3.2.3. The Grünwald–Letnikov Derivative

Instead of the two previous fractional derivatives, it is possible to use the Grünwald-Letnikov (G.L.) derivative, with appropriate initial conditions. In fact, it is preferable to consider the G.L. integrator that corresponds to the discretization of the Riemann–Liouville integral.

The Nth integer order Euler derivative of *x*(*t*) is defined as

$$\left(D^N(\mathbf{x}(t))\right)\_{t=kT\_\varepsilon} = \lim\_{T\_\varepsilon \to 0} \frac{\left(1 - q^{-1}\right)^N}{T\_\varepsilon^N} \mathbf{x}\_{k\prime} \tag{18}$$

where *Te* is the sample time, *xk* = *x*(*kTe*), and *q*−<sup>1</sup> is the delay operator.

The generalization to the fractional order case provides the Grünwald–Letnikov derivative

$$\left( \, ^{GL}D^n(\mathbf{x}(t)) \right)\_{t=kT\_\varepsilon} = \lim\_{T\_\varepsilon \to 0} \frac{\left(1 - q^{-1}\right)^n}{T\_\varepsilon^n} \mathbf{x}\_k \; \; 0 < n < 1. \tag{19}$$

Since *L q*−<sup>1</sup> = *e*−*Tes* , we obtain

$$L\left\{{}^{GL}D^{n}(\mathbf{x}(t))\right\} = \lim\_{T\_{\varepsilon}\to 0} \frac{\left(1 - e^{-T\_{\varepsilon}s}\right)^{n}}{T\_{\varepsilon}^{n}} L\{\mathbf{x}(t)\} = s^{n}X(s). \tag{20}$$

Notice that

$$\frac{\left(1-q^{-1}\right)^{n}}{T\_{\varepsilon}^{\;n}} = \frac{1}{T\_{\varepsilon}^{\;n}} \left[1 + \sum\_{i=0}^{\infty} \alpha\_{i,GL} q^{-i}\right]\_{\prime} \tag{21}$$

with *αi*,*GL* = (−1) *i n* 1 *n*−1 <sup>2</sup> *<sup>n</sup>*−<sup>2</sup> <sup>3</sup> ...... *<sup>n</sup>*−(*i*+1) *<sup>i</sup>* ,

$$\text{soo}\left(^{GL}D^{\eta}(\mathbf{x}(t))\right)\_{t=kT\_{\varepsilon}} = \frac{1 + \sum\_{i=0}^{\infty} a\_{i,GL}q^{-i}}{T\_{\varepsilon}^{\eta^n}} \mathbf{x}\_{k\prime} \tag{22}$$

which is the Moving Average formulation of the Grünwald–Letnikov derivative.

Reciprocally, we can define the Grünwald–Letnikov integral operator [34] as

$${}^{GL}I^{n}(f\_{k}) = \frac{T\_{\varepsilon}{}^{n}q^{-1}}{1 + \sum\_{i=0}^{\infty} \mathfrak{a}\_{i,GL}q^{-i}} f\_{k\prime} \tag{2.3}$$

which is the Auto-Regressive formulation of the Grünwald–Letnikov integrator. Notice that

$$L\left\{{}^{GL}I^{\mathfrak{n}}(f(t))\right\} = \lim\_{T\_t \to 0} \frac{T\_t{}^{\mathfrak{n}}e^{-T\_t \mathfrak{s}}}{\left(1 - e^{-T\_t \mathfrak{s}}\right)^{\mathfrak{n}}} L\{f(t)\} = \frac{1}{\mathfrak{s}^{\mathfrak{n}}} F(\mathfrak{s}),\tag{24}$$

which means that the Grünwald–Letnikov integrator is the time discretization of the Riemann–Liouville integral.

Consider now the elementary FDE initial value problem (6):

$$D\_t^n(\mathfrak{x}(t)) = f(\mathfrak{x}(t), \mathfrak{u}(t)) \quad 0 < n < 1.$$

Using the Grünwald–Letnikov integrator, we can express {*x*(*k*)} as:

$$\{\mathbf{x}(k+1)\} = {}^{GL}I^{n}(f(\{\mathbf{x}k\}, \{\mathbf{u}k\})) + \mathbf{g}\{\mathbf{x}\_{init}\},\tag{25}$$

where {*xinit*} = {*x*(0), *x*(−1),......, *x*(−*i*),......, *x*(−∞)}.

This means that the initial conditions are composed of all the past values of *x*(−*i*), since *k* = −∞.

Practically, this technique is used for the numerical simulation of the FDE/FDS problem.

The interested reader can refer to chapter 3 volume 1 of [34], where different initializations of the G.L. integral and the short memory principle [3] are analyzed.
