# Quadratic Model-Based Dynamically Updated PID Control of CSTR System with Varying Parameters +
## 3 Results
### Simulation Results and Discussion
Adding bias during the transient state can provoke an unwanted overshoot in the plant response, or in the extreme case, it can destabilize the plant. The possible downside of calculating the other incremental values in each simulation time steps is having the added computational time burden. On the other hand, this can be justified by achieving a smaller IAE value and a better overall control performance. In a real case scenario, the number of time steps needed to pass for the SM to start calculating the incremental value will be determined based on the plant itself, and on the experience of the process technician (the operator) on how much time is needed for the plant output to settle in steady-state. However, to omit the possible subjective human decision, in future modifications of the DUPID algorithm the moment of the first incremental value calculation should be determined automatically by accounting for the proximity of the PV with respect to the SP over time. The simulation time step is assumed to be \(T_{i}=0.1\) min. In the first ten simulation time steps (until minute 1) the value of \(\Delta u\) is kept on zero since the first calculated incremental value by SM is carried out at the tenth time step. The termination criterion was defined with respect to the maximum permitted time steps \(i_{max}=300\). The total time of the plant operation is \(i_{max}\cdot T_{i}=30\) min. In Figure 7, the resulting plant response under PID control in all three scenarios is given. It is indicative that the error area increases over time in all scenarios, (d), (e) and (f). This is a consequence of the changing plant parameters over time. Since the PID controller coefficients are kept constant the controller is unable to adapt to the change in plant operating conditions and is therefore unable to steer the controlled variable (\(T\)) to the set point. The 1D and 2D DUPID controllers proved to be more agile in detecting and countering the effects caused by the time-varying parameters. From the Figures 8d-f and 9d-f, it can be seen that the DUPID anticipates the change in plant parameters and pushes the PID control value in the right direction to compensate for its stationary coefficients. Figure 8: The plant response under 1D DUPID control in all three scenarios. The figure is structured in three rows of three pictures, in the first row (**a–c**) the plant output (blue line) and the set point (red dashed line) are plotted; in the second row (**d–f**) the incremental value is given (blue markers); in the third row (**g–i**) the resultant control error (blue markers) is plotted against the zero error line (black dashed line). Figure 7: The plant response under PID control in all three scenarios. The figure is structured in two rows of three pictures, in the first row (**a–c**) show the plant output (blue line) and the set point (red dashed line); in the second row (**d–f**) show the resultant control error (blue markers) is plotted against the zero error line (black dashed line). The control performance of each controller was quantified by the IAE metric, Table 2b. The relative differences (The relative difference is calculated as \(d_{r}=|x-y|/\max(|x|,|y|)\)) of the obtained IAE values, Table 2c, were calculated for each scenario. On average, the 1D DUPID and 2D DUPID controllers achieved improvement over PID control performance of around 62% and 59%, respectively. The direct comparison between 1D and 2D DUPID controllers showed that 1D DUPID is performing slightly better, on average of around 7% (Table 2c) first column). Moreover, we compared 1D and 2D DUPID in terms of how much time is needed on average for each controller to calculate the incremental value in each scenario, Table 1. From the values of the calculated relative differences, it is clear that 2D DUPID needs around twice as much time on average to produce the next incremental value compared to the 1D DUPID.