# Comparative Real-Time Study of Three Enhanced Control Strategies Applied to Dynamic Process Systems
## 4 Proposed Controller Method for Coupled Tank System
### Fractional Order Adaptive Model-Based PID-Type Sliding Mode Control (FO-APIDSMC)
\[\mathbf{s}(\mathbf{t})_{2\texttt{x}\texttt{t}}=\mathbf{k}_{p_{2\texttt{x} \texttt{t}}}\mathbf{e}(\mathbf{t})_{2\texttt{x}\texttt{t}}+\mathbf{k}_{i_{2\texttt{x} \texttt{t}}}\mathbf{\mathcal{D}}^{-\alpha}\big{\{}\mathbf{e}(\mathbf{t})_{2\texttt{x} \texttt{t}}\big{\}}+\mathbf{k}_{d_{2\texttt{x}\texttt{t}}}\mathbf{\int}\mathbf{e}(\mathbf{t})_ {2\texttt{x}\texttt{t}}\big{\}} \tag{53}\]  
If we take the derivative of Equation (53) twice in time, we obtain Equation (54) below:  
\[\tilde{\mathbf{s}}(\mathbf{t})_{2\texttt{x}\texttt{t}}=\mathbf{k}_{p_{2\texttt{x} \texttt{t}}}\tilde{\mathbf{e}}\big{(}\mathbf{t})_{2\texttt{x}\texttt{t}}+\mathbf{k}_{i_{2 \texttt{x}\texttt{t}}}\mathbf{\mathcal{D}}^{2-\alpha}\big{\{}\mathbf{e}(\mathbf{t})_{2 \texttt{x}\texttt{t}}\big{\}}+\mathbf{k}_{d_{2\texttt{x}\texttt{t}}}\mathbf{\int}\mathbf{e }(\mathbf{t})_{2\texttt{x}\texttt{t}}\big{\}} \tag{54}\]  
By employing the two-times derivative of the error function, Equation (54) can be reformulated as follows:  
\[\tilde{\mathbf{s}}(\mathbf{t})_{2\texttt{x}\texttt{t}}=\mathbf{k}_{p_{2\texttt{x} \texttt{t}}}\big{\{}\tilde{\mathbf{q}}_{d_{2\texttt{x}\texttt{t}}}-\tilde{\mathbf{q}}_{2 \texttt{x}\texttt{t}}\big{\}}+\mathbf{k}_{i_{2\texttt{x}\texttt{t}}}\mathbf{\mathcal{D }}^{2-\alpha}\Big{\{}\mathbf{e}(\mathbf{t})_{2\texttt{x}\texttt{t}}\big{\}}+\mathbf{k}_{d_{ 2\texttt{x}\texttt{t}}}\mathbf{\mathcal{D}}^{\alpha+2}\big{\{}\mathbf{e}(\mathbf{t})_{2 \texttt{x}\texttt{t}}\big{\}} \tag{55}\]  
By substituting the value of \(\tilde{\mathbf{q}}_{2\texttt{x}\texttt{t}}\) from Equation (35) into Equation (55), we obtain  
\[\mathcal{D}^{-\alpha}f(t)=\mathcal{L}^{\alpha}f(t)=\frac{1}{\Gamma(\alpha)} \int\limits_{0}^{t}\frac{f(\tau)}{(t-\tau)^{1-\alpha}}d\tau \tag{49}\]0  
When the values of \(\mathbf{s}(\mathbf{t})_{2\texttt{x}\texttt{t}}\), \(\tilde{\mathbf{s}}(\mathbf{t})_{2\texttt{x}\texttt{t}}\), and \(\tilde{\mathbf{s}}(\mathbf{t})_{2\texttt{x}\texttt{t}}\) are all equal to zero in the second-order sliding surface condition, the tracking error \(\mathbf{e}(\mathbf{t})_{2\texttt{x}\texttt{t}}\) becomes zero. Hence, by utilizing the \(\tilde{\mathbf{s}}(\mathbf{t})_{2\texttt{x}\texttt{t}}=0\), the \(\mathbf{u}_{\eta\texttt{p}_{2\texttt{x}\texttt{t}}}\) (the reaching phase control law) is calculated with the following uncertainties:  
\[\begin{split}\mathbf{u}_{\eta\texttt{p}_{2\texttt{x} \texttt{t}}}=&\,g(\mathbf{x})_{2\texttt{x}\texttt{t}}{}^{-1}(\tilde{ \mathbf{q}}_{d_{2\texttt{x}\texttt{t}}1}-\mathbf{\zeta}(\mathbf{t},\mathbf{u}(\mathbf{t}))_{2\texttt{ x}\texttt{t}}+\frac{\mathbf{k}_{d_{2\texttt{x}\texttt{t}}}}{k_{p_{2\texttt{x} \texttt{t}}}}\mathbf{\mathcal{D}}^{\alpha+2}\big{\{}\mathbf{e}(\mathbf{t})_{2\texttt{x} \texttt{t}}\big{\}}\\ &\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,By applying the adaptation law, the controller function of FO-APIDSMC can be reformulated as follows:  
\[ \begin{split}\mathbf{u}_{total_{2\text{xt}}}&=\mathbf{g}(x)_{2 \text{x}2}{}^{-1}(\ddot{q}_{d2\text{xt}}-\mathbf{f}(x)_{2\text{xt}}-\xi(t,\mathbf{u}(t)) _{2\text{xt}}+\frac{k_{\text{zx}2}}{k_{\text{zx}2}}\mathbf{\mathcal{D}}^{\mathbf{\alpha} +2}\{\mathbf{e}(t)_{2\text{xt}}\}\\ &\quad+\frac{k_{\text{zx}2}}{k_{\text{zx}2}}\mathbf{\mathcal{D}}^{2- \mathbf{\alpha}}\{\mathbf{e}(t)_{2\text{xt}}\})+\mathbf{g}(x)_{2\text{xt}}^{-1}\Big{\{} \lambda_{2\text{x}2}\mathbf{s}(t)_{2\text{xt}}+\hat{\mu}_{2\text{x}2}\mathbf{s}\mathbf{n} \big{(}\hat{\mathbf{s}}(t)_{2\text{xt}}\big{)}\Big{\}}\end{split}  \tag{60}\]  
The chattering effect can be amplified by the signum function by increasing the switching gain value, as expressed in Equation (60). To avoid this scenario, the function (\(\mathbf{u}_{total_{2\text{xt}}}\)) is defined as follows:  
\[