7.3.1. Modeling of an N-derivative FDS

As any linear FDE can be written as a FDSs, we present the general case of an N -derivative FDS.

Let

[7.9]

Note that we directly consider the non-commensurate order case.

More specifically, the order nⁱ can be equal to 1, as in the induction motor example presented in Chapter 5.

Note also that it is possible to consider , with (so 0 < n_i < 1). Thus, it is another reason to introduce integer orders in the model of [7.9].

Obviously, this general model also corresponds to the commensurate order case with .

The fractional integrator associated with the pseudo-state x_i (t) corresponds to the equation:

[7.10]

For n_i = 1, these equations are replaced by:

[7.11]

Using the Laplace transform, we can write:

[7.12]

Let us define

[7.13]

and the matrix

[7.14]

Then

[7.15]

In the Laplace domain, we obtain:

[7.16]

Note that

[7.17]

and

[7.18]

Thus, we can write

[7.19]

Left multiplication of [7.19] by and simple calculations yield

[7.20]

As noted previously, the first term corresponds to the free response initialized by Z(ω, 0), whereas the second term represents the forced response.

Obviously, equation [7.20] is more simple in the commensurate order case (i.e. ). Then

[7.21]

Therefore

[7.22]

The analytical expression of X(t) will be formulated in Chapter 9.

Previous modeling can be qualified as closed-loop modeling, thanks to the closed-loop nature shown in Figure 7.1. However, we can also call it the internal modeling of the FDS, since the state variables z_i (ω, t) are those of associated integrators “inside” the system. The external (or diffusive) representation of FDS will be presented in section 7.7.

7.3.2. Distributed state

Equations [7.8] and [7.22] can be used to express the distributed state.

For example, consider again the elementary system [7.1] and its step response x(t) caused by u(t) = U H(t) with initial conditions equal to 0, i.e. .

[7.23]

[7.24]

[7.25]

Therefore, we can calculate the components of z(ω, t) at the steady state, i.e. z(ω, ∞):

[7.26]

Thus:

[7.27]

and

[7.28]

Therefore, z(0, ∞) is not defined.

Nevertheless, x(∞) is perfectly defined because

[7.29]

Hence, z(0, ∞) is indirectly characterized by the integral equation:

[7.30]

REMARK 1.– With the frequency discretized model (Chapter 2):

[7.31]

and:

[7.32]

We can also express the components of z(ω, t) when the system is at rest.

Consider the free response caused by z(ω, 0) ≠ 0,

then

[7.33]

which represents the contribution of all the elementary z(ω, 0) frequency components.

Therefore,

[7.34]

Of course, the system is at rest for t → ∞ and:

[7.35]

so we obtain

[7.36]

7.4.1. Numerical simulation

Consider again the elementary system [7.1]:

with a₀ = 1 and n = 0.5.

Simulation is performed using equations [7.3].

Practically, a numerical algorithm is used to perform the simulation, where the distributed state variable z(ω, t) is frequency discretized into components z_j (t):

[7.37]

Finally, using the Z-transform technique, the complete numerical algorithm is:

[7.38]

where α_j, β_j and c_j have been defined in Chapter 2 (equations [2.37]–[2.40]).

We consider the following input:

[7.39]

The reference output, using the Mittag-Leffler function (Appendix A.3), is:

[7.40]

where E_{n, 1}(−atⁿ) is the Mittag-Leffler function.

The system is simulated with the following parameters:

[7.41]

The reference and simulated outputs are shown in Figure 7.2: as expected and in accordance with Chapter 3, the two curves fit exactly.

7.4.2. Initialization at

In Figure 7.2, we note that x(t) is equal to 0 at the instant t₁ = 5.25 s.

Therefore, we propose to initialize the system at t = t₁ by imposing u(t) = 0 for t > t₁.

Thus, we modify the computation of x_mit (t) for t > t₁ , according to:

[7.42]

which corresponds to the equivalent input:

[7.43]

As u(t) = 0 for t > t₁, the reference x_mit(t) represents the free response of the system for t > t₁ , starting from x(t₁) = 0 . Hence, we create a reference free response as presented in Figure 7.3.

Moreover, at t = t₁, we measure the state z(ω,t₁) of the fractional integrator (see Figure 7.5).

Therefore, it is possible to initialize the system, with z(ω, t₁) and u(t) = 0 for t > t₁.

The initialized response is plotted in Figure 7.3: we can note that the two responses fit perfectly.

The conclusions are as follows:

• – the initial state z(ω, t₁) perfectly summarizes the past behavior for t < t₁, so the initialized response fits exactly the reference free response for t > t₁;
• – the initial state z(ω, t₁) corresponds to x(t₁) = 0: therefore, we verify that the only knowledge of the pseudo-state variable does not allow the prediction of the fractional system’s free response; it is necessary to refer to its internal state z(ω, t₁).

7.4.3. Initialization at different instants

Finally, we consider the following modified input:

[7.44]

The corresponding response x(t) is plotted in Figure 7.4.

[7.45]

Figure 7.5 shows the frequency distribution of z(ω, t_i ) for these three instants (the index of ω is equal to the index of the discretized frequency ω_j , with j varying from 0 to J).

Although the output x(t) has the same value, the three distributions are completely different.

Therefore, the system initialization at each of these instants is represented in Figure 7.4: since the distributed state variables are different, the free responses are of course completely different.

We can conclude that the output x(t) of a fractional integrator is unable to characterize its true state; on the contrary, knowledge of its internal state z(ω,t_i ) is required to predict the future behavior.

REMARK 2.– Obviously, these three initializations starting at x(t_i ) = 0 are counter examples to the usual belief that the behavior of a fractional system can be characterized by x(t_i ) , according to Caputo’s derivative definition (see Chapter 8).

7.6.1. Introduction

To the best of our knowledge, a non-commensurate order FDS can be modeled only by integrators, so it is necessarily a closed-loop or internal representation. On the contrary, for a commensurate order FDE or FDS, we can calculate the roots of the characteristic equation and the possible poles of the system. Therefore, using the inverse Laplace transform, we obtain a stable multimode and a possible integer order mode, corresponding to each root. Then, the impulse response is the sum of all these modes, and we obtain an external or open-loop representation of the fractional system, also called the diffusive representation of the system [MON 98, HEL 98].

7.6.2. External model of an elementary FDE

Consider again the one-derivative FDE

[7.48]

or equivalently when the system is at rest at t = 0.

[7.49]

Let us define

[7.50]

and

[7.51]

The equation D(s) = sⁿ + a = 0 has a real root λ which verifies

[7.52]

Note that λ is the root of D(s), but it is not the pole of H_n,a(s), as in the integer order case.

Indeed, let us define

[7.53]

and

[7.54]

Then

[7.55]

The function sⁿ − λ = 0 is multiform [LEP 80, POD 99]; it requires a cut in the complex plane because we have to respect the constraint:

[7.56]

Recall that λ = −a is real:

• – if a > 0 , then λ < 0 so θ = π + 2kπ and with 0 < n < 1

so and the constraint [7.56] cannot be respected: therefore, there is no pole for a stable FDE;

• – on the contrary, if a < 0 , then λ > 0 so θ = π + 2kπ and

which implies : therefore, there is a positive unstable pole α.

Then, let us calculate the impulse response using the inverse Laplace transform (see Appendix A.7). We obtain the general expression:

[7.57]

The first term corresponds to the response of the integer order unstable mode e^αt, whereas the second term is a stable aperiodic multimode.

Because there is no pole if a > 0, the impulse response h_n,a(t) is only composed of the stable multimode:

[7.58]

where

[7.59]

REMARK 3.– If a = 0, then , i.e. we recover the fractional integrator.

It is easy to verify that:

Expressions [7.57] and [7.58] are the external impulse responses [ORT 15] of the FDE; they are also known as the diffusive representation of the FDE [MON 98, HEL 98].

Now, consider the response x(t) to any input u(t):

[7.60]

Let ξ(ω, t) be the distributed variable associated with h_n , _a (t). Then, as with the fractional integrator, we can write:

[7.61]

Let ξ(ω, 0) be the initial condition at t = 0 of system [7.61]. Then

[7.62]

The first term represents the free response, whereas the second term corresponds to the forced response.

Model [7.61] is called external or open-loop because it is not referred to an internal integrator.

Obviously, the distributed variables z(ω, t) (internal state variable) and ξ(ω, t) (external state variable) are completely different.

7.6.3. External representation of a two-derivative FDE

Consider the system

[7.63]

Let λ₁ and λ₂ be the roots of .

Let , we can consider the following cases:

• – if a₀ < 0 , then and the roots λ₁ and λ₂ are real with opposite signs;
• – if a₀ < 0, then Δ > 0 if a₁ < 2a₀, so λ₁ and λ₂ are real with the same signs;
• – if a₀ < 0, then Δ < 0 if −2a₀ < a₁ < 2a₀.

Then the roots λ₁ and λ₂ are complex conjugate numbers:

[7.64]

We can also write

[7.65]

With λ₁ and λ₂, we can write

[7.66]

Now consider the possible poles of .

If λ₁ and λ₂ are real, it is the same case as noted previously (section 7.2.2).

However, if λ₁ and λ₂ are complex, there are two conjugate poles:

[7.67]

These poles can be expressed as s₁ = α+ jβ and s₂ = α− jβ with α = cos(ψ) and β = sin(ψ).

They have a negative real part (i.e. they are stable poles) if , i.e. if or . Therefore, we can express the impulse response using the inverse Laplace transform (see Appendix A.7):

[7.68]

This important case will be used in Chapter 8 Volume 2 to analyze system stability.

7.6.4. External representation of an N-derivative FDE

Consider the N -derivative commensurate order FDE (or FDS):

[7.69]

In this case, the characteristic equation D(s) has N roots λ_i that can be real or complex and

[7.70]

These roots λ_i can correspond to possible poles s_i.

The impulse response h_{n, H} (t) is the sum of elementary impulse responses as defined in sections 7.6.2 and 7.6.3.

Therefore:

[7.71]

The sum of multimodes gives an equivalent aperiodic multimode, with an equivalent weighting function μ_{n, H} (ω).

Therefore, we can write:

[7.72]

Let ξ(ω, t) be the distributed state variable associated with the equivalent multimode. Then, for any excitation u(t):

[7.73]

Moreover, a response is associated with each possible integer order pole s_i.

Therefore, the global forced response x(t) is expressed as:

[7.74]

7.8.1. Introduction

We propose a technique for the computation of the Mittag-Leffler function:

Let us recall the definition of this function (see Appendix A.3):

[7.75]

Truncation of this sum to a large value K >> 1 enables direct computation of this function, as with the exponential function:

[7.76]

However, whereas this computation is independent of t for the exponential function, we note a divergence of the sum for the Mittag-Leffler function occurring for large values of t.

Thus, it is a challenge to propose a computation technique independent of t values. Several techniques have been proposed, for example approaches based on an integral formulation [GOR 02, ORT 17].

We propose to use the equation

[7.77]

where x(t) is the unit step response of the elementary system:

[7.78]

This technique was already proposed by Heleschewitz [HEL 00]; with the same principle, we propose an improved formulation.

7.8.2. Divergence of direct computation

Direct computation is compared to a technique based on the erf function [KOR 68] for n = 0.5 [HEL 00]:

[7.79]

Figure 7.9 presents the graphs of the two functions for a = 1.

We verify whether t is limited to a value t_max depending on a (see Table 7.1).

Note that the computation with the erf function is also divergent for t > t_max.

7.8.3. Step response approach

We first use the computation of x(t) based on the closed-loop representation for the simulation of . This technique is very easy to implement; however, there is an obvious drawback: in order to obtain E_n,₁(−atⁿ) for the desired value t, it is necessary to recursively compute all the previous values .

The graphs of E_n,1(−atⁿ) and are plotted in Figure 7.10 with the simulation parameters:

We note that the proposed technique enables the computation of the Mittag-Leffler function for t > t_max.

7.8.4. Improved step response approach

We use the same step response technique, but with another formulation of H(s). With the open-loop (or diffusive) representation, we can express H(s) as:

[7.80]

Let u(t) = H(t), then:

[7.81]

As , we obtain:

[7.82]

Let us recall that , thus:

[7.83]

Therefore,

[7.84]

Thus,

[7.85]

with

[7.86]

It is essential to note that this approach directly provides the value of the Mittag-Leffler function for the desired value t , without the recursive computation as noted previously.

This technique is based on the computation of the integral . The function μ_n,a(ω) is precomputed as it does not depend on t . The integral is computed from ω_min to ω_max , with frequencies varying according to a geometric recursion ω_j = rω_j−1.

Figure 7.11 presents the weighting function μ_n,a(ω) for n = 0.5, a = 1.

We note that ω_min must be chosen close to whereas ω_max must be very large with r = 1.04.

Figure 7.12 presents the computation error:

[7.87]

This error is maximum for t → 0, and then it decreases ε(t)→0 . The sudden increase at t = 25 s corresponds to the divergence of E_n,1(−atⁿ).

Figure 7.13 presents the computation of on the interval [0;100s] for three values of a (0.1 1 10) and for n = 0.5 .

All of these graphs highlight that the improved step response technique allows the computation of the Mittag-Leffler function with good accuracy for a > 0, 0 < n < 1 and .