# Inverse Optimal Zero Effort Miss Guidance Based on Disturbance Observer

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## Abstract

**:**

## 1. Introduction

## 2. Problem Statement and Modeling

**Assumption 1.**

**Assumption 2.**

**Assumption 3.**

**Assumption 4.**

**Remark 1.**

**Remark 2.**

**Remark 3.**

**Lemma 1**

**.**Consider a nonlinear affine in the control system as

**Lemma 2.**

## 3. Design of the Disturbance Observer

## 4. Design of the Inverse Optimal Guidance

**Theorem 1.**

**Proof of Theorem 1.**

## 5. Simulation

#### 5.1. Constant Maneuver

#### 5.2. Square Wave Maneuver

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**The target’s acceleration ${a}_{{T}_{\alpha}}$ and its estimation ${\widehat{a}}_{{T}_{\alpha}}$ with constant maneuver.

**Figure 6.**The normal acceleration of the missile ${a}_{M}$ and the target ${a}_{T}$ with constant maneuver.

**Figure 7.**Time evolution of the relative distance between the missile and the target with constant maneuver.

**Figure 11.**Time evolution of the relative distance between the missile and the target with white noise and constant maneuver.

**Figure 15.**The target’s acceleration ${a}_{{T}_{\alpha}}$ and its estimation ${\widehat{a}}_{{T}_{\alpha}}$ with square wave maneuver.

**Figure 16.**The normal acceleration of the missile ${a}_{M}$ and the target ${a}_{T}$ with square wave maneuver.

**Figure 17.**Time evolution of the relative distance between the missile and the target with square wave maneuver.

**Figure 21.**Time evolution of the relative distance between the missile and the target with white noise and square wave maneuver.

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

${V}_{M}$ | 544 | m/s |

${V}_{T}$ | 408 | m/s |

$r\left(0\right)$ | 5000 | m |

$\left({x}_{M}\left(0\right),{y}_{M}\left(0\right)\right)$ | $(0,0)$ | m |

$\alpha \left(0\right)$ | 0 | Degree |

${\theta}_{M}\left(0\right)$ | 45 | Degree |

${\theta}_{T}\left(0\right)$ | 60 | Degree |

${\alpha}_{0}$ | 0 | Degree |

${k}_{1}$ | 20 | |

${k}_{2}$ | 5 | |

${k}_{3}$ | 5 |

Guidance Accuracy/m | Gidance Time/s | |
---|---|---|

ZEM guidance | 0.55 | 8.7 |

APN | 3.84 | 8.8 |

PAG | 2.83 | 8.8 |

${\mathit{k}}_{1}$ | ${\mathit{k}}_{2}$ | ${\mathit{k}}_{3}$ | Gidance Time/s | Guidance Accuracy/m |
---|---|---|---|---|

25 | 5 | 5 | 8.77 | 0.41 |

30 | 5 | 5 | 8.77 | 0.40 |

35 | 5 | 5 | 8.77 | 0.39 |

40 | 5 | 5 | 8.77 | 0.40 |

45 | 5 | 5 | 8.77 | 0.37 |

20 | 5 | 5 | 8.77 | 0.29 |

20 | 10 | 5 | 8.77 | 0.12 |

20 | 15 | 5 | 8.77 | 0.02 |

20 | 20 | 5 | 8.77 | 0.14 |

20 | 25 | 5 | 8.77 | 0.26 |

20 | 5 | 10 | 8.77 | 0.41 |

20 | 5 | 15 | 8.77 | 0.41 |

20 | 5 | 20 | 8.77 | 0.41 |

20 | 5 | 25 | 8.77 | 0.41 |

20 | 5 | 30 | 8.77 | 0.41 |

${\mathit{V}}_{\mathit{M}}$ | ${\mathit{V}}_{\mathit{T}}$ | $\mathit{r}\left(0\right)$ | ${\mathit{\theta}}_{\mathit{M}}\left(0\right)$ | ${\mathit{\theta}}_{\mathit{T}}\left(0\right)$ | Gidance Time/s | Guidance Accuracy/m |
---|---|---|---|---|---|---|

510 | 408 | 5000 | 45 | 60 | 9.21 | 0.16 |

544 | 374 | 5000 | 45 | 60 | 8.41 | 0.03 |

544 | 408 | 6000 | 45 | 60 | 9.83 | 0.53 |

544 | 408 | 5000 | 30 | 60 | 8.78 | 0.85 |

544 | 408 | 5000 | 45 | 50 | 9.41 | 0.46 |

Guidance Accuracy/m | Guidance Time/s | |
---|---|---|

ZEM guidance | 0.22 | 27.85 |

APN | 1.03 | 27.85 |

PAG | 0.87 | 27.88 |

${\mathit{k}}_{1}$ | ${\mathit{k}}_{2}$ | ${\mathit{k}}_{3}$ | Gidance Time/s | Guidance Accuracy/m |
---|---|---|---|---|

25 | 5 | 5 | 27.85 | 0.49 |

30 | 5 | 5 | 27.85 | 0.49 |

35 | 5 | 5 | 27.85 | 0.42 |

40 | 5 | 5 | 27.85 | 0.0.43 |

45 | 5 | 5 | 27.85 | 0.41 |

20 | 5 | 5 | 27.85 | 0.41 |

20 | 10 | 5 | 27.85 | 0.41 |

20 | 15 | 5 | 27.85 | 0.44 |

20 | 20 | 5 | 27.85 | 0.48 |

20 | 25 | 5 | 27.85 | 0.54 |

20 | 5 | 10 | 27.85 | 0.47 |

20 | 5 | 15 | 27.85 | 0.49 |

20 | 5 | 20 | 27.85 | 0.46 |

20 | 5 | 25 | 27.85 | 0.41 |

20 | 5 | 30 | 27.85 | 0.42 |

${\mathit{V}}_{\mathit{M}}$ | ${\mathit{V}}_{\mathit{T}}$ | $\mathit{r}\left(0\right)$ | ${\mathit{\theta}}_{\mathit{M}}\left(0\right)$ | ${\mathit{\theta}}_{\mathit{T}}\left(0\right)$ | Gidance Time/s | Guidance Accuracy/m |
---|---|---|---|---|---|---|

510 | 408 | 5000 | 45 | 60 | 36.18 | 0.54 |

544 | 374 | 5000 | 45 | 60 | 23.23 | 0.34 |

544 | 408 | 6000 | 45 | 60 | 33.33 | 0.24 |

544 | 408 | 5000 | 30 | 60 | 27.85 | 0.40 |

544 | 408 | 5000 | 45 | 50 | 31.04 | 0.18 |

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**MDPI and ACS Style**

Ma, B.; Chen, M.; Shen, Y.; Lungu, M.
Inverse Optimal Zero Effort Miss Guidance Based on Disturbance Observer. *Aerospace* **2022**, *9*, 767.
https://doi.org/10.3390/aerospace9120767

**AMA Style**

Ma B, Chen M, Shen Y, Lungu M.
Inverse Optimal Zero Effort Miss Guidance Based on Disturbance Observer. *Aerospace*. 2022; 9(12):767.
https://doi.org/10.3390/aerospace9120767

**Chicago/Turabian Style**

Ma, Biao, Mou Chen, Yaohua Shen, and Mihai Lungu.
2022. "Inverse Optimal Zero Effort Miss Guidance Based on Disturbance Observer" *Aerospace* 9, no. 12: 767.
https://doi.org/10.3390/aerospace9120767