# Comparative Real-Time Study of Three Enhanced Control Strategies Applied to Dynamic Process Systems
## 4 Proposed Controller Method for Coupled Tank System  
### Adaptive Model-Based PID-Type Sliding Mode Control (APIDSMC)  
This section presents the adaptive model-based PID-type SMC (APIDSMC) control system. The purpose of this technique is to provide robust trajectory tracking in the presence of uncertainties and external disturbances. Figure 5 shows the PID-SMC method block diagram. This figure compares the reference liquid level trajectory with the coupled tank system's real-time output to calculate the execution error. In this case, we will use a computed error in the sliding surface function (S) and feed this function into the PID-SMC. In the present scenario, the state variables may be precisely described as follows:  
\[\dot{q}_{1}(t)=q(t)=L_{1}(t), \tag{31}\]  
\[\dot{q}_{3}(t)=q_{4}(t)=L_{2}(t) \tag{32}\]  
Therefore, by applying the inverse Laplace transformation to Equations (17) and (18), we may derive the differential equation of the system in the following manner:  
\[\ddot{q}_{1}(t)=-\frac{1}{\tau_{1}}\dot{q}_{1}(t)+\frac{K_{1}}{\tau_{1}}u_{2_ {1}}(t), \tag{33}\]  
\[\ddot{q}_{3}(t)=-\frac{1}{\tau_{2}}\dot{q}_{3}(t)+\frac{K_{2}}{\tau_{2}}u_{2_ {2}}(t) \tag{34}\]  
The outputs of the process are determined by the variables \(q_{2}(t)\) and \(q_{4}(t)\). The control input of Tank 1 is denoted as \(u_{2_{1}}(t)\), while the control inputs of Tank 2 are represented by \(u_{2_{2}}(t)=L_{1}(t)\). Therefore, it is possible to derive the second-order equation of motion in matrix form as follows:  
\[\ddot{q}_{2x1}=f(\mathbf{q})_{2x1}+g(\mathbf{q})_{2y2}\mathbf{u}(\mathbf{t})_{2x1}+\zeta(\mathbf{ t},\mathbf{u}(t))_{2x1} \tag{35}\]  
Figure 5: The PID-SMC block diagram for industrial process system. Let the bounded lumped uncertainty be expressed as \(\xi(t,\mathbf{u}(t))_{2x1}\). Furthermore, the tracking error \(\mathbf{e}(\mathbf{t})_{2x1}\) is provided as follows:  
\[\mathbf{e}(\mathbf{t})_{2x1}=\mathbf{q}_{d2x1}-\mathbf{q}_{2x1} \tag{36}\]  
The APIDSMC control approach is illustrated in Figure 6. 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 process system. Subsequently, the error was utilized by the sliding surface function to calculate the S parameter. The resulting output of the sliding surface function was then inputted into the adaptive model-based PID-type sliding mode controller. The APIDSMC produces the control action as its output. The developed APIDSMC controller is conceptually separated into three subsections to facilitate comprehension of the control approach. The initial section can be represented as the phase control law, the subsequent section establishes the control of the switching feedback law, and ultimately, an adaptive control law is provided to estimate the gain parameters of the switching controller. The APIDSMC approach utilizes a PID-type sliding surface to enhance the resilience and efficiency of the system. The associated surface function is defined as  
\[\dot{\mathbf{s}}(\mathbf{t})_{2x1}=\mathbf{k}_{p_{2x2}}\mathbf{e}(\mathbf{t})_{2x1}+\mathbf{k}_{i_{2x2 }}\mathbf{e}(\mathbf{t})_{2x1}+\mathbf{k}_{d_{2x2}}\dot{\mathbf{e}}(\mathbf{t})_{2x1} \tag{37}\]  
where \(\mathbf{k}_{p_{2x2}}\), \(\mathbf{k}_{d_{2x2}}\), and \(\mathbf{k}_{i_{2x2}}\) are the gain matrices for positive control.