A new continuous adaptive finite time guidance law against highly maneuvering targets

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A new continuous adaptive finite time guidance law against highly maneuvering targets

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Institute of Precision Guidance and Control, Northwestern Polytechnical University, Xi’an, Shannxi, 710072, People’s Republic of China

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a r t i c l e

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a b s t r a c t

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Article history: Received 25 January 2018 Received in revised form 18 July 2018 Accepted 17 August 2018 Available online xxxx

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Jianguo Guo, Yifei Li, Jun Zhou

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Keywords: Guidance law Sliding mode control Adaptive Finite time convergence Lyapunov stability theory

A novel continuous adaptive finite time guidance (CAFTG) law is proposed for homing missiles. Firstly, three-dimensional nonlinear dynamics describing the pursuit situation of the missile and the target are introduced to obtain the mathematical model of engagement. Secondly, in order to improve the accuracy of interception, a nonlinear disturbance observer with finite time convergence is employed to estimate the acceleration of a target and compensate the guidance law. A continuous guidance scheme with robustness is constructed via sliding mode control theory, which guarantees finite time convergence by Lyapunov stability theory. Finally, simulations are conducted on the nonlinear dynamic models and results demonstrate the effectiveness of proposed guidance method. © 2018 Elsevier Masson SAS. All rights reserved.

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1. Introduction

36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Proportional navigation guidance law (PNGL) has been the most widely applied interception strategy for homing missiles in the past decades due to its computational simplicity, robustness and implementability [1,2]. Many variations of proportional navigation (PN), such as pure proportional navigation (PPN), true proportional navigation (TPN), ideal proportional navigation (IPN) and generalized true proportional navigation (GTPN) are also developed and their merits and demerits are discussed in [3–6]. Scholars have proposed lots of modified forms of PN for different needs and constraints, especially for the need against a maneuvering target and the impact angle constraint. However, PN is more applicable for the task of intercepting a non-maneuvering target. The performance of PN will be degraded when intercepting a target with powerful maneuvering capability. This disadvantage motivates the researches on advanced control methods and their applications in missile guidance system design. Recently, the research on three-dimensional guidance laws design has made a progress, such as a robust PN guidance law based on input–output linearization method in [7], a feedbacklinearization-based guidance law that achieves all-aspect interception for an acceleration-constrained missile in [8] and an adaptive reaching law based finite time guidance law in [9]. Furthermore, a three-dimensional relative kinematic equation set established in a rotating line of sight coordinate system is applied in guidance

61 62 63 64 65 66

E-mail address: [email protected] (J. Guo). https://doi.org/10.1016/j.ast.2018.08.018 1270-9638/© 2018 Elsevier Masson SAS. All rights reserved.

law design [10], which is demonstrated helpful to mitigate the cross coupling issues. In [11], the command filtered back-stepping scheme is utilized to design a nonlinear guidance law, which considers both acceleration command constraint and the second-order autopilot dynamics. Theoretically speaking, sliding mode control (SMC) method for general linear/nonlinear systems offers better robustness to external disturbances and parameter perturbations than other methods [12]. SMC has been widely applied in the process of designing missile guidance and control systems. Unknown target acceleration is viewed as a type of external disturbance in missile–target engagement. Lots of disturbance-estimation/attenuation techniques are employed to eliminate the effects caused by target maneuvering. For example, in [13] the target acceleration is dealt through an adaptive procedure to select the gain of the switching controller. In [14] SMC and time-to-go estimation method are synthesized in missile guidance scheme. In [15] nonlinear disturbance observer technique is used to estimate target acceleration and integral sliding mode guidance is proposed. In [16,17] target acceleration bound is estimated by an adaptive law without the requirement of information from the target. Nevertheless, the chattering of sliding mode is intractable and harms SMC-based guidance systems. This undesirable phenomenon is caused by the discontinuous switching function in guidance law. To address this issue, the switching function is usually approximated by a sigmoid function [18–20], but the disadvantages still exist as follows. (1) The sigmoid function cannot provide finite time convergence of the sliding variable to zero in the presence of the external disturbances. (2) The sigmoid function is not ca-

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pable of driving the sliding variable to zero [21]. (3) The sigmoid function is not considered directly at the level of stability analysis. Up to now, the disturbance/uncertainty estimation technique has made essential contributions to dealing with numerous uncertainties and disturbances in guidance and control systems [22–26]. Several mostly employed strategies are studied, such as disturbance observer (DOB) [27,28], uncertainty and disturbance estimator (UDE) [29], extended state observer (ESO) [30]. In the targetinterception scene, an extended high-gain observer is employed designing the guidance law in [31]. A new form of extended state observer is considered in [32]. The disturbance observers with finite time convergence drive the estimation error to zero in a settling time [33]. For example, based on the super-twisting method, an exact robust sliding mode differentiator is firstly proposed in [34], which provides accurate estimation of bounded input signal and its arbitrary-order derivatives. This idea is further researched in controller design [35,36]. This differentiator is also considered in estimating unknown disturbances in linear/nonlinear dynamics with available input states [36]. The main issues in designing a three-dimensional guidance law have to be addressed as follows. (1) The finite time convergence of line-of-sight (LOS) angle rates; (2) The interception of highly maneuvering targets; (3) The nonlinear cross couplings in missile– target engagement. In order to overcome the problem of the cross couplings, it is usually assumed that the LOS angles and the LOS angle rates are small enough so that they can be completely neglected [37]. However, under this approach the performance of the interceptor may is degraded [38]. Mainly contributions of this paper are summarized as follows:

31 32 33 34 35 36 37 38 39 40 41

1) A new form of continuous adaptive function is introduced to replace the switching function. As a result, chattering of sliding mode is removed and robustness is guaranteed. 2) A new continuous adaptive finite time guidance method is proposed to overcome the above mentioned issues in threedimensional engagement of intercepting highly maneuvering targets. 3) Nonlinear disturbance observer with finite time convergence is utilized to estimate the acceleration of target and compensate the guidance law.

42 43 44 45 46 47 48 49 50

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81

Fig. 1. Geometry in a three-dimensional space.

82 83

r¨ = r q˙ 2 + r θ˙ 2 cos2 q + atx − amx r q¨ = −2r˙q˙ − r θ˙ 2 cos q sin q + at y − amy

84

(1)

−r θ¨ cos q = −2r q˙ θ˙ sin q + 2r˙θ˙ cos q + atz − amz

88

where r (m), r˙ (m/s) are the target-to-missile relative range and closing velocity along LOS. q, θ (rad) are elevation andazimuth angles respectively. atx , at y , atz (m/s2 ) denote target accelerations along three axes in LOS coordinate system and amx , amy , amz (m/s2 ) denote missile accelerations. Usually, the acceleration along LOS cannot be controlled in the terminal phase because the thrustor is shut off, which indicates that amx = 0. Meanwhile, according to assumption 3), the value of target acceleration along LOS is assumed to be zero, which indicates atx = 0. For better illustration, r q˙ , r θ˙ cos q are chosen as the states and missile–target relative dynamics (1) can be rearranged as follows:

55 56 57

x˙ = f (x) − u + w

(2)

where states are x = [x1 , x2 ] = [r q˙ , r θ˙ cos q] T , controls are u = [u 1 , u 2 ] T = [amy , −amz ] T , and disturbances are w = [ w 1 , w 2 ] T = [at y , −atz ] T , respectively. f (x) = [ f 1 (x), f 2 (x)] T = [−˙r q˙ − r θ˙ 2 × sin q cos q, r q˙ θ˙ sin q − r˙ θ˙ cos q] T . T

The remaining part of this paper is organized as follows. In section 2, a three-dimensional missile–target engagement is formulated. A new continuous adaptive guidance law is proposed and the finite time convergence is proven strictly in section 3. In section 4, a nonlinear disturbance observer is applied in target acceleration estimation, CAFTG law is presented and the advantages are analyzed. Numerical simulations are carried out to demonstrate the effectiveness of proposed CAFTG method in section 5.

Remark 1. It is clear that, in the three-dimensional missile–target dynamics (2), the cross couplings exist between the pitch and yaw planes which degrade the performance of missile guidance system. In conclusion, the control interest here is to design a guidance law for system (2) in such a way that the missile can capture a maneuvering target in three-dimensional space in the presence of external disturbances.

2. Missile-target geometric engagement

3. Continuous guidance law design

60 61 62 63

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91 92 93 94 95 96 97 98 99 100 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 118 119

In this section the three-dimensional engagement of a missile intercepting a target is constructed in LOS coordinate, which is described by relative positions (r , q, θ) dynamics (see Fig. 1). Some assumptions are given for the sake of simplicity as follows: 1) Both the missile and target are regarded as point mass models. 2) The effects caused by earth turning are ignored. 3) Missile and target velocities υ M , υ T are constant and satisfy υM ≥ υT . 4) Target acceleration a T and its second derivative are bounded.

64 65

90

117

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89

101

53 54

86 87

51 52

85

Then, the corresponding equations describing the missile–target relative motion dynamics are formulated as follows:

In this section a new nonlinear guidance law with finite time convergence is developed to intercept a non-maneuvering target in three-dimensional space. Stability of guidance system is proven via Lyapunov stability theory and the dynamic characteristics are also analyzed. Following the idea of parallel navigation, LOS angle rates q˙ , θ˙ are desired to be zero. The sliding mode surfaces are chosen as

s1 = q˙ s2 = θ˙ cos q Continuous adaptive reaching laws are selected as

120 121 122 123 124 125 126 127 128

(3)

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s˙ 1 = − N 1 s1 −

14

17 18 19 20 21 22 23 24 25 26 27 28 29 30

(4)

δ˙1 (t ) = − δ˙2 (t ) = −

kε1 δ1 |˙q| (5)

p ε2 δ2 |θ˙ cos q|

|θ˙ cos q| + δ22 e −α2 t

33 34 35 36 37 38 39

42 43 44 45

(6)

s˙ 2 = θ˙ θ¨ cos2 q − θ˙ 2 q sin q cos q It is obtained from Eqs. (2), (4), (6) that

q˙

u 1 = − N 1 r˙q˙ + k|˙q|γ

|˙q| + δ12 e −α1 t θ˙ cos q u 2 = − N 2 r˙ θ˙ + p |θ˙ cos q|γ |θ˙ cos q| + δ 2 e −α2 t

48 49 50 51 52 53 54 55

58 59 60 61

2

The continuous nonlinear guidance laws against a nonmaneuvering target are derived as Eq. (7). Remark 2. In Eq. (4), the term

q˙ |˙q|+δ12 e −α1 t

replaces switching func-

tion and removes the chattering of the sliding variables. Besides, the sliding variables s1 , s1 are able to reach s1 = 0, s1 = 0 in finite time under designed reaching laws. Before conducting the stability analysis of designed guidance law, the following lemma is introduced, which will be used in the subsequent analysis.

66

r

Lemma 1. [39] A general nonlinear system is considered as

x˙ = f (x), f (0) = 0, x ∈ R

n

(8)

Suppose that there exists a continuous function V (x) : U ⊂ R such that the following conditions hold: 1) V (x) is positive definite. 2) There exist real numbers a > 0, b > 0, γ ∈ (0, 1) and an open neighborhood U0 ⊂ U of the origin such that

V˙ (x) + aV + bV γ (x) ≤ 0, x ∈ U {0}

(9)

If U = U0 = R , the origin is a globally finite time stable equilibrium with n

setting time t e ≤

1 a(1−γ )

ln

aV (0)1−γ +b . b

Theorem 1. For nonlinear dynamics described by Eq. (2), a continuous nonlinear guidance law against non-maneuvering targets is designed as Eq. (7). If the parameters satisfy N 1 ≥ 2, N 2 ≥ 2, k > 0, p > 0, 0 < γ < 1 and δ1 (t 0 ) > 0, δ2 (t 0 ) > 0, it is established in finite time that q˙ = 0, θ˙ = 0. Proof. A Lyapunov function candidate is chosen as

V =

1 2

q˙ 2 +

1 2

(θ˙ cos q)2

69

˙ ˙2

(10)

70

2

r θ cos q

71

r

72

|˙q|γ +2 atz − r |˙q| + δ12 e −αt k

73 74 75

|θ˙ cos q|γ +2 r |θ˙ cos q| + δ22 e −αt − q˙ θ˙2 sin q cos q + 2q˙ θ˙2 sin q cos q − q˙ θ˙2 sin q cos q r˙q˙ 2 r˙ θ˙ 2 cos2 q = ( N 1 − 2) + ( N 2 / cos q − 2) p

θ˙ cos q r

76 77

(11)

80 81

r

atz −

k

78 79

82

|˙q|γ +2

83

r |˙q| + δ12 e −αt

84

p |θ˙ cos q|γ +2 − r |θ˙ cos q| + δ22 e −αt

85 86 87

Because at y = 0, atz = 0, it works out by Eq. (11) that

88

V˙ = q˙ q¨ + θ˙ θ¨ cos2 q − θ˙ 2 q sin q cos q

90

r˙q˙ 2 r

+ ( N 2 / cos q − 2)

89

r˙ θ˙ 2 cos2 q

91

r

93

92

|˙q|γ +2 p |θ˙ cos q|γ +2 − − r |˙q| + δ12 e −αt r |θ˙ cos q| + δ22 e −αt

94

k

95

(12)

1

≤ − M ( q˙ + (θ˙ cos q)2 ) − N (|˙q|γ +2 + |θ˙ cos q|γ +2 ) 2

2 1

2 1

2

2

≤ − M ( q˙ 2 + (θ˙ cos q)2 ) − N (|˙q|2 + |θ˙ cos q|2 ) ≤ − MV − NV

97 98 100 101



,

96

99

2

2



k/r |˙q|+δ12 e −αt

γ +2

γ +2

where 0 < M ≤ min 2( N 1 − 2) N ≤ min

64 65

r

1

62 63

θ˙ cos q

= ( N 1 − 2)

56 57

+ ( N 2 / cos q − 2)

q˙

(7)

46 47

r

q˙ + at y +

40 41

rq

r

s˙ 1 = q˙

68

r

where ε1 > 0, ε2 > 0. Taking the first derivatives of Eq. (3) yields

31 32

˙ ˙2

+ at y + −

|˙q| + δ12 e −α1 t

67

V˙ = q˙ q¨ + θ˙ θ¨ cos2 q − θ˙ 2 q sin q cos q

= ( N 1 − 2)

where N 1 > 0, N 2 > 0, δ1 (0) > 0, δ2 (0) > 0, k > 0, p > 0, α1 > 0, α2 > 0, 0 < γ < 1. Adaptive terms δ1 (t ), δ2 (t ) are chosen to satisfy the equations as

15 16

Taking its first derivative with respect to time yields

|s1 | + δ12 e −α1 t p | s 2 |γ s 2 s˙ 2 = − N 2 s2 − |s2 | + δ22 e −α2 t

12 13

k | s 1 |γ s 1

3

p /r

|˙q cos θ|+δ22 e −αt

|˙r | r



, 2( N 2 / cos q − 2) |˙rr |

102



103

and 0 <

.

According to Lemma 1, one can conclude that q˙ , θ˙ converge to zero in a finite time. 2 Remark 3. In Eq. (11), the cross coupling terms are eliminated by choosing a Lyapunov function. Consequently, the finite time convergence of guidance system is guaranteed.

104 105 106 107 108 109 110 111 112 113 114

4. Continuous guidance law based on nonlinear disturbance observer

115 116 117

In this section, the main objective is to design a guidance law so that a missile can intercept highly maneuvering targets. To this end, a nonlinear disturbance observer is employed to estimate the acceleration of target firstly. Then the estimate is used to compensate the proposed guidance law (7).

118 119 120 121 122 123

4.1. Nonlinear disturbance observer application

124 125

The external disturbances in missile guidance system bring a lot of troubles to the missile guidance and control system. Inspired from [33,35,36], a nonlinear disturbance observer convergence property is introduced. One of the advantages of utilizing the nonlinear disturbance observer is its capability of achieving finite time convergence and providing an adequate way to estimate the acceleration of the target.

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Lemma 2. [33] For the nonlinear missile–target dynamics described by (2), a three-order disturbance observer is constructed as follows:

z˙ 0i = υ0i + f (x) − u

11

υ2i = −λ2 L i sign(z2i − υ1i ) (i = 1, 2)

12 13 14 15 16

(13)

υ1i = −λ1 L 1i /2 |z1i − υ0i |1/2 sign(z1i − υ0i ) + z2i z˙ 2i = υ2i

19 20 21 22

Combining Eqs. (2) and (13), the estimation errors of ith subobserver are given as 1/3

r

+ ( N 2 / cos q − 2)

74 75

r˙ θ˙ 2 cos2 q

76

r

78

77

|˙q|γ +2 p |θ˙ cos q|γ +2 − − r |˙q| + δ12 e −αt r |θ˙ cos q| + δ22 e −αt 1

1

2 1

2 1

79 80

(19)

≤ − M ( q˙ 2 + (θ˙ cos q)2 ) − N (|˙q|γ +2 + |θ˙ cos q|γ +2 ) ≤ − M ( q˙ + (θ˙ cos q) ) − N (|˙q| + |θ˙ cos q| ) 2

2

71 73

k

2

2

2

81 82 83

γ +2

84

2

85

2

≤ − MV − NV

|e 0i |2/3 sign(e 0i ) + e 1i

1/2 −λ1 L i |e 1i

r˙q˙ 2

70 72

Substituting the second equation of (18) into (17) yields

≤ ( N 1 − 2)

where [ z01 , z02 ] T are the estimates of [x1 , x2 ] T , [ z10 , z11 ] T are the ˙ 1, w ˙ 2 ]T . estimates of [ w 1 , w 2 ] T , [ z21 , z22 ] T are the estimates of [ w λ0 > 0, λ1 > 0, λ2 > 0. L i > 0 is a group of known Lipshitz constants for second derivatives of xi (i = 1, 2).

e˙ 0i = −λ0 L i

(18)

V˙ = q˙ q¨ + θ˙ θ¨ cos2 q − θ˙ 2 q sin q cos q

17 18

69

a˙ t y = z21 , a˙ tz = z22

z˙ 1i = υ1i

67 68

x1 = z01 , x2 = z02 at y = z11 , atz = z12

υ0i = −λ0 L 1i /3 |z0i − xi |2/3 sign(z0i − xi ) + z1i

10

According to Lemma 2, following equations are established.

86

γ +2 2

87



|˙r | 

88

23

e˙ 1i =

24

¨ i (t ) e˙ 2i = −λ2 L i sign(e 2i − e˙ 1i ) − w

min

where the estimation errors are defined as e 0i = z0i − xi , e 1i = ˙ i respectively. It is proved in [33,40] that z1i − w i , e 2i = z2i − w the estimation errors converge to zero in a finite time.

Lemma 1, the origin q˙ = 0, θ˙ = 0 is globally finite-time stable. 2

91

Remark 6. The terms − N 1 r˙q˙ and − N 2 r˙ θ˙ can be regarded as PNGL, which show that the proposed CAFTG law is composed of the proportional term, the nonlinear compensation term and the acceleration estimation term.

93

25 26 27 28 29 30 31 32 33 34 35 36 37 38

− e˙ 0i |

1/2

sign(e 1i − e˙ 0i ) + e 2i

(14)

Remark 4. Coefficients λ0 , λ1 , λ2 are selected recursively so that only λ0 needs to be assigned. Simulations have demonstrated candidate values λ0 = 2, λ1 = 1.5, λ2 = 1.1 provide a reasonable fast convergence and high accuracy [40]. The estimation precision can be further enhanced by increasing the order number. Remark 5. The super twisting theory is employed to achieve the finite time convergence of estimations errors. More content of super twisting theory can be seen in [33,34].

where



M = min 2( N 1 − 2)

k/r |˙q|+δ12 e −αt

,



p /r

|˙q cos θ|+δ22 e −αt

|˙r |

, 2( N 2 /cos q − 2) r , N = and M > 0, N > 0. According to r

43

Theorem 2. For the missile–target engagement (2), continuous adaptive finite time guidance laws are considered as follows.

44 45 46 47 48 49 50 51

u 1 = − N 1 r˙q˙ + k|˙q|γ

q˙

+ z11

|˙q| + δ12 e −α1 t θ˙ cos q u 2 = − N 2 r˙ θ˙ + p |θ˙ cos q|γ + z12 ˙ |θ cos q| + δ 2 e −α2 t

(15)

2

where N 1 > 2, N 2 ≥ 2, k > 0, p > 0, 0 < γ < 1 and δ1 (t 0 ) > 0, δ2 (t 0 ) > 0. It is achieved that q˙ = 0, θ˙ = 0 in a finite time.

52 53

56 57 58 59 60 61

V =

1 2

q˙ 2 +

64 65 66

1 2

(θ˙ cos q)2

(16)

Taking its first derivative with respect to time yields

V˙ = q˙ q¨ + θ˙ θ¨ cos2 q − θ˙ 2 q sin q cos q

= ( N 1 − 2)

62 63

94 95 96 97 99 100 101 102 103

5. Numerical simulations

104

In this section, numerical simulations are performed to validate the proposed CAFTG law in three-dimensional interception cases. The missile and the target are modeled by relative dynamics. Two scenarios will be considered and simulation results are being computed. The robustness of guidance law against internal and external disturbances, such as flurry, measurement errors of r , q˙ , θ˙ and initial measurement errors of q˙ , θ˙ , is demonstrated by Monte Carlo method. Random disturbances considered in this paper are regarded to accord the normal distribution: r ∼ N (0, 100), ˙q ∼ N (0, 0.0001), θ˙ ∼ N (0, 0.0001), q(0) ∼ N (0, 0.0001), θ(0) ∼ N (0, 0.0001). The mathematical expectations and variances are calculated by

Proof. A Lyapunov function candidate is considered as

54 55

92

105

4.2. Observer-based guidance law design

41 42

90

98

Remark 7. Compared to the guidance law (7), the observer-based guidance law (15) takes the target maneuverability into account. Consequently, the robustness against the maneuvering target is enhanced and high precision is guaranteed.

39 40

89

+

r˙q˙ 2

θ˙ cos q r

r

+ ( N 2 / cos q − 2)

(atz − z12 ) −

|θ˙ cos q|γ +2 − r |θ˙ cos q| + δ22 e −αt p

k

r˙ θ˙ 2 cos2 q

|˙q|γ +2

r

r |˙q| + δ12 e −αt

x=

300  xi i =1

300

,

y=

300  yi i =1

300

,

z=



300

 σx =  (xi − x)2 /(300 − 1)

300  zi i =1

300

+ (at y − z11 ) r

(17)



300

 σ y =  ( y i − y )2 /(300 − 1) i =1



300

 σz =  (zi − z)2 /(300 − 1) i =1

107 108 109 110 111 112 113 114 115 116 117 118 119

(20)

120 121 122 123 124

i =1

q˙

106

125 126

(21)

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Table 1 Initial conditions for missile and target.

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68

Value

Target

Value

Guidance law

Miss distance (m)

Energy (m2 /s4 )

C E P (m)

x M (m) y M (m) z M (m) υM (m/s)

0 0 0 1000 60 0

x T (m) y T (m) z T (m) υT (m/s)

0 30000 30000 700 0 −90

CAFTG PNGL FNTSMG

0.87 1150.9 2.41

260690.9 534215.5 318203.5

0.62 – 1.85

ψ M (◦ )

9

67

Table 3 Results in scenario 1.

Missile

ϕM (◦ )

8

5

ϕT (◦ ) ψ T (◦ )

12 13 14 15

70 71 72 73 74 75 76

10 11

69

Table 2 Initial relative information. Data r (m) r˙ (m/s)

77 78

Value

Data

Value

42426.40 −1460.90

q (◦ ) q˙ (◦ /s)

45 0.32

Data

(◦ )

θ θ˙ (◦ /s)

Value

79

90 0

80 81

16

82

17

83

18

84

19

85

20

86 87

21

Fig. 3. Monte carlo shooting results.

22

88

23

89

24

90

25

91

26

92

27

93

28

94

29

95

30

96

Fig. 2. Interception plan.

97

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55



CEP =

0.615σx + 0.562σ y ,

σx ≤ σ y

0.615σ y + 0.562σx ,

σx > σ y

60 61 62 63 64 65 66

100 101 102

Fig. 4. Elevation angle rate.

103

The classical performance indices miss distance, intercepting time and total control energy are computed for comparison. Miss distance represents the minimum distance between missile and target. Intercepting time represents the time achieving miss disfrom t = 0 to impact time tance. Total control energy

t needed 2 2 t e is calculated by E = 0 e (amy + amz )dt. As a comparison, PNGL and FNTSMG law are also discussed. PNGL is a classical guidance method to intercept non-maneuvering targets, FNTSMG law in [9] has been verified effective to intercept maneuvering targets. The guidance command of PNGL is represented as

104

u 1 = − N 1 r˙q˙

115

u 2 = − N 2 r˙ θ˙

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(23)

Simulation uses the Runge–Kutta method with a time step of 0.001 s. Guidance update rate is 100 Hz. It is assumed that the knowledge of the range r and range rate r˙ is perfect. The initial information of missile and target is given in Table 1 and Table 2. 5.1. Scenario 1

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The circle of equal probability (CEP) is utilized to analyze Monte Carlo shooting practice and it is represented as

In this scenario, target accelerations normal to LOS are chosen as constants at y = 60 m/s2 , atz = 60 m/s2 . Although the parameters in PNGL is usually selected as N i = 3 ∼ 5 (i = 1, 2) to acquire optimal intercepting property against non-maneuvering targets. Parameters in CAFTG law are selected as N 1 = 5, N 2 = 5, k = 10, p = 10, γ = 0.3, λ0 = 2, λ1 = 1.5, λ2 = 1.1, L 1 = 1, L 2 = 1. As is shown in Fig. 2, CAFTG law and FNTSMG law can guarantee the successful interception while PNGL can not (see also

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Fig. 5. Azimuth angle rate.

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Fig. 3). Besides, CAFTG law leads to a smaller miss distance than FNTSMG law, which is clearly seen in Table 3. From Figs. 4, 5 it is obviously that both CAFTG law and FNTSMG law can drive q˙ , θ˙ to zero in finite time, but PNGL is not able to achieve the convergence of elevation and azimuth angle rates. In Figs. 6, 7 the accelerations of the missile produced by the proposed CAFTG law are smaller than those under FNTSMG law and PNGL, the conclusion is more clearly made by total energies in Table 3. Additionally, it is shown in Table 3 that the C E P of CAFTG law is smaller than that of FNTSMG law, which indicates that CAFTG law exhibits better robustness against random disturbances than FNTSMG law. It is known from Figs. 8, 9 that disturbance estimation errors under

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Fig. 7. Missile acceleration along Z axis.

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Table 4 Results in scenario 2.

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Guidance law

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nonlinear disturbance observer converge to zero in less than 2 seconds.

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5.2. Scenario 2

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Fig. 8. Target acceleration estimation error along Y axis.

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In this scenario, accelerations of target are described as at y = 60 cos(0.1π t ) m/s2 atz = 60 cos(0.1π t ) m/s2 . The main purpose discussing this scenario is to further verify the performance of the proposed CAFTG law. The capability of the nonlinear disturbance observer is challenged in this case because target maneuvering is more complicated. Values of parameters are selected as N 1 = 5, N 2 = 5, k = 10, p = 10, γ = 0.3, λ0 = 2, λ1 = 1.5, λ2 = 1.1, L 1 = 65, L 2 = 65. It is shown from Fig. 10 that three guidance strategies all achieve successful interception (see also Fig. 11). From Table 4, CAFTG law leads to the smallest miss distance (0.83 m). It is shown from Figs. 12, 13 that elevation angle rate and azimuth angle rate are convergent under CAFTG law and FNTSMG law (see also Figs. 14, 15). In addition, it is shown in Table 4 that the C E P of CAFTG law is the smallest one among three strategies, indicating that CAFTG method provides robustness against random external disturbances. Figs. 16 and 17 show that the estimation accuracy of nonlinear disturbance observer is satisfactory in the case of intercepting the target of high maneuverability.

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Fig. 9. Target acceleration estimation error along Z axis.

Novel continuous adaptive finite time guidance laws are investigated and the finite time convergences are proved via Lyapunov

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Fig. 12. Elevation angle rate.

Fig. 16. Target acceleration estimation error along Y axis.

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Fig. 17. Target acceleration estimation error along Z axis.

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None declared.

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Acknowledgement

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This work is supported by the National Natural Science Foundation of China under Grant 6170339.

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Fig. 14. Missile acceleration along Y axis.

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theory in three-dimensional space. A continuous adaptive guidance law against non-maneuvering target is proposed by using SMC method, while an improved continuous adaptive guidance law is presented against maneuvering target by introducing the nonlinear disturbance observer. It has been demonstrated by simulations that CAFTG law achieves a high accuracy and robustness against highly maneuvering target. Conflict of interest statement

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Fig. 15. Missile acceleration along Z axis.

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