Impact Time Guidance Law Considering Autopilot Dynamics Based on Variable Coefficients Strategy for Maneuvering Target

This paper investigates the terminal guidance problem for themissile intercepting amaneuvering targetwith impact time constraint. An impact time guidance law based on finite time convergence control theory is developed regarding the target motion as an unknown disturbance. To further improve the performance of the guidance law, an autopilot dynamics which is considered as a first-order lag is taken into consideration. In the proposed method, the coefficients change with the relative distance between missile and target. This variable coefficient strategy ensures that the missile impacts the target at the desired time with little final miss distance. Then it is proved that states of the guidance system converge to sliding mode in finite time under the proposed guidance law. Numerical simulations are presented to demonstrate the effectiveness of the impact time guidance law with autopilot dynamics (ITGAD).


Introduction
When missiles attack the target with defense system or strong maneuverability, an effective way is salvo attack.Salvo attack means that several missiles, launched from different platforms or a single platform consecutively, hit the target at the same time.Consequently, the impact time is particularly important for salvo attack.
To improve the effectiveness of the warfare system, the guidance law with impact time constraint has been developed [1].Jeon et al. [2] combined the well-known PNG law and the feedback of the impact time error.They proposed an impact time control guidance law to hit a stationary target simultaneously at a desirable impact time.A flight vehicle's homing guidance problem has been solved by Lee et al. [3] with the consideration of both impact time and impact angle.Harl and Balakrishnan [4] adopted the second-order sliding mode control method and designed a guidance law concerning impact time and angle constraints.Similarly, Zhang et al. [5] derived a close-form guidance law with the same constraints on the basis of biased proportional navigation guidance (BPNG).Then, with the impact time control guidance law, some papers deliberated the multimissile attack strategy [6][7][8].
Furthermore, in practical applications, the autopilot lag of missile exerts bad influence on guidance precision.To deal with the problem, [9][10][11][12] considered the autopilot dynamics in the process of designing guidance laws, which can compensate for the effects of the autopilot dynamics.Moosapour et al. [13] proposed a novel robust proportional navigation guidance (RPNG) law by integrating sliding mode control and proportional navigation guidance law which assumed the autopilot dynamics as first-order lags system.Formulating a fourth-order state equation for integrated guidance and control loop with target uncertainties and control loop dynamics, Chwa and Choi [14] derived an adaptive nonlinear guidance law.Qu and Zhou [15] presented a dimension reduction observer-based guidance law based on the dynamic surface control-method regarding the missile autopilot as second-order dynamics.Taking both the autopilot dynamics and uncertainties into consideration, Li et al. [16] developed a finite time convergent guidance law for homing missile to intercept a maneuvering target.

Mathematical Problems in Engineering
To the best of our knowledge, we find that the existing impact time control guidance laws are mainly for hitting the stationary or weak maneuverability target.Based on this observation, we proposed an impact time guidance law for attacking a maneuvering target, in which the impact time and the light-of-sight (LOS) angular rate are chosen as the sliding variables while the motion of target is taken as disturbance.In addition, the existing impact time control guidance laws do not concern the dynamics of the autopilot.To further improve the effectiveness of our proposed method, the autopilot regarded as first-order dynamics is taken into account based on the finite time convergence theory.To ensure that the missile impacts the target at desired time, a strategy making the coefficient vary with relative distance which can be easily measured by a missile guided seeker is designed in the guidance law.Then, an impact time guidance law considering autopilot dynamics based on variable coefficients strategy (ITGAD) is derived.
Compared with the previous works on impact time control, the advantages of ITGAD are that, first, it deals with the problem of missiles impacting a maneuvering target at the desired impact time.Then ITGAD considered the autopilot dynamics which may induce time lags during the terminal guidance process.The remainder of this paper is organized as follows.Section 2 describes the relative motion models for missile and target.In Section 3, an impact time guidance law considering autopilot dynamics is derived.Variable coefficient strategy is proposed and the stability of the guidance system is verified in Section 4. Finally, simulation results and analysis of the guidance law are provided.

Equations of Motion
This section formulates the mathematic model of the guidance system for intercepting.The planar relative motion of missile and target is shown in Figure 1, where the missile and the target are denoted by subscripts  and , respectively.In this scenario, the missile is traveling at a constant velocity.To simplify the guidance law design, missile and target are assumed to be point masses, and only kinematics is considered.The corresponding equations of guidance are given by where  is the relative distance, ṙ is the relative velocity of the missile and the target, V  is the missile velocity,   denotes the missile lateral acceleration which is normal to missile velocity, and   represents the missile flight-path angle.V  is the target velocity,   is the target lateral acceleration that is normal to target velocity,   is the target flight-path angle, and  and q are the light-of-sight (LOS) angle and LOS angular rate of missile and target, respectively.
To simplify the transformation from the missile dynamics, we assume that the velocities of the missile and the target are constant.Then differentiating (1) and ( 2) with respect to time yields where   = ṙ ,   =  q ,   =   cos   , and   =   sin   .  and   are the target's tangential acceleration and normal acceleration, respectively.
A useful variable when analyzing a homing guidance is the time-to-go,  go , which indicates the time required for the missile to hit the target on its present course.In this work, the time-to-go parameter is approximated as In the terminal guidance process, usually only the acceleration normal to missile's velocity can be adjusted and so we will just discuss the relative motion normal to the LOS while ṙ < 0. Equation ( 9) can be written as To make the LOS angular rate converge to zero in finite time, define two state variables as From (8), we can obtain Differentiating ( 1) and (10) with respect to time, we immediately get The design objective here is to develop a guidance law.This guidance law not only can ensure the missile has a short miss distance, but can intercept the maneuvering target at the desired impact time.That is, we develop a guidance law such that the LOS angular rate q will converge to zero and the impact time  go will converge to the value of   −  in finite time in the presence of maneuvering targets.

Finite Time Convergence Guidance Law with Impact Time and Autopilot Dynamics
3.1.Finite Time Convergence System.For designing the finite time convergence guidance law, we make a definition on the finite time stability of SISO system and a lemma first.
Definition 1 (see [17]).Consider a SISO system: where () is the state variables of the system,  is a smooth control input, () and () are known functions, and  is smooth disturbance of system.The function is interpreted as the sliding variable dynamics.For any given initial time  0 and initial state  0 , there will exist a settling time  ≥ 0 so that the every solution of the system satisfies lim () = 0,  > . (15) When the system is Lyapunov stable with finite time convergence, then the system is called finite time stable.
Lemma 2 (see [18]).Consider the nonlinear system described by (14).Suppose there exists an open neighborhood Θ of the origin, a  1 positive-definite function  : Θ → Ω, and real numbers  > 0, such that V +   is negative semidefinite on Θ.Then the origin is a finite time stable equilibrium of (14).Moreover, if  is the settling time, then  ≤ (1/(1 − )) 1− for all  in some open neighborhood of the origin.

Impact Time Guidance Law Based on Finite Time Convergence Control with Integral Manifold.
Considering the process of the terminal guidance of missile, the following sliding mode manifold is introduced: where  1 ,  2 ,  3 are positive constants and   is the desired impact time.The initial value is denoted as  2 (0) =  1 (0) + ( 1 V  /(0)) 3 (0) and  3 (0) =  2 (0) −   which mean the LOS angular rate error q and impact time error  2 − (  − ) on the sliding mode manifold starting from the initial time instance.
Theorem 3. Consider the guidance system (1)∼( 6) and ( 10)∼ (11).In the following time convergence guidance law: the switching term amplitude  satisfies  > 0 and  satisfies  > max(), where  is target motion which is considered as disturbance of the guidance system.Then, the guidance law (20) guarantees that the LOS angular rate q converges to zero and the impact time   converges to the value of   −  in finite time.
Proof.Differentiating ( 16), (17), and (19) with respect to time yields Substituting ( 12) and ( 13) into (21) and with (20), where  =   / +  1 V    / 2  is denoted as target motion.Define a Lyapunov function Mathematical Problems in Engineering The derivative of  1 along the trajectories of (22) satisfies This implies that  1 () ≤  1 (0).Now, by Lemma 2, the LOS angular rate converges to zero and the impact time converges to the value of   −  in finite time, and the settling time is given by The proof is finished.

Impact Guidance
Law with Autopilot Dynamics.The acceleration command generated by the guidance law is achieved by a missile's autopilot which tracks the expected acceleration command by modulating fins or thrusters.
There is a lag between the expected acceleration and the achieved acceleration.The lag will induce bad influence on the guidance law, especially with the time limited.In practical application, designing the guidance law considering autopilot dynamics is the best way to improve the performance of the missile.The autopilot can be well approximated as the following first-order dynamics: where   is the actual acceleration command,  1 is the time constant of the autopilot, and  is the missile acceleration command.
The acceleration command  is easy to be obtained, if the actual acceleration command   and its differential form ȧ  can be known.However, the differential form of actual acceleration command includes the first-order derivative of the LOS angular rate which is hard to be accurately estimated, thus making it impossible to implement the guidance law.Thus, low-pass filter is adopted to solve this problem.
Pass guidance law (20) through a first-order filter as follows: where  *  is a control law obtained from (20), â is the output of first-order filter, and  is a time constant.
Considering the autopilot dynamics and the first-order filter, the following sliding mode manifold and the resolution error of the first-order filter are introduced: Theorem 4. Consider (20), (26), and (27).In the following control law: the time constant  satisfies  ≤ ||/(| ȧ *  | + ), || = , and  < 0, where  is a positive constant.The control law ensures that the lag between the expected acceleration and the achieved acceleration converges to zero in finite time.
Proof.Differentiating (28) with respect to time yields gives Define a Lyapunov function With ( 29) and (30), the derivative of  2 is deduced as The elements of guidance system are all bounded, so ȧ *  ≤ | ȧ * max | is a nonnegative function and has a maximum value.With  ≤ ||/(| ȧ * max | + ), (32) can be obtained as Since || > 0 and  1/2 2 > 0 for  1 ̸ = 0, there is V 1 () < 0. The system satisfies the Lyapunov stability.By Lemma 2, the expected acceleration converges to the achieved acceleration in finite time, and the settling time is given by The ITGAD law involves a signum function which induces the chattering effect.So, to remove the chattering, it can be smoothened with a saturation function sat().The saturation function is expressed as where  is a small positive constant.

Impact Time Guidance Law Based on Variable Coefficient Strategy
To prove the stability of the guidance system, we need the following theorem.Proof.For Δ  ∈ Ω and Δ  → 0, (Δ  ) is nearly a positive constant.Thus, it is clear that In the area of Δ  ,  is finite time stable.And there exists where  0 is the initial state and   is the final state in the area of Δ  .
Moreover, by Lemma 2, if  is the settling time,  can be written as Equation (39) implies that the function is globally finite time stable.The proof is finished.
In the guidance law (20) and (29),  1 ,  2 , and  3 are selected mainly based on the desired time and initial states of missile.To avoid this problem, the variable strategy is proposed to achieve the objective that the missile impacts the target in the desired time.From the sliding mode manifold ( 16)∼( 19),  1 and  3 are coefficients concerning impact time which determine the impact time of the missile and  2 is coefficient concerning LOS angular rate which determines the final miss distance of the missile.The guidance law with variable strategy is given as where  1 (),  2 (), and  3 () are functions with respect to the system parameters.The focus of this strategy is to adjust  1 (),  3 () to a small value and  2 () to a large value when the missile is close to the target to ensure that the missile impacts the target with little final miss distance.As the relative distance between missile and target can be detected easily by the seekers, an available approach in this paper is to make the coefficients vary with the relative distance.The functions of  1 (),  2 (), and  3 () are defined as where  1 ,  2 , and  3 are positive constant. 0 is the initial relative distance., , and  are positive constants which determine whether the missile impacts the target in the desired time.  is short durations for the variation of  1 (),  2 (), and  3 ().
Theorem 6.In the guidance law (40), the coefficients of ( 41) are positive constants. satisfies  = || and  1 = 0.The guidance system with autopilot dynamics is globally finite time stable and the guidance law ensures that the LOS angular rate converges to zero and the rate of the impact time converges to the value of   −  in finite time.
Choosing small clips of the guidance law,  1 (Δ  ),  2 (Δ  ), and  3 (Δ  ) are nearly a constant.Define a Lyapunov function The derivative of  is Since  > 0 and  1/2 > 0 for  ̸ = 0, there is V() < 0. The system satisfies the Lyapunov stability.By Lemma 2, the settling time is given by By Theorem 5, the settling time of the guidance system is where Δ  is the time of th clip of the guidance law and set to be Δ  = Δ  / ṙ  .Then the guidance system is globally finite time stable.The proof is finished.

Simulation
In this section, simulations of the proposed ITGAD guidance law are presented for a variety of scenarios.The missile's maximum acceleration is 80, where  is the gravity acceleration.
The parameters of ITGAD are set to be  = 1,  1 = 0,  1 = 0.05,  = −1,  = 0.01, and  = 300.For each simulation, it is assumed that the desired time is chosen to be larger than the impact time based on finite time convergence guidance (FTCG) [17].
Case 1 (impact a maneuvering target with constant velocity).For comparison, we apply the proposed guidance laws ITGAD and FTCG in this simulation.The missile speed is 1000 m/s with an initial heading angle of 60 deg.The target speed is 400 m/s with an initial heading angle of 0 deg.target's normal acceleration is set to be 20cos0.7.The initial positions of the missile and target are taken as (0,0) km and (15,13) km, respectively.Figures 2 and 3 show the trajectories and relative distance of missile and target under FTCG.The impact time is observed in simulation to be about 28.94 s as shown in Figure 3.The following simulation uses the same conditions, only now a designated impact time of 35 s is specified.The results of ITGAD are represented in Figures 4-8.
Figure 4 shows that the missile impacts the target at 34.95 s with the impact time error of 0.05 s.Meanwhile, the trajectories of missile and target with 0.02823 m final miss distance are shown in Figures 5 and 9. To ensure that the missile impacts the target at desired impact time, the missile turns slightly away from the target.Because of the variable coefficient strategy, at the end of the terminal guidance, the missile requires a high maneuverability to guarantee that the impact time is equal to the desired impact time as shown in Figure 8. Figures 6 and 7 illustrate that, under the ITGAD,  the LOS angular rate and the rate of time-to-go converge to a small neighborhood of sliding surface.
Case 2 (impact a maneuvering target with variable velocity).Different from Case 1, the velocity of target in this case is set to be varied.As  < 18 s, the acceleration components of target's tangential acceleration are set to be 5 m/s 2 .As  ≥ 18 s, the tangential acceleration becomes −15 m/s 2 .The initial speed of target is 400 m/s with an initial heading angle of 0 deg.The acceleration component of the target normal to the velocity is set to be 20cos0.7.The remaining conditions are the same as Case 1. Figure 10 demonstrates the trajectories of missile and target under FTCG.Figure 11 shows the relative distance of missile and impact time of the terminal guidance process under FTCG.and 17, despite the variation of target's absolute speed and its direction, the missile hits the target at 35.05 s with the impact time error of 0.05 s.Figures 13 and 17 reveal that the missile impacts the target with 0.00476 m final miss distance.The acceleration of missile shown in Figure 16 guarantees that the LOS angular rate and the rate of time-to-go converge to a small neighborhood of sliding surface as shown in Figures 14  and 15.
Case 3 (multimissile attacks a maneuvering target).This case was performed with the ITGAD applied to a salvo attack to improve the probability of hitting the target.In this simulation three missiles with different conditions are required to impact a maneuvering target at a desired impact time.The initial conditions are given in Table 1.The initial velocity and acceleration are the same as Case 1.The desired impact time for each missile is set to be 35 s. Figure 10 shows the trajectories of the three missiles under FTCG.The impact time under different initial conditions is 27.08 s, 30.62 s, and 28.05 s, respectively.So the three missiles impact the target at the different positions as shown in Figure 18. Figure 19 demonstrates that the three missiles under ITGAD impact the target at the same position.The impact time and the final miss distance are shown in Table 2.In this simulation, ITGAD reduced the impact time dispersion within about 0.1 s around the designated value.

Conclusions
In this paper, to impact a maneuvering target at a desired impact time, an impact time guidance law based on finite time convergence control theory has been designed.In ITGAD, the autopilot dynamics is also taken into consideration to compensate the lag between the expected command and the achieved command.Through the use of ITGAD, the states of guidance system including impact time and LOS angular rate converge to sliding surface in finite time.And the ITGAD achieves the guidance objective taking the target motion as disturbance since the target motion cannot be known in advance.The performance and the feasibility of the ITGAD are demonstrated through several simulations against the maneuvering target.

Figure 1 :
Figure 1: Relative motion of missile and target.

Figure 2 :
Figure 2: Trajectories of missile and target under FTCG.

Figure 3 :
Figure 3: Relative distance of missile and target under FTCG.

Figure 8 :
Figure 8: The acceleration of missile.

Figure 9 :
Figure 9: Relative distance of missile and target.

Figure 19 :
Figure 19: Trajectories of missiles and target under ITGAD.
The designed impact time is 35 s.Simulations under ITGAD are given in Figures 12-17.From Figures 12

Table 1 :
Initial condition of each missile.

Table 2 :
Impact time and final miss distance of each missile.