# A Hybrid MPC-PID Control System Design for the Continuous Purification and Processing of Active Pharmaceutical Ingredients

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## Abstract

**:**

## 1. Introduction and Objective

#### 1.1. Objectives

- Present a system-wide hybrid MPC-PID control scheme.
- Quantify the importance of a cascade control scheme in case of pharmaceutical unit operations.
- Present a comparative study between PID-only control and MPC-PID control schemes based on set point tracking and disturbance rejection criteria.

## 2. Background

## 3. Flowsheet Model Description

#### 3.1. Numerical Technique

## 4. A Systematic Framework for Hybrid Control Design

- Identify the critical control variables.
- Identify suitable actuators or manipulating variables to be paired with the control variables.
- Obtain the response of the control variables by implementing a step change to the manipulating variables (from the actual flowsheet model, which has been developed in gPROMS™).
- Fit a transfer function model to relate the control and manipulating variables from the response obtained in the previous step. This has been done using the system identification tool box of MATLAB™.
- Group the control loops into two categories (i.e., MPC controlled and PID controlled)
- Design the hybrid MPC-PID control loops in Simulink™.
- Evaluate the performance of the designed control system in terms of set point tracking and disturbance rejection ability.

## 5. Design Strategy of the Control System

#### 5.1. Selection of the Control Variables and Pairing with Suitable Actuators

**Table 1.**List of control and manipulated variables for each unit operation. API, active pharmaceutical ingredient.

Process | Control variable | Actuators (Manipulating variable) |
---|---|---|

Crystallization | Temperature cooling schedule | Coolant temperature |

Saturation concentration | ||

Drying | Drying gas temperature | Super-heated steam temperature |

Blending | API composition | Excipient flow rate |

Holdup | Weir height |

#### 5.2. Design of Controller

## 6. Results and Discussion

**Figure 2.**Pictorial representation of the closed-loop continuous process. PID (proportional-integral-derivative); MPC (model predictive control); MV (manipulated variable/actuator); CV (control variable).

#### 6.1. Control System Design

PID parameters | |||
---|---|---|---|

Control loop | Gain $\left({K}_{c}\right)$ | Reset Time $\left({\tau}_{I}\right)$ | Rate $\left({\tau}_{D}\right)$ |

Crystallization (slave loop) | −0.166 | 2.95 × ${10}^{4}$ s | 1.09 × ${10}^{4}$ s |

MPC parameters | |||

Control loop | Input Weight | Rate Weight | Output Weight |

Crystallization (master loop) | 0 | 0.1 | 1 |

Drying | 0 | 0.1 | 1 |

Mixing (API composition) | 0 | 0.1 | 1 |

Mixing (holdup) | 0 | 0.1 | 1 |

#### 6.2. Comparison of the Hybrid MPC-PID Scheme with PID-only Scheme

**Table 3.**List of integral time averaged error (ITAE) values for the MPC vs. PID performance assessment.

Set point tracking | ||
---|---|---|

Control loop | MPC-PID | PID-only |

Saturation concentration | 6.322 × ${10}^{5}$ | 2.935 × ${10}^{6}$ |

Drying gas temperature | 7.371 × ${10}^{5}$ | 2.068 × ${10}^{6}$ |

API composition | 1.584 × ${10}^{3}$ | 1.892 × ${10}^{3}$ |

Disturbance rejection | ||

Control loop | MPC-PID | PID only |

Saturation concentration | 3.317 × ${10}^{5}$ | 4.540 × ${10}^{5}$ |

Drying gas temperature | 1.556 × ${10}^{7}$ | 6.193 × ${10}^{7}$ |

**Figure 6.**Controller performance evaluation for the set point tracking of the saturation concentration.

**Figure 8.**Controller performance evaluation for the set point tracking of the drying gas temperature.

## 7. Conclusions

## Acknowledgments

## Nomenclature

Symbol | Description | Units |

A_{c} | Area of heat transfer | m^{2} |

B_{0} | Primary nucleation term | particles/(m^{3}/s) |

b | Kinetic parameter for crystallization | Dimensionless |

C | Solute concentration in crystallization | moles/m^{3} |

C_{sat} | Saturation concentration of solute | moles/m^{3} |

C_{pw} | Specific heat constant for water | J/K |

D_{p} | Crystal diameter | μm |

F | Particle density | particles/m^{3} |

G_{1} | Growth rate | m/s |

G_{2} | Growth rate | m/s |

g_{1} | Kinetic parameter for crystallization | Dimensionless |

g_{2} | Kinetic parameter for crystallization | Dimensionless |

k_{g1} | Kinetic parameter for crystallization | m/s |

k_{g2} | Kinetic parameter for crystallization | m/s |

k_{b} | Kinetic parameter for crystallization | particles/(m^{3}/s) |

L_{1} | Internal coordinate for length of solid | m |

L_{2} | Internal coordinate for length of solid | m |

L_{3} | Internal coordinate for length of liquid | m |

M_{w} | Cooling water flow rate | kg/s |

T | Temperature (cooling schedule) | Kelvin |

T_{c} | Temperature of cooling water | Kelvin |

T_{in} | Inlet temperature of water | Kelvin |

U | Overall heat transfer coefficient | W/(m^{2}·K) |

ϵ | Porosity of the cake | Dimensionless |

ρ_{avg} | Average density of wet particles | kg/m^{3} |

ρ_{s} | Density of solid | kg/m^{3} |

∆P | Filter pressure difference | kPa |

μ | Fluid viscosity | kg/(m·s) |

α | Specific cake resistance | m/kg |

A | Filter surface area | m^{2} |

C_{F} | Concentration of solutes in slurry | moles/m^{3} |

c | Mass of solute deposited on filter per unit volume of filtrate | kg/m^{3} |

m_{F} | Mass of wet cake | kg |

m_{c} | Mass of dry cake | kg |

m_{v} | Rate of evaporation during drying | kg/s |

N_{a} | Avogadro number | Dimensionless |

R_{m} | Filter medium resistance | 1/m |

V | Filtrate volume | m^{3} |

V_{p} | Particle volume | m^{3} |

A_{s} | Area of heat transfer | m^{2} |

C_{ps} | Specific heat capacity | J/(kg·K) |

C_{psteam} | Specific heat constant for steam | J/K |

k | Mass transfer coefficient | m/s |

h_{fg} | Specific heat of evaporation | J/kg |

h | Heat transfer coefficient | W/(M^{2}·K) |

T_{s} | Temperature of steam | Kelvin |

T_{p} | Temperature of particle | Kelvin |

T_{g} | Drying gas temperature | Kelvin |

T_{in-steam} | Inlet temperature of steam | Kelvin |

mass_{out} | Outlet flow rate of API crystals from dryer | kg/s |

M_{s} | Steam flow rate | kg/s |

N_{u} | Nusselt’s Number | Dimensionless |

P_{r} | Prandtl Number | Dimensionless |

R_{e} | Reynolds Number | Dimensionless |

S_{h} | Sherwood Number | Dimensionless |

U | Overall heat transfer coefficient | W/(m^{2}·K) |

x_{p} | Liquid content of solid particle | Dimensionless |

x_{eql} | Liquid content of solid particle at equilibrium | Dimensionless |

n | Counter for number of components | Dimensionless |

n_{max} | Maximum number of components | - |

t | Time | s |

V_{f} | Forward axial velocity | m/s |

V_{b} | Backward axial velocity | m/s |

V_{r} | Radial velocity | m/s |

x | Spatial coordinate in axial direction | - |

x_{max} | Maximum number of axial compartments | - |

y | Spatial coordinate in axial direction | - |

y_{max} | Maximum number of radial compartments | - |

y_{API} | Fractional API composition at mixer outlet | Dimensionless |

y_{avg} | Average spatial composition of component A | moles/m^{3} |

y_{i} | Composition of component A in i-th compartment | moles/m^{3} |

## Appendix: Process Model Integrated with the Hybrid MPC-PID Scheme

## Crystallizer

## Filtration Process

## Drying Process

## Mixer

## MPC-PID Controller Equations

## Conflicts of Interest

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**MDPI and ACS Style**

Sen, M.; Singh, R.; Ramachandran, R.
A Hybrid MPC-PID Control System Design for the Continuous Purification and Processing of Active Pharmaceutical Ingredients. *Processes* **2014**, *2*, 392-418.
https://doi.org/10.3390/pr2020392

**AMA Style**

Sen M, Singh R, Ramachandran R.
A Hybrid MPC-PID Control System Design for the Continuous Purification and Processing of Active Pharmaceutical Ingredients. *Processes*. 2014; 2(2):392-418.
https://doi.org/10.3390/pr2020392

**Chicago/Turabian Style**

Sen, Maitraye, Ravendra Singh, and Rohit Ramachandran.
2014. "A Hybrid MPC-PID Control System Design for the Continuous Purification and Processing of Active Pharmaceutical Ingredients" *Processes* 2, no. 2: 392-418.
https://doi.org/10.3390/pr2020392