# Fuzzy Adaptive Dynamic Surface Control with Constant Gain for Non-Affine Pure-Feedback Systems
## 3 Controller Design and Stability Analysis
The temporal derivative of the tracking error \(e_{1}\) is obtained as  
\[e_{1}=\hat{x}_{1}-\hat{y}_{1d}. \tag{17}\]  
The definition of \(e_{2}\) can be obtained from (10):  
\[e_{2}=x_{2}-w_{2} \tag{18}\]  
By adding \(z_{2}=w_{2}-\alpha_{1}\) and \(\alpha_{1}\) at both ends of the Equation (18), the state \(x_{2}\) can be obtained:  
\[x_{2}=e_{2}+z_{2}+\alpha_{1} \tag{19}\]  
\[\alpha_{1}=\frac{-a_{1}e_{1}}{\psi_{1}}-\frac{e_{1}\dot{\theta}\bar{W}_{1}^{T }\bar{W}_{1}}{\psi_{1}2b_{1}^{2}}-\frac{g_{1}}{\psi_{1}}-e_{2}+x_{2}^{0}+ \frac{\dot{y}_{1d}}{\psi_{1}} \tag{12}\]0 A schematic diagram of the closed-loop control system. Considering the strict-feedback structure in (3) and the new form of \(x_{2}\) in (19), the derivative \(\dot{x}_{1}\) is formulated as  
\[\alpha_{1}=\frac{-a_{1}e_{1}}{\psi_{1}}-\frac{e_{1}\dot{\theta}\bar{W}_{1}^{T }\bar{W}_{1}}{\psi_{1}2b_{1}^{2}}-\frac{g_{1}}{\psi_{1}}-e_{2}+x_{2}^{0}+ \frac{\dot{y}_{1d}}{\psi_{1}} \tag{12}\]1  
Substitution of (20) into (17) yields the error dynamics:  
\[\alpha_{1}=\frac{-a_{1}e_{1}}{\psi_{1}}-\frac{e_{1}\dot{\theta}\bar{W}_{1}^{T }\bar{W}_{1}}{\psi_{1}2b_{1}^{2}}-\frac{g_{1}}{\psi_{1}}-e_{2}+x_{2}^{0}+ \frac{\dot{y}_{1d}}{\psi_{1}} \tag{12}\]2  
To ensure rapid convergence of the first subsystem, the virtual control law (12) is introduced. Its incorporation into (21) results in  
\[\alpha_{1}=\frac{-a_{1}e_{1}}{\psi_{1}}-\frac{e_{1}\dot{\theta}\bar{W}_{1}^{T }\bar{W}_{1}}{\psi_{1}2b_{1}^{2}}-\frac{g_{1}}{\psi_{1}}-e_{2}+x_{2}^{0}+ \frac{\dot{y}_{1d}}{\psi_{1}} \tag{12}\]3  
Determining how to construct a suitable Lyapunov function can refer to the following experience: On the one hand, in order to ensure the positive definiteness of the Lyapunov function, the square terms of the key variables can be constructed. In addition, the proposed controller must strictly control the convergence trend of tracking error and filtering error, so they can be considered as important components of the Lyapunov function. Therefore, consider the following quadratic Lyapunov function candidate:  
\[\alpha_{1}=\frac{-a_{1}e_{1}}{\psi_{1}}-\frac{e_{1}\dot{\theta}\bar{W}_{1}^{T }\bar{W}_{1}}{\psi_{1}2b_{1}^{2}}-\frac{g_{1}}{\psi_{1}}-e_{2}+x_{2}^{0}+ \frac{\dot{y}_{1d}}{\psi_{1}} \tag{12}\]4  
whose time derivative is given by  
\[\alpha_{1}=\frac{-a_{1}e_{1}}{\psi_{1}}-\frac{e_{1}\dot{\theta}\bar{W}_{1}^{T }\bar{W}_{1}}{\psi_{1}2b_{1}^{2}}-\frac{g_{1}}{\psi_{1}}-e_{2}+x_{2}^{0}+ \frac{\dot{y}_{1d}}{\psi_{1}} \tag{12}\]5  
After substituting (22) into (24), the derivative of Lyapunov can be further expressed in the following form:  
\[\alpha_{1}=\frac{-a_{1}e_{1}}{\psi_{1}}-\frac{e_{1}\dot{\theta}\bar{W}_{1}^{T }\bar{W}_{1}}{\psi_{1}2b_{1}^{2}}-\frac{g_{1}}{\psi_{1}}-e_{2}+x_{2}^{0}+ \frac{\dot{y}_{1d}}{\psi_{1}} \tag{12}\]6  
Choose a continuous real-valued bounded function \(f_{1}(\bar{X}_{1})\) as the input function of the first FLS (8):  
\[\alpha_{1}=\frac{-a_{1}e_{1}}{\psi_{1}}-\frac{e_{1}\dot{\theta}\bar{W}_{1}^{T }\bar{W}_{1}}{\psi_{1}2b_{1}^{2}}-\frac{g_{1}}{\psi_{1}}-e_{2}+x_{2}^{0}+ \frac{\dot{y}_{1d}}{\psi_{1}} \tag{12}\]7  
with \(\bar{X}_{1}=[x_{1},y_{1d},t]^{T}\). Substituting (26) into (25), one has  
\[\alpha_{1}=\frac{-a_{1}e_{1}}{\psi_{1}}-\frac{e_{1}\dot{\theta}\bar{W}_{1}^{T }\bar{W}_{1}}{\psi_{1}2b_{1}^{2}}-\frac{g_{1}}{\psi_{1}}-e_{2}+x_{2}^{0}+ \frac{\dot{y}_{1d}}{\psi_{1}} \tag{12}\]8  
Based on Lemma 2, the FLS can be adopted to approximate the \(f_{1}(\bar{X}_{1})\) such that for any accuracy \(\widetilde{\xi}_{1}\in\Re^{+}\), \(f_{1}\) satisfies the following equation:  
\[\alpha_{1}=\frac{-a_{1}e_{1}}{\psi_{1}}-\frac{e_{1}\dot{\theta}\bar{W}_{1}^{T }\bar{W}_{1}}{\psi_{1}2b_{1}^{2}}-\frac{g_{1}}{\psi_{1}}-e_{2}+x_{2}^{0}+ \frac{\dot{y}_{1d}}{\psi_{1}} \tag{12}\]9  
where \(\xi_{1}(\bar{X}_{1})\) is error function with upper bound \(\widetilde{\xi}_{1}\).