A new integrated guidance and control (IGC) law is investigated for a homing missile with an impact angle against a ground target. Firstly, a control-oriented model with impact angle error of the IGC system in the pitch plane is formulated by linear coordinate transformation according to the motion kinematics and missile dynamics model. Secondly, an IGC law is proposed to satisfy the impact angle constraint and to improve the rapidity of the guidance and control system by combining the sliding mode control method and nonlinear extended disturbance observer technique. Thirdly, stability of the closed-loop guidance and control system is proven based on the Lyapunov stability theory, and the relationship between the accuracy of the impact angle and the estimate errors of nonlinear disturbances is derived from stability of the sliding mode. Finally, simulation results confirm that the proposed IGC law can improve the performance of the missile guidance and control system against a ground target.
It is well known that the guidance and control system plays a key role in realizing the flight mission of missiles, and it becomes more and more important to improve the whole performance of the flight control system. The traditional guidance and control systems are separately designed based on the principle of separate frequency spectrum [
In order to improve the whole performance of the guidance and control system, an IGC method was proposed; that is, a whole model combining the guidance system and the autopilot system can be directly designed [
The first category method mainly focuses on the two-loop control structure in an IGC framework, and the outer loop and inner loop are separated in the guidance and control system [
The second category method for the IGC system combines the backstepping technique and the other control methods [
In order to overcome the delay problem, a new IGC law is presented for a homing missile with an impact angle against a ground target, motivated by the sliding surface technique consisting of the system states and the estimated states [
The contributions of this paper lie in the following aspects.
A new control-oriented model with impact angle error of the IGC system in the pitch plane is built by combining the missile dynamics model and motion kinematics between the missile and target. A novel IGC law is proposed for a homing missile to satisfy the impact angle constraint and to improve rapidity of the IGC system by using the sliding mode control and the nonlinear disturbance observer technique. The stability of the closed-loop system is proven by using the Lyapunov stability theory, and the relationship between the accuracy of the impact angle and the estimate error of nonlinear disturbances is derived from the stability of the sliding mode. The proposed IGC law is evaluated by comparison with the classic guidance and control law and the other IGC law for a homing missile with an impact angle against a ground target.
The rest of this paper is organized as follows: in Section
The two-dimensional engagement dynamics model is described in Figure
Two-dimensional engagement geometry.
By differentiating (
The longitude model of a missile in the pitch plane is described as [
The control-oriented model of the IGC system is built to design the IGC law. Assuming that
Noting that
Let
Differentiation (
The acceleration of the missile is described as
It is obtained from (
According to the above analysis, the control-oriented model of the IGC system for a homing missile against a ground target in the pitch plane can be described as
The disturbance
Note that the control-oriented model of the IGC system for a homing missile is affected by the mismatched uncertainties; a new sliding mode controller is designed for the IGC system to exploit the synergistic relationship between guidance and autopilot subsystems. Meanwhile, the mismatched disturbances in the control-oriented model are estimated by using the nonlinear disturbance observer to compensate the sliding mode controller.
Considering the control-oriented models (
In order to obtain the abovementioned estimates, three nonlinear disturbance observers based on [
The estimate errors are also defined as
Meanwhile, it can be also obtained from the disturbance observers (
Computing the derivative of
Accordingly, the new sliding controller for the control-oriented models (
In order to illustrate the effectiveness of the proposed IGC, the stability of a closed system is analyzed as follows.
Choosing a Lyapunov function candidate as
According to (
It is also obtained from (
It is obtained that
A corresponding Lyapunov function candidate is chosen as
Differentiating
Due to (
From inequality (
It is worth noting that the state
In this section, the effectiveness of the proposed IGC law is verified by the nonlinear numerical simulations. For the nonlinear numerical simulations, the original nonlinear motion model of the missile given in [
The simulation step is 0.001 s; the initial values of the simulation are assumed to be
In addition, the uncertainties of missile-related aerodynamic parameters can include 20% variations with respect to their normal values; some requirements for the guidance and control system are listed as follows:
The blind area for the homing guidance seeker is 50 m. The miss distance is no more than 0.5 m. The absolute error of terminal impact angle distance is no more than 5 deg. The angle of attack is less than 10 deg. The fin deflection limit and fin rate limit are restricted as The states of the missile are bounded.
For comparison studies, two control laws are introduced to the nonlinear numerical simulations. The first is the classical guidance and control (CGC) law [
In order to demonstrate the performance of the proposed IGC law in the presence of uncertainties in the parameters, three cases are considered for a homing missile against a fixed target, a moving target, and a maneuvering target on the ground.
Variable response curves of the guidance and control system under the three control laws against a fixed target on the ground are shown in Figure
As presented in Figure
Moreover, the time under the IGC law is about 0.1 s shorter than that of the PIGC law and about 0.6 s shorter than that of the CGC law against a fixed target on the ground.
To make the work more challenging, the target moves at 20 m/s on the ground in this case. Missile/target trajectories under the three control laws are shown in Figure
From the simulation results among the three control laws, the states are stabilized most rapidly under the IGC; both miss distance and impact angle under the IGC law also achieve satisfactory performance. The time in the simulation experiment under the IGC law is about 0.1 s shorter than that of the PIGC law and about 0.5 s shorter than that of the CGC law against a moving target on the ground.
In this case, the target on the ground moves with a maneuver of
In short, it is easily obtained that the states are stabilized most rapidly; both the miss distance and the impact angle can be satisfied under the IGC law, while only partial index can be satisfied under the CGC law and the PIGC law. Meanwhile, the flight time of engagement with a fixed target, a moving target, and a maneuvering target is shortened by utilizing the proposed IGC law, compared with the CGC law and the PIGC law.
Variable response curves under the three control laws against a fixed target.
Estimate curves of uncertainties under the nonlinear disturbance observer.
Missile/target trajectories under the three control laws against a fixed target and a moving target.
Miss distance under the three control laws against a fixed target.
CGC law | PIGC law | IGC law |
---|---|---|
0.09943 m | 5.611 m | 0.2059 m |
Impact angle under the three control laws against a fixed target.
CGC law | PIGC law | IGC law |
---|---|---|
−62.8 deg | −89.7 deg | −88.7 deg |
Variable response curves under the three control laws against a moving target.
Miss distance under the three control laws against a moving target.
CGC law | PIGC law | IGC law |
---|---|---|
0.093 m | 11.4 m | 0.4426 m |
Impact angle under the three control laws against a moving target.
CGC law | PIGC law | IGC law |
---|---|---|
−68.1 deg | −89.9 deg | −86.2 deg |
Variable response curves under the three control laws against a maneuvering target.
Miss distance under the three control laws against a maneuvering target.
CGC law | PIGC law | IGC law |
---|---|---|
0.067 m | 3.176 m | 0.4221 m |
Impact angle under the three control laws against a maneuvering target.
CGC law | PIGC law | IGC law |
---|---|---|
−67.6 deg | −89.4 deg | −84.9 deg |
A novel IGC law for a homing missile with impact angle constraint is proposed against a ground target to improve the rapidity of the missile guidance and control system. A new control-oriented model with impact angle error of the IGC system in the pitch plane is built, and an IGC law is designed by utilizing the sliding mode control and the nonlinear disturbance observer. The relationship between the accuracy of impact angle and the estimate error of mismatched uncertainties can be obtained from the stability of the system. Simulation results have shown that the proposed IGC approach achieves good performance and shortens the time of engagement.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
This work is supported by the National Nature Science Foundation of China under Grant 61703339.