# Fuzzy Adaptive Dynamic Surface Control with Constant Gain for Non-Affine Pure-Feedback Systems
## 4 Simulation Results
### Simulation Example 1
Therefore, in order to ensure sufficient approximation accuracy, a smaller \(\varphi\) can be prioritized._  
_Based on the same FLSs and initial conditions of state variables (\(x_{1}(0)=0.5\), \(x_{2}(0)=0.5\)), to facilitate comparison research, we provide the initial control signal, final actual control signal, and adaptive law of existing fuzzy control scheme (see [31]):_  
\[\alpha_{1}=-A_{1}e_{1}-\frac{e_{1}}{2}-\frac{e_{1}\hat{\mathrm{O}}_{1}\hat{ \mathrm{W}}_{1}^{T}(\hat{X}_{1}^{*})\hat{\mathrm{W}}_{1}(\hat{X}_{1}^{*})}{2B _{1}^{2}}+x_{2}^{0} \tag{82}\]  
\[u=-A_{2}e_{2}-\frac{e_{2}}{2}-\frac{e_{2}}{2}\frac{\hat{\mathrm{O}}_{2}\hat{ \mathrm{W}}_{2}^{T}(\hat{X}_{2}^{*})\hat{\mathrm{W}}_{2}(\hat{X}_{2}^{*})}{2B _{2}^{2}}+u^{0} \tag{83}\]  
\[\dot{\hat{\mathrm{O}}}_{i}=\varsigma_{i}\frac{e_{i}^{2}\hat{\mathrm{W}}_{i}^{ T}(\hat{X}_{i}^{*})\hat{\mathrm{W}}_{i}(\hat{X}_{i}^{*})}{2B_{i}^{2}}-\varphi_{i}^{*} \hat{\mathrm{O}}_{i},\ i=1,2 \tag{84}\]  
_where \(\hat{\mathrm{O}}_{1}(0)=0.1\), \(\hat{\mathrm{O}}_{2}(0)=0.1\), \(A_{1}=9\), \(A_{2}=9\), \(B_{1}=4\), \(B_{2}=4\), \(\varsigma_{1}^{*}=10\), \(\varsigma_{2}^{*}=10\), \(\varphi_{1}^{*}=0.5\), \(\varphi_{2}^{*}=0.5\), \(u^{0}=0\), \(x_{2}^{0}=0\)._  
**Remark 7**. : _It should be pointed out that two adaptive parameters (\(\hat{\mathrm{O}}_{1}\) and \(\hat{\mathrm{O}}_{2}\)) are required to update the weight vectors of all FLSs in the existing fuzzy control approach. However, the proposed control scheme needs only one adaptive parameter to achieve the above functions regardless of the system order, thus, for an \(n\)-order pure-feedback system, the online learning time of the FLSs of the proposed scheme can be reduced by \((n-1)/n\), compared with the literature [31]._  
However, \(\kappa_{1}^{T}\hat{\mathrm{W}}_{1}\) is used to approximate unknown function \(F_{1}(\hat{X}_{1}^{*})\)=\(g_{1}+\frac{e_{1}}{2}-\dot{y}_{1d}+A_{1}e_{1}\), where \(\hat{X}_{1}^{*}=[x_{1},y_{1d},\dot{y}_{1d}]^{T}\) is the input vector of the first FLS. \(\hat{\kappa}_{2}^{T}\hat{\mathrm{W}}_{2}\) is utilized to approximate unknown function \(F_{2}(\hat{X}_{2}^{*})\), which can be described as  
\[ \begin{split} F_{2}(\hat{X}_{2}^{*})=&\psi_{1}e_{1} +g_{2}+A_{2}e_{2}+\frac{e_{2}}{2}-\frac{\partial\alpha_{1}}{\partial x_{1}}g_ {1}+\frac{1}{2}e_{2}\bigg{(}\frac{\partial\alpha_{1}}{\partial x_{1}}\bigg{)} ^{2}+\frac{e_{2}\psi_{2}^{2}}{2}+\frac{\partial\alpha_{1}}{\partial y_{1d}} \dot{y}_{1d}\\ &+\frac{\partial\alpha_{1}}{\partial\dot{y}_{1d}}\dot{y}_{1d}- \frac{\partial\alpha_{1}}{\partial\dot{\mathrm{O}}_{1}}\dot{\mathrm{O}}_{1}- \frac{\varsigma_{i}e_{2}^{2}\hat{\mathrm{W}}_{2}^{T}\dot{W}_{2}}{2B_{2}^{2}} \frac{\partial\alpha_{1}}{\partial\dot{\mathrm{O}}_{1}}\end{split}  \tag{85}\]  
where \(\hat{X}_{2}^{*}=[x_{1},x_{2},y_{1d},\dot{y}_{1d},\dot{y}_{1d},\dot{O}_{1}]^{T}\). In addition to state errors and disturbance terms, \(F_{1}(\hat{X}_{1}^{*})\), \(F_{2}(\hat{X}_{2}^{*})\) also contain system functions (\(g_{1},g_{2}\)) and the derivative term of \(\alpha_{1}\) such that the degree of nonlinearity of \(F_{1}(\hat{X}_{1}^{*})\) and \(F_{2}(\hat{X}_{2}^{*})\) are significantly higher than \(F_{1}(\hat{X}_{1})\) and \(F_{2}(\hat{X}_{2})\), respectively. As we are using FLSs with the same approximation ability in this simulation, the tracking performance of the existing control proposal [31] will be affected. Figures 2-6 show that the developed control method not only guarantees that tracking errors converge to an arbitrarily small region of zero, but it also ensures that state variables, control signal, and adaptive updating function are bounded, which proves the correctness of Theorem 1. Figure 2 displays the following performance of the presented controller. It is obviously observed that the output of the proposed controller is more closely tracking the \(y_{1d}\). The range of variation between the maximum and minimum values of the steady-state tracking error (RTE) is regarded as a quantitative index for evaluating tracking performance.