# A Combined Feed-Forward/Feed-Back Control System for a QbD-Based Continuous Tablet Manufacturing Process

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## Abstract

**:**

## 1. Introduction

## 2. Continuous Tablet Manufacturing Process

#### 2.1. Process Description

#### 2.2. Control Relevant Process Model

Transfer Functions | Models | Inputs | Outputs |
---|---|---|---|

G_{p1}(S) | $\frac{0.3782\left(77.3204S+1\right)}{43.6110{S}^{2}+25.5342S+1}$ | Fill depth | Main compression force |

G_{p2}(S) | $\frac{1.4459\left(19.4813S+1\right)}{13.1562S+1}$ | Main compression force | Tablet hardness |

G_{p3}(S) | $\frac{10.1868\left(0.2595S+1\right)}{10.9218{S}^{2}+0.07510S+1}$ | Main compression force | Tablet weight |

G_{p4}(S) | $\frac{1.1701}{106.6667{S}^{2}+4.8672S+1}$ | Punch displacement | Main compression force |

G_{d}(S) | $\frac{34.6667\left(247.1219S+1\right)}{453.5147{S}^{2}+86.4399S+1}$ | Powder bulk density | Main compression force |

**Figure 1.**Poles-zeros plot and phase diagram of transfer function model relating fill depth with main compression force. (

**a**) Poles-zeros plot; (

**b**) Frequency response.

Control Loops | Controllers | Control Variables | Inputs | Outputs |
---|---|---|---|---|

1: Feed-back | G_{c1}(S) | Main compression force (y_{1}) | Deviation of main compression force from set point | Fill depth (u_{11}) |

2: Feed-forward | G_{c2}(S) | - | Powder bulk density | Fill depth (u_{12}) |

3: Feed-back | G_{c3}(S) | Tablet weight (y_{2}) | Deviation of tablet weight from set point | Main compression force set point |

4: Feed-back | G_{c4}(S) | Tablet hardness (y_{3}) | Deviation of tablet hardness from set point | Punch displacement (u_{2}) |

## 3. Combined Feed-Forward/Feed-Back Control of Tablet Press

#### 3.1. Architecture of the Combined Feed-Forward/Feed-Back Control System

**Figure 4.**Architecture of the feed-forward/feed-back control system and decoupled tablet weight and hardness control loops. SP: Set point; MCF: Main compression force.

_{d}(s)) relating the input variable (or disturbances) to the control variable has been developed. Then, a process model (G

_{p1}(s)) relating the actuator with the control variable has been identified. Subsequently, the characteristic equation for the feed-forward control loop has been derived. For a perfect controller, the characteristic equation can be equated to zero and thereby the model (-G

_{d}(s)/G

_{p1}(s)) for feed-forward controller can be generated.

#### 3.2. Controller Parameters Tuning

^{®}(Natick, MA, USA) algorithm for tuning PID controllers is based on insuring the closed-loop stability, adequate performance and adequate robustness. This algorithm meets these objectives by tuning the PID parameters to achieve a good balance between performance and robustness. The algorithm designs an initial controller by choosing a bandwidth to achieve that balance, based upon the open-loop frequency response of linearized model. Upon interactively changing the response time, bandwidth, transient response, or phase margin using the PID Tuner interface, the algorithm computes new PID gains. The controller tuning ensures that the closed-loop system tracks reference changes and suppresses disturbances as rapidly as possible. The tuning method uses the rise time, settling time, overshoot, gain margin phase margin and closed-loop stability as an index to assess the performance of the controller parameters tuning. The process flowsheet model simulated in Simulink has been utilized to tune the controller parameters. The anti-windup reset algorithm, which ensures that the controller output lies within the specified upper and lower bounds, has been included [23]. Because of anti-windup, if the bounds are violated, the time derivative of the integral error is set to zero, and the controller output is clipped to the bounds. Once the controller output is back in the range of the bounds, the integral error will change according to the current error. The controllers need to be re-tuned if there are any changes in the control architecture and/or process. The tuned controller parameters are given in Table 3. The parameters reported in Table 3 correspond to the PID controller form given in Equation 1. A standard PID form, inbuilt in Simulink (Mathworks) has been adapted to represent in terms of controller gain, integral and derivative time constants, as given in Equation 1. Controller gain (K

_{C}), Integral time constant (${\tau}_{I}$), derivative time constant (${\tau}_{D}$) and filter coefficient (N) are the parameters of Equation 1. The filter coefficient sets the location of the poles in the derivative filter. Loop 1 is slave controller and loop 3 is master controller. As given in Table 3, the reset time of loops 3 is 3.52 times that of loop 1 meaning that loop 1 (slave) is faster than loop 3 (master). Note that in a cascade arrangement, the slave loop dynamics needs to be faster than the master loop dynamics. The derivative action has been found to be zero for all loops; therefore, the controller actions are based on proportional and integral actions only.

Controllers | Control Variables | Controller Gain (K_{C}) | Integral Time Constant (${\tau}_{I}$) | Derivative Time Constant (${\tau}_{D}$) |
---|---|---|---|---|

Gc1 | Main compression force | 0.06224 cm/KN | 0.361027 s | 0 s |

Gc_{3} | Tablet weight | 0.010000 KN/g | 1.272265 s | 0 s |

Gc_{4} | Tablet hardness | 0.000010 cm/Kp | 0.029412 s | 0 s |

## 4. Results and Discussion

**Figure 5.**Comparison of combined feed-forward/feed-back control strategy with feed-back only control strategy for hardness control.

**Figure 6.**Comparison of actuators obtained from combined feed-forward/feed-back control strategy with feed-back only control strategy for hardness control.

**Figure 7.**Comparison of the performance of the combined feed-forward/feed-back control scheme with feed-back only control scheme for tablet weight control.

**Figure 9.**Comparison of actuators obtained from combined feed-forward/feed-back control strategy with feed-back only control strategy for main compression force control.

## 5. Conclusions

## Acknowledgements

## Author Contributions

## Nomenclature

Abbreviations | |

API | Active Pharmaceutical Ingredient |

APAP | Acetyl-Para-Aminophenol |

CPP | Critical Process Parameter |

CQA | Critical Quality Attribute |

DT | Dead Time |

MgSt | Magnesium Stearate |

MCF | Main Compression Force |

NIR | Near Infrared |

OPC | OLE (Object linked and embedding) for process control |

PAT | Process Analytical Technology |

PID | Proportional Integral Derivative |

QbD | Quality by Design |

QbT | Quality by Testing |

RSD | Relative Standard Deviation |

SMCC | Silicified Microcrystalline Cellulose |

SP | Set point |

Symbols | Variables |

G_{d}(s) | Disturbance transfer function model |

G_{p}(s) | Process transfer function model |

G_{c}(s) | Controller transfer function model |

d | Disturbances |

u | Actuator |

y | Control variable |

Subscript | Description |

d | disturbance |

p | process |

c | controller |

1,2,3,4 | Process or controller numbers |

## Conflicts of Interest

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**MDPI and ACS Style**

Singh, R.; Muzzio, F.J.; Ierapetritou, M.; Ramachandran, R.
A Combined Feed-Forward/Feed-Back Control System for a QbD-Based Continuous Tablet Manufacturing Process. *Processes* **2015**, *3*, 339-356.
https://doi.org/10.3390/pr3020339

**AMA Style**

Singh R, Muzzio FJ, Ierapetritou M, Ramachandran R.
A Combined Feed-Forward/Feed-Back Control System for a QbD-Based Continuous Tablet Manufacturing Process. *Processes*. 2015; 3(2):339-356.
https://doi.org/10.3390/pr3020339

**Chicago/Turabian Style**

Singh, Ravendra, Fernando J. Muzzio, Marianthi Ierapetritou, and Rohit Ramachandran.
2015. "A Combined Feed-Forward/Feed-Back Control System for a QbD-Based Continuous Tablet Manufacturing Process" *Processes* 3, no. 2: 339-356.
https://doi.org/10.3390/pr3020339