## 5 AFSMC Design
### Adaptive Fuzzy Sliding Mode Controller  
In the actual control process, since \(g(x,t)\) and \(f(x,t)\) are unknown, Equation (10) is difficult to implement. Using the estimated terms \(\hat{g}(x|\mathbf{\theta}_{g})\) and \(\hat{f}(x|\mathbf{\theta}_{f})\) to approximate \(g(x,t)\) and \(f(x,t)\), Equation (10) can be expressed as follows:  
\[\left\{ \begin{array}{l}u=\frac{R-f(x|\mathbf{\theta}_{f})}{\hat{g}(x|\mathbf{\theta} _{f})}\\ \hat{f}(x|\mathbf{\theta}_{f})=\mathbf{\theta}_{f}^{\text{T}}\xi(x)\\ \hat{g}(x|\mathbf{\theta}_{g})=\mathbf{\theta}_{g}^{\text{T}}\xi(x)\end{array} \right. \tag{37}\]  
**Theorem 2**. : _For the nonlinear system (27), if the parameter vectors \(\mathbf{\theta}_{f}\) and \(\mathbf{\theta}_{g}\) are adjusted according to Equation (38), the closed-loop system signal is bounded and the tracking error asymptotically converges to zero._  
\[\left\{ \begin{array}{l}\dot{\mathbf{\theta}}_{f}=r_{1}s\xi(x)\\ \dot{\mathbf{\theta}}_{g}=r_{1}s\xi(x)u\end{array} \right. \tag{38}\]_The optimal parameters of the fuzzy system can be defined as follows:_  
\[\left\{ \begin{array}{l}\mathbf{\theta}_{f}^{*}=\arg\min_{\mathbf{\theta}_{f}\in\mathbf{ \Omega}_{f}}\left[\sup_{\mathbf{x}\in\ \mathbb{R}^{n}}\left|\hat{f}(\mathbf{x}|\mathbf{\theta}_{f})-\hat{f}(\mathbf{x},t)\right| \right]\\ \mathbf{\theta}_{g}^{*}=\arg\min_{\mathbf{\theta}_{g}\in\ \mathbf{\Omega}_{g}}\left[\sup_{\mathbf{x}\in \ \mathbb{R}^{n}}\left|\hat{f}(\mathbf{x}|\mathbf{\theta}_{g})-\hat{f}(\mathbf{x},t)\right| \right]\end{array} \right. \tag{39}\]  
_where \(\mathbf{\Omega}_{f}\) and \(\mathbf{\Omega}_{g}\) are the constraint sets of \(\mathbf{\theta}_{f}\) and \(\mathbf{\theta}_{g}\), respectively._  
The minimum approximation error can be defined as follows:  
\[\left\{ \begin{array}{l}\omega=f(\mathbf{x},t)-\hat{f}(\mathbf{x}|\ \mathbf{\theta}_{f}^{*})+(g(\mathbf{x},t)-\hat{g}(\mathbf{x}|\ \mathbf{\theta}_{g}^{*}))u\\ \dot{s}=c_{1}\dot{e}+\ddot{e}=c_{1}\dot{e}+\ddot{x}-\ddot{x}_{\rm d}=f(\mathbf{x}, t)+g(\mathbf{x},t)u-\varsigma(\mathbf{x},t)\end{array} \right. \tag{40}\]  
Substituting Equation (37) into Equation (40), Equation (41) can be obtained. \[\dot{s} =\hat{f}(\mathbf{x}|\ \mathbf{\theta}_{f}^{*})-f(\mathbf{x},t)+(\hat{g}(\mathbf{x}| \ \mathbf{\theta}_{g}^{*})-\hat{g}(\mathbf{x},t))u-k\ \mathrm{sgn}\,s-\tau s+\omega \tag{41}\] \[=f(\mathbf{x},t)+[g(\mathbf{x},t)+\hat{g}(\mathbf{x}|\ \mathbf{\theta}_{g}^{*})]u+ \hat{g}(\mathbf{x}|\ \mathbf{\theta}_{g}^{*})[\beta^{-1}(\mathbf{x}|\ \mathbf{\theta}_{g}^{*}) \cdot((-\hat{f}(\mathbf{x}|\ \mathbf{\theta}_{f}^{*})+R)]-\varsigma(\mathbf{x},t)\] \[=[f(\mathbf{x},t)-\hat{f}(\mathbf{x}|\ \mathbf{\theta}_{f}^{*})]+[g(\mathbf{x},t)+ \hat{g}(\mathbf{x}|\ \mathbf{\theta}_{g}^{*})]u+R-\varsigma(\mathbf{x},t)\] \[=[f(\mathbf{x},t)-\hat{f}(\mathbf{x}|\ \mathbf{\theta}_{f}^{*})]+[g(\mathbf{x},t)+ \hat{g}(\mathbf{x}|\ \mathbf{\theta}_{g}^{*})]u-k\ \mathrm{sgn}\,s-\tau s\]  
According to Equation (40) and Equation (41), Equation (42) can be obtained,  
\[\dot{s} =\hat{f}(\mathbf{x}|\ \mathbf{\theta}_{f}^{*})-f(\mathbf{x},t)+(\hat{g}(\mathbf{x}| \ \mathbf{\theta}_{g}^{*})-\hat{g}(\mathbf{x},t))u-k\ \mathrm{sgn}\,s-\tau s+\omega \tag{42}\] \[=\mathbf{\varphi}_{f}^{\mathrm{T}}\varsigma(\mathbf{x})+\mathbf{\varphi}_{f}^ {\mathrm{T}}\varsigma(\mathbf{x})u-k\ \mathrm{sgn}\,s-\tau s+\omega\]  
where \(\mathbf{\varphi}_{f}=\mathbf{\theta}_{f}^{*}-\mathbf{\theta}_{f}\) and \(\mathbf{\varphi}_{g}=\mathbf{\theta}_{g}^{*}-\mathbf{\theta}_{g}\).