## 3 Tool Design
### Tool for Optimizing the RTD-A Controller Parameter (Tool-IV)
#### 3.4.5 Grey Wolf Algorithm (GWA)  
This algorithm is one of the efficient and simple algorithms present in the research field. It mimics the technique of the grey wolf, which depends on leadership and a hunting mechanism. There are preys that are the food of wolves. The algorithm employs four grey wolves that are simultaneously working on three tasks that are searching, encircling and attacking prey [36].  
All these tuning parameters will affect the final controller output. The optimal tuned values for the four parameters can be obtained by minimizing the suitable cost function. The controlled process response can be altered by adjusting the tuning parameters. The challenge in the optimal tuning of the RTD-A controller parameters is succeeded by the proposed tool.  
The researchers have developed different methods for dealing with the controller tuning parameters. One such technique is the Ogunnaike tuning rule. It has one notable parameter called the composite parameter \(\rho\). It is given in (26):  
\[\rho=\lambda\left(\frac{1+\lambda}{1-\lambda}\right)\!\!\frac{\alpha}{\tau}, \tag{26}\]  
where \(\lambda\) is the multiplicative factor, which is associated with process uncertainty and its value lies in the range of 0 to 1 for the FOPDT process model.  
When the uncertainty parameter \(\lambda\) is added to the nominal plant model in (1), the plant parameters is defined in (27).  
\[K^{0}=K(1+\lambda),\,\tau\,^{0}=\tau(1+\lambda)\text{ and }\alpha^{0}=\alpha(1+\lambda). \tag{27}\]  
The equivalent discrete plant parameters is given in (28):  
\[a^{0}=e^{-\Delta t/\tau 0},\,b^{0}=K^{0}(1-\text{a}^{0})\text{ and }m^{0}= \text{round}(\alpha^{0}/\Delta t). \tag{28}\]  
The plant output can be written as (29), where \(d\) is the disturbance.  
\[y_{p}(k)=a^{0}y_{p}(k-1)+b^{0}(u(k-m^{0}-1)+d(k-m^{0}-1)). \tag{29}\]  
Similarly, the model output from (2) can be written as (30).  
\[\overline{y}(k)=a\overline{y}(k-1)+bu(k-m-1). \tag{30}\]  
The model error \(e(k)\) in (31) is used in (7) to calculate the control action.  
\[e(k)=y_{p}(k)-\overline{y}(k). \tag{31}\]  
There are definite constraints on the tuning parameters with respect to the composite parameter. The Mukati-Ogunnaike tuning rule is presented in Table 1.  
The Kariwala, tuning rule is shown in Table 2. It is a semi-analytical tuning method. The tuning rules are obtained with the extensive experimental study on the different FOPDT transfer function [11].  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Composite Parameter** & \(\theta_{R}\) & \(\theta_{T}\) & \(\theta_{D}\) & \(\theta_{A}\) \\ \hline \(\rho<1\) & 0.5 & 0.8 & \(\geq 0.5\) & 0.9 \(\rho\) \\ \(1\leq\rho\leq 1.6\) & 0.9 & 0.8 & \(\geq 0.1\) & \(1.25(1-e^{-\rho})\) \\ \(\rho\geq 1.6\) & \(>0.95\) & 0.8 & \(\geq 1-The variety of the first order transfer function models is tested with robust stability analysis and the plant model mismatch using the Kariwala tuning method in the literature. Here, the researcher can observe that the tuning rule is transparent with respect to the controller performance features. The Kariwala tuning rule demonstrates the decent performance for a variety of processes. In some cases, Mukati-Ogunnaike tuning rules may lead to instability because of its inherent aggressiveness characteristics Mukati and Ogunnaike (2018).  
The model order reduction techniques, tuning rules and optimization algorithms discussed in this section were used to design a GUI tool in the tool description section. The proposed four tools are described with GUI in Section 4.