Impact-Time-Control Guidance Law for Missile with Time-Varying Velocity

The problem of impact-time-control guidance (ITCG) for the homingmissile with time-varying velocity is addressed. First, a novel ITCG law is proposed based on the integral sliding mode control (ISMC) method.Then, a salvo attack algorithm is designed based on the proposed guidance law.The performances of the conventional ITCG laws strongly depend on the accuracy of the estimated time-to-go (TTG). However, the accurate estimated TTG can be obtained only if the missile velocity is constant. The conventional ITCG laws were designed under the assumption that the missile velocity is constant.The most attractive feature of this work is that the newly proposed ITCG law relaxes the constant velocity assumption, which only needs the variation range of themissile velocity. Finally, the numerical simulation demonstrates the effectiveness of the proposed method.


Introduction
Proportional navigation guidance law (PNGL) [1][2][3][4] has been widely used in the area of homing guidance, which can achieve excellent performance in the presence of a nonmaneuvering target.However, in recent years, with the development of defense systems such as space defense antimissile system [5], electronic countermeasure system (ECMS) [6], and close-in weapon system (CIWS) [7], the survivability of attack missile with conventional guidance scheme has been intimidated.Fortunately, most defense systems have "one-toone" feature; thus the salvo attack of multiple missiles can be one of the most effective countermeasures for missiles against the threats of defense systems.The salvo attack can be realized if all missiles hit the target simultaneously; this is called impact-time-control guidance (ITCG).
In 2006, Jeon et al. [8] first proposed an ITCG law to realize the salvo attack.The solution is a combination of the PNGL and the feedback of the impact time error.In [9], a cooperative-proportional navigation guidance (CPNG) law was designed to realize the salvo attack by decreasing the time-to-go (TTG) variance of missiles.In [10], a bias proportional navigation guidance (BPNG) law was developed, in which the desired impact time and angle were achieved simultaneously.In [11], the authors proposed an optimal guidance law for controlling the impact time and angle.In [12], an ITCG law based on a two-step control strategy was proposed.In [13], the authors proposed a polynomial guidance law to control the impact time and angle.
Actually, the above-mentioned ITCG laws in [8][9][10][11][12][13] were designed based on linearized-engagement-dynamics.However, the linearized-engagement-dynamic-based method can achieve high precision only if the missile's flight-path angle is small.It is well known that the sliding mode control (SMC) method is powerful in controlling nonlinear system [14][15][16][17].So far, some SMC-based ITCG laws have been developed in [18][19][20].And they were derived based on the nonlinear engagement dynamics.In [18], a guidance law based on the second-order SMC method and the backstepping scheme was developed.In order to satisfy the impact time constraint, one coefficient of the guidance law was searched for by using the off-line simplex algorithm.In [19], a SMC-based guidance law was developed to meet the requirement of ITCG, in which the sliding mode was defined as the combination of the line-ofsight (LOS) angle rate and the impact time error.However, if the initial LOS angle is zero, the guidance law proposed in [19] will generate zero-acceleration command during the whole guidance process.To overcome this problem, in 2015, The remaining parts of this paper are as follows.In Section 2, the problems of existing ITCG laws are formulated.The main results are presented in Section 3. In Section 3, an ITCG law based on ISMC method is proposed and the permissible set of the desired impact times is discussed.Section 4 shows a salvo attack algorithm based on the proposed guidance law.In Section 5, the numerical simulations verify the effectiveness of the proposed method in comparison with the methods presented in [8,20].In Section 6, the conclusions of the whole paper are presented.

Notations.
The following notations will be used in this paper. denotes the elapsed time after launching the missile,  0 denotes  at initial time  0 , and (  ) denotes  at time   .

Problem Formulation
As shown in Figure 1,  represents the horizontal direction,  represents the longitudinal direction,  represents the stationary target,  represents the missile,  represents the LOS angle,  represents the relative distance between the missile and the target,  represents the heading angle,   represents the missile acceleration, and  represents the missile velocity.From Figure 1, the following equalities can be established [20]: where (, ) are the coordinates of missile.The equations for relative motion can be expressed as [20] where   denotes the aerodynamic coefficients,  denotes the air density,   denotes the reference area,  denotes the missile mass, and   denotes the lateral acceleration.It is assumed that air resistance plays a major role in the change of the missile velocity; thus we have Note that if condition (4) is satisfied, then the objective of ITCG is realized [8][9][10][11][12][13][18][19][20]: where   denotes the impact time and    denotes the desired impact time.  can be rewritten as where  is the elapsed time after launching the missile and  go is the time-to-go (TTG).Combining (4) with (5), condition ( 4) is equivalent to the following condition: In order to satisfy (6), the conventional ITCG laws in [8][9][10][11][12][13][18][19][20] need  go .However,  go can not be measured directly.These literatures design estimation algorithm to estimate the TTG.The performances of these ITCG laws strongly depend on the accuracy of the estimated TTG.The length of the trajectory of missile, , is given by If  is constant ( V = 0), then we have Combining ( 5) with ( 8), we have Based on ( 9), the estimated TTG tgo is obtained in [8][9][10][11][12][13][18][19][20].For example, in [8], the estimated TTG is ) .
In [9], the estimated TTG is where  is a constant parameter.However, if  is time-varying, (8) cannot be derived from (7).In other word, the estimation error of TTG may be very large if we still use the estimation methods in [8][9][10][11][12][13][18][19][20].And the requirement of the ITCG cannot be satisfied with their guidance laws (in Section 5 of this paper, the simulation result also demonstrates that the performance of existing ITCG laws is poor when the missile velocity is timevarying).This motivates the research topic of this paper, that is, designing a new ITCG law for missile with time-varying velocity to satisfy condition (4).
The following assumptions should be assumed to be valid throughout this paper: (A1) Missile velocity  satisfies  min ≤  ≤  max , where  min and  max are the known constants.

New Impact-Time-Control Guidance Law Based on ISMC
3.1.Design of ITCG Law.Condition (4) is equivalent to the following conditions: Conditions (12) mean that the missile can attack the target only if  =    .In this section, a novel ITCG law is developed to satisfy (12).
A novel integral sliding mode (ISM) is constructed as where , , and  are the guidance parameters, which are all constants.Then, using the following guidance law: where  is a small positive constant and sgn (⋅) denotes the signum function, we have the following results.
Remark 2. From Theorem 1, it is clear that the proposed guidance law (17) can guarantee each missile attacks the target only when  =    .
Remark 3. Equation (17) shows that the control law contains (   − ) in the denominator, which may bring the singularity when  =    .Fortunately, the singularity can be eliminated; combining (27) with (28) yields From (39), we can know that the singularity brought by (   −) is eliminated as long as  1 ≥ 2 and  2 ≥ 2. (17) shows that the guidance command  → ∞, which means it is impossible to realize the guidance law.In what follows, Theorem 4 shows the condition which can ensure () ̸ = 0 during the guidance process.
Proof.From the demonstration of Theorem 4, we known that the guidance law (17) can ensure the state variables of system (15) satisfy the following equations: where  1 and  2 are constants and  1 and  2 are the solutions to the equation  2 − ( + ) +  = 0.
Since  min / ≤  min / < 1 and  0 cos Then, we have where  is a sufficiently small time.The proof is finished.
Remark 5.In the practical engineering, the available missile acceleration is bounded: where  max is the maximum available acceleration of missile and is determined by the performance of missile.Let  = − −1 ()(() + ((   − ) 2 +  1 )/(   − ) 2 +  sgn ( ISM )).To satisfy (62) in the practical engineering, the guidance law (17) must be modified as In the most ideal case ( max = ∞), we can expect the relative distance (   ) = 0.However,  max is bounded in engineering.Fortunately, from [8], it can be known that the objective of ITCG can be realized if the impact time error is smaller than 0.1 s.Thus, if the following conditions (63), (64), and (65) can be satisfied simultaneously, the objective of ITCG also can be obtained: where  is a sufficiently small time and  < 0.1. set is a small distance that can guarantee the precision of attack (e.g., if the target is ship,  set = 1 m can guarantee the precision of attack).
From (2), we have Substituting (39) into guidance law (17) and considering (67) yield Since  > 0,  > 0,  1 ≥ 2, and  2 ≥ 2, it is clear that  is bounded when  0 ≤  ≤    −.Thus, if the maximum available acceleration  max is large enough, then condition (66) can be satisfied even if  is very small.Since condition (66) can be satisfied, we can know that That is, the guidance law ( 17) is valid when  0 ≤  ≤    − .Thus the mathematical derivations from ( 12) to (61) are still valid when  0 ≤  ≤    − .From (37), it is clear that (64) is satisfied.Substituting  =    −  into (45) yields From ( 72), it can be known that condition (65) can be satisfied if  is small enough.In short, for the given small  and  set , (64), (65), and (66) can be satisfied simultaneously if  max is large enough and bounded.Actually, from the simulation result in Section 5 of this paper, it can be known that the modified guidance law (63) can guarantee (64), (65), and (66) are satisfied simultaneously in the case that  set = 0.1 m,  = 10 −3 s, and  max = 100 m/s 2 (from the results in [8][9][10][11][12][13][18][19][20], it can be known that 100 m/s 2 is a reasonable maximum available acceleration for missile; and the objective of ITCG can be accomplished when  set = 0.1 m and  = 10 −3 s).

The Permissible Set of the Desired Impact Times.
In this section, the permissible set of the desired impact times is discussed.From the above Theorems 1 and 4, the desired impact time and the guidance parameters must satisfy where  max is the maximum value of  1 and  2 and  min is the minimum value of  1 and  2 .From (73), the bound of    can be defined as where  is variable.If  =  max ,  is the lower bound.If  =  min ,  is the upper bound, taking the following partial differential operation as From ( 75), we know that  reduces monotonically with increasing .Then, we get the following.
(a) The lower bound of the permissible set of the desired impact times    is   min , which is described as If  max → ∞, the lower bound can be achieved.(b) The upper bound of the permissible set of the desired impact times    is   max , which is described as If  min = 2, we can get the upper bound.From (a) and (b), the permissible set of the desired impact times    can be given as If we select    ∈  and assume that (   −  0 ) min −  0 > 0, to meet condition (73), the permissible sets of  max and  min can be described as Because  max and  min are the solutions to the equation  2 − ( + ) +  = 0, the guidance parameters can be obtained as (82)

Salvo Attack
In this section, an algorithm will be designed based on the proposed guidance law to realize the salvo attack.Consider  missiles   ( = 1, . . ., ) engaged in a salvo attack against a stationary target as shown in Figure 2.   denotes the LOS angle,   denotes the relative distance between the missile   and the target,   denotes the heading angle,   denotes the missile acceleration, and   denotes the missile velocity.The permissible set of the desired impact times    for missile   is   = (    min ,     max ], and the intersection set can be described as  = ∩  =1   .It is assumed that  is not a null set.Each missile can utilize the guidance law (17) to achieve the desired impact time at each time step.Inspired by the salvo attack strategy introduced in [24], the detailed algorithm is given in Algorithm 1.

Simulation
In this section, to illustrate the effectiveness of the proposed guidance law, the mathematical simulation is presented.To remove the chattering, sgn () is replaced by the sigmoid function given as [20] sgmf where  is selected as 30.In addition, the maximum limit of the missile acceleration command is selected as 100 m/s 2 .

Performance of the Proposed Guidance
Law.This subsection shows the performance of the proposed guidance law.The simulation result for the missile with constant velocity is shown in Case 1.And the simulation result for the missile with time-varying velocity is provided in Case 2.
For the comparison, the traditional proportional navigation guidance law (PNGL), the impact-time-control guidance law (ITCGL) in [8], and the nonsingular sliding mode guidance law (NSMGL) in [20] are also considered.The PNGL [8] is defined as The parameter is chosen as  = 3.The ITCGL [8] is defined as where the parameter is chosen as  = 3.The NSMGL [20] is defined as where where the parameters are chosen as  = 50,  = 10,  = 2,  1 = 0.001,  2 = 0.015, and  = 3.In this paper, the values of , , , , ,  1 ,  2 , and  are the same as that in [8,20] and used here to ensure the fairness of comparison.
Case 1 (missile with constant velocity).In this case, we consider that the missile velocity is constant; that is, V = 0.
The initial conditions used in Case 1 are listed in Table 1.
From Table 1, the permissible set of the desired impact times can be calculated as  = (26.1008,42.4381] by using (80).
Choose the desired impact time as    = 37s ∈ .The permissible set of  max and  min can be calculated from (81) as  max ∈ (2.6137, ∞) ∩ ( min , ∞) and  min ∈ [2, ∞).From the permissible sets,  max and  min are selected as  max = 4 and  min = 2.5.Using  max and  min , the guidance parameters , , and  can be obtained from (82) as  = 1,  = 5.5, and  = 10.The simulation results of Case 1 are shown in Figures 3(a), 3(b), 3(c), and 3(d) and Table 2, respectively.From Figures 3(a) and 3(b) and Table 2, it can be seen that the proposed law, NSMGL, and ITCGL all can guarantee that the impact time errors are less than 4 × 10 −2 s and the miss distances are less than 1 m, which means that the impact time constraint can be satisfied by using these laws when the missile velocity is constant.The PNGL makes the missile have a large impact time error −6.39 s, which means that the mission of impact time constraint cannot be accomplished under PNGL.Moreover, in order to control the impact time, it can be seen from Figure 3(c) that the proposed law, NSMGL, and ITCGL employ more control energy compared to PNGL.
Case 2 (missile with time-varying velocity).In this case, we consider that the missile velocity is varying.The aerodynamic coefficients are given by The initial conditions used in Case 1 are listed in Table 1.
From Table 1, the permissible set of the desired impact times can be calculated as  = (29.8294,47.2385] by using (80).
The simulation results of Case 2 are shown in Figures 4(a), 4(b), 4(c), and 4(d) and Table 2, respectively.From Figures 4(a) and 4(b) and Table 2, it can be seen that the proposed law can guarantee that the impact time error is only 6×10 −4 s and the miss distance is 0.2 m, which means that the impact time constraint (37 s) can be satisfied by using the proposed law.However, for NSMGL and ITCGL, the impact time errors are greater than 0.48 s.As mentioned before, this is because the NSMGL and ITCGL are designed under the assumption that the missile velocity is constant.Compared with the impacttime-control guidance laws based on the assumption that the missile velocity is constant, the proposed law can achieve smaller impact time errors when the velocity is varying.

Application of Proposed Guidance Law to Salvo Attack
Scenario.This subsection is performed with the proposed guidance law for a salvo attack.The air density is 1.293 kg/m 3 .The aerodynamic coefficients for each missile are given by The initial conditions used in this subsection are listed in Table 3.
From Table 3, the permissible sets of the desired impact times for each missile can be calculated as (90) Choose the desired impact time as    = 37 s ∈  1 ∩  2 ∩  3 .Then, the permissible set of  max and  min can be calculated from (81) as From the permissible sets, we select that  1max =  2max =  3max = 4 and  1min =  2min =  3min = 2.5.Then, the guidance parameters can be obtained from (82) as       4, it is clear that the proposed law can guarantee the miss distances are less than 0.5 m, and the impact time errors are less than 4 × 10 −4 s, which means that the missiles can accomplish salvo attack even if the missile velocities are time-varying.

Conclusions
In this paper, a novel ITCG law has been proposed for the homing missile with time-varying velocity.The main contribution here is that the proposed ITCG law relaxed the constant velocity assumption which is needed for the existing ITCG laws.Since most missiles have time-varying velocity, the proposed method is more feasible than the existing ITCG  laws in engineering.In addition, the proposed law also exhibited one attractive feature.The proposed guidance law was derived based on the nonlinear engagement dynamic rather than the linearized-engagement-dynamics used in many traditional ITCG laws.A salvo attack algorithm has been developed based on the proposed ITCG law.The theoretical derivations and simulation results all demonstrated that the proposed guidance law achieved a better performance than the existing ITCG laws when the missile velocity is timevarying and that the proposed salvo attack algorithm worked well.

Figure 5 :
Figure 5: Responses in the salvo attack scenario.

Table 1 :
Initial condition for Cases 1 and 2.

Table 2 :
Performance of guidance laws in Section 5.1.

Table 3 :
Initial conditions for salvo attack.