# Robust Controller Decorated by Nonlinear S Function and Its Application to Water Tank
## 3 Controller Design
\tag{24}\]  
Then, \(\omega\) is a bounded interference term and \(\left\|\omega\right\|_{\infty}\leq\rho\). \[ \begin{array}{c}\dot{V}(x)=-\big{(}1-{T_{1}}^{2}\big{)}{x_{1}}^{2}-\big{(}1 -{T_{1}}^{2}\big{)}{x_{2}}^{2}+\frac{w^{2}}{2}\\ \leq-\big{(}1-{T_{1}}^{2}\big{)}{x_{1}}^{2}-\big{(}1-k^{2}\big{)}{x_{2}}^{2}+ \frac{\rho^{2}}{2}\end{array}  \tag{25}\]  
To define \(a=\max(T_{1},k)\), then  
\[\dot{V}(x)\leq-\Big{(}1-a^{2}\Big{)}\textbf{X}^{T}\textbf{X}+\frac{\rho^{2}}{2} \tag{26}\]  
Formula (26), indicated that when state variable of the system \(\left\|\textbf{X}\right\|_{\infty}>\rho/\sqrt{2(1-a^{2})}\), then \(\dot{V}(x)<0\). For further discussing the robustness for the external distraction, \(L_{2}\) which is the performance index of gain robust is interpreted as  
\[\int_{0}^{t}\left\|\textbf{X}\right\|^{2}\text{d}t\leq\mu_{1}\int_{0}^{t}w^{2} (t)\text{d}t+\mu_{2} \tag{27}\]  
where \(\mu_{1}\) and \(\mu_{2}\) are positive and small, and the following theorem is obtained. Theorem: in the controller design for the water tank level control system, as shown in Formula (14) by applying the closed-loop gain algorithm, and the parameter \(T_{1}\) of controller satisfying the condition of \(0<T_{1}\leq 1\), the state variable of the system \(\|\mathbf{X}\|_{\infty}>\rho/\sqrt{2(1-a^{2})}\) ensures the whole controller uniform ultimate boundness and also achieves the performance index of gain robust \(L_{2}\), which is related to the controller parameter \(T_{1}\). This theorem can be proved by integrating Formula (26) from \(t=0\) to \(t=t_{0}\), to obtain  
\[V(t_{0})+(1-a^{2})\int_{0}^{t_{0}}\mathbf{X}^{\mathsf{T}}\mathbf{X}\mathrm{d}t -\frac{1}{2}\int_{0}^{t_{0}}w(t)^{\mathsf{T}}w(t)\mathrm{d}t\leq V(0) \tag{28}\]  
On the basis of the definition of, Formula (28) can be derived as  
\[\int_{0}^{t_{0}}\|\mathbf{X}\|^{2}\mathrm{d}t\leq\frac{1}{2(1-a^{2})}\int_{0}^ {t_{0}}w^{2}(t)\mathrm{d}t+\frac{1}{1-a^{2}}\varepsilon \tag{29}\]  
After discussing the bounds of \(a\) as shown in Formula (29), \(a\) can be stated as [22]  
\[a= \begin{cases}T_{1},&T_{0}\geq 1\\ \frac{T_{1}(1+T_{0}T_{1})}{T_{1}+T_{0}},&0<T_{0}<1\end{cases}  \tag{30}\]  
The formula \(1-a^{2}>0\) is a requisite, for the purpose of keeping the negative definite of Formula (26), and then the range of controller parameters is derived according to Formula (30)  
\[0<T_{1}\leq 1 \tag{31}\]  
To summarize, based on the result of the Formula (27) about the performance index of gain robust and Formula (29), when the state variable of water tank control system satisfies the requirement of \(\|\mathbf{X}\|_{\infty}>\rho/\sqrt{2(1-a^{2})}\), if the parameter \(T_{1}\) of the controller of closed-loop gain control law can meet \(0<T_{1}\leq 1\), then the water tank control system shown in Formula (14) will obtain the performance index of gain robust \(L_{2}\), as shown in Formula (27), which is \(1/2(1-a^{2})\) related to the parameter of controller parameter \(T_{1}\). If the S function is applied to decorate the output \((u)\) of the existing controller, then  
\[K=\frac{1-e^{-\gamma u}}{1+e^{-\gamma u}} \tag{32}\]  
Based on the Taylor series expansion, the Formula (33) can be obtain  
\[e^{-\gamma u}\approx 1-\gamma u \tag{33}\]  
Then  
\[\frac{1-\mathrm{e}^{-\gamma u}}{1+\mathrm{e}^{-\gamma u}}\approx\frac{\gamma u }{2-\gamma u} \tag{34}\]  
When \(u\) is not too large, then  
\[\frac{1-\mathrm{e}^{-\gamma u}}{1+\mathrm{e}^{-\gamma u}}\approx 0.5\gamma u \tag{35}\]  
When \(\gamma=1.5\), the above derivation is still valid. The above proof is based on the reference [14] using the \(L_{2}\) gain robustness performance index. In the actual simulation, it is found that the requirement \(0<T_{1}<1\) is conservative and the \(H_{\infty}\) robust theorem is used to re-prove. According to Formulas (11) and (14), if the output matrix \(\mathbf{C}=\left[ \begin{array}{cc}0&1\end{array}