# Dynamic Analysis of Fuzzy Systems
## 4 Dynamic Analysis of Non-Linear Fuzzy Systems: Second-Order Model
### Second-Order Fuzzy Model with Linguistic Hedges  
A model with triangular membership functions was developed, which will now be modified with the linguistic hedges type mentioned above, such as dilation and concentration. We will use a non-linear model that is represented in this way: \(\tilde{x}=-2\zeta\omega_{n}\tilde{x}-\omega_{n}^{2}x|x|+K\omega_{n}^{2}u\), in a fuzzy system. It will be a concentration operation \(\mu x^{2}(x)\), and it is shown in Figure 13. If the model is \(\tilde{x}=-2\zeta\omega_{n}\tilde{x}-\omega_{n}^{2}\sqrt{|x|}+K\omega_{n}^{2}u\), this  
Figure 11: Second-order inference map.  
Figure 12: Scaling factors for a second-order system.  
Figure 10: Fuzzy sets; (**a**) input set \(x\), (**b**) input set \(\tilde{x}\), \(u\), (**c**) output set \(\tilde{x}\) centre of gravity and (**d**) output surface between \(\tilde{x}\), \(\tilde{x}\) and \(x\).  
represents a dilation \(\mu_{\mathrm{x}}{}^{0.5}(x)\) (see Figure 14). The linguistic hedges used for the concentration at input x will be \(X_{1}^{(1)}=VN\), \(X_{1}^{(2)}=VZ\), \(X_{1}^{(3)}=VP\). The dilation of x is expressed by linguistic hedges \(X_{1}^{(1)}=MN\), \(X_{1}^{(2)}=MZ\), \(X_{1}^{(3)}=MP\).  
The resultant Mamdani model:  
\[R^{(1)} : \mathrm{IF}\ x\ \mathrm{is}\ X_{1}^{(1)}\ \mathrm{AND}\ \dot{x}\ \mathrm{is}\ N\ \mathrm{THEN}\ \ddot{x}=1\] \[R^{(2)} : \mathrm{IF}\ x\ \mathrm{is}\ X_{1}^{(1)}\ \mathrm{AND}\ \dot{x}\ \mathrm{is}\ Z\ \mathrm{THEN}\ \ddot{x}=0.5\] \[R^{(3)} : \mathrm{IF}\ x\ \mathrm{is}\ X_{1}^{(1)}\ \mathrm{AND}\ \dot{x}\ \mathrm{is}\ P\ \mathrm{THEN}\ \ddot{x}=0\] \[R^{(4)} : \mathrm{IF}\ x\ \mathrm{is}\ X_{1}^{(2)}\ \mathrm{AND}\ \dot{x}\ \mathrm{is}\ N\ \mathrm{THEN}\ \ddot{x}=0.5\] \[R^{(5)} : \mathrm{IF}\ x\ \mathrm{is}\ X_{1}^{(2)}\ \mathrm{AND}\ \dot{x}\ \mathrm{is}\ Z\ \mathrm{THEN}\ \ddot{x}=0\] \[R^{(6)} : \mathrm{IF}\ x\ \mathrm{is}\ X_{1}^{(2)}\ \mathrm{AND}\ \dot{x}\ \mathrm{is}\ P\ \mathrm{THEN}\ \ddot{x}=-0.5\] \[R^{(7)} : \mathrm{IF}\ x\ \mathrm{is}\ X_{1}^{(3)}\ \mathrm{AND}\ \dot{x}\ \mathrm{is}\ N\ \mathrm{THEN}\ \ddot{x}=0\] \[R^{(8)} : \mathrm{IF}\ x\ \mathrm{is}\ X_{1}^{(3)}\ \mathrm{AND}\ \dot{x}\ \mathrm{is}\ Z\ \mathrm{THEN}\ \ddot{x}=-0.5\] \[R^{(9)} : \mathrm{IF}\ x\ \mathrm{is}\ X_{1}^{(3)}\ \mathrm{AND}\ \dot{x}\ \mathrm{is}\ P\ \mathrm{THEN}\ \ddot{x}=-1\] \[R^{(10)} : \mathrm{IF}\ u\ \mathrm{is}\ N\ \mathrm{THEN}\ \ddot{x}=-0.5\] \[R^{(11)} : \mathrm{IF}\ u\ \mathrm{is}\ Z\ \mathrm{THEN}\ \ddot{x}=0\] \[R^{(12)} : \mathrm{IF}\ u\ \mathrm{is}\ P\ \mathrm{THEN}\ \ddot{x}=0.5\]  
The scaling factors for the concentration will be as follows for the inputs \(x=\frac{1}{(4K)^{2}}\), \(\dot{x}=\frac{\zeta}{2K\omega_{n}}\) and \(u=\frac{1}{4}\), at the output \(\ddot{x}=16K\omega_{n}^{2}\).  
The scaling factors for the dilation will be as follows for the inputs \(x=\frac{1}{(2K)^{2}}\), \(\dot{x}=\frac{\zeta}{K\omega_{n}}\) and \(u=\frac{1}{2}\), at the output \(\ddot{x}=8K\omega_{n}^{2}\).  
The response of a unit step using linguistic operators, concentration and dilation is compared to a second-order linear system in Figure 15. This graph indicates that the peak  
Figure 14: The dilation of the input membership functions x and the resultant inference map.  
Figure 13: The concentration of the input membership functions x and the resulting inference map.  
time (tp) is less than that of the second-order linear system in concentration, which implies that it is faster than the second-order linear system. On the contrary, the tp value is higher in dilation than that of the second-order linear system. Therefore, the system is slower compared to the second-order linear system. This is valid only for the first 20 s. After that, it is observed that the dilation is faster than the concentration.