1.2.1. Introduction

There are two distinct problems: the initial values of an ordinary differential equation (known as the initial value problem) [KOR 68] and the response of this ODE to any excitation u(t) that takes into account initial conditions at t = t₀, summarizing the past behavior of the system. As the solution of the first problem is straightforward, only the second problem will be discussed.

1.2.2. Response of a linear system

Consider the linear system characterized by the transfer function H(s) = L{h(t)} (where h(t) is the impulse response) with input u(t) and output y(t). The model H(s) corresponds to an input/output representation, i.e.

[1.1]

A state space model can be associated with this system

[1.2]

where is the system state.

Assume that we consider the behavior of this system since t = –a (the system is assumed to be initially at rest at t = –a).

The objective is to express the system response y(t) for t ≥ 0 without the explicit use of past behavior, corresponding to –a < t < 0.

According to the linear system theory, y(t) must take into account this past behavior using initial conditions at t = 0, thanks to a free response or an initialization function ψ(t), and a forced response caused by the excitation u(t) for t > 0 [KAI 80]; therefore

[1.3]

This problem can be summarized by the graphs presented in Figure 1.1.

Let us call history functions (or past behavior, or pre-history) the graphs of u(t) and y(t) for –a < t < 0.

Usually, the solution y(t) (t > 0) is expressed using a state space representation, with the explicit use of .

However, it is also possible to express y(t) with a convolution integral (i.e. with an input/output approach), without the explicit use of the initial state . This second solution is used by Lorenzo and Hartley in the history function approach, whereas the infinite state approach refers to a state space representation, i.e. to the first solution.

1.2.3. Input/output solution

Let us express the convolution y(t) = h(t)*u(t) since t = –a :

[1.4]

y(t) can be expressed as the sum of two integrals, from τ = –a to 0 and from τ = 0 to t.

Therefore

[1.5]

Then, let us define

[1.6]

and

[1.7]

1.2.4. State space solution

Let be the system state at t = 0, summarizing the past behavior from τ = –a to τ = 0.

Then, on the one hand

[1.8]

and on the other hand

[1.9]

REMARK 1.– The input/output solution, expressed with the impulse response, refers only to the observable and controllable part of the system state [KAI 80, ZAD 08]. On the contrary, the state space solution takes into account all state components; therefore, it represents the more complete solution.

REMARK 2.– As for t ≤ –a, we can write

[1.10]

Therefore

[1.11]

Using [1.10], we obtain

[1.12]

i.e.

[1.13]

Thus, we can verify the fundamental equality

[1.14]

1.2.5. First-order system example

Consider the first-order system

[1.15]

and its impulse response

[1.16]

Let us apply a step input U at instant t = –a.

Therefore

[1.17]

Then (see Figure 1.2)

[1.18]

Obviously

[1.19]

Then, the free response is expressed as

[1.20]

We can also write

[1.21]

i.e.

[1.22]

Thus, we verify again that the input/output and the state space representations are equivalent:

[1.23]

1.3.1. Introduction

As mentioned in the introduction, the initialization of an FDE cannot be performed as in the integer order case, using the initial value of the pseudo-state variable . It has been proved that the knowledge of is unable to give information on the future behavior of the FDE (see Chapters 7 and 8 of Volume 1). In fact, the future behavior depends on the initial value of the internal distributed state variables z_i(ω, 0), with

[1.24]

Let us remember that an infinity of different dynamics (each one depending on a particular frequency distribution z_i(ω, 0)) can correspond to the same pseudo-initial value x_i(0) [TAR 16a, TAR 16c]. As the general problem was treated previously (see Chapter 7 of Volume 1), we only present thereafter some simulation examples for a one-derivative FDE, in order to illustrate the dependence of dynamics on the initial distributed state z(ω, 0).

1.3.2. Free response of a simple FDE

Consider the elementary FDE

[1.25]

The following distributed differential system is associated with this FDE:

[1.26]

The distributed initial condition is z(ω, 0), which verifies the initial value

[1.27]

In order to emphasize the dependence of x(t) (t > 0) on z(ω, 0), we consider a numerical simulation of [1.25, 1.26] with three different distributions z(ω, 0) corresponding to the same value x(0) [1.27].

Therefore, system [1.25] is frequency discretized (see Chapter 2 of Volume 1):

[1.28]

The simulation parameters are

[1.29]

Each initial condition z(ω, 0) corresponds to the discretized distribution

[1.30]

with the constraint

[1.31]

Three distributions are considered: they are designed to correspond to the same initial value x(0):

1) z_j(0) = 0 for j = 1 to J, with

[1.32]

2) with

[1.33]

3) with

[1.34]

The distribution [1.32] corresponds to the pseudo-initial condition of Caputo, requiring x(t) = cte = x(0) for t = –∞ to 0 (see Chapter 3 of Volume 1).

Therefore, x(0) depends only on z₀ (0) (z_j(0) = 0). It corresponds to the slower decrease of x(t), as observed in Figure 1.3.

Distributions [1.33] and [1.34] have components at high frequency. Their amplitude decrease with frequency; however, the high-frequency components of distribution [1.33] are relatively more important than those of distribution [1.34] (see Figure 1.4). Consequently, the decrease of x(t) with distribution [1.33] is faster than with distribution [1.34] (see Figure 1.3).

Although the considered distributions are essentially theoretical, the main conclusion is that the corresponding transients characterized by the same initial value x(0) are completely different. Obviously, the knowledge of x(0) is of no help to explain the different dynamics of x(t), because these dynamics depend essentially on z(ω, 0).

1.4.1. Introduction

The initialization of an FDS is closely related to the initialization of its integer order counterpart (see section 1.2). There are two approaches: one deals with the input/output formulation, and the other deals with the state space representation. However, the analysis of this problem is more complex, mainly caused by the duality between the pseudo-state variables and the distributed internal state variables. Moreover, in contrast with the integer order case, analytical solutions can be formulated only in a limited number of particular cases.

The principles of the two approaches are presented later in a theoretical framework. Some examples are used to illustrate these two methodologies in the next section.

1.4.2. State space representation

The presentation is limited to the case of an N-derivative commensurate order FDS.

Consider

[1.35]

A distributed integer order differential system is associated with [1.35]:

[1.36]

Let Ẕ(ω, 0) be the distributed initial state at t = 0.

As demonstrated in Chapter 9 of Volume 1, the general response of [1.35, 1.36] to the excitation u(t) with the initial state Ẕ(ω, 0) can be expressed using

the Mittag-Leffler technique as

[1.37]

where

[1.38]

represents the free responses of the N integrators associated with [1.35, 1.36]. ũ(t) is an equivalent input [MON 10] expressed as

[1.39]

E_n,1 (Atⁿ) is the matrix Mittag-Leffler function.

The first term of [1.37] represents the free response or the initialization function of the pseudo-state .

Thus, the initialization function of y(t) is expressed as

[1.40]

Obviously, it will be very difficult to analytically express ψ(t) in the general case.

1.4.3. Input/output formulation

Let H_n(s) be the transfer function of [1.35]:

[1.41]

Let h_n(t) be its impulse response:

[1.42]

Therefore, we can write

[1.43]

As noted previously, the convolution integral is calculated since t = –a, and the system is assumed to be at rest for t < –a.

Therefore

[1.44]

which corresponds to

[1.45]

Thus, the general expression of the initialization function is

[1.46]

and the forced response is expressed as

[1.47]

The history function approach of Lorenzo and Hartley corresponds to the application of this input/output methodology [LOR 01, HAR 11].

The authors’ main objective was to express ψ(t) using the Mittag-Leffler function and the incomplete Gamma function [HAR 07].

Moreover, they were forced to use simplifying assumptions, such as constant and ramp history functions, in order to formulate analytical expressions of ψ(t) [HAR 09b, HAR 09c].

1.5.1. Introduction

A comparative analysis of the two approaches was published in [HAR 13]. Two simple examples were used: initialization of the fractional integrator and the fractional differentiator.

They are presented later. A third example that deals with an elementary FDS is also presented to illustrate the complexity of analytical fractional order initialization.

1.5.2. Initialization of the fractional integrator

Consider the elementary integrator

[1.48]

excited by a Dirac impulse at instant t = –a.

Therefore

[1.49]

and

[1.50]

1.5.2.1. State space approach

Let z(ω,t) be the distributed variable of defined by:

[1.51]

[1.52]

Thus, at instant t = 0, we obtain

[1.53]

Hence, for t > 0

[1.54]

and

[1.55]

As (Appendix A.1. Chapter 1 Volume 1)

[1.56]

we obtain

[1.57]

1.5.2.2. Input/output approach

Recall that

[1.58]

Then

[1.59]

Let

[1.60]

Therefore, we obtain [HAR 13]

[1.61]

where Γ(n, as) is the incomplete gamma function [HAR 07].

As

[1.62]

We finally obtain

[1.63]

REMARK 3.– We can directly obtain ψ(t) without using its Laplace transform ψ(s).

Since

[1.64]

Using [1.59], we directly obtain

1.5.3. Initialization of the Riemann-Liouville derivative

The Riemann-Liouville derivative is defined by the relation

[1.65]

As stated previously, we apply a Dirac impulse at t = –a, i.e. u(t) = δ(t + a).

1.5.3.1. State space approach

Let z_RL(ω, t) be the distributed variable associated with :

[1.66]

Therefore

[1.67]

Then, for t > 0

[1.68]

Therefore

[1.69]

As

[1.70]

we obtain

[1.71]

thus

[1.72]

1.5.3.2. Input/output approach

Let us define α = 1 – n and q = n; Lorenzo and Hartley demonstrate [HAR 13] that

[1.73]

Thus

[1.74]

REMARK 4.– Again, we can directly express ψ(t) without the Laplace transform ψ(s).

Using the definition of x(t)

[1.75]

Then

[1.76]

1.5.4. Initialization of an elementary FDS

1.5.4.1. Introduction

Consider the initialization of the elementary FDS at t = 0:

[1.77]

excited by a step input U at t = –a, so u(t) = UH(t + a); this system is assumed to be at rest at t = –a.

This example is used to illustrate the complexity of the initialization problem for a system less basic than the integrator or the differentiator.

Two fractional system representations will be used: the closed-loop formulation and the open-loop formulation (see Chapter 7 of Volume 1).

1.5.4.2. Open-loop model formulation

Consider again

[1.78]

Using the inverse Laplace transform, we can express its impulse response h_n(t) as

[1.79]

with (see Chapter 7 of Volume 1)

[1.80]

The open-loop model is characterized by the distributed variable ξ(ω, t):

[1.81]

where

[1.82]

Our objective is to initialize this system at t = 0 using ξ(ω, t) and ξ(ω, 0).

1) State space approach

Then

[1.83]

and

[1.84]

Therefore

[1.85]

and

[1.86]

2) Input/output approach

x(t) is defined as

[1.87]

Therefore

[1.88]

Obviously, equation [1.88] exactly corresponds to [1.86].

1.5.4.3. Closed-loop model, Mittag-Leffler formulation

The system [1.78] corresponds to the elementary FDS:

[1.89]

and to the closed-loop distributed model (recall that ξ(ω, t) ≠ z(ω, t)):

[1.90]

1) Input/output approach

Let us recall that if u(t) = UH(t) or , we can express

[1.91]

As (see Chapter 3 of Volume 1)

[1.92]

we can write

[1.93]

As

[1.94]

we obtain

[1.95]

If we use a step input applied at t = –a, it is necessary to translate the time scale; thus

[1.96]

This result corresponds to

[1.97]

and

[1.98]

Note that if we use the fictive excitation ũ(t)

[1.99]

we can write

[1.100]

therefore

[1.101]

Using [1.99], we obtain the expression of ψ(t)

[1.102]

i.e. for t > 0

[1.103]

We can conclude that the history function approach (input/output approach) provides an analytic formulation of the initialization function ψ(t).

2) State space approach

With u(t) = UH(t) and no initial condition, the Laplace transform of [1.90] gives

[1.104]

As

[1.105]

we obtain

[1.106]

Therefore, as [1.94]

we obtain

[1.107]

For u(t) = UH(t + a) (i.e. a step input at t = –a), we can write

[1.108]

If we want to initialize the system at t = 0, we obtain

[1.109]

Thus, the free response of [1.89] for t ≥ 0, using z(ω, 0), is expressed as

[1.110]

where x₀(t) is the free response of :

[1.111]

Therefore

[1.112]

An expression of ψ(t) can be formulated as

[1.113]

where

[1.114]

Obviously, it will be difficult to obtain an analytical formulation of ψ(t) in order to compare it to [1.103].

However, we can verify the identity of equations [1.103] and [1.113] that perform a numerical simulation.

Therefore, we use the frequency discretized model of [1.90], i.e.

[1.115]

The following simulation parameters are used:

J = 20; ω_b = 10^–3 rd/s; ω_h = 10⁺³ rd/s; T_e = 10^–3 s; a = 2.5s; U = 1; α = 1

The simulation of [1.115] provides z(ω, 0) at t = 0.

Then, the initialization function ψ(t) is provided by the distributed model [1.90] initialized by z(ω, 0) with u(t) = 0:

[1.116]

Figure 1.5 presents x(t) provided by equation [1.115], ψ(t) provided by [1.116] and ψ(t) provided by [1.103].

Thus, we numerically verify that the two expressions of ψ(t) are identical.

1.5.4.4. Closed-loop model and distributed exponential formulation

In Chapter 9 of Volume 1, we expressed FDS transients using two approaches: the Mittag-Leffler technique and the distributed exponential formulation. Thus, we intend to express ψ(t) with this new formulation, in order to highlight its potentialities.

Consider again the elementary FDS [1.89]:

corresponding to the distributed differential system [1.90].

The step response of the distributed state z(ω, t) with z(ω, –a) = 0 ∀ ω and u(t) = UH(t + a) is expressed as

[1.117]

Therefore, we can define

[1.118]

where

[1.119]

is the distributed exponential with

[1.120]

(see the definitions of Chapter 9 of Volume 1).

Finally

[1.121]

1) State space formulation

Let us define

[1.122]

We can express

[1.123]

Therefore

[1.124]

Note that

[1.125]

Thus

[1.126]

and

[1.127]

REMARK 5.– If n = 1, then μ₁(ω) = δ(ω); therefore, f(ω, ξ) = –αδ(ω).

Then, and exp{t g(0, α)} = e^–αt.

Therefore,

and

[1.128]

This result corresponds of course to [1.22].

2) Input/output formulation

As mentioned previously, using the fictive input ũ(t) [1.99]

we can write

[1.129]

where

[1.130]

and

[1.131]

since z(ω, 0) = 1 ∀ ω when system [1.89] is excited by a Dirac impulse δ(t).

Then

[1.132]

[1.133]

therefore

[1.134]

Since

[1.135]

we obtain

[1.136]

i.e. the same result [1.127] as obtained previously.

REMARK 6.– Using the Mittag-Leffler formulation:

[1.137]

therefore

[1.138]

If n =1, the right side of [1.138] is equal to

[1.139]

[1.140]

REMARK 7.– Using the distributed exponential formulation, we obtain results that are equivalent to those of the integer order case (section 1.2.5).

This means that this formulation makes it possible to generalize the integer order case: this is obviously interesting to derive mathematical proofs. However, we have to keep in mind that the apparent simplicity of the distributed exponential formulation hides, in fact, major computational complexity (see Chapter 9 of Volume 1).

1.5.5. Conclusion

The previous comparative analysis has made it possible to formally verify the equivalence of the history function and infinite state approaches, each of which provides a solution to the initialization of linear fractional differential systems. However, these theoretical approaches cannot provide an analytical solution for systems more complex than the one-derivative FDS.

Moreover, the theoretical assumptions of this comparative analysis are too restrictive, particularly the assumption that the system is at rest at instant t = –a is not realistic.

In fact, this analysis has proved the influence of past behavior on initialization (or long memory phenomenon). Theoretically, we have to take into account all the dynamical history of the system, since t = –∞.

Practically, it will be necessary to truncate past dynamical behavior (or pre-history). Therefore, the system cannot be considered at rest at instant t = –a.

Thus, the practical initialization based on an arbitrary truncated dynamical past behavior is studied in Chapter 3.