# Quadratic Model-Based Dynamically Updated PID Control of CSTR System with Varying Parameters +
## 3 Results  
### Case Study  
In this paper, as a benchmark plant model for the controllers the highly nonlinear continuous stirred tank reactor (CSTR) system, Figure 6, with varying parameters is considered.  
The model equations were taken from [20]. The CSTR system is a second-order control system which is modelled by the Equations (12) and (13). In spite of the simplicity of this model, it is well known by its strong nonlinear dynamics which is often manifested by: strong dependence of particular parameters, steady-state multiplicity, limit cycle etc. [23].  
\[\frac{dT}{dt}=\frac{q}{V}\Big{(}T_{f}-T\Big{)}+\frac{\Delta H}{\rho C_{p}}k_{0 }C_{A}e^{-\frac{E}{RT}}+\frac{hA_{Ia}}{V\rho C_{p}}(T_{c}-T), \tag{12}\]  
\[\frac{dC_{A}}{dt}=\frac{q}{V}\Big{(}C_{Af}-C_{A}\Big{)}-k_{0}C_{A}e^{-\frac{E} {RT}}. \tag{13}\]  
The parameters of the model are defined as: \(q\)--Volumetric flow-rate; \(V\)--Volume of the CSTR; \(\rho\)--Density of the mixture; \(C_{p}\)--Heat capacity; \(\Delta H\)--Heat of the reaction; \(E\)--Activation energy; \(R\)--Universal gas constant; \(k_{0}\)--Pre-exponential factor; \(h\)--Overall heat transfer coefficient; \(A_{Ia}\)--Heat transfer area; \(T_{f}\)--Feed fluid temperature; \(C_{Af}\)--Feed concentration of reactant A; \(T_{c}\)--Temperature of the cooling jacket; \(C_{A}\)--Concentration of reactant A; \(T\)--Temperature in the CSTR. The nominal values assumed for the plant parameters are given in Table 1. The manipulated variable is assumed to be the cooling jacket fluid temperature \(T_{c}\), which is operated in the interval \([250,\ 350]\) K, and the reactor temperature \(T\) is assumed to be the controlled variable, and the only measured state.  
Figure 6: Graphical representation of the Continuous Stirred Tank Reactor (CSTR) system.  
For relevant comparison between the discussed control algorithms, it is assumed that the heat transfer coefficient \(h\) and the feed fluid temperature \(T_{f}\) change over time. These changes are realistic in practice as they often occur as a result of material deposition (fouling film) on the heat transfer surfaces, over the period of the plant operation. The process of fouling film development on the heat transfer surfaces is a slow and gradual process, which as a consequence adds resistance to the flow of heat. There are two types of fouling film build-up: asymptotic and linear. In the former, the resistance due to the fouling film increases very quickly at the beginning of the operation and becomes asymptotic to a steady-state value at the end. In the latter, the fouling film develops linearly over the entire period of plant operation [24].  
Here, linear deposit contamination was assumed, and the parameters \(h\) and \(T_{f}\) were assumed to be subjected to linear change. The heat transfer coefficient \(h\) is replaced in Equation (12) by \(h_{d}\), which is defined as:  
\[h_{d}=\phi_{h}(t)h=(1-\alpha_{h}t)h, \tag{14}\]  
where \(t\) is time, \(h\) is the value of the heat transfer coefficient, \(h_{d}\) is the scaled version of the heat transfer coefficient, \(\phi_{h}(t)\in(0,1)\) is the fouling coefficient (the scaling coefficient), and \(\alpha_{h}\) is the fouling constant. The feed fluid temperature \(T_{f}\) was assumed to be constant for a certain period of time, before changing into a ramp function in a negative direction. To broaden the discussion further, it was assumed that \(T_{f}\) changes as ramp function in positive direction also:  
\[T_{f}=\left\{\begin{array}{c}T_{fcons};1\leq i<lb\\ \pm T_{d\frac{i-lb}{i_{max}-lb}}+T_{fcons};\;lb\leq i\leq i_{max}\end{array},\right. \tag{15}\]  
where \(T_{fcons}\) is the initial value of the feed temperature, \(lb\) defines the interval of simulation time steps \(i\) while \(T_{f}\) is kept constant, \(T_{d}\) defines how much \(T_{f}\) changes in the positive or negative direction, and \(i_{max}\) is the maximal number of simulation time steps. The model equations are numerically solved in MATLAB by the function ode23t with simulation time step \(T_{i}=0.1\) min.