Integrated guidance and control with L2 disturbance attenuation for hypersonic vehicles

Available online at www.sciencedirect.com

ScienceDirect Advances in Space Research 57 (2016) 2519–2528 www.elsevier.com/locate/asr

Integrated guidance and control with L2 disturbance attenuation for hypersonic vehicles Tun Zhao ⇑, Peng Wang, Luhua Liu, Jie Wu 1303 Lab, College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China Received 7 November 2015; received in revised form 3 March 2016; accepted 28 March 2016 Available online 9 April 2016

Abstract A robust integrated guidance and control (IGC) approach with L2 gain performance is addressed for a hypersonic vehicle that operates in the dive phase and attacks a fixed target with a terminal angular constraint. A full-state hypersonic vehicle model that adopts the bank-to-turn technique is developed by combining relative motion equations, expressed in the line-of-sight coordinate system, between the vehicle and the target with rotational motion equations. For the proposed model in a strict feedback system, a novel IGC law with L2 gain performance is developed based on the backstepping design procedure by recursively constructing Lyapunov functions of the model subsystems. Numerical simulations conducted for a six degrees of freedom model of the general hypersonic vehicle show that the proposed IGC law is robust against existing uncertainties and satisfies performance requirements. Ó 2016 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: Integrated guidance and control; Hypersonic vehicles; L2 gain performance; Backstepping design

1. Introduction The integrated guidance and control (IGC) approach for a general hypersonic vehicle (GHV) operating in the dive phase considers the translational and rotational dynamics in entirety so that the performance of the vehicle can be fully exploited. The rudder commands, which control the vehicle, can be obtained directly from the relative dynamics between the vehicle and the target with respect to the given constraints and miss distance (Williams et al., 1983). In general, GHVs fly in a complex environment; therefore, hypersonic aerodynamic parameters predicted by ground tests or theoretical computational methods do not reflect the actual flight parameters, and there are signif⇑ Corresponding author.

E-mail addresses: [email protected] (T. Zhao), wonderful2020@ 163.com (P. Wang), [email protected] (L. Liu), [email protected] (J. Wu). http://dx.doi.org/10.1016/j.asr.2016.03.042 0273-1177/Ó 2016 COSPAR. Published by Elsevier Ltd. All rights reserved.

icant uncertainties in the parameter values required for control system design (Coleman and Faruqi, 2009). Consequently, the IGC design approach should be sufficiently robust to accomplish a stable sustained flight. The performance of the guidance and control system of the GHV can be improved by attacking the vulnerable area of a target. Thus, the control object of the GHV should satisfy the very small miss distance as well as the desired terminal angle (Li et al., 2015). Numerous control methods have been adopted to synthesize the IGC design problem. Menon and Ohlmeyer (1999) and Palumbo and Jackson (1999) used the state dependent Riccati equation (SDRE) technique to formulate nonlinear optimal or suboptimal IGC laws for a homing missile. Vaddi and Menon (2007) developed a numerical SDRE based IGC formulation for an internally actuated missile. Moreover, Xin et al. (2006) employed a suboptimal control method, h
D method, to obtain an approximate closed-form solution to the IGC problem. However, to solve the Hamilton–Jaccobi–Bellman (HJB)

2520

T. Zhao et al. / Advances in Space Research 57 (2016) 2519–2528

equation online, these two methods need massive numerical computations. Moreover, these methods cannot ensure robustness of the closed-loop system. The sliding mode control (SMC) method is most commonly employed to design the IGC system. Shtessel and Shkolnikov (2003) designed an IGC law with the secondorder SMC and backstepping method. They designed an integrated two loop guidance and control system in the combined state space of engagement kinematics and vehicle dynamics. Idan et al. (2007) defined the zero-effort miss distance as a sliding surface to derive a sliding mode controller for an integrated missile autopilot and guidance loop. Hou and Duan (2008) developed a scheme for integrated guidance and an autopilot design for homing missiles released against fixed ground targets in the pitch plane using a SMC-based approach. Then, Hou et al. (2013) proposed an IGC-based design method using an adaptive block dynamic surface control-based approach, which can avoid the ‘‘explosion of complexity” problem compared with the backstepping method, in threedimensional space for homing missiles. Guo and Zhou (2010) used the H1 theory to synthesize an IGC law for homing missiles in the pitch plane. In this paper, we propose a robust integrated guidance and control method with L2 gain performance for a GHV that adopts the bank-to-turn (BTT) technique and operates in the dive phase to attack a fixed target with a terminal angular constraint. This paper is organized as follows. First, the rotational kinematic equations are transformed using a diffeomorphism to obtain an IGCbased model suitable for BTT technology with a terminal angular constraint in Section 2. Then, in Section 3, using the backstepping method, we develop a robust IGC law for the IGC based model in a strict feedback system; this law ensures that the states of the closed-loop system are uniformly ultimate bounded and achieve disturbance attenuation with L2 gain performance. Finally, six degrees of freedom (6DOF) numerical simulations in Section 4 show that the IGC method can satisfy the accuracy of the miss distance and the constraints with respect to the uncertainties in the IGC model. 2. Model serivation 2.1. Aerodynamic model of the vehicle A GHV (Keshmiri and Colgren, 2007) model is used in this paper. Therefore, the fitted aerodynamic forces and moments, drag D, lift L, side force N, rolling moment Mx, yawing moment My, and pitching moment Mz, can be modeled as D ¼ qSC D L ¼ qSC L N ¼ qSC N M x ¼ qS
bC M x M y ¼ qS
bC M y M z ¼ qS
lC M z ;

ð1Þ

where q ¼ 0:5qV 2 is the dynamic pressure; q is the density of air; V is the velocity of the vehicle; S is the reference

area; CD, CL, and CN are the total drag, total lift, and total side force coefficients, respectively;
b is the lateral–directional reference length;
l is the longitudinal reference length; and C M x , C M y , and C M z are the total rolling, yawing, and pitching moment coefficients, respectively. In the following derivation, we assume that the lift, L, primarily depends on the angle of attack, a, and the damping moments can be considered secondary factors. Therefore, L, Mx, My, and Mz can be simplified to L ¼ qSC aL a þ DL
 d M x ¼ qS
b C M x0 þ C dMx x dx þ C My x dy þ C dMz x dz þ DM x
 d M y ¼ qS
b C M y0 þ C dMx y dx þ C My y dy þ C dMz y dz þ DM y
 d M z ¼ qS
l C M z0 þ C dMx z dx þ C My z dy þ C dMz z dz þ DM z

ð2Þ

where C aL is the lift increment coefficient for the angle of attack; C M x0 is the rolling moment increment coefficient for a basic vehicle; dx, dy, and dz are the rolling, yawing, d and pitching deflection angles, respectively; C dMx x , C My x , and C dMz x are the rolling moment increment coefficients for the rolling, yawing, and pitching deflection angles, respectively; C M y0 is the yawing moment increment coefficient for a basic d

vehicle; C dMx y , C My y , and C dMz y are the yawing moment increment coefficients for the rolling, yawing, and pitching deflection angles, respectively; C M z0 is the pitching moment d increment coefficient for a basic vehicle; C dMx z , C My z , and C dMz z are the pitching moment increment coefficients for the rolling, yawing, and pitching deflection angles, respectively; and DL , DM x , DM y , and DM z are uncertain terms. 2.2. Rotational motion equations Assuming that the Earth is non-rotating and spherical and the products of inertia I xy ¼ I yz ¼ I xz ¼ 0, the rotational motion equations of the vehicle can be described as follows: x_ 02 ¼ f 02 ðx02 Þ þ g 02 ðx02 Þx3 þ D02 ;

ð3Þ

x_ 3 ¼ f 3 ðx3 Þ þ g 3 ðtÞu þ D3 ;

ð4Þ

where D02 and D3 are uncertain vectors. D02 is caused by the side force N. D3 is caused by the biases between the real and computational moments. T T T x02 ¼ ½ a cV b  ; x3 ¼ ½ xx xy xy  ; u ¼ ½ dx dy dz  ;

2 f 02 ðx02 Þ ¼

1 mV


sec bðmgHz sin cV þ mgHy coscV þ LÞ

3

6 tan b cos c mg þ ðtan h þ tan b sin c Þmg 7 V Hy V Hz 7 6 6 7; 4 5 þðtan h sin cV þ tan bÞL mgHz cos cV
mgHy sin cV

T. Zhao et al. / Advances in Space Research 57 (2016) 2519–2528

2


cos a tan b

sin a tan b

6 g 02 ðx02 Þ ¼ 4 cos a sec b
sin a sec b sin a cos a 2 ðI y
I z Þxy xz þqS
bCM 3

1

3

The equations for the relative motion between the vehicle and the target (Yan, 2013) are 8 €r ¼ rk_ 2D þ rk_ 2T cos2 kD þ ar > > < €kD ¼
2_rk_ D
k_ 2 sin kD cos kD þ akD ; ð5Þ T r r > > akT :€
2_rk_ T _ _ kT ¼ þ 2kD kT tan kD


7 0 5; 0

x0

Ix 6 7 6 ðI z
I x Þxx xz þqS
bCM y0 7 f 3 ðx3 Þ ¼ 6 7; Iy 4 5 ðI x
I y Þxx xy þqS
lC M z 0 Iz

2


bC dMx

x

6 Ix 6
dx 6 bC g 3 ðtÞ ¼ qS 6 M y 6 Iy 4 dx
lC M Iz

z

dy
bC M

x


bC dMz

y


bC dMz

z


lC M

Ix

Iy
lC dy M Iz

r cos kD

r

where kD and kT are the elevation and azimuth angles of the LOS; k_ D and k_ T are the LOS angle rates; r is the distance between the vehicle and the target, and T as ¼ ½ ar akD akT  is the acceleration vector of the vehicle expressed in the LOS coordinate frame. We suppose that the vehicle attacks the target when k_ D and k_ T are driven to zero. Therefore, we only choose the LOS equations of motion in Eq. (5) that represent relative motion. They are as follows: ( €kD ¼
2_rk_ D
k_ 2 sin kD cos kD þ akD T r r : ð6Þ €kT ¼
2_rk_ T þ 2k_ D k_ T tan kD
akT

3 x

7 7 7 y 7; Iy 7 5 dz Ix

dy
bC M

2521

z

Iz

where a is the angle of attack; cV is the bank angle; b is the sideslip angle; h is the flight path angle; xx, xy, and xz are the rolling, yawing, and pitching rates, respectively; m is the vehicle’s mass; Ix, Iy, and Iz are the main inertia of the three axes; gHx, gHy, and gHz are the components of gravity in the ballistic coordinate frame.

2.4. Control-oriented IGC model

2.3. Relative motion equations between the vehicle and the target

The components, akD and akT , of the acceleration vector, as, are  akD ¼ S H2;1 aV þ S H2;2 ah þ S H2;3 ar ; ð7Þ akT ¼ S H3;1 aV þ S H3;2 ah þ S H3;3 ar

As shown in Fig. 1, we calculate the line of sight (LOS) in reference to the East–North–Up (ENU) coordinate system, which is formed by a plane tangential to the Earth’s surface and is attached to the target. The East axis is labeled X, the North axis is labeled Y, and the up axis is labeled Z. Then, we define the line-of-sight coordinate system so that its origin is on the target: the Sx-axis points to the vehicle, the Sy-axis is in the horizontal plane of the target, and the Sz-axis is determined using the right-hand rule. gD is the angle between the direction of velocity and the LOS, and cD is the longitudinal velocity azimuth in the dive plane.

where S Hi;j i; j ¼ 1; 2; 3 are the elements of the transformation between the ballistic and LOS coordinate frames, and T aH ¼ ½ aV ah ar  is the acceleration vector of the vehicle expressed in the ballistic coordinate frame: 8 D > < aV ¼ gHx
m ah ¼ gHy þ m1 ðL cos cV
N sin cV Þ : ð8Þ > : 1 ar ¼ gHz þ m ðL sin cV þ N cos cV Þ

Z(Up)

λD

γD

sz

sx

Vehicle

ηD

r cos kD

r

Y(North)

v

O Target

r

Line-of-sight coordinates

λT

O Target

ENU coordinates

sy

X(East)

Fig. 1. East–North–Up coordinate system and the line of sight coordinate system.

2522

T. Zhao et al. / Advances in Space Research 57 (2016) 2519–2528

Because the GHV adopts the BTT control strategy during the dive phase, the side force, N, is so small that it can be considered an uncertain term. Moreover, we consider the major part of the lift, L, to be due to the angle of attack term. Thus, Eq. (8) can be simplified as 8 D > < aV ¼ gHx
m ah ¼ gHy þ m1 qSC aL a cos cV þ Dh ; ð9Þ > : ar ¼ gHz þ m1 qSC aL a sin cV þ Dr where Dh and Dr are uncertain terms. From Eqs. (6), (7), and (9), we obtain the controloriented LOS motion equations as follows: 8 _ € kD ¼
2_rrkD
k_ 2T sinkD coskD þ 1r ðS H 2;1 aV þS H2;2 gHy þS H 2;3 gHz Þ > > > > > > > qSC a > < þ L ðS H2;2 acosc þS H asinc ÞþD€ mr

V

2;3

V

kD

_ > 1 € > ðS H3;1 aV þS H3;2 gHy þS H3;3 gHz Þ kT ¼
2_rrkT þ2k_ D k_ T tankD
rcosk > > D > > > > a : qSC
mrcoskL D ðS H3;2 acoscV þS H 3;3 asincV ÞþD€kT

;

ð10Þ

where D€kD and D€kT are uncertain terms that include Dh and Dr . To establish relative motion equations with a constrained terminal impact angle, we restrict the local velocity elevation angle cDF. As shown in Fig. 1, the relative motion between the vehicle and the fixed target can be described as  r_ ¼
v cos gD : ð11Þ rk_ D ¼ v sin gD When the vehicle arrives at the target, the following equation and inequality hold true ( rk_ D ¼ v sin gD ¼ v sinðkD þ cDF Þ ¼ 0 : ð12Þ jkD þ cDF j < p2 Thus, kD þ cDF ¼ 0: Let 8 x ¼ kD þ cDF > < F T x1 ¼ ½ x11 x12  ¼ ½ k_ D þ k F xF > :  T x2 ¼ ½ a cos cV a sin cV 

ð13Þ

where

2 
2_r r

 þ k F k_ D
k_ 2T sin kD cos kD þ 1r ðS H 2;1 aV þ S H2;2 gHy þ S H 2;3 gHz Þ

3

6 7 6 7 7 f 1 ðx1 Þ ¼ 6 6
2_rk_ T 7 4 r þ 2k_ D k_ T tan kD 5
r cos1 kD ðS H3;1 aV þ S H3;2 gHy þ S H3;3 gHz Þ " # S H 2;3 S H2;2 qSC aL g1 ðtÞ ¼ mr
S H3;2 = cos kD
S H 3;3 = cos kD  T D1 ¼ D€kD D€kT : To connect the relative motion equations (Eq. (15)) and the rotational kinematic equations (Eq. (3)), we establish a diffeomorphism, U : x02 ¼ ½ a cV

T

b  ! x2 ¼ ½ a cos cV

a sin cV

T

b ; ð16Þ

on the set X0 :¼ fða; b; cV Þja – 0g:

ð17Þ

Then, the rotational kinematic equation expressed in the form of x2 is as follows: x_ 2 ¼ f 2 ðx2 Þ þ g 2 ðx2 Þx3 þ D2 ; where

2

ð18Þ

cos cV f 02 ð1; 1Þ
a sin cV f 02 ð2; 1Þ

3

6 7 f 2 ðx2 Þ ¼ 4 sin cV f 02 ð1; 1Þ þ a cos cV f 02 ð2; 1Þ 5: f 02 ð3; 1Þ The elements of g2 ðx2 Þ are g 2 ð1; 1Þ ¼
cos aðtan b cos cV þ a sec b sin cV Þ g 2 ð1; 2Þ ¼ sin aðtan b cos cV þ a sec b sin cV Þ g 2 ð1; 3Þ ¼ cos cV g 2 ð2; 1Þ ¼ cos aða sec b cos cV
tan b sin cV Þ g 2 ð2; 2Þ ¼
sin aða sec b cos cV
tan b sin cV Þ g 2 ð2; 3Þ ¼ sin cV g 2 ð3; 1Þ ¼ sin a g 2 ð3; 2Þ ¼ cos a

k_ T  : T

ð14Þ

Remark. xF is the terminal angle error. k F is a positive gain constant and determines the weight of xF . When hðsÞ ¼ k F þ s is Hurwitz: (1) if x11 is asymptotically converging to zero, xF and k_ D are asymptotically converging to zero; (2) if x11 is bounded, xF and k_ D are bounded.

g2 ð3; 3Þ ¼ 0

and D2 is the uncertain vector whose elements are combinations of Dh and Dr . Combining Eqs. (15), (18), and (4) yields the following integrated model of the guidance and control system: 8  > < x_ 1 ¼ f 1 ðx1 Þ þ g1 ðtÞx2 þ D1 ð19Þ x_ 2 ¼ f 2 ðx2 Þ þ g2 ðx2 Þx3 þ D2 : > : x_ 3 ¼ f 3 ðx3 Þ þ g3 ðtÞu þ D3 Because the determinant of g2 ðx2 Þ is detðg 2 ðx2 ÞÞ ¼ a sec b;

ð20Þ

By combining Eqs. (14) and (10), we rewrite the relative motion equations with the constrained terminal impact angle as follows:

we find that g2 ðx2 Þ is invertible for arbitrary values of cV and the set X1

x_ 1 ¼ f 1 ðx1 Þ þ g1 ðtÞx2 þ D1 ;

X1 :¼ fða; bÞjamin 6 a 6 amax ; jbj 6 bmax g;

ð15Þ

ð21Þ

T. Zhao et al. / Advances in Space Research 57 (2016) 2519–2528

where amin, amax, and bmax are positive constants that satisfy p 0 < amin < amax < ; 2

p 0 < bmax < : 2

In accordance with Eqs. (17) and (21), we make the following assumption: Assumption 1. (a, b) is a member of the set X1 throughout the dive phase. Obviously, g1 ðtÞ and g3 ðtÞ are non-singular during the dive phase. Therefore, under Assumption 1, we can use the block backstepping method to synthesize the controller for the strict feedback system (Eq. (19)). 3. IGC law design First of all, we introduce some guidelines for the IGC law design. They are: (1) The IGC law must enable the GHV to impact the fixed target with a small miss distance and a desired terminal angle. (2) The IGC law must maintains the sideslip angle near zero throughout the diving phase. (3) The IGC law must stabilize all the states of the vehicle. (4) The IGC law must attenuate the disturbances in the vehicle dynamics and in the external environment, i.e., it is extremely robust against the uncertainties in the vehicle model and disturbances from aerodynamic coefficients and air density. (5) The circular error probable (CEP) of the results of Monte Carlo simulation experiments is less than 100 m. And the CEP (Wei and Wang, 2005) can be defined as 8 0:615rN þ 0:562rE > < CEP ¼ 1:177rN ðrN Þ > : 0:615rE þ 0:562rN

rN < rE rN ¼ r E

ð22Þ

rN > rE

where rN and rE are standard deviations of north errors and east errors respectively. For simplicity, the functions f j ðÞ and gj ðÞ are denoted by f j and g j , j ¼ 1; 2; 3, in the following derivation. Choosing z ¼ x1 as the output signals to be minimized, the system equation can be described as 8 x_ 1 ¼ f 1 þ g 1 x2 þ gw1 D > > > > > < x_ 2 ¼ f 2 þ g 2 x3 þ gw2 D : ð23Þ > > > x_ 3 ¼ f 3 þ g 3 u þ g w3 D > > : z ¼ x1

2523

D2

where D ¼ ½ D1

T

D3  , gw1 ¼

2

3 0 0 1 0 0 0 0 0 4 g w2 ¼ 0 0 0 1 0 0 0 0 5; 0 0 0 0 1 0 0 0

1 0 0 0 0 0 0 0 , 0 1 0 0 0 0 0 0

2

3 0 0 0 0 0 1 0 0 4 g w3 ¼ 0 0 0 0 0 0 1 0 5: 0 0 0 0 0 0 0 1

This section recursively designs the control law with a prescribed L2 performance for guaranteeing that all closed-loop signals are bounded. Moreover, given the parameter c > 0 for a desired level of disturbance attenuation, a robust adaptive controller with L2 gain performance is designed as follows: Z T Z T 2 2 2 kzk dt 6 c kDk dt; 8D 2 L2 ½0;T ; 8T > 0; ð24Þ 0

0

When D1 ¼ 0, D2 ¼ 0, D3 ¼ 0, the closed-loop system is asymptotically stable at the origin. When D1 – 0, D2 – 0, D3 – 0, the L2 gain from the disturbance to the output signals of the system is smaller than or equal to c. First, we introduce a lemma that will be used repeatedly in the backstepping design process. Lemma (Jiao and Guan, 2008). Consider a noise perturbed nonlinear system x_ ¼ f ðxÞ þ gw ðxÞw þ gðxÞu

ð25Þ

z ¼ hðxÞ such that three conditions are satisfied:

(1) ff ðxÞ; hðxÞg is zero state detectable. (2) g w ðxÞ satisfies the matched condition g w ðxÞ ¼ gðxÞ~ gw ðxÞ. (3) There exists a stabilization control law pðxÞ, and the Lyapunov function V ðxÞ satisfies @V 1 ½f ðxÞ þ gðxÞpðxÞ 6
hT ðxÞhðxÞ: @x 2

ð26Þ

Then, given arbitrary c > 0, the control law for the system (Eq. (24)) with the L2 performance criteria can be designed as follows: u ¼ pðxÞ þ qðxÞ;

ð27Þ

gTw ðxÞgT ðxÞ @@xV . where qðxÞ ¼
2c12 g~w ðxÞ~ Step 1: For the first subsystem of the system in Eq. (22), let s1 ¼ x1 and s2 ¼ x2
x2v , where x2v is a function to be designed. Choosing z1 ¼ s1 as the regulation output, we obtain the first closed-loop subsystem: T

s_ 1 ¼ f 1 þ g 1 s2 þ g 1 x2v þ gw1 D z1 ¼ s 1

ð28Þ

where gw1 ¼ g1 g~w1 , i.e., gw1 satisfies the matched condition for the subsystem (Eq. (27)). Considering the first partial Lyapunov function 1 V 1 ¼ sT1 s1 ; 2

ð29Þ

2524

T. Zhao et al. / Advances in Space Research 57 (2016) 2519–2528

we use Lemma 1 to obtain x2v x2v ¼ p1 þ q1 ;

ð30Þ

where p1 ¼
g
1 1 ðf 1 þ k1 s1 Þ; q1 ¼


T 1 1 T T @ V ~ ~ g ¼
2 g
1 s1 ; g g w1 w1 1 2 2c1 @s1 2c1 1

ð31Þ ð32Þ

where k1 is a symmetrical positive definite matrix to be designed, and p1 satisfies the following inequality @V 1 1 ðf 1 þ g1 p1 Þ 6
sT1 s1 : 2 @s1

ð33Þ

Thus, from Eqs. (30)–(32), we have the control law with the L2 performance criteria of the subsystem (28) as follows:  1 
1 x2v ¼
g 1 f 1 þ k1 s1 þ 2 s1 ; ð34Þ 2c1 and the L2 gain of the first closed-loop subsystem (28) is c1 . Step 2: Because x1v only includes the angle of attack and T the bank angle, we extend x2v to x2v ¼ ½ x2v 0  to match the rotational kinematic equation. Let s2 ¼ x2
x2v and s3 ¼ x3
x3v . Then, the second closed-loop subsystem is 8   > < s_ 1 ¼ f 1 þ g1 s2 þ g1 x2v þ g w1 D s_ 2 ¼ f 2
x_ 2v þ g2 s3 þ g2 x3v þ g w2 D ; > : T z2 ¼ ½ s 1 s 2 

ð35Þ

where gw2 ¼ g2 g~w2 , i.e., gw2 satisfies the matched condition for the subsystem (Eq. (34)). According to Lemma 1, we can find a feedback law p2 and a Lyapunov function V2 such that the following inequality is satisfied: h i f þ g s þ g p

1 1 1 2 1 1 @V 2 @V 2 ð36Þ 6
zT2 z2 : @s1 @s2 2 f 2
x_ 2v þ g2 p2 In addition, x3v can be designed as x3v ¼ p2 þ q2

ð37Þ

Considering the Lyapunov function 1 V 2 ¼ V 1 þ sT2 s2 ; 2 we can obtain p2 as follows:  T g s1
1 1 s2 þ f 2
x_ 2v þ 1 p2 ¼
g 2 2 0 Moreover, we derive the time derivative of V2 as

ð38Þ

ð39Þ

@V 2 ðf 2
x_ 2v þ g 2 s3 þ g 2 x3v þ gw2 DÞ V_ 2 ¼ V_ 1 þ @s2 1 @V 1 6 ðc21 kDk2
kz1 k2 Þ þ g s 2 @s1 1 2 @V 2 þ ðf 2
x_ 2v þ g2 p2 Þ @s2 @V 2 @V 2 þ g ðq þ g~w2 DÞ þ g s3 @s2 2 2 @s2 2 1 1 2 2 2 ¼ ðc21 kDk
kz1 k Þ
ks2 k 2 2 @V 2 @V 2 þ g ðq þ g~w2 DÞ þ g s3 @s2 2 2 @s2 2 1 1 2 2 ¼ ðc21 kDk
kz2 k Þ þ 2 sT2 g2 g~w2 g~Tw2 gT2 s2 2 2c2 1 T c2 2
2 ks2 g2 g~w2
c22 Dk þ 2 kDk þ sT2 g2 q2 þ sT2 g2 s3 2c2 2 1 2 2 6 ½ðc21 þ c22 ÞkDk
kz2 k  2  1 T T ~ ~ þ sT2 g2 g s þ q g g þ sT2 g2 s3 w2 2 2 w2 2 2c22 ð40Þ Thus, with q2 ¼


1 1 s2 g~ g~T gT s ¼
2 g
1 2 w2 w2 2 2 2c2 2c2 2

ð41Þ

and s3 ¼ 0, the L2 gain of the second closed-loop subsyspffiffiffiffiffiffiffiffiffiffiffiffiffiffi tem (35) is c21 þ c22 . From Eqs. (37), (39), and (41), we have T 

g 1 s1 1 1
1 þ x3v ¼
g2 f 2
x_ 2v þ þ s2 ð42Þ 2 2c22 0 Step 3: The derivative of s3 is s_ 3 ¼ f 3
x_ 3v þ g3 u þ g w3 D: Then, we get the following closed-loop system 8 s_ 1 ¼ f 1 þ g 1 s2 þ g 1 x2v þ gw1 D > > > < s_ ¼ f
x_ þ g s þ g x þ g D 2 2 2v 2 3 2 3v w2 > _ _ s ¼ f
x þ g u þ g D 3 3 3v > 3 w3 > : z3 ¼ ½ s1 s2 s3 T

ð43Þ

ð44Þ

where gw3 ¼ g3 g~w3 , i.e., gw3 satisfies the matched condition for the system (Eq. (43)). Likewise, we can design a feedback law p3 and a Lyapunov function V3 such that the following inequality is satisfied with Lemma 1. 2 3 f 1 þ g1 s2 þ g1 p1 h i 1 7 @V 3 @V 3 @V 3 6 _ 2v þ g2 s3 þ g2 p2 5 6
zT3 z3 @s1 @s2 @s3 4 f 2
x 2 f 3
x_ 3v þ g3 p3 ð45Þ Eventually, the control law of the overall system (Eq. (22)) can be obtained as

T. Zhao et al. / Advances in Space Research 57 (2016) 2519–2528

u ¼ p3 þ q3

ð46Þ

Choosing the Lyapunov function 1 V 3 ¼ V 1 þ V 2 þ sT3 s3 ; 2

ð47Þ

we can get  1
1 T p3 ¼
g3 f 3
x_ 3v þ g 2 s2 þ s3 2

ð48Þ

Then, the time derivative of V3 is @V 3 V_ 3 ¼ V_ 2 þ ðf 3
x_ 3v þ g 3 u þ g w3 DÞ @s3 1 2 2 6 ½ðc21 þ c22 ÞkDk
kz2 k  þ sT2 g 2 s3 2 @V 3 @V 3 þ ðf 3
x_ 3v þ g 3 p3 Þ þ g ðq þ g~w3 DÞ @s3 @s3 3 3 1 1 @V 3 ¼ ½ðc21 þ c22 ÞkDk2
kz2 k2 
ks3 k2 þ g ðq þ g~w3 DÞ 2 2 @s3 3 3 1 1 2 2 ¼ ½ðc21 þ c22 ÞkDk
kz3 k  þ 2 sT3 g3 g~w3 g~Tw3 gT3 s3 2 2c3 1 c2
2 ksT3 g3 g~w3
c23 Dk þ 3 kDk2 þ sT3 g3 q3 2c3 2 1 2 2 6 ½ðc21 þ c22 þ c23 ÞkDk
kz3 k  2  1 T T T þ s3 g 3 g~w3 g~w3 g3 s3 þ q3 2c23 ð49Þ When choosing q3 as q3 ¼


1 1 s3 ; g~ g~T gT s ¼
2 g
1 2 w3 w3 3 3 2c3 2c3 3

ð50Þ

the L2 gain of the overall closed-loop system (Eq. (43)) is ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c21 þ c22 þ c23 . When D ¼ 0, the overall closed-loop system T is asymptotically stable at s ¼ ½ s1 s2 s3  ¼ 0. From Eqs. (46), (48), and (50), the control law with the L2 gain performance of the system (44) is 

1 1 T _ þ u ¼
g
1 f
x þ g s þ s3 : ð51Þ 3 3v 2 3 2 2 2c23 To sum up, the robust control law for disturbance attenuation of the system in Eq. (23) can be written as follows: 8 s1 ¼ x1 > >
 > > > 
1 1 > x ¼
g f þ k s þ s 1 1 1 1 2 > 2v 1 2c1 > > > > > x ¼ ½ x 0  T > 2v > 2v > < s2 ¼ x2
x2v : ð52Þ T


> s g > 1 1 > > x3v ¼
g
1 f 2
x_ 2v þ þ 12 þ 2c12 s2 > 2 > 2 > 0 > > > ¼ x
x s > 3 3 3v > h
 i > > > : u ¼
g
1 f 3
x_ 3v þ gT2 s2 þ 12 þ 2c12 s3 3 3

2525

4. Numerical simulations To verify the effectiveness of the proposed nonlinear robust integrated guidance and control law (Eq. (52)), numerical simulations are conducted using the 6DOF GHV model in reference (Keshmiri and Colgren, 2007). The initial conditions are listed in Table 1. In Table 1, k0 and /0 are the longitude and latitude of the launch point, and kT and /T are the longitude and latitude of the target. The design parameters of the IGC law (Eq. (51)) are k F ¼ 0:8, k1 ¼ diagð0:001; 0:1Þ, c1 ¼ 8, c2 ¼ 0:2, and c3 ¼ 1. The deflection angles are limited as follows: 8   > <
20 6 dx 6 20 ð53Þ
20 6 dy 6 20 : > :  
20 6 dz 6 20 In order to verify the performance of the IGC law, comparative simulation experiments are constructed under the following two cases: Case 1: The terminal angle is set to
60°. Case 2: The terminal angle is set to
70°. The simulation results using the nominal model, that is, no aerodynamic coefficients uncertainties and air density uncertainty, are shown in Figs. 2–7. In Case 1, the miss distances is 12.89 m, and the terminal angle is
60.01°. In Case 2, the miss distance is 4.70 m and the terminal angle is restricted to
69.47°. From Fig. 3, it can be seen that in both cases a can be kept in a reasonable domain which satisfies Assumption 1 and b is near zero throughout the dive phase. From the bank angle curve, we know that the vehicle adopts BTT-180 steering in these two cases. Furthermore, there are wider ranges of a and b in Case 2 than Case 1. Fig. 4 shows that the states of the vehicle are bounded in both cases. Fig. 5 shows that the deflection angles are saturated for a longer time in Case 2 than Case 1 when cV turns to
180 . Fig. 6 and Fig. 7 show that the lateral maneuver amplitude is larger in Case 2 than Case 1. In order to verify the robustness of the IGC method, one hundred times of Monte Carlo simulation experiments are conducted. The aerodynamic coefficients with uncertainties are defined as

Table 1 Initial states of the vehicle and target. Name

Value

Name

Value

V0/(m/s) h0/° r0/° xx0/(°/s) cV0/° x/m y/m z/m

3500 0 0 0 0 0 30,000 0

xy0/(°/s) xz0/(°/s) a0/° b0/° /0/° k0/° //° kT/°

0 0 5 0 0 0 1 1

2526

T. Zhao et al. / Advances in Space Research 57 (2016) 2519–2528

Fig. 2. Curves representing the height, velocity, and terminal angle.

Fig. 5. Curves representing the fin deflections.

Fig. 3. Curves representing the attack, sideslip, and bank angles. Fig. 6. Curves representing the vehicle’s position projected onto the launch coordinate system.

Fig. 4. Curves representing the rolling, yawing, and pitching rates.

~ D ð1 þ DC D Þ C L ¼ C ~ L ð1 þ DC L Þ CD ¼ C ~ N ð1 þ DC N Þ C M x ¼ C ~ M x ð1 þ DC M x Þ CN ¼ C ~ M y ð1 þ DC M y Þ C M z ¼ C ~ M z ð1 þ DC M z Þ CMy ¼ C

Fig. 7. Three-dimensional vehicle trajectory.

ð54Þ

and the air density with uncertainty are defined as ~ð1 þ DqÞ q¼q

ð55Þ

~ D, C ~ L, C ~N, C ~ Mx , C ~ My , C ~ M z and q ~ are nominal values; where C DC D , DC L , DC N , DC M x , DC M y and DC M z are uncertainties, respectively. And these uncertainties obey the normal distributions as follows:

T. Zhao et al. / Advances in Space Research 57 (2016) 2519–2528

Fig. 8. Curves representing the height, velocity, and terminal angle.

2527

Fig. 11. Curves representing the fin deflections.

Fig. 9. Curves representing the attack, sideslip, and bank angles. Fig. 12. Curves representing the vehicle’s position projected onto the launch coordinate system.

Fig. 10. Curves representing the rolling, yawing, and pitching rates. Fig. 13. Three-dimensional vehicle trajectory.

DC D  N ð0; 0:2Þ DC L  N ð0; 0:2Þ DC N  N ð0; 0:2Þ DC M x  N ð0; 0:2Þ DC M y  N ð0; 0:2Þ DC M z  N ð0; 0:2Þ Dq  N ð0; 0:5Þ

ð56Þ

where N ð0; 0:2Þ can generate a number from a normal distribution with mean 0 and standard deviation 0.2; N ð0; 0:5Þ can generate a number from a normal distribution with mean 0 and standard deviation 0.5.

2528

T. Zhao et al. / Advances in Space Research 57 (2016) 2519–2528

law reacts when faced with a target that is beyond the capability of the vehicle to reach. References

Fig. 14. Impact point deviation distributions.

When the terminal angle is set to
70°. The results of the Monte Carlo simulation experiments are shown in Figs. 8–14. The mean miss distance is 41.70 m with standard deviation 32.66 m. And the CEP is 41.29 m. Fig. 9 shows that b is maintained near zero throughout the diving phase. Fig. 10 shows that the states of the vehicle are bounded, that is, the IGC law can stabilize all the states of the vehicle when considering uncertainties in the vehicle dynamics and in the external environment. Therefore, we have reasons to believe that the proposed IGC law is robust with respect to the uncertainties, and it achieves the design objectives mentioned in Section 3.

5. Conclusions An IGC model with a terminal angular constraint is proposed for GHVs that adopt the BTT technique. We apply L2 gain theory in the backstepping design process to obtain a controller with L2 gain performance for the IGC model. The controller ensures that the states of the overall closed-loop system asymptotically converge to zero when the uncertainties do not exist, and when the uncertainties appear, the L2 gain from the uncertainties to the output signals of the system is smaller than or equal to pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c21 þ c22 þ c23 . Numerical simulations of a GHV adopting the IGC law proposed in the paper show that the design objectives are achieved completely. Future research will consider the dynamic performances of the fins and will conduct some research on how the IGC

Coleman, C, Faruqi, F, 2009. On Stability and Control of Hypersonic Vehicles. Defence Science and Technology Organisation. Guo, J.G., Zhou, J., 2010. Integrated guidance-control system design based on h-infinity control. In: 2010 International Conference on Electrical and Control Engineering. IEEE, Wuhan, China, pp. 1204– 1207. Hou, M.Z., Duan, G.R., 2008. Integrated guidance and control for homing missiles against ground fixed targets. Chin. J. Aeronaut. 21, 162–168. Hou, M.Z., Liang, X.L., Duan, G.R., 2013. Adaptive block dynamic surface control for integrated missile guidance and autopilot. Chin. J. Aeronaut. 26, 741–750. Idan, M., Shima, T., Golan, O.M., 2007. Integrated sliding mode autopilot-guidance for dual-control missiles. J. Guidance Control Dyn. 30, 1081–1089. Jiao, X., Guan, X., 2008. Nonlinear System Analysis and Design. Publishing House of Electronics Industry, Beijing. Keshmiri, S., Colgren, R., 2007. Six DoF nonlinear equations of motion for a generic hypersonic vehicle. In: AIAA Atmospheric Flight Mechanics Conference and Exhibit. Hilton Head, South Carolina, pp. 1–28. Li, G.H., Zhang, H.B., Tang, G.J., 2015. Maneuver characteristics analysis for hypersonic glide vehicles. Aerosp. Sci. Technol. 43, 321– 328. Menon, P.K., Ohlmeyer, E.J., 1999. Integrated design of agile missile guidance and control system. In: Proceedings of the 7th Mediterranean Conference on Control and Automation. Haifa, Israel, pp. 1469–1494. Palumbo, N.F., Jackson, T.D., 1999. Integrated missile guidance and control: a state dependent riccati differential equation approach. In: Proceedings of the 1999 IEEE International Conference on Control Applications. IEEE, Kohala Coast, HI, USA, pp. 243–248. Shtessel, Y.B., Shkolnikov, I.A., 2003. Integrated guidance and control of advanced interceptors using second order sliding modes. In: Proceedings of the 42nd IEEE Conference on Decision and Control, pp. 4587– 4592, Maui, Hawaii USA. Vaddi, S.S., Menon, P.K., 2007. Numerical SDRE approach for missile integrated guidance – control. In: AIAA Guidance, Navigation and Control Conference and Exhibit.. Hilton Head, South Carolina. Wei, S.H., Wang, H.M., 2005. Research on the missile operational use CEP assessment method. Flight Dyn. 23, 52–54. Williams, D.E., Richman, J., Friedland, B., 1983. Design of an integrated strapdown guidance and control system for a tactical missile. In: Proceedings of AIAA Guidance and Control Conference. AIAA, Gatlinburg, USA, pp. 57–66. Xin, M., Balakrishnan, S.N., Ohlmeyer, E.J., 2006. Integrated guidance and control of missiles with theta-D method. IEEE Trans. Control Syst. Technol. 14, 981–992. Yan, H., 2013. Research on Robust Nonlinear Integrated Guidance and Control Design. University of Science and Technology of China, Hefei, pp. 17–18.