# A Novel Hybrid Data-Driven Modeling Method for Missiles

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Analysis of Problems in Missile Mathematical Model

#### Mechanism Modeling and Analysis

- The earth is a homogeneous sphere, and the rotation of the earth is neglected;
- Missile has an axisymmetric layout form with constant mass and rotational inertia;
- This paper studies the case of missile without engine thrust.

## 3. Hybrid Data-Driven Modeling Method

#### 3.1. Introduction to Usage Scenario

#### 3.2. Traditional Neural Network Modeling

#### 3.3. A Novel Hybrid Data-Driven Modeling Method

## 4. Hybrid Data-Driven Modeling for Missle

- Step 1:
- Flight test data Preprocessing.
- Step 2:
- Establish acceleration model.
- Step 3:
- Establish angular acceleration model.
- Step 4:
- Establish a missile hybrid model.

#### 4.1. Acceleration and Angular Acceleration Modeling

- (i)
- Identify, eliminate, and correct the outliers to eliminate signal anomalies, which are caused by external disturbances during flight test;
- (ii)
- Data smoothing filtering are used to filter out measurement noise;
- (iii)
- Correct the time delay between each measurement data.

#### 4.2. Establishment of Missile Hybrid Model

## 5. Simulation Result and Analysis

#### 5.1. Feasibility Analysis

#### 5.2. Credibility Analysis

- Step 1:
- Establish the mathematical model of the missile by the mechanism modeling method, and the data collected by the mathematical model (pseudo-missile model) is taken as the flight test data of realistic missile.
- Step 2:
- Determine the main random disturbance factors encountered during the missile flight and determine its distribution law;
- Step 3:
- Simulation target practice of missile is done with Monte Carlo method, and the data of the n collected random trajectories is taken as the flight test data of n ballistic trajectory;
- Step 4:
- According to the hybrid data-driven modeling method proposed in this paper, the data of each random ballistic trajectory collected in step 3 is used to establish a missile hybrid model.
- Step 5:
- Compare the output results of n missile hybrid models with the pseudo-missile model, and statistical analysis to verify the accuracy and credibility of missile hybrid model.

**Remark 1:**The Monte Carlo method is a kind of numerical method for solving approximate solutions of mathematical, physical and engineering problems through statistical experiments or stochastic simulations of random variables, also known as statistical experiment or stochastic simulation [5].

- (i)
- Internal factors: missile structural parameter errors, mainly including mass deviation, moment of inertia deviation, and aerodynamic coefficient deviation.
- (ii)
- External factors: the influence of ballistic wind.

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## References

- Siouris, G.M. Missile Guidance and Control Systems; Springer Science & Business Media: New York, NY, USA, 2004; pp. 53–61. [Google Scholar]
- Zarchan, P. Tactical and Strategic Missile Guidance; American Institute of Aeronautics and Astronautics, Inc.: Reston, VA, USA, 2012; pp. 1–50. [Google Scholar]
- Yanushevsky, R. Modern Missile Guidance; CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]
- Blakelock, J.H. Automatic Control of Aircraft and Missiles; John Wiley & Sons: Hoboken, NJ, USA, 1991. [Google Scholar]
- Xingfang, Q.; Ruixiong, L.; Yanan, Z. Missile Flight Dynamics; Beijing Institute of Technology Press: Beijing, China, 2011; pp. 28–48. [Google Scholar]
- Xu, Y.; Wang, Z.; Gao, B. Six-degree-of-freedom digital simulations for missile guidance and control. Math. Probl. Eng.
**2015**, 2015, 829473. [Google Scholar] [CrossRef] [Green Version] - Yuan, G.; Liangxian, G.; Lei, P. Modeling and simulating dynamics of missiles with deflectable nose control. Chin. J. Aeronaut.
**2009**, 22, 474–479. [Google Scholar] [CrossRef] [Green Version] - Tan, W.; Packard, A.K.; Balas, G.J. Quasi-LPV modeling and LPV control of a generic missile. In Proceedings of the 2000 American Control Conference, Chicago, IL, USA, 28–30 June 2000; pp. 3692–3696. [Google Scholar]
- Li, K.; Liu, P.; Chen, N. Modeling and Simulation of Engagement Process of Antiaircraft Missile for Training Simulation. Appl. Mech. Mater.
**2014**, 541, 1304–1308. [Google Scholar] [CrossRef] - Fancheng, K.; Wei, C.; Zhang, D.; Zhenzhou, B. Research on the Missile Ballistic System Simulation Based on Matlab. In Proceedings of the 2017 International Conference on Electronic Industry and Automation (EIA 2017); Atlantis Press: Paris, France, 2017. [Google Scholar]
- Chen, J.; Li, S.G.; Lei, J.W. Modeling and Simulation of Standard II Missiles Intercepting a Low Target. Adv. Mater. Res.
**2014**, 846, 1505–1508. [Google Scholar] [CrossRef] - Jianguo, G.; Yuanjun, H.; Jun, Z. Modeling and simulation research of missile with deflectable nose. In Proceedings of the 2009 IEEE International Conference on Intelligent Computing and Intelligent Systems, Shanghai, China, 20–22 November 2009; pp. 86–89. [Google Scholar]
- Wei, W. Development of an Effective System Identification and Control Capability for Quad-Copter UAVs. Ph.D. Thesis, University of Cincinnati, Engineering and Applied Science, Cincinnati, OH, USA, 2015. [Google Scholar]
- Tang, S.; Zhao, X.; Zhang, L.; Zheng, Z. System identification of heave-yaw dynamics for small-scale unmanned helicopter using genetic algorithm. In Proceedings of the 32nd Chinese Control Conference, Xi’an, China, 26–28 July 2013; pp. 1797–1802. [Google Scholar]
- Qing, L.; Yahui, L.; Wenshuai, Z. Analysis method of aerodynamic model of simulator based on flight test. Flight Dyn.
**2015**, 3, 265–268. [Google Scholar] - Dimitriadis, G.; Cooper, J.E. Flutter prediction from flight flutter test data. J. Aircr.
**2001**, 38, 355–367. [Google Scholar] [CrossRef] - Nissim, E.; Gilyard, G. Method for experimental determination of flutter speed by parameter identification. In Proceedings of the 30th Structures, Structural Dynamics and Materials Conference, Mobile, AL, USA, 3–5 April 1989; p. 1324. [Google Scholar]
- Takens, F. Detecting strange attractors in turbulence. In Dynamical Systems and Turbulence; Springer: Berlin/Heidelberg, Germany, 1981; pp. 366–381. [Google Scholar]
- Narendra, K.S.; Parthasarathy, K. Identification and control of dynamical systems using neural networks. IEEE Trans. Neural Netw.
**1990**, 1, 4–27. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Barreto, G.A.; Araujo, A.F. Identification and control of dynamical systems using the self-organizing map. IEEE Trans. Neural Netw.
**2004**, 15, 1244–1259. [Google Scholar] [CrossRef] - Chen, S.; Billings, S.; Grant, P. Non-linear system identification using neural networks. Int. J. Control
**1990**, 51, 1191–1214. [Google Scholar] [CrossRef] - Patra, J.C.; Pal, R.N.; Chatterji, B.; Panda, G. Identification of nonlinear dynamic systems using functional link artificial neural networks. IEEE Trans. Syst. Man, Cybern. Part (Cybernetics)
**1999**, 29, 254–262. [Google Scholar] [CrossRef] - Benyassi, M.; Brouri, A. Nonlinear Systems Identification With Discontinuous Nonlinearity. In Proceedings of the 2017 European Conference on Electrical Engineering and Computer Science (EECS), Bern, Switzerland, 17–19 November 2017; pp. 311–313. [Google Scholar]
- Brewick, P.T.; Masri, S.F.; Carboni, B.; Lacarbonara, W. Enabling reduced-order data-driven nonlinear identification and modeling through naive elastic net regularization. Int. J. -Non-Linear Mech.
**2017**, 94, 46–58. [Google Scholar] [CrossRef] - Guoqiang, Y.; Weiguang, L.; Hao, W. Study of RBF neural network based on PSO algorithm in nonlinear system identification. In Proceedings of the 2015 8th International Conference on Intelligent Computation Technology and Automation (ICICTA), Nanchang, China, 14–15 June 2015; pp. 852–855. [Google Scholar]
- Yan, L.; Zhang, R. Research on aircraft modeling based on neural network. In Proceedings of the Tactical Missile Control Technology, Hangzhou, China, 23–25 March 2012; pp. 11–15. [Google Scholar]
- San Martin, R.; Barrientos, A.; Gutierrez, P.; del Cerro, J. Unmanned aerial vehicle (UAV) modelling based on supervised neural networks. In Proceedings of the 2006 IEEE International Conference on Robotics and Automation, Orlando, FL, USA, 15–19 May 2006; pp. 2497–2502. [Google Scholar]
- San Martin, R.; Barrientos, A.; Gutierrez, P.; del Cerro, J. Neural networks training architecture for uav modelling. In Proceedings of the 2006 World Automation Congress, Budapest, Hungary, 24–26 July 2006; pp. 1–6. [Google Scholar]
- Meng, Y.; Yu, S.; Zhang, J.; Qin, J.; Dong, Z.; Lu, G.; Pang, H. Hybrid modeling based on mechanistic and data-driven approaches for cane sugar crystallization. J. Food Eng.
**2019**, 257, 44–55. [Google Scholar] [CrossRef] - Shitong, Y.; Pu, H.; Ming, S. Research on multivariable system modelingmethod based on big data. J. Syst. Simul.
**2014**, 26, 1454–1459. [Google Scholar] - Hunt, K.J.; Sbarbaro, D.; Żbikowski, R.; Gawthrop, P.J. Neural networks for control systems-a survey. Automatica
**1992**, 28, 1083–1112. [Google Scholar] [CrossRef] - Hou, Z.-S.; Wang, Z. From model-based control to data-driven control: Survey, classification and perspective. Inf. Sci.
**2013**, 235, 3–35. [Google Scholar] [CrossRef] - Tang, K.; Wang, W.; Meng, Y.; Zhang, M. Flight control and airwake suppression algorithm for carrier landing based on model predictive control. Trans. Inst. Meas. Control.
**2019**, 41, 2205–2213. [Google Scholar] [CrossRef] - Xiao, L.; Xu, M.; Chen, Y.; Chen, Y. Hybrid Grey Wolf Optimization Nonlinear Model Predictive Control for Aircraft Engines Based on an Elastic BP Neural Network. Appl. Sci.
**2019**, 9, 1254. [Google Scholar] [CrossRef] [Green Version] - Ming, C.; Wang, X.; Sun, R. A novel non-singular terminal sliding mode control-based integrated missile guidance and control with impact angle constraint. Aerosp. Sci. Technol.
**2019**, 94, 105368. [Google Scholar] [CrossRef] - Chen, C.B.; Liu, B.; He, N.; Gao, S.; Pan, Q. PID Control Based Missile Sub-channel Simulation. Proc. Adv. Mater. Res.
**2012**, 433, 7011–7016. [Google Scholar] [CrossRef] - Bingwei, Y. Formulaization expression of standard atmospheric parameters. J. Astronaut.
**1983**, 1, 86–89. [Google Scholar] - Rumelhart, D.E.; Hinton, G.E.; Williams, R.J. Learning representations by back-propagating errors. Nature
**1986**, 323, 533–536. [Google Scholar] [CrossRef] - Rafiq, M.; Bugmann, G.; Easterbrook, D. Neural network design for engineering applications. Comput. Struct.
**2001**, 79, 1541–1552. [Google Scholar] [CrossRef] - Hairer, E.; Lubich, C.; Roche, M. The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods; Springer: Berlin/Heidelberg, Germany, 2006; Volume 1409. [Google Scholar]
- Butcher, J.C. A history of Runge-Kutta methods. Appl. Numer. Math.
**1996**, 20, 247–260. [Google Scholar] [CrossRef]

**Figure 12.**Simulation results of missile hybrid model and mathematical model: (

**a**) Position in Y-X direction; (

**b**) Position in Z-X direction; (

**c**) Mach number; (

**d**) Pitch angle; (

**e**) Yaw angle; (

**f**) Pitching angular velocity.

Output of Missile | MSE | MAPE |
---|---|---|

x | $8.83{e}^{-5}$ | $6.62{e}^{-7}$ |

y | $1.99{e}^{-4}$ | $3.80{e}^{-5}$ |

z | $5.17{e}^{-5}$ | $1.36{e}^{-2}$ |

${v}_{x}$ | $3.66{e}^{-7}$ | $2.27{e}^{-6}$ |

${v}_{y}$ | $7.10{e}^{-6}$ | $-3.96{e}^{-5}$ |

${v}_{z}$ | $8.78{e}^{-7}$ | $-9.87{e}^{-4}$ |

${\omega}_{x}$ | $1.19{e}^{-7}$ | $0.19$ |

${\omega}_{y}$ | $4.89{e}^{-6}$ | $3.96{e}^{-5}$ |

${\omega}_{z}$ | $4.7{e}^{-3}$ | $0.26$ |

$\phi $ | $2.38$ | $6.28{e}^{-2}$ |

$\psi $ | $9.40{e}^{-7}$ | $2.30{e}^{-3}$ |

$\gamma $ | $8.88{e}^{-11}$ | $0.24{e}^{-4}$ |

Category | Disturb | Distribution Characteristics | Distributed Parameter |
---|---|---|---|

Missile body | Error of missile mass m | Normal distribution | $N(0,{0.007}^{2})$ |

Error of rotational inertias $\overrightarrow{I}$ | Normal distribution | $N(0,{0.02}^{2})$ | |

Aerodynamic coefficient | Error of axial force coefficient ${C}_{x}1$ | Normal distribution | $N(0,{0.067}^{2})$ |

Error of derivative of lateral force coefficient ${C}_{z1}^{\beta}$ | Normal distribution | $N(0,{0.1}^{2})$ | |

Error of derivative of moment coefficient ${m}_{z1}^{\alpha}$, ${m}_{\mathrm{z}1}^{{\omega}_{\mathrm{z}1}}$, ${m}_{x1}^{\delta}$, ${m}_{y1}^{\delta}$, ${m}_{z1}^{\delta}$ | Normal distribution | $N(0,{0.1}^{2})$ | |

Error of Wind disturbance | Wind speed ${V}_{w}$ | Normal distribution | $N(0,{1}^{2})$ |

Serial Number | MAPE of x | MAPE of y | MAPE of z | MAPE of Mach | MAPE of $\mathit{\phi}$ | Miss Distance | Operational Result |
---|---|---|---|---|---|---|---|

1 | $0.23\%$ | $0.99\%$ | $0.79\%$ | $0.35\%$ | $6.00\%$ | $0.8m$ | destroying |

2 | $0.33\%$ | $1.75\%$ | $6.35\%$ | $0.87\%$ | $3.56\%$ | $2m$ | destroying |

3 | $0.19\%$ | $1.10\%$ | $0.55\%$ | $0.75\%$ | $4.21\%$ | $1.4m$ | destroying |

4 | $0.15\%$ | $2.27\%$ | $4.21\%$ | $0.80\%$ | $2.84\%$ | $1.1m$ | destroying |

5 | $0.38\%$ | $0.54\%$ | $0.84\%$ | $0.81\%$ | $3.02\%$ | $2.3m$ | destroying |

6 | $0.12\%$ | $0.71\%$ | $2.21\%$ | $0.77\%$ | $2.90\%$ | $0.1m$ | destroying |

7 | $0.9\%$ | $4.19\%$ | $4.97\%$ | $1.54\%$ | $2.53\%$ | $2.8m$ | destroying |

8 | $0.08\%$ | $0.38\%$ | $5.15\%$ | $0.08\%$ | $3.83\%$ | $0.01m$ | destroying |

9 | $0.23\%$ | $1.02\%$ | $0.97\%$ | $0.40\%$ | $3.43\%$ | $1.6m$ | destroying |

10 | $0.25\%$ | $1.15\%$ | $1.36\%$ | $0.64\%$ | $3.15\%$ | $0.05m$ | destroying |

© 2019 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**

He, Y.; Guo, H.; Han, Y.
A Novel Hybrid Data-Driven Modeling Method for Missiles. *Symmetry* **2020**, *12*, 30.
https://doi.org/10.3390/sym12010030

**AMA Style**

He Y, Guo H, Han Y.
A Novel Hybrid Data-Driven Modeling Method for Missiles. *Symmetry*. 2020; 12(1):30.
https://doi.org/10.3390/sym12010030

**Chicago/Turabian Style**

He, Yongxiang, Hongwu Guo, and Yang Han.
2020. "A Novel Hybrid Data-Driven Modeling Method for Missiles" *Symmetry* 12, no. 1: 30.
https://doi.org/10.3390/sym12010030