## 3 Controller Design for WWTP
### Fast Integral Terminal Sliding Mode Surface  
There are two phases to the SMC design. Creating the sliding mode surface is the first step in forcing the system state to reach the equilibrium point. Creating the reaching law is the second step in making sure the system's motion trajectory is driven to and maintained on the sliding mode surface. The robustness of the system during the arrival stage cannot be guaranteed by a standard SMC. To solve this problem, integral sliding mode control (ISMC) is introduced to eliminate the reaching phase. Describing the tracking error of the system \(e=y_{d}-y\), where \(y_{d}\) is the desired output, \(y\) is the system output, then ISMC's sliding mode surface is created as:  
\[s(t)=e(t)+k\int_{0}^{t}e(\tau)d\tau \tag{9}\]  
where \(k>0\) is a coefficient that requires designing. Assuming the initial tracking error is \(e(0)\), when the integral in Equation (9) has the initial value \(-e(0)/k\), \(s(t)\) starts at zero and stays there for the duration of the system response. On the manifold \(s(t)=0\) of ISMC, we have  
\[e(t)=e(0)\exp(-kt) \tag{10}\]  
which suggests that there is no guarantee that the tracking error will converge to zero in a finite amount of time. To solve this problem, the ISMC is combined with the terminal sliding mode control (TSMC) which makes finite-time tracking possible. The sliding mode surface of the integral terminal sliding mode control (ITSMC) is intended to be:  
\[s(t)=e(t)+\alpha\int_{0}^{t}e^{p_{1}/p_{2}}(\tau)d\tau \tag{11}\]  
where \(\alpha>0\) is a coefficient that requires designing, \(p_{1}\) and \(p_{2}\) are positive odd integers satisfying \(p_{2}>p_{1}>0\). For Equation (11), define the integral term \(e_{I}(t)=\int_{0}^{t}e^{p_{1}/p_{2}}(\tau)d\tau\), and ensure \(e_{I}(0)=-e(0)/\alpha\), when \(s(t)=0\) (i.e., \(e(t)=-\alpha e_{I}(t)\)), we have  
\[\dot{e}_{I}(t)=-\alpha^{p1/p2}e_{I}^{p1/p2}(t) \tag{12}\]  
The convergent time of \(e_{I}(t)\) can be obtained by solving the error dynamic Equation (12) as follows:  
\[T_{f1}=\frac{|e_{I}(0)|^{1-p1/p2}}{\alpha^{p1/p2}(1-p1/p2)}=\frac{|e(0)|^{1-p_ {1}/p_{2}}}{\alpha(1-p_{1}/p_{2})} \tag{13}\]  
which indicates that in a limited amount of time \(T_{f1}\), the tracking error, can converge to zero. Since \(p1/p2<1\), when far away from the equilibrium point, i.e., \(|e_{I}(t)|\gg 1\), the value of \(\left|\dot{e}_{I}(t)\right|\) will significantly decrease according to Equation (12); thus, the convergence speed of \(e_{I}(t)\) will be greatly reduced. To solve this problem, according to the research of [PERSON] [26], a FITMS with fast convergence properties is intended to be:  
\[s(t)=e(t)+\int_{0}^{t}\Bigl{[}\alpha|e(\tau)|^{p1/p_{2}}\text{sgn}(e(\tau))+ \beta\text{sgn}(e(\tau))\Bigr{]}d\tau \tag{14}\]  
where \(e(t)\), \(\alpha\), \(p_{1}\), and \(p_{2}\) have the same definition as in Equation (11), and \(\beta>0\) is a coefficient that requires designing. The difference between the fractional order term in the integral term and Equation (11) is to ensure that when the error is negative, the calculation result is a real number. It is essentially the same as Equation (11). **Theorem 1**. : _For the WWTP system shown in Equation (8), using FITMS based on Equation (14), when the tracking error is on the sliding mode surface, the tracking error can converge to zero in a finite time to make the WWTP system stable._  
**Proof of Theorem 1.