# 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)  
In this part, initial information regarding the FO calculus is presented. The Riemann-Liouville definition is the most used among the fractional derivative definitions. We defined the integral and fractional derivative of the \(\alpha\)-th, order of function f(t), in the time domain, as shown in [43]. \[\mathcal{D}^{\alpha}f(t)=\frac{d^{\alpha}f(t)}{dt^{\alpha}}=\frac{1}{\Gamma(n- \alpha)}\bigg{(}\frac{d^{n}}{dt^{n}}\bigg{)}\int\limits_{0}^{t}\frac{f(\tau) }{(t-\tau)^{\alpha+1-n}}d\tau, \tag{48}\]  
\[\mathcal{D}^{-\alpha}f(t)=\mathcal{L}^{\alpha}f(t)=\frac{1}{\Gamma(\alpha)} \int\limits_{0}^{t}\frac{f(\tau)}{(t-\tau)^{1-\alpha}}d\tau \tag{49}\]  
where \(n-1<\alpha<n\), and \(n\) denotes an integer value; the fractional derivative and integral are shown by \(D^{n}\) and \(\mathcal{L}^{n}\), respectively. [PERSON]'s Gamma function, \(\Gamma(. )\), can be shown by  
\[\Gamma(\lambda)=\int\limits_{0}^{\infty}e^{-t}t^{\alpha-1}dt \tag{50}\]  
To express the \(n\)th order derivative \((d^{n}/dt^{n})\) of the FO derivative operator, \(\mathcal{D}^{\alpha}f(t)\), is turned into the following:  
\[\frac{d^{n}}{dt^{n}}(\mathcal{D}^{\alpha}f(t))=\mathcal{D}^{\alpha}\bigg{(} \frac{d^{n}f(t)}{dt^{n}}\bigg{)}=\mathcal{D}^{\alpha+n}f(t) \tag{51}\]  
This section introduces a novel adaptive model-based PID-type SMC control technique that is specifically developed to achieve reliable trajectory tracking even in the presence of uncertainties and external disturbances. The control approach being presented is illustrated in Figure 7. This figure presents the reference trajectory as the intended trajectory for the experimental application. An analysis will be conducted to identify any errors in the implementation by comparing the output of the reference trajectory with the actual trajectory of the industrial process system. Subsequently, the error introduced by the sliding surface function was utilized to determine the S parameter. The resulting output of the sliding surface function was then inputted into the PID-type sliding mode controller based on the FO-adaptive model. As the control action, the output of the FO-APIDSMC is provided. Figure 7: The proposed FO-APIDSMC block diagram for wheeled mobile robot. In order to comprehend the intended controller, the FO-APIDSMC control approach is formally segmented into three subsections. The initial section can be represented as the phase control law. The subsequent section suggests the implementation of the switching feedback law to address the finite uncertainty and disturbances in the system. Lastly, an adaptation component of the control law is executed to estimate the gain parameters of the switching controller. In order to enhance the efficiency and robustness of the FO-APIDSMC approach, the PID-type sliding surface is utilized. The surface function is defined as follows:  
\[\mathbf{s}(\mathbf{t})_{2\texttt{x}\texttt{t}}=\mathbf{k}_{p_{2\texttt{x} \texttt{t}}}\mathbf{e}(\mathbf{t})_{2\texttt{x}\texttt{t}}+\mathbf{k}_{i_{2\texttt{x} \texttt{t}}}\mathbf{\int}\mathbf{e}(\mathbf{\tau})_{2\texttt{x}\texttt{t}}d\mathbf{\tau}+\mathbf{ k}_{d_{2\texttt{x}\texttt{t}}}\dot{\mathbf{e}}(\mathbf{t})_{2\texttt{x}\texttt{t}} \tag{52}\]  
Let \(\mathbf{k}_{p_{2\texttt{x}\texttt{t}}},\mathbf{k}_{d_{2\texttt{x}\texttt{t}}}\), and \(\mathbf{k}_{i_{2\texttt{x}\texttt{t}}}\) represent the positive control gain matrix. Given the PID-type sliding surface function in Equation (52), an FO-PID-type sliding surface is developed utilizing an FO calculus to achieve accurate, resilient, and robust control performance and rapid finite-time convergence for an industrial process system.