## 2 Methodology
### Proposed Controller Design
#### 2.4.2 Stability Analysis
For nonpositive values of \(\cos\theta\) we have that  
\[\dot{\mathbf{e}}=\mathbf{A}\mathbf{e}+\mathbf{b}[L\mathbf{\xi}_{1}{}^{T}\tilde{\mathbf{ \theta}}_{f_{1}}+\mathbf{\xi}_{2}{}^{T}\tilde{\mathbf{\theta}}_{f_{2}}+L\mathbf{\eta}_{1}{}^ {T}\tilde{\mathbf{\theta}}_{g_{1}}+\mathbf{\eta}_{2}{}^{T}\tilde{\mathbf{\theta}}_{g_{2}}\]5  
On the other hand, for positive values of \(\cos\theta\), \(\kappa\) bounds the following expression: \(\kappa\geq ml^{2}-mLl\), then by selecting the design parameter  
\[\dot{\mathbf{e}}=\mathbf{A}\mathbf{e}+\mathbf{b}[L\mathbf{\xi}_{1}{}^{T}\tilde{\mathbf{ \theta}}_{f_{1}}+\mathbf{\xi}_{2}{}^{T}\tilde{\mathbf{\theta}}_{f_{2}}+L\mathbf{\eta}_{1}{}^ {T}\tilde{\mathbf{\theta}}_{g_{1}}+\mathbf{\eta}_{2}{}^{T}\tilde{\mathbf{\theta}}_{g_{2}}\]6  
we guarantee that \(\kappa\geq ml^{2}-mLl\geq 0\ \forall\ \theta\in\mathbb{R}\), which implies also that  
\[\dot{\mathbf{e}}=\mathbf{A}\mathbf{e}+\mathbf{b}[L\mathbf{\xi}_{1}{}^{T}\tilde{\mathbf{ \theta}}_{f_{1}}+\mathbf{\xi}_{2}{}^{T}\tilde{\mathbf{\theta}}_{f_{2}}+L\mathbf{\eta}_{1}{}^ {T}\tilde{\mathbf{\theta}}_{g_{1}}+\mathbf{\eta}_{2}{}^{T}\tilde{\mathbf{\theta}}_{g_{2}}\]7  
Now we examine the third term of the right-hand side of (34). Note that the control objective implies \(\theta_{d}=0\), then \(\tilde{y}=x_{d}-x-L\theta\) and therefore \(e^{T}\mathbf{Pbk_{d}}^{T}\tilde{\mathbf{x}}\) can be expressed as  
\[\dot{\mathbf{e}}=\mathbf{A}\mathbf{e}+\mathbf{b}[L\mathbf{\xi}_{1}{}^{T}\tilde{\mathbf{ \theta}}_{f_{1}}+\mathbf{\xi}_{2}{}^{T}\tilde{\mathbf{\theta}}_{f_{2}}+L\mathbf{\eta}_{1}{}^ {T}\tilde{\mathbf{\theta}}_{g_{1}}+\mathbf{\eta}_{2}{}^{T}\tilde{\mathbf{\theta}}_{g_{2}}\]8  
where \(\mathbf{M}\in\mathbb{R}^{4\times 4}\) is given by  
\[\mathbf{M}=\left[ \begin{array}{ccc}-L&0&1&0\\ 0&-L&0&1\end{array} \right]^{T}\mathbf{Pbk_{d}}^{T}\]  
Let us define  
\[\mathbf{B}=\frac{\mathbf{M}+\mathbf{M}^{T}}{2}\]  
Then, \(\mathbf{M}\) is a positive semidefinite, i.e., \(\mathbf{M}\geq 0\) if and only if all principal minors \(\Delta_{i}\) (\(i=1,\cdots,4\)) of \(\mathbf{B}\) are nonnegative [25]. Principal minors of \(\mathbf{B}\) are the followings:  
\[\Delta_{1}=\det[b_{11}]=-Lk_{d_{1}}p_{12},\]  
and by selecting \(k_{d_{1}}<0\) we have that \(\Delta_{1}>0\). The second principal minor is computed as  
\[\Delta_{2}=\frac{-L^{2}k_{d_{1}}^{2}p_{22}^{2}}{4}+\frac{L^{2}k_{d_{1}}k_{d_{ 2}}p_{12}p_{22}}{2}-\frac{L^{2}k_{d_{2}}^{2}p_{12}^{2}}{4}.\]  
Then selecting \(k_{d_{2}}<0\) and \(k_{d_{1}}p_{22}=k_{d_{2}}p_{12}\) we ensure that \(\Delta_{2}=0\). Finally we have that  
\[\Delta_{3}=0\]  
and  
\[\Delta_{4}=0.\]  
Under above considerations it may be concluded that  
\[e^{T}\mathbf{Pbk_{d}}^{T}\tilde{\mathbf{x}}=\tilde{\mathbf{x}}^{T}\mathbf{M}\tilde{\mathbf{x}}\geq 0 \tag{39}\]By combining results (37) and (39) it is achieved that  
\[-G(\mathbf{x})\mathbf{e}^{T}\mathbf{Pbk_{d}}^{T}\mathbf{\tilde{x}}\leq 0 \tag{40}\]  
Next, the Rayleigh-Ritz Theorem is invoked to bound the first term on the right-hand side of (34) as  
\[-\mathbf{e}^{T}\mathbf{Q}\mathbf{e}\leq-\lambda_{m}\{\mathbf{Q}\}\|\mathbf{e}\|^{2} \tag{41}\]  
where \(\lambda_{m}\{\mathbf{Q}\}\) denotes the smallest eigenvalue of \(\mathbf{Q}\).