# An Easily Understandable Grey Wolf Optimizer and Its Application to Fuzzy Controller Tuning
## 3 Grey Wolf Optimizer, Analysis, and Optimal Tuning Approach
### Grey Wolf Optimizer and Analysis
 \tag{32}\]  
and since the right-hand term in Equation (32) is negative, this indicates that  
\[x_{i}^{f}(k+1)-x_{i}^{f}(k)\leq 0,\ f=1...q,\ i=1...N,\ k=1...k_{\max}. \tag{33}\]  
Using \(k=k_{\max}\) in Equation (22) and then Equation (21), the coefficients obtain the expressions  
\[a^{f}(k_{\max})=a_{\alpha}^{f}(k_{\max})=a_{\beta}^{f}(k_{\max})=a_{\delta}^{f }(k_{\max})=0,\ f=1...q,\ i=1...N. \tag{34}\]  
Using Equation (34) in Equation (32) for \(k=k_{\max}\), this gives the condition for the final values of the agents' positions  
\[x_{i}^{f}(k_{\max}+1)=x_{i}^{f}(k_{\max})\leq 0,\ f=1...q,\ i=1...N. \tag{35}\]  
In addition, Equation (35) used in combination with Equation (33) gives the following condition related to the dynamics of the agents' positions:  
\[x_{i}^{f}(k+1)-x_{i}^{f}(k)<0,\ f=1...q,\ i=1...N,\ k=1...k_{\max}-1. \tag{36}\]  
Concluding, Equation (36) indicates that \(\left\{x_{i}^{f}(k)\right\}_{k=1...k_{\max}}\) is a monotonic strictly decreasing sequence as this will guarantee the convergence of GWO. Therefore, it is recommended to initialize as large as possible values of \(x_{i}^{f}(1),\ f=1...q,\ i=1...N\). The stability analysis of the dynamics of the agents' positions starts with the definition of the Lyapunov function candidate  
\[V(x_{i}^{f}(k))=\left(x_{i}^{f}(k)\right)^{2},\ f=1...q,\ i=1...N,\ k=1...k_{ \max}-1 \tag{37}\]  
that fulfils the conditions  
\[ \begin{array}{ll}V(x_{i}^{f}(k))>0,\ \forall x_{i}^{f}(k)\
eq 0,\ f=1...q,\ i=1...N,\ k=1...k_{\max}-1,\\ V(0)=0.\end{array}  \tag{38}\]  
The expression of the increment of \(V\) is  
\[ \begin{array}{ll}\Delta V(x_{i}^{f}(k))&=V(x_{i}^{f}(k+1))-V(x_{i}^{f}(k))= \left(x_{i}^{f}(k+1)\right)^{2}-\left(x_{i}^{f}(k)\right)^{2}\\ &=[x_{i}^{f}(k+1)-x_{i}^{f}(k)][x_{i}^{f}(k+1)+x_{i}^{f}(k)],\ f=1...q,\ i=1...N,\ k=1...k_{\max}-1.\end{array}  \tag{39}\]  
Using Equation (36) in Equation (37), the sufficient condition \(\Delta V(x_{i}^{f}(k))<0\) for the global asymptotic stability of the origin of dynamics of the agents' positions is  
\[x_{i}^{f}(k+1)+x_{i}^{f}(k)>0,\ f=1...q,\ i=1...N,\ k=1...k_{\max}-1. \tag{40}\]Using \(x_{i}^{f}(k+1)\) taken from Equation (31) in Equation (40), Equation (40) is then transformed into  
\[ \begin{array}{ll}2x_{i}^{f}(k)>&[a_{\alpha}^{f}(k)|c_{\alpha}^{f}(k)x^{\alpha f }(k)-x_{i}^{f}(k)|+a_{\beta}^{f}(k)|c_{\beta}^{f}(k)x^{\beta f}(k)-x_{i}^{f}(k) |\\ &+a_{\beta}^{f}(k)|c_{\beta}^{f}(k)x^{\delta f}(k)-x_{i}^{f}(k)|]/3,\;f=1...q,\;i =1...N,\;k=1...k_{\max}-1.\end{array}  \tag{41}\]  
Since the right-hand term in Equation (41) is strictly positive, Equation (41) leads to the first sufficient condition for the agents' positions:  
\[x_{i}^{f}(k)>0,\;f=1...q,\;i=1...N,\;k=1...k_{\max}-1, \tag{42}\]  
which also implies that  
\[x^{\alpha f}(k)>0,\;x^{\beta f}(k)>0,\;x^{\delta f}(k)>0,\;f=1...q,\;i=1...N,\; k=1...k_{\max}-1. \tag{43}\]  
Using Equations (42) and (43) and the properties of the modulus, the right-hand term in Equation (41) is upper bounded in terms of  
\[ \begin{array}{ll}&[a_{\alpha}^{f}(k)|c_{\alpha}^{f}(k)x^{\alpha f}(k)-x_{i }^{f}(k)|+a_{\beta}^{f}(k)|c_{\beta}^{f}(k)x^{\beta f}(k)-x_{i}^{f}(k)|\\ &+a_{\beta}^{f}(k)|c_{\beta}^{f}(k)x^{\delta f}(k)-x_{i}^{f}(k)|]/3\leq[a_{ \alpha}^{f}(k)(c_{\alpha}^{f}(k)x^{\alpha f}(k)+x_{i}^{f}(k))\\ &+a_{\beta}^{f}(k)(c_{\beta}^{f}(k)x^{\beta f}(k)+x_{i}^{f}(k))+a_{\beta}^{f} (k)(c_{\delta}^{f}(k)x^{\delta f}(k)+x_{i}^{f}(k))]/3,\\ &f=1...q,\;i=1...N,\;k=1...k_{\max}-1.\end{array}  \tag{44}\]  
To sum up, in order to fulfill the condition \(\Delta V(x_{i}^{f}(k))<0\) for the global asymptotic stability of the origin of dynamics of the agents' positions, it is sufficient to fulfill the following condition, as it results from Equations (41) and (44) and the transitive property of inequalities:  
\[