## 3 Global Stability Analysis
### Stability Analysis for the First Tank  
The stability analysis is based on determining the non-positive nature of \(dV_{1}/dt\), where \(V_{1}\) is the dead-zone quadratic form for \(x_{1}\). The function \(V_{1}\) and its gradient \(dV_{1}/dx_{1}\) are defined so as to achieve this goal. The procedure comprises the following tasks: (i) determination of the expression for \(dV_{1}/dt\); (ii) definition of the gradient \(dV_{1}/dx_{1}\) and the dead zone quadratic form \(V_{1}\) for the state \(x_{1}\); (iii) arrangement of the \(dV_{1}/dt\) expression in terms of a non-positive function of \(dV_{1}/dx_{1}\); (iv) integration of the \(dV_{1}/dt\) expression. The definition of the gradient \(dV_{1}/dx_{1}\) and \(V_{1}\) requires the determination of the \(dV_{1}/dt\) expression and the properties of the terms involved. **Theorem 1**. : _Consider the model in Equations (11) and (12), subject to assumptions 1 to 3. (Ti) The state \(x_{1}\) converges asymptotically to \(\Omega_{x1}\), \(\Omega_{x1}=\left\{x_{1}:x_{1}^{l}\leq x_{1}\leq x_{1}^{u}\right\}\), where \(x_{1}^{u}=\left\{x_{1}:g_{x1}-max\{\delta_{sin}\}=0\right\}\), \(x_{1}^{u}>0\), \(x_{1}^{l}=\left\{x_{1}:g_{x1}-min\{\delta_{sin}\}=0\right\}\), \(x_{1}^{l}<0\), and \(g_{x1}=x_{1}+\frac{r_{x1}}{D}\), and \(S_{1}\) converges asymptotically to \(\Omega_{S1}\) where \(\Omega_{S1}=\left[S_{1}^{l}\ \ S_{1}^{u}\right]\), \(S_{1}^{l}=x_{1}^{l}+S_{1}^{cq}\), \(S_{1}^{u}=x_{1}^{u}+S_{1}^{cq}\) and \(S_{1}^{cq}\) is provided by Equation (5). (Tii) The sets \(\Omega_{x1}\) and \(\Omega_{S1}\) are invariant._  
**Proof. ****Task 1. ** Determination of the \(dV_{1}/dt\) expression. The time derivative of \(V_{1}\) can be expressed as:  
\[\frac{dV_{1}}{dt}=f_{v1}\frac{dx_{1}}{dt}, \tag{16}\]  
\[f_{v1}=\frac{dv_{1}}{dx_{1}}. \tag{17}\]  
Incorporating the \(x_{1}\) dynamics (11) and arranging, yields:  
\[\frac{dV_{1}}{dt}=(-1)Df_{v1}(g_{x1}-\delta_{sin}), \tag{18}\]  
where,  
\[g_{x1}=x_{1}+\frac{r_{x1}}{D}, \tag{19}\]  
\[\frac{dg_{x1}}{dx_{1}}=1+\frac{1}{D}\frac{dr_{x1}}{dx_{1}}\geq 1. \tag{20}\]  
\(\square\)  
**Task 2. ** Definition of the gradient \(dV_{1}/dx_{1}\) and the dead zone quadratic form \(V_{1}\) for the state \(x_{1}\). At what follows, we examine the properties of the term \(g_{x1}-\delta_{sin}\) appearing in Equation (18). From Equations (12), (19) and (20) and Assumption 3 we have:  
\[\frac{dg_{x1}}{dx_{1}}\geq 1;\ g_{x1}=0\ for\ x_{1}=0;\ sgn(g_{x1})=sgn(x_{1}), \tag{21}\]  
\[g_{x1}-\delta_{sin}>0\ \ \ for\ x_{1}>x_{1}^{u}, \tag{22}\]  
\[g_{x1}-\delta_{sin}<0\ \ \ for\ x_{1}<x_{1}^{l}, \tag{23}\]  
where,  
\[x_{1}^{u}=\{x_{1}:g_{x1}-max\{\delta_{sin}\}=0\ \},x_{1}^{u}>0, \tag{24}\]  
\[x_{1}^{l}=\{x_{1}:g_{x1}-min\{\delta_{sin}\}=0\ \},\ x_{1}^{l}<0. \tag{25}\]  
Therefore,  
\[f_{v1}=\frac{dv_{1}}{dx_{1}}. \tag{17}\]0To obtain non-positive \(dV_{1}/dt\) in Equation (18), we need to choose \(f_{v1}\) such that:  
\[f_{v1}=\frac{dv_{1}}{dx_{1}}. \tag{17}\]1  
\[f_{v1}=\frac{dv_{1}}{dx_{1}}. \tag{17}\]2  
In view of properties (26) and to fulfill conditions (27) and (28), we choose:  
\[f_{v1}=\left\{ \begin{array}{c}x_{1}-x_{1}^{u}\ for\ x_{1}>x_{1}^{u}\\ 0\ for\ \ x_{1}\ \in\ \Big{[}x_{1}^{l},x_{1}^{u}\Big{]}.\\ x_{1}-x_{1}^{l}\ for\ x_{1}<x_{1}^{l}\end{array} \right.