# Fuzzy Adaptive Dynamic Surface Control with Constant Gain for Non-Affine Pure-Feedback Systems
## 4 Simulation Results
### Simulation Example 1  
A comparison research between the presented approach and the fuzzy dynamic control method with unknown system dynamics [31] is provided to demonstrate the advantages of tracking performances, based on the same dynamics system and FLS. The same dynamic system and fuzzy logic system (FLS) are utilized for fair comparison. In this section, a nonlinear pure-feedback system subject to external disturbances (\(d_{1}(t)\), \(d_{2}(t)\)) is considered: [32]:  
\[\dot{x}_{1} =x_{1}+x_{2}+x_{2}^{3}/5\ +d_{1}(t)\] \[\dot{x}_{2} =x_{1}x_{2}+u+u^{3}/7\ +d_{2}(t) \tag{78}\] \[y =x_{1}\]  
where \(d_{1}(t)=0.1\sin(100t)\), \(d_{2}(t)=0.01\sin(100t)\), and the desired tracking trajectory is selected as \(y_{1d}=0.5\sin(t)+0.5\sin(0.5t)\). Based on the aforementioned practical system (78), the control signal and the updating law are designed in the following forms:  
\[\alpha_{1}=\frac{\dot{y}_{1d}-a_{1}e_{1}-g_{1}}{\dot{\psi}_{1}}-\frac{e_{1} \dot{\theta}\bar{W}_{1}^{T}(\bar{X}_{1})\bar{W}_{1}(\bar{X}_{1})}{\psi_{1}2b_{ 1}^{2}}-e_{2}+x_{2}^{0} \tag{79}\]  
\[u=\frac{-a_{2}e_{2}-g_{2}+\dot{w}_{2}}{\dot{\psi}}-\frac{e_{2}\dot{\theta}\bar {W}_{2}^{T}(\bar{X}_{2})\bar{W}_{2}(\bar{X}_{2})}{\dot{\psi}2b_{2}^{2}}+u^{0} \tag{80}\]  
\[\dot{\theta}=\varsigma\sum_{i=1}^{2}\frac{e_{i}^{2}\bar{W}_{i}^{T}\bar{W}_{i}} {2b_{i}^{2}}-\varphi\dot{\theta} \tag{81}\]  
where \(\bar{X}_{1}=[x_{1},y_{1d},t]^{T}\), \(\bar{X}_{2}=[x_{1},x_{2},y_{1d},w_{2},t]^{T}\); therefore, the number of input neurons of \(\bar{\kappa}_{1}^{\tau}\bar{W}_{1}\) and \(\bar{\kappa}_{2}^{\tau}\bar{W}_{2}\) are 3 and 5, respectively. Furthermore, \(\bar{\kappa}_{1}^{\tau}\bar{W}_{1}\) and \(\bar{\kappa}_{2}^{\tau}\bar{W}_{2}\) have the same center position vector \([-3,-2,-1,0,1,2,3]^{T}\), seven nodes, and Gaussian width value 2. The initial conditions and design parameters of this simulation are given as follows: \(x_{1}(0)=0.5\), \(x_{2}(0)=0.5\), \(\dot{\theta}(0)=0.1\), \(w_{2}(0)=0\), \(a_{1}=5\), \(a_{2}=57\), \(b_{1}=0.8\), \(b_{2}=0.2\), \(\psi=11\), \(\varsigma=10\), \(\varphi=1\), \(u^{0}=0\), \(x_{2}^{0}=0\). Then, we set the filter positive constant \(\Phi_{2}=0.01\) of the first-order filter. **Remark 6**. : _The recommended value of \(\psi\) can be calculated using the theoretical Formula (2). In addition, \(\psi\) is regarded as the negative gain coefficient of tracking error. The smaller the value of \(\psi\), the more significant its amplification effect on tracking error. In order to improve the convergence speed, a smaller \(\psi\) can be selected while ensuring the stability of the closed-loop system. In addition, through parameter debugging, it can be found that the design parameter \(\varphi\) has the greatest impact on the estimated value \(\hat{\theta}\). The recommended range for setting the parameter \(\varphi\) is 0.5-20. The larger the value of \(\varphi\), the faster the convergence speed of the estimated value \(\hat{\theta}\). However, too fast a convergence speed can easily lead to estimation values falling into local optima.