## 2 Theoretical Background and Preliminaries
### Fractional Calculus and Fractional-Order Systems
In control engineering, FOPID controllers are widely expressed in transfer function form [12]  
\[C(s)=k_{p}+\frac{k_{i}}{s^{\lambda}}+k_{d}s^{\mu}, \tag{7}\]  
where parameters \(k_{p}k_{d}\) and \(k_{i}\) are controller coefficients, and \(\lambda\in R\) and \(\mu\in R\) represent fractional-orders of the controller function; if \(\lambda=\mu=1\), then a conventional PID controller is obtained. Fractional-order derivative is not a local operator. For this reason, ideal digital realization of fractional-order elements uses growing computational resources in time as a result of the long memory effect. Therefore, for the practical realization of this controller, approximate fractional-order models are used to implement fractional-order derivative and integral elements [43; 44; 45; 46]. In practice, FOPID controllers are commonly implemented by using these approximate fractional-order models. A major advantage of fractional order derivative and integral element comes from the property that fractional-order elements provide flexibility for tuning of dynamic responses in a range from local to non-local operation. Let us consider this benefit for the closed-loop FOPID control system: Integer order derivative is originally a local operator and it considers very recent changes in the control error while generating a control action. However, fractional order derivative exhibits non-locality in differentiation operation. By adjusting the fractional order of the derivative element, FOPID controller can be tuned to consider all previous changes in control error while generating a control action [3]. Such adjustability option in the locality of fractional order elements can contribute to better modeling of real systems, which presents the property of non-locality in time and space, and to generate a better response to such a real system dynamics in control applications. In this study, the Oustaloup approximation method, which is widely preferred in fractional-order control system simulations, was utilized in the implementation of FOPID controller in simulation and experimental systems. To replace existing PID controllers in control loops with FOPID-based dynamics without breaking the existing loop, a retuning FOPID controller structure was proposed for the practical implementation of closed-loop FOPID controllers [17; 18]. The transfer function of the retuning FOPID controller is expressed in the form of  
\[C(s)=(C_{R}(s)+1)C_{PID}(s), \tag{8}\]  
where, \(C_{PID}(s)\) stands for the existing PID controller. The retuning controller function is found as  
\[C_{R}(s)=\frac{C(s)}{C_{PID}(s)}-1. \tag{9}\]  
Figure 1 depicts the integration of the retuning controller function \(C_{R}(s)\) to an existing PID control loop to realize a FOPID control system. This method is of high industrial interest because it allows us to replace existing, suboptimally tuned PID control loops with optimally tuned PID or FOPID controllers essentially without interrupting the control process since the dynamics are introduced through manipulating the original reference signal. This means that the controlled process ideally does not experience any downtime and hence this change in control strategy should not result in additional expenses related to restarting the process after the change of controller has taken place. As it can be seen from Figure 1, when discussing retuning control, in an ideal scenario one can just speak about the control loop consisting of a FOPID controller and plant in a negative unity feedback configuration because, at least from a purely theoretical perspective, the retuning method ensures that the complete control loop will indeed have the dynamics introduced by the external FOPID controller instead of those stemming from the original PID controller. In other words, it is equivalent to discussing a simpler FOPID control loop rather than the retuning control loop, at least for the sake of the simplicity of exposition.