# Three-dimensional impact angle guidance law based on robust repetitive control

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## Abstract

This paper presents a robust repetitive control (RC) design applied to three-dimensional homing guidance of missiles with impact angle constraint. The proposed guidance law is substantially a composite control method, which is constructed through a combination of RC and sliding mode control. More specifically, the RC exerts advantages to drive the state tracking error converge to zero, then sliding mode control is triggered, making the system be robust in terms of noise and disturbance. The effectiveness of the proposed guidance law is validated through simulation.

## Keywords

Guidance law Impact angle Repetitive control## List of symbols

*R*Relative distance between the missile and the target

- \(\phi\)
Pitch line-of-sight angle (PLOS)

- \(\theta\)
Yaw line-of-sight angle (YLOS)

- \(\vec{e}_{r}\)
Unit vector along the LOS

- \(\vec{e}_{\phi }\)
Unit vector along the PLOS

- \(\vec{e}_{\theta }\)
Unit vector along the YLOS

- \(\vec{a}_{T} = w_{r} \vec{e}_{r} + w_{\theta } \vec{e}_{\theta } + w_{\phi } \vec{e}_{\phi }\)
Acceleration vector of the target

- \(\vec{a}_{M} = u_{\theta } \vec{e}_{\theta } + u_{\phi } \vec{e}_{\phi }\)
Acceleration vector of the missile

- \(\ddot{R}\)
Relative acceleration along to LOS

- \(\ddot{\phi }\)
Angular acceleration along to LOS

- \(\ddot{\theta }\)
Angular acceleration of \(\theta\)

- \(\dot{R}\)
Relative velocity between the missile and the target

- \(\dot{\theta }\)
Angular velocity of \(\theta\)

- \(\dot{\phi }\)
Angular velocity of \(\phi\)

## 1 Introduction

Intercepting maneuvering targets with a small miss-distance is not the only task of the guidance law design in some applications, for example, antitank or antiship missiles, which are also required to approach the target from a predetermined impact angle in order to increase the warhead effectiveness [1, 2]. Hence, it is necessary to design guidance law with impact angle constraint.

During the guidance process, the guidance system continuously measures the relative position information, and sends command to the flight control system. The kinematics equation of the missile-target pursuit dynamic behavior is found to be uncertain nonlinear multiple-input multiple-output (MIMO) system with cross-coupling [3]. In the past, Proportional Navigation Guidance law (PNG) was widely used in homing guidance area [4]. Along with the progress of computer science and mathematics, a lot of nonlinear control methods have been applied to this issue [5, 6, 7, 8]. Among them, sliding mode control (SMC) was widely adopted by researchers for its unique properties, for example, it is robust to parameter variations and external disturbance [9]. But SMC suffers from some drawbacks, which are: the upper bound of uncertainties must be known, and the existence of chattering phenomenon, which may cause the excitation of unmodeled dynamics [10].

Recently, a new guidance law design based on Iterative Learning Control (ILC) is proposed in [11], and the numerical experiments show that the proposed method is capable of reducing the time to reach the head-on condition to interception. However, impact angle constraint is not taken into consideration in this paper. Besides, there exists robustness problem of ILC [12].

In this paper, we propose a robust repetitive control strategy for guidance mission of homing missiles. The RC is combined with sliding mode control in order to acquire both of their advantages. Specifically, the RC is utilized to guarantee the reachability of the sliding mode, and then the sliding mode control is committed to enhance the robustness of the system. Simulations under different scenarios are performed, and the validation of the proposed method is verified.

This paper is organized as follows: in Sect. 2, the dynamics of target-missile relative motion is illustrated, and the object of the guidance law with impact angle constraint is addressed. In Sect. 3, the robust repetitive control is designed in the framework of sliding mode control. Numerical experiments are performed to demonstrate the effectiveness of the proposed method in Sect. 4. At last, concluding remarks are summarized in Sect. 5.

## 2 Problem formulation

In fact, only the accelerations normal to the missile’s velocity are available in the terminal guidance phase. Therefore, only Eqs. (2) and (3) are used in guidance law design.

### **Assumption 1**

[14] Assume that the missile intercepting the target by impact happens when \(R = R_{0} \ne 0\), and there exist two positive constants \(R_{min}\) and \(R_{max}\), which satisfy \(R_{min} < R < R_{max}\).

Let \(\theta_{d}\) and \(\phi_{d}\) be the desired final LOS angles in elevation and azimuth, respectively. By accepting the concept that zeroing the LOS angle rate will lead a perfect interception and taking the terminal angle constraint into consideration, the control object is to design a guidance law in such a way that \(\theta \to \theta_{d}\), \(\phi \to \phi_{d}\), \(\dot{\theta } \to 0,\dot{\phi } \to 0\) can be fulfilled asymptotically [15].

## 3 Composite guidance law design

### 3.1 Derivation of sliding surface

### 3.2 Robust RC guidance law design

### 3.3 Stability analysis

In this section, we will prove the system represented by Eq. (11) is stable by Lyapunov stable theory [16]. Because the dynamic models of \(\sigma_{\theta }\) and \(\sigma_{\phi }\) have the similar forms, here we only take \(\sigma_{\theta }\) as an example to show the process of proof.

Firstly, the tracking error of \(\sigma_{\theta }\) is defined as \(e_{1} \left( t \right) = y_{1} - \sigma_{\theta }\), and the initial condition of \(e_{\theta }\) can be characterised by the following assumption:

### **Assumption 2**

\(e_{1} \left( 0 \right)\) is random and bounded by a constant C.

*t*of \(V\left( {\sigma_{\theta } ,\phi \left( t \right),\phi \left( {t - T} \right),t} \right)\) is

According to Lyapunov stable theory and taking Assumption 2 into consideration, \(e_{1}\) is convergent, which means \(\left| {e_{1} \left( {t + T} \right)} \right|\) is less than \(\left| {e_{1} \left( t \right)} \right|\). If \(t\) is large enough, \(e_{1}\) will be able to approach to origin, which means \(y_{1} - \sigma_{\theta } = 0\) will be guaranteed. Using Eq. (12), \(\sigma_{\theta }\) is able to converge to zero. In other words, the sliding mode is reachable. As previously mentioned, the constants \(c_{i}\) are Hurwitz polynomial, hence \(e_{\theta }\) and \(e_{\phi }\) as well as their derivatives converge to zero asymptotically during the sliding mode, and the system is able to keep invariant to noise and disturbance. Therefore, the system (11) is stable.

### *Remark 1*

## 4 Simulation results

### 4.1 Numerical experiments under different scenarios

To verify the effectiveness of the proposed robust repetitive control based guidance law (RRCGL), the following two cases for the different target accelerations are considered as follows:

**Case 1**

**Case 2**

Initial conditions for the two cases

\(r\left( 0 \right)\) | 12,000 m |

\(\theta \left( 0 \right)\) | \(\uppi/4\) rad |

\(\phi \left( 0 \right)\) | \(\uppi/4\) rad |

\(V_{r} \left( 0 \right)\) | − 900 |

\(V_{\theta } \left( 0 \right)\) | 400 |

\(V_{\phi } \left( 0 \right)\) | 500 |

For performance comparison, the ILCGL in Ref. [11] and NTSMGL in Ref. [15] are simulated under the same conditions.

#### 4.1.1 Simulation results for case 1

#### 4.1.2 Simulation results for case 2

## 5 Conclusion

In this paper, a novel composite guidance law based on RC and sliding mode control is proposed. The RC is integrated to guarantee the reachability of the sliding mode, during which the variables of interest converge to the demanded values asymptotically. Numerical simulations show that the proposed guidance law is robust to extreme forms of target maneuver, not only the target can be intercepted, but also the demanded impact angles are satisfied.

## Notes

### Funding

This work was supported in part of China Postdoctoral Science Foundation under Grant of 2019M651838.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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