# Quadratic Model-Based Dynamically Updated PID Control of CSTR System with Varying Parameters +
## 3 Results
### Simulation Results and Discussion  
In this section, the control performances of the 1D and 2D DUPID controllers are compared against a benchmark PID controller. The benchmark PID controller is tuned by trial and error, as it is carried out in numerous industrial plants. To support this fact, in [25] the authors are referring to a statistic indicating that 30% of the PID controllers in the industry operate in manual mode and require continuous fine-tuning and supervising by the process technologist. Additionally, in [26], it is reported that 25% of the PID applications use coefficients that are pre-set by the manufacturer with no update of their values with respect to the particular process. Therefore, to showcase a realistic scenario, the parameters of the PID controller are tuned by trial and error assuming non-varying plant parameters. The PID controller parameters were assumed to be constant over the plant operation and they were selected so that the plant output achieves smooth transient response and zero \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(q\) & 100 L/min & \(hA_{ta}\) & 5\(\cdot 10^{4}\) J/min K \\ \(V\) & 100 L & \(T_{f}\) & 350 K \\ \(\rho\) & 1000 g/L & \(C_{Af}\) & 0.5 mol/L \\ \(C_{p}\) & 0.239 J/g K & \(T_{c}(0)\) & 300 K \\ \(\Delta H\) & 5\(\cdot 10^{4}\) J/mol & \(C_{A}(0)\) & \(\approx\)0.46 mol/L \\ \(E/R\) & 8750 K & \(T(0)\) & \(\approx\)318.9 K \\ \(k_{0}\) & 7.2\(\cdot 10^{10}\) 1/min & \([T_{c,\;min},\;T_{c,\;max}]\) & [250, 350] K \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The considered nominal values for the parameters of the CSTR plant. steady-state error, without breaching the bounds of the interval of admissible control values. As a quantifying measure for the performance assessment of the controllers, the Integral of Absolute Error (IAE) metric is used (Equation (16)). \[IAE=\frac{1}{i_{max}}\sum_{i=1}^{i_{max}}\Bigl{|}e^{(i)}\Bigr{|}. \tag{16}\]  
For the sake of relevant comparison, three scenarios were addressed when the controllers were applied to control the CSTR system. These three scenarios are defined as follows:  
* In the first scenario (Scenario 1), it is assumed that the heat transfer coefficient linearly drops to 50% of its initial value. Therefore, the value of \(\alpha_{h}\) was assumed to be 0.0167. * In the second scenario (Scenario 2), it is assumed that the feed temperature \(T_{f}\) linearly increases for \(T_{d}=+40\)\(K\) of its initial value, \(T_{fconst}=350\)\(K\). The value of \(lb\) is 30. * In the third scenario (Scenario 3), it is assumed that the feed temperature \(T_{f}\) linearly drops for \(T_{d}=-40\)\(K\) of its initial value \(T_{fconst}=350\)\(K\). The value of \(lb\) is 30. The PID parameters were tuned to be \(K_{p}=4.5\), \(K_{i}=3.31\) and \(K_{d}=0.01\). All DUPID parameters are pre-set and are constant in all scenarios. The DUPID parameters in both versions are assumed to be constant in all scenarios, \(N_{pp}=12\) and \(\sigma_{tol}=0.1\). The simplest guideline for tuning these two parameters would be to select the value of the first parameter (\(N_{pp}\)) to be at least three times larger (we used here four times larger number) than the minimum needed points for establishing the model given with Equation (3), while selecting the value of 0.1 for the latter parameter (\(\sigma_{tol}\)) will be suitable in most of the cases. Regarding the rest of the DUPID parameters, the \(k_{ss}\) value is set to be 1 and \(DV\) is given as:  
\[DV=\left\{ \begin{array}{c}DV^{(i)}=10;\;i\;\in[1,10]\\ DV^{(i)}=1;\;i\;\in[11,i_{max}]\end{array} \right.. \tag{17}\]  
The SM calculates the first incremental value after ten simulation time steps (one minute), whereas the other incremental values are calculated in each simulation time step. The idea behind the calculation of the first incremental value after ten simulation time steps is to start adding bias after the plant is settled in the steady-state.