# Dynamic Simulation of Multiple Launch Rocket System Marching Fire Based on the Fuzzy Adaptive Sliding Mode Control

^{*}

## Abstract

**:**

## 1. Introduction

- The co-simulation dynamic control model of the motor-mechanism coupling MLRS was established.
- The FASMC controller was developed considering the systematic nonlinearity, in which the FASMC was built to adapt to the uncertainties of the system.
- The proposed controller is first introduced and successfully applied to the field of the dynamic control for MLRS marching fire considering the occurrence of uncertainties.

## 2. Nonlinear Dynamic Model of MLRS Marching Fire

#### 2.1. Mechanical System Dynamic Modeling

#### 2.2. 3-D Road Roughness Model

#### 2.3. Permanent Synchronous Motor Modeling

#### 2.4. The Co-Simulation Model of the MLRS

## 3. Control System Design and Stability Analysis

#### 3.1. Fuzzy Logic System

#### 3.2. Design of Fuzzy Adaptive Sliding Mode Controller

## 4. Simulation and Analysis

## 5. Conclusions

- (1)
- FASM demonstrates superior robustness and accuracy in commanding signals. In comparison to PID control, the adjustment time was reduced by 30% and compared to SMC, it was reduced by 6.2%. Additionally, the steady-state error and shock disturbance were decreased by 49% and 67%, respectively, in comparison to PID control and by 34% and 39%, respectively, in comparison to SMC.
- (2)
- FASMC significantly improved the chatter characteristics of SMC, reducing the frequency of chatter and decreasing the amplitude by 75% compared to SMC.
- (3)
- FASMC also significantly improved the tracking accuracy of MRLS, controlling the tracking error under F-level pavement excitation within 10 mils, resulting in a performance improvement of 74% over PID control and 50% over SMC control. The study found that under Class D and F pavements, pavement excitation exceeded impact disturbance as the main factor affecting accuracy.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Three dimensional road excitation. (

**a**–

**c**) The magnitude of the roads in classes A, D, and F.

**Figure 6.**Simulated results of different controllers in step signal. (

**a**–

**c**) Angular displacements of azimuth for class A, D, and F roads.

**Figure 7.**Simulated results of different controllers in step signal. (

**a**–

**c**) Angular displacements of pitch for class A, D, and F roads.

**Figure 8.**Comparison of SMC and FASMC chattering amplitude values. (

**a**) Chattering amplitude value of SMC, (

**b**) Chattering amplitude value of FASMC.

**Figure 9.**Simulated results of different controllers in sine signal. (

**a**–

**c**) Angular displacements of azimuth for class A, D, and F roads.

**Figure 10.**Simulated results of different controllers in sine signal. (

**a**–

**c**) Angular error of azimuth for class A, D, and F roads.

**Figure 11.**Simulated results of different controllers in sine signal. (

**a**–

**c**) Angular displacements of pitch for class A, D, and F roads.

**Figure 12.**Simulated results of different controllers in sine signal. (

**a**–

**c**) Angular error of pitch for class A, D, and F roads.

Road Grade | ${\mathit{G}}_{\mathit{q}}\left(0\right)/{10}^{-6}{\mathbf{m}}^{3}$ | ||
---|---|---|---|

Upper Limit | Mean Value | Lower Limit | |

A | 8 | 16 | 32 |

D | 512 | 1024 | 2048 |

F | 8192 | 16,384 | 32,768 |

Parameter of PMSM | Value of Pitch/Azimuth |
---|---|

Inertia(converted to motor output shaft) (kg∙m^{2}) | J = 3.569 × 10^{−3}/4.369 × 10^{−2} |

Electromagnetic torque coefficient (N∙m/A) | K_{t} = 1.11/1.34 |

Damping coefficient (N∙m/s) | B = 3.34 × 10^{−3} |

Stator resistor $(\mathrm{\Omega})$ | R_{S} = 2.875 |

Winding inductance (H) | L_{d} = L_{q} = 8.5 × 10^{−3} |

Rated current (A) | I_{e} = 6.4/9.9 |

Rated rotation speed (RPM) | n = 3000/2500 |

Maximum allowable current (A) | I_{max} = 12.8/19.8 |

Polar logarithm | P_{n} = 4 |

PID | SMC | FASMC | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

K_{p} | K_{i} | K_{d} | c | k | ε | D | C_{1} | C_{2} | ${\mathsf{\eta}}_{1}$ | ${\mathsf{\eta}}_{2}$ |

520 | 0.05 | 13 | 30 | 15 | 0.05 | 15 | 0.01 | 15 | 200 | 0.5 |

Class of Road | Direction | Adjustment Time (s) | Maximum Steady State Error (mil) | Impact Disturbance (mil) | ||||||
---|---|---|---|---|---|---|---|---|---|---|

A | D | F | A | D | F | A | D | F | ||

PID | Pitch | 1.03 | 1.10 | 1.15 | 0.67 | 4.78 | 9.55 | 4.78 | 4.82 | 14.30 |

Azimuth | 1.01 | 1.03 | 1.08 | 0.47 | 2.87 | 7.64 | 5.73 | 5.84 | 8.62 | |

SMC | Pitch | 0.73 | 0.78 | 0.82 | 0.12 | 4.05 | 7.35 | 2.35 | 2.30 | 7.64 |

Azimuth | 0.72 | 0.75 | 0.80 | 0.08 | 1.53 | 2.05 | 1.85 | 2.03 | 2.32 | |

FASMC | Pitch | 0.71 | 0.75 | 0.77 | 0.10 | 3.65 | 4.86 | 2.05 | 2.20 | 4.65 |

Azimuth | 0.71 | 0.74 | 0.75 | 0.07 | 1.35 | 2.02 | 1.73 | 2.01 | 1.05 |

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**MDPI and ACS Style**

Qu, P.; Sun, Z.; Li, Q.; Zhang, J.; Liu, P.; Zhou, D. Dynamic Simulation of Multiple Launch Rocket System Marching Fire Based on the Fuzzy Adaptive Sliding Mode Control. *Machines* **2023**, *11*, 427.
https://doi.org/10.3390/machines11040427

**AMA Style**

Qu P, Sun Z, Li Q, Zhang J, Liu P, Zhou D. Dynamic Simulation of Multiple Launch Rocket System Marching Fire Based on the Fuzzy Adaptive Sliding Mode Control. *Machines*. 2023; 11(4):427.
https://doi.org/10.3390/machines11040427

**Chicago/Turabian Style**

Qu, Pu, Zhiqun Sun, Qiang Li, Jiabo Zhang, Pengzhan Liu, and Dongmo Zhou. 2023. "Dynamic Simulation of Multiple Launch Rocket System Marching Fire Based on the Fuzzy Adaptive Sliding Mode Control" *Machines* 11, no. 4: 427.
https://doi.org/10.3390/machines11040427