# Field-to-View Constrained Integrated Guidance and Control for Hypersonic Homing Missiles Intercepting Supersonic Maneuvering Targets

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## Abstract

**:**

## 1. Introduction

- In contrast to the literature [17,24], based on the BLOS, LOS angular rate, angle of attack, pitch angle rate, and rudder angle, a strict-feedback nonlinear IGC pursuit model under the variable-speed condition is established, which considers the aerodynamic drag and omits the assumption of a small angle of attack.
- Based on the BC, BLF, and DSC, a new IGC controller is proposed to achieve precise hitting of a supersonic maneuvering target by a hypersonic missile, where the BLF is introduced to ensure that the FOV constraint is realized. The lumped disturbances in each loop of the system model were estimated using the reduced-order ESO and compensated in the controller to enhance robustness.
- The stability of the closed-loop system was strictly proven using the Lyapunov theory, and the boundedness of the FOV angle was theoretically proven. The simulation results further verified the effectiveness and robustness of the proposed IGC scheme.

## 2. Problem Formulation

#### 2.1. IGC Model Derivation

**Assumption**

**1.**

**Remark**

**1.**

**Assumption**

**2.**

**Remark**

**2.**

#### 2.2. IGC Controller Design Objective

- The FOV constraint is satisfied during the entire interception; that is, $\left|{x}_{1}\right|<{q}_{B}^{\mathrm{max}}$, where ${q}_{B}^{\mathrm{max}}$ is the maximum seeker FOV angle.
- The desired terminal FOV angle converges to 0, i.e., ${q}_{B}\to {q}_{Bd}=0$.
- The actuator dynamic characteristics are considered, and a small miss distance is achieved.
- All signals of the closed-loop system are uniform ultimately bounded.
- There is strong robustness in the face of multiple uncertainties.

## 3. IGC Controller Design

**Step****1:**- The first equation in Equation (17) is the seeker subsystem that guarantees that the BLOS angle does not exceed the maximum FOV angle by the BLF, that is, $\left|{x}_{1}\right|<{k}_{c1}$. To ensure the precise interception of a maneuvering target, the desired FOV angle is ${x}_{1d}=0$.The first sliding mode surface is defined as$${s}_{1}={x}_{1}-{x}_{1d}$$

**Step****2:**- For the guidance subsystem (the second equation in Equation (17)), to ensure the tracking of ${x}_{2}$ to the virtual command ${x}_{2d}$, the second sliding mode surface is defined as

**Step****3:**- For the angle-of-attack subsystem (the third equation in Equation (17)), to ensure the tracking of ${x}_{3}$ to the virtual command ${x}_{3d}$, the third sliding mode surface is defined as

**Step****4:**- For the attitude subsystem (the fourth equation in Equation (17)), to ensure the tracking of ${x}_{4}$ to the virtual command ${x}_{4d}$, the fourth sliding mode surface is defined as

**Step****5:**- For the rudder subsystem (the fifth equation in Equation (17)), to ensure the tracking of ${x}_{5}$ to the virtual command ${x}_{5d}$, the fifth sliding mode surface is defined as

## 4. Stability Analysis of the Closed-Loop System

**Lemma**

**1.**

**Lemma**

**2.**

**Lemma**

**3.**

**Assumption**

**3.**

**Assumption**

**4.**

**Assumption**

**5.**

**Remark**

**3.**

## 5. Simulation Results and Analysis

**Case**

**1:**

**Case**

**2:**

**Case**

**3:**

- (1)
- The FOV angle constraint is generated randomly and satisfies ${k}_{c1}=10\xb0~20\xb0$.
- (2)
- The normal acceleration of the target is generated randomly and satisfies ${a}_{T}=10\mathrm{sin}(0.25t)\text{}\sim 50\mathrm{sin}(0.25\mathrm{t})\text{}\mathrm{m}/{\mathrm{s}}^{2}$.
- (3)
- Other biased parameters are listed in Table 5. The coefficients of the aerodynamic forces and moments are assumed to change randomly in the range of −20% to 20% of their respective nominal values.
- (4)
- The initial values of the missile are assumed to change randomly in the range of −5% to 5%.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Liu, X.; Huang, W.; Du, L. An integrated guidance and control approach in three-dimensional space for hypersonic missile constrained by impact angles. Isa Trans.
**2017**, 66, 164–175. [Google Scholar] [CrossRef] [PubMed] - Yang, H.; Bai, X.; Zhang, S. Layout Design of Strapdown Array Seeker and Extraction Method of Guidance Information. Aerospace
**2022**, 9, 373. [Google Scholar] [CrossRef] - Shuang, Y.; Yanzheng, Z. A New Measurement Method for Unbalanced Moments in a Two-axis Gimbaled Seeker. Chin. J. Aeronaut.
**2010**, 23, 117–122. [Google Scholar] [CrossRef] [Green Version] - He, S.; Lin, D.; Wang, J. Coning motion stability of spinning missiles with strapdown seekers. Aeronaut. J.
**2016**, 120, 1566–1577. [Google Scholar] [CrossRef] - Wang, P.; Zhang, K. Research on Line-of-Sight Rate Extraction of Strapdown Seeker. In Proceedings of the 2014 33rd Chinese Control Conference (CCC), Nanjing, China, 1 January 2014; pp. 859–863. [Google Scholar]
- Kim, H.G.; Kim, H.J. Field-of-View Constrained Guidance Law for a Maneuvering Target with Impact Angle Control. IEEE T. Aero. Elec. Sys.
**2020**, 56, 4974–4983. [Google Scholar] [CrossRef] - Kim, H.; Lee, J. Generalized Guidance Formulation for Impact Angle Interception with Physical Constraints. Aerospace
**2021**, 8, 307. [Google Scholar] [CrossRef] - Tian, J.; Bai, X.; Yang, H.; Zhang, S. Time-Varying Asymmetric Barrier Lyapunov Function-Based Impact Angle Control Guidance Law with Field-of-View Constraint. IEEE Access
**2020**, 8, 185346–185359. [Google Scholar] [CrossRef] - Liu, B.; Hou, M.; Feng, D. Nonlinear mapping based impact angle control guidance with seeker’s field-of-view constraint. Aerosp. Sci. Technol.
**2019**, 86, 724–736. [Google Scholar] [CrossRef] - Zhou, S.; Hu, C.; Wu, P.; Zhang, S. Impact Angle Control Guidance Law Considering the Seeker’s Field-of-View Constraint Applied to Variable Speed Missiles. IEEE Access
**2020**, 8, 100608–100619. [Google Scholar] [CrossRef] - Ma, S.; Wang, X.; Wang, Z. Field-of-View Constrained Impact Time Control Guidance via Time-Varying Sliding Mode Control. Aerospace
**2021**, 8, 251. [Google Scholar] [CrossRef] - Han, T.; Hu, Q.; Xin, M. Analytical solution of field-of-view limited guidance with constrained impact and capturability analysis. Aerosp. Sci. Technol.
**2020**, 97, 105586. [Google Scholar] [CrossRef] - Ai, X.; Wang, L.; Yu, J.; Shen, Y. Field-of-view constrained two-stage guidance law design for three-dimensional salvo attack of multiple missiles via an optimal control approach. Aerosp. Sci. Technol.
**2019**, 85, 334–346. [Google Scholar] [CrossRef] - Park, B.; Kim, T.; Tahk, M. Optimal impact angle control guidance law considering the seeker’s field-of-view limits. Proc. Inst. Mech. Eng. G J. Aerops. Eng.
**2012**, 227, 1347–1364. [Google Scholar] [CrossRef] - 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] - Wang, J.; Liu, L.; Zhao, T.; Tang, G. Integrated guidance and control for hypersonic vehicles in dive phase with multiple constraints. Aerosp. Sci. Technol.
**2016**, 53, 103–115. [Google Scholar] [CrossRef] - Guo, J.; Xiong, Y.; Zhou, J. A new sliding mode control design for integrated missile guidance and control system. Aerosp. Sci. Technol.
**2018**, 78, 54–61. [Google Scholar] [CrossRef] - Jiang, S.; Tian, F.; Sun, S.; Liang, W. Integrated guidance and control of guided projectile with multiple constraints based on fuzzy adaptive and dynamic surface. Def. Technol.
**2019**, 6, 1130–1141. [Google Scholar] [CrossRef] - Li, Z.; Dong, Q.; Zhang, X.; Gao, Y. Impact angle-constrained integrated guidance and control for supersonic skid-to-turn missiles using backstepping with global fast terminal sliding mode control. Aerosp. Sci. Technol.
**2022**, 122, 107386. [Google Scholar] [CrossRef] - Zhao, B.; Xu, S.; Guo, J.; Jiang, R.; Zhou, J. Integrated strapdown missile guidance and control based on neural network disturbance observer. Aerosp. Sci. Technol.
**2019**, 84, 170–181. [Google Scholar] [CrossRef] - Tian, J.; Chen, H.; Liu, X.; Yang, H.; Zhang, S. Integrated Strapdown Missile Guidance and Control with Field-of-View Constraint and Actuator Saturation. IEEE Access
**2020**, 8, 123623–123638. [Google Scholar] [CrossRef] - Tian, J.; Xiong, N.; Zhang, S.; Yang, H.; Jiang, Z. Integrated guidance and control for missile with narrow field-of-view strapdown seeker. Isa Trans.
**2020**, 106, 124–137. [Google Scholar] [CrossRef] [PubMed] - Zhang, D.; Ma, P.; Du, Y.; Chao, T. Integral barrier Lyapunov function-based three-dimensional low-order integrated guidance and control design with seeker’s field-of-view constraint. Aerosp. Sci. Technol.
**2021**, 116, 106886. [Google Scholar] [CrossRef] - Zhao, B.; Feng, Z.; Guo, J. Integral barrier Lyapunov functions-based integrated guidance and control design for strap-down missile with field-of-view constraint. Trans. Inst. Meas. Control
**2021**, 43, 1464–1477. [Google Scholar] [CrossRef] - Ames, A.D.; Coogan, S.; Egerstedt, M.; Notomista, G.; Sreenath, K.; Tabuada, P. Control Barrier Functions: Theory and Applications. In Proceedings of the 18th European Control Conference (ECC), Naples, Italy, 25–28 June 2019; IEEE: New York, NY, USA; pp. 3420–3431. [Google Scholar]
- Shao, X.; Wang, H. Back-stepping active disturbance rejection control design for integrated missile guidance and control system via reduced-order ESO. Isa T.
**2015**, 57, 10–22. [Google Scholar] [CrossRef] - Chen, M.; Shao, S.; Jiang, B. Adaptive Neural Control of Uncertain Nonlinear Systems Using Disturbance Observer. IEEE Trans. Cybern.
**2017**, 47, 3110–3123. [Google Scholar] [CrossRef] - Chang, J.; Guo, Z.; Cieslak, J.; Chen, W. Integrated guidance and control design for the hypersonic interceptor based on adaptive incremental backstepping technique. Aerosp. Sci. Technol.
**2019**, 89, 318–332. [Google Scholar] [CrossRef]

**Figure 3.**Simulation results under Case 1. (

**a**) Missile–target pursuit trajectory. (

**b**) The BLOS angle. (

**c**) The LOS angle. (

**d**) The velocity of missile. (

**e**) The angle of attack. (

**f**) The pitch angular rate. (

**g**) The relative missile-target distance. (

**h**) The deflection angle. (

**i**) The control input.

**Figure 4.**Simulation results under Case 2. (

**a**) Missile–target pursuit trajectory. (

**b**) The BLOS angle. (

**c**) The LOS angle. (

**d**) The velocity of missile. (

**e**) The angle of attack. (

**f**) The pitch angular rate. (

**g**) The relative missile-target distance. (

**h**) The deflection angle. (

**i**) The control input.

$({\mathit{x}}_{\mathit{M}},\text{}{\mathit{y}}_{\mathit{M}})$ | ${\mathit{V}}_{\mathit{M}}$ | ${\mathit{\theta}}_{\mathit{M}}$ | $\mathit{\vartheta}$ | $\mathit{\alpha}$ | ${\mathit{\omega}}_{\mathit{z}}$ | ${\mathit{\delta}}_{\mathit{z}}$ |
---|---|---|---|---|---|---|

$(0,0)$ | $2000\mathrm{m}/\mathrm{s}$ | $35\xb0$ | $35\xb0$ | $0\xb0$ | $0\xb0/s$ | $0\xb0$ |

$({\mathit{x}}_{\mathit{T}},\text{}{\mathit{y}}_{\mathit{T}})$ | ${\mathit{V}}_{\mathit{T}}$ | ${\mathit{\theta}}_{\mathit{T}}$ |
---|---|---|

$(2500\sqrt{3},2500)$ | $800\mathrm{m}/\mathrm{s}$ | ${0}^{\xb0}$ |

FOV Constraint | ${\mathit{k}}_{\mathit{c}1}=10\xb0$ | ${\mathit{k}}_{\mathit{c}1}=15\xb0$ | ${\mathit{k}}_{\mathit{c}1}=20\xb0$ |
---|---|---|---|

Miss distance | 0.81 m | 0.85 m | 0.88 m |

Interception time | 5.136 s | 5.092 s | 5.08 s |

FOV Constraint | ${\mathit{k}}_{\mathit{c}1}=10\xb0$ | ${\mathit{k}}_{\mathit{c}1}=15\xb0$ | ${\mathit{k}}_{\mathit{c}1}=20\xb0$ |
---|---|---|---|

Miss distance | 0.76 m | 0.77 m | 0.78 m |

Interception time | 5.106 s | 5.062 s | 5.05 s |

Biased Parameters | Bias Value | |
---|---|---|

Initial values | Pitch angle $\vartheta $ | $\pm 5\%$ |

Flight path angle ${\theta}_{M}$ | ||

Aerodynamic coefficients | Drag coefficient ${c}_{x0},{c}_{x}^{{\alpha}^{2}}$ | $\pm 20\%$ |

Lift coefficient ${c}_{y}^{\alpha},{c}_{y}^{{\delta}_{z}}$ | ||

Pitch moment coefficient ${m}_{z}^{\alpha},{m}_{z}^{{\omega}_{z}},{m}_{z}^{{\delta}_{z}}$ |

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**MDPI and ACS Style**

Li, Z.; Dong, Q.; Zhang, X.; Zhang, H.; Zhang, F.
Field-to-View Constrained Integrated Guidance and Control for Hypersonic Homing Missiles Intercepting Supersonic Maneuvering Targets. *Aerospace* **2022**, *9*, 640.
https://doi.org/10.3390/aerospace9110640

**AMA Style**

Li Z, Dong Q, Zhang X, Zhang H, Zhang F.
Field-to-View Constrained Integrated Guidance and Control for Hypersonic Homing Missiles Intercepting Supersonic Maneuvering Targets. *Aerospace*. 2022; 9(11):640.
https://doi.org/10.3390/aerospace9110640

**Chicago/Turabian Style**

Li, Zhibing, Quanlin Dong, Xiaoyue Zhang, Huanrui Zhang, and Feng Zhang.
2022. "Field-to-View Constrained Integrated Guidance and Control for Hypersonic Homing Missiles Intercepting Supersonic Maneuvering Targets" *Aerospace* 9, no. 11: 640.
https://doi.org/10.3390/aerospace9110640