## 2 Problem Formulation
: _A nonlinear function \(\eta(\cdot)\) and real matrices \(m_{1}\) and \(m_{2}\) are given, and \(m_{2}-m_{1}>0\)._  
_If the particular condition_  
\[(\eta(\chi)-m_{1}\chi)^{\mathrm{T}}(\eta(\chi)-m_{2}\chi)\leq 0, \tag{4}\]  
_holds, then \(\eta(\cdot)\) is defined such that it satisfies the sector-bounded condition and belongs to the sector \(m_{1}\) and \(m_{2}\)._According to Definition 2 and Figure 1, the real matrices \(\tilde{U}_{1}\) and \(\tilde{U}\) are given and \(0\leq\tilde{U}_{1}<1\leq\tilde{U}\), which guarantees  
\[\Big{(}\Gamma_{t}(\tilde{\phi}_{t,u})-\tilde{U}_{1}\tilde{\phi}_{t,u}\Big{)}^{ \mathsf{T}}\Big{(}\Gamma_{t}(\tilde{\phi}_{t,u})-\tilde{U}\tilde{\phi}_{t,u} \Big{)}\leq 0, \tag{5}\]  
where \(\Gamma_{t}(\cdot)\) is the sector-bounded nonlinear function, which belongs to \([\tilde{U}_{1},\tilde{U}]\). In practical engineering, there generally exists a vector \(\phi_{t,\max}\geq\tilde{\eta}_{t,u}\). The saturation function \(\Gamma_{t}(\cdot)\) can be decomposed into two components: the linear and nonlinear parts, that is,  
\[\Gamma_{t}(\tilde{\mathcal{C}}\delta_{u})=\tilde{U}_{1}\tilde{\mathcal{C}} \delta_{u}+\theta(\tilde{\mathcal{C}}\delta_{u}), \tag{6}\]  
where \(\theta(\tilde{\mathcal{C}}\delta_{u})\) is the sector-bounded nonlinear function belonging to \([0,\tilde{U}_{2}]\), in which \(\tilde{U}_{2}\triangleq\tilde{U}-\tilde{U}_{1}\). Then, \(\theta(\tilde{\mathcal{C}}\delta_{u})\) satisfies:  
\[\theta^{\mathsf{T}}(\tilde{\mathcal{C}}\delta_{u})\Big{(}\theta(\tilde{ \mathcal{C}}\delta_{u})-\tilde{U}_{2}(\tilde{\mathcal{C}}\delta_{u})\Big{)} \leq 0. \tag{7}\]  
Next, according to [2], we can obtain the state-dependent functions:  
\[a_{i}(\tilde{\sigma}_{i,u})= \begin{cases}\widetilde{s}_{i}\mid\tilde{\sigma}_ {i,u}\rvert>r_{i}\\ \widetilde{s}_{i}\mid\tilde{\sigma}_{i,u}\rvert\leq r_{i}\end{cases} ,\,b_{ij}( \widetilde{\sigma}_{i,u})=\begin{cases}\widetilde{b}_{ij}\mid\tilde{\sigma}_ {i,u}\rvert>r_{i  
We can obtain that \(\bar{A}(\bar{\delta}_{u})\in[A^{-},A^{+}]\), \(\bar{B}(\bar{\delta}_{u})\in[\ \&\ ^{-},\ \&\ ^{+}]\), \(\bar{D}(\bar{\delta}_{u})\in[v^{-},v^{+}]\) and \(\bar{E}(\bar{\delta}_{u})\in[E^{-},E^{+}]\). Next, \(\bar{A}\), \(\bar{B}\), \(\bar{D}\) and \(\bar{E}\) are defined as follows:  
\[\bar{A}\triangleq(A^{+}+A^{-})/2,\ \ \bar{B}\triangleq(\ \&\ ^{+}+\ \&\ ^{-})/2,\]  
\[\bar{D}\triangleq(v^{+}+v^{-})/2,\ \ \bar{E}\triangleq(E^{+}+E^{-})/2.\]  
Furthermore, the matrices \(\bar{A}(\bar{\delta}_{u})\), \(\bar{B}(\bar{\delta}_{u})\), \(\bar{D}(\bar{\delta}_{u})\) and \(\bar{E}(\bar{\delta}_{u})\) can be denoted as  
\[\bar{A}(\bar{\delta}_{u})=\bar{A}+\Delta\bar{A}_{u},\ \bar{B}(\bar{\delta}_{u})= \bar{B}+\Delta\bar{B}_{u},\]  
\[\bar{D}(\bar{\delta}_{u})=\bar{D}+\Delta\bar{D}_{u},\ \bar{E}(\bar{\delta}_{u})= \bar{E}+\Delta\bar{E}_{u},\]  
where \(\Delta\bar{A}_{u}=\sum_{i=1}^{n_{x}}e_{i}\delta_{i,i}e_{i}^{\mathsf{T}}\), \(\Delta\bar{B}_{u}=\sum_{i,j=1}^{n_{x}}e_{i}e_{j,i}e_{j,i}^{\mathsf{T}}\), \(\Delta\bar{D}_{u}=\sum_{i,j=1}^{n_{x}}e_{i}p_{ij,i}e_{j}^{\mathsf{T}}\), \(\Delta\bar{E}_{u}=\sum_{i,j=1}^{n_{x}}e_{i}r_{ij,i}e_{j}^{\mathsf{T}}\), \(e_{k}\in R^{n_{x}}\) is the column vector with the k-th element being 1 and others being 0, \(\delta_{i,u}\), \(t_{ij,u}\), \(p_{ij,u}\) and \(r_{ij,u}\) are unknown scalars and the unknown parameter matrices \(\Delta\bar{A}_{u}\), \(\Delta\bar{B}_{u}\), \(\Delta\bar{D}_{u}\) and \(\Delta\bar{E}_{u}\) can be represented as follows:  
\[\Delta\bar{A}_{u}=\bar{T}F_{1,u}\bar{X}_{1}\in[-(A^{+}-A^{-})/2 \,\ (A^{+}-A^{-})/2\ ], \tag{8}\] \[\Delta\bar{B}_{u}=\bar{T}F_{2,u}\bar{X}_{2}\in[-(\ \&\ ^{+}-\ \&\ ^{-})/2\,\ (\&\ ^{+}-\ \&\ ^{-})/2\ ],\] \[\Delta\bar{D}_{u}=\bar{T}F_{3,u}\bar{X}_{3}\in[-(v^{+}-v^{-})/2 \,\ (v^{+}-v^{-})/2\ ],\] \[\Delta\bar{E}_{u}=\bar{T}F_{4,u}\bar{X}_{4}\in[-(E^{+}-E^{-})/2 \,\ (E^{+}-E^{-})/2\ ],\]  
where \(\bar{T}\) and \(\bar{X}_{i}\) (\(i=1,\ 2,\ 3,\ 4\)) are known real-valued matrices.