# Comparative Real-Time Study of Three Enhanced Control Strategies Applied to Dynamic Process Systems
## 2 Dynamic Model of Liquid Level and System Description  
Figure 1 depicts the closed recirculating test system used for a connected tank. The system comprises two interconnected liquid tanks, an electrically operated pump, a level monitoring aperture at the bottom of each tank, and a pressure sensor. Tank 1 is supplied by the electric pump, and Figure 1 shows how both liquid tanks are mounted on the front plate. Tank 2 receives the outflow from Tank 1. Two pressure-sensitive sensors located at the base of each tank measure the liquid levels in both reservoirs. The mathematical basis of the interconnected tank system can be found in [41]. The rate of change of the liquid level in each separate reservoir is as follows:  
\[\dot{L}_{1}(t)=\frac{1}{A_{1}}\big{(}F_{IN_{1}}-F_{OUT_{1}}\big{)}, \tag{1}\]  
The variable \(L_{i}(t)\) represents the liquid level in the reservoir, measured in centimeters. \(A_{i}\) represents the cross-sectional area of the reservoir, measured in square centimetres. \(F_{IN_{i}}\) and \(F_{OUT_{i}}\) represent the inflow and outflow rates of the \(i\)-th tank, measured in cubic centimetres per second. The rate at which liquid flows into tank 1 is precisely proportional to the voltage that is applied to the pump. This means that as the voltage increases, the inflow rate into the tank also increases. \[F_{IN_{1}}=K_{p}V_{p}(t), \tag{2}\]  
The constant \(K_{p}\) represents the pump's conversion factor in cubic centimetres per volt-second (cm\({}^{3}\)/Vs). The control input, \(V_{p}(t)\), also known as the voltage pump \(u(t)\) in volts (\(Volt\)), is utilized to attain the desired levels in either tank 1 or tank 2, as follows:  
\[u(t)=u_{1}(t)+u_{2}(t), \tag{3}\]  
The contributions of the cascade rule and the feedforward action are denoted by \(u_{1}(t)\) and \(u_{2}(t)\), respectively, in Equation (3). \[v_{OUT_{i}}(t)=\sqrt{2 gL_{i}(t)}, \tag{4}\]  
Moreover, Equation (4) provides the outflow velocity from the orifice at the bottom of each unique tank using Bernoulli's law. \[F_{OUT_{i}}(t)=a_{i}\sqrt{2 gL_{i}(t)}, \tag{5}\]  
Next, using Equation (5), the outflow percentage of a separate liquid tank is calculated, where \(a_{i}\) is the cross-sectional area of the outflow orifice (cm\({}^{2}\)) at the bottom of the \(i\)-th tank, and \(g\) is the gravitational acceleration (981 cm/s\({}^{2}\)). Tank 2's inflow is determined as follows:  
\[F_{IN_{2}}(t)=F_{OUT_{1}}(t), \tag{6}\]  
Figure 1: The coupled-tank recirculating system. Thus, the following mathematical expression for the liquid level in the connected tanks can be produced by applying the mass balance principle and Equations (1)-(6):  
\[\dot{L}_{1}(t)=\frac{K_{p}u(t)-a_{1}\sqrt{2 gL_{1}(t)}}{A_{1}}, \tag{7}\]  
\[\dot{L}_{2}(t)=\frac{a_{1}}{A_{2}}\sqrt{2 gL_{1}(t)}-\frac{a_{2}}{A_{2}}\sqrt{2 gL _{2}(t)}, \tag{8}\]  
The steady-state pump voltage, \(u_{1}(t)\), that leads to the steady-state level, \(L_{10}\), in reservoir 1 is determined by using Equation (7) when the system is at equilibrium (\(\dot{L}_{1}(t)=0\)). Equation (8) is employed to determine the value of \(L_{10}\) in reservoir 1 when the system is at equilibrium (\(\dot{L}_{2}(t)=0\)). This value represents the desired constant level in reservoir 2, referred to as the planned steady-state level. \[u_{1}(t)=a_{1}\frac{\sqrt{2 gL_{10}}}{K_{p}}, \tag{9}\]  
\[F_{IN_{1}}=K_{p}V_{p}(t), \tag{2}\]0  
The relationships are specified in Equations (9) and (10). Deviation variables can be defined by utilizing the operational range that corresponds to small departure voltages (\(u_{1}(t)\)) and heights (\(L_{1}(t),\ L_{2}(t)\)):  
\[ \begin{array}{l}L_{11}(t)=L_{1}(t)-L_{10},\\ L_{21}(t)=L_{2}(t)-L_{20},\end{array}  \tag{11}\]  
The cascade control is denoted by \(u_{2}(t)\) in Equation (11).