## 2 The Preliminary Background of the Proposed Control Method
### Fractional Calculus Definitions  
Fractional derivatives include the Caputo and Riemann-Liouville (RL) types. All of them are extensions of the standard differential and integral operators. Nonetheless, the features of the fractional derivatives are less than those of the comparable classical ones. Because of this, these derivatives are highly helpful in explaining the abnormal happenings [32; 33]. This study makes use of the RL derivative, one of the most often utilized fractional derivatives. The RL derivative outperforms Caputo in that it permits the function in question to have discontinuity at the origin. However, it prohibits the use of conventional initial conditions; instead, the initial conditions in the RL case must either be weighted initial conditions or in the integral form. More numerical aspect/advantages can be found in [34; 35; 36].  
The FO controllers are described by differential equations with fractional-order integrals and derivatives. An FO operator \(vB_{t}^{x}\) is denoted by the following:  
\[B_{t}^{x}=\begin{cases}\frac{dx}{dt},&R(x)>0\\ 1,&R(x)=0\\ \int_{b}^{t}{(d\lambda)}^{-x},&R(x)<0\end{cases} \tag{12}\]  
with \(x\in\mathbb{R}\). According to [PERSON] et al. [31], the three definitions for fractional derivatives that are most frequently employed are the Caputo definition, the Riemann-Liouville definition, and the Grunwald-Letnikov definition. However, the Riemann-Liouville definition is more common and is given by  
\[vB_{t}^{x}f(x)=\frac{1}{\gamma(m-x)}\frac{d^{a}}{dt^{a}}\int_{l}^{u}\frac{f(x )}{(t-\tau)x^{-a+1}}dx,\ a-1<x<a \tag{13}\]  
where \(\gamma(.)\) stands for the Euler gamma function, and \(l\) and \(u\), respectively, are the lower and upper bounds. The order of the integral or derivative is \(x\), which could be a complex number or a non-integer, and \(vB_{t}^{x}\) is the fractional differentiation or integration. To determine the transfer-function of integer-order systems, the Laplace transform is used. The signal's \(a^{th}\) order derivative's Laplace transformation is \(x(t)\).  
\[\mathcal{L}B^{a}x(t)=\int_{0}^{\infty}e^{-st}{}_{0}B_{t}^{a}x(t)dt=S^{a}X(s)- \sum_{k=0}^{-1}s^{k}{}_{0}B_{t}^{n-k-1}\left.x(t)\right|_{t=0} \tag{14}\]  
with \((m-1)<a<m\) and \(m\in Z\) as explained in [37]. One method to be applied in simulations, models, or controllers is to approximate fractional orders with integer-order transfer functions. To replicate a fractional transfer function accurately, an integer-order transfer function must have an unlimited number of poles and zeros. However, a finite number \((N)\) of zeros and poles allows for reliable estimation. The Oustaloup strategy, which takes advantage of the recursive distribution of zeros and poles, is one of the many well-known approximation techniques. The deeper analysis and optimal continuous approximation Oustaloup method can be found in [38; 39; 40]. The transfer function in the Oustaloup method is written as  
\[s^{v}\approx k\prod_{n=1}^{N}\frac{1+(s/v_{z,a})}{1+\left(s/v_{p,a}\right)} \tag{15}\]  
In \([\omega_{1},\omega_{h}]\), the approximation is defined. To guarantee that Equation (15) will have a unity gain at 1 rad/s, gain k is modified. The order of approximation affects the chosen value of N. Gain and phase ripples are the results of low-order approximations. With a higher order of N, ripples can be eliminated; however, the computing process is very challenging. Equation (15) describes the frequencies of zeros and poles, which are given by  
\[\omega_{z,l}=\omega_{l}\sqrt{\beta} \tag{16}\]  
\[\omega_{p,a}=\omega_{z}\zeta,\;\;a=1,\ldots,A \tag{17}\]  
\[\omega_{z,a+1}=\omega_{p,a},\;\;a=1,\ldots,A-1 \tag{18}\]  
\[\zeta=\left(\omega_{h}/\omega_{l}\right)^{v/A} \tag{19}\]  
\[\beta=\left(\omega_{h}/\omega_{l}\right)^{(1-v)/A} \tag{20}\]  
To take into account the condition \(v<0\), Equation (15) might be reversed. If \(|v|>1\), which typically leads to distinct fractional orders as seen below, the approximation becomes unacceptable.  
\[s^{v}=s^{a}s^{\delta},\;\;\;a\zeta Z,\;\;\;\delta\zeta[0,1] \tag{21}\]