# A Water Tank Level Control System with Time Lag Using CGSA and Nonlinear Switch Decoration
## 3 Control System Structure Design
### Improved Non-Linear Switch Modification  
The non-linear modifier switching technique is essentially a segmentation function. In this paper, the segmentation function is constructed using the arcsine function in the following form.  
\[f(u)=\begin{cases}\alpha a\sin(\beta u),&u\leq 2\\ u,&u>2\end{cases} \tag{9}\]  
where \(\alpha\beta<1\) and \(\beta\
eq 0,\;\alpha\
eq 0\).  
It indicates that the system output differs significantly from the set output when \(u\) is large. It is often the initial stage of the control process or a situation where a major disturbance occurs. At this time, the arcsine non-linear modification does not work, maintaining the dynamic performance of the control law (8). It means that the difference between the system output and the set output is little when \(u\) is small. It is often the stable phase of the control process or a situation where small disturbances occur. Now the arcsine non-linear modification takes effect, making the control input smaller with little impact on  
Figure 2: Typical S&T singular value curve.  
the dynamic performance of the control law (8) and thus reducing energy consumption. Since the dynamic performance of the control law (8) remains unchanged for \(u>2\). So, we list the following analysis for the case where \(u\leq 2\).  
When \(u\leq 2\), we choose to retain the first-order Taylor expansion of the arcsine function.  
\[\alpha a\sin(\beta u)\approx\alpha\beta u \tag{10}\]  
(1) Effect on the steady state of the system  
Assume that the input is a unit step signal and then analyze the output steady-state values using a modified model of the control object. According to the final value theorem of the Rasch transform, we obtain Equation (11).  
\[\begin{array}{ll}C(\infty)&=\underset{s\to 0}{\lim}\frac{GK\alpha\beta}{1+GK \alpha\beta}\frac{1}{s}\\ &=\underset{s\to 0}{\lim}\frac{\underset{s\to 0}{\lim}\frac{GK\alpha\beta}{1+GK \alpha\beta}\frac{s}{s}}{1+\underset{s\to 0}{\lim}\frac{GK\alpha\beta}{1+GK \alpha\beta}\frac{s}{s}}\\ &=\underset{s\to 0}{\lim}\frac{GK\alpha\beta}{1+GK\alpha\beta}\frac{s}{s} \end{array} \tag{11}\]  
Therefore, the non-linear switching modification of the arcsine function does not affect the final steady state of the system.  
(2) Effect on the dynamic performance  
\[\frac{C(s)}{R(s)}=\frac{GK\alpha\beta}{1+GK\alpha\beta} \tag{12}\]  
Equation (12) is the transfer function of the closed-loop system. According to the closed-loop gain-shaping theory, the open-loop transfer function \(GK\) of the system meets the requirements of high gain at low frequencies and low gain at high frequencies when \(\alpha\beta<1\). Therefore, in the low-frequency range of Equation (12) compared with the standard feedback system \(GK/(1+GK)\), adding \(\alpha\beta\) has little effect on the dynamic performance of the system.  
(3) Effect on controller output  
\[\frac{u}{u_{d}}=\frac{K\alpha\beta}{1+GK\alpha\beta} \tag{13}\]  
Equation (13) is the transfer function from the input to the controller output. The numerator of Equation (13) decreases more significantly than the denominator. So \(\alpha\beta\) will reduce the control output. Non-linear switching technology is the introduction of \(\alpha\beta\) to reduce energy consumption during the stabilization phase of the control, but at the same time to reduce the output.