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

Authors: Jun Zhou; Yang Wang; Bin Zhao
Journal: Mathematical Problems in Engineering
 (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.


---

## Body

## 1. Introduction

Proportional navigation guidance law (PNGL) [1–4] 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-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 [8–13] 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 [14–17]. So far, some SMC-based ITCG laws have been developed in [18–20]. 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-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 [8–13, 18–20] all need the TTG. However, the TTG cannot be measured directly. To overcome this problem, [8–13, 18–20] all design estimation algorithm to estimate the TTG. These estimation results in [8–13, 18–20] 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 [8–13, 18–20] 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 [8–13, 18–20], 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 [8–13].The remaining parts of this paper are as follows. In Section2, 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. t denotes the elapsed time after launching the missile, x
0 denotes x at initial time t
0, and x
t
k denotes x at time t
k.


## 2. Problem Formulation

As shown in Figure1, O
x represents the horizontal direction, O
y represents the longitudinal direction, T represents the stationary target, M represents the missile, λ represents the LOS angle, R represents the relative distance between the missile and the target, γ represents the heading angle, a
M represents the missile acceleration, and V represents the missile velocity. From Figure 1, the following equalities can be established [20]:(1)
x
˙
=
V
cos
⁡
γ
,
y
˙
=
V
sin
⁡
γ
,where x
,
y are the coordinates of missile. The equations for relative motion can be expressed as [20](2)
R
˙
=
-
V
cos
⁡
λ
-
γ
=
-
V
cos
⁡
ϕ
,
R
λ
˙
=
-
V
sin
⁡
λ
-
γ
=
-
V
sin
⁡
ϕ
,
a
M
=
V
γ
˙
,
V
˙
=
1
2
C
x
ρ
V
2
S
M
m
,where C
x denotes the aerodynamic coefficients, ρ denotes the air density, S
M denotes the reference area, m denotes the missile mass, and a
M denotes the lateral acceleration. It is assumed that air resistance plays a major role in the change of the missile velocity; thus we have(3)
C
x
<
0
.Figure 1
Missile-target engagement geometry.Note that if condition (4) is satisfied, then the objective of ITCG is realized [8–13, 18–20]: (4)
t
f
=
t
f
d
,where t
f denotes the impact time and t
f
d denotes the desired impact time. t
f can be rewritten as (5)
t
f
=
t
+
t
g
o
,where t is the elapsed time after launching the missile and t
g
o is the time-to-go (TTG).Combining (4) with (5), condition (4) is equivalent to the following condition:(6)
t
f
d
=
t
+
t
g
o
.In order to satisfy (6), the conventional ITCG laws in [8–13, 18–20] need t
g
o. However, t
g
o 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,S, is given by (7)
S
=
∫
t
0
t
f
V
d
t
=
∫
x
0
x
t
f
1
+
d
y
d
x
2
d
x
.If V is constant (V
˙
=
0), then we have(8)
t
f
=
1
V
∫
x
0
x
t
f
1
+
d
y
d
x
2
d
x
+
t
0
.Combining (5) with (8), we have(9)
t
g
o
=
1
V
∫
x
0
x
t
f
1
+
d
y
d
x
2
d
x
+
t
0
-
t
.Based on (9), the estimated TTG t
^
g
o is obtained in [8–13, 18–20]. For example, in [8], the estimated TTG is(10)
t
^
g
o
=
R
V
1
+
γ
-
λ
2
10
.In [9], the estimated TTG is(11)
t
^
g
o
=
R
V
1
+
γ
-
λ
2
2
2
N
-
1
,where N is a constant parameter.However, ifV 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 [8–13, 18–20]. 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 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 velocityV satisfies V
m
i
n
≤
V
≤
V
m
a
x, where V
m
i
n and V
m
a
x are the known constants.
(A2)
V
˙ is available (see [21–23]).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 [8–13, 18–20]. Because V
˙ can be measured by accelerometer, assumption (A2) is a commonly used assumption for the design of guidance law (see [21–23]).


## 3. New Impact-Time-Control Guidance Law Based on ISMC

### 3.1. Design of ITCG Law

Condition (4) is equivalent to the following conditions:(12)
R
t
>
0
,
∀
t
0
≤
t
<
t
f
d
,
R
t
f
d
=
0
.Conditions (12) mean that the missile can attack the target only if t
=
t
f
d. In this section, a novel ITCG law is developed to satisfy (12).A state variablez
1 is defined as(13)
z
1
=
R
-
V
m
i
n
t
f
d
-
t
.Differentiating (13) gives(14)
z
˙
1
=
R
˙
+
V
m
i
n
.Let z
˙
1
=
z
2, and the guidance system (2) can be rewritten as (15)
z
˙
1
=
z
2
,
z
˙
2
=
f
z
+
g
z
u
,
V
˙
=
1
2
C
x
ρ
V
2
S
M
m
,where u
=
a
M, f
z
=
-
V
˙
cos
⁡
λ
-
γ
+
V
sin
⁡
λ
-
γ
λ
˙ and g
z
=
-
sin
⁡
λ
-
γ.A novel integral sliding mode (ISM) is constructed as(16)
s
I
S
M
=
z
2
-
z
2
t
0
+
∫
t
0
t
b
t
f
d
-
t
z
2
+
c
z
1
a
t
f
d
-
t
2
d
τ
,where a, b, and c are the guidance parameters, which are all constants. Then, using the following guidance law:(17)
u
=
-
g
-
1
z
f
z
+
b
t
f
d
-
t
z
2
+
c
z
1
a
t
f
d
-
t
2
+
β
sgn
⁡
s
I
S
M
,where β is a small positive constant and sgn
⁡
· denotes the signum function, we have the following results.Theorem 1.
Consider system (15) with the guidance law (17), provided that q
1 and q
2 are the solutions to the equation a
q
2
-
a
+
b
q
+
c
=
0, where a, b, and c are the guidance parameters in (17). If t
f
d satisfies(18)
t
0
+
R
0
q
2
-
R
˙
0
-
1
-
q
2
V
m
i
n
<
t
f
d
<
t
0
+
R
0
q
1
-
R
˙
0
-
1
-
q
1
V
m
i
n
,
i
f
q
2
>
q
1
,
t
0
+
R
0
q
1
-
R
˙
0
-
1
-
q
1
V
m
i
n
<
t
f
d
<
t
0
+
R
0
q
2
-
R
˙
0
-
1
-
q
2
V
m
i
n
,
i
f
q
1
>
q
2
,
q
1
≥
2, q
2
≥
2, q
1
≠
q
2, and V
0
cos
⁡
λ
0
-
γ
0
/
V
m
i
n
<
1, then the following conditions will be satisfied:(19)
R
t
>
0
,
∀
t
0
≤
t
<
t
f
d
,
R
t
f
d
=
0
.Proof.
Differentiating (16) with respect to time yields(20)
s
˙
I
S
M
=
z
˙
2
+
b
t
f
d
-
t
z
2
+
c
z
1
a
t
f
d
-
t
2
.Combining (15) with (20), we have(21)
s
˙
I
S
M
=
f
z
+
g
z
u
+
b
t
f
d
-
t
z
2
+
c
z
1
a
t
f
d
-
t
2
.Construct a Lyapunov function V
1 as(22)
V
1
=
1
2
s
I
S
M
2
.Differentiating (22) and then using (21) and (17) yield(23)
V
˙
1
=
s
I
S
M
s
˙
I
S
M
=
s
I
S
M
f
z
+
g
z
u
+
b
t
f
d
-
t
z
2
+
c
z
1
a
t
f
d
-
t
2
=
-
s
I
S
M
β
sgn
⁡
s
I
S
M
<
0
,
when
s
I
S
M
≠
0
.Note that s
I
S
M
t
0
=
0, and one can imply that (24)
s
I
S
M
t
=
0
,
∀
t
0
≤
t
≤
t
f
d
.As s
I
S
M
t remains as zero, we get (25)
s
˙
I
S
M
t
=
0
,
∀
t
0
≤
t
≤
t
f
d
.The following equation can be obtained by combining (20) with (25):(26)
a
t
f
d
-
t
2
z
¨
1
+
b
t
f
d
-
t
z
˙
1
+
c
z
1
=
0
,
∀
t
0
≤
t
≤
t
f
d
.Provided that q
1 and q
2 are the solutions to the equation a
q
2
-
a
+
b
q
+
c
=
0, if q
1
≠
q
2, q
1
≥
2, and q
2
≥
2, the solutions to the differential equation (26) can be easily obtained as(27)
z
1
=
c
1
t
f
d
-
t
q
1
+
c
2
t
f
d
-
t
q
2
,
∀
t
0
≤
t
≤
t
f
d
,
(28)
z
˙
1
=
-
c
1
q
1
t
f
d
-
t
q
1
-
1
-
c
2
q
2
t
f
d
-
t
q
2
-
1
,
∀
t
0
≤
t
≤
t
f
d
,where c
1 and c
2 are constant.
Substituting (13) into (27) and substituting (14) into (28) yield(29)
R
-
V
m
i
n
t
f
d
-
t
=
c
1
t
f
d
-
t
q
1
+
c
2
t
f
d
-
t
q
2
,
∀
t
0
≤
t
≤
t
f
d
,
(30)
R
˙
+
V
m
i
n
=
-
c
1
q
1
t
f
d
-
t
q
1
-
1
-
c
2
q
2
t
f
d
-
t
q
2
-
1
,
∀
t
0
≤
t
≤
t
f
d
.Substituting the initial values R
0 and R
˙
0 into (29) and (30), we get(31)
R
0
-
V
m
i
n
t
f
d
-
t
0
=
c
1
t
f
d
-
t
0
q
1
+
c
2
t
f
d
-
t
0
q
2
,
∀
t
0
≤
t
≤
t
f
d
,
R
˙
0
+
V
m
i
n
t
f
d
-
t
0
=
-
c
1
q
1
t
f
d
-
t
0
q
1
-
c
2
q
2
t
f
d
-
t
0
q
2
,
∀
t
0
≤
t
≤
t
f
d
.From (31), we have(32)
c
1
=
R
˙
0
+
V
m
i
n
1
-
q
2
t
f
d
-
t
0
+
q
2
R
0
q
2
-
q
1
t
f
d
-
t
0
q
1
,
c
2
=
-
R
˙
0
-
V
m
i
n
1
-
q
1
t
f
d
-
t
0
-
q
1
R
0
q
2
-
q
1
t
f
d
-
t
0
q
2
.According to (18), V
0
cos
⁡
λ
0
-
γ
0
/
V
m
i
n
<
1, q
1
≥
2, and q
2
≥
2, we have(33)
c
1
<
0
,
c
2
<
0
.From (32), (33), and t
f
d
-
t
0
>
0, one can imply that(34)
R
˙
0
+
V
m
i
n
1
-
q
2
t
f
d
-
t
0
+
q
2
R
0
q
2
-
q
1
<
0
,
-
R
˙
0
-
V
m
i
n
1
-
q
1
t
f
d
-
t
0
-
q
1
R
0
q
2
-
q
1
<
0
.Substituting (32) into (29), we have(35)
R
-
V
m
i
n
t
f
d
-
t
=
R
˙
0
+
V
m
i
n
1
-
q
2
t
f
d
-
t
0
+
q
2
R
0
q
2
-
q
1
t
f
d
-
t
q
1
t
f
d
-
t
0
q
1
+
-
R
˙
0
-
V
m
i
n
1
-
q
1
t
f
d
-
t
0
-
q
1
R
0
q
2
-
q
1
t
f
d
-
t
q
2
t
f
d
-
t
0
q
2
,
∀
t
0
≤
t
≤
t
f
d
.Considering (34), 0
≤
t
f
d
-
t
/
t
f
d
-
t
0
≤
1, q
1
≥
2, and q
2
≥
2, we get (36) from (35). Consider(36)
R
-
V
m
i
n
t
f
d
-
t
≥
R
˙
0
+
1
-
q
2
V
m
i
n
t
f
d
-
t
0
+
q
2
R
0
q
2
-
q
1
t
f
d
-
t
t
f
d
-
t
0
+
-
R
˙
0
-
1
-
q
1
V
m
i
n
t
f
d
-
t
0
-
q
1
R
0
q
2
-
q
1
t
f
d
-
t
t
f
d
-
t
0
=
R
0
t
f
d
-
t
t
f
d
-
t
0
-
V
m
i
n
t
f
d
-
t
,
∀
t
0
≤
t
≤
t
f
d
.Then, we get(37)
R
≥
R
0
t
f
d
-
t
t
f
d
-
t
0
>
0
,
∀
t
0
≤
t
<
t
f
d
.
Moreover, according to (29), we get(38)
R
t
f
d
=
0
.The proof is finished.Remark 2.
From Theorem1, it is clear that the proposed guidance law (17) can guarantee each missile attacks the target only when t
=
t
f
d.Remark 3.
Equation (17) shows that the control law contains t
f
d
-
t in the denominator, which may bring the singularity when t
=
t
f
d. Fortunately, the singularity can be eliminated; combining (27) with (28) yields(39)
b
t
f
d
-
t
z
˙
1
+
c
z
1
a
t
f
d
-
t
2
=
c
1
c
a
-
c
1
q
1
b
a
t
f
d
-
t
q
1
-
2
+
c
2
c
a
-
c
2
q
2
b
a
t
f
d
-
t
q
2
-
2
,
∀
t
0
≤
t
≤
t
f
d
.From (39), we can know that the singularity brought by t
f
d
-
t is eliminated as long as q
1
≥
2 and q
2
≥
2.


### 3.2. Nonsingular Impact-Time-Control Guidance Law

Ifg
z
=
0, (17) shows that the guidance command u
→
∞, which means it is impossible to realize the guidance law. In what follows, Theorem 4 shows the condition which can ensure g
z
≠
0 during the guidance process.Theorem 4.
Consider system (15) with the guidance law (17). Provided that q
1 and q
2 are the solutions to the equation a
q
2
-
a
+
b
q
+
c
=
0, where a, b, and c are the parameters in (17), if t
f
d satisfies (18), q
1
≥
2, q
2
≥
2, q
1
≠
q
2, sin
⁡
λ
0
-
γ
0
≠
0, and V
0
cos
⁡
λ
0
-
γ
0
/
V
m
i
n
<
1
-
ψ
<
1, then g
z satisfies the following condition:(40)
g
z
≥
θ
-
>
0
,
∀
t
0
≤
t
≤
t
f
d
-
ε
,where ε is a sufficiently small time and (41)
θ
-
=
min
⁡
1
-
ψ
-
,
sin
⁡
λ
0
-
γ
0
,
(42)
ψ
-
=
max
⁡
V
m
i
n
V
-
2
,
-
1
+
ψ
2
,where V
- is the value of V at time t
=
t
f
d
-
ε.Proof.
From the demonstration of Theorem4, we known that the guidance law (17) can ensure the state variables of system (15) satisfy the following equations:(43)
z
1
=
c
1
t
f
d
-
t
q
1
+
c
2
t
f
d
-
t
q
2
,
∀
t
0
≤
t
≤
t
f
d
,
(44)
z
˙
1
=
-
c
1
q
1
t
f
d
-
t
q
1
-
1
-
c
2
q
2
t
f
d
-
t
q
2
-
1
,
∀
t
0
≤
t
≤
t
f
d
,where c
1 and c
2 are constants and q
1 and q
2 are the solutions to the equation a
q
2
-
a
+
b
q
+
c
=
0.
Substituting (13) into (43) and substituting (14) into (44) yield(45)
R
-
V
m
i
n
t
f
d
-
t
=
c
1
t
f
d
-
t
q
1
+
c
2
t
f
d
-
t
q
2
,
∀
t
0
≤
t
≤
t
f
d
,
(46)
R
˙
+
V
m
i
n
=
-
c
1
q
1
t
f
d
-
t
q
1
-
1
-
c
2
q
2
t
f
d
-
t
q
2
-
1
,
∀
t
0
≤
t
≤
t
f
d
.According to (18), q
1
≥
2, and q
2
≥
2, we can get the following results from the demonstration of Theorem 4:(47)
c
1
<
0
,
c
2
<
0
.Combining (47) with (46) yields(48)
R
˙
+
V
m
i
n
>
0
,
∀
t
0
≤
t
<
t
f
d
.Substituting (2) into (48) yields(49)
cos
⁡
λ
-
γ
<
V
m
i
n
V
,
∀
t
0
≤
t
<
t
f
d
.Differentiating (44) gives(50)
z
¨
1
=
c
1
q
1
q
1
-
1
t
f
d
-
t
q
1
-
2
+
c
2
q
2
q
2
-
1
t
f
d
-
t
q
2
-
2
,
∀
t
0
≤
t
<
t
f
d
.Then, considering q
1
≥
2, q
2
≥
2, c
1
<
0, and c
2
<
0, we have(51)
z
¨
1
<
0
,
∀
t
0
≤
t
<
t
f
d
.Equation (51) means that z
˙
1 reduces monotonically with increasing t, and we get(52)
z
˙
1
<
z
˙
1
t
0
,
∀
t
0
<
t
<
t
f
d
.From (52), we get(53)
cos
⁡
λ
-
γ
>
V
0
cos
⁡
λ
0
-
γ
0
V
,
∀
t
0
<
t
<
t
f
d
.Combining (49) and (53) yields(54)
V
0
cos
⁡
λ
0
-
γ
0
V
<
cos
⁡
λ
-
γ
<
V
m
i
n
V
,
∀
t
0
<
t
<
t
f
d
.From (3), it can be known that V
˙
<
0. Then, we have(55)
V
m
i
n
V
≤
V
m
i
n
V
-
<
1
,
∀
t
0
<
t
≤
t
f
d
-
ε
,where V
- is the value of V at time t
=
t
f
d
-
ε.
SinceV
m
i
n
/
V
≤
V
m
i
n
/
V
-
<
1 and V
0
cos
⁡
λ
0
-
γ
0
/
V
≥
-
V
0
cos
⁡
λ
0
-
γ
0
/
V
m
i
n
>
-
1
+
ψ
>
-
1, we have(56)
-
1
+
ψ
<
cos
⁡
λ
-
γ
≤
V
m
i
n
V
-
,
∀
t
0
<
t
≤
t
f
d
-
ε
.Let ψ
-
=
max
⁡
V
m
i
n
/
V
-
2
,
-
1
+
ψ
2, then we have(57)
cos
2
⁡
λ
-
γ
≤
ψ
-
,
∀
t
0
<
t
≤
t
f
d
-
ε
.It is clear that 0
<
ψ
-
<
1. Since sin
⁡
λ
-
γ
=
1
-
cos
2
⁡
λ
-
γ, we have(58)
sin
⁡
λ
-
γ
≥
1
-
ψ
-
,
∀
t
0
<
t
≤
t
f
d
-
ε
.Let θ
-
=
min
⁡
1
-
ψ
-
,
sin
⁡
λ
0
-
γ
0, we have(59)
sin
⁡
λ
-
γ
≥
θ
-
,
∀
t
0
≤
t
≤
t
f
d
-
ε
.Since sin
⁡
λ
0
-
γ
0
>
0 and 1
-
ψ
-
>
0, we have(60)
sin
⁡
λ
-
γ
≥
θ
-
>
0
,
∀
t
0
≤
t
≤
t
f
d
-
ε
.Then, we have(61)
g
z
≥
θ
-
>
0
,
∀
t
0
≤
t
≤
t
f
d
-
ε
,where ε is a sufficiently small time. The proof is finished.Remark 5.
In the practical engineering, the available missile acceleration is bounded:(62)
u
≤
u
m
a
x
,where u
m
a
x is the maximum available acceleration of missile and is determined by the performance of missile. Let u
-
=
-
g
-
1
z
f
z
+
(
b
t
f
d
-
t
z
2
+
c
z
1
)
/
a
t
f
d
-
t
2
+
β
sgn
⁡
s
I
S
M. To satisfy (62) in the practical engineering, the guidance law (17) must be modified as(63)
u
=
u
m
a
x
;
if
u
-
>
u
m
a
x
u
-
;
if
u
-
≤
u
m
a
x
-
u
m
a
x
;
if
u
-
<
-
u
m
a
x
.In the most ideal case (u
m
a
x
=
∞), we can expect the relative distance R
t
f
d
=
0. However, u
m
a
x 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:(64)
R
>
0
,
∀
t
0
≤
t
≤
t
f
d
-
ε
,
(65)
R
t
f
d
-
ε
≤
R
s
e
t
,
(66)
u
-
≤
u
m
a
x
,
∀
t
0
≤
t
≤
t
f
d
-
ε
,where ε is a sufficiently small time and ε
<
0.1. R
s
e
t is a small distance that can guarantee the precision of attack (e.g., if the target is ship, R
s
e
t
=
1 m can guarantee the precision of attack).From (2), we have(67)
λ
˙
=
-
V
sin
⁡
λ
-
γ
R
=
V
R
g
z
.Substituting (39) into guidance law (17) and considering (67) yield(68)
u
-
=
-
g
-
1
z
f
z
+
c
1
c
a
-
c
1
q
1
b
a
t
f
d
-
t
q
1
-
2
+
c
2
c
a
-
c
2
q
2
b
a
t
f
d
-
t
q
2
-
2
+
β
sgn
⁡
s
I
S
M
=
-
g
-
1
z
-
V
˙
cos
⁡
λ
-
γ
+
V
2
sin
⁡
λ
-
γ
g
z
R
+
c
1
c
a
-
c
1
q
1
b
a
t
f
d
-
t
q
1
-
2
+
c
2
c
a
-
c
2
q
2
b
a
t
f
d
-
t
q
2
-
2
+
β
sgn
⁡
s
I
S
M
=
V
˙
cos
⁡
λ
-
γ
g
z
-
V
2
sin
⁡
λ
-
γ
R
-
c
1
c
/
a
-
c
1
q
1
b
/
a
t
f
d
-
t
q
1
-
2
+
c
2
c
/
a
-
c
2
q
2
b
/
a
t
f
d
-
t
q
2
-
2
+
β
sgn
⁡
s
I
S
M
g
z
,
∀
t
0
≤
t
≤
t
f
d
-
ε
.From (68), we have(69)
u
-
≤
V
˙
cos
⁡
λ
-
γ
g
z
+
c
1
c
/
a
-
c
1
q
1
b
/
a
t
f
d
-
t
q
1
-
2
+
c
2
c
/
a
-
c
2
q
2
b
/
a
t
f
d
-
t
q
2
-
2
+
β
sgn
⁡
s
I
S
M
g
z
+
V
2
sin
⁡
λ
-
γ
R
,
∀
t
0
≤
t
≤
t
f
d
-
ε
.Combining (37), (61), and (69) yields(70)
u
-
≤
V
˙
cos
⁡
λ
-
γ
θ
-
+
c
1
c
/
a
-
c
1
q
1
b
/
a
t
f
d
-
t
q
1
-
2
+
c
2
c
/
a
-
c
2
q
2
b
/
a
t
f
d
-
t
q
2
-
2
+
β
sgn
⁡
s
I
S
M
θ
-
+
V
2
sin
⁡
λ
-
γ
t
f
d
-
t
0
R
0
·
t
f
d
-
t
≤
V
˙
cos
⁡
λ
-
γ
+
c
1
c
/
a
-
c
1
q
1
b
/
a
t
f
d
-
t
q
1
-
2
+
c
2
c
/
a
-
c
2
q
2
b
/
a
t
f
d
-
t
0
q
2
-
2
+
β
θ
-
+
V
2
sin
⁡
λ
-
γ
t
f
d
-
t
0
/
R
0
ε
,
∀
t
0
≤
t
≤
t
f
d
-
ε
.Since θ
-
>
0, ε
>
0, q
1
≥
2, and q
2
≥
2, it is clear that u
- is bounded when t
0
≤
t
≤
t
f
d
-
ε. Thus, if the maximum available acceleration u
m
a
x is large enough, then condition (66) can be satisfied even if ε is very small. Since condition (66) can be satisfied, we can know that (71)
u
=
u
-
=
-
g
-
1
z
f
z
+
b
t
f
d
-
t
z
2
+
c
z
1
a
t
f
d
-
t
2
+
β
sgn
⁡
s
I
S
M
,
∀
t
0
≤
t
≤
t
f
d
-
ε
.That is, the guidance law (17) is valid when t
0
≤
t
≤
t
f
d
-
ε. Thus the mathematical derivations from (12) to (61) are still valid when t
0
≤
t
≤
t
f
d
-
ε. From (37), it is clear that (64) is satisfied. Substituting t
=
t
f
d
-
ε into (45) yields(72)
R
t
f
d
-
ε
=
V
m
i
n
ε
+
c
1
ε
q
1
+
c
2
ε
q
2
.From (72), it can be known that condition (65) can be satisfied if ε is small enough.In short, for the given smallε and R
s
e
t, (64), (65), and (66) can be satisfied simultaneously if u
m
a
x 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 R
s
e
t
=
0.1 m, ε
=
10
-
3 s, and u
m
a
x
=
100 m/s2 (from the results in [8–13, 18–20], it can be known that 100
m/s2 is a reasonable maximum available acceleration for missile; and the objective of ITCG can be accomplished when R
s
e
t
=
0.1 m and ε
=
10
-
3 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 Theorems1 and 4, the desired impact time and the guidance parameters must satisfy(73)
t
0
+
R
0
q
m
a
x
-
R
˙
0
-
1
-
q
m
a
x
V
m
i
n
<
t
f
d
<
t
0
+
R
0
q
m
i
n
-
R
˙
0
-
1
-
q
m
i
n
V
m
i
n
,
q
m
a
x
>
q
m
i
n
≥
2
,where q
m
a
x is the maximum value of q
1 and q
2 and q
m
i
n is the minimum value of q
1 and q
2.From (73), the bound of t
f
d can be defined as(74)
ξ
=
t
0
+
R
0
q
-
R
˙
0
-
1
-
q
V
m
i
n
,
q
≥
2
,where q is variable. If q
=
q
m
a
x, ξ is the lower bound. If q
=
q
m
i
n, ξ is the upper bound, taking the following partial differential operation as (75)
∂
ξ
∂
q
=
-
R
0
R
˙
0
+
V
m
i
n
-
R
˙
0
-
1
-
q
V
m
i
n
2
,
q
≥
2
.Theorems 1 and 4 need the following condition:(76)
R
˙
0
V
m
i
n
=
V
0
cos
⁡
λ
0
-
γ
0
V
m
i
n
<
1
.Combining (75) and (76) yields(77)
∂
ξ
∂
q
<
0
,
q
≥
2
.From (75), we know that ξ reduces monotonically with increasing q. Then, we get the following.(a) The lower bound of the permissible set of the desired impact timest
f
d is t
f
m
i
n
d, which is described as(78)
t
f
m
i
n
d
=
t
0
+
R
0
V
m
i
n
.If q
m
a
x
→
∞, the lower bound can be achieved.(b) The upper bound of the permissible set of the desired impact timest
f
d is t
f
m
a
x
d, which is described as(79)
t
f
m
a
x
d
=
t
0
+
2
R
0
V
m
i
n
-
R
˙
0
.If q
m
i
n
=
2, we can get the upper bound.From (a) and (b), the permissible set of the desired impact timest
f
d can be given as(80)
F
=
t
f
m
i
n
d
,
t
f
m
a
x
d
.If we selectt
f
d
∈
F and assume that t
f
d
-
t
0
V
m
i
n
-
R
0
>
0, to meet condition (73), the permissible sets of q
m
a
x and q
m
i
n can be described as(81)
q
m
a
x
∈
t
f
d
-
t
0
V
m
i
n
+
R
˙
0
t
f
d
-
t
0
V
m
i
n
-
R
0
,
∞
∩
q
m
i
n
,
∞
,
q
m
i
n
∈
2
,
∞
.Because q
m
a
x and q
m
i
n are the solutions to the equation a
q
2
-
a
+
b
q
+
c
=
0, the guidance parameters can be obtained as(82)
a
=
1
,
b
=
q
m
a
x
+
q
m
i
n
-
1
,
c
=
q
m
a
x
q
m
i
n
.


## 3.1. Design of ITCG Law

Condition (4) is equivalent to the following conditions:(12)
R
t
>
0
,
∀
t
0
≤
t
<
t
f
d
,
R
t
f
d
=
0
.Conditions (12) mean that the missile can attack the target only if t
=
t
f
d. In this section, a novel ITCG law is developed to satisfy (12).A state variablez
1 is defined as(13)
z
1
=
R
-
V
m
i
n
t
f
d
-
t
.Differentiating (13) gives(14)
z
˙
1
=
R
˙
+
V
m
i
n
.Let z
˙
1
=
z
2, and the guidance system (2) can be rewritten as (15)
z
˙
1
=
z
2
,
z
˙
2
=
f
z
+
g
z
u
,
V
˙
=
1
2
C
x
ρ
V
2
S
M
m
,where u
=
a
M, f
z
=
-
V
˙
cos
⁡
λ
-
γ
+
V
sin
⁡
λ
-
γ
λ
˙ and g
z
=
-
sin
⁡
λ
-
γ.A novel integral sliding mode (ISM) is constructed as(16)
s
I
S
M
=
z
2
-
z
2
t
0
+
∫
t
0
t
b
t
f
d
-
t
z
2
+
c
z
1
a
t
f
d
-
t
2
d
τ
,where a, b, and c are the guidance parameters, which are all constants. Then, using the following guidance law:(17)
u
=
-
g
-
1
z
f
z
+
b
t
f
d
-
t
z
2
+
c
z
1
a
t
f
d
-
t
2
+
β
sgn
⁡
s
I
S
M
,where β is a small positive constant and sgn
⁡
· denotes the signum function, we have the following results.Theorem 1.
Consider system (15) with the guidance law (17), provided that q
1 and q
2 are the solutions to the equation a
q
2
-
a
+
b
q
+
c
=
0, where a, b, and c are the guidance parameters in (17). If t
f
d satisfies(18)
t
0
+
R
0
q
2
-
R
˙
0
-
1
-
q
2
V
m
i
n
<
t
f
d
<
t
0
+
R
0
q
1
-
R
˙
0
-
1
-
q
1
V
m
i
n
,
i
f
q
2
>
q
1
,
t
0
+
R
0
q
1
-
R
˙
0
-
1
-
q
1
V
m
i
n
<
t
f
d
<
t
0
+
R
0
q
2
-
R
˙
0
-
1
-
q
2
V
m
i
n
,
i
f
q
1
>
q
2
,
q
1
≥
2, q
2
≥
2, q
1
≠
q
2, and V
0
cos
⁡
λ
0
-
γ
0
/
V
m
i
n
<
1, then the following conditions will be satisfied:(19)
R
t
>
0
,
∀
t
0
≤
t
<
t
f
d
,
R
t
f
d
=
0
.Proof.
Differentiating (16) with respect to time yields(20)
s
˙
I
S
M
=
z
˙
2
+
b
t
f
d
-
t
z
2
+
c
z
1
a
t
f
d
-
t
2
.Combining (15) with (20), we have(21)
s
˙
I
S
M
=
f
z
+
g
z
u
+
b
t
f
d
-
t
z
2
+
c
z
1
a
t
f
d
-
t
2
.Construct a Lyapunov function V
1 as(22)
V
1
=
1
2
s
I
S
M
2
.Differentiating (22) and then using (21) and (17) yield(23)
V
˙
1
=
s
I
S
M
s
˙
I
S
M
=
s
I
S
M
f
z
+
g
z
u
+
b
t
f
d
-
t
z
2
+
c
z
1
a
t
f
d
-
t
2
=
-
s
I
S
M
β
sgn
⁡
s
I
S
M
<
0
,
when
s
I
S
M
≠
0
.Note that s
I
S
M
t
0
=
0, and one can imply that (24)
s
I
S
M
t
=
0
,
∀
t
0
≤
t
≤
t
f
d
.As s
I
S
M
t remains as zero, we get (25)
s
˙
I
S
M
t
=
0
,
∀
t
0
≤
t
≤
t
f
d
.The following equation can be obtained by combining (20) with (25):(26)
a
t
f
d
-
t
2
z
¨
1
+
b
t
f
d
-
t
z
˙
1
+
c
z
1
=
0
,
∀
t
0
≤
t
≤
t
f
d
.Provided that q
1 and q
2 are the solutions to the equation a
q
2
-
a
+
b
q
+
c
=
0, if q
1
≠
q
2, q
1
≥
2, and q
2
≥
2, the solutions to the differential equation (26) can be easily obtained as(27)
z
1
=
c
1
t
f
d
-
t
q
1
+
c
2
t
f
d
-
t
q
2
,
∀
t
0
≤
t
≤
t
f
d
,
(28)
z
˙
1
=
-
c
1
q
1
t
f
d
-
t
q
1
-
1
-
c
2
q
2
t
f
d
-
t
q
2
-
1
,
∀
t
0
≤
t
≤
t
f
d
,where c
1 and c
2 are constant.
Substituting (13) into (27) and substituting (14) into (28) yield(29)
R
-
V
m
i
n
t
f
d
-
t
=
c
1
t
f
d
-
t
q
1
+
c
2
t
f
d
-
t
q
2
,
∀
t
0
≤
t
≤
t
f
d
,
(30)
R
˙
+
V
m
i
n
=
-
c
1
q
1
t
f
d
-
t
q
1
-
1
-
c
2
q
2
t
f
d
-
t
q
2
-
1
,
∀
t
0
≤
t
≤
t
f
d
.Substituting the initial values R
0 and R
˙
0 into (29) and (30), we get(31)
R
0
-
V
m
i
n
t
f
d
-
t
0
=
c
1
t
f
d
-
t
0
q
1
+
c
2
t
f
d
-
t
0
q
2
,
∀
t
0
≤
t
≤
t
f
d
,
R
˙
0
+
V
m
i
n
t
f
d
-
t
0
=
-
c
1
q
1
t
f
d
-
t
0
q
1
-
c
2
q
2
t
f
d
-
t
0
q
2
,
∀
t
0
≤
t
≤
t
f
d
.From (31), we have(32)
c
1
=
R
˙
0
+
V
m
i
n
1
-
q
2
t
f
d
-
t
0
+
q
2
R
0
q
2
-
q
1
t
f
d
-
t
0
q
1
,
c
2
=
-
R
˙
0
-
V
m
i
n
1
-
q
1
t
f
d
-
t
0
-
q
1
R
0
q
2
-
q
1
t
f
d
-
t
0
q
2
.According to (18), V
0
cos
⁡
λ
0
-
γ
0
/
V
m
i
n
<
1, q
1
≥
2, and q
2
≥
2, we have(33)
c
1
<
0
,
c
2
<
0
.From (32), (33), and t
f
d
-
t
0
>
0, one can imply that(34)
R
˙
0
+
V
m
i
n
1
-
q
2
t
f
d
-
t
0
+
q
2
R
0
q
2
-
q
1
<
0
,
-
R
˙
0
-
V
m
i
n
1
-
q
1
t
f
d
-
t
0
-
q
1
R
0
q
2
-
q
1
<
0
.Substituting (32) into (29), we have(35)
R
-
V
m
i
n
t
f
d
-
t
=
R
˙
0
+
V
m
i
n
1
-
q
2
t
f
d
-
t
0
+
q
2
R
0
q
2
-
q
1
t
f
d
-
t
q
1
t
f
d
-
t
0
q
1
+
-
R
˙
0
-
V
m
i
n
1
-
q
1
t
f
d
-
t
0
-
q
1
R
0
q
2
-
q
1
t
f
d
-
t
q
2
t
f
d
-
t
0
q
2
,
∀
t
0
≤
t
≤
t
f
d
.Considering (34), 0
≤
t
f
d
-
t
/
t
f
d
-
t
0
≤
1, q
1
≥
2, and q
2
≥
2, we get (36) from (35). Consider(36)
R
-
V
m
i
n
t
f
d
-
t
≥
R
˙
0
+
1
-
q
2
V
m
i
n
t
f
d
-
t
0
+
q
2
R
0
q
2
-
q
1
t
f
d
-
t
t
f
d
-
t
0
+
-
R
˙
0
-
1
-
q
1
V
m
i
n
t
f
d
-
t
0
-
q
1
R
0
q
2
-
q
1
t
f
d
-
t
t
f
d
-
t
0
=
R
0
t
f
d
-
t
t
f
d
-
t
0
-
V
m
i
n
t
f
d
-
t
,
∀
t
0
≤
t
≤
t
f
d
.Then, we get(37)
R
≥
R
0
t
f
d
-
t
t
f
d
-
t
0
>
0
,
∀
t
0
≤
t
<
t
f
d
.
Moreover, according to (29), we get(38)
R
t
f
d
=
0
.The proof is finished.Remark 2.
From Theorem1, it is clear that the proposed guidance law (17) can guarantee each missile attacks the target only when t
=
t
f
d.Remark 3.
Equation (17) shows that the control law contains t
f
d
-
t in the denominator, which may bring the singularity when t
=
t
f
d. Fortunately, the singularity can be eliminated; combining (27) with (28) yields(39)
b
t
f
d
-
t
z
˙
1
+
c
z
1
a
t
f
d
-
t
2
=
c
1
c
a
-
c
1
q
1
b
a
t
f
d
-
t
q
1
-
2
+
c
2
c
a
-
c
2
q
2
b
a
t
f
d
-
t
q
2
-
2
,
∀
t
0
≤
t
≤
t
f
d
.From (39), we can know that the singularity brought by t
f
d
-
t is eliminated as long as q
1
≥
2 and q
2
≥
2.


## 3.2. Nonsingular Impact-Time-Control Guidance Law

Ifg
z
=
0, (17) shows that the guidance command u
→
∞, which means it is impossible to realize the guidance law. In what follows, Theorem 4 shows the condition which can ensure g
z
≠
0 during the guidance process.Theorem 4.
Consider system (15) with the guidance law (17). Provided that q
1 and q
2 are the solutions to the equation a
q
2
-
a
+
b
q
+
c
=
0, where a, b, and c are the parameters in (17), if t
f
d satisfies (18), q
1
≥
2, q
2
≥
2, q
1
≠
q
2, sin
⁡
λ
0
-
γ
0
≠
0, and V
0
cos
⁡
λ
0
-
γ
0
/
V
m
i
n
<
1
-
ψ
<
1, then g
z satisfies the following condition:(40)
g
z
≥
θ
-
>
0
,
∀
t
0
≤
t
≤
t
f
d
-
ε
,where ε is a sufficiently small time and (41)
θ
-
=
min
⁡
1
-
ψ
-
,
sin
⁡
λ
0
-
γ
0
,
(42)
ψ
-
=
max
⁡
V
m
i
n
V
-
2
,
-
1
+
ψ
2
,where V
- is the value of V at time t
=
t
f
d
-
ε.Proof.
From the demonstration of Theorem4, we known that the guidance law (17) can ensure the state variables of system (15) satisfy the following equations:(43)
z
1
=
c
1
t
f
d
-
t
q
1
+
c
2
t
f
d
-
t
q
2
,
∀
t
0
≤
t
≤
t
f
d
,
(44)
z
˙
1
=
-
c
1
q
1
t
f
d
-
t
q
1
-
1
-
c
2
q
2
t
f
d
-
t
q
2
-
1
,
∀
t
0
≤
t
≤
t
f
d
,where c
1 and c
2 are constants and q
1 and q
2 are the solutions to the equation a
q
2
-
a
+
b
q
+
c
=
0.
Substituting (13) into (43) and substituting (14) into (44) yield(45)
R
-
V
m
i
n
t
f
d
-
t
=
c
1
t
f
d
-
t
q
1
+
c
2
t
f
d
-
t
q
2
,
∀
t
0
≤
t
≤
t
f
d
,
(46)
R
˙
+
V
m
i
n
=
-
c
1
q
1
t
f
d
-
t
q
1
-
1
-
c
2
q
2
t
f
d
-
t
q
2
-
1
,
∀
t
0
≤
t
≤
t
f
d
.According to (18), q
1
≥
2, and q
2
≥
2, we can get the following results from the demonstration of Theorem 4:(47)
c
1
<
0
,
c
2
<
0
.Combining (47) with (46) yields(48)
R
˙
+
V
m
i
n
>
0
,
∀
t
0
≤
t
<
t
f
d
.Substituting (2) into (48) yields(49)
cos
⁡
λ
-
γ
<
V
m
i
n
V
,
∀
t
0
≤
t
<
t
f
d
.Differentiating (44) gives(50)
z
¨
1
=
c
1
q
1
q
1
-
1
t
f
d
-
t
q
1
-
2
+
c
2
q
2
q
2
-
1
t
f
d
-
t
q
2
-
2
,
∀
t
0
≤
t
<
t
f
d
.Then, considering q
1
≥
2, q
2
≥
2, c
1
<
0, and c
2
<
0, we have(51)
z
¨
1
<
0
,
∀
t
0
≤
t
<
t
f
d
.Equation (51) means that z
˙
1 reduces monotonically with increasing t, and we get(52)
z
˙
1
<
z
˙
1
t
0
,
∀
t
0
<
t
<
t
f
d
.From (52), we get(53)
cos
⁡
λ
-
γ
>
V
0
cos
⁡
λ
0
-
γ
0
V
,
∀
t
0
<
t
<
t
f
d
.Combining (49) and (53) yields(54)
V
0
cos
⁡
λ
0
-
γ
0
V
<
cos
⁡
λ
-
γ
<
V
m
i
n
V
,
∀
t
0
<
t
<
t
f
d
.From (3), it can be known that V
˙
<
0. Then, we have(55)
V
m
i
n
V
≤
V
m
i
n
V
-
<
1
,
∀
t
0
<
t
≤
t
f
d
-
ε
,where V
- is the value of V at time t
=
t
f
d
-
ε.
SinceV
m
i
n
/
V
≤
V
m
i
n
/
V
-
<
1 and V
0
cos
⁡
λ
0
-
γ
0
/
V
≥
-
V
0
cos
⁡
λ
0
-
γ
0
/
V
m
i
n
>
-
1
+
ψ
>
-
1, we have(56)
-
1
+
ψ
<
cos
⁡
λ
-
γ
≤
V
m
i
n
V
-
,
∀
t
0
<
t
≤
t
f
d
-
ε
.Let ψ
-
=
max
⁡
V
m
i
n
/
V
-
2
,
-
1
+
ψ
2, then we have(57)
cos
2
⁡
λ
-
γ
≤
ψ
-
,
∀
t
0
<
t
≤
t
f
d
-
ε
.It is clear that 0
<
ψ
-
<
1. Since sin
⁡
λ
-
γ
=
1
-
cos
2
⁡
λ
-
γ, we have(58)
sin
⁡
λ
-
γ
≥
1
-
ψ
-
,
∀
t
0
<
t
≤
t
f
d
-
ε
.Let θ
-
=
min
⁡
1
-
ψ
-
,
sin
⁡
λ
0
-
γ
0, we have(59)
sin
⁡
λ
-
γ
≥
θ
-
,
∀
t
0
≤
t
≤
t
f
d
-
ε
.Since sin
⁡
λ
0
-
γ
0
>
0 and 1
-
ψ
-
>
0, we have(60)
sin
⁡
λ
-
γ
≥
θ
-
>
0
,
∀
t
0
≤
t
≤
t
f
d
-
ε
.Then, we have(61)
g
z
≥
θ
-
>
0
,
∀
t
0
≤
t
≤
t
f
d
-
ε
,where ε is a sufficiently small time. The proof is finished.Remark 5.
In the practical engineering, the available missile acceleration is bounded:(62)
u
≤
u
m
a
x
,where u
m
a
x is the maximum available acceleration of missile and is determined by the performance of missile. Let u
-
=
-
g
-
1
z
f
z
+
(
b
t
f
d
-
t
z
2
+
c
z
1
)
/
a
t
f
d
-
t
2
+
β
sgn
⁡
s
I
S
M. To satisfy (62) in the practical engineering, the guidance law (17) must be modified as(63)
u
=
u
m
a
x
;
if
u
-
>
u
m
a
x
u
-
;
if
u
-
≤
u
m
a
x
-
u
m
a
x
;
if
u
-
<
-
u
m
a
x
.In the most ideal case (u
m
a
x
=
∞), we can expect the relative distance R
t
f
d
=
0. However, u
m
a
x 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:(64)
R
>
0
,
∀
t
0
≤
t
≤
t
f
d
-
ε
,
(65)
R
t
f
d
-
ε
≤
R
s
e
t
,
(66)
u
-
≤
u
m
a
x
,
∀
t
0
≤
t
≤
t
f
d
-
ε
,where ε is a sufficiently small time and ε
<
0.1. R
s
e
t is a small distance that can guarantee the precision of attack (e.g., if the target is ship, R
s
e
t
=
1 m can guarantee the precision of attack).From (2), we have(67)
λ
˙
=
-
V
sin
⁡
λ
-
γ
R
=
V
R
g
z
.Substituting (39) into guidance law (17) and considering (67) yield(68)
u
-
=
-
g
-
1
z
f
z
+
c
1
c
a
-
c
1
q
1
b
a
t
f
d
-
t
q
1
-
2
+
c
2
c
a
-
c
2
q
2
b
a
t
f
d
-
t
q
2
-
2
+
β
sgn
⁡
s
I
S
M
=
-
g
-
1
z
-
V
˙
cos
⁡
λ
-
γ
+
V
2
sin
⁡
λ
-
γ
g
z
R
+
c
1
c
a
-
c
1
q
1
b
a
t
f
d
-
t
q
1
-
2
+
c
2
c
a
-
c
2
q
2
b
a
t
f
d
-
t
q
2
-
2
+
β
sgn
⁡
s
I
S
M
=
V
˙
cos
⁡
λ
-
γ
g
z
-
V
2
sin
⁡
λ
-
γ
R
-
c
1
c
/
a
-
c
1
q
1
b
/
a
t
f
d
-
t
q
1
-
2
+
c
2
c
/
a
-
c
2
q
2
b
/
a
t
f
d
-
t
q
2
-
2
+
β
sgn
⁡
s
I
S
M
g
z
,
∀
t
0
≤
t
≤
t
f
d
-
ε
.From (68), we have(69)
u
-
≤
V
˙
cos
⁡
λ
-
γ
g
z
+
c
1
c
/
a
-
c
1
q
1
b
/
a
t
f
d
-
t
q
1
-
2
+
c
2
c
/
a
-
c
2
q
2
b
/
a
t
f
d
-
t
q
2
-
2
+
β
sgn
⁡
s
I
S
M
g
z
+
V
2
sin
⁡
λ
-
γ
R
,
∀
t
0
≤
t
≤
t
f
d
-
ε
.Combining (37), (61), and (69) yields(70)
u
-
≤
V
˙
cos
⁡
λ
-
γ
θ
-
+
c
1
c
/
a
-
c
1
q
1
b
/
a
t
f
d
-
t
q
1
-
2
+
c
2
c
/
a
-
c
2
q
2
b
/
a
t
f
d
-
t
q
2
-
2
+
β
sgn
⁡
s
I
S
M
θ
-
+
V
2
sin
⁡
λ
-
γ
t
f
d
-
t
0
R
0
·
t
f
d
-
t
≤
V
˙
cos
⁡
λ
-
γ
+
c
1
c
/
a
-
c
1
q
1
b
/
a
t
f
d
-
t
q
1
-
2
+
c
2
c
/
a
-
c
2
q
2
b
/
a
t
f
d
-
t
0
q
2
-
2
+
β
θ
-
+
V
2
sin
⁡
λ
-
γ
t
f
d
-
t
0
/
R
0
ε
,
∀
t
0
≤
t
≤
t
f
d
-
ε
.Since θ
-
>
0, ε
>
0, q
1
≥
2, and q
2
≥
2, it is clear that u
- is bounded when t
0
≤
t
≤
t
f
d
-
ε. Thus, if the maximum available acceleration u
m
a
x is large enough, then condition (66) can be satisfied even if ε is very small. Since condition (66) can be satisfied, we can know that (71)
u
=
u
-
=
-
g
-
1
z
f
z
+
b
t
f
d
-
t
z
2
+
c
z
1
a
t
f
d
-
t
2
+
β
sgn
⁡
s
I
S
M
,
∀
t
0
≤
t
≤
t
f
d
-
ε
.That is, the guidance law (17) is valid when t
0
≤
t
≤
t
f
d
-
ε. Thus the mathematical derivations from (12) to (61) are still valid when t
0
≤
t
≤
t
f
d
-
ε. From (37), it is clear that (64) is satisfied. Substituting t
=
t
f
d
-
ε into (45) yields(72)
R
t
f
d
-
ε
=
V
m
i
n
ε
+
c
1
ε
q
1
+
c
2
ε
q
2
.From (72), it can be known that condition (65) can be satisfied if ε is small enough.In short, for the given smallε and R
s
e
t, (64), (65), and (66) can be satisfied simultaneously if u
m
a
x 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 R
s
e
t
=
0.1 m, ε
=
10
-
3 s, and u
m
a
x
=
100 m/s2 (from the results in [8–13, 18–20], it can be known that 100
m/s2 is a reasonable maximum available acceleration for missile; and the objective of ITCG can be accomplished when R
s
e
t
=
0.1 m and ε
=
10
-
3 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 Theorems1 and 4, the desired impact time and the guidance parameters must satisfy(73)
t
0
+
R
0
q
m
a
x
-
R
˙
0
-
1
-
q
m
a
x
V
m
i
n
<
t
f
d
<
t
0
+
R
0
q
m
i
n
-
R
˙
0
-
1
-
q
m
i
n
V
m
i
n
,
q
m
a
x
>
q
m
i
n
≥
2
,where q
m
a
x is the maximum value of q
1 and q
2 and q
m
i
n is the minimum value of q
1 and q
2.From (73), the bound of t
f
d can be defined as(74)
ξ
=
t
0
+
R
0
q
-
R
˙
0
-
1
-
q
V
m
i
n
,
q
≥
2
,where q is variable. If q
=
q
m
a
x, ξ is the lower bound. If q
=
q
m
i
n, ξ is the upper bound, taking the following partial differential operation as (75)
∂
ξ
∂
q
=
-
R
0
R
˙
0
+
V
m
i
n
-
R
˙
0
-
1
-
q
V
m
i
n
2
,
q
≥
2
.Theorems 1 and 4 need the following condition:(76)
R
˙
0
V
m
i
n
=
V
0
cos
⁡
λ
0
-
γ
0
V
m
i
n
<
1
.Combining (75) and (76) yields(77)
∂
ξ
∂
q
<
0
,
q
≥
2
.From (75), we know that ξ reduces monotonically with increasing q. Then, we get the following.(a) The lower bound of the permissible set of the desired impact timest
f
d is t
f
m
i
n
d, which is described as(78)
t
f
m
i
n
d
=
t
0
+
R
0
V
m
i
n
.If q
m
a
x
→
∞, the lower bound can be achieved.(b) The upper bound of the permissible set of the desired impact timest
f
d is t
f
m
a
x
d, which is described as(79)
t
f
m
a
x
d
=
t
0
+
2
R
0
V
m
i
n
-
R
˙
0
.If q
m
i
n
=
2, we can get the upper bound.From (a) and (b), the permissible set of the desired impact timest
f
d can be given as(80)
F
=
t
f
m
i
n
d
,
t
f
m
a
x
d
.If we selectt
f
d
∈
F and assume that t
f
d
-
t
0
V
m
i
n
-
R
0
>
0, to meet condition (73), the permissible sets of q
m
a
x and q
m
i
n can be described as(81)
q
m
a
x
∈
t
f
d
-
t
0
V
m
i
n
+
R
˙
0
t
f
d
-
t
0
V
m
i
n
-
R
0
,
∞
∩
q
m
i
n
,
∞
,
q
m
i
n
∈
2
,
∞
.Because q
m
a
x and q
m
i
n are the solutions to the equation a
q
2
-
a
+
b
q
+
c
=
0, the guidance parameters can be obtained as(82)
a
=
1
,
b
=
q
m
a
x
+
q
m
i
n
-
1
,
c
=
q
m
a
x
q
m
i
n
.


## 4. Salvo Attack

In this section, an algorithm will be designed based on the proposed guidance law to realize the salvo attack. Considern missiles M
i
i
=
1
,
…
,
n engaged in a salvo attack against a stationary target as shown in Figure 2. λ
i denotes the LOS angle, R
i denotes the relative distance between the missile M
i and the target, γ
i denotes the heading angle, a
M
i denotes the missile acceleration, and V
i denotes the missile velocity.Figure 2
Missile-target relative motion relationship.The permissible set of the desired impact timest
f
d for missile M
i is F
i
=
t
f
i
m
i
n
d
,
t
f
i
m
a
x
d, and the intersection set can be described as H
=
∩
i
=
1
n
F
i. It is assumed that H 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.Algorithm 1:Salvo attack algorithm.
foreach missile M
i, i
=
1,2
,
…
,
n
do
Compute the permissible setF
i
=
t
f
i
m
i
n
d
,
t
f
i
m
a
x
d from (80).
end for
Find the intersection setH
=
∩
i
=
1
n
F
i.
Select at
f
d
∈
H.
foreach missile M
i, i
=
1,2
,
…
,
n
do
Selectq
i
m
a
x and q
i
m
i
n from (81) and obtain the
parametersa
i, b
i and c
i from (82).
end for
foreach time step do
foreach missile M
i, i
=
1,2
,
…
,
n
do
Usea
i, b
i and c
i to generate the guidance input u
i 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,sgn
⁡
x is replaced by the sigmoid function given as [20](83)
sgmf
x
=
2
1
1
+
e
x
p
-
α
x
-
1
,where α 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 Case1. 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(84)
a
P
N
G
L
=
W
V
λ
˙
.The parameter is chosen as W
=
3. The ITCGL [8] is defined as(85)
a
I
T
C
G
L
=
H
V
λ
˙
3
2
-
1
2
1
+
240
V
5
H
V
λ
˙
2
R
3
t
f
d
-
t
-
1
+
γ
-
λ
2
/
10
R
V
,where the parameter is chosen as H
=
3. The NSMGL [20] is defined as(86)
a
N
S
M
G
L
=
a
M
e
q
+
a
M
M
c
o
n
+
a
M
s
w
,where(87)
a
M
e
q
=
V
2
2
N
-
1
R
cos
⁡
γ
-
1
γ
+
0.5
V
2
R
γ
cos
⁡
γ
+
V
λ
˙
,
h
γ
=
sgn
⁡
γ
γ
i
f
γ
<
ε
1
1
-
ε
1
ε
1
-
ε
2
γ
+
ε
1
ε
2
-
1
ε
2
-
ε
1
i
f
ε
1
<
γ
<
ε
2
1
o
t
h
e
r
w
i
s
e
,
a
M
M
c
o
n
=
-
h
γ
γ
k
s
N
,
a
M
s
w
=
-
M
p
sgn
⁡
γ
+
1
sgn
⁡
s
N
,
s
N
=
R
V
1
+
0.5
2
N
-
1
γ
2
,where the parameters are chosen as k
=
50, M
=
10, p
=
2, ε
1
=
0.001, ε
2
=
0.015, and N
=
3. In this paper, the values of W, H, k, M, p, ε
1, ε
2, and N 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 F
=
26.1008,42.4381 by using (80). Choose the desired impact time as t
f
d
=
37
s
∈
F. The permissible set of q
m
a
x and q
m
i
n can be calculated from (81) as q
m
a
x
∈
2.6137
,
∞
∩
q
m
i
n
,
∞ and q
m
i
n
∈
2
,
∞. From the permissible sets, q
m
a
x and q
m
i
n are selected as q
m
a
x
=
4 and q
m
i
n
=
2.5. Using q
m
a
x and q
m
i
n, the guidance parameters a, b, and c can be obtained from (82) as a
=
1, b
=
5.5, and c
=
10.
The simulation results of Case1 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.Table 1
Initial condition for Cases1 and 2.
Parameters
Case1 (constant velocity)
Case2 (varying velocity)
Initial missile position
(−10000, −3000) m
Initial missile velocityV
0
400m
/
s
Initial missile heading angleγ
0
−60 deg
Target position
0
,
0 m
Desired impact timet
f
d
37 s
Missile massm
400 kg
Reference areaS
M
—
0.039m
2
Air densityρ
—
1.293k
g
/
m
3
Minimum speedV
m
i
n
400 m/s
350 m/sTable 2
Performance of guidance laws in Section5.1.
Case
Guidance law
t
f
-
t
f
d (s)
Miss distance (m)
1
Proposed law
1.24
×
10
-
4
0.07
NSMGL
1.45
×
10
-
4
0.05
ITCGL
3.12
×
10
-
2
0.33
PNGL
-
6.39
0.03
2
Proposed law
5.15
×
10
-
4
0.15
NSMGL
0.57
0.06
ITCGL
0.49
0.47
PNGL
-
3.095
0.09Figure 3
Responses in Case1 (constant velocity).
(a)
Miss distanceR
(b)
Flight trajectory
(c)
Acceleration command
(d)
Missile velocityCase 2 (missile with time-varying velocity).
In this case, we consider that the missile velocity is varying. The aerodynamic coefficients are given by(88)
C
x
=
-
0.37
t
<
6
-
0.112
t
≥
6
.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 F
=
29.8294,47.2385 by using (80). Choose the desired impact time as t
f
d
=
37
s
∈
F. Then, the permissible set of q
m
a
x and q
m
i
n can be calculated from (81) as q
m
a
x
∈
3.8033
,
∞
∩
q
m
i
n
,
∞ and q
m
a
x
∈
2
,
∞. From the permissible sets, q
m
a
x and q
m
i
n are selected as q
m
a
x
=
4 and q
m
i
n
=
2.5. Using q
m
a
x and q
m
i
n, the guidance parameters a, b, and c can be obtained from (82) as a
=
1, b
=
5.5, and c
=
10.
The simulation results of Case2 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 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.Figure 4
Responses in Case2 (time-varying velocity).
(a)
Miss distanceR
(b)
Flight trajectory
(c)
Acceleration command
(d)
Missile velocity


### 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 by(89)
Missile  1:
C
x
=
-
0.37
t
<
6
-
0.112
t
≥
6
Missile  2:
C
x
=
-
0.47
t
<
10
-
0.325
t
≥
10
Missile  3:
C
x
=
-
0.57
t
<
10
-
0.5
t
≥
10
.The initial conditions used in this subsection are listed in Table 3.Table 3
Initial conditions for salvo attack.
Object
x
0/m
y
0/m
γ
0/(deg)
V
0/(m/s)
V
m
i
n/(m/s)
Mass/(kg)
S
M/(m
2)
Target
0
0
0
0
0
0
0
Missile 1
-
10000
-
3000
-
60
400
350
400
0.0
39
Missile 2
-
2000
-
8000
-
20
410
330
395
0.040
Missile 3
-
5000
4000
70
420
310
410
0.04
1From Table3, the permissible sets of the desired impact times for each missile can be calculated as(90)
F
1
=
29.8294,47.2385
,
F
2
=
24.9885,57.3847
,
F
3
=
20.6552,72.9195
.Choose the desired impact time as t
f
d
=
37
s
∈
F
1
∩
F
2
∩
F
3. Then, the permissible set of q
m
a
x and q
m
i
n can be calculated from (81) as (91)
q
1
m
a
x
∈
3.8033
,
∞
∩
q
1
m
i
n
,
∞
,
q
1
m
i
n
∈
2
,
∞
,
q
2
m
a
x
∈
3.4780
,
∞
∩
q
2
m
i
n
,
∞
,
q
2
m
i
n
∈
2
,
∞
,
q
3
m
a
x
∈
3.2450
,
∞
∩
q
3
m
i
n
,
∞
,
q
3
m
i
n
∈
2
,
∞
.From the permissible sets, we select that q
1
m
a
x
=
q
2
m
a
x
=
q
3
m
a
x
=
4 and q
1
m
i
n
=
q
2
m
i
n
=
q
3
m
i
n
=
2.5. Then, the guidance parameters can be obtained from (82) as(92)
a
1
=
a
2
=
a
3
=
1
,
b
1
=
b
2
=
b
3
=
5.5
,
c
1
=
c
2
=
c
3
=
10
.The simulation results are shown in Figures4(a), 4(b), 4(c), and 4(d) and Table 4, respectively. From Figures 5(a), 5(b), 5(c), and 5(d) and Table 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.Table 4
Performance of guidance law in Section5.2.
Object
t
f
-
t
f
d (s)
Miss distance (m)
Missile 1
3.13
×
10
-
4
0.15
Missile 2
1.15
×
10
-
4
0.45
Missile 3
0.89
×
10
-
4
0.31Figure 5
Responses in the salvo attack scenario.
(a)
Flight trajectory
(b)
Miss distanceR
(c)
Missile velocity
(d)
Acceleration command


## 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 Case1. 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(84)
a
P
N
G
L
=
W
V
λ
˙
.The parameter is chosen as W
=
3. The ITCGL [8] is defined as(85)
a
I
T
C
G
L
=
H
V
λ
˙
3
2
-
1
2
1
+
240
V
5
H
V
λ
˙
2
R
3
t
f
d
-
t
-
1
+
γ
-
λ
2
/
10
R
V
,where the parameter is chosen as H
=
3. The NSMGL [20] is defined as(86)
a
N
S
M
G
L
=
a
M
e
q
+
a
M
M
c
o
n
+
a
M
s
w
,where(87)
a
M
e
q
=
V
2
2
N
-
1
R
cos
⁡
γ
-
1
γ
+
0.5
V
2
R
γ
cos
⁡
γ
+
V
λ
˙
,
h
γ
=
sgn
⁡
γ
γ
i
f
γ
<
ε
1
1
-
ε
1
ε
1
-
ε
2
γ
+
ε
1
ε
2
-
1
ε
2
-
ε
1
i
f
ε
1
<
γ
<
ε
2
1
o
t
h
e
r
w
i
s
e
,
a
M
M
c
o
n
=
-
h
γ
γ
k
s
N
,
a
M
s
w
=
-
M
p
sgn
⁡
γ
+
1
sgn
⁡
s
N
,
s
N
=
R
V
1
+
0.5
2
N
-
1
γ
2
,where the parameters are chosen as k
=
50, M
=
10, p
=
2, ε
1
=
0.001, ε
2
=
0.015, and N
=
3. In this paper, the values of W, H, k, M, p, ε
1, ε
2, and N 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 F
=
26.1008,42.4381 by using (80). Choose the desired impact time as t
f
d
=
37
s
∈
F. The permissible set of q
m
a
x and q
m
i
n can be calculated from (81) as q
m
a
x
∈
2.6137
,
∞
∩
q
m
i
n
,
∞ and q
m
i
n
∈
2
,
∞. From the permissible sets, q
m
a
x and q
m
i
n are selected as q
m
a
x
=
4 and q
m
i
n
=
2.5. Using q
m
a
x and q
m
i
n, the guidance parameters a, b, and c can be obtained from (82) as a
=
1, b
=
5.5, and c
=
10.
The simulation results of Case1 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.Table 1
Initial condition for Cases1 and 2.
Parameters
Case1 (constant velocity)
Case2 (varying velocity)
Initial missile position
(−10000, −3000) m
Initial missile velocityV
0
400m
/
s
Initial missile heading angleγ
0
−60 deg
Target position
0
,
0 m
Desired impact timet
f
d
37 s
Missile massm
400 kg
Reference areaS
M
—
0.039m
2
Air densityρ
—
1.293k
g
/
m
3
Minimum speedV
m
i
n
400 m/s
350 m/sTable 2
Performance of guidance laws in Section5.1.
Case
Guidance law
t
f
-
t
f
d (s)
Miss distance (m)
1
Proposed law
1.24
×
10
-
4
0.07
NSMGL
1.45
×
10
-
4
0.05
ITCGL
3.12
×
10
-
2
0.33
PNGL
-
6.39
0.03
2
Proposed law
5.15
×
10
-
4
0.15
NSMGL
0.57
0.06
ITCGL
0.49
0.47
PNGL
-
3.095
0.09Figure 3
Responses in Case1 (constant velocity).
(a)
Miss distanceR
(b)
Flight trajectory
(c)
Acceleration command
(d)
Missile velocityCase 2 (missile with time-varying velocity).
In this case, we consider that the missile velocity is varying. The aerodynamic coefficients are given by(88)
C
x
=
-
0.37
t
<
6
-
0.112
t
≥
6
.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 F
=
29.8294,47.2385 by using (80). Choose the desired impact time as t
f
d
=
37
s
∈
F. Then, the permissible set of q
m
a
x and q
m
i
n can be calculated from (81) as q
m
a
x
∈
3.8033
,
∞
∩
q
m
i
n
,
∞ and q
m
a
x
∈
2
,
∞. From the permissible sets, q
m
a
x and q
m
i
n are selected as q
m
a
x
=
4 and q
m
i
n
=
2.5. Using q
m
a
x and q
m
i
n, the guidance parameters a, b, and c can be obtained from (82) as a
=
1, b
=
5.5, and c
=
10.
The simulation results of Case2 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 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.Figure 4
Responses in Case2 (time-varying velocity).
(a)
Miss distanceR
(b)
Flight trajectory
(c)
Acceleration command
(d)
Missile velocity


## 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 by(89)
Missile  1:
C
x
=
-
0.37
t
<
6
-
0.112
t
≥
6
Missile  2:
C
x
=
-
0.47
t
<
10
-
0.325
t
≥
10
Missile  3:
C
x
=
-
0.57
t
<
10
-
0.5
t
≥
10
.The initial conditions used in this subsection are listed in Table 3.Table 3
Initial conditions for salvo attack.
Object
x
0/m
y
0/m
γ
0/(deg)
V
0/(m/s)
V
m
i
n/(m/s)
Mass/(kg)
S
M/(m
2)
Target
0
0
0
0
0
0
0
Missile 1
-
10000
-
3000
-
60
400
350
400
0.0
39
Missile 2
-
2000
-
8000
-
20
410
330
395
0.040
Missile 3
-
5000
4000
70
420
310
410
0.04
1From Table3, the permissible sets of the desired impact times for each missile can be calculated as(90)
F
1
=
29.8294,47.2385
,
F
2
=
24.9885,57.3847
,
F
3
=
20.6552,72.9195
.Choose the desired impact time as t
f
d
=
37
s
∈
F
1
∩
F
2
∩
F
3. Then, the permissible set of q
m
a
x and q
m
i
n can be calculated from (81) as (91)
q
1
m
a
x
∈
3.8033
,
∞
∩
q
1
m
i
n
,
∞
,
q
1
m
i
n
∈
2
,
∞
,
q
2
m
a
x
∈
3.4780
,
∞
∩
q
2
m
i
n
,
∞
,
q
2
m
i
n
∈
2
,
∞
,
q
3
m
a
x
∈
3.2450
,
∞
∩
q
3
m
i
n
,
∞
,
q
3
m
i
n
∈
2
,
∞
.From the permissible sets, we select that q
1
m
a
x
=
q
2
m
a
x
=
q
3
m
a
x
=
4 and q
1
m
i
n
=
q
2
m
i
n
=
q
3
m
i
n
=
2.5. Then, the guidance parameters can be obtained from (82) as(92)
a
1
=
a
2
=
a
3
=
1
,
b
1
=
b
2
=
b
3
=
5.5
,
c
1
=
c
2
=
c
3
=
10
.The simulation results are shown in Figures4(a), 4(b), 4(c), and 4(d) and Table 4, respectively. From Figures 5(a), 5(b), 5(c), and 5(d) and Table 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.Table 4
Performance of guidance law in Section5.2.
Object
t
f
-
t
f
d (s)
Miss distance (m)
Missile 1
3.13
×
10
-
4
0.15
Missile 2
1.15
×
10
-
4
0.45
Missile 3
0.89
×
10
-
4
0.31Figure 5
Responses in the salvo attack scenario.
(a)
Flight trajectory
(b)
Miss distanceR
(c)
Missile velocity
(d)
Acceleration command


## 6. 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 time-varying and that the proposed salvo attack algorithm worked well.



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*Source: 7951923-2016-06-30.xml*