# Quadratic Model-Based Dynamically Updated PID Control of CSTR System with Varying Parameters +
###### Abstract  
In this paper, we discuss an improved version of the conventional PID (Proportional-Integral-Derivative) controller, the Dynamically Updated PID (DUPID) controller. The DUPID is a control solution which preserves the advantages of the PID controller and tends to improve them by introducing a quadratic error model in the PID control structure. The quadratic error model is constructed over a window of past error points. The objective is to use the model to give the conventional PID controller the awareness needed to battle the effects caused by the variation of the parameters. The quality of the predictions that the model is able to deliver depends on the appropriate selection of data used for its construction. In this regard, the paper discusses two algorithms, named 1D (one dimensional) and 2D (two dimensional) DUPID. Appropriate to their names, the former selects data based on one coordinate, whereas the latter selects the data based on two coordinates. Both these versions of the DUPID controller are compared to the conventional PID controller with respect to their capabilities of controlling a Continuous Stirred Tank Reactor (CSTR) system with varying parameters in three different scenarios. As a quantifying measure of the control performance, the integral of absolute error (IAE) metric is used. The results from the performed simulations indicated that the two versions of the DUPID controller improved the control performance of the conventional PID controller in all scenarios. d 2021 14 31 11.11.  
identified, and subsequently the obtained model is used to adjust the parameters of the PID controller. In the direct methods, the estimation of the controller parameters is carried out online so that the error between the reference model output and plant output tends to be zero. In general, these methods face difficulties emanating from two sources; one of the difficulties arises from the fact that the controlled plants are black boxes and their identification is a tough and complex task, and the second one stems from the fact that it is difficult to guarantee that the controller parameters will converge to the desired values, and achieve the goal of zero error. Another common approach for countering the consequences of varying plant parameters found in industry is the robust control approach [10]. The ideas emerging from the development of the robust control paradigm were specialized for the case of the PID controllers, resulting in numerous tuning methods for robust PID control [11]. One of the most often used is the internal-model-based (IMC) PID robust tuning method [11; 12; 13]. A novel robust approach is presented in [14], to deal with the negative implications caused by the inherent feature of the IMC tuning of pole-zero cancelation. Lately, the robust control community has shifted its attention toward tuning methods based on multi-objective optimization [11; 15; 16]. The tuning parameters in these methods are obtained as a solution to an optimization problem which is often non-convex. The discussed robust tuning methods have proven effective in improving the robustness in a variety of control systems. However, they have certain downsides, and some of them are pinpointed as follows: firstly, in the tuning process it is necessary to account for all the parameters that can possibly vary and the bounds of their variation in advance; secondly, they usually need one or multiple linearized models of the nonlinear plant to design the controller; lastly, the methods based on optimization techniques come with a high computational burden as the optimization problem therein is non-convex. To tackle some of the aforementioned problems, in this paper we discuss a simple and computationally non-intensive, hands-on control algorithm which is an upgraded version of the conventional PID controller--a dynamically updated PID (DUPID) control scheme [17].