3.2.1. Euler’s technique

Let us remember that the derivative of a function f(t) is approximated

by:

[3.1]

Let q^–1 be the delay operator and {f_k} be the sequence .

We can write symbolically:

[3.2]

The z -transform [KRA 92] of this equation is:

[3.3]

where

[3.4]

Note that the complex variable z⁻¹ corresponds to the delay operator q⁻¹.

z⁻¹ is defined as z⁻¹ = e^−T_es which provides the relation between the discrete and continuous time domains.

Note that:

[3.5]

This result corresponds to:

[3.6]

which is the well-known Laplace transform of the derivative of a function f(t), where the initial condition is equal to 0 .

Then, consider the simulation of the first-order ODE:

[3.7]

which is performed as:

or

[3.8]

which corresponds to the numerical algorithm:

[3.9]

Obviously, there is a causality problem, because x_k on the left side depends on x_k on the right side.

This is solved by the inclusion of a time delay, i.e.

[3.10]

i.e.

[3.11]

which can be written as:

[3.12]

[3.13]

is Euler’s integer order integrator.

3.2.2. The Grünwald–Letnikov fractional derivative

Note that higher-order integer order time derivatives are approximated by the following equation, for order N :

[3.14]

As a generalization, the fractional order derivative of f(t) (for n real), known as the Grünwald–Letnikov derivative [POD 99, ORT 11], is obtained by replacing N with n in the previous equation.

Then

[3.15]

Let us verify the validity of this definition.

Note that:

[3.16]

Therefore

[3.17]

As in the integer order case, this is the well-known Laplace transform of the fractional derivative of f(t) , when the initial conditions (for more details, see Chapter 8) are equal to 0.

Now, let us express :

[3.18]

which is usually expressed as:

[3.19]

where

[3.20]

However, it is also possible to write:

[3.21]

where

[3.22]

We can see that:

[3.23]

i.e.

[3.24]

Therefore, coefficients α_i,GL are recursively calculated with α_0,GL = 1.

Equation [3.25] defines a moving average (MA) filter [LJU 87], with an infinite number of terms:

[3.25]

3.2.3. Numerical simulation with the Grünwald–Letnikov integrator

Consider the elementary one-derivative FDE:

[3.26]

therefore:

[3.27]

which is equivalent to

[3.28]

hence:

[3.29]

As mentioned previously, it is necessary to include a time delay q⁻¹ for causality:

[3.30]

where

[3.31]

is the fractional order Grünwald–Letnikov integrator.

This fractional integrator is an auto-regressive (AR) filter [LJU 87], with an infinite number of terms.

Note that can also be expressed as:

[3.32]

3.2.4. Some specificities of the Grünwald–Letnikov integrator

For a better understanding of simulation, let us expand equation [3.30]:

[3.33]

Assume that system [3.26] is initially at rest, i.e.

[3.34]

Moreover, assume that the input is a step of amplitude E.

Then

[3.35]

Since the sequence of α_i,GL terms is infinite, the term x_k depends on all the past values, since x₀. Of course, this dependence increases with time t (t = kT_e).

Therefore, it is necessary to memorize all the past values of the response x(t).

Then, if we modify the time origin, i.e. if k = 0, with a different input u(t), since the new origin, the response x(t) depends on the initial conditions which are the previous values .

Consequently, this means that the number of initial conditions is theoretically infinite. Of course, this is not surprising because a fractional system is characterized by infinite memory: the response at instant t depends on all the past values. On the contrary, an integer order system is characterized by a reduced dependence on the past, because the number of its initial conditions is equal to system order N. In other words, this means that the state vector of the sampled system [3.33] has an infinite dimension.

In conclusion, we can say that the simplicity of the simulation technique has to be paid by an important cost due to this infinite number of terms: the number of elementary arithmetic operations (additions, multiplications) and memory length (to memorize past values ) increase considerably with time.

3.2.5. Short memory principle

It is for this reason that Podlubny [POD 99] formulated the short memory principle. As the terms of sequence decrease with i, he proposed to limit the memory to a value I, i.e. to replace by:

[3.36]

The direct consequence is the limitation of the number of elementary arithmetic operations and the influence of cumulative numerical errors.

However, it is necessary to analyze its consequence on simulation error.

Consider again the step input u(t) = EH(t) applied to the elementary system [3.26].

For a system initially at rest, the Laplace transform provides the relation:

[3.37]

Therefore

[3.38]

Regardless of the simulation technique, we have to respect this asymptotic value .

Using the simulation technique, we can write:

[3.39]

i.e.

[3.40]

u(t) is a step input, so:

[3.41]

Note that:

[3.42]

Hence,

[3.43]

Note that

[3.44]

Hence,

[3.45]

i.e.

[3.46]

We can write

[3.47]

Let us define

[3.48]

Hence,

[3.49]

Conclusion: if we use the exact integrator (i.e. I → +∞), we obtain the exact asymptotic value , with no simulation error.

On the contrary, with the short memory principle, we obtain:

[3.50]

Therefore,

[3.51]

We can conclude

[3.52]

On the contrary, using the short memory principle, the truncation error ΔI causes a static simulation error.

3.5.1. Introduction

Numerical simulations are necessary to illustrate the previous theoretical analysis, in order to highlight the close similarities between the two algorithms concerning their simulation accuracy, and also their differences related to their ability to reject static errors and to take into account initial conditions.

3.5.2. Comparison of discrete impulse responses (DIRs)

Each fractional integrator, or , is characterized by its DIR. Hence, we can compare the two algorithms thanks to these DIRs.

The DIR corresponds to the response of each integrator to a discrete Dirac impulse δ_k:

[3.65]

Therefore, for :

[3.66]

and for :

[3.67]

Each DIR has an infinite length; moreover, γ_0,GL = γ_0,GL = 0 in order to respect the causality principle.

According to

we obtain:

[3.68]

and for

For the elementary cell, we can write:

[3.69]

Hence:

[3.70]

[3.71]

The graphs of RID_GL and RID_IS are presented in Figure 3.1 for n = 0.5 Te = 0.001 s.

For infinite state, we consider two cases:

1. 1)
2. 2)

For case 1, we note some difference between RID_GL and RID_IS. On the contrary, for case 2, the graphs of the two RIDs are identical.

These results demonstrate that:

[3.72]

3.5.3. Simulation accuracy

Consider the elementary FDE

[3.73]

We are interested in the unit step response of H(s).

Note that the response x(t) can be expressed thanks to the Mittag-Leffler function [POD 99] (see Appendix A.3.):

[3.74]

x_ML (t) is used as a reference response.

The responses x_GL(t) and x_IS(t) are provided by numerical simulation with and . Each simulation is characterized by its simulation error:

[3.75]

H(s) is defined by a₀ = 1 n = 0.5 .

The simulations are performed with Te = 0.001 s.

For the GL algorithm, memory increases with , i.e. memory becomes infinite as t → ∞.

For the IS algorithm, we consider three cases:

1. 1)
2. 2)
3. 3)

The graphs of x_ML(t), x_GL (t) and x_IS (t) are not presented because they are too close to each other. On the contrary, it is more significant to consider the graphs of ε_GL (t) and ε_IS(t) (see Figure 3.2).

The GL algorithm provides the lower value of the simulation error. The simulation error of the IS algorithm decreases as the frequency interval is broadened (from case 1 to case 3). Case 3 provides the minimal error, similar to the GL algorithm.

This result is equivalent to the previous one concerning DIRs, i.e. behaves asymptotically as .

3.5.4. Static error caused by the short memory principle

Using again the unit step response, we study the influence of the short memory index I_SM on static error. The response x_IS (t) that has no static error (thanks to its integer order integration action) is used as a reference. It is simulated with . Simulations are performed with Te = 0.001 s. Figure 3.3 presents the graphs x_GL (t) for increasing values of I_SM: 500, 1,000, 2,000, 3,000.

We verify that static error is maximum with I_SM = 500. This error decreases as I_SM is increased, according to the theoretical analysis. Of course, it is equal to 0 if I_SM = k max . With this example, .

3.5.5. Caputo’s initialization

The free responses x_GL (t) and x_IS (t) are compared to the theoretical free response expressed with the Mittag-Leffler function [LES 11] (see Appendix A.3).

For the elementary system , we obtain:

[3.76]

where x₀ is the theoretical Caputo’s initial condition.

We use the following parameters:

The IS algorithm is simulated with:

The corresponding initial condition is defined as:

The GL algorithm is characterized by the following two parameters:

• – k_g: memory length;
• – k_i: initialization index.

In the first step, with Te = 0.001 s , we impose k_g = k_max = 5000 (optimal value for the step response). On the contrary, we impose for initialization:

The graphs of x_ML (t), x_GL (t) and x_IS (t) are presented in Figure 3.4.

We note that the graphs of x_IS (t) and x_ML (t) are close to each other, i.e. the IS algorithm performs correct initialization.

According to the previous theoretical analysis, x_GL (t) corresponds to a bad initialization for k_i = 100 and k_i = 1000 . On the contrary, we note that initialization is not improved by the condition k_i = k_g = 5000 , which is a priori surprising.

In order to analyze this apparent anomaly, we increase k_g in the second step, with the condition k_i =k_g .

Moreover, we use T_e = 0.05 s and k_max = 500 . The corresponding graphs are presented in Figure 3.5.

For k_g =k _max = 500, we note the same phenomenon as shown previously. On the contrary, if we increase k_g (k_g = 5000 and k_g = 50000), we can improve initialization, but it remains imperfect.

Let us recall that x_GL (t) is expressed theoretically as:

[3.77]

In fact, the sum is truncated by k_g ; therefore, there is an error equal to:

[3.78]

For a step response initialized by x₀ = 0 , the terms x_GL(k − i) are equal to 0 for i > k_g. However, it is not the same for the free response, because the terms are equal to x₀ ≠ 0. Thus, the only way to reduce this initialization error is to increase k_g, which is verified when k_g varies from 500 to 50000.

We can conclude that Caputo’s initialization is not a simple problem for the GL algorithm, whereas it is straightforward for the IS algorithm.

3.5.6. Conclusion

The technique of the equivalent integrator has proved its interest for the comparison of the Grünwald–Letnikov and infinite state simulation algorithms. The equivalence of their DIRs has demonstrated that these algorithms are equivalent, which has been confirmed by the similarity of the unit step responses with the simulation of a one-derivative FDE. The GL algorithm performs the best accuracy; nevertheless, the IS algorithm exhibits reasonable precision. The main drawback of the GL algorithm is its increasing memory, which is an obvious limitation to fast computation. The short memory principle is an illusory solution because it creates static error. Another crucial problem concerns the formulation of initial conditions. The elementary example of Caputo’s initialization has demonstrated the inability of the GL algorithm to perform good initialization, whereas it is straightforward with the other algorithm.

The infinite state approach, although slightly less precise, performs a good compromise between precision and computation time. Moreover, this algorithm is characterized by its ability to reject static error and to perform any type of initialization.

A.3.1. Definition

Let us recall that the exponential function is defined as:

[3.79]

The Mittag-Leffler function [MIT 03] is the generalization of the exponential function:

[3.80]

There is a more general definition introduced by Agarwal [AGA 53]:

[3.81]

Note that:

[3.82]

A.3.2. Laplace transform

It is demonstrated that (see [POD 99]):

[3.83]

if β = 1 and α = n, then:

[3.84]

There is another relation:

[3.85]

we can write:

[3.86]

A.3.3. Unit step response of

Consider the elementary system and its unit step response (u(t) = H(t)).

Then:

[3.87]

[3.88]

[3.89]

[3.90]

A.3.4. Caputo’s initialization

Consider the elementary system:

[3.91]

and Caputo’s derivative ^CDⁿ(x(t)) with the ideal initial condition x(0), i.e. if the system is at rest since t = −∞ (see Chapter 8). Then:

[3.92]

Therefore,

[3.93]

Thus,

[3.94]