## 2 Systems Description  
### Matrix Second-Order System  
Consider a non-linear system, whose dynamics are given by  
\[\ddot{\mathbf{x}}(t)+\mathbf{F}(\dot{\mathbf{x}},\mathbf{x})\dot{\mathbf{x}}(t)+\mathbf{G}(\mathbf{x})=\bm {H}(\mathbf{x})\mathbf{u}(t)+\mathbf{z}(\ddot{\mathbf{x}},\dot{\mathbf{x}},\mathbf{x},t)\,, \tag{1}\]  
where \(\mathbf{x}(t)\in\mathbb{X}\subset\mathbb{R}^{n}\) represents the internal state of the system, \(\mathbf{u}(t)\in\mathbb{U}\subset\mathbb{R}^{r}\) is the input state, \(\mathbf{z}(\ddot{\mathbf{x}},\dot{\mathbf{x}},\mathbf{x},t)\in\mathbb{R}^{n}\) is a noise disturbance, \(t>0\) is the time, \(\mathbf{F}:\mathbb{R}^{n}\times\mathbb{R}^{n}\supset\Omega\times\Omega\to \mathbb{R}^{n\times n}\), \(\mathbf{G}:\mathbb{R}^{n}\supset\Omega\to\mathbb{R}^{n}\), \(\mathbf{H}:\mathbb{R}^{n}\supset\Omega\to\mathbb{R}^{n\times r}\), \(\Omega\subset\mathbb{R}^{n}\) is a neighborhood of zero (\(\mathbf{0}\in\mathbb{R}^{n}\)), \(\mathbb{X}\) is the internal state space, \(\mathbb{U}\) is the input state space, \(\mathbb{R}^{n}\) and \(\mathbb{R}^{r}\) are vector spaces of column vectors with real elements, and \(\mathbb{R}^{n\times n}\) (\(\mathbb{R}^{n\times r}\)) stands for a matrix with \(n\) rows and \(n\) columns (\(n\) rows and \(r\) columns), \(n\in\mathbb{N}\), \(r\in\mathbb{N}\). The functions \(\mathbf{F}\), \(\mathbf{G}\), and \(\mathbf{H}\) have the following form:  
\[\mathbf{F}(\mathbf{\tilde{\mathbf{x}}},\mathbf{\eta})=\begin{bmatrix}f_{11}(\mathbf{\tilde{\mathbf{x} }},\mathbf{\eta})&f_{12}(\mathbf{\tilde{\mathbf{x}}},\mathbf{\eta})&\ldots&f_{1n}(\mathbf{\tilde{ \mathbf{x}}},\mathbf{\eta})\\ f_{21}(\mathbf{\tilde{\mathbf{x}}},\mathbf{\eta})&f_{22}(\mathbf{\tilde{\mathbf{x}}},\mathbf{\eta})& \ldots&f_{2n}(\mathbf{\tilde{\mathbf{x}}},\mathbf{\eta})\\ \vdots&\vdots&\ddots&\vdots\\ f_{n1}(\mathbf{\tilde{\mathbf{x}}},\mathbf{\eta})&f_{n2}(\mathbf{\tilde{\mathbf{x}}},\mathbf{\eta})& \ldots&f_{nn}(\mathbf{\tilde{\mathbf{x}}},\mathbf{\eta})\end{bmatrix}\,, \tag{2}\]  
\[\mathbf{G}(\mathbf{\tilde{\mathbf{x}}})=\begin{bmatrix}g_{1}(\mathbf{\tilde{\mathbf{x}}})&g_{2}( \mathbf{\tilde{\mathbf{x}}})&\ldots&g_{n}(\mathbf{\tilde{\mathbf{x}}})\end{bmatrix}^{\rm T}\,, \tag{3}\]  
\[\mathbf{H}(\mathbf{\tilde{\mathbf{x}}})=\begin{bmatrix}h_{11}(\mathbf{\tilde{\mathbf{x}}})&h_{12} (\mathbf{\tilde{\mathbf{x}}})&\ldots&h_{1r}(\mathbf{\tilde{\mathbf{x}}})\\ h_{21}(\mathbf{\tilde{\mathbf{x}}})&h_{22}(\mathbf{\tilde{\mathbf{x}}})&\ldots&h_{2r}(\mathbf{ \tilde{\mathbf{x}}})\\ \vdots&\vdots&\ddots&\vdots\\ h_{n1}(\mathbf{\tilde{\mathbf{x}}})&h_{n2}(\mathbf{\tilde{\mathbf{x}}})&\ldots&h_{nr}(\mathbf{ \tilde{\mathbf{x}}})\end{bmatrix}\,. \tag{4}\]  
The system starts from an initial condition, denoted as  
\[\dot{\mathbf{x}}(0)=\mathbf{x}_{01}\in\mathbb{R}^{n}\,,\quad\mathbf{x}(0)=\mathbf{x}_{02}\in \mathbb{R}^{n}\,. \tag{5}\]  
**Assumption 1**.: _The functions \(f_{ij}(\cdot\,,\cdot)\), \(g_{i}(\cdot)\), and \(h_{ik}(\cdot)\), \(i,j=1,2,\ldots,n\), \(k=1,2,\ldots,r\) and their derivatives, with respect to each variable in the set \(\Omega\), are considered to be continuous._  
**Assumption 2**.: _The function \(\mathbf{z}(\ddot{\mathbf{x}},\dot{\mathbf{x}},\mathbf{x},t)\) represents an external noise disturbance, which is unknown but fulfils the following condition:_  
\[\mathbf{z}^{\rm max}=\max_{\ddot{\mathbf{x}},\dot{\mathbf{x}},\mathbf{x},t}\mathbf{z}(\ddot{\mathbf{x} },\dot{\mathbf{x}},\mathbf{x},t)\geq 0. \tag{6}\]  
**Assumption 3**.: _The matrix \(\mathbf{H}(\mathbf{x})\) is non-singular for every \(\mathbf{x}\in\Omega\)._