## 3 Global Stability Analysis
### Stability Analysis for the Second and Third Tanks
\begin{array}{c}g_{x2}-max\{\delta_{2}\}for\ x_{2}>x_{2}^{u}\\ 0\ for\ \ x_{2}\in\ \left[x_{2}^{l},\ x_{2}^{u}\right]\\ g_{x2}-min\{\delta_{2}\}for\ x_{2}<x_{2}^{l}\end{array} \right.. \tag{87}\]  
The main properties of \(g_{2v}\) are:  
\[(Pi)\ g_{2v}\ \textit{is continuous with respect to}\ x_{2}, \tag{88}\]  
\[(Pi)\ g_{2v}=0\ for\ x_{2}\ \in\ \left[x_{2}^{l},\ x_{2}^{u}\right], \tag{89}\]  
\[(Piii)\ g_{2v}\
eq 0\ for\ x_{2}\ \
otin\ \left[x_{2}^{l},\ x_{2}^{u}\right], \tag{90}\]  
\[(Piv)\ sgn(g_{2v})=sgn(x_{2})\
eq 0\ for\ x_{2}\ \
otin\ \left[x_{2}^{l},\ x_{2}^{u} \right]. \tag{91}\]  
From Equations (85) and (86), it follows that:  
\[f_{v2}(g_{x2}-\delta_{2})=|f_{v2}||g_{x2}-\delta_{2}|\geq|f_{v2}g_{2v}|\geq 0. \tag{92}\]  
From the definitions of \(g_{2v}\) (87) and \(f_{v2}\) (75) it follows that:  
\[|g_{2v}|\geq|f_{v2}|. \tag{93}\]  
Therefore, Equation (92) yields:  
\[f_{v2}(g_{x2}-\delta_{2})\geq f_{v2}^{2}\geq 0. \tag{94}\]  
Using this, Equation (62), yields:  
\[\frac{dV_{2}}{dt}+\frac{d}{dt}\bigg{(}\frac{v_{2}}{v_{2}}V_{3}\bigg{)}\leq(-1 )\frac{Q_{D}}{v_{2}}(x_{2}-x_{3})(f_{v2}-f_{v3})+(-1)\frac{Q}{v_{2}}f_{v2}^{2} +f_{v2}\frac{Q}{v_{2}}f_{v1}. \tag{95}\]  
\(\square\)  
**Task 5. ** Definition of gradient \(dV_{3}/dx_{3}\) and dead zone quadratic form \(V_{3}\). We choose \(f_{v3}\) with the structure of \(f_{v2}\):  
\[f_{v3}=\left\{ \begin{array}{c}x_{3}-x_{3}^{u}\ for\ \ x_{3}>x_{3}^{u}>0\\ 0\ for\ \ x_{3}\ \in\ \left[x_{3}^{l},\ \ x_{3}^{u}\right]\\ x_{3}-x_{3}^{l}\ for\ \ x_{3}<x_{3}^{l}<0\end{array} \right.. \tag{96}\]  
where the bounds \(x_{3}^{u}\), \(x_{3}^{l}\) are defined as:  
\[x_{3}^{u}=x_{2}^{u};\ x_{3}^{l}=x_{2}^{l}\, \tag{97}\]and \(x_{2}^{l}\), \(x_{2}^{u}\) are defined in Equations (70) and (71). A Lyapunov function that satisfies the gradient \(dV_{3}/dx_{3}=f_{v3}\) (Equation (54)) is:  
\[V_{3}=\frac{1}{2}f_{v3}^{2} \tag{98}\]  
The main properties of \(V_{3}\) are:  
\[V_{3}>0\;for\;\;x_{3}\;\
ot\in\;\;\left[x_{3}^{l},\;\;x_{3}^{u}\right], \tag{99}\]  
\[V_{3}=0\;for\;x_{3}\in\;\left[x_{3}^{l},\;\;x_{3}^{u}\right], \tag{100}\]  
\[V_{3}\;\text{is continuous with respect to }x_{3}. \tag{101}\]  
As a consequence of the definitions of \(f_{v2}\) (75) and \(f_{v3}\) (96), the property:  
\[(x_{2}-x_{3})(f_{v2}-f_{v3})\geq 0, \tag{102}\]  
holds true, as proved in Appendix A. Equation (95) can be arranged as:  
\[\tfrac{dV_{3}}{dt}+\tfrac{d}{dt}\big{(}\tfrac{v_{1}}{v_{2}}V_{3}\big{)}\leq(-1 )\tfrac{Q_{1}}{v_{2}}(x_{2}-x_{3})(f_{v2}-f_{v3})-\beta_{1}\tfrac{Q}{v_{2}}f_ {v2}^{2}+(-1)\beta_{2}\tfrac{Q}{v_{2}}f_{v2}^{2}+f_{v2}\tfrac{Q}{v_{2}}f_{v1}, \tag{103}\]  
where \(\beta_{1}\), \(\beta_{2}\) are positive constants that satisfy:  
\[1=\beta_{1}+\beta_{2},\;\;\beta_{1}\in(0,\;1),\;\;\beta_{2}\in(0,\;1). \tag{104}\]  
\(\square\)  
**Task 6. ** Definition of the overall Lyapunov function \(V\) and determination of the non-positive nature of the expression for \(dV/dt=dV_{1}/dt+kdV_{2}/dt+dV_{3}/dt\). The term \((-1)\beta_{2}\tfrac{Q}{v_{2}}f_{v2}^{2}+f_{v2}\tfrac{Q}{v_{2}}f_{v1}\) appearing in Equation (103) can be expressed in terms of \(f_{v1}^{2}\):  
\[(-1)\beta_{2}\frac{Q}{v_{2}}f_{v2}^{2}+f_{v2}\frac{Q}{v_{2}}f_{v1}=(-1)\beta_ {2}\frac{Q}{v_{2}}\bigg{(}f_{v2}-\frac{1}{2\beta_{2}}f_{v1}\bigg{)}^{2}+\frac{ 1}{4\beta_{2}}\frac{Q}{v_{2}}f_{v1}^{2}\leq\frac{1}{4\beta_{2}}\frac{Q}{v_{2} }f_{v1}^{2}. \tag{105}\]  
Substituting into Equation (103), we have:  
\[\frac{dV_{2}}{dt}+\frac{d}{dt}\bigg{(}\frac{v_{3}}{v_{2}}V_{3}\bigg{)}\leq(-1 )\frac{Q_{D}}{v_{2}}(x_{2}-x_{3})(f_{v2}-f_{v3})+(-1)\beta_{1}\frac{Q}{v_{2}} f_{v2}^{2}+\frac{1}{4\beta_{2}}\frac{Q}{v_{2}}f_{v1}^{2}. \tag{106}\]  
We tackle the effect of the \(f_{v1}^{2}\) term by incorporating the \(dV_{1}/dt\) expression.