, the more accurate the numerical integration method is. However, when the value of T s decreases, the number of the observed data M increases, hence the computation time becomes larger. In order to solve this problem, we introduce a non-zero integer value θ, and we only take one sample value per θ.

Hence, we fix τ j = j M with M = M θ , ( M , θ) ∈ (N * ) 2 , and we denote by w j ≥ 0, for j = 0, . . . , M , the weights of a given numerical integration method. Then, the θ-adaptive fractional order Jacobi differentiator

D (α),θ h,µ,κ,N y ̟ (a + hξ i ) is expressed in the discrete case as follows: ∀ ξ i ∈]0, 1], D (α),θ h,µ,κ,N y ̟ (a + hξ i ) := 1 h α M j=0 w j M Q µ,κ,α,N (τ j , ξ i ) y ̟ (a + hτ j ), (4)
with κ ≥ 0, µ ≥ 0. Secondly, one can remark that when κ is negative in (1) then the corresponding integral is an improper integral. Hence, if w 0 = 0 in (4), then there will be a singular value at τ 0 = 0.

In order to avoid this problem, we apply the following change of variable τ → τ 1 1+κ in (1) (see [START_REF] Press | Numerical Recipes in C, Second Edition[END_REF], p.

145). Thus, we get:

D (α) h,µ,κ,N y ̟ (a + hξ) = 1 h α 1 0 Qµ,κ,α,N (τ 1 1+κ , ξ) y ̟ (a + h τ 1 1+κ ) dτ, (5)
where -1 < κ < 0, µ ≥ 0, and Qµ,κ,α,N (τ

1 1+κ , ξ) = 1 1 + κ (1 -τ 1 1+κ ) µ N i=0 q µ,κ,i (ξ) P (µ,κ) i 2 µ,κ P (µ,κ) i (τ 1 1+κ
).

Since y ̟ is equidistantly given, let τj = j M 1+κ , i.e. τ 1 1+κ j = j M = τ j , be the new abscissas. Hence, the numerical integration steps are equal to 1 M 1+κ [j 1+κ -(j -1) 1+κ ], for j = 1, . . . , M . We denote by ŵj ≥ 0 their corresponding weights. Then, we have:

∀ ξ i ∈]0, 1], D (α),θ h,µ,κ,N y ̟ (a + hξ i ) = 1 h α M j=0 ŵj M 1+κ Qµ,κ,α,N (τ j , ξ i ) y ̟ (a + hτ j ), (6)
with -1 < κ < 0, µ ≥ 0. Finally, we can use a similar procedure in the case where µ is negative (see [START_REF] Press | Numerical Recipes in C, Second Edition[END_REF], p. 145 for more details). Then, we can apply a numerical integration method without any singular value.

The estimation error for the fractional order Jacobi differentiator in the discrete noisy case can be decomposed into three sources:

∀ ξ i ∈]0, 1], D (α),θ h,µ,κ,N y ̟ (a + hξ i ) -D α 0,hξi y(a + •) = D (α),θ h,µ,κ,N y ̟ (a + hξ i ) -D (α),θ h,µ,κ,N y(a + hξ i ) + D (α),θ h,µ,κ,N y(a + hξ i ) -D (α) h,µ,κ,N y(a + hξ i ) + D (α) h,µ,κ,N y(a + hξ i ) -D α 0,hξi y(a + •) , (7)
where we have:

1) the discrete noise error contribution:

e ̟,θ µ,κ,h,α,N (ξ i ) = D (α),θ h,µ,κ,N y ̟ (a + hξ i ) -D (α),θ h,µ,κ,N y(a + hξ i ), (8)
January 19, 2013 DRAFT which is due to the noise in the noisy signal y ̟ ;

2) the numerical error:

e θ µ,κ,h,α,N (ξ i ) = D (α),θ h,µ,κ,N y(a + hξ i ) -D (α) h,µ,κ,N y(a + hξ i ), (9)
which is due to the used numerical integration method;

3) the N th order truncated term error e ∞ µ,κ,h,α,N (ξ i ), which is studied in Proposition 3 in the first part [START_REF] Liu | Fractional order differentiation by integration and error analysis in noisy environment: Part 1 continuous case[END_REF].

By using (4), the corresponding discrete noise error contribution e ̟,θ µ,κ,h,α,N (ξ i ) can be written as follows:

∀ ξ i ∈]0, 1], e ̟,θ µ,κ,h,α,N (ξ i ) = 1 h α M j=0 w j M Q µ,κ,α,N (τ j , ξ i ) ̟(a + hτ j ), (10)
where µ ≥ 0, κ ≥ 0. According to the previous study, e ̟,θ µ,κ,h,α,N (ξ i ) can also be given for the other values of κ and µ.

In the next subsection, we are going to study the influence of the sampling period T s to this discrete noise error contribution.

B. Influence of the sampling period on the discrete noise error contribution

The convergence in mean square with respect to the sampling period of the discrete noise error contributions was studied for the integer order Jacobi differentiator and for some parameter estimators in [START_REF] Liu | Error analysis of Jacobi derivative estimators for noisy signals[END_REF] and [START_REF] Liu | Parameters estimation of a noisy sinusoidal signal with time-varying amplitude[END_REF] respectively. A study was also done in [START_REF] Fliess | Analyse non standard du bruit[END_REF] by using the non-standard framework. We can then extend these results in the following proposition.

Proposition 1 Let ̟(•) be a stochastic process satisfying conditions (C 1 ), (C 2 ), (C 3 ), and {̟(t i )} i=0,••• ,M be a sequence where t i = iT s . If the values of M and θ are fixed, then we have the following convergence in mean square of the discrete noise error contribution in the fractional order Jacobi differentiator

D (α) h,µ,κ,N y ̟ (a + hξ i ): e ̟,θ µ,κ,h,α,N (ξ i ) L 2 (I) = == =⇒ Ts→0 1 h α 1 0 Q µ,κ,α,N (τ, ξ i ) E [̟(a + hτ )] dτ, ( 11
)
where L 2 (I) refers to the set of the square-integrable functions defined on I. Moreover, we have:

• if ∀ t ∈ I, E [̟(t)] = 0, then R e ̟,θ µ,κ,h,α,N (ξ i ) L 2 (I) = == =⇒ Ts→0 0, (12)
• if ∀ t ∈ I, E [̟(t)] = l-1 j=0 ν j t j with ν j ∈ R, l ∈ N * and l -1 < α < l, then C e ̟,θ µ,κ,h,α,N (ξ i ) L 2 (I) = == =⇒ Ts→0 0, (13)
where R e ̟,θ µ,κ,h,α,N (ξ i ) and C e ̟,θ µ,κ,h,α,N (ξ i ) refer to the discrete noisy error contributions in the fractional order Jacobi differentiators R D [START_REF] Liu | Fractional order differentiation by integration and error analysis in noisy environment: Part 1 continuous case[END_REF].

(α) h,µ,κ,N y ̟ (a + hξ i ) and C D (α) h,µ,κ,N y ̟ (a + hξ i ) given in
According to the previous proposition, the discrete noise error contribution can be increasing with respect to the sampling period. As we can see, unlike the Riemann-Liouville fractional order differentiator, the Caputo fractional order differentiator C D (α) h,µ,κ,N y ̟ (a + h•) can cope with a large class of noises the mean value and variance functions of which are time-variant. When the sampling period is set, the discrete noise error contribution does not converge to zero. In the next subsection, we are going to study the discrete noise error contribution with a fixed sampling period.

C. Error bounds on the discrete noise error contribution

An error bound for the noise error contribution is given in Proposition 2 in the first part [START_REF] Liu | Fractional order differentiation by integration and error analysis in noisy environment: Part 1 continuous case[END_REF] in the continuous case where the noise is assumed to be integrable and locally essentially bounded. In this subsection, we are going to study the discrete noise error contribution e ̟,θ µ,κ,h,α,N (ξ i ) by using the stochastic properties of the noise. Similar to the integer order Jacobi differentiator case [START_REF] Liu | Error analysis of Jacobi derivative estimators for noisy signals[END_REF], [START_REF] Liu | Error analysis for a class of numerical differentiator: application to state observation[END_REF], sharper noise error bounds can be given, which permit to study the influence of the parameters on e ̟,θ µ,κ,h,α,N (ξ i ). To simplify our notations, we denote e ̟,θ µ,κ,h,α,N (ξ i ) by e ̟,θ (ξ i ). Since the noise is a stochastic process, it is generally not bounded. However, if the noise ̟(•) satisfies the conditions (C 1 ), (C 2 ), (C 3 ), then by applying the properties of the mean value and variance functions, E[e ̟,θ (ξ i )] and Var[e ̟,θ (ξ i )] can be obtained for any ξ i ∈]0, 1]. Consequently, by using the Bienaymé-Chebyshev inequality, we obtain: for any real number γ > 0,

∀ ξ i ∈]0, 1], Pr e ̟,θ (ξ i ) -E[e ̟,θ (ξ i )] < γ Var[e ̟,θ (ξ i )] > 1 - 1 γ 2 ,
i.e. the probability for e ̟,θ (ξ i ) to be within the interval

M γ l (ξ i ) , M γ h (ξ i ) is higher than 1 -1 γ 2
, where

M γ l (ξ i ) = E[e ̟,θ (ξ i )] -γ Var[e ̟,θ (ξ i )], M γ h (ξ i ) = E[e ̟,θ (ξ i )] + γ Var[e ̟,θ (ξ i )].
January 19, 2013 DRAFT Thus, we deduce two error bounds as follows:

∀ ξ i ∈]0, 1], M γ l (ξ i ) pγ < e ̟,θ (ξ i ) pγ < M γ h (ξ i ), (14)
where a pγ < b means that the probability for a real number b to be larger than an other real number a

is equal to p γ with p γ > 1 -1 γ 2 .
In particular, if ̟ is a white Gaussian noise, then according to the three-sigma rule, we have:

∀ ξ i ∈]0, 1], M γ l (ξ i ) pγ ≤ e ̟,θ (ξ i ) pγ ≤ M γ h (ξ i ), (15)
where p 1 = 68.26%, p 2 = 95.44% and p 3 = 99.73%, for γ = 1, 2, 3, respectively.

III. ANALYSIS OF THE PARAMETERS' INFLUENCE

As previously shown, the estimation error for the fractional order Jacobi differentiator D (α),θ h,µ,κ,N y ̟ (a + hξ i ) in the discrete noisy case can be decomposed into three sources: the numerical error e θ µ,κ,h,α,N (ξ i ), the truncated term error e ∞ µ,κ,h,α,N (ξ i ) and the discrete noise error contribution e ̟,θ µ,κ,h,α,N (ξ i ). We assume that the numerical integration steps used in D (α),θ h,µ,κ,N y ̟ (a+hξ i ) are small enough such that the numerical error can be negligible with respect to the truncated term error and the discrete noise error contribution.

We are going to study the influence of parameters on the truncated term error and the discrete noise error contribution.

An error bound for the truncated term error is given in Proposition 3 in the first part [START_REF] Liu | Fractional order differentiation by integration and error analysis in noisy environment: Part 1 continuous case[END_REF] and denoted by h n+1-α M n+1 C µ,κ,α,n,N (ξ) for any ξ ∈]0, 1]. Let us recall that the value of h depends on the interval where we want to estimate the fractional order derivative, and M n+1 depends on the original signal which is unknown. Hence, we are going to study the influence of parameters on the term C µ,κ,α,n,N (ξ) with ξ ∈]0, 1]. This can help us to generally characterize these parameters' influence on the truncated error, independently of the signal to be differentiated. For doing so, we only consider the case of the Riemann-Liouville fractional derivative. The results for the Caputo fractional derivative case can be similarly obtained.

Firstly, we show in Figure 1(a) the variation of log 10 C µ,κ,α,n,N (ξ i ) for N = 5, 6, . . . , 15 and ξ i = 0.01, 0.1, 0.2, . . . , 1 in the case where n = N , µ = κ = 0 and α = 0.5. Hence, we can deduce that C µ,κ,α,n,N (ξ i ) is decreasing with respect to N , and the values of C µ,κ,α,n,N (•) obtained near the extremities are larger than the other values.

Secondly, we show in Figure 2 the variations of log 10 C µ,κ,α,n,N (ξ i ) for µ = -0.9, -0.8, . . . , 1 and κ = -0.9, -0.8, . . . , 1 in the case where α = 0.5, n = N = 5, ξ i = 0.01 and 1, respectively. Hence, we Fig. 2. Variations of log 10 Cµ,κ,α=0.5,n=5,N=5(ξi) for µ = -0.9, -0.8, . . . , 1 and κ = -0.9, -0.8, . . . , 1.

can see that the contour level set of log 10 C µ,κ,α,n,N (ξ i ) varies differently with respect to κ and µ for different values of ξ i . For this, we study the norm C µ,κ,α,n,N (ξ i ) 2 .

Finally, we show in Figure 3(a) (resp. the error bounds obtained in [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF]. This study is also independent of the original signal to be differentiated, and it only depends on the probability properties of the noise. To simplify this study, we assume that the noises satisfy the conditions (C 1 ), (C 2 ), (C 3 ) and the following condition:

(C 4 ) : ∀ t ∈ I, E[̟(t)] = 0 and Var[̟(t)] = σ 2 ,
with σ ∈ R + . Hence, we only need to study the variance of e ̟,θ µ,κ,h,α,N (ξ i ), which can be obtained in the for ξi = 0.01, 0.1, 0.2, . . . , 1, µ = -0.9, -0.8, . . . , 1 and κ = -0.9, -0.8, . . . , 1.

case of µ ≥ 0 and κ ≥ 0 by using [START_REF] Liu | Parameters estimation of a noisy sinusoidal signal with time-varying amplitude[END_REF]:

∀ ξ i ∈]0, 1], Var e ̟,θ µ,κ,h,α,N (ξ i ) = 1 h 2α σ 2 M 2 M j=0 w 2 j Q 2 µ,κ,α,N (τ j , ξ i ). (16)
We use the trapezoidal rule in [START_REF] Saadatmandi | A new operational matrix for solving fractional-order differential equations[END_REF]. Consequently, we study the variation of Var e ̟,θ µ,κ,h,α,N (ξ i ) with respect to the parameters µ, κ and N by taking θ = h = 1, M = 10 3 and α = 0.5.

Firstly, we show in Figure 1(b) the variation of log 10 Var e ̟,θ µ,κ,h,α,N (ξ i ) for N = 5, . . . , 15 and ξ i = 0.01, 0.1, 0 . . . , 1 in the case where µ = κ = 0. Hence, Var e ̟,θ µ,κ,h,α,N (ξ i ) can be increasing with respect to N , and Var e ̟,θ µ,κ,h,α,N (•) is decreasing (resp. increasing) with respect to ξ i when 0.01 ≤ ξ i ≤ 0.5 (resp. 0.5 ≤ ξ i ≤ 1).

Secondly, we show in Figure 4 the variations of log 10 Var e ̟,θ µ,κ,h,α,N (ξ i ) for µ = -0.9, -0.8, . . . , 1 and κ = -0.9, -0.8, . . . , 1 in the case where N = 5, ξ i = 0.01 and 1, respectively. Hence, we can see that the variation of Var e ̟,θ µ,κ,h,α,N (ξ i ) has a local minimum at κ = µ = 0 in each case. Finally, we show in Figure 5(a) (resp. Figure 5(b)) the variation of log 10 Var e ̟,θ µ,κ,h,α,N (ξ i ) 2 for ξ i = 0.01, 0.1, 0.2, . . . , 1, µ = -0.9, -0.8, . . . , 1 and κ = -0.9, -0.8, . . . , 1 in the case where N = 5 We conclude the parameters' influence on different error bounds in Table I. The first row means that in order to reduce the truncated term error, according to the analysis on the term C µ,κ,α,n,N (ξ i ) 2 in January 19, 2013 DRAFT its associated error bound, we can reduce the value of h and κ. We can also increase the value of N .

Moreover, there exists an optimal value of µ. According to the previous analysis, we should take a trade-off between the parameters so as to do not produce neither a large truncated term error nor a large noise error contribution. Generally, if the value of h = ba is small, then we can take a small value of N . On the one hand, this value can give a relative small truncated term error. On the other hand, a small value of N can avoid a large noise error contribution. But, when the value of h increases with time, we increase the value of N so as to reduce the truncated term error. Nevertheless this strategy consisting of increasing the value of N can increase the noise error contribution. This effect can be thwarted since a larger value of h reduces the noise error contribution. After setting the value of N , we can take κ = µ = 0 to reduce the discrete noisy error contribution. This explains why we have increased the value of N when h was increased in the numerical examples of the frit part [START_REF] Liu | Fractional order differentiation by integration and error analysis in noisy environment: Part 1 continuous case[END_REF]. Consequently, the problem is reduced to how to choose the value of N .

This choice needs the information on the noise level of the noisy signal and on the derivatives of the original signal, which is usually unknown in practical work. Hence, a criterion should be considered in order to choose an appropriate value of N . This is beyond the scope of this paper.

IV. FRACTIONAL ORDER LEGENDRE DIFFERENTIATOR IN DISCRETE NOISY CASE

In this section, we are going to show how to apply the fractional order Jacobi differentiator for online applications in discrete noisy case. According to the analysis done in the previous section, we set κ = µ = 0 in the differentiator so as to reduce the discrete noisy error contribution. This parameters' choice implies the use of the Legendre orthogonal polynomials.

The n th order shifted Legendre orthogonal polynomial defined on [0, 1] is given as follows (see [START_REF] Saadatmandi | A new operational matrix for solving fractional-order differential equations[END_REF]):

∀ τ ∈ [0, 1], P n (τ ) := n k=0 (-1) n+k (k!) 2 (n + k)! (n -k)! τ k . ( 17
)
In order to simplify our notations, we denote D (α) h,0,0,N y ̟ (a + hξ) and Q 0,0,α,N (τ, ξ) by D (α) h,N y ̟ (a + hξ) and Q α,N (τ, ξ) respectively. Moreover, we denote P (1), we obtain:

∀ ξ ∈]0, 1], D (α) h,N y ̟ (a + hξ) = 1 h α 1 0 Q α,N (τ, ξ) y ̟ (a + hτ ) dτ, (18)
where

Q α,N (τ, ξ) = N i=0 (2i + 1)P i (τ ) P (α) i (ξ).
Thus, we call the differentiator D (α) h,N y ̟ (a + hξ) fractional order Legendre differentiator. In next subsection, we are going to consider this differentiator in discrete noisy case so as to introduce a FIRtype digital fractional order Legendre differentiator.

A. A FIR-type digital fractional order Legendre differentiator

We assume that the signal y is observed in a discrete noisy case:

y ̟ (t i ) = y(t i ) + ̟(t i ), for i = 0, 1, . . . , (19)
where t i = iT s , T s is an equidistant sampling period, and ̟ is a noise.

Let us consider the interval

I M = [0, t M ] with M ∈ N * . By taking a = 0, h = t M and ξ = 1 in (18),
we can use the fractional order Legendre differentiator to estimate the value of the α th order derivative of y at t M . Then, we apply the right rectangle rule to approximate the integral in (18):

D α 0,tM y(•) ≈ D (α) h,N y ̟ (t M ) ≈ T -α s M α+1 M j=1 Q α,N (1, τ j ) y ̟ (t j ), (20)
where τ j = j M are the associated abscissas. Then, by applying a change of indices j → Mj in (20), we get:

D α 0,tM y(•) ≈ T -α s M α+1 M j=1 Q α,N (1, τ M -j ) y ̟ (t M -j ). (21)
Consequently, by denoting y ̟ (t M -j ) by y ̟ M -j and using the expression of Q α,N (•, •) given in (18), we obtain the following FIR digital fractional order differentiator for D α 0,tM y(•):

ỹ(α) M (N ) := M j=1 H M j y ̟ M -j , for M = 1, 2, . . . , (22)
where

H M j = T -α s M α+1 N i=0 (2i + 1)P (α) i (1) P i ( M -j M ).
January 19, 2013 DRAFT

B. A time-delayed digital fractional order Legendre differentiator

The digital fractional order Legendre differentiator ỹ(α) M (N ) is obtained by taking ξ = 1 in the fractional order Legendre differentiator given in (18) in the discrete noisy case. However, it was shown in Figure 1(a) and Figure 1(b) in Section VI that if we take ξ = 1 or ξ = 0.01 in (18), then the obtained estimation errors, both in noise-free and in noisy cases, can be much larger than the ones obtained by taking the other values of ξ.

Let us recall that an important contribution of the integer order Jacobi differentiator was to introduce a time-delay in the differentiator, which improves significatively both the truncated term error and the noise error contribution (see [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF], [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF], [START_REF] Liu | Convergence Rate of the Causal Jacobi Derivative Estimator, Curves and Surfaces[END_REF] for more details). Bearing this idea in mind, we consider the following digital fractional order differentiator for D α 0,tM -ϑ y(•) with ϑ ∈ R * + :

ỹ(α) M,ϑ (N ) := M j=1 H M,ϑ j y ̟ M -j , for M = 1, 2, . . . , (23)
where

H M,ϑ j = T -α s M α+1 N i=0 (2i + 1)P (α) i (1 -
ϑ t M ) P i ( M -j M ) with t M = M T s . If we use ỹ(α) M,ϑ (N ) to
estimate the value D α 0,tM y(•), then we introduce a time-delay of value ϑ. As we can see, if we take ϑ = 0 in (23), then we can obtain (22). Hence, (23) gives a general expression of the digital fractional order Legendre differentiator ỹ(α) M,ϑ (N ) with ϑ ∈ R + .

C. A recursive algorithm

Let us remark that the coefficients H M,ϑ j given in (23) depend on the value of M . Then, we need to calculate them for each values of M . Since these coefficients do not depend on the samples of y, we can calculate them in an off-line manner. Hence, according to (23) we need M -1 additions to calculate the digital fractional order differentiator ỹ(α) M,ϑ (N ) (ϑ ∈ R + ) in an on-line application. However, when the value of M increases, the computation time becomes larger and larger. In order to solve this problem, we give a recursive algorithm in the following proposition.

Proposition 2

The digital fractional order Legendre differentiator ỹ(α) M,ϑ (N ) given in ( 23) with ϑ ∈ R + can be calculated by:

ỹ(α) M,ϑ (N ) = N k=0 λ α,ϑ k (N, M ), for M = 1, 2, . . . , (24)
where the coefficients are given as follows:

λ α,ϑ k (N, M ) = 1 T α s l α,ϑ k (N ) ϕ α k (M ), (25)
with

ϕ α k (M ) = 1 M α+k+1 M j=1 j k y ̟ j , (26)
l α,ϑ k (N ) = N i=k (-1) i+k (k!) 2 (i + k)! (i -k)! (2i + 1)P (α) i (1 - ϑ t M ). (27)
Moreover, ϕ α k (M ) can be given by the following recursive formula:

   ϕ α k (1) = y ̟ 1 , for M = 1, ϕ α k (M ) = (M -1) α+k+1 M α+k+1 ϕ α k (M -1) + 1 M α+1 y ̟ M , for M = 2, 3, . . . . (28)
In particular, if ϑ = 0 for any M ∈ N * , then the coefficients can be given in the following recursive formula:

   λ α,0 k (N, 1) = l α,0 k (N ) T α s y ̟ 1 , for M = 1, λ α,0 k (N, M ) = (M -1) α+k+1 M α+k+1 λ α,0 k (N, M -1) + l α,0 k (N ) T α s M α+1 y ̟ M , for M = 2, 3, . . . . (29) Consequently, ỹ(α)
M,ϑ (N ) can be considered as a sum of N + 1 coefficients. When the value of M increases, it is sufficient to recalculate these coefficients using the recursive algorithm. When ϑ = 0, after setting the value of N , we can calculate the coefficients

l α,0 k (N ) T α s , l α,0 k (N ) T α s M α+1
, and (M -1) α+k+1 M α+k+1 , for k = 0, 2, . . . , N , in an off-line work. Then, when the signal passes form t M -1 to t M , by using the old coefficients λ α,0 k (N, M -1), and the new sample y ̟ M in (29), we only need 2(N + 1) multiplications and N + 1 additions to construct all the new coefficients λ α,0 k (N, M ). Consequently, we only need 4N + 3 operations to calculate ỹ(α) M (N ) in an on-line application for each value of M ∈ N * . Thus, comparing to (22) the calculation time is significantly improved. If ϑ = 0, according to (28) we then need 2(N + 1) multiplications and N +1 additions to construct all new coefficients ϕ α k (M ). Then, by using (25) we need N + 1 multiplications to calculate λ α,ϑ k (N, M ). Consequently, we need 5N + 4 operations to calculate the time-delayed fractional order differentiator ỹ(α) M,ϑ (N ). Although the accuracy of the digital fractional order Legendre differentiator can be improved by admitting a time-delay, the price to pay is that the computation time is increased.

D. An adaptive algorithm

According to Table I in Section VI, when the length of the interval, where we estimate the fractional order derivative of a noisy signal, increases, we need to increase the value of N so as to reduce the estimation error. Hence, when the value of M increases in the estimation procedure, we should increase the value of N in the digital fractional order Legendre differentiator.

January 19, 2013 DRAFT Let us substitute N in (24) by N + 1, then we obtain:

ỹ(α) M,ϑ (N + 1) = N +1 k=0 λ α,ϑ k (N + 1, M ). (30)
Hence, we need to calculate the coefficients λ α,ϑ k (N + 1, M ) for k = 0, 1, . . . , N + 1. By using (27), we get: for k = 0, 1, . . . , N + 1,

l α,ϑ k (N + 1) = l α,ϑ k (N ) + d α,ϑ N +1,k , (31)
where

d α,ϑ N +1,k = (-1) N +1+k (k!) 2 (N +1+k)! (N +1-k)! (2N + 3)P (α) N +1 (1 -ϑ tM )
. Then, we deduce from ( 25) and (31) a recursive relation for the coefficients λ α,ϑ k (•, M ):

λ α,ϑ k (N + 1, M ) = λ α,ϑ k (N, M ) + ϕ α k (M ) T α s d α,ϑ N +1,k , (32)
for k = 0, 1, . . . , N . Moreover, by using ( 25) and ( 27) we get:

λ α,ϑ N +1 (N + 1, M ) = 1 T α s l α,ϑ N +1 (N + 1) ϕ α N +1 (M ) = 1 T α s d α,ϑ N +1,N +1 ϕ α N +1 (M ). (33)
Consequently, according to (32) and (33), when the value of N passes from N to N + 1, we only need to calculate the following terms: ϕ α N +1 (M ), and d α,ϑ N +1,k for k = 0, 1, . . . , N + 1, where ϕ α N +1 (M ) can be given by using (26).

V. SIMULATION RESULTS

In the first part of this work [START_REF] Liu | Fractional order differentiation by integration and error analysis in noisy environment: Part 1 continuous case[END_REF], numerical simulations have been given to show the accuracy and the robustness with respect to corrupting noises of the fractional order Jacobi differentiators for off-line applications. In this section, we are going to show the properties of these differentiators for on-line applications.

As done in Example 2 in the first part [START_REF] Liu | Fractional order differentiation by integration and error analysis in noisy environment: Part 1 continuous case[END_REF], we take y ̟ (t i ) = sin(5t i ) + σ̟(t i ), where Example 1. In this example, the noise ̟(t i ) is assumed to be a zero-mean white Gaussian noise with σ = 0.074. The discrete noisy signal is given in Figure 6(a). Then, we use ỹ(α) i (N ) and ỹ(α) i,ϑ (N ) with ϑ = 0.16 to estimate R D 0.5 0,ti y(•) in the noise-free and noisy cases respectively. We can see the obtained estimations in Figure 7 Example 2. In this example, the noise σ̟(t i ) + 1 is assumed to be a biased Poisson noise with σ = 0.052. The discrete noisy signal is given in Figure 6(b). According to Proposition 1, we can use the fractional order differentiator ỹ(α) i,ϑ (N ) (ϑ = 0.16) corresponding to the Caputo fractional derivative to cope with this non-centered noise. We obtain the estimation and the corresponding estimation error with a correction of time-delay in Figure 8. Consequently, we can see that the time-delayed digital fractional order differentiator ỹ(α) i,ϑ (N ) is robust with respect to a Poisson noise.

t i = iT s ∈ I = [0, h] = [0, 4], T s = I M , for i = 0, • • • , M ∈ N * ,

VI. CONCLUSION

In the first part of this work [START_REF] Liu | Fractional order differentiation by integration and error analysis in noisy environment: Part 1 continuous case[END_REF], two kinds of fractional order differentiators have been proposed and studied in continuous case. In this paper, we have studied these differentiators in discrete case. Hence, they can be used both for continuous-time and discrete-time models in noisy environment. Firstly, the noise error contribution due to a large class of stochastic processes has been studied. In particular, it has been shown that the differentiator based on the Caputo fractional order derivative can cope with a class of noncentered noises. Secondly, the analysis on the obtained error bounds has given us a guideline on how to January 19, 2013 DRAFT choose the design parameters on which the fractional order Jacobi differentiators depend. Thirdly, a FIRtype digital fractional order Legendre differentiator has been given. Then, according to the knowledge of the design parameters' influence, an improved digital fractional order Legendre differentiator has been proposed by introducing a time-delay. A recursive algorithm has also been obtained, which is useful for reducing the computation time for on-line applications. Finally, numerical simulations have been performed to evaluate the proposed differentiators, and the results show their efficiency. In these numerical examples, the choice of the values of N was done by experience. The objective was to show the efficiency and stability of our fractional order differentiators with these values of N , and to give a general idea on how to choose the value of N . In order to choose appropriate values of N in the case where the noisy signal is unknown, a criterion should be considered in the adapted algorithm. In a future work, the generalization for the fractional order Jacobi differentiators to multivariate case will be done for applications in image enhancement.

  ANALYSIS IN DISCRETE CASE Let y ̟ = y + ̟ be a noisy observation of y defined on an interval I = [a, b] ⊂ R of length h = ba.

Fig. 1 .

 1 Fig. 1. N = 5, 6, . . . , 15 and ξi = 0.01, 0.1, 0.2, . . . , 1, n = N , µ = κ = 0 and α = 0.5.

Figure 3 (

 3 b)) the variation of log 10 C µ,κ,α,n,N (ξ i ) 2 for ξ i = 0.01, 0.1, 0.2, . . . , 1, µ = -0.9, -0.8, . . . , 1 and κ = -0.9, -0.8, . . . , 1 in the case where α = 0.5, and n = N = 5 (reps. n = N = 15). Hence, C µ,κ,α,n,N (ξ i ) 2 can be increasing with respect to κ. Moreover, C µ,κ,α,n,N (ξ i ) 2 can have a local minimum. Now, we study the parameters' influence on the discrete noise error contribution e ̟,θ µ,κ,h,α,N (ξ i ) by using January 19

Fig. 3 .Fig. 4 .

 34 Fig. 3.Variations of log 10 Cµ,κ,α=0.5,n=N,N (ξi) 2 for ξi = 0.01, 0.1, 0.2, . . . , 1, µ = -0.9, -0.8, . . . , 1 and κ = -0.9, -0.8, . . . , 1.

Fig. 5 .

 5 Fig. 5. Variations of log 10 Var e ̟,θ=1 µ,κ,h=1,α=0.5,N (ξi) 2

(

  reps. N = 15). Hence, Var e ̟,θ µ,κ,h,α,N (ξ i ) 2 has a local minimum at κ = µ = 0 (resp. κ = µ = -0.1) when N = 5 (resp. N = 15).

  (α) i (ξ) = D α 0,ξ P i (•). Then, by taking κ = µ = 0 in January 19, 2013 DRAFT

  and σ ∈ R * + is adjusted in such a way that the signal-tonoise ratio is equal to SN R = 20dB. In the following examples, we fix the value of T s to T s = 4 × 10 -3 .Then, we use the digital fractional order Legendre differentiator ỹ(α) i (N ) (DFOLD) given in (22) and the time-delayed digital fractional order Legendre differentiator ỹ(α) i,ϑ (N ) (DFOLD-TD) given in (24) to estimate the values of D 0.5 0,ti y(•) on I. Moreover, we take different values of N for different interval's lengths. Hence, we apply the adaptive algorithm given in Subsection IV-D to these differentiators. For this purpose, we take the following values of N in different intervals:N = 5 for t i ∈ [50T s , 1[; N = 8 for t i ∈ [1, 2[; N = 11 for t i ∈ [2, 3[; N = 15 for t i ∈[START_REF] Oldham | The Fractional Calculus[END_REF][START_REF] Oustaloup | From fractal robustness to the CRONE control[END_REF]. y ̟ (ti) = sin(5ti) + 0.074̟(ti).

  y ̟ (ti) = sin(5ti) + 0.052̟(ti) + 1.

Fig. 6 .

 6 Fig. 6. Signal y and two different noisy signals y ̟ ((a) E[̟] = 0 and (b) E[̟] = 1).

  (a) and Figure 7(c), where the dot lines (black) present the estimations obtained by ỹ(α) i (N ), and the dot-dash lines (red) present the ones obtained by ỹ(α) i,ϑ (N ). Moreover, since the timedelay for ỹ(α) i,ϑ (N ) is equal to ϑ, by shifting the obtained estimations, we can calculate the estimation errors for ỹ(α) i,ϑ (N ). The obtained estimation errors are shown in Figure 7(b) and Figure 7(d). Consequently, the digital fractional order Legendre differentiators ỹ(α) i (N ) is significantly improved by ỹ(α) i,ϑ (N ) by admitting a time-delay.

  i y(•) , ỹ(α=0.5) i,ϑ=0.16 (N ) -R D 0.5 0,t i -ϑ y(•) .

  -R D 0.5 0,t i y(•) , ỹ(α=0.[START_REF] Pommier | CRONE control of a nonlinear hydraulic actuator[END_REF] i,ϑ=0.16 (N ) -R D 0.5 0,t i -ϑ y(•) .

Fig. 7 .

 7 Fig. 7. Example 1: ̟(ti) is a zero-mean Gaussian noise.

  C D 0.5 0,t i y(•) and Caputo DFOLD-TD: ỹ(α=0.5) i,ϑ=0.16 (N ). .16 (N ) -C D 0.5 0,t i -ϑ y(•) .

Fig. 8 .

 8 Fig. 8. Example 2: ̟(ti) is a biased Poisson noise.

1 .

 1 If κ ≥ 0 and µ ≥ 0, then according to (2) we have:∀ ξ ∈]0, 1], Q µ,κ,α,N (•, ξ) ∈ L 2 ([0, 1]) .January 19, 2013 DRAFT

January 19, 2013 DRAFT

If -1 < κ < 0 and µ ≥ 0, then according to [START_REF] Pommier | CRONE control of a nonlinear hydraulic actuator[END_REF] we have:

Consequently, by using Theorem 1 in [START_REF] Liu | Parameters estimation of a noisy sinusoidal signal with time-varying amplitude[END_REF], [START_REF] Fliess | Analyse non standard du bruit[END_REF] can be obtained in the case where -1 < κ and µ ≥ 0. It can also be similarly obtained in the case where -1 < µ and κ ≥ 0. Then, (12) can be deduced. Finally, by using Lemma 2 given in the first part [START_REF] Liu | Fractional order differentiation by integration and error analysis in noisy environment: Part 1 continuous case[END_REF] and the definition of the Caputo fractional order derivative, (13) can be obtained.

Proof of Proposition 2. By applying a change of indices j → Mj in (23) and using (17), the digital fractional order Legendre differentiator ỹ(α) M,ϑ (N ) can also be given as follows:

where c α,ϑ i = (2i + 1)P

, and τ j = j M . Let us regroup the terms in the following sums:

Then, by denoting

where

Thus, (25) is obtained. Moreover, ϕ α k (•) satisfies the recursive relation given in (28). Finally, if ϑ = 0, then l α,ϑ k (N ) does not depend on M any more. Consequently, (29) can be obtained by using (25) and (28).