This paper investigates the problem of designing angle constraint guidance law against unknown maneuvering targets based on discretetime sliding mode control theory. Invoking the fact that the future course of action of the target, an independent entity, cannot be predicted beforehand due to its complexity and unpredictability, a modelassisted discretetime disturbance observer in cooperation with a singularityfree strategy is proposed first to estimate the target maneuver. Based on the reconstructed signal and a fast convergence timevarying sliding surface, a new chatteringmitigated supertwistinglike discretetime impact angle constraint guidance law is then synthesized. Stability analysis shows that the closedloop system trajectory can be forced to enter into a small region around the sliding surface. Simulations and comparisons with classical discretetime sliding mode guidance law validate the effectiveness of the proposed guidance law.
Proportional navigation guidance (PNG) law has been widely used in missile guidance system in the past few decades due to its easy implementation and might suffice for many typical cases of applications [
During the development of modern guidance law, the concept of imposing hard constraint on intercept angle or impact angle provides engineers a new view to design guidance laws for tactical missiles to exploit the weak points of the target and increase the overall kill probability [
With the development of modern control theory, sliding mode control (SMC) methodology was widely accepted in angle constraint guidance law design for intercepting maneuvering targets [
Motivated by the above analysis, this paper considers designing discretetime SMC impact angle guidance law for unknown maneuvering target interception. A robust discretetime guidance SMC guidance law with a timedelay observer was designed by Wang [
An exponentially stable modelassisted disturbance observer is proposed to estimate the target maneuver online. Different from the existing target maneuver estimators [
A discretetime timevarying sliding surface is designed to accelerate the convergence speed of the LOS angle tracking error
Based on the reconstructed target maneuver information, a new chatteringmitigated supertwistinglike discretetime guidance law is proposed for accurate impact angle control. Theoretical analysis shows that the closedloop system trajectory can converge into a small region around the designed sliding surface under the proposed guidance law
The rest of the paper is organized as follows. The problem formulation stated in Sec. II. In Sec. III, the proposed guidance law and convergence analysis are presented in details, followed by the simulation results provided in Sec. IV. Finally, some conclusions are offered in Sec. V.
In this section, engagement kinematics of maneuvering target interception with impact angle constraint is formulated for the analysis and the design of guidance law. The following general assumptions in guidance law design are considered for convenient analysis.
both the interceptor and the target are considered as point masses, e.g., only kinematics is considered in this paper.
the interceptor is considered as an ideal point mass such that the achieved acceleration is the same as the command signal generated by guidance law.
only aerodynamically controlled missiles are considered in this paper, which means that the missile acceleration along the LOS usually cannot be adjusted.
both the interceptor and targets fly with constant speeds.
Consider a typical twodimensional engagement scenario shown in Figure
Planar homing engagement geometry.
The corresponding equations describing the missiletarget relative kinematics are formulated as
Let
Compared with other variables, the relative range and its rate are slowvarying variables [
The change rate of the target maneuver between two sampling intervals is bounded by a constant
To ensure successful target interception and kill probability, the pursuer should have to impose hard constraints on terminal miss distance as well as terminal impact angle. In this regard, the following two general definitions are introduced for the convenience of analysis.
The term zeroeffort miss, denoted by
According to Eq. (
The quantity impact angle, denoted as
Assuming a perfect interception is achieved, i.e.,
Based on Eq. (
In this section, the proposed composite guidance law is derived in details, and the convergence analysis is also presented. A discretetime modelassisted disturbance observer is firstly proposed to estimate the target maneuver. Then, a new discretetime chatteringmitigated supertwistinglike guidance law is synthesized based on the designed sliding variable.
Generally, by viewing the target maneuver as an external disturbance for the guidance system, classical disturbance observer can then be applied for target maneuver estimation, for instance, continuous timedelay estimator [
Consider error dynamics (
With error dynamics (
Solving inequality (
If the upper limit when
This completes the proof. It follows from Eq. (
Similarly, the target maneuver perpendicular to the LOS can also be estimated via
With the above two target maneuver projection estimations, the real target maneuver estimation can be obtained by the following singularityfree strategy
Comparison of two target maneuver estimation strategy: (a) estimation algorithm [
To begin with, let
Let
Next, consider the following timevarying sliding surface
If the closedloop system enters a small region around the sliding manifold, e.g.,
Using Euler discretization approach, one can obtain that
Following the preceding equation, it can be deduced that
Inequality (
Theorem
For guidance law design, deriving the sliding manifold dynamics as
For sliding variable dynamics (
It follows from Eq. (
For discretetime sliding mode dynamics (
Substituting guidance law (
Let
Suppose that the gains
For system (
Substituting Eq. (
Applying lambda inequality
Substituting inequality (
Note that
By choosing
Following the previous result, one can imply that
Solving inequality (
If the upper limit when
Inequality (
In this section, the effectiveness of the proposed guidance law is demonstrated through numerical simulations, in which a surfacetoair missile is considered to intercept a maneuvering headon target during its terminal guidance phase.
The simulations are performed in Matlab/Simulink platform with a sampling rate 0.01 s. Here, the missile is supposed to be equipped with an active radar seeker, providing LOS angle, LOS angular rate, range, and range rate measurement. The required simulation data for Eqs. (
Required data for initializing simulations.
Parameter  Value 

Initial relative range 

Initial flight path angle 

Initial LOS angle 

Flying velocity 

Target maneuver  Case 1: 
The desired impact angles are set as
Mean miss distances and impact angle errors of 50 runs MonteCarlo simulations.





Case 1  Miss distance  0.749 m  0.659 m  0.713 m 
Impact angle error 




Case 2  Miss distance  0.758 m  0.788 m  0.695 m 
Impact angle error 



Simulation results under the proposed guidance law for Case 1: (a) interception trajectory; (b) missile achieved acceleration; (c) LOS angle profile; (d) target maneuver projection estimation; (e) impact angle profile; and (f) target maneuver estimation.
Simulation results under the proposed guidance law for Case 2: (a) interception trajectory; (b) missile achieved acceleration; (c) LOS angle profile; (d) target maneuver projection estimation; (e) impact angle profile; and (f) target maneuver estimation.
One can note from these two figures that accurate interception is achieved under the proposed guidance law whatever the desired impact angle is, that is, the miss distances in these scenarios are less than
To illustrate the effectiveness of the proposed guidance law, the existing discretetime sliding mode control guidance (DSMCG) law [
The simulation results, including interception trajectory, achieved acceleration, target maneuver estimation performance, with
Simulation results under the proposed guidance law for Case 1: (a) interception trajectory; (b) achieved acceleration; and (c) target maneuver estimation.
Simulation results under the proposed guidance law for Case 2: (a) interception trajectory; (b) achieved acceleration; and (c) target maneuver estimation.
Under the framework of sliding mode control, the problem of intercepting unknown maneuvering targets with impact angle constraint is investigated by a new discretetime guidance law enhanced by a disturbance observer. A modelassisted discretetime disturbance observer is first proposed to estimate the target maneuver, and theoretical analysis shows that the estimation error is exponentially stable with a small upper bound determined by the sampling rate. Based on the reconstructed target maneuver information, a chatteringmitigated supertwistinglike discretetime guidance law is proposed to force the system trajectory to converge into a small region around the sliding surface, and therefore, accurate interception with impact angle constraint is satisfied. Simulation and comparison studies of a surfacetoair missile have been carried out to show the efficiency of the proposed approach.
The data used to support the findings of this study are included within the article.
The authors declare that there is no conflict of interest regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China under grant U1613225.