# Dynamic Analysis of Fuzzy Systems
## 4 Dynamic Analysis of Non-Linear Fuzzy Systems: Second-Order Model
### Second-Order Approach  
A fuzzy dynamic model equivalent to the linear second-order dynamic model can be obtained by using the following logical operators in the fuzzy inference process; the product for conjunction, the product for implication, the sum for aggregation and the centroid for defuzzification. These operators are used so that the output of the fuzzy system is linear. In the second-order model, we obtain that \(\ddot{x}^{(1)}=8K\omega_{n}^{2}\), \(\ddot{x}^{(2)}=0\), \(\ddot{x}^{(3)}=0\), \(\ddot{x}^{(4)}=-8K\omega_{n}^{2}\), \(\ddot{x}^{(5)}=4K\omega_{n}^{2}\) and \(\ddot{x}^{(6)}=-4K\omega_{n}^{2}\) can be approximated. Another important thing to consider is the universe of discourse, which can be obtained from the phase plane. Using the Mamdani model:  
\[R^{(1)}:\text{IF $x$ is $N$ AND $\dot{x}$ is $N$ THEN $\dot{x}$ =8K\omega_{n}^{2}$}\] \[R^{(2)}:\text{IF $x$ is $N$ AND $\dot{x}$ is $P$ THEN $\dot{x}$ =0$}\] \[R^{(3)}:\text{IF $x$ is $P$ AND $\dot{x}$ is $N$ THEN $\dot{x}$ =0$}\] \[R^{(4)}:\text{IF $x$ is $P$ AND $\dot{x}$ is $P$ THEN $\ddot{x}$ =-8K\omega_{n}^{2}$}\] \[R^{(5)}:\text{IF $u$ is $P$ THEN $\dot{x}$ =4K\omega_{n}^{2}$}\] \[R^{(6)}:\text{IF $u$ is $N$ THEN $\ddot{x}$ =-4K\omega_{n}^{2}$}\]  
These rules proved that the fuzzy system is equivalent to a linear system. Since the membership functions for \(x\) and \(\dot{x}\) are:  
\[\mu_{N}(x) =\frac{2K-x}{4K},\ \forall\ x\ \in\ [-2K,2K]\] \[\mu_{P}(x) =\frac{x+2K}{4K},\ \forall\ x\ \in\ [-2K,2K]\] \[\mu_{N}(\dot{x}) =\frac{K\omega_{n}-\zeta\dot{x}}{2K\omega_{n}},\ \forall\ \dot{x}\ \in\ [-\frac{K\omega_{n}}{\zeta},\frac{K\omega_{n}}{\zeta}]\] \[\mu_{P}(\dot{x}) =\frac{\zeta\dot{x}+K\omega_{n}}{2K\omega_{n}},\ \forall\ \dot{x}\ \in\ [-\frac{K\omega_{n}}{\zeta},\frac{K\omega_{n}}{\zeta}]\] \[R^{(2)}:\text{IF $x$ is $N$ AND $\dot{x}$ is $P$ THEN $\dot{x}$ =0$}\]0 \[R^{(2)}:\text{IF $x$ is $N$ AND $\dot{x}$ is $P$ THEN $\dot{x}$ =0$}\]1  
The output of the system can be computed as follows:  
\[R^{(2)}:\text{IF $x$ is $N$ AND $\dot{x}$ is $P$ THEN $\dot{x}$ =0$}\]2  
by applying\[R^{(2)}:\text{IF $x$ is $N$ AND $\dot{x}$ is $P$ THEN $\dot{x}$ =0$}\]3 \[R^{(2)}:\text{IF $x$ is $N$ AND $\dot{x}$ is $P$ THEN $\dot{x}$ =0$}\]4 \[R^{(2)}:\text{IF $x$ is $N$ AND $\dot{x}$ is $P$ THEN $\dot{x}$ =0$}\]5 \[R^{(2)}:\text{IF $x$ is $N$ AND $\dot{x}$ is $P$ THEN $\dot{x}$ =0$}\]6 \[R^{(2)}:\text{IF $x$ is $N$ AND $\dot{x}$ is $P$ THEN $\dot{x}$ =0$}\]7 \[R^{(2)}:\text{IF $x$ is $N$ AND $\dot{x}$ is $P$ THEN $\dot{x}$ =0$}\]8 \[R^{(2)}:\text{IF $x$ is $N$ AND $\dot{x}$ is $P$ THEN $\dot{x}$ =0$}\]9  
where:  
\[R^{(3)}:\text{IF $x$ is $P$ AND $\dot{x}$ is $N$ THEN $\dot{x}$ =0$}\]0 \[R^{(3)}:\text{IF $x$ is $P$ AND $\dot{x}$ is $N$ THEN $\dot{x}$ =0$}\]1 \[R^{(3)}:\text{IF $x$ is $P$ AND $\dot{x}$ is $N$ THEN $\dot{x}$ =0$}\]2 \[R^{(3)}:\text{IF $x$ is $P$ AND $\dot{x}$ is $N$ THEN $\dot{x}$ =0$}\]3 \[R^{(3)}:\text{IF $x$ is $P$ AND $\dot{x}$ is $N$ THEN $\dot{x}$ =0$}\]4 \[R^{(3)}:\text{IF $x$ is $P$ AND $\dot{x}$ is $N$ THEN $\dot{x}$ =0$}\]5 \[R^{(3)}:\text{IF $x$ is $P$ AND $\dot{x}$ is $N$ THEN $\dot{x}$ =0$}\]6  
Which matches the polynomial in Equation ((23))  
\[R^{(3)}:\text{IF $x$ is $P$ AND $\dot{x}$ is $N$ THEN $\dot{x}$ =0$}\]7  
To allow the adjustments of \(K,\omega_{n}\) and \(\zeta\), input and output scaling factors will be used. For this purpose, the universe of discourse will be left normalised to the interval \([-1,1]\) (see Figure 10). For the output, the centre of gravity will be \(\{-1,-0.5,0.0,5,1\}\) and will be multiplied by the maximum value of the output. In this case, it becomes \(8K\omega_{n}^{2}\). The input variable \(x\), which is within the interval \([-2K,2K]\), will be normalised to \(\frac{1}{2K}\), and then the variable \(\dot{x}\) will be normalised to \(\frac{\zeta}{K\omega_{n}}\). Next, \(u\) will be normalised to \(\frac{1}{2}\). The number of sets used in the input can be modified by applying scaling factors. For example, from 2 to 3 fuzzy sets, as shown in the output inference map in Figure 11 below.