# Construction and Evaluation of a Control Mechanism for Fuzzy Fractional-Order PID
## 5 Controller Design
### Fractional-Order PID Controller  
The differentiator of fractional-order can be characterized as a continuous differential operator, which is expressed as:  
\[\mathrm{a_{n}D_{t}^{r}=\left\{\begin{array}{c}\frac{d^{r}}{dt^{r}}\colon r>0 \\ 1:r=0\\ \int\limits_{a}^{t}(d\tau)^{-r}\colon r<0\end{array}\right. \tag{7}\]  
The well-known derivatives of Liouville, Riemann-Liouville, and Caputo naturally extend into complex function spaces, establishing interesting connections between them and the derivatives of Grunwald-Letnikov. The boundaries of the non-integer order basic operatorare a and t. The Grunwald-Letnikov (GL), the Riemann-Liouville (RL), and the Caputo definitions are all fractional differ-integral techniques. The following is the GL definition:  
\[{}_{\rm a}D_{\rm r}^{\rm t}({\rm t})=\underset{\rm h\to 0}{\rm limh-r}\sum_{j=0}^{ \left[\frac{t-a}{h}\right]}(-1)^{j}\binom{r}{j}f(t-jh) \tag{8}\]  
The definition of RL is as follows:  
\[{}_{\rm a}D_{\rm r}^{\rm t}f(t)=\frac{1}{\Gamma(n-1)}\frac{d^{n}}{dt^{n}}\int \limits_{a}^{t}\frac{f(\tau)}{f(t-\tau)^{r-n+1}}d\tau \tag{9}\]  
The Caputo definition is as follows:  
\[{}_{\rm a}D_{\rm r}^{\rm t}f(t)=\frac{1}{\Gamma(r-n)}\int\limits_{a}^{t}\frac{ f^{(n)}(\tau)}{f(t-\tau)^{r-n+1}}d\tau \tag{10}\]  
(\(n-1<r<n\)) and \(\Gamma(m)\), the well-known Gamma function is explained for positive real m by the following equation.  
\[\Gamma(m)=\int\limits_{0}^{\infty}e^{-u}u^{m-1}du \tag{11}\]  
In Caputo derivatives, the beginning conditions for integer-order and fractional-order differential equations are the same. Take an appropriate n order integer derivative of m order non-integer integral to get a \(n-m=q\) order one to get the definition of fractional derivative:  
\[\frac{d^{q}f(t)}{dt^{q}}=\frac{d^{n-m}f(t)}{dt^{n-m}}=\frac{1}{\Gamma(m)}\int \limits_{0}^{t}(t-y)^{m-1}f(y)dy \tag{12}\]  
The equation above becomes a canonical first-order derivative. If q = 1 (n = 2, m = 1). Therefore, the FOPID regulator is comprised of partial administrators with regulator gains. The FOPID regulator move capacity might be communicated as utilizing fragmentary vital and partial subordinate terms.  
\[G_{\rm c}(s)=\frac{u(s)}{e(s)}=K_{P}+K_{1^{S-\lambda}}+K_{D^{SH}} \tag{13}\]  
where (\(\lambda\), \(\mu\)) is greater than zero. This regulator offers higher tuning adaptability, permitting it to cover a more extensive scope of boundaries to settle the plant taken care of and further develop control circle power. The same research has been conducted to corroborate this feature (see, for example, [38; 39; 40]), and in the following sections, the research will look at several innovative pieces of literature addressing fractal resilience. Gc(s) is the transfer function of controller, e(s) is the error, and u(s) is the yield in Equation (13). The constants for proportional, integral, and derivative expressions are KP, KI, and KD. The fractional component of integral parts is \(\lambda\), while the fractional component of derivative parts is \(\mu\). In time-domain, representation (13) is written as:  
\[u(t)=K_{\rm pe}(t)+K_{\rm I}D^{-\lambda}e(t)+K_{\rm D}D^{\mu}e(t) \tag{14}\]  
Apart from the typical three parameters KP, KI, and KD, it is self-evident that two additional parameters derivative order and integral order should be addressed in the design of FOPID controller. To get optimum controller settings, we use the Nelder-mead approach [41] with the performance metric of Integral of Absolute Error (IAE). The estimated FOPID controller values are shown in Table 1.