# Analytical MPC Algorithm Using Steady-State Process Model
## 3 Results  
### Control Plant  
The control plant is a nonlinear, isothermal CSTR in which the van de Vusse reaction takes place--a popular benchmark with difficult dynamics, used in much research to test newly developed control algorithms (see, e.g., [26; 27; 44; 45; 46; 47]) as well as in MIMO configuration, where not only concentration, but also temperature [48; 49], is stabilized. Also, other CSTRs with less demanding dynamics that are controlled in the MIMO configuration can be found in the literature; see, e.g., [50; 51]. The process model of the reactor is as follows [52]:  
\[ \begin{array}{l}\dfrac{dC_{\rm A}}{dt}=-k_{1}\cdot C_{\rm A}-k_{3}\cdot C_{ \rm A}^{2}+\frac{F}{V}(C_{\rm Af}-C_{\rm A})\ \,\\ \dfrac{dC_{\rm B}}{dt}=k_{1}\cdot C_{\rm A}-k_{2}\cdot C_{\rm B}-\frac{F}{V}C_ {\rm B}\ \,\end{array}  \tag{32}\]  
where \(C_{\rm A}\), \(C_{\rm B}\)--the concentrations of components A and B, respectively; \(F\)--the flow rate, where it is assumed that its upper bound is equal to \(F_{max}=60\) L/h; \(V\)--the volume in which the reaction takes place (assumed constant and \(V=1\) L); and \(C_{\rm Af}\)--the concentration of component A in the inlet flow stream (\(C_{\rm Af}=10\) mol/L if not declared otherwise). The parameters of the model have the following values: \(k_{1}=50\) L/h, \(k_{2}=100\) L/h, and \(k_{3}=10\) L/(h\(\cdot\) mol). The output variable is the concentration \(C_{\rm B}\) of substance B, the manipulated variable is the flow rate \(F\), and \(C_{\rm Af}\) is the disturbance variable. The control plant under consideration is highly nonlinear. Its step responses obtained for different flow rate values (\(F_{1}=54.3\) L/h, \(F_{2}=44.3\) L/h, \(F_{3}=34.3\) L/h, \(F_{4}=24.3\) L/h, \(F_{5}=14.3\) L/h) are shown in Figure 1. Although the difference between the subsequent values of the flow rate is the same (\(\Delta F=10\) L/h) the obtained responses are asymmetric; it is clear that the control plant has both nonlinear dynamics and statics. Moreover, inverse response is present in the step responses obtained. The same control plant was also used during the tests in [26; 27], where advanced numerical versions of the Fuzzy MPC algorithms were proposed and tested; the nonlinear steady-state characteristic of the process is depicted in Figure 2. The algorithm proposed in the current article exploits this characteristic. To calculate the current gain \(K_{k}\), the equation that describes the steady-state characteristic was obtained from Equation (32). The equations were rewritten for the steady-state; the constant \(V=1\) L was also taken into account. Moreover, it is important to note that all variables representing the steady-state are denoted by an overline to clearly distinguish them from their dynamic counterparts. The first equation from (32), therefore, transforms into the following equation:  
\[0=-k_{1}\cdot\overline{\mathrm{C}}_{\mathrm{A}}-k_{3}\cdot\overline{\mathrm{C} }_{\mathrm{A}}^{2}+\overline{F}\big{(}\overline{\mathrm{C}}_{\mathrm{A}}- \overline{\mathrm{C}}_{\mathrm{A}}\big{)}\;\;. \tag{33}\]  
And the second one transforms into the following equation:  
\[0=k_{1}\cdot\overline{\mathrm{C}}_{\mathrm{A}}-k_{2}\cdot\overline{\mathrm{C} }_{\mathrm{B}}-\overline{F}\overline{\mathrm{C}}_{\mathrm{B}}\;\;. \tag{34}\]  
Figure 1: Step responses of control plant. Figure 2: Steady-state characteristic of control plant.