# Quadratic Model-Based Dynamically Updated PID Control of CSTR System with Varying Parameters +
## 4 Conclusions  
In this paper, we discussed an improved version of the PID controller, the Dynamically Updated PID controller. The DUPID controller is constructed based on the assumption that the parameters in the plant vary with time in a smooth and gradual fashion. Its objective is to track the variation of the parameters and to intervene in the PID control value to reduce the increasing control error, which is a direct consequence of their variation. This controller resides on the paradigm of using recent history error data to construct a quadratic model that models the error caused by the varying parameters. This model is then exploited for the iterative calculation of a sequence of incremental values, which are applied directly to the PID control value, without changing the controller parameters, for the purpose of lessening the effect caused by the varying parameters in plant's output. The control performance of the DUPID controller is dependent on the precision of the quadratic model. To ensure the quadratic model describes the data well, the regression data used for its construction must fulfil certain criteria. In this regard, we discussed two versions of DUPID, the 1D and 2D DUPID. These versions differ in the approach of how the data are selected before the model coefficients are determined: the 1D DUPID uses only one coordinate in the process of regression data selection, whereas the 2D DUPID uses two coordinates for data selection. We compared the control performances of the conventional parallel-form PID controller and the DUPID controllers (1D and 2D DUPID). Both controllers were applied to the CSTR system with varying parameters. We observed the control performance of the controllers in three scenarios which were derived from the assumption that the parameters \(h\) and \(T_{f}\) vary linearly over the time the plant is controlled. The results showed that the DUPID controller was capable of reducing the control error to a satisfactory level, whereas the PID control performance dropped with time. Finally, we performed a direct comparison between 1D and 2D DUPID controllers, and the results showed that their control performances do not differ by much. To illustrate that the DUPID controller has the potential of broader use, different than the one discussed here, a meaningful control scenario is pinpointed in the Supplementary Materials. Namely, the DUPID controller is used to attribute the problem of the poorly tuned integral term in the PID controller. In this scenario, it was assumed that the integral term (\(K_{i}\)) of the PID controller is seriously mistuned and the set-point is changing as a ramp function. The results indicated that the DUPID controller accomplished set-point tracking with satisfactory steady-state error, and therefore it was able to make up for the mistuned integral term. To show that the improvement is genuine, we simulated the plant for 10 different uniformly distributed values of \(K_{i}\) drawn from the interval \([0.01,1]\). For all the realizations of \(K_{i}\) the DUPID showed superior performance with respect to the mistuned PID controller. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**(b)** & \multicolumn{3}{c}{**IAE values**} \\ \hline
**Seen.** & **1D** & **2D** & **PID** \\ \hline
1 & 0.1002 & 0.1005 & 0.2529 \\ \hline
2 & 0.1066 & 0.1084 & 0.2781 \\ \hline
3 & 0.1097 & 0.1337 & 0.2960 \\ \hline
**(c)** & \multicolumn{3}{c}{**RD values**} \\ \hline
**Seen.** & **1D/2D** & **1D/PID** & **2D/PID** \\ \hline
1 & 0.0030 & 0.6038 & 0.6026 \\ \hline
2 & 0.0166 & 0.6167 & 0.6102 \\ \hline
3 & 0.1795 & 0.6294 & 0.5483 \\ \hline Avg. & 0.0664 & 0.6166 & 0.5870 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: \(Cont.\)There are few open questions that will be addressed in the future versions of the DUPID controller:  
First of all, to prove that the DUPID controller is multifaceted, it should be tested on the same plant discussed here, but when the concentration of the reactant A (\(C_{Af}\)) or the volumetric flow rate (\(q\)) are subject to change.