## 3 Results and Discussion  
In this paper, the application to ratio control has been investigated for the specific pharmaceutical process units (i.e., feeder and blender). A benchmark simulator has been proposed in [8] describing tableting production using wet/dry granulation for simulation, analysis and control design and validation purposes. The continuous manufacturing using the wet and dry granulation flowcharts is depicted in Figure 5. In our open-source MATLAB/Simulink simulation platform the wet and dry granulation route is implemented [34], however, a model library is available for simulating the user-specified sub-processes production line. The entire manufacturing process is described in [8]. Of particular interest is the blender unit, where the powder and excipient are mixed to produce a material that is granular in form. A population balance modeling approach has been used to model the blending process [8]:  
\[ \begin{array}{l}\frac{\partial m_{ij}}{\partial t}=F_{f}[m_{i-1,j}-m_{i,j}]+ F_{b}[m_{i+1,j}-m_{i,j}]+\\ F_{r}[m_{i,j+1}+m_{i,j-1}-m_{i,j}]\end{array}  \tag{24}\]  
which represents the powder distribution of both the active pharmaceutical ingredient (API) and the excipient in each compartment. The corresponding flows are obtained from:  
\[ \begin{array}{l}F_{f}=av_{b}+b\\ F_{b}=cv_{b}+d\\ F_{r}=e\end{array}  \tag{25}\]where \(v_{b}\) represents the speed of the blender in rpm and the \(a,b,c,d,e\) coefficients are parameters estimated from experimental data to calibrate the model. The combined flow is given by:  
\[F_{blender}=\sum_{j=1}^{N_{v}}F_{f,API}m_{API,i=N_{a},j}+\sum_{j=1}^{N_{v}}F_{f, Exc}m_{Exc,i=N_{a},j} \tag{26}\]  
Consequently, the outlet concentration (\(C_{API}\)) is given by:  
\[C_{API}=\frac{\sum_{j=1}^{N_{v}}F_{f,API}m_{API,j=N_{a},j}}{F_{blender}} \tag{27}\]  
and the relative standard deviation (\(RSD_{API}\)):  
\[RSD_{API}=\frac{\sqrt{\frac{1}{N_{v}-1}\sum_{j=1}^{N_{v}}\left(\frac{F_{f,API} m_{API,i=N_{a},j}}{F_{f,API}m_{API,i=N_{a},j}+F_{f,Exc}m_{Exc,i=N_{a},j}}-C_{API} \right)^{2}}}{C_{API}} \tag{28}\]  
Essentially, this process requires ratio control as a natural solution for mixing flows. A proposed solution is given in Figure 6.  
All transfer functions implemented in the benchmark pharma simulator are based on literature models. Every process unit is characterized by a specific transfer function representing the process dynamics. For the feeder unit, a simplified semi-empirical first-order model (FOPDT) is utilized for modeling the feeder unit [35]. The relationship between the screw speed and the mass flow rate of the feeder is accurately predicted. It requires  
Figure 5: Continuous pharmaceutical tablet manufacturing processes: wet granulation (**left**) and direct compaction (**right**) routes. Figure 6: A ratio control structure solution for the blender unit. extra parameters, such as the vertical stress (\(\sigma_{v}\)), the density constant (\(\beta\)), the mass and radius of the blade (\(M_{blade}\) and \(R_{blade}\)), etc. The paper [36] presents a FOPDT model as well, like in [37]. The difference is that the mass flow rate is calculated via a physics-based approach from the bulk density of the powder, the screw speed, the volumetric efficiency and the twin screws geometry. It would be more interesting to use this model since it points out where the time constant and delay are coming from. Unfortunately, no parameter data is available. These models are relatively new since no significant work has been done before. Because of that, they used a FOPDT model where the parameters were purely determined experimentally like in [38]. The work in [39] even uses the same loss-in-weight (LIW) model as in [40]. All models described in these papers are very similar, making them more or less even accurate. The availability of parameter data was the decisive factor for choosing the model in [40]. For the blender unit, transfer function models have been derived from the population balance models given in Equations (24)-(28).