#
A Nonlinear Systems Framework for Cyberattack Prevention for Chemical Process Control Systems ^{†}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Notation

#### 2.2. Class of Systems

**Assumption**

**1.**

#### 2.3. Model Predictive Control

#### 2.4. Lyapunov-Based Model Predictive Control

## 3. A Nonlinear Dynamic Systems Perspective on Cyberattacks

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Remark**

**1.**

**Remark**

**2.**

## 4. Defining Cyberattack Resilience Against Specific Attack Types: Sensor Measurement Falsification in Feedback Control Loops

**Definition**

**4.**

## 5. Control Design Concepts for Deterring Sensor Measurement Falsification Cyberattacks on Safety: Benefits, Limitations, and Perspectives

#### 5.1. Motivating Example: The Need for Cyberattack-Resilient Control Designs

^{®}.

#### 5.2. Deterring Sensor Measurement Falsification Cyberattacks on Safety: Creating Non-Intuitive Controller Outputs

**Remark**

**3.**

#### 5.2.1. Problems with Creating Non-Intuitive Controller Outputs

**Remark**

**4.**

**Remark**

**5.**

#### 5.3. Deterring Sensor Measurement Falsification Cyberattacks on Safety: Creating Unpredictable Controller Outputs

#### 5.3.1. Creating Unpredictable Controller Outputs: Incorporating Randomness in LMPC Design

**Remark**

**6.**

#### 5.3.1.1. Stability Analysis of Randomized LMPC

**Proposition**

**1.**

**Proposition**

**2.**

**Proposition**

**3.**

**Theorem**

**1.**

**Proof.**

**Remark**

**7.**

**Remark**

**8.**

#### 5.3.2. Problems with Incorporating Randomness in LMPC Design

**Remark**

**9.**

#### 5.3.3. Creating Unpredictable Controller Outputs: Other Types of Randomness in MPC Design

#### 5.4. Deterring Sensor Measurement Falsification Cyberattacks on Safety: Using Open-Loop Controller Outputs

#### 5.4.1. Using Open-Loop Controller Outputs: Integration with LMPC

#### Stability Analysis of Open-Loop Control Integrated with LMPC

**Proposition**

**4.**

**Theorem**

**2.**

**Proof.**

**Remark**

**10.**

#### 5.4.2. Problems with Integrating Open-Loop Control and LMPC

#### 5.5. Deterring Sensor Measurement Falsification Cyberattacks on Safety: Perspectives

## 6. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**State-space trajectory showing the state trajectory in 10 sampling periods with the falsified state measurements determined through optimization applied at every sampling time, in the absence of disturbances.

**Figure 3.**Intersecting stability regions with two different potential initial conditions $x\left({t}_{k}\right)={x}_{a}$ and $x\left({t}_{k}\right)={x}_{b}$.

**Figure 6.**State-space trajectories under the single LMPC for the CSTR of Equations (46) and (47). The figure indicates that the closed-loop trajectory settled on the boundary of ${\mathrm{\Omega}}_{{\rho}_{e,1}}$ to optimize the objective function while meeting the constraints. For simplicity, only one level set for each of the ${n}_{p}$ potential LMPC’s is shown (${\mathrm{\Omega}}_{{\rho}_{i}}$ is shown if ${V}_{i}\ne {V}_{1}$, and ${\mathrm{\Omega}}_{{\rho}_{e,i}}$ is shown if ${V}_{i}={V}_{1}$, $i>1$).

**Figure 9.**State-space trajectories under the randomized LMPC implementation strategy for the CSTR of Equations (46) and (47). For simplicity, only one level set for each of the ${n}_{p}$ potential LMPC’s is shown (${\mathrm{\Omega}}_{{\rho}_{i}}$ is shown if ${V}_{i}\ne {V}_{1}$, and ${\mathrm{\Omega}}_{{\rho}_{e,i}}$ is shown if ${V}_{i}={V}_{1}$, $i>1$).

**Figure 10.**Scatter plot showing the control law chosen (i in Table 3) in each sampling period by the randomized LMPC implementation strategy.

**Figure 11.**State-space trajectories for all of the situations in Table 4. The numbers in the caption represent the seed values for rng. ‘S’ represents the single LMPC.

**Figure 12.**Trajectories of ${u}_{1}$, ${u}_{2}$, and ${V}_{1}$ under the randomized LMPC implementation strategy for rng(20) (denoted by ‘Randomized’ in the figure) and under the single LMPC (denoted by ‘Single’ in the figure). The value of ${\rho}_{1}$ is denoted by the horizontal line in the plot for the value of ${V}_{1}$. The bottom plot indicates the controller selected by the randomized LMPC implementation strategy at each of the 10 sampling times in the simulation.

**Table 1.**Steady-state values for the states of the Tennessee Eastman Process [78].

Parameter | Value | Unit |
---|---|---|

${N}_{A,s}$ | 44.49999958429348 | kmol |

${N}_{B,s}$ | 13.53296996509594 | kmol |

${N}_{C,s}$ | 36.64788062995841 | kmol |

${N}_{D,s}$ | 110.0 | kmol |

${X}_{1,s}$ | 60.95327313484253 | % |

${X}_{2,s}$ | 25.02232231706676 | % |

${X}_{3,s}$ | 39.25777017606444 | % |

${X}_{4,s}$ | 47.03024823457651 | % |

${u}_{1,s}$ | 60.95327313484253 | % |

${u}_{2,s}$ | 25.02232231706676 | % |

${u}_{3,s}$ | 39.25777017606444 | % |

${V}_{\%,sp}$ | 44.17670682730923 | % |

${F}_{1,s}$ | 201.43 | kmol/h |

${F}_{2,s}$ | 5.62 | kmol/h |

${F}_{3,s}$ | 7.05 | kmol/h |

${F}_{4,s}$ | 100 | kmol/h |

${P}_{s}$ | 2700 | kPa |

${y}_{A3,s}$ | 0.47 | - |

${y}_{B3,s}$ | 0.1429 | - |

${y}_{C3,s}$ | 0.3871 | - |

${K}_{c,1}$ | 0.1 | % h/kmol |

${\tau}_{I,1}$ | 1 | h |

${K}_{c,2}$ | 2 | % |

${\tau}_{I,2}$ | 3 | h |

${K}_{c,3}$ | –0.25 | %/kPa |

${\tau}_{I,3}$ | 1.5 | h |

${K}_{c,4}$ | 0.7 | kmol/kPa·h |

${\tau}_{I,4}$ | 3 | h |

Parameter | Value | Unit |
---|---|---|

V | 1 | m${}^{3}$ |

${T}_{0}$ | 300 | K |

${C}_{p}$ | $0.231$ | kJ/kg·K |

${k}_{0}$ | $8.46\times {10}^{6}$ | m${}^{3}$/h·kmol |

F | 5 | m${}^{3}$/h |

${\rho}_{L}$ | 1000 | kg/m${}^{3}$ |

E | $5\times {10}^{4}$ | kJ/kmol |

${R}_{g}$ | $8.314$ | kJ/kmol·K |

$\Delta H$ | $-1.15\times {10}^{4}$ | kJ/kmol |

i | ${\mathit{P}}_{11}$ | ${\mathit{P}}_{12}$ | ${\mathit{P}}_{22}$ | ${\mathit{\rho}}_{\mathit{i}}$ | ${\mathit{\rho}}_{\mathit{e},\mathit{i}}$ |
---|---|---|---|---|---|

1 | 1200 | 5 | 0.1 | 180 | 144 |

2 | 2000 | –20 | 1 | 180 | 144 |

3 | 1500 | –20 | 10 | 180 | 144 |

4 | 0.2 | 0 | 2000 | 180 | 144 |

5 | 1200 | 5 | 0.1 | 180 | 100 |

6 | 1200 | 5 | 0.1 | 180 | 130 |

7 | 1200 | 5 | 0.1 | 180 | 30 |

**Table 4.**Approximate time after which ${x}_{2}>55$ K for various seed values of rng for the randomized LMPC design subjected to a cyberattack on the sensors determined in Section 5.2.1.

Seed | Time ${\mathit{x}}_{2}>55$ K (h) |
---|---|

5 | 0.0143 |

10 | 0.0148 |

15 | 0.0146 |

20 | 0.0324 |

25 | 0.0146 |

30 | 0.0142 |

35 | 0.0143 |

40 | 0.0147 |

45 | 0.0248 |

50 | 0.0231 |

**Table 5.**Approximate time after which ${x}_{2}>55$ K for various seed values of rng for the randomized LMPC design subjected to a cyberattack on the sensors with ${x}_{1}=0.0632$ kmol/m${}^{3}$ and ${x}_{2}=21.2056\phantom{\rule{3.33333pt}{0ex}}\mathrm{K}$.

Seed | Time ${\mathit{x}}_{2}>55$ K (h) |
---|---|

5 | 0.0674 |

10 | 0.0458 |

15 | 0.0555 |

20 | 0.0767 |

25 | 0.0569 |

30 | 0.0418 |

35 | 0.0457 |

40 | 0.0874 |

45 | 0.0580 |

50 | 0.0950 |

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Durand, H.
A Nonlinear Systems Framework for Cyberattack Prevention for Chemical Process Control Systems ^{†}. *Mathematics* **2018**, *6*, 169.
https://doi.org/10.3390/math6090169

**AMA Style**

Durand H.
A Nonlinear Systems Framework for Cyberattack Prevention for Chemical Process Control Systems ^{†}. *Mathematics*. 2018; 6(9):169.
https://doi.org/10.3390/math6090169

**Chicago/Turabian Style**

Durand, Helen.
2018. "A Nonlinear Systems Framework for Cyberattack Prevention for Chemical Process Control Systems ^{†}" *Mathematics* 6, no. 9: 169.
https://doi.org/10.3390/math6090169