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Robust intercept guidance law with predesigned zero-effort miss distance convergence for capturing maneuvering targets
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Robust Intercept Guidance Law with Predesigned Zero-Effort Miss Distance Convergence for Capturing Maneuvering Targets Xiaodong Yan , Shi Lyu PII: DOI: Reference:
S0016-0032(19)30753-7 https://doi.org/10.1016/j.jfranklin.2019.10.021 FI 4219
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
15 August 2019 29 September 2019 20 October 2019
Please cite this article as: Xiaodong Yan , Shi Lyu , Robust Intercept Guidance Law with Predesigned Zero-Effort Miss Distance Convergence for Capturing Maneuvering Targets, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.10.021
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Robust Intercept Guidance Law with Predesigned Zero-Effort Miss Distance Convergence for Capturing Maneuvering Targets Xiaodong Yan a,b , Shi Lyua,b a
School of Astronautics, Northwestern Polytechnical University, Xi’an, 710072, China Shaanxi Key Laboratory of Aerospace Flight Vehicle Technology, Northwestern Polytechnical University, Xi’an, 710072, China b
Abstract: In this paper, a homing guidance in a two-dimensional target-interceptor scenario is studied with an unknown target maneuver strategy. In order to capture the target with acceptable accuracy, a robust guidance law with a predesigned zero-effort miss (ZEM) distance convergence for the interceptor is proposed. In order to guarantee the ZEM distance to converge into a small region around the predesigned terminal value within a finite time, a novel function of ZEM distance is proposed. The ZEM distance convergence profile can be reasonably controlled according to the interceptor’s initial state. Based on the designed profile, an initial delay control and prescribed performance control are utilized to achieve a high terminal miss distance accuracy and account for uncertainties caused by the target’s unknown maneuverability. Four maneuvering strategies of the target are tested to verify the performance of the proposed guidance law.
Keywords: Endgame; Zero-effort miss distance; Predesigned performance; Finite time convergence; Guidance law.
1. Introduction Modern defense missiles (interceptors) present a significant threat to today’s maneuvering targets. In order to increase the survivability of targets, plenty of maneuvering strategies have been created. An example of such a strategy is the barrel roll and sinusoidal/weaving maneuver [1]. This weaving maneuver is easily designed and has been widely adopted for targets in both low and high altitude applications [2]. Especially, the moving mass control for the target’s maneuver has become a research hotspot [3, 4]. When a target adopts maneuvering strategies to improve its penetration, an effective interception guidance control law for the interceptor is a very important countermeasure. In order to capture a high-speed maneuvering target, there are three key challenges for the design of a successful interception guidance law: 1) Since the remaining time of interception is quite short, the miss distance caused by an interceptor system lag can be significant. Therefore, a finite-time convergence of the guidance law should be guaranteed. 2) The unknown maneuvering
strategies of the target can have a significant impact on the terminal interception precision. As a result, the robustness of the guidance law under unknown maneuvering information should be fully considered in order to yield a high interception precision. 3) The interceptor’s maneuverability can vary greatly under changing flight altitudes. Therefore, the guidance commands must be adjustable depending on the interceptor’s available accelerations. In order to address the first and second challenges described above, several guidance laws have been developed. Some of these strategies include sliding mode control [5-9], differential game theory [10, 11], prescribed performance control (PPC) [12-15], and adaptive dynamic programming [16]. Each type of guidance laws can achieve the terminal interception precision as expected. In particular, the PPC-based interception guidance laws [12-15] have been shown to have flexible controller gains and can guarantee the asymptotic convergence of the desired parameter to a predetermined small residual and exhibits acceptable transient performance. Even though these guidance laws are effective in many interception situations, the variability of interceptor maneuverability is generally not fully considered. Recently, Ref. [17] presented a command shaping guidance law for intercepting high-speed targets. This guidance law has the potential to obtain the optimal guidance commands to account for varying acceleration profiles by taking advantage of the linearized optimal control. However, the guidance law in Ref. [17] focuses on a non-maneuvering target. Therefore, its performance may not behave as expected without precise acceleration information in advance. In the present work, a new robust interception guidance law is presented to address the aforementioned three challenges. The present guidance law is designed to perform against maneuvering targets with a high terminal interception precision. Firstly, relative equations between the interceptor and target with uncertainties caused by the target’s unknown maneuvers are constructed. Subsequently, a segmented finite time convergence function is established to design a zero-effort miss (ZEM) distance convergence profile directly. Compared with the existing methods [18-20] and previous work by the present authors [13-15], two advantages exist in the proposed guidance law. Firstly, the ZEM distance convergence property can be manipulated directly. Therefore, the convergence can be flexibly configured based on the interceptors’ available accelerations due to variations in flight altitude. Secondly, a new prescribed performance function is established. As a result, the continuity and finite-time convergence of the guidance commands
can be yielded simultaneously.
2. Target-interceptor endgame kinematics
Initial LOS
Fig. 1
Schematics of target-interceptor engagement
In this paper, the two-player homing guidance is studied in the X Y plane, as shown in Fig. 1. The scenario involves a high-speed target with an unknown maneuvering strategy and an agile defending missile (i.e., the interceptor). In Fig. 1, X I OYI denotes a Cartesian inertial reference frame with OX I representing the local horizon. The subscripts M and T represent the interceptor and target, respectively. In addition, without loss of any generality, the gravitational forces are neglected in the homing phase, and the speed of each of the entities is assumed to be time-invariant throughout the engagement. The system dynamics of the target is assumed to be captured by the first-order lag system as follows: AT
where T
AT
ATc
T T c is a constant time, and AT is the guidance command normal to velocity.
(1)
In addition, the kinematics between the target and interceptor are governed by R Vr VT cos( T ) VM cos( M )
(2)
VMT / R [VT sin( T ) VM sin( M )] / R
(3)
where R and Vr are the relative distance and velocity between the target and interceptor, respectively. VM is the velocity of the interceptor and M is the path angle of the interceptor. VT is velocity of the target and T is the path angle of target. denotes the angle between the
line of sight (LOS) and OX I , where LOS is defined as the instantaneous LOS between the target and interceptor. VMT is the relative velocity normal to LOS.
The interceptor’s time-to-go can be calculated as according to the initial states is calculated as
t go R / Vr
t f R t 0 Vr
t 0
. As such, the total flight time
, and the time derivatives of Vr ,
VMT , t go and are given by[21] 2 Vr VMT / R ATL AML
(4)
VMT VrVMT / R ATN AMN
(5)
t go R Vr / Vr2 1
(6)
VMD / R VMDVr / R 2
(7)
where the interceptor’s accelerations along and normal to the LOS are: L AM AM sin( M ) N AM AM cos( M )
(8)
Similarly, the target’s accelerations along and normal to the LOS are: ATL AT sin( T ) N AT AT cos( T )
(9)
Taking into account the dynamics of the interceptor tracking the reference guidance command, the derivative of its acceleration becomes
AM ( AMc AM ) / M
(10)
where AMc is the reference guidance command to be specified, and M is a time constant.
3. Zero-effort Miss Distance of Interceptor By instigating a head-on engagement scenario, the error angle between M 0 and T 0 is relatively small. In this case, fluctuates in the neighborhood of zero during the homing phase [22]. Consequently, the linearized kinematics can be written as follows:
X MT AMT X MT BT ATNc BM AMNc where X MT [ yMT , yMT , ATN , AMN ]T , and
AMT
0 0 0 0
1 0
0 1
0
1 / T
0
0
0 1 0 1/M
0 0 BT 1 / T 0
0 0 BM 0 1 / M
(11)
where ATNc and AMNc are target’s and interceptor’s guidance commands normal to initial LOS, respectively. These commands are given by Nc c AT AT cos T 0 Nc c AM AM cos M 0
The state transformation function,
MT
1 t go 0 1 tgo 0 0 0 0
MT (t go )
, of Eq. (11) is given by
T2 t go T T2 e T e
t go / T
e
(12)
t go / T
M2 t go M M2 e
T
M e
t go / M
t go / T
t go / M
M
0
0
e
t go / M
(13)
and
DMT [1,0,0,0]
(14)
As such, the interceptor’s ZEM distance is given by ZEM DMT MT t go X MT
yMT t go yMT T2 tgo T T2 e
M2 t go M M2 e
t go / T
t go / M
A
A
N T
(15)
N M
Due to the fact that the angle between R and the initial LOS is relatively small, yMT can be linearized such that
yMT R sin 0 R 0
(16)
yMT R( 0 ) Vr tgo
(17)
and yMT is given by
Combining Eqs. (15), (16) and (17), ZEM can be rewritten as
ZEM Vr t go T2 t go T T2 e 2
M2 t go M M2 e
t go / M
A
t go
A
N T
(18)
N M
Similarly, the time derivative of ZEM can be written as ZEM
Vr R Vr2
V AN 1 e tgo / T A N 1 e tgo / M M M MT T T
T t go T e
t go / T
A Nc T
M
t go M e
t go / M
A
(19)
Nc M
In practice, the target’s dynamic information is often unknown, or only partially known, to the interceptor. In this case, the proposed guidance command for the interceptor will struggle to calculate ZEM, and its time derivative is desired. In this case, Eq.(18) and Eq.(19) can be
rewritten as follows.
ZEM u Vr tgo M2 tgo M M2 e 2
ZEM u
tgo / M
A
N M
(20)
Vr R t / t / VMT M AMN 1 e go M M t go M e go M AMNc h Vr2
(21)
where h is the uncertainty caused by the target.
4. The Robust Guidance Law with Expected ZEM Convergence As per the analysis of the prescribed performance control in Ref.[18, 19], it is critically to design a prescribed performance function to guarantee the convergence property of an appointed state. Meanwhile, to improve the performance of the control, a suitable prescribed performance function depended on the system should be constructed. Correspondingly, to guarantee the continuity and finite-time convergence of the ZEM-based PPF, a new multi-segment function u is defined as follows: a0 a1t a2t 2 a3t 3 , 1 1 u t2 t t2 t1 a0 a1t1 a2t12 a3t13 u u t t2
t t2
if t t1 (a ) u t t2
if t1 t t2 (b) if t t2
(22)
(c )
where a0 , a1 , a2 , a3 are determined by the task requirement. t1 , t 2 are predesigned parameters. p q is a designed parameter. As well, p q and both are odd values. u
value of u and must satisfy the relation, sign u
t 0
sign
u t t2
t t2
is the terminal
0 , as per [19].
Accordingly, the derivative of prescribed performance function u is given by a1t 2a2 t 3a3t 2 , u u u t t sign u u 2 0
t t2
if t t1
(a )
if t1 t t2 (b) if t t2
(23)
(c )
where is defined as
a0 a1t1 a2t12 a3t13 u
If t1 , t2 , ，and u
t t1
1
t t2
t2 t1 1
(24)
are predesigned, can be determined using Eq. (24) directly.
Meanwhile, the values of parameters
a0 , a1 , a2 , a3 can be determined by
a0 ZEM u t 0 ˆ a1 ZEM u t 0 2 3 a a t a 0 1 1 2 t1 a3t1 u t t1 a1` 2a2 t1 3a3t12 u t t u 1
where u
t td
ˆ is a predefined value, and ZEM u
ˆ ZEM u
t 0
(25) t t2
sign u
t t1
u
is the estimate of t 0
t t2
ZEM u
t 0
from Eq. (26).
V R t / r 2 VMT M AMN 1 e go M Vr t 0
(26)
Using this strategy, the new PPF u is designed completely. To demonstrate the effectiveness of the present PPF, a comparison with existing methods is performed here. In this demonstration case, the exponential decay function proposed by [18] and the finite-time convergence of PPF proposed by [20] are introduced. As well, it is assumed that ZEM u
t 0
ˆ 1.0 , ZEM u
t 0
0.1 , and u
t 10 s
0.1 . The results for the comparison are shown in
Figs. 2 and 3. 1.4
Proposed PPF PPF with exponential convergence PPF with Finite time convergence
1.2 1
u
0.8 0.6 0.4 0.2 0
0
5
10
15
20
Time(s)
Fig. 2 PPFs convergence phase
Fig. 3 PPFs convergence rate
Remark 1: As shown in Fig. 2 and Fig. 3, compared with the exponential decay function proposed by [18] and the PPF with finite-time convergence proposed by [20], the derivative of the PPF proposed by the present method can be assigned according to ZEM u , but not by the existing methods. In addition, the segmented u and u continually change from their assigned initial values to required instantaneous values at t1 and eventually converge to the required u
t t2
within a finite time. Therefore, the proposed method exhibits first-order continuity and finite-time convergence. Correspondingly, these advantages in the proposed PPF will benefit in the guidance
law design for improving the performance guidance commands of the interceptor. Accordingly, to guarantee the ZEM distance of the interceptor to a predefined bound, a transform error Z u is defined as ZEM u M min ZEM 1 u u Zu S ln ZEM u u M max u
(27)
Clearly, the initial value of Z u is zero, and the time derivative of Z u becomes Z u ( ZEM u u u ZEM u )CZu / u2 ( M max M min ) CZu ZEM u ZEM u M min M max u u
(28)
AMNc u G11CZu1 U1 U 2 t / G1 (t ) M t go M e go M
(29)
U1 u1CZu F2 (t ) u2CZu u ZEM u t go Vr R M N F ( t ) V A 1 e 2 MT M M Vr2
(30)
Next, AMNc is defined as
where U 1 is given by
and
h1 u1CZu h
(31)
Combining Eqs. (21), (28)-(31), the following relationship is found as Z u U 2 h1
(32)
Let h1 pass through a first-order low-pass filter G f s [23]. G f ( s)
1
f s 1
(33)
with f as a time constant. Subsequently, the dynamics of the estimated h1 is given by
f hˆ1 hˆ1 Z u U 2 Similarly, let U 2 be
(34)
U 2 kZu Zu hˆ1 ,
k Zu 0
(35)
Combining Eqs. (34) and (35), a revised hˆ1 is given by f hˆ1 Z u +k Z Z u
(36)
u
Integrating both sides of Eq. (36) yields hˆ1 Zu + hˆ1
t
t 0
kZu Zu dt
(37)
0
To guarantee AMNc to be continuous at the initial phase, hˆ1
t 0
0 should be enforced.
Therefore, the proposed guidance law for the interceptor without known target’s maneuvering information is yielded as
t AMNc u G11CZu1 U1 kZu f 1 Zu f 1 kZu Zu dt 0
(38)
As a result, the guidance command of an interceptor is found by integrating Eqs.(12) and (38). Next, the robustness of the guidance law given in Eq. (38) is demonstrated. Substituting Eqs. (31), (37) and (38) into Eq. (28), yields Zu kZu Zu (h1 hˆ1 )
(39)
Let h1 h1 hˆ1 . The dynamics of h1 is [23], h1 f 1h1 h1
(40)
Consider the following candidate Lyapunov function V Z u , h1 as Eq. (41).
V Z u , h1
1 2 1 2 Z u h1 2 2
(41)
Then, the time derivative of V Z u , h1 is
V Zu , h1 kZu Zu2 Zu h1 f 1h12 h1h1
(42)
1 1 kZu Z u2 f 1 1 h12 h12 2 2
It is clear that, if h12 is upper bounded, k Z 1 2 and 0 f 1 . As such, V Z u , h1 is u
bounded from above, indicating that Z u is also bounded. As proven in Ref. [19], ZEM u
t t f
will
converge to the neighborhood of a region defined by ZEM u
t t f
0, M max u
t t f
(43)
Therefore, it can be safely concluded that the effectiveness of the proposed guidance law is not comprised due to a lack of target maneuvering information. Therefore, the robustness of this method in the underlying scenario is retained. Remark 2: The proposed guidance law guarantees a high terminal ZEM distance accuracy, such that: 1) At the terminal interception point, G1 t f 0 , the guidance command may diverge in the case of a continuously maneuvering target. 2) Even though the target’s information is unknown to the interceptor, the proposed information filter is feasible to provide a continuous estimate. Consequently, the guidance command by the proposed PPF (22) can be guaranteed continuously.
5. Simulations and Discussions In this section, the performance of the proposed guidance law is verified by using numerical simulations. In particular, the verification will be conducted in three ways. First, a comparison is conducted among guidance laws based on the proposed PPF and two existing PPFs. Second, four maneuver strategies are adopted by the target and performed to demonstrate the efficiency and accuracy of the guidance law. Finally, the robustness of the proposed guidance law is tested using a Monte Carlo simulation.
5.1 Comparison with existing PPFs The target information in this example is given as Table 1. Table 1 Initial States of the target State xT
t 0
VT
AT
t 0
T
Value
State
Value
65000m
yT
t 0
15000m
1200m / s
T
t 0
00
ATc
t 0
0 0.2s
The parameters of the interceptor are given as Table 2.
0
Table 2 Initial States of the interceptor State
Value
VM
1500m / s
M AMc
State
0.1s
Value
M
t 0
0
AM
t 0
0
0
t 0
AM 20 g and
The acceleration of the interceptor and its derivative are limited by AM 40 g s 1 , respectively, where g is the gravity acceleration of 9.8m / s 2 .
In order to verify that the proposed guidance law is adaptive to different available acceleration profiles, two scenarios are established here using the conclusions of the pursuit-evasion game [11]. The first scenario is a diving pursuit mode, in which the interceptor’s trajectory varies from high to
low.
x
M t 0
In
this
30000m, yM
case, t 0
the
initial
positions
of
the
interceptor
are
set
to
18000m , and the target assumes a constant maneuver command of
5g . The second scenario is an ascending pursuit mode. In this case, the initial positions are
x
M t 0
30000m, yM
t 0
12000m , and the target takes a constant maneuver command of 5g .
Three types of PPFs are to be compared here and are stated as follows: (1) Type 1: An exponentially decaying function [18], referred to here as exp-PPF:
u ZEMu t 0 u t t sign ZEM u t 0 exp 1t u t t sign ZEM u t 0 2
(44)
2
(2) Type 2: A function with finite-time convergence [20], referred to here as finite-PPF:
t t t 1 12 ZEM 2 u 2 u
t 0
u
u
t t2
t t2
sign ZEM u
sign ZEM u
t 0
t 0
u t t
2
sign ZEM u
t 0
0 t t2 t t2
(45) (3) Type 3: The proposed PPF given in Eq. (22). The parameters of the guidance laws are set to: M min 4 , M max 6 , u k Zu 5 ,
and
t 0
2 ,
t2 t go
t 0
t 0
4 , t2 t go
t 0
t1 t go t1 t go
t t1
ZEM u
t 0
3 .
u
t t1
ZEM u
t 0
20 .
In
the
ascending
pursuit
mode,
0.5m , 7 9 ,
f 0.1s . In the diving pursuit mode,
u
t t2
, ,
In order to analyze tracking error between acceleration and guidance command of the interceptor caused by a lag in Eq.(10), an accumulated tracking error is defined as Ve
tf
0
AMc AM dt
(46)
The simulation is stopped when R 200m , and the time step for the simulation is chosen as tstep 2ms .
Simulation results for the diving pursuit mode are shown in Fig. 4 to Fig. 7. Here, the available acceleration at the beginning of the simulation is relatively small due to the thin atmosphere at high altitude. However, the available acceleration then increases as the interceptor approaches the target. Therefore, the reasonable guidance command should first increase from a feasible value, and subsequently converges to a small value at the terminal engagement position. Fig. 4 shows that the ZEM distance obtained by the three guidance laws can converge to the predesigned small values. However, Type 1 and Type 2 induce excessively large commands for feasible tracking at the beginning of the simulation, as shown in Figs. 5 and 6. By comparison, the proposed guidance law achieves a less acceleration consumption. The accumulated tracking error of the proposed approach is much smaller than the other two methods, as shown in Fig.7.
500
0
0
-2
exp-PPF finite-PPF Proposed-PPF
-4 -500 -8 AM (g)
ZEMu(m)
-6 -1000 -1500
-12
-2000
exp-PPF finite-PPF proposed-PPF
-2500
-14 -16
-3000 -3500
-10
-18 0
2
4
6
8
10
12
-20
14
0
2
4
Fligt Time(s)
6
8
10
12
14
Fligt Time(s)
Fig. 4 ZEMs under different PPF types
Fig. 5 Interceptor’s guidance commands
5
140
0
120 Tracking Error(m/s)
-5 -10
c
AM (g)
-15 -20 -25
exp-PPF finite-PPF proposed-PPF
-30 -35 -40 -45
0
2
4
6
8
10
12
100 80
exp-PPF finite-PPF proposed-PPF
60 40 20
14
0
0
Fligt Time(s)
Fig. 6 Interceptor’s accelerations
2
4
6
8
10
12
14
Fligt Time(s)
Fig. 7 Accumulated tracking error
Figs. 8 to 11 show the simulation results for the ascending mode. In this mode, the available acceleration rises as the interceptor approaches target. Therefore, the reasonable guidance command should be large initially in order to eliminate the ZEM distance as soon as possible, and eventually converge to a small value. Fig. 8 shows that the ZEM distance from the proposed method converges faster than the other methods. Fig. 9 and Fig. 10 show that the command acceleration from the proposed method is much larger than the other two methods at the beginning of the simulation and converges faster to a small value. From Fig. 11, it can be concluded that the proposed method has the smallest accumulated tracking error.
3500
20
exp-PPF finite-PPF proposed-PPF
3000 2500
exp-PPF finite-PPF Proposed-PPF
18 16
12 AM (g)
ZEMu(m)
14 2000 1500
10 8
1000
6 500 4 0 -500
2 0
2
4
6
8
10
12
0
14
0
2
4
Fligt Time(s)
Fig. 8 ZEMs under different PPF types
10
12
14
140
exp-PPF finite-PPF proposed-PPF
35
120 Tracking Error(m/s)
40
30 25 c
8
Fig. 9 Interceptor’s accelerations
45
AM (g)
6
Fligt Time(s)
20 15 10
100 80
exp-PPF finite-PPF proposed-PPF
60 40
5 20 0 -5
0
2
4
6
8
10
12
0
14
0
Fligt Time(s)
2
4
6
8
10
12
14
Fligt Time(s)
Fig. 10 Interceptor’s guidance commands
Fig. 11 Accumulated tracking error
5.2 Efficiency and Accuracy of the Proposed Guidance Law In this section, the initial states of the interceptor are the same as Condition 2 in Section 5.1. Four cases of target maneuver-types are tested here: Case 1: A sinusoidal maneuver-type, given by: ATc 5 g sin t 2
(47)
Case 2: A positive constant maneuver-type, given by: ATc 5 g
(48)
Case 3: A constant negative maneuver-type, given by: ATc 5 g
(49)
Case 4: A bang-bang maneuver-type, given by: t 4s 5 g ATc 5 g t 4s
(50)
The parameters for the design of the PPF in the proposed guidance law are t1 t go t 0 3 , t2 t go
t 0
, u
t t1
ZEM u
t 0
5.
The simulation results are shown in Fig. 12 to Fig. 15. 3500
14
Case 1 Case 2 Case 3 Case 4
3000
2000
10 8 6 AM (g)
ZEMu(m)
2500
Case 1 Case 2 Case 3 Case 4
12
1500
4 2
1000
0 500 -2 0 -500
-4 0
2
4
6
8
10
12
-6
14
0
2
4
Fligt Time(s)
Fig. 12
ZEMs for four Cases
Fig. 13
14
2
12
8
c
8
10
12
14
1.7
4
Accelerations of the interceptor
4
Target in Case 1 Attacker in Case 1 Target in Case 2 Attacker in Case 2 Target in Case 3 Attacker in Case 3 Target in Case 4 Attacker in Case 4
1.8
y(m)
6
x 10
1.9
Case 1 Case 2 Case 3 Case 4
10
AM (g)
6
Fligt Time(s)
1.6 1.5
2 1.4
0 -2
1.3
-4
1.2
-6
0
2
4
6
8
10
12
1.1
14
Fig.
14
Guidance
commands
of
3
3.5
4
4.5
5
x(m)
Fligt Time(s)
the
5.5
6
6.5 x 10
4
Fig. 15 Trajectories of target and interceptor
interceptor
The resulting ZEM distance are shown in Fig. 12. According to the same convergence property of the PPF, the results show that the convergence properties of the ZEM distance in each of the four cases have similar characteristics. Additionally, the terminal ZEM distance of the interceptor can be guaranteed within a small region around the predesigned terminal value. The accelerations and the guidance commands of the interceptor are shown in Fig. 13 and Fig. 14, respectively. During the interception time, the accelerations of the interceptor are not saturated and continuous, similar to the guidance commands. This demonstrates that the guidance commands can be effectively tracked under a delayed model. The trajectories of the target and interceptor are shown
in Fig. 15. Despite the fact that different maneuver-types are adopted by the target, and that these maneuvers are unknown to the interceptor, the interceptor is able to capture target precisely.
5.3 Performance Evaluation Using Monte Carlo Simulation In this section, a Monte Carlo test using 200 runs is conducted to evaluate the performance of the proposed guidance law. The maneuver-type of the target adopted here is the sinusoidal type shown in Eq.(47). Uncertainties and dispersions were implemented in the initial position, velocity, and flight path angle of the target. Additionally, the distributions of these uncertainties were assumed to be Gaussian. The details are listed in Table 3. Table 3 Dispersions in the initial state of the interceptor States
xT
t 0
yT
t 0
VT
t 0
T
t 0
Mean
3 dispersion
65000m
500m,500m
14500m
500m,500m
1200m/s
20m/s, 20m/s
0
0.5 , 0.5
The Monte-Carlo simulation results are shown in Fig. 16 and Fig. 17. Even though the uncertainties are distributed over a wide range, the terminal ZEM distance of the interceptor are scattered within a range of [0.05m,0.3m] . From the cumulative distribution of the terminal ZEM distance shown in Fig. 17, it can be seen that the precision and robustness of the proposed guidance law are verified perfectly.
1
0.3
0.9 Cumulative Distribution
Terminal ZEMu(m)
0.35
0.25 0.2 0.15 0.1 0.05 0 -0.05 12.7
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
12.8
12.9
13
13.1
13.2
Fligt Time(s)
0 -0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Terminal ZEMu(m)
Fig. 16 Dispersion of terminal ZEMs of the
Fig. 17 Cumulative distribution of terminal
interceptor
ZEMs of the interceptor
6.
Conclusion
In this paper, a two-dimensional target-interceptor endgame was investigated. A robust guidance law was proposed for an interceptor to achieve high interception accuracy with continuity and finite-time convergence of guidance commands. In addition to sharing the advantages of the prescribed performance control (PPC) method, in the present control approach, continuous guidance commands are yielded even when the target’s maneuvering accelerations are unknown in advance. Simulations were conducted to demonstrate the preciseness and robustness of proposed guidance law. Declaration of Interest Statement
The authors declared that they have no conflicts of interest to this work.
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Robust Intercept Guidance Law with Predesigned Zero-Effort Miss Distance Convergence for Capturing Maneuvering Targets Xiaodong Yan , Shi Lyu PII: DOI: Reference:
S0016-0032(19)30753-7 https://doi.org/10.1016/j.jfranklin.2019.10.021 FI 4219
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
15 August 2019 29 September 2019 20 October 2019
Please cite this article as: Xiaodong Yan , Shi Lyu , Robust Intercept Guidance Law with Predesigned Zero-Effort Miss Distance Convergence for Capturing Maneuvering Targets, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.10.021
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Robust Intercept Guidance Law with Predesigned Zero-Effort Miss Distance Convergence for Capturing Maneuvering Targets Xiaodong Yan a,b , Shi Lyua,b a
School of Astronautics, Northwestern Polytechnical University, Xi’an, 710072, China Shaanxi Key Laboratory of Aerospace Flight Vehicle Technology, Northwestern Polytechnical University, Xi’an, 710072, China b
Abstract: In this paper, a homing guidance in a two-dimensional target-interceptor scenario is studied with an unknown target maneuver strategy. In order to capture the target with acceptable accuracy, a robust guidance law with a predesigned zero-effort miss (ZEM) distance convergence for the interceptor is proposed. In order to guarantee the ZEM distance to converge into a small region around the predesigned terminal value within a finite time, a novel function of ZEM distance is proposed. The ZEM distance convergence profile can be reasonably controlled according to the interceptor’s initial state. Based on the designed profile, an initial delay control and prescribed performance control are utilized to achieve a high terminal miss distance accuracy and account for uncertainties caused by the target’s unknown maneuverability. Four maneuvering strategies of the target are tested to verify the performance of the proposed guidance law.
Keywords: Endgame; Zero-effort miss distance; Predesigned performance; Finite time convergence; Guidance law.
1. Introduction Modern defense missiles (interceptors) present a significant threat to today’s maneuvering targets. In order to increase the survivability of targets, plenty of maneuvering strategies have been created. An example of such a strategy is the barrel roll and sinusoidal/weaving maneuver [1]. This weaving maneuver is easily designed and has been widely adopted for targets in both low and high altitude applications [2]. Especially, the moving mass control for the target’s maneuver has become a research hotspot [3, 4]. When a target adopts maneuvering strategies to improve its penetration, an effective interception guidance control law for the interceptor is a very important countermeasure. In order to capture a high-speed maneuvering target, there are three key challenges for the design of a successful interception guidance law: 1) Since the remaining time of interception is quite short, the miss distance caused by an interceptor system lag can be significant. Therefore, a finite-time convergence of the guidance law should be guaranteed. 2) The unknown maneuvering
strategies of the target can have a significant impact on the terminal interception precision. As a result, the robustness of the guidance law under unknown maneuvering information should be fully considered in order to yield a high interception precision. 3) The interceptor’s maneuverability can vary greatly under changing flight altitudes. Therefore, the guidance commands must be adjustable depending on the interceptor’s available accelerations. In order to address the first and second challenges described above, several guidance laws have been developed. Some of these strategies include sliding mode control [5-9], differential game theory [10, 11], prescribed performance control (PPC) [12-15], and adaptive dynamic programming [16]. Each type of guidance laws can achieve the terminal interception precision as expected. In particular, the PPC-based interception guidance laws [12-15] have been shown to have flexible controller gains and can guarantee the asymptotic convergence of the desired parameter to a predetermined small residual and exhibits acceptable transient performance. Even though these guidance laws are effective in many interception situations, the variability of interceptor maneuverability is generally not fully considered. Recently, Ref. [17] presented a command shaping guidance law for intercepting high-speed targets. This guidance law has the potential to obtain the optimal guidance commands to account for varying acceleration profiles by taking advantage of the linearized optimal control. However, the guidance law in Ref. [17] focuses on a non-maneuvering target. Therefore, its performance may not behave as expected without precise acceleration information in advance. In the present work, a new robust interception guidance law is presented to address the aforementioned three challenges. The present guidance law is designed to perform against maneuvering targets with a high terminal interception precision. Firstly, relative equations between the interceptor and target with uncertainties caused by the target’s unknown maneuvers are constructed. Subsequently, a segmented finite time convergence function is established to design a zero-effort miss (ZEM) distance convergence profile directly. Compared with the existing methods [18-20] and previous work by the present authors [13-15], two advantages exist in the proposed guidance law. Firstly, the ZEM distance convergence property can be manipulated directly. Therefore, the convergence can be flexibly configured based on the interceptors’ available accelerations due to variations in flight altitude. Secondly, a new prescribed performance function is established. As a result, the continuity and finite-time convergence of the guidance commands
can be yielded simultaneously.
2. Target-interceptor endgame kinematics
Initial LOS
Fig. 1
Schematics of target-interceptor engagement
In this paper, the two-player homing guidance is studied in the X Y plane, as shown in Fig. 1. The scenario involves a high-speed target with an unknown maneuvering strategy and an agile defending missile (i.e., the interceptor). In Fig. 1, X I OYI denotes a Cartesian inertial reference frame with OX I representing the local horizon. The subscripts M and T represent the interceptor and target, respectively. In addition, without loss of any generality, the gravitational forces are neglected in the homing phase, and the speed of each of the entities is assumed to be time-invariant throughout the engagement. The system dynamics of the target is assumed to be captured by the first-order lag system as follows: AT
where T
AT
ATc
T T c is a constant time, and AT is the guidance command normal to velocity.
(1)
In addition, the kinematics between the target and interceptor are governed by R Vr VT cos( T ) VM cos( M )
(2)
VMT / R [VT sin( T ) VM sin( M )] / R
(3)
where R and Vr are the relative distance and velocity between the target and interceptor, respectively. VM is the velocity of the interceptor and M is the path angle of the interceptor. VT is velocity of the target and T is the path angle of target. denotes the angle between the
line of sight (LOS) and OX I , where LOS is defined as the instantaneous LOS between the target and interceptor. VMT is the relative velocity normal to LOS.
The interceptor’s time-to-go can be calculated as according to the initial states is calculated as
t go R / Vr
t f R t 0 Vr
t 0
. As such, the total flight time
, and the time derivatives of Vr ,
VMT , t go and are given by[21] 2 Vr VMT / R ATL AML
(4)
VMT VrVMT / R ATN AMN
(5)
t go R Vr / Vr2 1
(6)
VMD / R VMDVr / R 2
(7)
where the interceptor’s accelerations along and normal to the LOS are: L AM AM sin( M ) N AM AM cos( M )
(8)
Similarly, the target’s accelerations along and normal to the LOS are: ATL AT sin( T ) N AT AT cos( T )
(9)
Taking into account the dynamics of the interceptor tracking the reference guidance command, the derivative of its acceleration becomes
AM ( AMc AM ) / M
(10)
where AMc is the reference guidance command to be specified, and M is a time constant.
3. Zero-effort Miss Distance of Interceptor By instigating a head-on engagement scenario, the error angle between M 0 and T 0 is relatively small. In this case, fluctuates in the neighborhood of zero during the homing phase [22]. Consequently, the linearized kinematics can be written as follows:
X MT AMT X MT BT ATNc BM AMNc where X MT [ yMT , yMT , ATN , AMN ]T , and
AMT
0 0 0 0
1 0
0 1
0
1 / T
0
0
0 1 0 1/M
0 0 BT 1 / T 0
0 0 BM 0 1 / M
(11)
where ATNc and AMNc are target’s and interceptor’s guidance commands normal to initial LOS, respectively. These commands are given by Nc c AT AT cos T 0 Nc c AM AM cos M 0
The state transformation function,
MT
1 t go 0 1 tgo 0 0 0 0
MT (t go )
, of Eq. (11) is given by
T2 t go T T2 e T e
t go / T
e
(12)
t go / T
M2 t go M M2 e
T
M e
t go / M
t go / T
t go / M
M
0
0
e
t go / M
(13)
and
DMT [1,0,0,0]
(14)
As such, the interceptor’s ZEM distance is given by ZEM DMT MT t go X MT
yMT t go yMT T2 tgo T T2 e
M2 t go M M2 e
t go / T
t go / M
A
A
N T
(15)
N M
Due to the fact that the angle between R and the initial LOS is relatively small, yMT can be linearized such that
yMT R sin 0 R 0
(16)
yMT R( 0 ) Vr tgo
(17)
and yMT is given by
Combining Eqs. (15), (16) and (17), ZEM can be rewritten as
ZEM Vr t go T2 t go T T2 e 2
M2 t go M M2 e
t go / M
A
t go
A
N T
(18)
N M
Similarly, the time derivative of ZEM can be written as ZEM
Vr R Vr2
V AN 1 e tgo / T A N 1 e tgo / M M M MT T T
T t go T e
t go / T
A Nc T
M
t go M e
t go / M
A
(19)
Nc M
In practice, the target’s dynamic information is often unknown, or only partially known, to the interceptor. In this case, the proposed guidance command for the interceptor will struggle to calculate ZEM, and its time derivative is desired. In this case, Eq.(18) and Eq.(19) can be
rewritten as follows.
ZEM u Vr tgo M2 tgo M M2 e 2
ZEM u
tgo / M
A
N M
(20)
Vr R t / t / VMT M AMN 1 e go M M t go M e go M AMNc h Vr2
(21)
where h is the uncertainty caused by the target.
4. The Robust Guidance Law with Expected ZEM Convergence As per the analysis of the prescribed performance control in Ref.[18, 19], it is critically to design a prescribed performance function to guarantee the convergence property of an appointed state. Meanwhile, to improve the performance of the control, a suitable prescribed performance function depended on the system should be constructed. Correspondingly, to guarantee the continuity and finite-time convergence of the ZEM-based PPF, a new multi-segment function u is defined as follows: a0 a1t a2t 2 a3t 3 , 1 1 u t2 t t2 t1 a0 a1t1 a2t12 a3t13 u u t t2
t t2
if t t1 (a ) u t t2
if t1 t t2 (b) if t t2
(22)
(c )
where a0 , a1 , a2 , a3 are determined by the task requirement. t1 , t 2 are predesigned parameters. p q is a designed parameter. As well, p q and both are odd values. u
value of u and must satisfy the relation, sign u
t 0
sign
u t t2
t t2
is the terminal
0 , as per [19].
Accordingly, the derivative of prescribed performance function u is given by a1t 2a2 t 3a3t 2 , u u u t t sign u u 2 0
t t2
if t t1
(a )
if t1 t t2 (b) if t t2
(23)
(c )
where is defined as
a0 a1t1 a2t12 a3t13 u
If t1 , t2 , ，and u
t t1
1
t t2
t2 t1 1
(24)
are predesigned, can be determined using Eq. (24) directly.
Meanwhile, the values of parameters
a0 , a1 , a2 , a3 can be determined by
a0 ZEM u t 0 ˆ a1 ZEM u t 0 2 3 a a t a 0 1 1 2 t1 a3t1 u t t1 a1` 2a2 t1 3a3t12 u t t u 1
where u
t td
ˆ is a predefined value, and ZEM u
ˆ ZEM u
t 0
(25) t t2
sign u
t t1
u
is the estimate of t 0
t t2
ZEM u
t 0
from Eq. (26).
V R t / r 2 VMT M AMN 1 e go M Vr t 0
(26)
Using this strategy, the new PPF u is designed completely. To demonstrate the effectiveness of the present PPF, a comparison with existing methods is performed here. In this demonstration case, the exponential decay function proposed by [18] and the finite-time convergence of PPF proposed by [20] are introduced. As well, it is assumed that ZEM u
t 0
ˆ 1.0 , ZEM u
t 0
0.1 , and u
t 10 s
0.1 . The results for the comparison are shown in
Figs. 2 and 3. 1.4
Proposed PPF PPF with exponential convergence PPF with Finite time convergence
1.2 1
u
0.8 0.6 0.4 0.2 0
0
5
10
15
20
Time(s)
Fig. 2 PPFs convergence phase
Fig. 3 PPFs convergence rate
Remark 1: As shown in Fig. 2 and Fig. 3, compared with the exponential decay function proposed by [18] and the PPF with finite-time convergence proposed by [20], the derivative of the PPF proposed by the present method can be assigned according to ZEM u , but not by the existing methods. In addition, the segmented u and u continually change from their assigned initial values to required instantaneous values at t1 and eventually converge to the required u
t t2
within a finite time. Therefore, the proposed method exhibits first-order continuity and finite-time convergence. Correspondingly, these advantages in the proposed PPF will benefit in the guidance
law design for improving the performance guidance commands of the interceptor. Accordingly, to guarantee the ZEM distance of the interceptor to a predefined bound, a transform error Z u is defined as ZEM u M min ZEM 1 u u Zu S ln ZEM u u M max u
(27)
Clearly, the initial value of Z u is zero, and the time derivative of Z u becomes Z u ( ZEM u u u ZEM u )CZu / u2 ( M max M min ) CZu ZEM u ZEM u M min M max u u
(28)
AMNc u G11CZu1 U1 U 2 t / G1 (t ) M t go M e go M
(29)
U1 u1CZu F2 (t ) u2CZu u ZEM u t go Vr R M N F ( t ) V A 1 e 2 MT M M Vr2
(30)
Next, AMNc is defined as
where U 1 is given by
and
h1 u1CZu h
(31)
Combining Eqs. (21), (28)-(31), the following relationship is found as Z u U 2 h1
(32)
Let h1 pass through a first-order low-pass filter G f s [23]. G f ( s)
1
f s 1
(33)
with f as a time constant. Subsequently, the dynamics of the estimated h1 is given by
f hˆ1 hˆ1 Z u U 2 Similarly, let U 2 be
(34)
U 2 kZu Zu hˆ1 ,
k Zu 0
(35)
Combining Eqs. (34) and (35), a revised hˆ1 is given by f hˆ1 Z u +k Z Z u
(36)
u
Integrating both sides of Eq. (36) yields hˆ1 Zu + hˆ1
t
t 0
kZu Zu dt
(37)
0
To guarantee AMNc to be continuous at the initial phase, hˆ1
t 0
0 should be enforced.
Therefore, the proposed guidance law for the interceptor without known target’s maneuvering information is yielded as
t AMNc u G11CZu1 U1 kZu f 1 Zu f 1 kZu Zu dt 0
(38)
As a result, the guidance command of an interceptor is found by integrating Eqs.(12) and (38). Next, the robustness of the guidance law given in Eq. (38) is demonstrated. Substituting Eqs. (31), (37) and (38) into Eq. (28), yields Zu kZu Zu (h1 hˆ1 )
(39)
Let h1 h1 hˆ1 . The dynamics of h1 is [23], h1 f 1h1 h1
(40)
Consider the following candidate Lyapunov function V Z u , h1 as Eq. (41).
V Z u , h1
1 2 1 2 Z u h1 2 2
(41)
Then, the time derivative of V Z u , h1 is
V Zu , h1 kZu Zu2 Zu h1 f 1h12 h1h1
(42)
1 1 kZu Z u2 f 1 1 h12 h12 2 2
It is clear that, if h12 is upper bounded, k Z 1 2 and 0 f 1 . As such, V Z u , h1 is u
bounded from above, indicating that Z u is also bounded. As proven in Ref. [19], ZEM u
t t f
will
converge to the neighborhood of a region defined by ZEM u
t t f
0, M max u
t t f
(43)
Therefore, it can be safely concluded that the effectiveness of the proposed guidance law is not comprised due to a lack of target maneuvering information. Therefore, the robustness of this method in the underlying scenario is retained. Remark 2: The proposed guidance law guarantees a high terminal ZEM distance accuracy, such that: 1) At the terminal interception point, G1 t f 0 , the guidance command may diverge in the case of a continuously maneuvering target. 2) Even though the target’s information is unknown to the interceptor, the proposed information filter is feasible to provide a continuous estimate. Consequently, the guidance command by the proposed PPF (22) can be guaranteed continuously.
5. Simulations and Discussions In this section, the performance of the proposed guidance law is verified by using numerical simulations. In particular, the verification will be conducted in three ways. First, a comparison is conducted among guidance laws based on the proposed PPF and two existing PPFs. Second, four maneuver strategies are adopted by the target and performed to demonstrate the efficiency and accuracy of the guidance law. Finally, the robustness of the proposed guidance law is tested using a Monte Carlo simulation.
5.1 Comparison with existing PPFs The target information in this example is given as Table 1. Table 1 Initial States of the target State xT
t 0
VT
AT
t 0
T
Value
State
Value
65000m
yT
t 0
15000m
1200m / s
T
t 0
00
ATc
t 0
0 0.2s
The parameters of the interceptor are given as Table 2.
0
Table 2 Initial States of the interceptor State
Value
VM
1500m / s
M AMc
State
0.1s
Value
M
t 0
0
AM
t 0
0
0
t 0
AM 20 g and
The acceleration of the interceptor and its derivative are limited by AM 40 g s 1 , respectively, where g is the gravity acceleration of 9.8m / s 2 .
In order to verify that the proposed guidance law is adaptive to different available acceleration profiles, two scenarios are established here using the conclusions of the pursuit-evasion game [11]. The first scenario is a diving pursuit mode, in which the interceptor’s trajectory varies from high to
low.
x
M t 0
In
this
30000m, yM
case, t 0
the
initial
positions
of
the
interceptor
are
set
to
18000m , and the target assumes a constant maneuver command of
5g . The second scenario is an ascending pursuit mode. In this case, the initial positions are
x
M t 0
30000m, yM
t 0
12000m , and the target takes a constant maneuver command of 5g .
Three types of PPFs are to be compared here and are stated as follows: (1) Type 1: An exponentially decaying function [18], referred to here as exp-PPF:
u ZEMu t 0 u t t sign ZEM u t 0 exp 1t u t t sign ZEM u t 0 2
(44)
2
(2) Type 2: A function with finite-time convergence [20], referred to here as finite-PPF:
t t t 1 12 ZEM 2 u 2 u
t 0
u
u
t t2
t t2
sign ZEM u
sign ZEM u
t 0
t 0
u t t
2
sign ZEM u
t 0
0 t t2 t t2
(45) (3) Type 3: The proposed PPF given in Eq. (22). The parameters of the guidance laws are set to: M min 4 , M max 6 , u k Zu 5 ,
and
t 0
2 ,
t2 t go
t 0
t 0
4 , t2 t go
t 0
t1 t go t1 t go
t t1
ZEM u
t 0
3 .
u
t t1
ZEM u
t 0
20 .
In
the
ascending
pursuit
mode,
0.5m , 7 9 ,
f 0.1s . In the diving pursuit mode,
u
t t2
, ,
In order to analyze tracking error between acceleration and guidance command of the interceptor caused by a lag in Eq.(10), an accumulated tracking error is defined as Ve
tf
0
AMc AM dt
(46)
The simulation is stopped when R 200m , and the time step for the simulation is chosen as tstep 2ms .
Simulation results for the diving pursuit mode are shown in Fig. 4 to Fig. 7. Here, the available acceleration at the beginning of the simulation is relatively small due to the thin atmosphere at high altitude. However, the available acceleration then increases as the interceptor approaches the target. Therefore, the reasonable guidance command should first increase from a feasible value, and subsequently converges to a small value at the terminal engagement position. Fig. 4 shows that the ZEM distance obtained by the three guidance laws can converge to the predesigned small values. However, Type 1 and Type 2 induce excessively large commands for feasible tracking at the beginning of the simulation, as shown in Figs. 5 and 6. By comparison, the proposed guidance law achieves a less acceleration consumption. The accumulated tracking error of the proposed approach is much smaller than the other two methods, as shown in Fig.7.
500
0
0
-2
exp-PPF finite-PPF Proposed-PPF
-4 -500 -8 AM (g)
ZEMu(m)
-6 -1000 -1500
-12
-2000
exp-PPF finite-PPF proposed-PPF
-2500
-14 -16
-3000 -3500
-10
-18 0
2
4
6
8
10
12
-20
14
0
2
4
Fligt Time(s)
6
8
10
12
14
Fligt Time(s)
Fig. 4 ZEMs under different PPF types
Fig. 5 Interceptor’s guidance commands
5
140
0
120 Tracking Error(m/s)
-5 -10
c
AM (g)
-15 -20 -25
exp-PPF finite-PPF proposed-PPF
-30 -35 -40 -45
0
2
4
6
8
10
12
100 80
exp-PPF finite-PPF proposed-PPF
60 40 20
14
0
0
Fligt Time(s)
Fig. 6 Interceptor’s accelerations
2
4
6
8
10
12
14
Fligt Time(s)
Fig. 7 Accumulated tracking error
Figs. 8 to 11 show the simulation results for the ascending mode. In this mode, the available acceleration rises as the interceptor approaches target. Therefore, the reasonable guidance command should be large initially in order to eliminate the ZEM distance as soon as possible, and eventually converge to a small value. Fig. 8 shows that the ZEM distance from the proposed method converges faster than the other methods. Fig. 9 and Fig. 10 show that the command acceleration from the proposed method is much larger than the other two methods at the beginning of the simulation and converges faster to a small value. From Fig. 11, it can be concluded that the proposed method has the smallest accumulated tracking error.
3500
20
exp-PPF finite-PPF proposed-PPF
3000 2500
exp-PPF finite-PPF Proposed-PPF
18 16
12 AM (g)
ZEMu(m)
14 2000 1500
10 8
1000
6 500 4 0 -500
2 0
2
4
6
8
10
12
0
14
0
2
4
Fligt Time(s)
Fig. 8 ZEMs under different PPF types
10
12
14
140
exp-PPF finite-PPF proposed-PPF
35
120 Tracking Error(m/s)
40
30 25 c
8
Fig. 9 Interceptor’s accelerations
45
AM (g)
6
Fligt Time(s)
20 15 10
100 80
exp-PPF finite-PPF proposed-PPF
60 40
5 20 0 -5
0
2
4
6
8
10
12
0
14
0
Fligt Time(s)
2
4
6
8
10
12
14
Fligt Time(s)
Fig. 10 Interceptor’s guidance commands
Fig. 11 Accumulated tracking error
5.2 Efficiency and Accuracy of the Proposed Guidance Law In this section, the initial states of the interceptor are the same as Condition 2 in Section 5.1. Four cases of target maneuver-types are tested here: Case 1: A sinusoidal maneuver-type, given by: ATc 5 g sin t 2
(47)
Case 2: A positive constant maneuver-type, given by: ATc 5 g
(48)
Case 3: A constant negative maneuver-type, given by: ATc 5 g
(49)
Case 4: A bang-bang maneuver-type, given by: t 4s 5 g ATc 5 g t 4s
(50)
The parameters for the design of the PPF in the proposed guidance law are t1 t go t 0 3 , t2 t go
t 0
, u
t t1
ZEM u
t 0
5.
The simulation results are shown in Fig. 12 to Fig. 15. 3500
14
Case 1 Case 2 Case 3 Case 4
3000
2000
10 8 6 AM (g)
ZEMu(m)
2500
Case 1 Case 2 Case 3 Case 4
12
1500
4 2
1000
0 500 -2 0 -500
-4 0
2
4
6
8
10
12
-6
14
0
2
4
Fligt Time(s)
Fig. 12
ZEMs for four Cases
Fig. 13
14
2
12
8
c
8
10
12
14
1.7
4
Accelerations of the interceptor
4
Target in Case 1 Attacker in Case 1 Target in Case 2 Attacker in Case 2 Target in Case 3 Attacker in Case 3 Target in Case 4 Attacker in Case 4
1.8
y(m)
6
x 10
1.9
Case 1 Case 2 Case 3 Case 4
10
AM (g)
6
Fligt Time(s)
1.6 1.5
2 1.4
0 -2
1.3
-4
1.2
-6
0
2
4
6
8
10
12
1.1
14
Fig.
14
Guidance
commands
of
3
3.5
4
4.5
5
x(m)
Fligt Time(s)
the
5.5
6
6.5 x 10
4
Fig. 15 Trajectories of target and interceptor
interceptor
The resulting ZEM distance are shown in Fig. 12. According to the same convergence property of the PPF, the results show that the convergence properties of the ZEM distance in each of the four cases have similar characteristics. Additionally, the terminal ZEM distance of the interceptor can be guaranteed within a small region around the predesigned terminal value. The accelerations and the guidance commands of the interceptor are shown in Fig. 13 and Fig. 14, respectively. During the interception time, the accelerations of the interceptor are not saturated and continuous, similar to the guidance commands. This demonstrates that the guidance commands can be effectively tracked under a delayed model. The trajectories of the target and interceptor are shown
in Fig. 15. Despite the fact that different maneuver-types are adopted by the target, and that these maneuvers are unknown to the interceptor, the interceptor is able to capture target precisely.
5.3 Performance Evaluation Using Monte Carlo Simulation In this section, a Monte Carlo test using 200 runs is conducted to evaluate the performance of the proposed guidance law. The maneuver-type of the target adopted here is the sinusoidal type shown in Eq.(47). Uncertainties and dispersions were implemented in the initial position, velocity, and flight path angle of the target. Additionally, the distributions of these uncertainties were assumed to be Gaussian. The details are listed in Table 3. Table 3 Dispersions in the initial state of the interceptor States
xT
t 0
yT
t 0
VT
t 0
T
t 0
Mean
3 dispersion
65000m
500m,500m
14500m
500m,500m
1200m/s
20m/s, 20m/s
0
0.5 , 0.5
The Monte-Carlo simulation results are shown in Fig. 16 and Fig. 17. Even though the uncertainties are distributed over a wide range, the terminal ZEM distance of the interceptor are scattered within a range of [0.05m,0.3m] . From the cumulative distribution of the terminal ZEM distance shown in Fig. 17, it can be seen that the precision and robustness of the proposed guidance law are verified perfectly.
1
0.3
0.9 Cumulative Distribution
Terminal ZEMu(m)
0.35
0.25 0.2 0.15 0.1 0.05 0 -0.05 12.7
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
12.8
12.9
13
13.1
13.2
Fligt Time(s)
0 -0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Terminal ZEMu(m)
Fig. 16 Dispersion of terminal ZEMs of the
Fig. 17 Cumulative distribution of terminal
interceptor
ZEMs of the interceptor
6.
Conclusion
In this paper, a two-dimensional target-interceptor endgame was investigated. A robust guidance law was proposed for an interceptor to achieve high interception accuracy with continuity and finite-time convergence of guidance commands. In addition to sharing the advantages of the prescribed performance control (PPC) method, in the present control approach, continuous guidance commands are yielded even when the target’s maneuvering accelerations are unknown in advance. Simulations were conducted to demonstrate the preciseness and robustness of proposed guidance law. Declaration of Interest Statement
The authors declared that they have no conflicts of interest to this work.
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