*6.1. Integration of a FDE*

We demonstrated in the previous section that it is necessary to use the frequencydistributed model of the fractional integrator to take into account the transients of the free response. Thus, we have to apply the same approach for the integration of any FDE/FDS initial value problem (6,7). As we have shown, the solution is provided by the fractional integral equation, which is equivalent to the integer order case (2):

$$\mathbf{x}(t) = I\_t^n(f(\mathbf{x}(t), \boldsymbol{\mu}(t))) + \mathbf{x}\_0(t), \tag{42}$$

where *x*0(*t*) is the initialization function of the Riemann–Liouville integral

$$\mathbf{x}\_{0}(t) = \int\_{0}^{\infty} \mu\_{n}(\omega) z(\omega, 0) e^{-\omega t} d\omega. \tag{43}$$

Notice that (42) is a Volterra integral equation.

Fundamentally, Figure 3 displays the graphical representation of the integral Equation (42), which underlines the closed-loop behavior of the FDE/FDS based on the fractional integrator <sup>1</sup> *<sup>s</sup><sup>n</sup>* with the initialization function *x*0(t).

**Figure 3.** Closed-loop model of the FDE/FDS.

However, we can also use the frequency-distributed model of the fractional integrator, where its input is *v*(*t*) = *Dn*(*x*(*t*)) = *f*(*x*(*t*), *u*(*t*)), which leads to the distributed representation of the FDE:

$$\begin{cases} \frac{\partial z(\omega, t)}{\partial t} = -\omega z(\omega, t) + f(\mathbf{x}(t), \boldsymbol{\mu}(t)) \\ \qquad \boldsymbol{x}(t) = \int \mu\_n(\omega) z(\omega, t) d\omega \\ \quad \boldsymbol{\mu}(\omega) = \frac{\sin(n\pi)}{\pi} \omega^{-n} \quad 0 < n < 1 \end{cases} \tag{44}$$

In this case, the solution *z*(*ω*, *t*) is provided by the distributed integer-order integral equation:

$$\begin{aligned} z(\omega, t) &= \int\_0^t [-\omega z(\omega, t) + f(\mathbf{x}(\tau), \boldsymbol{\mu}(\tau))] d\tau + z(\omega, 0) \quad \forall \boldsymbol{\omega} \in [0, \infty) \\ &= I\_t^1(-\omega z(\omega, t) + f(\mathbf{x}(t), \boldsymbol{\mu}(t))) + z(\omega, 0) \end{aligned} \tag{45}$$

where *z*(*ω*, 0) is the initial condition of the integer order integral.

We can represent (see Figure 4) this frequency distributed system graphically, where frequency varies from *ω* = 0 to *ω* = +∞, according to its Laplace transform. This graph corresponds to Figure 3, where the fractional integrator is replaced by its distributed graph of Figure 1.

**Figure 4.** Closed-loop model of the FDE/FDS based on the fractional integrator distributed-frequency model.

Equations (42) and (43) and the graph of Figure 3 focus on the pseudo-state variable *x*(*t*), whereas Equation (44) and the graph of Figure 4 focus on the internal state variables *z*(*ω*, *t*) of the integrator, which are in fact those of the FDE/FDS.

Notice that, for the isolated integrator (Figure 1), with input *v*(*t*) and output *x*(*t*), the state variables are decoupled and evolve independently. On the other hand, in system (44), i.e., in the graph of Figure 4, the state variables are coupled by the relation *v*(*t*) = *f*(*x*(*t*), *u*(*t*)). This means that the evolution of the state variable *z*(*ω*, *t*) (for the particular value) depends on all the other state variables *z*(*ξ*, *t*) *ξ* ∈ [0, ∞). Namely, the

original FDE/FDS (6) has been transformed into an infinite-dimension system of first-order differential Equations (44).

These are the two "faces" of the same problem:


In Figures 3 and 4, there is no hypothesis about the nature of *f*(*x*(*t*), *u*(*t*)) which can be either linear or nonlinear. In the nonlinear case, the integral formulation of the FDE/FDS initial-value problem leads to (42) or (45). The solution of these integral equations can be obtained with Picard's method, which is currently used to solve nonlinear fractional order problems (see for instance [3]).

Of course, the nonlinear case is a very wide topic, and our objective is not to treat it in this paper. In fact, the linear case is also of fundamental interest, and we propose to revisit it, essentially to formulate free responses.
