# 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)
Upon differentiating Equation (37) with regard to time, the subsequent equation is derived:  
\[\ddot{\mathbf{s}}(\mathbf{t})_{2x1}=\mathbf{k}_{p_{2x2}}\dot{\mathbf{e}}(\mathbf{t})_{2x1}+\mathbf{k}_ {i_{2x2}}\mathbf{e}(\mathbf{t})_{2x1}+\mathbf{k}_{d_{2x2}}\ddot{\mathbf{e}}(\mathbf{t})_{2x1} \tag{38}\]  
By employing the two-time derivative of the error function, Equation (38) can be reformulated as follows:  
\[\dot{q}_{3}(t)=q_{4}(t)=L_{2}(t) \tag{32}\]0  
By substituting the value of \(\ddot{\mathbf{q}}_{2x1}\) from Equation (35) into Equation (39), we obtain  
\[\dot{q}_{3}(t)=q_{4}(t)=L_{2}(t) \tag{32}\]1  
Given that \(\mathbf{s}(\mathbf{t})_{2x1}\), \(\dot{\mathbf{s}}(\mathbf{t})_{2x1}\), and \(\ddot{\mathbf{s}}(\mathbf{t})_{2x1}\) are all null under the second-order sliding surface condition, it follows that the tracking error \(\mathbf{e}(\mathbf{t})_{2x1}\) becomes zero. Hence, by utilizing the \(\ddot{\mathbf{s}}(\mathbf{t})_{2x1}=0\), \(\mathbf{u}_{\mathbf{p}_{2x1}}\) (the reaching phase control law) is calculated with the following uncertainties:  
\[\dot{q}_{3}(t)=q_{4}(t)=L_{2}(t) \tag{32}\]2  
\[\dot{q}_{3}(t)=q_{4}(t)=L_{2}(t) \tag{32}\]3 The APIDSMC block diagram for industrial process system. Nevertheless, considering the uncertainties that are limited but unknown, it is not appropriate to regulate the system merely with the reaching phase control law. This may give rise to significant complications during the functioning of the process system. Hence, to enhance the dependability of the system against external or system disruptions, it is necessary to include the switching control law, \(\mathbf{u}_{\mathbf{sc_{22\text{t}}}}\), expressed by Equation (42) into the control signal. \[\dot{q}_{3}(t)=q_{4}(t)=L_{2}(t) \tag{32}\]4  
Let \(\lambda_{2\text{x2}}\) and \(\mu_{2\text{x2}}\) represent the switching gains. Therefore, the feedback control law, denoted as \((\mathbf{u}_{total_{2\text{x1}}})\), for the system without uncertainty may be expressed as  
\[ \begin{split}\mathbf{u}_{total_{2\text{x1}}}=& g(\mathbf{x})_{2\text{x2}}^{-1}(\bar{q}_{d2\text{x1}}-f(\mathbf{x})_{2 \text{x1}}+\zeta(t,\mathbf{u}(\mathbf{t}))_{2\text{x1}}+\frac{k_{2\text{x2}}}{k_{d_{2 \text{x2}}}}\dot{\mathbf{e}}(\mathbf{t})_{2\text{x1}}+\frac{k_{2\text{x2}}}{k_{d_{2 \text{x2}}}}e(\mathbf{t})_{2\text{x1}}\\ &+\lambda_{2\text{x2}}\mathbf{s}(\mathbf{t})_{2\text{x1}}+\mu_{2\text{x2 }}\mathbf{sign}\big{(}\dot{\mathbf{s}}(\mathbf{t})_{2\text{x1}}\big{)})\end{split}  \tag{43}\]  
In Equation (43), \(\mathbf{g}(\mathbf{x})\) is typically a matrix or function that describes how control inputs influence the system's states. If \(g(\mathbf{x})\) were to be zero (or non-invertible), it would be impossible to apply the necessary control input to achieve the desired system behaviour, as there would be no well-defined way to map the control law to the actual control signal. Essentially, \(g(\mathbf{x})\) being zero would mean that the system cannot respond to control inputs, resulting in no control authority over the system. Also, control systems often rely on feedback mechanisms to maintain stability. If the inverse of \(g(\mathbf{x})\) is not well-defined (i.e., if \(g(\mathbf{x})\) = 0 or is singular), it could result in instability or uncontrollable oscillations in the system. This is particularly critical in systems where precise control over external disturbances and uncertainties is required, as is evident in the sliding mode control method discussed in the paper. Sliding mode controllers, such as those described in the equations, are highly sensitive to discontinuities and rapid changes in the control law. A zero or undefined \(g(\mathbf{x})\) would introduce significant challenges in adapting the system's switching gains, leading to potential chattering issues that are difficult to mitigate. Thus, ensuring that the inverse of \(g(\mathbf{x})\) is non-zero is crucial for guaranteeing that the control input has the intended effect on the system and for maintaining the stability and reliability of the controller.