# Forecast-Triggered Model Predictive Control of Constrained Nonlinear Processes with Control Actuator Faults

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 3. An Auxiliary Model-Based Fault-Tolerant Controller

#### 3.1. Controller Synthesis and Analysis under Continuous State Measurements

**Remark**

**1.**

**Remark**

**2.**

#### 3.2. Characterization of Closed-Loop Stability under Discretely Sampled State Measurements

**Theorem**

**1.**

**Proof.**

**Remark**

**3.**

**Remark**

**4.**

## 4. Design and Analysis of Lyapunov-Based Fault-Tolerant MPC

**Theorem**

**2.**

**Proof.**

**Remark**

**5.**

**Remark**

**6.**

**Remark**

**7.**

## 5. Fault-Tolerant MPC Implementation Using Forecast-Triggered Communication

Algorithm 1: Forecast-triggered sensor–controller communication strategy |

Initialize $\widehat{\mathbf{x}}\left({t}_{0}\right)=\mathbf{x}\left({t}_{0}\right)\in \mathsf{\Omega}$ and set $k=0$, $p=0$ |

Solve Equation (28) for $[{t}_{0},{t}_{1})$ and implement the first step of the control sequence |

if $\widehat{\mathbf{x}}\left({t}_{k+1}\right)\in \mathsf{\Omega}\backslash {\mathsf{\Omega}}^{\prime}$ then |

Calculate $\overline{V}\left(\mathbf{x}\left({t}_{k+2}\right)\right)$ (estimate of $V\left(\mathbf{x}\left({t}_{k+2}\right)\right)$) using Equation (40a) and $V\left(\mathbf{x}\left({t}_{k+1}\right)\right)$ |

else |

Calculate $\overline{V}\left(\mathbf{x}\left({t}_{k+2}\right)\right)$ (estimate of $V\left(\mathbf{x}\left({t}_{k+2}\right)\right)$) using Equation (40b) and $\mathbf{e}\left({t}_{k+1}\right)$ |

end if |

if $\overline{V}\left(\mathbf{x}\left({t}_{k+2}\right)\right)<V\left(\mathbf{x}\left({t}_{k+1}\right)\right)$ then |

Solve Equation (28) without Equation (28d) for $[{t}_{k+1},{t}_{k+2})$ |

else if $\overline{V}\left(\mathbf{x}\left({t}_{k+2}\right)\right)\ge V\left(\mathbf{x}\left({t}_{k+1}\right)\right)$ and $\overline{V}\left(\mathbf{x}\left({t}_{k+2}\right)\right)\le {\delta}^{\prime}$ then |

Solve Equation (28) without Equation (28d) for $[{t}_{k+1},{t}_{k+2})$ |

else |

Solve Equation (28) for $[{t}_{k+1},{t}_{k+2})$ and set $p=k+1$ |

end if |

Implement the first step of the control sequence on $[{t}_{k+1},{t}_{k+2})$ |

Set $k=k+1$ and go to step 3 |

**Remark**

**8.**

**Remark**

**9.**

**Remark**

**10.**

**Remark**

**11.**

## 6. Simulation Case Study: Application to a Chemical Process

^{3}, ${T}^{s}=395.3$ K) in the presence of input constraints, control actuator faults and limited sensor–controller communication. The manipulated input is chosen as the inlet reactant concentration, i.e., $\mathbf{u}={C}_{A0}-{C}_{A0}^{s}$, subject to the constraint $\parallel \mathbf{u}\parallel \le 0.5\phantom{\rule{0.166667em}{0ex}}\mathrm{mol}/{\mathrm{m}}^{3}$, where ${C}_{A0}^{s}$ is the nominal steady state value of ${C}_{A0}$, and control actuator faults. We define the displacement variables $\mathbf{x}={\left[{x}_{1}\phantom{\rule{0.277778em}{0ex}}{x}_{2}\right]}^{T}={[{C}_{A}-{C}_{A}^{s}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}T-{T}^{s}]}^{T}$, where the superscript s denotes the steady state value, which places the nominal equilibrium point of the system at the origin. A quadratic Lyapunov function candidate of the form $V\left(\mathbf{x}\right)={\mathbf{x}}^{T}\mathbf{P}\mathbf{x}$, where:

#### 6.1. Characterization of the Fault-Tolerant Stabilization Region

#### 6.2. Active Fault Accommodation in the Implementation of MPC

#### 6.3. Implementation of Fault-Tolerant MPC Using Forecast-Triggered Sensor–Controller Communication

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Flowchart of implementation of the forecast-triggered communication strategy. MPC: model predictive control.

**Figure 2.**Estimates of the region of guaranteed fault-tolerant stabilization under MPC: (

**a**) ${\mathsf{\Omega}}_{1}$ represents the estimate when $\widehat{\theta}=1$ (blue level set); (

**b**) ${\mathsf{\Omega}}_{2}$ represents the estimate when $\widehat{\theta}=0.8$ (green level set); (

**c**) ${\mathsf{\Omega}}_{3}$ represents the estimate when $\widehat{\theta}=0.5$ (purple level set).

**Figure 3.**Comparison of the evolutions of: (

**a**) the closed-loop state trajectory; (

**b**) the closed-loop reactor temperature T; (

**c**) the closed-loop reactant concentration ${C}_{A}$; and (

**d**) the manipulated input, ${C}_{A0}$, for three different operating scenarios: one in the absence of any faults (black profiles); one in the presence of a fault but without implementing any fault accommodation (red profiles); and one in the presence of a fault and implementing fault accommodation (blue profiles).

**Figure 4.**Illustration of how the forecast-triggered sensor–controller communication strategy is implemented. The top plot depicts current values of the Lyapunov function (red squares), projected values of the Lyapunov function (blue circles) and update events (solid blue dots) at different sampling times. The bottom plot depicts the time instances when the model state is updated.

**Figure 5.**Closed-loop reactor temperature profiles (

**a**); and model state update instances (

**b**) under the conventional (blue) and forecast-triggered MPC schemes (red).

**Figure 6.**Comparison of the performance of forecast-triggered MPC scheme under fault-free conditions (black), an accommodated fault scenario (blue) and an unaccommodated fault scenario (red): (

**a**) closed-loop temperature profiles; (

**b**) reactant concentration profiles; (

**c**): manipulated input profile; and (

**d**): model update frequency.

**Table 1.**Process and model parameter values for the continuous stirred tank reactor (CSTR) example in Equation (41).

Parameter | Process | Model |
---|---|---|

F (m^{3}/h) | $3.34\times {10}^{-3}$ | $3.34\times {10}^{-3}$ |

V (m^{3}) | $0.1$ | $0.1$ |

${k}_{0}$ (h^{−1}) | $1.2\times {10}^{9}$ | $1.2\times {10}^{9}$ |

E (KJ/Kmol) | $8.314\times {10}^{4}$ | $8.30\times {10}^{4}$ |

R (KJ/Kmol/K) | $8.314$ | $8.314$ |

$\rho $ (Kg/m^{3}) | 1000 | 1010 |

${C}_{p}$ (KJ/Kg/K) | $0.239$ | $0.24$ |

$\Delta H$ (KJ/Kmol) | $-4.78\times {10}^{4}$ | $-4.8\times {10}^{4}$ |

${C}_{A0}^{s}$ (Kmol/m^{3}) | $0.79$ | $0.79$ |

${T}_{0}^{s}$ (K) | $352.6$ | $352.6$ |

${Q}^{s}$ (KJ/h) | 0 | 0 |

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**MDPI and ACS Style**

Xue, D.; El-Farra, N.H.
Forecast-Triggered Model Predictive Control of Constrained Nonlinear Processes with Control Actuator Faults. *Mathematics* **2018**, *6*, 104.
https://doi.org/10.3390/math6060104

**AMA Style**

Xue D, El-Farra NH.
Forecast-Triggered Model Predictive Control of Constrained Nonlinear Processes with Control Actuator Faults. *Mathematics*. 2018; 6(6):104.
https://doi.org/10.3390/math6060104

**Chicago/Turabian Style**

Xue, Da, and Nael H. El-Farra.
2018. "Forecast-Triggered Model Predictive Control of Constrained Nonlinear Processes with Control Actuator Faults" *Mathematics* 6, no. 6: 104.
https://doi.org/10.3390/math6060104