## 2 Theoretical Background and Preliminaries  
### Fractional Calculus and Fractional-Order Systems  
Fractional calculus introduces non-integer order operator and allows the orders of the operator to be rational, real, and complex numbers. This property offers a more coherent and flexible mathematical modeling approach of real-world system dynamics. The fractional derivative operator was presented in [12; 13] as  
\[{}_{a}D_{t}^{\alpha}=\left\{ \begin{array}{rl}\frac{\mathrm{d}^{\alpha}}{ \mathrm{d}t^{\alpha}}&\alpha>0\\ 1&\alpha=0\\ \int_{\alpha}^{t}(\mathrm{d}\tau)^{(-\alpha)}&\alpha<0\end{array} \right., \tag{1}\]  
where (generally) \(\alpha\in\mathbb{R}\) is the fractional order (noninteger order). For \(\alpha>0\), \({}_{a}D_{t}^{\alpha}\) operator expresses a fractional-order derivative, and for \(\alpha<0\), \({}_{a}D_{t}^{\alpha}\) operator expresses a fractional-order integral operator. The parameters \(a\) and \(t\) determine the lower and upper limits of the operation. Several mathematical definitions of fractional-order derivative operator have been proposed over some 300 years of the development of fractional calculus.Widely preferred definitions are the Grunwald-Letnikov definition, the Riemann-Liouville definition and the Caputo definition [12; 13; 14]. The Riemann-Liouville definition is preferred for system modeling and control engineering practice because the Laplace transform of this definition can be expressed as \(s^{\alpha}\) in the \(s\)-domain assuming zero initial conditions. This simplifies fractional-order transfer function modeling. Fractional-order differential equations are written in general form as [12; 13; 14]:  
\[a_{n}D^{\alpha_{n}}y(t)+a_{n-1}D^{\alpha_{n-1}}y(t)+...+a_{0}D^{\alpha_{0}}y(t )=b_{m}D^{\beta_{m}}u(t)+b_{m-1}D^{\beta_{m-1}}u(t)+...+b_{0}D^{\beta_{0}}u(t), \tag{2}\]where, \(\alpha_{i}\), \(\beta_{j}\) stands for fractional-orders of the system model and \(\alpha_{0}<.....<\alpha_{n-1}<\alpha_{n}\) and \(\beta_{0}<.....<\beta_{m-1}<\beta_{m}\) are real numbers. The parameters \(a_{i}\), \(b_{j}\) are constants and denotes coefficients of time invariant fractional-order system models. This fractional-order differential equation can model fractional-order dynamics in the time domain. Fractional-order control system design commonly requires transfer function models to perform \(s\)-domain design approaches. The transfer function models can be derived by applying a Laplace transform to the fractional-order differential equation model given by Equation (2). Due to providinga Laplace transform, the Riemann-Liouville definition of fractional-order derivative is frequently preferred in control theory and it was defined as  
\[{}_{a}D_{t}^{a}f(t)=\frac{1}{\Gamma(n-\alpha)}\frac{\mathrm{d}^{n}}{\mathrm{d} ^{n}}\int_{a}^{t}\frac{f(\tau)}{(t-\tau)^{\alpha-n+1}}\mathrm{d}\tau, \tag{3}\]  
where \(\Gamma(. )\) is Gamma function and \(n-1<\alpha<n\). By assuming zero initial conditions, Laplace transform of Riemann-Liouville fractional derivative is expressed as follows [12; 37]:  
\[L\{D^{a}f(t)\}=s^{a}F(s) \tag{4}\]  
By taking Laplace transform of Equation (2), again assuming zero initial condition, fractional-order transfer functions are written in a general form by [12; 37; 38]  
\[G(s)=\frac{b_{m}s^{\beta_{m}}+b_{m-1}s^{\beta_{m-1}}+...+b_{0}s^{\beta_{0}}}{ a_{n}s^{\alpha_{n}}+a_{n-1}s^{\alpha_{n-1}}+...+a_{0}s^{\alpha_{0}}}=\frac{\sum_{j= 0}^{m}b_{j}s^{\beta_{j}}}{\sum_{i=0}^{n}a_{i}s^{\alpha_{i}}}. \tag{5}\]  
After obtaining the transfer function model, the characteristic equation of fractional-order transfer function can be used for stability analysis. The fractional-order characteristic equation of \(G(s)\) is written by  
\[\Delta(s)=\sum_{i=0}^{n}a_{i}s^{\alpha_{i}}=0. \tag{6}\]  
Root locus of this characteristic equation is analyzed for stability property evaluation of fractional-order systems [39; 40; 41; 42].