## 2 Methodology
### Proposed Controller Design
#### 2.4.2 Stability Analysis  
Let \(\mathbf{\theta}_{\hat{f}_{1}}^{*}\in\mathbb{R}^{\prod_{j=1}^{2}p_{ij}}\) and \(\mathbf{\theta}_{g_{i}}^{*}\in\mathbb{R}^{\prod_{j=1}^{2}p_{ij}}\) (\(i=1,2\)) be the vector of optimal parameters such that the fuzzy systems  
\[\hat{f}_{1}^{*}(\theta,\dot{\theta}\,|\,\mathbf{\theta}_{f_{1}}^{*}) =\mathbf{\zeta}_{1}{}^{T}\mathbf{\theta}_{f_{1}}^{*},\ \ \hat{f}_{2}^{*}(x,\dot{x}\,|\,\mathbf{\theta}_{f_{2}}^{*})=\mathbf{\zeta}_{2}{}^{T}\mathbf{ \theta}_{f_{2}}^{*}\] \[\hat{g}_{1}^{*}(\theta,\dot{\theta}\,|\,\mathbf{\theta}_{g_{1}}^{*}) =\mathbf{\eta}_{1}{}^{T}\mathbf{\theta}_{g_{1}}^{*},\ \ \hat{g}_{2}^{*}(x,\dot{x}\,|\,\mathbf{\theta}_{g_{2}}^{*})=\mathbf{\eta}_{2}{}^{T}\mathbf{ \theta}_{g_{2}}^{*}\]  
are the best approximators of their corresponding nominal functions \(f_{1},f_{2},g_{1}\) and \(g_{2}\). Then, we define the minimum approximation error as  
\[w=[\hat{F}^{*}-F]+[\hat{G}^{*}-G]u+[G-\hat{G}]\mathbf{k}_{d}{}^{T}\mathbf{\tilde{x}}\]  
where \(\hat{F}^{*}=L\hat{f}_{1}^{*}(\theta,\dot{\theta}\,|\,\mathbf{\theta}_{f_{1}}^{*}) +\hat{f}_{2}^{*}(x,\dot{x}\,|\,\mathbf{\theta}_{f_{2}}^{*})\) and \(\hat{G}^{*}=L\hat{g}_{1}^{*}(\theta,\dot{\theta}\,|\,\mathbf{\theta}_{g_{1}}^{*}) +\hat{g}_{2}^{*}(x,\dot{x}\,|\,\mathbf{\theta}_{g_{2}}^{*})\). By using \(w\) we can now express (27) as  
\[\dot{\mathbf{e}}=\mathbf{A}\mathbf{e}+\mathbf{b}[L(\hat{f}_{1}-\hat{f}_{1}^{*})+( \hat{f}_{2}-\hat{f}_{2}^{*})+L(\hat{g}_{1}-\hat{g}_{1}^{*})\] \[\qquad\qquad+(\hat{g}_{2}-\hat{g}_{2}^{*})-G\mathbf{k}_{d}{}^{T}\mathbf{ \tilde{x}}+w] \tag{29}\]  
or well  
\[\dot{\mathbf{e}}= \mathbf{A}\mathbf{e}+\mathbf{b}\left[L(\mathbf{\xi}_{1}{}^{T}\mathbf{\theta}_{f_{1}} -\mathbf{\xi}_{1}{}^{T}\mathbf{\theta}_{f_{1}}^{*})\right.\] \[\left.+(\mathbf{\zeta}_{2}{}^{T}\mathbf{\theta}_{f_{2}}-\mathbf{\xi}_{2}{}^{T }\mathbf{\theta}_{f_{2}}^{*})+L(\mathbf{\eta}_{1}{}^{T}\mathbf{\theta}_{g_{1}}-\mathbf{\eta}_{ 1}{}^{T}\mathbf{\theta}_{g_{1}}^{*})\right.\] \[\left.+(\mathbf{\eta}_{2}{}^{T}\mathbf{\theta}_{g_{2}}-\mathbf{\eta}_{2}{}^{ T}\mathbf{\theta}_{g_{2}}^{*})-G\mathbf{k}_{d}{}^{T}\mathbf{\tilde{x}}+w\right] \tag{30}\]  
Finally, let us define  
\[\tilde{\mathbf{\theta}}=\left[ \begin{array}{cccc}\tilde{\mathbf{\theta}}_{f_{1}}& \tilde{\mathbf{\theta}}_{f_{2}}&\tilde{\mathbf{\theta}}_{g_{1}}&\tilde{\mathbf{\theta}}_{ g_{2}}\end{array} \right]^{T} \tag{31}\]  
Figure 5: Block diagram of the proposed controller. where every element \(\tilde{\mathbf{\theta}}_{(\cdot)}=\mathbf{\theta}_{(\cdot)}-\mathbf{\theta}_{(\cdot)}^{*}\). Then, by considering these elements into (30), we can obtain a closed-loop dynamic equation given by  
\[\dot{\mathbf{e}}=\mathbf{A}\mathbf{e}+\mathbf{b}[L\mathbf{\xi}_{1}{}^{T}\tilde{\mathbf{ \theta}}_{f_{1}}+\mathbf{\xi}_{2}{}^{T}\tilde{\mathbf{\theta}}_{f_{2}}+L\mathbf{\eta}_{1}{}^ {T}\tilde{\mathbf{\theta}}_{g_{1}}+\mathbf{\eta}_{2}{}^{T}\tilde{\mathbf{\theta}}_{g_{2}}\] \[\qquad\qquad-GK_{\mathbf{d}}{}^{T}\tilde{\mathbf{x}}+w] \tag{32}\]  
which specifies explicitly the relationship between the generalized error \(\mathbf{e}\) and the fuzzy parameters estimation error \(\tilde{\mathbf{\theta}}\). In order to achieve the analysis of boundedness of the solutions, the following Lyapunov function candidate is proposed  
\[V(\mathbf{e},\tilde{\mathbf{\theta}})=\frac{1}{2}\mathbf{e}^{T}\mathbf{P}\mathbf{e}+\frac{1}{2} \tilde{\mathbf{\theta}}^{T}\Gamma^{-1}\tilde{\mathbf{\theta}} \tag{33}\]  
which is globally positive definite and radially unbounded since \(\Gamma\in\mathbb{R}^{4\times 4}\) is a diagonal matrix of which non-zero elements \(\gamma_{k}\) (\(k=1,\cdots,4\)) are positive constants also known as adaptive gains, \(\mathbf{P}\in\mathbb{R}^{2\times 2}\) is a positive definite matrix.