## 3 Control Design  
### Fractional Order Fuzzy PID Controller  
The control scheme of the FOFPID shown in Figure 4 is a combination of fuzzy PI, and fuzzy PD controllers [19; 28], taking \(K_{e}\) and \(K_{de}\) as the input scaling factors, and \(\alpha\) and \(\beta\) as output factors. In addition, the differentiation and integration for D and I control can be of fractional orders, such that the control system can be tailored to fulfill the performance requirements. The parameters \(K_{e}\), \(K_{de}\), \(\alpha\), and \(\beta\) are the PID controller gains, while \(\mu\) and \(\lambda\) are the orders of differentiation and integration, respectively. The two inputs of the fuzzy logic controller are from the PD control in the front part of this scheme. One is the gain \(K_{e}\) multiplied by the control error \(e(t)\), and the other is \(K_{de}\), multiplied by the derivative \(\frac{d^{\mu}e}{dt^{\mu}}\). The fuzzy controller output \(c(t)\) is subsequently input to the PI control in the rear part. Then the control of the FOFPID can be obtained by  
\[u(t)=\alpha\cdot c(t)+\beta\cdot\int_{0}^{t}c(\tau)d\tau^{\lambda} \tag{13}\]  
where \(u(t)\) is the control output. The controller parameters \(K_{e}\), \(K_{de}\), \(\alpha\), \(\beta\), \(\mu\), and \(\lambda\) need tuning for precise tracking. However, it is hard to tune these gains, especially when using fractional order values. The parameter tuning in this study utilizes an evolutionary computation algorithm, discussed in the next chapter. In the fuzzy inference system [19], the input and output signals are transformed into fuzzy linguistic variables: PVB (Positive Very Big), NVB (Negative Very Big), PB (Positive Big), NB (Negative Big), PM (Positive Medium), NM (Negative Medium), PS (Positive Small), NS (Negative Small), and Z (Zero), respectively. The membership functions and rule base are shown in Figure 5 and Table 1. The triangular-shaped membership functions are chosen for the input variables, and singleton functions for the output. The two-dimensional linear rule base shown in Table 1 is used in the fuzzy controller. The inference mechanism is based on the Mamdani algorithm [19] and the rules are expressed by  
\[\text{IF }e(t)\text{ is A}^{\text{i}}\text{ AND }\frac{d^{\mu}e}{dt^{\mu}}\text{ is B}^{\text{i}},\text{ THEN }c(t)\text{ is C}^{\text{i}} \tag{14}\]  
where A\({}^{\text{i}}\), B\({}^{\text{i}}\), and C\({}^{\text{i}}\) are the linguistic labels of these linguistic variables \(e(t)\), \(\frac{d^{\mu}e}{dt^{\mu}}\) and \(c(t)\), and are characterized by membership functions for each variable. Here the max-min composition is used, and a fuzzy set \(u(F)\) can be inferred from the following relation:  
\[u(F_{i})=\overset{K}{\cup}\left(u^{\text{i}}(\text{A})\cap u^{\text{i}}(\text {B})\right) \tag{15}\]  
where \(K\) denotes the number of fuzzy output sets, and \(u^{\text{i}}(\text{A})\) and \(u^{\text{i}}(\text{B})\) are the MFs of the fuzzy sets A and B, respectively. \(\cap\) and \(\cup\) denote the min function (T-norm) and max function (T-conorm), respectively. \(F_{i}\) represents the value of the \(i\)-th output MF, and \(u(F_{i})\) is  
Figure 4: The block diagram of the fractional order fuzzy PID controller. its membership value at the \(i\)-th quantization level. The center-average defuzzification is utilized here to obtain the output \(c(t)\), as follows:  
\[c(t)=\frac{\sum_{i=1}^{K}F_{i}u(F_{i})}{\sum_{i=1}^{K}u(F_{i})} \tag{16}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**c(t)**} & \multirow{2}{*}{**c(t)**} & \multicolumn{5}{c}{**e(t)**} \\ \cline{3-8}  & & **NB** & **NS** & **Z** & **PS** & **PB** \\ \hline \multirow{4}{*}{\(\frac{d^{2}\epsilon}{d^{2}}\)} & NB & NVB & NB & NM & NS & Z \\ \cline{2-8}  & NS & NB & NM & NS & Z & PS \\ \cline{2-8}  & Z & NM & NS & Z & PS & PM \\ \cline{2-8}  & PS & NS & Z & PS & PM & PB \\ \cline{2-8}  & PB & Z & PS & PM & PB & PVB \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Rule base of the fuzzy logic controller.