Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2016 / Article

Research Article | Open Access

Volume 2016 |Article ID 7951923 | 14 pages | https://doi.org/10.1155/2016/7951923

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

Academic Editor: Andrzej Swierniak
Received25 Jan 2016
Accepted05 Jun 2016
Published30 Jun 2016

Abstract

The problem of impact-time-control guidance (ITCG) for the homing missile 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 the missile velocity. Finally, the numerical simulation demonstrates the effectiveness of the proposed method.

1. Introduction

Proportional navigation guidance law (PNGL) [14] 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-to-one” 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 [813] 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 [1417]. So far, some SMC-based ITCG laws have been developed in [1820]. 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-of-sight (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, Cho et al. [20] proposed an ITCG law based on a nonsingular sliding mode. Compared with the sliding mode in [19], the sliding mode in [20] was defined as only the impact time error.

Note that the aforementioned ITCG laws in [813, 1820] all need the TTG. However, the TTG cannot be measured directly. To overcome this problem, [813, 1820] all design estimation algorithm to estimate the TTG. These estimation results in [813, 1820] are obtained under the assumption that the missile velocity is invariant. However, the assumption is too restrictive for many cases. In practical missile systems, the missile velocity is affected inevitably by various forces such as the driving force of engine and the aerodynamic forces. When the constant velocity assumption is invalid, the estimation error of TTG may be large. Thus, these guidance laws in [813, 1820] may not work well in practical situations.

In this paper, a novel ITCG law is proposed for missile with time-varying velocity, based on integral sliding mode control (ISMC). The main contributions of this paper are the following:(1)As far as we know, the ITCG law for the missile with time-varying velocity is achieved for the first time. Compared with the conventional ITCG laws in [813, 1820], the constant velocity assumption is relaxed in this paper. In other words, the proposed law is more practical than the conventional ITCG laws.(2)The proposed guidance law is derived based on the nonlinear engagement dynamic rather than the linearized dynamics used in [813].

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, denotes at initial time , and denotes at time .

2. 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 [813, 1820]: 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 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 [813, 1820] need . However, 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 (), then we haveCombining (5) with (8), we haveBased on (9), the estimated TTG is obtained in [813, 1820]. For example, in [8], the estimated TTG isIn [9], the estimated TTG iswhere 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 [813, 1820]. 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 time-varying). 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 , where and are the known constants.(A2) is available (see [2123]).Assumption (A1) implies that we only need the bound of the time-varying velocity. Obviously, assumption (A1) is a relaxed version of the constant velocity assumption used in [813, 1820]. Because can be measured by accelerometer, assumption (A2) is a commonly used assumption for the design of guidance law (see [2123]).

3. 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 state variable is defined asDifferentiating (13) givesLet , and the guidance system (2) can be rewritten as where , and .

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 denotes the signum function, we have the following results.

Theorem 1. Consider system (15) with the guidance law (17), provided that and are the solutions to the equation , where , , and are the guidance parameters in (17). If satisfies, , , and , then the following conditions will be satisfied:

Proof. Differentiating (16) with respect to time yieldsCombining (15) with (20), we haveConstruct a Lyapunov function asDifferentiating (22) and then using (21) and (17) yieldNote that , and one can imply that As remains as zero, we get The following equation can be obtained by combining (20) with (25):Provided that and are the solutions to the equation , if , , and , the solutions to the differential equation (26) can be easily obtained aswhere and are constant.
Substituting (13) into (27) and substituting (14) into (28) yieldSubstituting the initial values and into (29) and (30), we getFrom (31), we haveAccording to (18), , , and , we haveFrom (32), (33), and , one can imply thatSubstituting (32) into (29), we haveConsidering (34), , , and , we get (36) from (35). ConsiderThen, we getMoreover, according to (29), we getThe proof is finished.

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) yieldsFrom (39), we can know that the singularity brought by is eliminated as long as and .

3.2. Nonsingular Impact-Time-Control Guidance Law

If , (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 during the guidance process.

Theorem 4. Consider system (15) with the guidance law (17). Provided that and are the solutions to the equation , where , , and are the parameters in (17), if satisfies (18), , , , , and , then satisfies the following condition:where is a sufficiently small time and where is the value of at time .

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 and are constants and and are the solutions to the equation .
Substituting (13) into (43) and substituting (14) into (44) yieldAccording to (18), , and , we can get the following results from the demonstration of Theorem 4:Combining (47) with (46) yieldsSubstituting (2) into (48) yieldsDifferentiating (44) givesThen, considering , , , and , we haveEquation (51) means that reduces monotonically with increasing , and we getFrom (52), we getCombining (49) and (53) yieldsFrom (3), it can be known that . Then, we havewhere is the value of at time .
Since and , we haveLet , then we haveIt is clear that . Since , we haveLet , we haveSince and , we haveThen, we havewhere is a sufficiently small time. The proof is finished.

Remark 5. In the practical engineering, the available missile acceleration is bounded:where is the maximum available acceleration of missile and is determined by the performance of missile. Let . To satisfy (62) in the practical engineering, the guidance law (17) must be modified asIn the most ideal case (), we can expect the relative distance . However, 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 . is a small distance that can guarantee the precision of attack (e.g., if the target is ship,  m can guarantee the precision of attack).

From (2), we haveSubstituting (39) into guidance law (17) and considering (67) yieldFrom (68), we haveCombining (37), (61), and (69) yieldsSince , , , and , it is clear that is bounded when . Thus, if the maximum available acceleration 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 . Thus the mathematical derivations from (12) to (61) are still valid when . From (37), it is clear that (64) is satisfied. Substituting into (45) yieldsFrom (72), it can be known that condition (65) can be satisfied if is small enough.

In short, for the given small and , (64), (65), and (66) can be satisfied simultaneously if 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  m,  s, and  m/s2 (from the results in [813, 1820], it can be known that /s2 is a reasonable maximum available acceleration for missile; and the objective of ITCG can be accomplished when  m and  s).

3.3. 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 satisfywhere is the maximum value of and and is the minimum value of and .

From (73), the bound of can be defined aswhere is variable. If , is the lower bound. If , is the upper bound, taking the following partial differential operation as Theorems 1 and 4 need the following condition:Combining (75) and (76) yieldsFrom (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 , which is described asIf , the lower bound can be achieved.

(b) The upper bound of the permissible set of the desired impact times is , which is described asIf , 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 , to meet condition (73), the permissible sets of and can be described asBecause and are the solutions to the equation , the guidance parameters can be obtained as

4. Salvo Attack

In this section, an algorithm will be designed based on the proposed guidance law to realize the salvo attack. Consider missiles 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 , and the intersection set can be described as . 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.

for each missile , do
Compute the permissible set from (80).
end for
Find the intersection set .
Select a .
for each missile , do
Select and from (81) and obtain the
parameters , and from (82).
end for
for each time step do
 for each missile , do
 Use , and to generate the guidance input of (17).
end for
end for

5. Simulation

In this section, to illustrate the effectiveness of the proposed guidance law, the mathematical simulation is presented. To remove the chattering, is replaced by the sigmoid function given as [20]where is selected as 30. In addition, the maximum limit of the missile acceleration command is selected as 100 m/s2.

5.1. 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 asThe parameter is chosen as . The ITCGL [8] is defined aswhere the parameter is chosen as . The NSMGL [20] is defined aswherewhere the parameters are chosen as , , , , , and . In this paper, the values of , , , , , , , 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, . 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 by using (80). Choose the desired impact time as . The permissible set of and can be calculated from (81) as and . From the permissible sets, and are selected as and . Using and , the guidance parameters , , and can be obtained from (82) as , , and .
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  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.


ParametersCase 1 (constant velocity)Case 2 (varying velocity)

Initial missile position(−10000, −3000) m
Initial missile velocity 400 
Initial missile heading angle −60 deg
Target position m
Desired impact time 37 s
Missile mass 400 kg
Reference area 0.039 
Air density 1.293 
Minimum speed 400 m/s350 m/s


Case Guidance law (s)Miss distance (m)

1Proposed law0.07
NSMGL0.05
ITCGL0.33
PNGL0.03

2Proposed law0.15
NSMGL0.06
ITCGL0.47
PNGL0.09

Case 2 (missile with time-varying velocity). In this case, we consider that the missile velocity is varying. The aerodynamic coefficients are given byThe 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 by using (80). Choose the desired impact time as . Then, the permissible set of and can be calculated from (81) as and . From the permissible sets, and are selected as and . Using and , the guidance parameters , , and can be obtained from (82) as , , and .
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 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 impact-time-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.

5.2. 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/m3. The aerodynamic coefficients for each missile are given byThe initial conditions used in this subsection are listed in Table 3.


Object/m/m/(deg)/(m/s)/(m/s)Mass/(kg)/()

Target0000000
Missile 1
Missile 2