8.3.1. Definitions

The modeling of FDSs was presented in Chapter 7, with fractional orders n_i verifying 0 < n_i ≤ 1.

Therefore, we are interested principally in the properties of Riemann–Liouville and Caputo derivatives for 0 < n < 1.

The fractional derivative of a function f(t) was defined as Dⁿ(f(t)) in Chapter 1, using the Laplace transform and for initial conditions equal to 0:

[8.9]

Equation [8.4] can be written as:

[8.10]

or

[8.11]

Equation [8.10] corresponds to:

[8.12]

where is the Caputo derivative, whereas equation [8.11] corresponds to:

[8.13]

where is the Riemann–Liouville derivative.

For each derivative, is the Riemann–Liouville integral of the order 1 − n associated with integer order differentiation.

REMARK 1.– These definitions are easily generalized to any value n:

N − 1 < n < N, where N is an integer.

As we can write:

[8.14]

therefore

[8.15]

and

[8.16]

We can note that these two derivatives are based on the fractional integration of the order where

Thus, as noted previously, the fractional integration is of the order whereas associated integer order differentiation is of the order N.

8.3.2. Theoretical prerequisites

The thereafter analysis of the different fractional derivatives relies on a fundamental assumption: we assume the existence of which is the exact fractional derivative of f(t).

We define three derivatives, in order to compute or calculate

Due to initial conditions at t = 0 which will be specified afterwards, implicit and explicit fractional derivatives provide different values of which is one of the major problems of fractional calculus [ORT 08, ORT 11].

In order to get rid of these initial conditions at t = 0 (or at t = t₀), we define these derivatives since t = −∞, i.e. the associated integrals are defined since t = −∞.

Let us recall that

Therefore, for the implicit derivative:

[8.17]

note that is the input of the fractional integrator delivering since t = −∞ with:

[8.18]

For the Caputo derivative:

[8.19]

and for the Riemann–Liouville derivative:

[8.20]

Because these three expressions of Dⁿ(f) are calculated since t = −∞, the possible transients caused by initial conditions at t = −∞ have completely vanished at instant t. Therefore, these derivatives are equal at instant t to the fractional derivative Dⁿ(f):

[8.21]

Of course, this equality is not verified for

8.3.3. Comments

Fractional differentiation obeys different definitions, unlike the integer order case, where the definition is unique. Moreover, it is important to note that each of these definitions is related to fractional integration, of the order n for the implicit derivative and of the order 1 − n for the Caputo and Riemann–Liouville derivatives.

The role of this fractional integration in the usual approach to fractional differentiation is untold. On the contrary, this fractional integration plays an essential role in the analysis of transients. It is one of the paradoxes of fractional calculus where focus is put on fractional differentiation, whereas it will be necessary to concentrate on fractional integration. Moreover, it is the cause of wrong formulations of fractional derivative initial conditions, as we will prove it in the next section.

8.4.1. Introduction

It has been demonstrated that provide the same value at any instant t.

Practically, these derivatives are calculated since an instant t₀ (t₀ = 0 afterwards), so we will consider Because these derivatives depend on a specific Riemann–Liouville integral, this integration process introduces distributed states z_I(ω,t₀), z_C(ω,t₀) or z_RL(ω,t₀). Equality of the three derivatives at any instant t is necessarily related to these distributed states.

Practically, because the Laplace transform is defined for t ≥ 0, the equality of the derivative Laplace transforms will be expressed at initial instant t₀ = 0.

Moreover, because the initial conditions of the derivatives are formulated in the framework of FDS, we first consider the one-derivative case, which we will generalize to N -derivative FDSs in the next section.

Let

[8.22]

be the one-derivative FDS (or FDE).

We assume that u(t) is applied to the elementary system since t = −∞

8.4.2. Implicit derivative

Let x(t) be the output of the fractional integrator (see Figure 8.1). The input of the integrator provides the implicit derivative and its internal state z(ω,0) = z_I(ω,0) at instant t₀ = 0 . z_I(ω,0) summarizes the past of the system, i.e. the influence at t = 0 of u(t) since t = −∞.

Now, if we start the integration process at t₀ = 0, we obtain The dynamics of the fractional integrator verify the distributed differential system:

[8.23]

where z_I(ω,0) is the distributed initial state.

If the fractional integrator is initialized with z_I(ω,0), we obtain the initialized implicit derivative

Using the Laplace transform, we can write:

[8.24]

Therefore

[8.25]

Thus, we finally obtain:

[8.26]

8.4.3. Caputo derivative

The Caputo derivative, calculated since t = −∞, i.e. is provided by a fractional integrator of the order (1 − n) excited by the integer order derivative according to Figure 8.2.

z_c(ω,t) is the distributed state associated with note that it is different from z_I(ω,t) (or z_RL(ω,t)).

Let z_c(ω,0) be the value at t = 0, summarizing the past behavior of the integrator (t < 0). The dynamics of verify the distributed differential system:

[8.27]

Let be the initialized Caputo derivative calculated since t = 0, where the integrator has been initialized by z_c(ω,0).

The Laplace transform provides:

[8.28]

[8.29]

we obtain:

[8.30]

As we finally obtain:

[8.31]

REMARK 2.– The commonly accepted definition of Caputo’s initial conditions [POD 99] is expressed as:

[8.32]

There is a serious problem with this definition because it does not take into account the distributed state of the fractional integrator at t = 0.

x(0) cannot be interpreted as the initial condition of the Caputo derivative: equation [8.31] clearly exhibits that these initial conditions are x(0) and z_c(ω,0). The same problem exists with the Riemann–Liouville derivative.

8.4.4. Riemann–Liouville derivative

The Riemann–Liouville derivative, calculated since t = −∞, i.e. is provided by the integer order derivative of the output of the fractional integrator according to Figure 8.3.

z_RL(ω,t) is the distributed state associated with and z_RL(ω,0) summarizes the past behavior of the integrator (for t < 0).

The dynamics of verify the distributed differential system:

[8.33]

Let be the initialized Riemann–Liouville derivative calculated since t = 0, where the integrator is initialized by z_RL(ω,0).

The Laplace transform provides:

[8.34]

[8.35]

where

Therefore, we finally obtain:

[8.36]

REMARK 3.– The initial conditions of Riemann–Liouville and Caputo derivatives have motivated several interpretations (see, for example, [POD 02a, TRI 10b]).

8.4.5. Relations between initial conditions

The previous three equations correspond to the initialized fractional derivatives. These initialized derivatives provide the exact value of since the influence of the past (t < 0) is summarized by the initial conditions.

Therefore, by equality of the Laplace transforms of the three derivatives, we can formulate the relations between the different initial conditions.

8.4.5.1. Relation between the implicit and Caputo initial conditions

[8.37]

Therefore, we obtain

[8.38]

8.4.5.2. Relation between the implicit and Riemann–Liouville initial conditions

Equality of Laplace transforms provides:

[8.39]

8.5.1. Introduction

Let us consider the N -derivative FDS:

[8.40]

We use the same procedure as with N = 1. The three derivatives are calculated since t = −∞, and we use the Laplace transform to infer relations between the initial conditions at t = 0.

8.5.2. Implicit derivatives

The implicit derivative corresponds to the input of the fractional integrator operating since t = 0.

Let Z_I(ω,t) be the distributed state variables of the integrators and Z_I(ω,0) their initial conditions at t₀ = 0 .

Then:

[8.41]

Therefore

[8.42]

Thus, we obtain:

[8.43]

were defined in Chapter 7.

8.5.3. Caputo derivatives

is defined as:

[8.44]

Let Z_C(ω,t) be the distributed state variable of the integrators (i = 1 to N) and Z_C(ω,0) be their initial conditions at t₀ = 0 .

Then:

[8.45]

Therefore:

[8.46]

Thus:

[8.47]

8.5.4. Riemann–Liouville derivatives

is defined as:

[8.48]

Let Z_RL(ω,t) be the distributed state variable of the integrators (i = 1 to N) and Z_RL(ω,0) be their initial conditions at t₀ = 0 .

Then:

[8.49]

Therefore:

[8.50]

Thus:

[8.51]

8.5.5. Relations between initial conditions

As in the one-derivative case, we obtain the following.

8.5.5.1. Relation between the implicit and Caputo initial conditions

[8.52]

8.5.5.2. Relation between the implicit and Riemann–Liouville initial conditions

[8.53]

In the commensurate order case (i.e. n₁ = ... = n_i = ... = n_N), these relations are simplified since:

[8.54]

8.6.1. Transients of the elementary FDE

Consider the elementary FDE:

[8.55]

We can express the free response using each of the fractional derivatives and their respective initial conditions. Therefore, we obtain three expressions of x(t): x_I(t), x_C(t), x_RL(t):

l) For the implicit derivative: the initial condition is z_I(ω,0).

Then, we obtain for the free response (u(t) = 0):

[8.56]

Using equation [8.26], we finally obtain:

[8.57]

Note that this result corresponds to equation [7.8] because z_I(ω,0) = z(ω,0).

2) For the Caputo derivative: the initial conditions are x_C(0) and z_C(ω,0). Then, we obtain for the free response:

[8.58]

Using [8.31], we finally obtain:

[8.59]

3) For the Riemann–Liouville derivative: and the initial conditions are Then, we obtain for the free response:

[8.60]

Using [8.36], we finally obtain:

[8.61]

8.6.2. Unicity of transients

The true “physical” free response of the FDE depends only on its initial condition and obviously cannot depend on the choice of the fractional derivative used to analyze the FDE.

However, the three responses x_I(t), x_C(t) and x_RL(t) seem to be different because of specific initial conditions.

In fact, we have determined relations between these initial conditions.

Consider, for example, x_I(t) and x_C(t):

[8.62]

Using relation [8.38] between the implicit and Caputo initial conditions, we obtain:

[8.63]

This means that

[8.64]

where x(t) is the free response expressed in Chapter 7.

Applying the same approach to x_RL(t), it is straightforward to obtain

• x_I(t) = x_RL(t) = x(t)

Therefore, we can conclude that: x_I(t) = x_C(t) = x_RL(t) = x(t). i.e. there is unicity of the free responses calculated with the different fractional derivatives.

Obviously, this result applies directly to an N -derivative FDS, using relations [8.52] and [8.53].

8.6.3. Conclusion

We can use either to express the free response of:

[8.65]

since we have proved the equivalence of X_I(t), X_C(t) or X_RL(t).

Nevertheless, there is a fundamental difficulty to express X_C(t) or X_RL(t).

Note that Z_I(ω,0) = Z(ω,0) is a “natural” initial condition because it is intrinsically linked to fractional integrators i.e. Z_I(ω,0) is directly available and its interpretation is straightforward.

On the contrary, Z_C(ω,0) and Z_RL(ω,0) are not directly available because they require the calculation of the corresponding fractional derivatives with integrators Moreover, their interpretation in terms of dynamics, i.e. of system dynamics, is not straightforward (see section 8.7).

It is necessary to remember that the Caputo derivative has motivated an unjustified interest due to the apparent simplicity of equation [8.32]. It has been interpreted as a strong similitude between the integer order derivative and the Caputo derivative, where x(0) has been interpreted as a physical initial condition of this derivative.

Consequently, the free response of is expressed as:

[8.66]

This expression is obviously wrong, as it is illustrated by the numerical simulation of section 8.7.

Expression [8.26] using Z_I(ω,0) = Z(ω,0) is not as simple as its erroneous homolog [8.32], but it is the price to pay in fractional calculus, where dynamical transients are more complex than in the integer order case.

Finally, let us highlight that Z_I(ω,t) = Z(ω,t) generalizes the concept of the integer order state variable to the fractional order case:

– X(t) is the state vector for integer order systems;

– X(t) is the pseudo-state vector, whereas Z(ω,t) is the distributed state vector for fractional order systems.

8.7.1. Introduction

Unicity of FDS transients was proved in the previous section. Nevertheless, it is interesting to verify this fundamental result by numerical simulations.

Therefore, the purpose is to illustrate that the transients of an elementary one-derivative FDE, expressed either with the Caputo derivative or the Riemann–Liouville derivative, are identical to the transients expressed with the implicit derivative, i.e. with the distributed variable z(ω,t).

8.7.2. Simulation of Caputo derivative initialization

Consider the elementary system [8.22]:

The input of the system is:

[8.67]

The system internal state z_I(ω,t) is equal to 0 at t = 0 .

Then, at t = t₀, this internal state is z_I (ω,t₀).

As u(t) = 0 for t > t₀, we observe the free response of the system, initialized by z_I(ω,t₀): the response x(t) is presented in Figure 8.5.

In order to initialize this system with the Caputo derivative, we must compute

[8.68]

At t = t₀, we obtain x(t₀) and z_C(ω,t₀) (internal state of I¹⁻ⁿ(s)).

Let us recall equation [8.59] with t₀ = 0 :

[8.69]

Let us define:

[8.70]

and

[8.71]

• Then, for t ≥ t₀:

[8.72]

This term is usually considered as the free response of the system, initialized by x(t₀) [LES 11].

Of course, this solution does not work because there is a transient, even though x(t₀) = 0!

Therefore, it is necessary to use the supplementary term:

[8.73]

is the free response of the I¹⁻ⁿ(s) integrator, with the initial condition z_C(ω,0), and x_z,Cap(t) is the response of to this free response, according to Figure 8.4.

Finally

[8.74]

These two signals are plotted in Figure 8.5.

Obviously, x_Cap(t) is different from x(t), although they share the same starting point x(t₀).

Finally, if we add −x_z, _Cap(t) to x_Cap(t), we exactly obtain x(t).

Therefore, it is possible to correctly initialize the FDS with the Caputo derivative, but the procedure is more complex than the direct one, based only on z_I(ω,t₀) = z(ω,t₀).

8.7.3. Simulation of Riemann–Liouville initialization

We consider the same system [8.22] and use the same procedure as described previously.

First, we must compute I¹⁻ⁿ(x(t)) and differentiate this integral to obtain

[8.75]

At t = t₀, we obtain and z_RL(ω,t₀) (state of I¹⁻ⁿ(s)).

Let us recall equation [8.61] with t₀ = 0 :

[8.76]

Let us define:

[8.77]

and

[8.78]

Then

[8.79]

This term is called the Riemann–Liouville free response [LES 11].

Of course, it cannot fit the true free response because x_RL(t) → ∞ when t → t₀.

Therefore, it is necessary to use the supplementary term:

[8.80]

is the free response of the I¹⁻ⁿ(s) integrator, with the initial condition z_RL(ω,0), and x_{z, RL}(t) is the integer derivative of the response of to this free response, according to Figure 8.6.

Note that x_{z, RL}(t) → ∞ when t → t₀ because of the integer derivative action.

These two signals are plotted in Figure 8.7 with x(t).

First, we note an important change in the amplitude scale in comparison with Figure 8.5: because of the integer derivative action, the signals x_RL(t) and x_{z, RL}(t) are larger than x(t), particularly for Nevertheless, when we add −x_z, _RL(t) to x_RL(t), we exactly obtain x(t).

Therefore, it is possible to initialize the FDS with the Riemann–Liouville derivative, with a similar procedure to the Caputo derivative and the conclusion is the same as summarized previously. We can use either the Caputo derivative or the Riemann–Liouville derivative to initialize the FDS, but the two procedures are obviously more complex than the direct one, based only on z_I(ω,t₀) = z(ω,t₀)!