# On the Use of Nonlinear Model Predictive Control without Parameter Adaptation for Batch Processes

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation—Model Predictive Control without Parameter Adaptation

## 3. Variational Analysis of Model Predictive Control without Parameter Adaptation

**Proposition 1.**

**Proof.**

**Proposition 2.**

**Proof.**

## 4. Results and Discussion

#### 4.1. Illustrative Example

^{−1}), which are obtained using the Arrhenius equation:

#### 4.2. Results

## 5. Conclusions

## Acknowledgment

## Author Contributions

## Conflicts of Interest

## Abbreviations

MPC | Model predictive control |

NMPC | Nonlinear MPC |

D-RTO | Dynamic real-time optimization |

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Parameter | Value | Units |
---|---|---|

${c}_{A0}$ | 5 | mol/L |

${c}_{B0}$ | 0 | mol/L |

${k}_{10}$ | 5 × 10^{3} | h^{−1} |

${k}_{20}$ | 7 × 10^{16} | - |

${E}_{1}$ | 2 × 10^{4} | J/mol |

${E}_{2}$ | 1 × 10^{5} | J/mol |

$R$ | 8.314 | J/mol.K |

$\mathsf{\alpha}$ | 5 | - |

${\overline{k}}_{1\text{}0}$ | 1 | - |

${\overline{k}}_{2\text{}0}$ | 0.0224 | - |

${t}_{f}$ | 2 | H |

${u}_{min}$ | 1.25 | - |

${c}_{A{f}_{max}}$ | 0.1 | mol/L |

$\mathsf{\gamma}$ | 0.001 | - |

$\mathsf{\beta}$ | 0.999 | - |

**Table 2.**Comparison of offline, re-optimization and plant optimum solutions for the six cases with parametric errors. Cost is maximized for Cases 1–3 and minimized for Cases 4–6.

Case | Parametric Error | Cost | ||
---|---|---|---|---|

Offline | Re-Optimization | Plant Optimum | ||

1. Unconstrained, Economic cost | -- | 4.03 | 4.03 | 4.03 |

${\overline{k}}_{10}$: −20%; $\alpha $: −20% | 3.70 | 3.71 | 3.72 | |

${\overline{k}}_{10}$: −20%; $\alpha $: +20% | 3.691 | 3.686 | 3.697 | |

2. Terminal constraint Economic cost | -- | 3.71 | 3.71 | 3.71 |

${\overline{k}}_{10}$: −20%; $\alpha $: −20% | 1.29 | 2.03 | 3.35 | |

${\overline{k}}_{10}$: −20%; $\alpha $: +20% | 3.01 | 2.93 | 3.06 | |

3. Path constraints Economic cost | -- | 3.80 | 3.80 | 3.80 |

${\overline{k}}_{10}$: −50%; $\alpha $: −50% | 3.079 | 3.079 | 3.17 | |

${\overline{k}}_{10}$: −50%; $\alpha $: +50% | 2.39 | 2.37 | 2.40 | |

4. Unconstrained Trajectory cost | -- | 0.00 | 0.00 | 0.00 |

${\overline{k}}_{10}$: −20%; $\alpha $: −20% | 1.16 | 0.303 | 0.03 | |

${\overline{k}}_{10}$: −20%; $\alpha $: +20% | 1.10 | 0.30 | 0.00 | |

5. Terminal constraint Trajectory cost | -- | 0.00 | 0.00 | 0.00 |

${\overline{k}}_{10}$: −20%; $\alpha $: −20% | 2.84 | 1.49 | 0.03 | |

${\overline{k}}_{10}$: −20%; $\alpha $: +20% | 1.80 | 1.38 | 0.36 | |

6. Path constraint Trajectory cost | -- | 0.21 | 0.21 | 0.21 |

${\overline{k}}_{10}$: −20%; $\alpha $: −20% | 0.20 | 0.16 | 0.02 | |

${\overline{k}}_{10}$: −20%; $\alpha $: +20% | 2.060 | 1.48 | 0.47 |

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**MDPI and ACS Style**

Binette, J.-C.; Srinivasan, B.
On the Use of Nonlinear Model Predictive Control without Parameter Adaptation for Batch Processes. *Processes* **2016**, *4*, 27.
https://doi.org/10.3390/pr4030027

**AMA Style**

Binette J-C, Srinivasan B.
On the Use of Nonlinear Model Predictive Control without Parameter Adaptation for Batch Processes. *Processes*. 2016; 4(3):27.
https://doi.org/10.3390/pr4030027

**Chicago/Turabian Style**

Binette, Jean-Christophe, and Bala Srinivasan.
2016. "On the Use of Nonlinear Model Predictive Control without Parameter Adaptation for Batch Processes" *Processes* 4, no. 3: 27.
https://doi.org/10.3390/pr4030027