# Safeness Index-Based Economic Model Predictive Control of Stochastic Nonlinear Systems

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Notations

#### 2.2. Class of Systems

**Definition**

**1.**

**Definition**

**2.**

**Proposition**

**1.**

#### 2.3. Stabilizability Assumptions

**Definition**

**3.**

**Remark**

**1.**

## 3. Main Results

#### 3.1. Process Safeness Index

#### 3.2. Safeness Index-Based LEMPC

**Remark**

**2.**

#### 3.3. Safeness Index-Based LEMPC Using Multiple Level Sets

**Remark**

**3.**

**Remark**

**4.**

#### 3.4. Stochastic Safeness Index-Based LEMPC

#### 3.5. Sample-And-Hold Implementation

**Theorem**

**1.**

**Proof.**

#### 3.6. Stability in Probability

**Theorem**

**2.**

**Proof.**

**Remark**

**5.**

#### 3.7. Feasibility in Probability

**Theorem**

**3.**

**Proof.**

**Remark**

**6.**

## 4. Application to a Chemical Process Example

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**A schematic representing the safe operating region $\mathcal{S}$ (the gray region) with an example closed-loop trajectory under the Safeness Index-based Lyapunov-based economic model predictive control (LEMPC) design of Equation (5) for the initial condition (0, 0).

**Figure 2.**A schematic representing the unsafe region $\mathcal{D}$ (dark gray) and the safe operating region $\mathcal{S}:={\mathcal{S}}_{1}\cup {\mathcal{S}}_{2}$ (light gray).

**Figure 3.**A schematic representing the unsafe region $\mathcal{D}$ (dark gray) and the region ${\mathcal{S}}_{e}:={\mathcal{S}}_{1e}\cup {\mathcal{S}}_{2e}$ (light gray), which is the safe operating region $\mathcal{S}$ subtracting the risk margins $\rho -{\rho}_{e}$ and $s-{s}_{e}$.

**Figure 4.**Closed-loop trajectory under the Safeness Index-based LEMPC of Equation (6) for the initial condition (0, 0) (in deviation variable form) with the additional material constraint: $\frac{1}{{t}_{s}}{\int}_{0}^{{t}_{s}}{u}_{1}(\tau )d\tau =0\phantom{\rule{3.33333pt}{0ex}}\mathrm{kmol}/{\mathrm{m}}^{3}$.

**Figure 5.**An example closed-loop trajectory under the Safeness Index-based LEMPC of Equation (7) for the initial condition (0, 0) that leaves the safe operating region $\mathcal{S}$, in which ${\rho}_{e}=320$ and ${s}_{e}=6800$.

**Figure 6.**The production rate ${L}_{e}={k}_{0}{e}^{-E/RT}{C}_{A}^{2}$ within the safe operating region $\mathcal{S}$.

${T}_{0}=300$ K | $F=5$ m${}^{3}$/h |

${V}_{L}=1$ m${}^{3}$ | $E=5\times {10}^{4}$ kJ/kmol |

${k}_{0}=8.46\times {10}^{6}$ m${}^{3}$/kmol h | $\Delta H=-1.15\times {10}^{4}$ kJ/kmol |

${C}_{p}=0.231$ kJ/kg K | $R=8.314$ kJ/kmol K |

$\rho =1000$ kg/m${}^{3}$ | ${C}_{A{0}_{s}}=4$ kmol/m${}^{3}$ |

${Q}_{s}=0.0$ kJ/h | ${C}_{{A}_{s}}=1.22$ kmol/m${}^{3}$ |

${T}_{s}=438$ K |

${\mathit{\rho}}_{\mathit{e}}/\mathit{\rho}$ | ${\mathit{s}}_{\mathit{e}}/\mathit{s}$ | $\mathit{P}({\mathit{A}}_{\mathit{V}})$ |
---|---|---|

$0.98$ | $0.99$ | $14.0\%$ |

$0.95$ | $0.99$ | $63.1\%$ |

$0.92$ | $0.99$ | $82.0\%$ |

$0.92$ | $0.97$ | $82.8\%$ |

$0.92$ | $0.95$ | $83.6\%$ |

$0.92$ | $0.92$ | $85.8\%$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wu, Z.; Durand, H.; Christofides, P.D.
Safeness Index-Based Economic Model Predictive Control of Stochastic Nonlinear Systems. *Mathematics* **2018**, *6*, 69.
https://doi.org/10.3390/math6050069

**AMA Style**

Wu Z, Durand H, Christofides PD.
Safeness Index-Based Economic Model Predictive Control of Stochastic Nonlinear Systems. *Mathematics*. 2018; 6(5):69.
https://doi.org/10.3390/math6050069

**Chicago/Turabian Style**

Wu, Zhe, Helen Durand, and Panagiotis D. Christofides.
2018. "Safeness Index-Based Economic Model Predictive Control of Stochastic Nonlinear Systems" *Mathematics* 6, no. 5: 69.
https://doi.org/10.3390/math6050069