# Parallel Solution of Robust Nonlinear Model Predictive Control Problems in Batch Crystallization

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulations

#### 2.1. NMPC Formulation

#### 2.2. MHE Formulation

#### 2.3. Robust NMPC Formulation

#### 2.4. Efficient Optimization via the Simultaneous Approach

## 3. Efficient Parallel Schur Complement Method for Stochastic Programs

## 4. Performance of Robust NMPC on Batch Crystalization

#### 4.1. Case Study: Multidimensional Unseeded Batch Crystallization Model

#### 4.2. Numerical Results

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Optimal temperature profile for nominal NMPC (nonlinear model predictive control) and robust NMPC.

**Table 1.**The robust performance (value of $cost$) of different control strategies evaluated using 100 test scenarios and exact information.

Control Strategies | Nominal | Average | Standard Deviation | Worst-Case |
---|---|---|---|---|

Ideal | $2\times {10}^{-4}$ | 30 | 66 | 499 |

Open-loop | $2\times {10}^{-4}$ | 167 | 223 | 1339 |

Nominal NMPC | 0.2 | 93 | 159 | 955 |

Exact Min–max NMPC | 32 | 78 | 113 | 677 |

Exact Min–expected NMPC | 12 | 99 | 169 | 1076 |

**Table 2.**The robust performance of the robust NMPC using different numbers of scenarios evaluated using 100 test scenarios.

Type | S | Nominal | Average | Standard Deviation | Worst-Case |
---|---|---|---|---|---|

Min–max | 25 | 15 | 99 | 170 | 1062 |

50 | 13 | 102 | 178 | 1129 | |

75 | 13 | 95 | 156 | 946 | |

100 | 25 | 80 | 120 | 767 | |

Min–expected | 25 | 21 | 89 | 138 | 902 |

50 | 11 | 100 | 172 | 1085 | |

75 | 12 | 99 | 169 | 1064 | |

100 | 12 | 99 | 169 | 1074 |

**Table 3.**The solution time of solving a robust optimization problem with 100 optimization scenarios.

# Processors | Full Factorization | Schur Complement Method | ||
---|---|---|---|---|

Time(s) | Time(s) | Speedup | ||

Building Model | 1 | 44.3 | 64.2 | - |

2 | - | 34.8 | 1.8 | |

5 | - | 14.9 | 4.3 | |

10 | - | 8.6 | 7.5 | |

20 | - | 6.3 | 10.2 | |

25 | - | 4.7 | 13.7 | |

Solving NLP | 1 | 406 | 426.9 | - |

2 | - | 216.3 | 2.0 | |

5 | - | 90.8 | 4.7 | |

10 | - | 51.0 | 8.4 | |

20 | - | 35.8 | 11.9 | |

25 | - | 30.0 | 14.2 |

**Table 4.**Robust performance of min–max NMPC with different numbers of optimization scenarios from Bayesian inference evaluated using 100 simulations.

Type | S | Nominal | Average | Standard Deviation | Worst-Case |
---|---|---|---|---|---|

Min–max | 12 | 18 | 74 | 120 | 744 |

25 | 13 | 61 | 96 | 584 | |

50 | 11 | 71 | 114 | 655 | |

Min–expected | 12 | 17 | 81 | 141 | 943 |

25 | 12 | 84 | 145 | 949 | |

50 | 11 | 84 | 145 | 934 |

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**MDPI and ACS Style**

Cao, Y.; Kang, J.; Nagy, Z.K.; Laird, C.D.
Parallel Solution of Robust Nonlinear Model Predictive Control Problems in Batch Crystallization. *Processes* **2016**, *4*, 20.
https://doi.org/10.3390/pr4030020

**AMA Style**

Cao Y, Kang J, Nagy ZK, Laird CD.
Parallel Solution of Robust Nonlinear Model Predictive Control Problems in Batch Crystallization. *Processes*. 2016; 4(3):20.
https://doi.org/10.3390/pr4030020

**Chicago/Turabian Style**

Cao, Yankai, Jia Kang, Zoltan K. Nagy, and Carl D. Laird.
2016. "Parallel Solution of Robust Nonlinear Model Predictive Control Problems in Batch Crystallization" *Processes* 4, no. 3: 20.
https://doi.org/10.3390/pr4030020