wang fei* and liu gang**past.ijass.org/on_line/admin/files/13.(278~294)14-087.pdf · 2015-07-15 ·...

Copyright ⓒ The Korean Society for Aeronautical & Space SciencesReceived: December 29, 2014 Revised: June 13, 2015 Accepted: June 15, 2015

278 http://ijass.org pISSN: 2093-274x eISSN: 2093-2480

PaperInt’l J. of Aeronautical & Space Sci. 16(2), 278–294 (2015)DOI: http://dx.doi.org/10.5139/IJASS.2015.16.2.278

Integrated Guidance and Control Design for the Near Space Interceptor

WANG Fei* and LIU Gang**Information Engineering College, Henan University of Science and Technology, Luoyang, China

LIANG Xiao-Geng***Luoyang Optoelectro Technology Development Center, Luoyang, China

Abstract

Considering the guidance and control problem of the near space interceptor (NSI) during the terminal course, this paper

proposes a three-channel independent integrated guidance and control (IGC) scheme based on the backstepping sliding

mode and finite time disturbance observer (FTDO). Initially, the three-channel independent IGC model is constructed based

on the interceptor-target relative motion and nonlinear dynamic model of the interceptor, in which the channel coupling term

and external disturbance are regarded as the total disturbances of the corresponding channel. Then, the FTDO is introduced

to estimate the target acceleration and control system loop disturbances, and the feed-forward compensation term based

on the estimated values is employed to effectively remove the effect of disturbances in finite time. Subsequently, the IGC

algorithm based on the backstepping sliding mode is also given to obtain the virtual control moment. Furthermore, a robust

least-squares weighted control allocation (RLSWCA) algorithm is employed to distribute the previous virtual control moment

among the corresponding aerodynamic fins and reaction jets, which also takes into account the uncertainty in the control

effectiveness matrix. Finally, simulation results show that the proposed IGC method can obtain the small miss distance and

smooth interceptor trajectories.

Key words: Integrated Guidance and Control (IGC), Finite Time Disturbance Observer (FTDO), Backstepping Sliding Mode,

Robust Least-Squares weighted Control Allocation (RLSWCA)

1. Introduction

The guidance and control algorithm of the near space

interceptor (NSI) is one of the most critical technologies to

effectively deal with the threat of near space air-breathing

hypersonic vehicles. The traditional design method is usually

based on the assumption that the spectral separation between

the guidance loop and control loop is satisfied, which may be

invalid during the terminal interception phase [1]-[2]. Since

this scheme cannot strictly guarantee the stability of the

overall guidance and control system, the modifications of each

subsystem are generally required to obtain overall system

performance. By viewing the guidance and control loops as

an integrated system and taking into account the coupling

terms between these two subsystems, the integrated guidance

and control (IGC) exploits the states of the interceptor and

the interceptor-target motion to directly generate the actuator

commands, which can efficiently reduce the cost of the

required sensors and increase the guidance accuracy and

system reliability [3]-[5].

The past few years have witnessed rapidly growing interest

in the IGC design, and many approaches have been reported

[6]-[13]. It is well known that sliding mode control is an effective

technology to deal with system disturbance and parameter

uncertainty. In [6] and [7], smooth second-order sliding mode

(SSOSM) control is proposed to study the IGC design problem

of the interceptor. The SSOSM-based guidance law is first

developed to obtain the desired overload command, which can

This is an Open Access article distributed under the terms of the Creative Com-mons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduc-tion in any medium, provided the original work is properly cited.

* Associate Professor, Corresponding author: [email protected] ** Associate Professor *** Professor

(278~294)14-087.indd 278 2015-07-03 오전 5:06:20

279

WANG Fei Integrated Guidance and Control Design for the Near Space Interceptor

http://ijass.org

be transformed into the desired value of pitch angular rate in

the inner loop. Subsequently, the SOSM-based control law

is constructed to track the desired pitch angular rate in finite

time and obtain the IGC algorithm. In fact, the methods in [6]

and [7] belong to the partial IGC design, which still cannot

break away from the constraint of the traditional separation

design. A novel IGC design method based on the robust

higher-order sliding mode is also given in [8]. Based on the

principle of zeroing the line-of-sight (LOS) angular rate, the

IGC design problem can be transformed into the stabilization

of a third-order integral chain system. Furthermore, a global

finite-time stabilization control law is proposed for the above

system based on geometric homogeneity theory, and the

compensation term based on the super twisting algorithm is

given to obtain the strong robustness against the uncertainties

caused by target maneuvers and interceptor aerodynamic

parameter perturbation.

Some nonlinear optimal control methods have also been

applied to the IGC design. For example, the state-dependent

Riccati equation (SDRE) technique is introduced to solve

the Hamilton-Jacobi-Bellman (HJB) equation online in the

nonlinear optimal control for the IGC design [9], but it is

time consuming. Furthermore, the θ-D method is used to

approximately solve the HJB equation, which did not require

the online numerical computation [10]. Furthermore, the

methods in [9] and [10] cannot ensure the robustness of the

closed-loop system in the presence of system disturbances

and parameter uncertainties. In [11], a novel IGC design

approach based on the small-gain theorem and input-to-

state stability (ISS) theory is proposed for missiles steered

by both canard and tail controls without the assumption

that the angle between LOS and missile velocity is almost

constant. Theoretical analysis shows that both the LOS

angular rate and the tracking error of the attitude angle

(rate) are practically stable with respect to target maneuvers

and missile model uncertainties. However, most of the

above design methods [6]-[13] have not considered the

uncertainties and disturbances or dealt with the effect of

system disturbances at the price of sacrificing the normal

control performance.

On the other hand, disturbance observer-based control

(DOBC) provides an active approach to deal with system

disturbances and uncertainties (see e.g., [14]-[16] and the

references therein for a survey of recent development), and

has been successfully applied in many fields, such as missiles

[17]-[18] and hypersonic vehicle [19]. In order to completely

remove the effect of system disturbances in finite time, finite

time disturbance observer (FTDO) has also drawn much

attention in the past few years [20]. Due to the redundancy

of the NSI actuators (aerodynamic fins and reaction jets),

control allocation is an effective scheme to distribute the

virtual control command among the individual actuators

[21]. Therefore, a novel robust least-squares weighted control

allocation (RLSWCA) algorithm is proposed to deal with the

above control allocation problem while taking into account

the uncertainties in the control effectiveness matrix and the

uneven distribution problem of the NSI actuator inputs [22]-

[23].

Motivated by the above analysis, we will study the three-

channel independent IGC design problem for the NSI based

on the backstepping sliding mode and FTDO. By viewing the

channel coupling term and external disturbance as the total

disturbances of the corresponding channel, a three-channel

independent IGC model is given for the NSI. Then, the FTDO

is introduced to estimate the target acceleration and control

system loop disturbances, and the backstepping sliding mode

control law based on the estimated value is given to obtain

the virtual control moment, in which the first-order low-pass

filters are introduced to compute the differentiations of the

virtual control variables at each step of the backstepping

control. Furthermore, the RLSWCA algorithm is employed to

distribute the previously virtual control moment among the

corresponding actuators. Finally, simulation results show

the effectiveness of the proposed design method.

Briefly, the rest of this paper is organized as follows.

Section 2 introduces the three-channel independent IGC

model for the NSI. Section 3 presents the corresponding IGC

design method and RLSWCA algorithm. Section 4 gives the

simulation results for the NSI guidance and control system.

Finally, a brief conclusion is drawn in Section 5.

2. IGC model description

The interceptor-target motion in the three-dimensional

space can be seen in Fig. 1, and the kinematical equations

are described as follows [24]:

4

term and external disturbance as the total disturbances of the corresponding channel, a three-channel

independent IGC model is given for the NSI. Then, the FTDO is introduced to estimate the target

acceleration and control system loop disturbances, and the backstepping sliding mode control law

based on the estimated value is given to obtain the virtual control moment, in which the first-order

low-pass filters are introduced to compute the differentiations of the virtual control variables at each

step of the backstepping control. Furthermore, the RLSWCA algorithm is employed to distribute the

previously virtual control moment among the corresponding actuators. Finally, simulation results

show the effectiveness of the proposed design method.

Briefly, the rest of this paper is organized as follows. Section 2 introduces the three-channel

independent IGC model for the NSI. Section 3 presents the corresponding IGC design method and

RLSWCA algorithm. Section 4 gives the simulation results for the NSI guidance and control system.



q

q

4y y

4x

x

4z

z

'x( )O M

T

r

Fig. 1 Interceptor-target motion in three-dimensional space

The interceptor-target motion in the three-dimensional space can be seen in Fig. 1, and the

kinematical equations are described as follows [24]:

2 2 24cos tr m rr rq rq q a a (1)

242 sin cos t mrq rq rq q q a a (2)

Fig. 1. Interceptor-target motion in three-dimensional space

(278~294)14-087.indd 279 2015-07-03 오전 5:06:20

DOI: http://dx.doi.org/10.5139/IJASS.2015.16.2.278 280

Int’l J. of Aeronautical & Space Sci. 16(2), 278–294 (2015)

4














q

q

4y y

4x

x

4z

z

'x( )O M

T

r





242 sin cos t mrq rq rq q q a a (2)

(1)

4














q

q

4y y

4x

x

4z

z

'x( )O M

T

r





242 sin cos t mrq rq rq q q a a (2) (2)

5

4cos 2 cos 2 sin t mrq q rq q rq q q a a (3)

where r is the relative distance between the interceptor and target, q and q are the azimuth

angle and elevation angle in the LOS coordinate system, 4 4 4[ , , ]Tm r m ma a a and [ , , ]T

tr t ta a a are

the interceptor and target accelerations in the LOS coordinate system, respectively.

The dynamic equations of the NSI in the three independent channels (pitch, yaw and roll) can be

written as follows:

3

tan sin tan cos

z

z

yz y x

m

x yz z z zz x y

z z z z

ym

qSCd

mVJ JqSLm qSLm M d

J J J JqSC

amg

(4)

3

sin sinsin

y

y

yzy x

m m

y y y y z xy x z

y y y y

zm

qSCqSC dmV mV

qSLm qSLm M J J dJ J J J

qSCamg

(5)

cos tan sin tan

x

x y z

y zxx y z

x x

dJ JM d

J J

(6)

where is q the dynamic pressure; , and are the angle of attack, sideslip angle and roll

angle; yC , zm , zzm , zC , ym and y

ym are the corresponding aerodynamic coefficients; d ,

zd , d ,

yd , d and

xd are the external disturbances; 3ma and 3ma are the interceptor

accelerations in the velocity coordinate system, respectively; zM , yM and xM are the moments

in the pitch, yaw and roll directions, which are mainly produced by the aerodynamic fins and reaction

jet devices. It is also assumed that the lateral thrusts produced by the reaction jet devices are treated as

(3)

where r is the relative distance between the interceptor

and target, qε and qβ are the azimuth angle and elevation

angle in the LOS coordinate system,

5




tr t ta a a are



written as follows:

3

tan sin tan cos

z

z

yz y x

m

x yz z z zz x y

z z z z

ym

qSCd

mVJ JqSLm qSLm M d

J J J JqSC

amg

(4)

3

sin sinsin

y

y

yzy x

m m

y y y y z xy x z

y y y y

zm

qSCqSC dmV mV


qSCamg

(5)

cos tan sin tan

x

x y z

y zxx y z

x x

dJ JM d

J J

(6)




zd , d ,

yd , d and





and

5




tr t ta a a are



written as follows:

3

tan sin tan cos

z

z

yz y x

m

x yz z z zz x y

z z z z

ym

qSCd

mVJ JqSLm qSLm M d

J J J JqSC

amg

(4)

3

sin sinsin

y

y

yzy x

m m

y y y y z xy x z

y y y y

zm

qSCqSC dmV mV


qSCamg

(5)

cos tan sin tan

x

x y z

y zxx y z

x x

dJ JM d

J J

(6)




zd , d ,

yd , d and





are the interceptor and target accelerations in

the LOS coordinate system, respectively.

The dynamic equations of the NSI in the three independent

channels (pitch, yaw and roll) can be written as follows:

5




tr t ta a a are



written as follows:

3

tan sin tan cos

z

z

yz y x

m

x yz z z zz x y

z z z z

ym

qSCd

mVJ JqSLm qSLm M d

J J J JqSC

amg

(4)

3

sin sinsin

y

y

yzy x

m m

y y y y z xy x z

y y y y

zm

qSCqSC dmV mV


qSCamg

(5)

cos tan sin tan

x

x y z

y zxx y z

x x

dJ JM d

J J

(6)




zd , d ,

yd , d and





(4)

5




tr t ta a a are



written as follows:

3

tan sin tan cos

z

z

yz y x

m

x yz z z zz x y

z z z z

ym

qSCd

mVJ JqSLm qSLm M d

J J J JqSC

amg

(4)

3

sin sinsin

y

y

yzy x

m m

y y y y z xy x z

y y y y

zm

qSCqSC dmV mV


qSCamg

(5)

cos tan sin tan

x

x y z

y zxx y z

x x

dJ JM d

J J

(6)




zd , d ,

yd , d and





(5)

5




tr t ta a a are



written as follows:

3

tan sin tan cos

z

z

yz y x

m

x yz z z zz x y

z z z z

ym

qSCd

mVJ JqSLm qSLm M d

J J J JqSC

amg

(4)

3

sin sinsin

y

y

yzy x

m m

y y y y z xy x z

y y y y

zm

qSCqSC dmV mV


qSCamg

(5)

cos tan sin tan

x

x y z

y zxx y z

x x

dJ JM d

J J

(6)




zd , d ,

yd , d and





(6)

where q is the dynamic pressure; α, β and γ are the angle of

attack, sideslip angle and roll angle;

5




tr t ta a a are



written as follows:

3

tan sin tan cos

z

z

yz y x

m

x yz z z zz x y

z z z z

ym

qSCd

mVJ JqSLm qSLm M d

J J J JqSC

amg

(4)

3

sin sinsin

y

y

yzy x

m m

y y y y z xy x z

y y y y

zm

qSCqSC dmV mV


qSCamg

(5)

cos tan sin tan

x

x y z

y zxx y z

x x

dJ JM d

J J

(6)




zd , d ,

yd , d and





and

5




tr t ta a a are



written as follows:

3

tan sin tan cos

z

z

yz y x

m

x yz z z zz x y

z z z z

ym

qSCd

mVJ JqSLm qSLm M d

J J J JqSC

amg

(4)

3

sin sinsin

y

y

yzy x

m m

y y y y z xy x z

y y y y

zm

qSCqSC dmV mV


qSCamg

(5)

cos tan sin tan

x

x y z

y zxx y z

x x

dJ JM d

J J

(6)




zd , d ,

yd , d and





are the corresponding aerodynamic coefficients;

5




tr t ta a a are



written as follows:

3

tan sin tan cos

z

z

yz y x

m

x yz z z zz x y

z z z z

ym

qSCd

mVJ JqSLm qSLm M d

J J J JqSC

amg

(4)

3

sin sinsin

y

y

yzy x

m m

y y y y z xy x z

y y y y

zm

qSCqSC dmV mV


qSCamg

(5)

cos tan sin tan

x

x y z

y zxx y z

x x

dJ JM d

J J

(6)




zd , d ,

yd , d and





5




tr t ta a a are



written as follows:

3

tan sin tan cos

z

z

yz y x

m

x yz z z zz x y

z z z z

ym

qSCd

mVJ JqSLm qSLm M d

J J J JqSC

amg

(4)

3

sin sinsin

y

y

yzy x

m m

y y y y z xy x z

y y y y

zm

qSCqSC dmV mV


qSCamg

(5)

cos tan sin tan

x

x y z

y zxx y z

x x

dJ JM d

J J

(6)




zd , d ,

yd , d and





and

5




tr t ta a a are



written as follows:

3

tan sin tan cos

z

z

yz y x

m

x yz z z zz x y

z z z z

ym

qSCd

mVJ JqSLm qSLm M d

J J J JqSC

amg

(4)

3

sin sinsin

y

y

yzy x

m m

y y y y z xy x z

y y y y

zm

qSCqSC dmV mV


qSCamg

(5)

cos tan sin tan

x

x y z

y zxx y z

x x

dJ JM d

J J

(6)




zd , d ,

yd , d and





are the external disturbances;

5




tr t ta a a are



written as follows:

3

tan sin tan cos

z

z

yz y x

m

x yz z z zz x y

z z z z

ym

qSCd

mVJ JqSLm qSLm M d

J J J JqSC

amg

(4)

3

sin sinsin

y

y

yzy x

m m

y y y y z xy x z

y y y y

zm

qSCqSC dmV mV


qSCamg

(5)

cos tan sin tan

x

x y z

y zxx y z

x x

dJ JM d

J J

(6)




zd , d ,

yd , d and





and

5




tr t ta a a are



written as follows:

3

tan sin tan cos

z

z

yz y x

m

x yz z z zz x y

z z z z

ym

qSCd

mVJ JqSLm qSLm M d

J J J JqSC

amg

(4)

3

sin sinsin

y

y

yzy x

m m

y y y y z xy x z

y y y y

zm

qSCqSC dmV mV


qSCamg

(5)

cos tan sin tan

x

x y z

y zxx y z

x x

dJ JM d

J J

(6)




zd , d ,

yd , d and





are the interceptor accelerations in the velocity

coordinate system, respectively; Mz, My and Mx are the

moments in the pitch, yaw and roll directions, which are mainly

produced by the aerodynamic fins and reaction jet devices. It is

also assumed that the lateral thrusts produced by the reaction

jet devices are treated as the continuous variables [25]. Then,

define the following equivalent control surface deflections of

the lateral thrusts in the pitch and yaw directions [25]:

6

the continuous variables [25]. Then, define the following equivalent control surface deflections of the

lateral thrusts in the pitch and yaw directions [25]:

max/sy y sF F , max/sz z sF F (7)

where yF and zF denote the lateral thrusts in the pitch and yaw directions, maxsF is the maximum

steady thrust of the reaction jets.

Define the following virtual control moment:

( )c c v B B u (8)

where cB is the uncertainty in the control effectiveness matrix, and

[ , , ] [ , , ]T Tr y p x y zv v v M M M v , [ , , , , ]T

x y z sy sz u , 1 2[ , ]c c cB B B ,

1

0 0

0 00 0

x

y

z

x

c y

z

qSLm

qSLmqSLm

B , 2 max

max

0 00 (1 )

(1 ) 0y

z

c M s m

M s m

K F L

K F L

B .

It is assumed that the angles between LOS and the interceptor velocity coordinate system are small

[1], i.e., 4 3m ma a and 4 3m ma a . Based on equations (1)~(6), we can obtain the IGC model

in the pitch channel:

22 22 2 23 3 2 1 1

3 33 3 4 3

4 43 3 44 4 4 4

2 2

sin cos tp p p p p y p p

p p p p p

p p p p p p p p

p p

ax a x a x x x xr

x a x x dx a x a x b v dy x

(9)

where

1px q , 2px q , 3px , 4p zx , p zv M ,

222

pra

r

, 23

yp

qSCa

mr

, 33y

pm

qSCa

mV

, 43z

pz

qSLmaJ

,

44

zz

pz

qSLmaJ

, 41

pz

bJ

, 3 tan sin tan cosp y xd d ,

(7)

where Fy and Fz denote the lateral thrusts in the pitch and yaw

directions, Fsmax is the maximum steady thrust of the reaction jets.


6







( )c c v B B u (8)



x y z sy sz u , 1 2[ , ]c c cB B B ,

1

0 0

0 00 0

x

y

z

x

c y

z

qSLm

qSLmqSLm

B , 2 max

max

0 00 (1 )

(1 ) 0y

z

c M s m

M s m

K F L

K F L

B .




22 22 2 23 3 2 1 1

3 33 3 4 3

4 43 3 44 4 4 4

2 2


p p p p p

p p p p p p p p

p p

ax a x a x x x xr


(9)

where


222

pra

r

, 23

yp

qSCa

mr

, 33y

pm

qSCa

mV

, 43z

pz

qSLmaJ

,

44

zz

pz

qSLmaJ

, 41

pz

bJ


(8)

where

6







( )c c v B B u (8)



x y z sy sz u , 1 2[ , ]c c cB B B ,

1

0 0

0 00 0

x

y

z

x

c y

z

qSLm

qSLmqSLm

B , 2 max

max

0 00 (1 )

(1 ) 0y

z

c M s m

M s m

K F L

K F L

B .




22 22 2 23 3 2 1 1

3 33 3 4 3

4 43 3 44 4 4 4

2 2


p p p p p

p p p p p p p p

p p

ax a x a x x x xr


(9)

where


222

pra

r

, 23

yp

qSCa

mr

, 33y

pm

qSCa

mV

, 43z

pz

qSLmaJ

,

44

zz

pz

qSLmaJ

, 41

pz

bJ


is the uncertainty in the control effectiveness

matrix, and

6







( )c c v B B u (8)



x y z sy sz u , 1 2[ , ]c c cB B B ,

1

0 0

0 00 0

x

y

z

x

c y

z

qSLm

qSLmqSLm

B , 2 max

max

0 00 (1 )

(1 ) 0y

z

c M s m

M s m

K F L

K F L

B .




22 22 2 23 3 2 1 1

3 33 3 4 3

4 43 3 44 4 4 4

2 2


p p p p p

p p p p p p p p

p p

ax a x a x x x xr


(9)

where


222

pra

r

, 23

yp

qSCa

mr

, 33y

pm

qSCa

mV

, 43z

pz

qSLmaJ

,

44

zz

pz

qSLmaJ

, 41

pz

bJ


6







( )c c v B B u (8)



x y z sy sz u , 1 2[ , ]c c cB B B ,

1

0 0

0 00 0

x

y

z

x

c y

z

qSLm

qSLmqSLm

B , 2 max

max

0 00 (1 )

(1 ) 0y

z

c M s m

M s m

K F L

K F L

B .




22 22 2 23 3 2 1 1

3 33 3 4 3

4 43 3 44 4 4 4

2 2


p p p p p

p p p p p p p p

p p

ax a x a x x x xr


(9)

where


222

pra

r

, 23

yp

qSCa

mr

, 33y

pm

qSCa

mV

, 43z

pz

qSLmaJ

,

44

zz

pz

qSLmaJ

, 41

pz

bJ


6







( )c c v B B u (8)



x y z sy sz u , 1 2[ , ]c c cB B B ,

1

0 0

0 00 0

x

y

z

x

c y

z

qSLm

qSLmqSLm

B , 2 max

max

0 00 (1 )

(1 ) 0y

z

c M s m

M s m

K F L

K F L

B .




22 22 2 23 3 2 1 1

3 33 3 4 3

4 43 3 44 4 4 4

2 2


p p p p p

p p p p p p p p

p p

ax a x a x x x xr


(9)

where


222

pra

r

, 23

yp

qSCa

mr

, 33y

pm

qSCa

mV

, 43z

pz

qSLmaJ

,

44

zz

pz

qSLmaJ

, 41

pz

bJ


6







( )c c v B B u (8)



x y z sy sz u , 1 2[ , ]c c cB B B ,

1

0 0

0 00 0

x

y

z

x

c y

z

qSLm

qSLmqSLm

B , 2 max

max

0 00 (1 )

(1 ) 0y

z

c M s m

M s m

K F L

K F L

B .




22 22 2 23 3 2 1 1

3 33 3 4 3

4 43 3 44 4 4 4

2 2


p p p p p

p p p p p p p p

p p

ax a x a x x x xr


(9)

where


222

pra

r

, 23

yp

qSCa

mr

, 33y

pm

qSCa

mV

, 43z

pz

qSLmaJ

,

44

zz

pz

qSLmaJ

, 41

pz

bJ


It is assumed that the angles between LOS and the

interceptor velocity coordinate system are small [1], i.e.,

6







( )c c v B B u (8)



x y z sy sz u , 1 2[ , ]c c cB B B ,

1

0 0

0 00 0

x

y

z

x

c y

z

qSLm

qSLmqSLm

B , 2 max

max

0 00 (1 )

(1 ) 0y

z

c M s m

M s m

K F L

K F L

B .




22 22 2 23 3 2 1 1

3 33 3 4 3

4 43 3 44 4 4 4

2 2


p p p p p

p p p p p p p p

p p

ax a x a x x x xr


(9)

where


222

pra

r

, 23

yp

qSCa

mr

, 33y

pm

qSCa

mV

, 43z

pz

qSLmaJ

,

44

zz

pz

qSLmaJ

, 41

pz

bJ


and

6







( )c c v B B u (8)



x y z sy sz u , 1 2[ , ]c c cB B B ,

1

0 0

0 00 0

x

y

z

x

c y

z

qSLm

qSLmqSLm

B , 2 max

max

0 00 (1 )

(1 ) 0y

z

c M s m

M s m

K F L

K F L

B .




22 22 2 23 3 2 1 1

3 33 3 4 3

4 43 3 44 4 4 4

2 2


p p p p p

p p p p p p p p

p p

ax a x a x x x xr


(9)

where


222

pra

r

, 23

yp

qSCa

mr

, 33y

pm

qSCa

mV

, 43z

pz

qSLmaJ

,

44

zz

pz

qSLmaJ

, 41

pz

bJ


. Based on equations (1)~(6), we

can obtain the IGC model in the pitch channel:

6







( )c c v B B u (8)



x y z sy sz u , 1 2[ , ]c c cB B B ,

1

0 0

0 00 0

x

y

z

x

c y

z

qSLm

qSLmqSLm

B , 2 max

max

0 00 (1 )

(1 ) 0y

z

c M s m

M s m

K F L

K F L

B .




22 22 2 23 3 2 1 1

3 33 3 4 3

4 43 3 44 4 4 4

2 2


p p p p p

p p p p p p p p

p p

ax a x a x x x xr


(9)

where


222

pra

r

, 23

yp

qSCa

mr

, 33y

pm

qSCa

mV

, 43z

pz

qSLmaJ

,

44

zz

pz

qSLmaJ

, 41

pz

bJ


(9)

where

6







( )c c v B B u (8)



x y z sy sz u , 1 2[ , ]c c cB B B ,

1

0 0

0 00 0

x

y

z

x

c y

z

qSLm

qSLmqSLm

B , 2 max

max

0 00 (1 )

(1 ) 0y

z

c M s m

M s m

K F L

K F L

B .




22 22 2 23 3 2 1 1

3 33 3 4 3

4 43 3 44 4 4 4

2 2


p p p p p

p p p p p p p p

p p

ax a x a x x x xr


(9)

where


222

pra

r

, 23

yp

qSCa

mr

, 33y

pm

qSCa

mV

, 43z

pz

qSLmaJ

,

44

zz

pz

qSLmaJ

, 41

pz

bJ


6







( )c c v B B u (8)



x y z sy sz u , 1 2[ , ]c c cB B B ,

1

0 0

0 00 0

x

y

z

x

c y

z

qSLm

qSLmqSLm

B , 2 max

max

0 00 (1 )

(1 ) 0y

z

c M s m

M s m

K F L

K F L

B .




22 22 2 23 3 2 1 1

3 33 3 4 3

4 43 3 44 4 4 4

2 2


p p p p p

p p p p p p p p

p p

ax a x a x x x xr


(9)

where


222

pra

r

, 23

yp

qSCa

mr

, 33y

pm

qSCa

mV

, 43z

pz

qSLmaJ

,

44

zz

pz

qSLmaJ

, 41

pz

bJ


6







( )c c v B B u (8)



x y z sy sz u , 1 2[ , ]c c cB B B ,

1

0 0

0 00 0

x

y

z

x

c y

z

qSLm

qSLmqSLm

B , 2 max

max

0 00 (1 )

(1 ) 0y

z

c M s m

M s m

K F L

K F L

B .




22 22 2 23 3 2 1 1

3 33 3 4 3

4 43 3 44 4 4 4

2 2


p p p p p

p p p p p p p p

p p

ax a x a x x x xr


(9)

where


222

pra

r

, 23

yp

qSCa

mr

, 33y

pm

qSCa

mV

, 43z

pz

qSLmaJ

,

44

zz

pz

qSLmaJ

, 41

pz

bJ


7

4 z

x yp x y

z

J Jd d

J

;

the IGC model in the yaw channel:

2 22 2 23 3 2 2 11

3 33 3 4 3

4 43 3 44 4 4 4

2 2

2 tancos

ty y y y y p y p

p

y y y y y

y y y y y y y y

y y

ax a x a x x x x

r xx a x x dx a x a x b v dy x

(10)

where

1yx q , 2yx q , 3yx , 4y yx , y yv M ,

222

yra

r

, 23

1cosz

yp

qSCamr x

, 33z

ym

qSCamV

, 43y

yy

qSLma

J

,

44

yy

yy

qSLma

J

, 41

yy

bJ

, 3

sin sinsin y

y xm

qSCd d

mV

,

4 y

z xy x z

y

J Jd dJ

;

and the IGC model in the roll channel:

3 4 3

4 4 4

r r r

r r r r

x x dx b v d

(11)

where

3rx , 4r xx , r xv M , 3 cos tan sin tanr y zd d ,

41

rx

bJ

, 4 x

y zr y z

x

J Jd d

J

.

Therefore, the IGC design objective is to derive the LOS angular rate from zero so that it is as small

as possible, and guarantee the stability of the NSI dynamic simultaneously — i.e., the design an

appropriate controller for the time-varying systems (9)~(11) to make the absolute values of the

outputs as small as possible.


7

4 z

x yp x y

z

J Jd d

J

;


2 22 2 23 3 2 2 11

3 33 3 4 3

4 43 3 44 4 4 4

2 2

2 tancos

ty y y y y p y p

p

y y y y y

y y y y y y y y

y y

ax a x a x x x x


(10)

where


222

yra

r

, 23

1cosz

yp

qSCamr x

, 33z

ym

qSCamV

, 43y

yy

qSLma

J

,

44

yy

yy

qSLma

J

, 41

yy

bJ

, 3

sin sinsin y

y xm

qSCd d

mV

,

4 y

z xy x z

y

J Jd dJ

;


3 4 3

4 4 4

r r r

r r r r

x x dx b v d

(11)

where


41

rx

bJ

, 4 x

y zr y z

x

J Jd d

J

.





(10)

(278~294)14-087.indd 280 2015-07-03 오전 5:06:23

281


http://ijass.org

where

7

4 z

x yp x y

z

J Jd d

J

;


2 22 2 23 3 2 2 11

3 33 3 4 3

4 43 3 44 4 4 4

2 2

2 tancos

ty y y y y p y p

p

y y y y y

y y y y y y y y

y y

ax a x a x x x x


(10)

where


222

yra

r

, 23

1cosz

yp

qSCamr x

, 33z

ym

qSCamV

, 43y

yy

qSLma

J

,

44

yy

yy

qSLma

J

, 41

yy

bJ

, 3

sin sinsin y

y xm

qSCd d

mV

,

4 y

z xy x z

y

J Jd dJ

;


3 4 3

4 4 4

r r r

r r r r

x x dx b v d

(11)

where


41

rx

bJ

, 4 x

y zr y z

x

J Jd d

J

.






7

4 z

x yp x y

z

J Jd d

J

;


2 22 2 23 3 2 2 11

3 33 3 4 3

4 43 3 44 4 4 4

2 2

2 tancos

ty y y y y p y p

p

y y y y y

y y y y y y y y

y y

ax a x a x x x x


(10)

where


222

yra

r

, 23

1cosz

yp

qSCamr x

, 33z

ym

qSCamV

, 43y

yy

qSLma

J

,

44

yy

yy

qSLma

J

, 41

yy

bJ

, 3

sin sinsin y

y xm

qSCd d

mV

,

4 y

z xy x z

y

J Jd dJ

;


3 4 3

4 4 4

r r r

r r r r

x x dx b v d

(11)

where


41

rx

bJ

, 4 x

y zr y z

x

J Jd d

J

.





(11)

where

7

4 z

x yp x y

z

J Jd d

J

;


2 22 2 23 3 2 2 11

3 33 3 4 3

4 43 3 44 4 4 4

2 2

2 tancos

ty y y y y p y p

p

y y y y y

y y y y y y y y

y y

ax a x a x x x x


(10)

where


222

yra

r

, 23

1cosz

yp

qSCamr x

, 33z

ym

qSCamV

, 43y

yy

qSLma

J

,

44

yy

yy

qSLma

J

, 41

yy

bJ

, 3

sin sinsin y

y xm

qSCd d

mV

,

4 y

z xy x z

y

J Jd dJ

;


3 4 3

4 4 4

r r r

r r r r

x x dx b v d

(11)

where


41

rx

bJ

, 4 x

y zr y z

x

J Jd d

J

.





7

4 z

x yp x y

z

J Jd d

J

;


2 22 2 23 3 2 2 11

3 33 3 4 3

4 43 3 44 4 4 4

2 2

2 tancos

ty y y y y p y p

p

y y y y y

y y y y y y y y

y y

ax a x a x x x x


(10)

where


222

yra

r

, 23

1cosz

yp

qSCamr x

, 33z

ym

qSCamV

, 43y

yy

qSLma

J

,

44

yy

yy

qSLma

J

, 41

yy

bJ

, 3

sin sinsin y

y xm

qSCd d

mV

,

4 y

z xy x z

y

J Jd dJ

;


3 4 3

4 4 4

r r r

r r r r

x x dx b v d

(11)

where


41

rx

bJ

, 4 x

y zr y z

x

J Jd d

J

.





7

4 z

x yp x y

z

J Jd d

J

;


2 22 2 23 3 2 2 11

3 33 3 4 3

4 43 3 44 4 4 4

2 2

2 tancos

ty y y y y p y p

p

y y y y y

y y y y y y y y

y y

ax a x a x x x x


(10)

where


222

yra

r

, 23

1cosz

yp

qSCamr x

, 33z

ym

qSCamV

, 43y

yy

qSLma

J

,

44

yy

yy

qSLma

J

, 41

yy

bJ

, 3

sin sinsin y

y xm

qSCd d

mV

,

4 y

z xy x z

y

J Jd dJ

;


3 4 3

4 4 4

r r r

r r r r

x x dx b v d

(11)

where


41

rx

bJ

, 4 x

y zr y z

x

J Jd d

J

.





Therefore, the IGC design objective is to derive the LOS

angular rate from zero so that it is as small as possible, and

guarantee the stability of the NSI dynamic simultaneously —

i.e., the design an appropriate controller for the time-varying

systems (9)~(11) to make the absolute values of the outputs

as small as possible.

3. IGC design and stability analysis

In this section, a novel IGC algorithm is developed for the

time-varying systems (9)~(11) based on the backstepping

sliding mode and FTDO, and the stability analysis is also

given for the closed-loop IGC system.

3.1 Control design in the pitch channel

Define the variable

8


In this section, a novel IGC algorithm is developed for the time-varying systems (9)~(11) based on

the backstepping sliding mode and FTDO, and the stability analysis is also given for the closed-loop

IGC system.


Define the variable v rq , and based on equations (2) and (9), we can obtain:

21

3

tan p yrp t

v x qSCv vv x ar r m

(12)

Then, the following FTDO is proposed to estimate the target acceleration ta :

2 2

20 20 2 21 21 22 22

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

20 21 22

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆ ˆ, ,

p p

p p p p p p p

p p p p p p

p p p p p p p p

q pp p p p p p p

p p t p

z v f z v z v

v L z v z v z

v L z v z v z

v L z v z v

z v z a z

ˆ

ta

(13)

where2

12 3

tan p yrp p

v x qSCv vf xr r m

, v̂ and ˆta are the estimated values of v and

ta , respectively; 20p , 21p and 22p are the observer gain coefficients to be determined.

When we define the error variables 20 20p pe z v , 21 21p p te z a and 22 22p p te z a , we

can obtain the following error system:

2 2

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

| | sgn( )

| | sgn( )

| | sgn( )p p

p p p p p p

p p p p p p p p

q pp p p p p p p t

e L e e e

e L e e e e e

e L e e e e a

(14)

Based on reference [20], we can determine that the error system (14) is stable in terms of finite

time. Then, the following FTDOs are also introduced to estimate the disturbances 3pd and 4pd in

the angle of the attack loop and pitch angular rate loop:

, and based on equations (2)

and (9), we can obtain:

8




IGC system.



21

3

tan p yrp t


(12)


2 2

20 20 2 21 21 22 22

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

20 21 22

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆ ˆ, ,

p p

p p p p p p p

p p p p p p

p p p p p p p p

q pp p p p p p p

p p t p

z v f z v z v

v L z v z v z

v L z v z v z

v L z v z v

z v z a z

ˆ

ta

(13)

where2

12 3

tan p yrp p

v x qSCv vf xr r m





2 2

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

| | sgn( )

| | sgn( )

| | sgn( )p p

p p p p p p

p p p p p p p p

q pp p p p p p p t

e L e e e

e L e e e e e

e L e e e e a

(14)




(12)

Then, the following FTDO is proposed to estimate the

target acceleration :

8




IGC system.



21

3

tan p yrp t


(12)


2 2

20 20 2 21 21 22 22

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

20 21 22

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆ ˆ, ,

p p

p p p p p p p

p p p p p p

p p p p p p p p

q pp p p p p p p

p p t p

z v f z v z v

v L z v z v z

v L z v z v z

v L z v z v

z v z a z

ˆ

ta

(13)

where2

12 3

tan p yrp p

v x qSCv vf xr r m





2 2

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

| | sgn( )

| | sgn( )

| | sgn( )p p

p p p p p p

p p p p p p p p

q pp p p p p p p t

e L e e e

e L e e e e e

e L e e e e a

(14)




(13)

where

8




IGC system.



21

3

tan p yrp t


(12)


2 2

20 20 2 21 21 22 22

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

20 21 22

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆ ˆ, ,

p p

p p p p p p p

p p p p p p

p p p p p p p p

q pp p p p p p p

p p t p

z v f z v z v

v L z v z v z

v L z v z v z

v L z v z v

z v z a z

ˆ

ta

(13)

where2

12 3

tan p yrp p

v x qSCv vf xr r m





2 2

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

| | sgn( )

| | sgn( )

| | sgn( )p p

p p p p p p

p p p p p p p p

q pp p p p p p p t

e L e e e

e L e e e e e

e L e e e e a

(14)




,

8




IGC system.



21

3

tan p yrp t


(12)


2 2

20 20 2 21 21 22 22

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

20 21 22

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆ ˆ, ,

p p

p p p p p p p

p p p p p p

p p p p p p p p

q pp p p p p p p

p p t p

z v f z v z v

v L z v z v z

v L z v z v z

v L z v z v

z v z a z

ˆ

ta

(13)

where2

12 3

tan p yrp p

v x qSCv vf xr r m





2 2

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

| | sgn( )

| | sgn( )

| | sgn( )p p

p p p p p p

p p p p p p p p

q pp p p p p p p t

e L e e e

e L e e e e e

e L e e e e a

(14)




and

8




IGC system.



21

3

tan p yrp t


(12)


2 2

20 20 2 21 21 22 22

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

20 21 22

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆ ˆ, ,

p p

p p p p p p p

p p p p p p

p p p p p p p p

q pp p p p p p p

p p t p

z v f z v z v

v L z v z v z

v L z v z v z

v L z v z v

z v z a z

ˆ

ta

(13)

where2

12 3

tan p yrp p

v x qSCv vf xr r m





2 2

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

| | sgn( )

| | sgn( )

| | sgn( )p p

p p p p p p

p p p p p p p p

q pp p p p p p p t

e L e e e

e L e e e e e

e L e e e e a

(14)




are

the estimated values of

8




IGC system.



21

3

tan p yrp t


(12)


2 2

20 20 2 21 21 22 22

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

20 21 22

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆ ˆ, ,

p p

p p p p p p p

p p p p p p

p p p p p p p p

q pp p p p p p p

p p t p

z v f z v z v

v L z v z v z

v L z v z v z

v L z v z v

z v z a z

ˆ

ta

(13)

where2

12 3

tan p yrp p

v x qSCv vf xr r m





2 2

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

| | sgn( )

| | sgn( )

| | sgn( )p p

p p p p p p

p p p p p p p p

q pp p p p p p p t

e L e e e

e L e e e e e

e L e e e e a

(14)




and

8




IGC system.



21

3

tan p yrp t


(12)


2 2

20 20 2 21 21 22 22

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

20 21 22

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆ ˆ, ,

p p

p p p p p p p

p p p p p p

p p p p p p p p

q pp p p p p p p

p p t p

z v f z v z v

v L z v z v z

v L z v z v z

v L z v z v

z v z a z

ˆ

ta

(13)

where2

12 3

tan p yrp p

v x qSCv vf xr r m





2 2

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

| | sgn( )

| | sgn( )

| | sgn( )p p

p p p p p p

p p p p p p p p

q pp p p p p p p t

e L e e e

e L e e e e e

e L e e e e a

(14)




, respectively;

8




IGC system.



21

3

tan p yrp t


(12)


2 2

20 20 2 21 21 22 22

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

20 21 22

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆ ˆ, ,

p p

p p p p p p p

p p p p p p

p p p p p p p p

q pp p p p p p p

p p t p

z v f z v z v

v L z v z v z

v L z v z v z

v L z v z v

z v z a z

ˆ

ta

(13)

where2

12 3

tan p yrp p

v x qSCv vf xr r m





2 2

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

| | sgn( )

| | sgn( )

| | sgn( )p p

p p p p p p

p p p p p p p p

q pp p p p p p p t

e L e e e

e L e e e e e

e L e e e e a

(14)




,

8




IGC system.



21

3

tan p yrp t


(12)


2 2

20 20 2 21 21 22 22

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

20 21 22

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆ ˆ, ,

p p

p p p p p p p

p p p p p p

p p p p p p p p

q pp p p p p p p

p p t p

z v f z v z v

v L z v z v z

v L z v z v z

v L z v z v

z v z a z

ˆ

ta

(13)

where2

12 3

tan p yrp p

v x qSCv vf xr r m





2 2

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

| | sgn( )

| | sgn( )

| | sgn( )p p

p p p p p p

p p p p p p p p

q pp p p p p p p t

e L e e e

e L e e e e e

e L e e e e a

(14)




and

8




IGC system.



21

3

tan p yrp t


(12)


2 2

20 20 2 21 21 22 22

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

20 21 22

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆ ˆ, ,

p p

p p p p p p p

p p p p p p

p p p p p p p p

q pp p p p p p p

p p t p

z v f z v z v

v L z v z v z

v L z v z v z

v L z v z v

z v z a z

ˆ

ta

(13)

where2

12 3

tan p yrp p

v x qSCv vf xr r m





2 2

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

| | sgn( )

| | sgn( )

| | sgn( )p p

p p p p p p

p p p p p p p p

q pp p p p p p p t

e L e e e

e L e e e e e

e L e e e e a

(14)




are the observer gain coefficients to be determined.

When we define the error variables

8




IGC system.



21

3

tan p yrp t


(12)


2 2

20 20 2 21 21 22 22

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

20 21 22

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆ ˆ, ,

p p

p p p p p p p

p p p p p p

p p p p p p p p

q pp p p p p p p

p p t p

z v f z v z v

v L z v z v z

v L z v z v z

v L z v z v

z v z a z

ˆ

ta

(13)

where2

12 3

tan p yrp p

v x qSCv vf xr r m





2 2

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

| | sgn( )

| | sgn( )

| | sgn( )p p

p p p p p p

p p p p p p p p

q pp p p p p p p t

e L e e e

e L e e e e e

e L e e e e a

(14)




,

8




IGC system.



21

3

tan p yrp t


(12)


2 2

20 20 2 21 21 22 22

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

20 21 22

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆ ˆ, ,

p p

p p p p p p p

p p p p p p

p p p p p p p p

q pp p p p p p p

p p t p

z v f z v z v

v L z v z v z

v L z v z v z

v L z v z v

z v z a z

ˆ

ta

(13)

where2

12 3

tan p yrp p

v x qSCv vf xr r m





2 2

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

| | sgn( )

| | sgn( )

| | sgn( )p p

p p p p p p

p p p p p p p p

q pp p p p p p p t

e L e e e

e L e e e e e

e L e e e e a

(14)




and

8




IGC system.



21

3

tan p yrp t


(12)


2 2

20 20 2 21 21 22 22

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

20 21 22

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆ ˆ, ,

p p

p p p p p p p

p p p p p p

p p p p p p p p

q pp p p p p p p

p p t p

z v f z v z v

v L z v z v z

v L z v z v z

v L z v z v

z v z a z

ˆ

ta

(13)

where2

12 3

tan p yrp p

v x qSCv vf xr r m





2 2

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

| | sgn( )

| | sgn( )

| | sgn( )p p

p p p p p p

p p p p p p p p

q pp p p p p p p t

e L e e e

e L e e e e e

e L e e e e a

(14)




, we can obtain the

following error system:

8




IGC system.



21

3

tan p yrp t


(12)


2 2

20 20 2 21 21 22 22

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

20 21 22

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆ ˆ, ,

p p

p p p p p p p

p p p p p p

p p p p p p p p

q pp p p p p p p

p p t p

z v f z v z v

v L z v z v z

v L z v z v z

v L z v z v

z v z a z

ˆ

ta

(13)

where2

12 3

tan p yrp p

v x qSCv vf xr r m





2 2

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

| | sgn( )

| | sgn( )

| | sgn( )p p

p p p p p p

p p p p p p p p

q pp p p p p p p t

e L e e e

e L e e e e e

e L e e e e a

(14)




(14)

Based on reference [20], we can determine that the

error system (14) is stable in terms of finite time. Then,

the following FTDOs are also introduced to estimate the

disturbances dp3 and dp4 in the angle of the attack loop and

pitch angular rate loop:

9

3 3

30 30 3 31 31 32 32

1/3 2/330 30 3 30 3 30 3 31

1/2 1/231 31 3 31 30 31 30 32

/32 32 3 32 31 32 31

30 3 31 3 3

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆˆ , ,

p p

p p p p p p p

p p p p p p p p

p p p p p p p p

q pp p p p p p p

p p p p p

z v f z v z v

v L z x z x z

v L z v z v z

v L z v z v

z x z d z

2 3ˆ

pd

(15)

4 4

40 40 4 41 41 42 42

1/3 2/340 40 4 40 4 40 4 41

1/2 1/241 41 4 41 40 41 40 42

/42 42 4 42 41 42 41

40 4 41 4 4

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆˆ , ,

p p

p p p p p p p

p p p p p p p p

p p p p p p p p

q pp p p p p p p

p p p p p

z v f z v z v

v L z x z x z

v L z v z v z

v L z v z v

z x z d z

2 4ˆ

pd

(16)

where 3 33 3 4p p p pf a x x , 4 43 3 44 4 4p p p p p p pf a x a x b v , 3ˆ

pd and 4ˆ

pd are the estimated

values of the disturbances 3pd and 4pd , respectively; the corresponding disturbance estimated

errors are defined as 31 31 3p p pe z d and 41 41 4p p pe z d .

Based on the FTDOs (13) and (15)~(16), the following backstepping sliding mode control law is

given for the IGC system (9):

Step 1: Define the first error surface as follows:

12 23 2 2( )p p p p ds a x x (17)

where 2p dx is the command signal for the system (9), and the derivative of 2ps is:

2 1 22 23 23 2 2 23 22 2 2 1 1 2 3( ) ( sin cos / )p p p p p d p p p y p p t p d ps a a x x a a x x x x a r x x (18)

Then, the virtual control *3px is constructed as follows:

* 2 1 23 23 23 2 2 23 22 2 2 1 1 2 2 2ˆ( ) ( sin cos / )p p p p p d p p p y p p t p d p px a a x x a a x x x x a r x k s (19)

where 2 0pk . In addition, the following first-order low-pass filter is introduced to obtain the

filtered virtual control *3px :

* * *3 3 3 3p p p px x x , * *

3 3(0) (0)p px x (20)

(15)

9

3 3

30 30 3 31 31 32 32

1/3 2/330 30 3 30 3 30 3 31

1/2 1/231 31 3 31 30 31 30 32

/32 32 3 32 31 32 31

30 3 31 3 3

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆˆ , ,

p p

p p p p p p p

p p p p p p p p

p p p p p p p p

q pp p p p p p p

p p p p p

z v f z v z v

v L z x z x z

v L z v z v z

v L z v z v

z x z d z

2 3ˆ

pd

(15)

4 4

40 40 4 41 41 42 42

1/3 2/340 40 4 40 4 40 4 41

1/2 1/241 41 4 41 40 41 40 42

/42 42 4 42 41 42 41

40 4 41 4 4

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆˆ , ,

p p

p p p p p p p

p p p p p p p p

p p p p p p p p

q pp p p p p p p

p p p p p

z v f z v z v

v L z x z x z

v L z v z v z

v L z v z v

z x z d z

2 4ˆ

pd

(16)


pd and 4ˆ







12 23 2 2( )p p p p ds a x x (17)







* * *3 3 3 3p p p px x x , * *

3 3(0) (0)p px x (20)

(16)

where

9

3 3

30 30 3 31 31 32 32

1/3 2/330 30 3 30 3 30 3 31

1/2 1/231 31 3 31 30 31 30 32

/32 32 3 32 31 32 31

30 3 31 3 3

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆˆ , ,

p p

p p p p p p p

p p p p p p p p

p p p p p p p p

q pp p p p p p p

p p p p p

z v f z v z v

v L z x z x z

v L z v z v z

v L z v z v

z x z d z

2 3ˆ

pd

(15)

4 4

40 40 4 41 41 42 42

1/3 2/340 40 4 40 4 40 4 41

1/2 1/241 41 4 41 40 41 40 42

/42 42 4 42 41 42 41

40 4 41 4 4

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆˆ , ,

p p

p p p p p p p

p p p p p p p p

p p p p p p p p

q pp p p p p p p

p p p p p

z v f z v z v

v L z x z x z

v L z v z v z

v L z v z v

z x z d z

2 4ˆ

pd

(16)


pd and 4ˆ







12 23 2 2( )p p p p ds a x x (17)







* * *3 3 3 3p p p px x x , * *

3 3(0) (0)p px x (20)

,

9

3 3

30 30 3 31 31 32 32

1/3 2/330 30 3 30 3 30 3 31

1/2 1/231 31 3 31 30 31 30 32

/32 32 3 32 31 32 31

30 3 31 3 3

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆˆ , ,

p p

p p p p p p p

p p p p p p p p

p p p p p p p p

q pp p p p p p p

p p p p p

z v f z v z v

v L z x z x z

v L z v z v z

v L z v z v

z x z d z

2 3ˆ

pd

(15)

4 4

40 40 4 41 41 42 42

1/3 2/340 40 4 40 4 40 4 41

1/2 1/241 41 4 41 40 41 40 42

/42 42 4 42 41 42 41

40 4 41 4 4

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆˆ , ,

p p

p p p p p p p

p p p p p p p p

p p p p p p p p

q pp p p p p p p

p p p p p

z v f z v z v

v L z x z x z

v L z v z v z

v L z v z v

z x z d z

2 4ˆ

pd

(16)


pd and 4ˆ







12 23 2 2( )p p p p ds a x x (17)







* * *3 3 3 3p p p px x x , * *

3 3(0) (0)p px x (20)

and

9

3 3

30 30 3 31 31 32 32

1/3 2/330 30 3 30 3 30 3 31

1/2 1/231 31 3 31 30 31 30 32

/32 32 3 32 31 32 31

30 3 31 3 3

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆˆ , ,

p p

p p p p p p p

p p p p p p p p

p p p p p p p p

q pp p p p p p p

p p p p p

z v f z v z v

v L z x z x z

v L z v z v z

v L z v z v

z x z d z

2 3ˆ

pd

(15)

4 4

40 40 4 41 41 42 42

1/3 2/340 40 4 40 4 40 4 41

1/2 1/241 41 4 41 40 41 40 42

/42 42 4 42 41 42 41

40 4 41 4 4

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆˆ , ,

p p

p p p p p p p

p p p p p p p p

p p p p p p p p

q pp p p p p p p

p p p p p

z v f z v z v

v L z x z x z

v L z v z v z

v L z v z v

z x z d z

2 4ˆ

pd

(16)


pd and 4ˆ







12 23 2 2( )p p p p ds a x x (17)







* * *3 3 3 3p p p px x x , * *

3 3(0) (0)p px x (20)

are the estimated values of the disturbances

dp3 and dp4, respectively; the corresponding disturbance

estimated errors are defined as ep31=zp31-dp3 and ep41=zp41-dp4.

Based on the FTDOs (13) and (15)~(16), the following

backstepping sliding mode control law is given for the IGC

system (9):


(278~294)14-087.indd 281 2015-07-03 오전 5:06:25



9

3 3

30 30 3 31 31 32 32

1/3 2/330 30 3 30 3 30 3 31

1/2 1/231 31 3 31 30 31 30 32

/32 32 3 32 31 32 31

30 3 31 3 3

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆˆ , ,

p p

p p p p p p p

p p p p p p p p

p p p p p p p p

q pp p p p p p p

p p p p p

z v f z v z v

v L z x z x z

v L z v z v z

v L z v z v

z x z d z

2 3ˆ

pd

(15)

4 4

40 40 4 41 41 42 42

1/3 2/340 40 4 40 4 40 4 41

1/2 1/241 41 4 41 40 41 40 42

/42 42 4 42 41 42 41

40 4 41 4 4

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆˆ , ,

p p

p p p p p p p

p p p p p p p p

p p p p p p p p

q pp p p p p p p

p p p p p

z v f z v z v

v L z x z x z

v L z v z v z

v L z v z v

z x z d z

2 4ˆ

pd

(16)


pd and 4ˆ







12 23 2 2( )p p p p ds a x x (17)







* * *3 3 3 3p p p px x x , * *

3 3(0) (0)p px x (20)

(17)

where xp2d is the command signal for the system (9), and the

derivative of sp2 is:

9

3 3

30 30 3 31 31 32 32

1/3 2/330 30 3 30 3 30 3 31

1/2 1/231 31 3 31 30 31 30 32

/32 32 3 32 31 32 31

30 3 31 3 3

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆˆ , ,

p p

p p p p p p p

p p p p p p p p

p p p p p p p p

q pp p p p p p p

p p p p p

z v f z v z v

v L z x z x z

v L z v z v z

v L z v z v

z x z d z

2 3ˆ

pd

(15)

4 4

40 40 4 41 41 42 42

1/3 2/340 40 4 40 4 40 4 41

1/2 1/241 41 4 41 40 41 40 42

/42 42 4 42 41 42 41

40 4 41 4 4

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆˆ , ,

p p

p p p p p p p

p p p p p p p p

p p p p p p p p

q pp p p p p p p

p p p p p

z v f z v z v

v L z x z x z

v L z v z v z

v L z v z v

z x z d z

2 4ˆ

pd

(16)


pd and 4ˆ







12 23 2 2( )p p p p ds a x x (17)







* * *3 3 3 3p p p px x x , * *

3 3(0) (0)p px x (20)

(18)

Then, the virtual control

9

3 3

30 30 3 31 31 32 32

1/3 2/330 30 3 30 3 30 3 31

1/2 1/231 31 3 31 30 31 30 32

/32 32 3 32 31 32 31

30 3 31 3 3

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆˆ , ,

p p

p p p p p p p

p p p p p p p p

p p p p p p p p

q pp p p p p p p

p p p p p

z v f z v z v

v L z x z x z

v L z v z v z

v L z v z v

z x z d z

2 3ˆ

pd

(15)

4 4

40 40 4 41 41 42 42

1/3 2/340 40 4 40 4 40 4 41

1/2 1/241 41 4 41 40 41 40 42

/42 42 4 42 41 42 41

40 4 41 4 4

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆˆ , ,

p p

p p p p p p p

p p p p p p p p

p p p p p p p p

q pp p p p p p p

p p p p p

z v f z v z v

v L z x z x z

v L z v z v z

v L z v z v

z x z d z

2 4ˆ

pd

(16)


pd and 4ˆ







12 23 2 2( )p p p p ds a x x (17)







* * *3 3 3 3p p p px x x , * *

3 3(0) (0)p px x (20)

is constructed as follows:

9

3 3

30 30 3 31 31 32 32

1/3 2/330 30 3 30 3 30 3 31

1/2 1/231 31 3 31 30 31 30 32

/32 32 3 32 31 32 31

30 3 31 3 3

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆˆ , ,

p p

p p p p p p p

p p p p p p p p

p p p p p p p p

q pp p p p p p p

p p p p p

z v f z v z v

v L z x z x z

v L z v z v z

v L z v z v

z x z d z

2 3ˆ

pd

(15)

4 4

40 40 4 41 41 42 42

1/3 2/340 40 4 40 4 40 4 41

1/2 1/241 41 4 41 40 41 40 42

/42 42 4 42 41 42 41

40 4 41 4 4

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆˆ , ,

p p

p p p p p p p

p p p p p p p p

p p p p p p p p

q pp p p p p p p

p p p p p

z v f z v z v

v L z x z x z

v L z v z v z

v L z v z v

z x z d z

2 4ˆ

pd

(16)


pd and 4ˆ







12 23 2 2( )p p p p ds a x x (17)







* * *3 3 3 3p p p px x x , * *

3 3(0) (0)p px x (20)

(19)

where kp2>0. In addition, the following first-order low-

pass filter is introduced to obtain the filtered virtual

control

9

3 3

30 30 3 31 31 32 32

1/3 2/330 30 3 30 3 30 3 31

1/2 1/231 31 3 31 30 31 30 32

/32 32 3 32 31 32 31

30 3 31 3 3

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆˆ , ,

p p

p p p p p p p

p p p p p p p p

p p p p p p p p

q pp p p p p p p

p p p p p

z v f z v z v

v L z x z x z

v L z v z v z

v L z v z v

z x z d z

2 3ˆ

pd

(15)

4 4

40 40 4 41 41 42 42

1/3 2/340 40 4 40 4 40 4 41

1/2 1/241 41 4 41 40 41 40 42

/42 42 4 42 41 42 41

40 4 41 4 4

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆˆ , ,

p p

p p p p p p p

p p p p p p p p

p p p p p p p p

q pp p p p p p p

p p p p p

z v f z v z v

v L z x z x z

v L z v z v z

v L z v z v

z x z d z

2 4ˆ

pd

(16)


pd and 4ˆ







12 23 2 2( )p p p p ds a x x (17)







* * *3 3 3 3p p p px x x , * *

3 3(0) (0)p px x (20)

:

9

3 3

30 30 3 31 31 32 32

1/3 2/330 30 3 30 3 30 3 31

1/2 1/231 31 3 31 30 31 30 32

/32 32 3 32 31 32 31

30 3 31 3 3

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆˆ , ,

p p

p p p p p p p

p p p p p p p p

p p p p p p p p

q pp p p p p p p

p p p p p

z v f z v z v

v L z x z x z

v L z v z v z

v L z v z v

z x z d z

2 3ˆ

pd

(15)

4 4

40 40 4 41 41 42 42

1/3 2/340 40 4 40 4 40 4 41

1/2 1/241 41 4 41 40 41 40 42

/42 42 4 42 41 42 41

40 4 41 4 4

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆˆ , ,

p p

p p p p p p p

p p p p p p p p

p p p p p p p p

q pp p p p p p p

p p p p p

z v f z v z v

v L z x z x z

v L z v z v z

v L z v z v

z x z d z

2 4ˆ

pd

(16)


pd and 4ˆ







12 23 2 2( )p p p p ds a x x (17)







* * *3 3 3 3p p p px x x , * *

3 3(0) (0)p px x (20) (20)

where τp3>0 denotes the filter time constant. Then, we can

obtain the differentiation of the filtered virtual control:

10

where 3 0p denotes the filter time constant. Then, we can obtain the differentiation of the filtered

virtual control:

* 1 * *3 3 3 3( )p p p px x x (21)

Step 2: Denote the second error surface as follows:

*3 3 3p p ps x x (22)

and its derivative is:

*3 33 3 4 3 3p p p p p ps a x x d x (23)


* *4 33 3 3 3 3 3

ˆp p p p p p px a x d x k s (24)

where 3 0pk . In addition, the following first-order low-pass filter is proposed to obtain the filtered

virtual control *4px :

* * *4 4 4 4p p p px x x , * *

4 4(0) (0)p px x (25)


virtual control:

* 1 * *4 4 4 4( )p p p px x x (26)

Step 3: Denote the third error surface as follows:

*4 4 4p p ps x x (27)


*4 43 3 44 4 4 4 4p p p p p p p p ps a x a x b v d x (28)

Then, we can construct the following control moment:

41 *4 43 3 44 4 4 4 4 4 5 4 4

ˆ[ | | sgn( )]pp p p p p p p p p p p p pv b a x a x d x k s k s s (29)

where 4 0pk , 5 0pk and 40 1p .

3.2 Control design in the yaw channel

(21)


10


virtual control:

* 1 * *3 3 3 3( )p p p px x x (21)


*3 3 3p p ps x x (22)


*3 33 3 4 3 3p p p p p ps a x x d x (23)


* *4 33 3 3 3 3 3




* * *4 4 4 4p p p px x x , * *

4 4(0) (0)p px x (25)


virtual control:

* 1 * *4 4 4 4( )p p p px x x (26)


*4 4 4p p ps x x (27)




41 *4 43 3 44 4 4 4 4 4 5 4 4




(22)


10


virtual control:

* 1 * *3 3 3 3( )p p p px x x (21)


*3 3 3p p ps x x (22)


*3 33 3 4 3 3p p p p p ps a x x d x (23)


* *4 33 3 3 3 3 3




* * *4 4 4 4p p p px x x , * *

4 4(0) (0)p px x (25)


virtual control:

* 1 * *4 4 4 4( )p p p px x x (26)


*4 4 4p p ps x x (27)




41 *4 43 3 44 4 4 4 4 4 5 4 4




(23)

Then, the virtual control

10


virtual control:

* 1 * *3 3 3 3( )p p p px x x (21)


*3 3 3p p ps x x (22)


*3 33 3 4 3 3p p p p p ps a x x d x (23)


* *4 33 3 3 3 3 3




* * *4 4 4 4p p p px x x , * *

4 4(0) (0)p px x (25)


virtual control:

* 1 * *4 4 4 4( )p p p px x x (26)


*4 4 4p p ps x x (27)




41 *4 43 3 44 4 4 4 4 4 5 4 4




is constructed as follows:

10


virtual control:

* 1 * *3 3 3 3( )p p p px x x (21)


*3 3 3p p ps x x (22)


*3 33 3 4 3 3p p p p p ps a x x d x (23)


* *4 33 3 3 3 3 3




* * *4 4 4 4p p p px x x , * *

4 4(0) (0)p px x (25)


virtual control:

* 1 * *4 4 4 4( )p p p px x x (26)


*4 4 4p p ps x x (27)




41 *4 43 3 44 4 4 4 4 4 5 4 4




(24)

where kp3>0. In addition, the following first-order low-pass

filter is proposed to obtain the filtered virtual control

10


virtual control:

* 1 * *3 3 3 3( )p p p px x x (21)


*3 3 3p p ps x x (22)


*3 33 3 4 3 3p p p p p ps a x x d x (23)


* *4 33 3 3 3 3 3




* * *4 4 4 4p p p px x x , * *

4 4(0) (0)p px x (25)


virtual control:

* 1 * *4 4 4 4( )p p p px x x (26)


*4 4 4p p ps x x (27)




41 *4 43 3 44 4 4 4 4 4 5 4 4




:

10


virtual control:

* 1 * *3 3 3 3( )p p p px x x (21)


*3 3 3p p ps x x (22)


*3 33 3 4 3 3p p p p p ps a x x d x (23)


* *4 33 3 3 3 3 3




* * *4 4 4 4p p p px x x , * *

4 4(0) (0)p px x (25)


virtual control:

* 1 * *4 4 4 4( )p p p px x x (26)


*4 4 4p p ps x x (27)




41 *4 43 3 44 4 4 4 4 4 5 4 4




(25)

where τp4>0 denotes the filter time constant. Then, we can

obtain the differentiation of the filtered virtual control:

10


virtual control:

* 1 * *3 3 3 3( )p p p px x x (21)


*3 3 3p p ps x x (22)


*3 33 3 4 3 3p p p p p ps a x x d x (23)


* *4 33 3 3 3 3 3




* * *4 4 4 4p p p px x x , * *

4 4(0) (0)p px x (25)


virtual control:

* 1 * *4 4 4 4( )p p p px x x (26)


*4 4 4p p ps x x (27)




41 *4 43 3 44 4 4 4 4 4 5 4 4




(26)


10


virtual control:

* 1 * *3 3 3 3( )p p p px x x (21)


*3 3 3p p ps x x (22)


*3 33 3 4 3 3p p p p p ps a x x d x (23)


* *4 33 3 3 3 3 3




* * *4 4 4 4p p p px x x , * *

4 4(0) (0)p px x (25)


virtual control:

* 1 * *4 4 4 4( )p p p px x x (26)


*4 4 4p p ps x x (27)




41 *4 43 3 44 4 4 4 4 4 5 4 4




(27)


10


virtual control:

* 1 * *3 3 3 3( )p p p px x x (21)


*3 3 3p p ps x x (22)


*3 33 3 4 3 3p p p p p ps a x x d x (23)


* *4 33 3 3 3 3 3




* * *4 4 4 4p p p px x x , * *

4 4(0) (0)p px x (25)


virtual control:

* 1 * *4 4 4 4( )p p p px x x (26)


*4 4 4p p ps x x (27)




41 *4 43 3 44 4 4 4 4 4 5 4 4




(28)


10


virtual control:

* 1 * *3 3 3 3( )p p p px x x (21)


*3 3 3p p ps x x (22)


*3 33 3 4 3 3p p p p p ps a x x d x (23)


* *4 33 3 3 3 3 3




* * *4 4 4 4p p p px x x , * *

4 4(0) (0)p px x (25)


virtual control:

* 1 * *4 4 4 4( )p p p px x x (26)


*4 4 4p p ps x x (27)




41 *4 43 3 44 4 4 4 4 4 5 4 4




(29)

where kp4>0, kp5>0 and 0<λp4>0<1.


Define the variable

11

Define the variable cosv rq q , and based on equations (3) and (10), we can obtain:

13

tan pr zy t

v v xv v qSCv x ar r m

(30)


2 2

20 20 2 21 21 22 22

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

20 21 22

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆ ˆ, ,

y y

y y y y y y y

y y y y y y

y y y y y y y y

q py y y y y y y

y y t y

z v f z v z v

v L z v z v z

v L z v z v z

v L z v z v

z v z a z

ˆ

ta

(31)

where 12 3

tan pr zy y

v v xv v qSCf xr r m

, 21yz is the estimated value of target acceleration

ta , and the estimated error is defined as 21 21y y te z a . The similar FTDOs are also

constructed to estimate the disturbances 3yd and 4yd in the sideslip angle loop and yaw angular

rate loop, and the estimated errors are defined as 31 31 3y y ye z d and 41 41 4y y ye z d .

Furthermore, we consider the following backstepping sliding mode control law for the system (10):

12 23 2 2

* 23 23 23 2 2 2 2

123 22 2 2 2 1 2

1

* * * * *3 3 3 3 3 3

*3 3 3

* *4 33 3 3 3 3 3

* *4 4 4 4

( )

( )

ˆ2 tan

cos

, (0) (0)

ˆ

y y y y d

y y y y y d y y

ty y y p y p y d

p

y y y y y y

y y y

y y y y y y y

y y y y

s a x x

x a a x x k s

aa a x x x x x

r x

x x x x x

s x x

x a x d x k s

x x x

4

* * *4 4

*4 4 4

1 *4 43 3 44 4 4 4 4 4 5 4 4

, (0) (0)

ˆ[ | | sgn( )]y

y y

y y y

y y y y y y y y y y y y y

x x

s x x

v b a x a x d x k s k s s

(32)

where 40 1y ; 2ys , 3ys and 4ys are the error surfaces of the guidance loop, sideslip angle

, and based on

equations (3) and (10), we can obtain:

11


13

tan pr zy t


(30)


2 2

20 20 2 21 21 22 22

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

20 21 22

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆ ˆ, ,

y y

y y y y y y y

y y y y y y

y y y y y y y y

q py y y y y y y

y y t y

z v f z v z v

v L z v z v z

v L z v z v z

v L z v z v

z v z a z

ˆ

ta

(31)

where 12 3

tan pr zy y







12 23 2 2

* 23 23 23 2 2 2 2

123 22 2 2 2 1 2

1

* * * * *3 3 3 3 3 3

*3 3 3

* *4 33 3 3 3 3 3

* *4 4 4 4

( )

( )

ˆ2 tan

cos

, (0) (0)

ˆ

y y y y d

y y y y y d y y

ty y y p y p y d

p

y y y y y y

y y y

y y y y y y y

y y y y

s a x x

x a a x x k s

aa a x x x x x

r x

x x x x x

s x x

x a x d x k s

x x x

4

* * *4 4

*4 4 4

1 *4 43 3 44 4 4 4 4 4 5 4 4

, (0) (0)

ˆ[ | | sgn( )]y

y y

y y y


x x

s x x


(32)


(30)

Then, the following FTDO is proposed to estimate the

target acceleration

11


13

tan pr zy t


(30)


2 2

20 20 2 21 21 22 22

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

20 21 22

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆ ˆ, ,

y y

y y y y y y y

y y y y y y

y y y y y y y y

q py y y y y y y

y y t y

z v f z v z v

v L z v z v z

v L z v z v z

v L z v z v

z v z a z

ˆ

ta

(31)

where 12 3

tan pr zy y







12 23 2 2

* 23 23 23 2 2 2 2

123 22 2 2 2 1 2

1

* * * * *3 3 3 3 3 3

*3 3 3

* *4 33 3 3 3 3 3

* *4 4 4 4

( )

( )

ˆ2 tan

cos

, (0) (0)

ˆ

y y y y d

y y y y y d y y

ty y y p y p y d

p

y y y y y y

y y y

y y y y y y y

y y y y

s a x x

x a a x x k s

aa a x x x x x

r x

x x x x x

s x x

x a x d x k s

x x x

4

* * *4 4

*4 4 4

1 *4 43 3 44 4 4 4 4 4 5 4 4

, (0) (0)

ˆ[ | | sgn( )]y

y y

y y y


x x

s x x


(32)


:

11


13

tan pr zy t


(30)


2 2

20 20 2 21 21 22 22

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

20 21 22

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆ ˆ, ,

y y

y y y y y y y

y y y y y y

y y y y y y y y

q py y y y y y y

y y t y

z v f z v z v

v L z v z v z

v L z v z v z

v L z v z v

z v z a z

ˆ

ta

(31)

where 12 3

tan pr zy y







12 23 2 2

* 23 23 23 2 2 2 2

123 22 2 2 2 1 2

1

* * * * *3 3 3 3 3 3

*3 3 3

* *4 33 3 3 3 3 3

* *4 4 4 4

( )

( )

ˆ2 tan

cos

, (0) (0)

ˆ

y y y y d

y y y y y d y y

ty y y p y p y d

p

y y y y y y

y y y

y y y y y y y

y y y y

s a x x

x a a x x k s

aa a x x x x x

r x

x x x x x

s x x

x a x d x k s

x x x

4

* * *4 4

*4 4 4

1 *4 43 3 44 4 4 4 4 4 5 4 4

, (0) (0)

ˆ[ | | sgn( )]y

y y

y y y


x x

s x x


(32)


(31)

where

11


13

tan pr zy t


(30)


2 2

20 20 2 21 21 22 22

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

20 21 22

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆ ˆ, ,

y y

y y y y y y y

y y y y y y

y y y y y y y y

q py y y y y y y

y y t y

z v f z v z v

v L z v z v z

v L z v z v z

v L z v z v

z v z a z

ˆ

ta

(31)

where 12 3

tan pr zy y







12 23 2 2

* 23 23 23 2 2 2 2

123 22 2 2 2 1 2

1

* * * * *3 3 3 3 3 3

*3 3 3

* *4 33 3 3 3 3 3

* *4 4 4 4

( )

( )

ˆ2 tan

cos

, (0) (0)

ˆ

y y y y d

y y y y y d y y

ty y y p y p y d

p

y y y y y y

y y y

y y y y y y y

y y y y

s a x x

x a a x x k s

aa a x x x x x

r x

x x x x x

s x x

x a x d x k s

x x x

4

* * *4 4

*4 4 4

1 *4 43 3 44 4 4 4 4 4 5 4 4

, (0) (0)

ˆ[ | | sgn( )]y

y y

y y y


x x

s x x


(32)


is the

estimated value of target acceleration -atβ, and the estimated

error is defined as ey21=-zy21-atβ. The similar FTDOs are also

constructed to estimate the disturbances dy3 and dy4 in

the sideslip angle loop and yaw angular rate loop, and the

estimated errors are defined as ey31=-zy31-dy3 and ey41=-zy41-dy4.

Furthermore, we consider the following backstepping

sliding mode control law for the system (10):

11


13

tan pr zy t


(30)


2 2

20 20 2 21 21 22 22

1/3 2/320 20 2 20 20 21

1/2 1/221 21 2 21 20 21 20 22

/22 22 2 22 21 22 21

20 21 22

, ,

| | sgn( )

| | sgn( )

| | sgn( )

ˆ ˆ, ,

y y

y y y y y y y

y y y y y y

y y y y y y y y

q py y y y y y y

y y t y

z v f z v z v

v L z v z v z

v L z v z v z

v L z v z v

z v z a z

ˆ

ta

(31)

where 12 3

tan pr zy y







12 23 2 2

* 23 23 23 2 2 2 2

123 22 2 2 2 1 2

1

* * * * *3 3 3 3 3 3

*3 3 3

* *4 33 3 3 3 3 3

* *4 4 4 4

( )

( )

ˆ2 tan

cos

, (0) (0)

ˆ

y y y y d

y y y y y d y y

ty y y p y p y d

p

y y y y y y

y y y

y y y y y y y

y y y y

s a x x

x a a x x k s

aa a x x x x x

r x

x x x x x

s x x

x a x d x k s

x x x

4

* * *4 4

*4 4 4

1 *4 43 3 44 4 4 4 4 4 5 4 4

, (0) (0)

ˆ[ | | sgn( )]y

y y

y y y


x x

s x x


(32)


(32)

where 0<λy4<1; sy2, sy3 and sy4 are the error surfaces of the

guidance loop, sideslip angle loop and yaw angular rate loop

in the yaw channel; ky2, ky3 and ky4 are the corresponding

error surface gains; τy3 and τy4 are the filter time constants;

12

loop and yaw angular rate loop in the yaw channel; 2yk , 3yk and 4yk are the corresponding error

surface gains; 3y and 4y are the filter time constants; *3yx and *

4yx are the virtual control of

the guidance loop and sideslip angle loop in the yaw channel; *3yx and *

4yx are the corresponding

filtered virtual control.

3.3 Control design in the roll channel

In this section, we firstly design the FTDOs to estimate the disturbances 3rd and 4rd in the roll

angle loop and roll angular rate loop, and the estimated errors are defined as 31 31 3r r re z d and

41 41 4r r re z d , respectively.

For the system (11), we consider the similar backstepping sliding mode control law:

4

3 3 3

*4 3 3 3 3

* * * * *4 4 4 4 4 4

*4 4 4

1 *4 4 4 4 4 5 4 4

ˆ

, (0) (0)

ˆ[ | | sgn( )]r

r r r d

r r r d r r

r r r r r r

r r r

r r r r r r r r r

s x x

x d x k sx x x x x

s x x

v b d x k s k s s

(33)

where 40 1r , 3rs and 4rs are the error surfaces of the roll angle loop and roll angular rate

loop, 3rk and 4rk are the corresponding error surface gains, 4r is the filter time constant.

3.4 Stability analysis

It is assumed that the estimated errors in the FTDO satisfy the following constraints [6]-[7]:

21 2 31 3 41 4

21 2 31 3 41 4

31 3 41 4

| | ,| | , | |

| | , | | , | |

| | , | |

p p p p p p

y y y y y y

r r r r

e N e N e Ne N e N e Ne N e N

(34)

where piN , yiN and rjN are positive constants, {2,3,4}i , {3,4}j .

Define the following filter errors:

* * * *3 3 3 4 4 4

* * * *3 3 3 4 4 4

* *4 4 4

,

,p p p p p p

y y y y y y

r r r

y x x y x x

y x x y x x

y x x

(35)

and

12












4

3 3 3

*4 3 3 3 3

* * * * *4 4 4 4 4 4

*4 4 4

1 *4 4 4 4 4 5 4 4

ˆ

, (0) (0)

ˆ[ | | sgn( )]r

r r r d

r r r d r r

r r r r r r

r r r

r r r r r r r r r

s x x

x d x k sx x x x x

s x x

v b d x k s k s s

(33)





21 2 31 3 41 4

21 2 31 3 41 4

31 3 41 4

| | ,| | , | |

| | , | | , | |

| | , | |

p p p p p p

y y y y y y

r r r r


(34)



* * * *3 3 3 4 4 4

* * * *3 3 3 4 4 4

* *4 4 4

,

,p p p p p p

y y y y y y

r r r

y x x y x x

y x x y x x

y x x

(35)

are the virtual control of the guidance loop and

sideslip angle loop in the yaw channel;

12












4

3 3 3

*4 3 3 3 3

* * * * *4 4 4 4 4 4

*4 4 4

1 *4 4 4 4 4 5 4 4

ˆ

, (0) (0)

ˆ[ | | sgn( )]r

r r r d

r r r d r r

r r r r r r

r r r

r r r r r r r r r

s x x

x d x k sx x x x x

s x x

v b d x k s k s s

(33)





21 2 31 3 41 4

21 2 31 3 41 4

31 3 41 4

| | ,| | , | |

| | , | | , | |

| | , | |

p p p p p p

y y y y y y

r r r r


(34)



* * * *3 3 3 4 4 4

* * * *3 3 3 4 4 4

* *4 4 4

,

,p p p p p p

y y y y y y

r r r

y x x y x x

y x x y x x

y x x

(35)

and

12












4

3 3 3

*4 3 3 3 3

* * * * *4 4 4 4 4 4

*4 4 4

1 *4 4 4 4 4 5 4 4

ˆ

, (0) (0)

ˆ[ | | sgn( )]r

r r r d

r r r d r r

r r r r r r

r r r

r r r r r r r r r

s x x

x d x k sx x x x x

s x x

v b d x k s k s s

(33)





21 2 31 3 41 4

21 2 31 3 41 4

31 3 41 4

| | ,| | , | |

| | , | | , | |

| | , | |

p p p p p p

y y y y y y

r r r r


(34)



* * * *3 3 3 4 4 4

* * * *3 3 3 4 4 4

* *4 4 4

,

,p p p p p p

y y y y y y

r r r

y x x y x x

y x x y x x

y x x

(35)

are the

corresponding filtered virtual control.


In this section, we firstly design the FTDOs to estimate the

(278~294)14-087.indd 282 2015-07-03 오전 5:06:27

283


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disturbances dr3 and dr4 in the roll angle loop and roll angular

rate loop, and the estimated errors are defined as er31=zr31-dr3

and er41=zr41-dr4, respectively.

For the system (11), we consider the similar backstepping

sliding mode control law:

12












4

3 3 3

*4 3 3 3 3

* * * * *4 4 4 4 4 4

*4 4 4

1 *4 4 4 4 4 5 4 4

ˆ

, (0) (0)

ˆ[ | | sgn( )]r

r r r d

r r r d r r

r r r r r r

r r r

r r r r r r r r r

s x x

x d x k sx x x x x

s x x

v b d x k s k s s

(33)





21 2 31 3 41 4

21 2 31 3 41 4

31 3 41 4

| | ,| | , | |

| | , | | , | |

| | , | |

p p p p p p

y y y y y y

r r r r


(34)



* * * *3 3 3 4 4 4

* * * *3 3 3 4 4 4

* *4 4 4

,

,p p p p p p

y y y y y y

r r r

y x x y x x

y x x y x x

y x x

(35)

(33)

where 0<λr4<1, sr3 and sr4 are the error surfaces of the roll

angle loop and roll angular rate loop, kr3 and kr4 are the

corresponding error surface gains, τr4 is the filter time

constant.


It is assumed that the estimated errors in the FTDO satisfy

the following constraints [6]-[7]:

12












4

3 3 3

*4 3 3 3 3

* * * * *4 4 4 4 4 4

*4 4 4

1 *4 4 4 4 4 5 4 4

ˆ

, (0) (0)

ˆ[ | | sgn( )]r

r r r d

r r r d r r

r r r r r r

r r r

r r r r r r r r r

s x x

x d x k sx x x x x

s x x

v b d x k s k s s

(33)





21 2 31 3 41 4

21 2 31 3 41 4

31 3 41 4

| | ,| | , | |

| | , | | , | |

| | , | |

p p p p p p

y y y y y y

r r r r


(34)



* * * *3 3 3 4 4 4

* * * *3 3 3 4 4 4

* *4 4 4

,

,p p p p p p

y y y y y y

r r r

y x x y x x

y x x y x x

y x x

(35)

(34)

where Npi, Nyi and Nri are positive constants,

12












4

3 3 3

*4 3 3 3 3

* * * * *4 4 4 4 4 4

*4 4 4

1 *4 4 4 4 4 5 4 4

ˆ

, (0) (0)

ˆ[ | | sgn( )]r

r r r d

r r r d r r

r r r r r r

r r r

r r r r r r r r r

s x x

x d x k sx x x x x

s x x

v b d x k s k s s

(33)





21 2 31 3 41 4

21 2 31 3 41 4

31 3 41 4

| | ,| | , | |

| | , | | , | |

| | , | |

p p p p p p

y y y y y y

r r r r


(34)



* * * *3 3 3 4 4 4

* * * *3 3 3 4 4 4

* *4 4 4

,

,p p p p p p

y y y y y y

r r r

y x x y x x

y x x y x x

y x x

(35)

,

12












4

3 3 3

*4 3 3 3 3

* * * * *4 4 4 4 4 4

*4 4 4

1 *4 4 4 4 4 5 4 4

ˆ

, (0) (0)

ˆ[ | | sgn( )]r

r r r d

r r r d r r

r r r r r r

r r r

r r r r r r r r r

s x x

x d x k sx x x x x

s x x

v b d x k s k s s

(33)





21 2 31 3 41 4

21 2 31 3 41 4

31 3 41 4

| | ,| | , | |

| | , | | , | |

| | , | |

p p p p p p

y y y y y y

r r r r


(34)



* * * *3 3 3 4 4 4

* * * *3 3 3 4 4 4

* *4 4 4

,

,p p p p p p

y y y y y y

r r r

y x x y x x

y x x y x x

y x x

(35)

.


12












4

3 3 3

*4 3 3 3 3

* * * * *4 4 4 4 4 4

*4 4 4

1 *4 4 4 4 4 5 4 4

ˆ

, (0) (0)

ˆ[ | | sgn( )]r

r r r d

r r r d r r

r r r r r r

r r r

r r r r r r r r r

s x x

x d x k sx x x x x

s x x

v b d x k s k s s

(33)





21 2 31 3 41 4

21 2 31 3 41 4

31 3 41 4

| | ,| | , | |

| | , | | , | |

| | , | |

p p p p p p

y y y y y y

r r r r


(34)



* * * *3 3 3 4 4 4

* * * *3 3 3 4 4 4

* *4 4 4

,

,p p p p p p

y y y y y y

r r r

y x x y x x

y x x y x x

y x x

(35) (35)

Then, the dynamics of the filter errors satisfy:

13


1 * 1 *3 3 3 3 4 4 4 4

1 * 1 *3 3 3 3 4 4 4 4

1 *4 4 4 4

,

,p p p p p p p p

y y y y y y y y

r r r r

y y x y y x

y y x y y x

y y x

(36)

From equations (17)~(29) and (32)~(33), we can obtain:

4

2 23 2 2

* *3 3 3 3 3 3

* *4 4 4 4 4 4

2 3 3 2 2 21

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )p

p p p p d

p p p p p p

p p p p p p

p p p p p p

p p p p p p

p p p p p p p

x a s x

x s x s y x

x s x s y xs s y k s es s y k s e

s k s k s s e

(37)

4

2 23 2 2

* *3 3 3 3 3 3

* *4 4 4 4 4 4

2 3 3 2 2 21

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )y

y y y y d

y y y y y y

y y y y y y

y y y y y y

y y y y y y

y y y y y y y

x a s x

x s x s y x


s k s k s s e

(38)

4

* *4 4 4 4 4 4

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )r

r r r r r r

r r r r r r

r r r r r r r

x s x s y xs s y k s es k s k s s e

(39)

where 21 21p py

me eqSC

, 21 21y y

z

me eqSC

. It is assumed that 21 2| |p pe N and 21 2| |y ye N .

For the system (9)~(11), consider the following Lyapunov function:

p y rV V V V (40)

where

2 2 2 2 22 3 4 3 40.5 0.5 0.5 0.5 0.5p p p p p pV s s s y y ,

2 2 2 2 22 3 4 3 40.5 0.5 0.5 0.5 0.5y y y y y yV s s s y y ,

(36)


13


1 * 1 *3 3 3 3 4 4 4 4

1 * 1 *3 3 3 3 4 4 4 4

1 *4 4 4 4

,

,p p p p p p p p

y y y y y y y y

r r r r

y y x y y x

y y x y y x

y y x

(36)


4

2 23 2 2

* *3 3 3 3 3 3

* *4 4 4 4 4 4

2 3 3 2 2 21

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )p

p p p p d

p p p p p p

p p p p p p

p p p p p p

p p p p p p

p p p p p p p

x a s x

x s x s y x


s k s k s s e

(37)

4

2 23 2 2

* *3 3 3 3 3 3

* *4 4 4 4 4 4

2 3 3 2 2 21

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )y

y y y y d

y y y y y y

y y y y y y

y y y y y y

y y y y y y

y y y y y y y

x a s x

x s x s y x


s k s k s s e

(38)

4

* *4 4 4 4 4 4

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )r

r r r r r r

r r r r r r

r r r r r r r


(39)

where 21 21p py

me eqSC

, 21 21y y

z

me eqSC



p y rV V V V (40)

where

2 2 2 2 22 3 4 3 40.5 0.5 0.5 0.5 0.5p p p p p pV s s s y y ,

2 2 2 2 22 3 4 3 40.5 0.5 0.5 0.5 0.5y y y y y yV s s s y y ,

(37)

13


1 * 1 *3 3 3 3 4 4 4 4

1 * 1 *3 3 3 3 4 4 4 4

1 *4 4 4 4

,

,p p p p p p p p

y y y y y y y y

r r r r

y y x y y x

y y x y y x

y y x

(36)


4

2 23 2 2

* *3 3 3 3 3 3

* *4 4 4 4 4 4

2 3 3 2 2 21

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )p

p p p p d

p p p p p p

p p p p p p

p p p p p p

p p p p p p

p p p p p p p

x a s x

x s x s y x


s k s k s s e

(37)

4

2 23 2 2

* *3 3 3 3 3 3

* *4 4 4 4 4 4

2 3 3 2 2 21

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )y

y y y y d

y y y y y y

y y y y y y

y y y y y y

y y y y y y

y y y y y y y

x a s x

x s x s y x


s k s k s s e

(38)

4

* *4 4 4 4 4 4

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )r

r r r r r r

r r r r r r

r r r r r r r


(39)

where 21 21p py

me eqSC

, 21 21y y

z

me eqSC



p y rV V V V (40)

where

2 2 2 2 22 3 4 3 40.5 0.5 0.5 0.5 0.5p p p p p pV s s s y y ,

2 2 2 2 22 3 4 3 40.5 0.5 0.5 0.5 0.5y y y y y yV s s s y y ,

(38)

13


1 * 1 *3 3 3 3 4 4 4 4

1 * 1 *3 3 3 3 4 4 4 4

1 *4 4 4 4

,

,p p p p p p p p

y y y y y y y y

r r r r

y y x y y x

y y x y y x

y y x

(36)


4

2 23 2 2

* *3 3 3 3 3 3

* *4 4 4 4 4 4

2 3 3 2 2 21

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )p

p p p p d

p p p p p p

p p p p p p

p p p p p p

p p p p p p

p p p p p p p

x a s x

x s x s y x


s k s k s s e

(37)

4

2 23 2 2

* *3 3 3 3 3 3

* *4 4 4 4 4 4

2 3 3 2 2 21

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )y

y y y y d

y y y y y y

y y y y y y

y y y y y y

y y y y y y

y y y y y y y

x a s x

x s x s y x


s k s k s s e

(38)

4

* *4 4 4 4 4 4

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )r

r r r r r r

r r r r r r

r r r r r r r


(39)

where 21 21p py

me eqSC

, 21 21y y

z

me eqSC



p y rV V V V (40)

where

2 2 2 2 22 3 4 3 40.5 0.5 0.5 0.5 0.5p p p p p pV s s s y y ,

2 2 2 2 22 3 4 3 40.5 0.5 0.5 0.5 0.5y y y y y yV s s s y y ,

(39)

where

13


1 * 1 *3 3 3 3 4 4 4 4

1 * 1 *3 3 3 3 4 4 4 4

1 *4 4 4 4

,

,p p p p p p p p

y y y y y y y y

r r r r

y y x y y x

y y x y y x

y y x

(36)


4

2 23 2 2

* *3 3 3 3 3 3

* *4 4 4 4 4 4

2 3 3 2 2 21

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )p

p p p p d

p p p p p p

p p p p p p

p p p p p p

p p p p p p

p p p p p p p

x a s x

x s x s y x


s k s k s s e

(37)

4

2 23 2 2

* *3 3 3 3 3 3

* *4 4 4 4 4 4

2 3 3 2 2 21

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )y

y y y y d

y y y y y y

y y y y y y

y y y y y y

y y y y y y

y y y y y y y

x a s x

x s x s y x


s k s k s s e

(38)

4

* *4 4 4 4 4 4

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )r

r r r r r r

r r r r r r

r r r r r r r


(39)

where 21 21p py

me eqSC

, 21 21y y

z

me eqSC



p y rV V V V (40)

where

2 2 2 2 22 3 4 3 40.5 0.5 0.5 0.5 0.5p p p p p pV s s s y y ,

2 2 2 2 22 3 4 3 40.5 0.5 0.5 0.5 0.5y y y y y yV s s s y y ,

,

13


1 * 1 *3 3 3 3 4 4 4 4

1 * 1 *3 3 3 3 4 4 4 4

1 *4 4 4 4

,

,p p p p p p p p

y y y y y y y y

r r r r

y y x y y x

y y x y y x

y y x

(36)


4

2 23 2 2

* *3 3 3 3 3 3

* *4 4 4 4 4 4

2 3 3 2 2 21

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )p

p p p p d

p p p p p p

p p p p p p

p p p p p p

p p p p p p

p p p p p p p

x a s x

x s x s y x


s k s k s s e

(37)

4

2 23 2 2

* *3 3 3 3 3 3

* *4 4 4 4 4 4

2 3 3 2 2 21

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )y

y y y y d

y y y y y y

y y y y y y

y y y y y y

y y y y y y

y y y y y y y

x a s x

x s x s y x


s k s k s s e

(38)

4

* *4 4 4 4 4 4

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )r

r r r r r r

r r r r r r

r r r r r r r


(39)

where 21 21p py

me eqSC

, 21 21y y

z

me eqSC



p y rV V V V (40)

where

2 2 2 2 22 3 4 3 40.5 0.5 0.5 0.5 0.5p p p p p pV s s s y y ,

2 2 2 2 22 3 4 3 40.5 0.5 0.5 0.5 0.5y y y y y yV s s s y y ,

. It is assumed

that

13


1 * 1 *3 3 3 3 4 4 4 4

1 * 1 *3 3 3 3 4 4 4 4

1 *4 4 4 4

,

,p p p p p p p p

y y y y y y y y

r r r r

y y x y y x

y y x y y x

y y x

(36)


4

2 23 2 2

* *3 3 3 3 3 3

* *4 4 4 4 4 4

2 3 3 2 2 21

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )p

p p p p d

p p p p p p

p p p p p p

p p p p p p

p p p p p p

p p p p p p p

x a s x

x s x s y x


s k s k s s e

(37)

4

2 23 2 2

* *3 3 3 3 3 3

* *4 4 4 4 4 4

2 3 3 2 2 21

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )y

y y y y d

y y y y y y

y y y y y y

y y y y y y

y y y y y y

y y y y y y y

x a s x

x s x s y x


s k s k s s e

(38)

4

* *4 4 4 4 4 4

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )r

r r r r r r

r r r r r r

r r r r r r r


(39)

where 21 21p py

me eqSC

, 21 21y y

z

me eqSC



p y rV V V V (40)

where

2 2 2 2 22 3 4 3 40.5 0.5 0.5 0.5 0.5p p p p p pV s s s y y ,

2 2 2 2 22 3 4 3 40.5 0.5 0.5 0.5 0.5y y y y y yV s s s y y ,

and

13


1 * 1 *3 3 3 3 4 4 4 4

1 * 1 *3 3 3 3 4 4 4 4

1 *4 4 4 4

,

,p p p p p p p p

y y y y y y y y

r r r r

y y x y y x

y y x y y x

y y x

(36)


4

2 23 2 2

* *3 3 3 3 3 3

* *4 4 4 4 4 4

2 3 3 2 2 21

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )p

p p p p d

p p p p p p

p p p p p p

p p p p p p

p p p p p p

p p p p p p p

x a s x

x s x s y x


s k s k s s e

(37)

4

2 23 2 2

* *3 3 3 3 3 3

* *4 4 4 4 4 4

2 3 3 2 2 21

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )y

y y y y d

y y y y y y

y y y y y y

y y y y y y

y y y y y y

y y y y y y y

x a s x

x s x s y x


s k s k s s e

(38)

4

* *4 4 4 4 4 4

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )r

r r r r r r

r r r r r r

r r r r r r r


(39)

where 21 21p py

me eqSC

, 21 21y y

z

me eqSC



p y rV V V V (40)

where

2 2 2 2 22 3 4 3 40.5 0.5 0.5 0.5 0.5p p p p p pV s s s y y ,

2 2 2 2 22 3 4 3 40.5 0.5 0.5 0.5 0.5y y y y y yV s s s y y ,

.

For the system (9)~(11), consider the following Lyapunov

function:

13


1 * 1 *3 3 3 3 4 4 4 4

1 * 1 *3 3 3 3 4 4 4 4

1 *4 4 4 4

,

,p p p p p p p p

y y y y y y y y

r r r r

y y x y y x

y y x y y x

y y x

(36)


4

2 23 2 2

* *3 3 3 3 3 3

* *4 4 4 4 4 4

2 3 3 2 2 21

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )p

p p p p d

p p p p p p

p p p p p p

p p p p p p

p p p p p p

p p p p p p p

x a s x

x s x s y x


s k s k s s e

(37)

4

2 23 2 2

* *3 3 3 3 3 3

* *4 4 4 4 4 4

2 3 3 2 2 21

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )y

y y y y d

y y y y y y

y y y y y y

y y y y y y

y y y y y y

y y y y y y y

x a s x

x s x s y x


s k s k s s e

(38)

4

* *4 4 4 4 4 4

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )r

r r r r r r

r r r r r r

r r r r r r r


(39)

where 21 21p py

me eqSC

, 21 21y y

z

me eqSC



p y rV V V V (40)

where

2 2 2 2 22 3 4 3 40.5 0.5 0.5 0.5 0.5p p p p p pV s s s y y ,

2 2 2 2 22 3 4 3 40.5 0.5 0.5 0.5 0.5y y y y y yV s s s y y ,

(40)

where

13


1 * 1 *3 3 3 3 4 4 4 4

1 * 1 *3 3 3 3 4 4 4 4

1 *4 4 4 4

,

,p p p p p p p p

y y y y y y y y

r r r r

y y x y y x

y y x y y x

y y x

(36)


4

2 23 2 2

* *3 3 3 3 3 3

* *4 4 4 4 4 4

2 3 3 2 2 21

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )p

p p p p d

p p p p p p

p p p p p p

p p p p p p

p p p p p p

p p p p p p p

x a s x

x s x s y x


s k s k s s e

(37)

4

2 23 2 2

* *3 3 3 3 3 3

* *4 4 4 4 4 4

2 3 3 2 2 21

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )y

y y y y d

y y y y y y

y y y y y y

y y y y y y

y y y y y y

y y y y y y y

x a s x

x s x s y x


s k s k s s e

(38)

4

* *4 4 4 4 4 4

3 4 4 3 3 31

4 4 4 5 4 4 41| | sgn( )r

r r r r r r

r r r r r r

r r r r r r r


(39)

where 21 21p py

me eqSC

, 21 21y y

z

me eqSC



p y rV V V V (40)

where

2 2 2 2 22 3 4 3 40.5 0.5 0.5 0.5 0.5p p p p p pV s s s y y ,

2 2 2 2 22 3 4 3 40.5 0.5 0.5 0.5 0.5y y y y y yV s s s y y ,

14

2 2 23 4 40.5 0.5 0.5r r r rV s s y .

The coefficients and variables for the system (9) and their derivatives are all bounded. By some

simple calculations, we have:

*3 3 2 3 3 21 2 2 2 2| | ( , , , , , , , )p p p p p p p p d p d p dx s s y e k x x x (41)

*4 4 2 3 4 3 4 21 31 2 3 2 2 2| | ( , , , , , , , , , , , )p p p p p p p p p p p p d p d p dx s s s y y e e k k x x x (42)

where 3p and 4p are both nonnegative continuous functions. For some given positive constants

p and *pR , the following sets

2 2 21 2 2 2 2 2 2{[ , , ] , }T

p p d p d p d p d p d p d pB x x x x x x (43)

*2 2 3 4 3 4 21 31{[ , , , , , , ] , }T

p p p p p p p p p pB s s s y y e e V R (44)

are compact. Obviously, 1 2p pB B is also compact. Therefore, 3p and 4p have maximum

values on the set 1 2p pB B , which satisfy the following conditions:

*3 3| |p px M , *

4 4| |p px M (45)

Based on the same method, we can also obtain:

*3 3| |y yx M , *

4 4| |y yx M , *4 4| |r rx M (46)

where 3pM , 4pM , 3yM , 4yM and 4rM are positive constants.

From equation (37), we have:

2 2 2 3 3 2 2 21( )p p p p p p p ps s s s y k s e

2 2 2 2 2 2 22 2 2 3 2 3 2 210.5( ) 0.5( ) 0.5( | | )p p p p p p p pk s s s s y s e

2 2 2 22 2 3 3 2(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (47)



2 2 2 23 3 4 4 3(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (48)

The coefficients and variables for the system (9) and their

derivatives are all bounded. By some simple calculations, we

have:

14

2 2 23 4 40.5 0.5 0.5r r r rV s s y .







2 2 21 2 2 2 2 2 2{[ , , ] , }T


*2 2 3 4 3 4 21 31{[ , , , , , , ] , }T




*3 3| |p px M , *

4 4| |p px M (45)


*3 3| |y yx M , *

4 4| |y yx M , *4 4| |r rx M (46)





2 2 2 22 2 3 3 2(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (47)



2 2 2 23 3 4 4 3(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (48)

(41)

14

2 2 23 4 40.5 0.5 0.5r r r rV s s y .







2 2 21 2 2 2 2 2 2{[ , , ] , }T


*2 2 3 4 3 4 21 31{[ , , , , , , ] , }T




*3 3| |p px M , *

4 4| |p px M (45)


*3 3| |y yx M , *

4 4| |y yx M , *4 4| |r rx M (46)





2 2 2 22 2 3 3 2(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (47)



2 2 2 23 3 4 4 3(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (48)

(42)

where

14

2 2 23 4 40.5 0.5 0.5r r r rV s s y .







2 2 21 2 2 2 2 2 2{[ , , ] , }T


*2 2 3 4 3 4 21 31{[ , , , , , , ] , }T




*3 3| |p px M , *

4 4| |p px M (45)


*3 3| |y yx M , *

4 4| |y yx M , *4 4| |r rx M (46)





2 2 2 22 2 3 3 2(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (47)



2 2 2 23 3 4 4 3(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (48)

and

14

2 2 23 4 40.5 0.5 0.5r r r rV s s y .







2 2 21 2 2 2 2 2 2{[ , , ] , }T


*2 2 3 4 3 4 21 31{[ , , , , , , ] , }T




*3 3| |p px M , *

4 4| |p px M (45)


*3 3| |y yx M , *

4 4| |y yx M , *4 4| |r rx M (46)





2 2 2 22 2 3 3 2(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (47)



2 2 2 23 3 4 4 3(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (48)

are both nonnegative continuous

functions. For some given positive constants

14

2 2 23 4 40.5 0.5 0.5r r r rV s s y .







2 2 21 2 2 2 2 2 2{[ , , ] , }T


*2 2 3 4 3 4 21 31{[ , , , , , , ] , }T




*3 3| |p px M , *

4 4| |p px M (45)


*3 3| |y yx M , *

4 4| |y yx M , *4 4| |r rx M (46)





2 2 2 22 2 3 3 2(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (47)



2 2 2 23 3 4 4 3(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (48)

and

14

2 2 23 4 40.5 0.5 0.5r r r rV s s y .







2 2 21 2 2 2 2 2 2{[ , , ] , }T


*2 2 3 4 3 4 21 31{[ , , , , , , ] , }T




*3 3| |p px M , *

4 4| |p px M (45)


*3 3| |y yx M , *

4 4| |y yx M , *4 4| |r rx M (46)





2 2 2 22 2 3 3 2(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (47)



2 2 2 23 3 4 4 3(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (48)

, the

following sets

14

2 2 23 4 40.5 0.5 0.5r r r rV s s y .







2 2 21 2 2 2 2 2 2{[ , , ] , }T


*2 2 3 4 3 4 21 31{[ , , , , , , ] , }T




*3 3| |p px M , *

4 4| |p px M (45)


*3 3| |y yx M , *

4 4| |y yx M , *4 4| |r rx M (46)





2 2 2 22 2 3 3 2(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (47)



2 2 2 23 3 4 4 3(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (48)

(43)

14

2 2 23 4 40.5 0.5 0.5r r r rV s s y .







2 2 21 2 2 2 2 2 2{[ , , ] , }T


*2 2 3 4 3 4 21 31{[ , , , , , , ] , }T




*3 3| |p px M , *

4 4| |p px M (45)


*3 3| |y yx M , *

4 4| |y yx M , *4 4| |r rx M (46)





2 2 2 22 2 3 3 2(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (47)



2 2 2 23 3 4 4 3(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (48)

(44)

are compact. Obviously, Bp1×Bp2 is also compact. Therefore,

14

2 2 23 4 40.5 0.5 0.5r r r rV s s y .







2 2 21 2 2 2 2 2 2{[ , , ] , }T


*2 2 3 4 3 4 21 31{[ , , , , , , ] , }T




*3 3| |p px M , *

4 4| |p px M (45)


*3 3| |y yx M , *

4 4| |y yx M , *4 4| |r rx M (46)





2 2 2 22 2 3 3 2(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (47)



2 2 2 23 3 4 4 3(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (48)

and

14

2 2 23 4 40.5 0.5 0.5r r r rV s s y .







2 2 21 2 2 2 2 2 2{[ , , ] , }T


*2 2 3 4 3 4 21 31{[ , , , , , , ] , }T




*3 3| |p px M , *

4 4| |p px M (45)


*3 3| |y yx M , *

4 4| |y yx M , *4 4| |r rx M (46)





2 2 2 22 2 3 3 2(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (47)



2 2 2 23 3 4 4 3(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (48)

have maximum values on the set Bp1×Bp2, which

satisfy the following conditions:

14

2 2 23 4 40.5 0.5 0.5r r r rV s s y .







2 2 21 2 2 2 2 2 2{[ , , ] , }T


*2 2 3 4 3 4 21 31{[ , , , , , , ] , }T




*3 3| |p px M , *

4 4| |p px M (45)


*3 3| |y yx M , *

4 4| |y yx M , *4 4| |r rx M (46)





2 2 2 22 2 3 3 2(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (47)



2 2 2 23 3 4 4 3(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (48)

(45)


(278~294)14-087.indd 283 2015-07-03 오전 5:06:30



14

2 2 23 4 40.5 0.5 0.5r r r rV s s y .







2 2 21 2 2 2 2 2 2{[ , , ] , }T


*2 2 3 4 3 4 21 31{[ , , , , , , ] , }T




*3 3| |p px M , *

4 4| |p px M (45)


*3 3| |y yx M , *

4 4| |y yx M , *4 4| |r rx M (46)





2 2 2 22 2 3 3 2(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (47)



2 2 2 23 3 4 4 3(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (48)

(46)

where Mp3, Mp4, My3, My4 and Mr4 are positive constants.


14

2 2 23 4 40.5 0.5 0.5r r r rV s s y .







2 2 21 2 2 2 2 2 2{[ , , ] , }T


*2 2 3 4 3 4 21 31{[ , , , , , , ] , }T




*3 3| |p px M , *

4 4| |p px M (45)


*3 3| |y yx M , *

4 4| |y yx M , *4 4| |r rx M (46)





2 2 2 22 2 3 3 2(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (47)



2 2 2 23 3 4 4 3(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (48)

(47)

14

2 2 23 4 40.5 0.5 0.5r r r rV s s y .







2 2 21 2 2 2 2 2 2{[ , , ] , }T


*2 2 3 4 3 4 21 31{[ , , , , , , ] , }T




*3 3| |p px M , *

4 4| |p px M (45)


*3 3| |y yx M , *

4 4| |y yx M , *4 4| |r rx M (46)





2 2 2 22 2 3 3 2(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (47)



2 2 2 23 3 4 4 3(1.5 ) 0.5 0.5 0.5p p p p pk s s y N (48)

(48)

15

44 4 4 4 4 5 4 4 41[ | | sgn( ) ]p

p p p p p p p p ps s s k s k s s e

4 12 2 24 4 4 41 5 40.5( | | ) | | p

p p p p p pk s s e k s

2 2 24 4 4 41 5 4 40.5( | | ) | |(1 | |)p p p p p p pk s s e k s s

2 2 2 2 24 4 4 41 5 4 40.5( | | ) (0.5 0.5 )p p p p p p pk s s e k s s

2 24 5 4 4 5(0.5 1.5 ) 0.5 0.5p p p p pk k s N k (49)

1 *3 3 3 3 3 3( )p p p p p py y y y x 1 2 2 * 2

3 3 3 30.5( | | )p p p py y x

1 2 2 23 3 3 30.5( )p p p py y M (50)

1 *4 4 4 4 4 4( )p p p p p py y y y x 1 2 2 * 2

4 4 4 40.5( | | )p p p py y x

1 2 2 24 4 4 40.5( )p p p py y M (51)

Based on the same method, from equations (38) and (39), we can obtain:

2 2 2 22 2 2 2 3 3 2

2 2 2 23 3 3 3 4 4 3

2 24 4 4 5 4 4 5

1 2 2 23 3 3 3 3 3

1 2 2 24 4 4 4 4 4

(1.5 ) 0.5 0.5 0.5

(1.5 ) 0.5 0.5 0.5

(0.5 1.5 ) 0.5 0.5

0.5( )

0.5( )

y y y y y y y

y y y y y y y

y y y y y y y

y y y y y y

y y y y y y

s s k s s y N

s s k s s y N

s s k k s N k

y y y y M

y y y y M

(52)

2 2 2 23 3 3 3 4 4 3

2 24 4 4 5 4 4 5

1 2 2 24 4 4 4 4 4

(1.5 ) 0.5 0.5 0.5

(0.5 1.5 ) 0.5 0.5

0.5( )

r r r r r r r

r r r r r r r

r r r r r r

s s k s s y Ns s k k s N ky y y y M

(53)

By taking the time derivative of ( )V t along the system (9)~(11), we have:

2 2 3 3 4 4 3 3 4 4 2 2p p p p p p p p p p y yV s s s s s s y y y y s s

3 3 4 4 3 3 4 4 3 3 4 4 4 4y y y y y y y y r r r r r rs s s s y y y y s s s s y y

2 2 2 1 22 2 3 3 4 5 4 3 3(1.5 ) (2 ) (1 1.5 ) (1 )p p p p p p p p pk s k s k k s y

1 2 2 2 2 2 24 4 2 3 4 3 4 5(1 ) 0.5 0.5 0.5 0.5 0.5 0.5p p p p p p p py N N N M M k

(49)

15

44 4 4 4 4 5 4 4 41[ | | sgn( ) ]p


4 12 2 24 4 4 41 5 40.5( | | ) | | p


2 2 24 4 4 41 5 4 40.5( | | ) | |(1 | |)p p p p p p pk s s e k s s

2 2 2 2 24 4 4 41 5 4 40.5( | | ) (0.5 0.5 )p p p p p p pk s s e k s s

2 24 5 4 4 5(0.5 1.5 ) 0.5 0.5p p p p pk k s N k (49)

1 *3 3 3 3 3 3( )p p p p p py y y y x 1 2 2 * 2

3 3 3 30.5( | | )p p p py y x

1 2 2 23 3 3 30.5( )p p p py y M (50)

1 *4 4 4 4 4 4( )p p p p p py y y y x 1 2 2 * 2

4 4 4 40.5( | | )p p p py y x

1 2 2 24 4 4 40.5( )p p p py y M (51)


2 2 2 22 2 2 2 3 3 2

2 2 2 23 3 3 3 4 4 3

2 24 4 4 5 4 4 5

1 2 2 23 3 3 3 3 3

1 2 2 24 4 4 4 4 4

(1.5 ) 0.5 0.5 0.5

(1.5 ) 0.5 0.5 0.5

(0.5 1.5 ) 0.5 0.5

0.5( )

0.5( )

y y y y y y y

y y y y y y y

y y y y y y y

y y y y y y

y y y y y y

s s k s s y N

s s k s s y N

s s k k s N k

y y y y M

y y y y M

(52)

2 2 2 23 3 3 3 4 4 3

2 24 4 4 5 4 4 5

1 2 2 24 4 4 4 4 4

(1.5 ) 0.5 0.5 0.5

(0.5 1.5 ) 0.5 0.5

0.5( )

r r r r r r r

r r r r r r r

r r r r r r


(53)






(50)

15

44 4 4 4 4 5 4 4 41[ | | sgn( ) ]p


4 12 2 24 4 4 41 5 40.5( | | ) | | p


2 2 24 4 4 41 5 4 40.5( | | ) | |(1 | |)p p p p p p pk s s e k s s

2 2 2 2 24 4 4 41 5 4 40.5( | | ) (0.5 0.5 )p p p p p p pk s s e k s s

2 24 5 4 4 5(0.5 1.5 ) 0.5 0.5p p p p pk k s N k (49)

1 *3 3 3 3 3 3( )p p p p p py y y y x 1 2 2 * 2

3 3 3 30.5( | | )p p p py y x

1 2 2 23 3 3 30.5( )p p p py y M (50)

1 *4 4 4 4 4 4( )p p p p p py y y y x 1 2 2 * 2

4 4 4 40.5( | | )p p p py y x

1 2 2 24 4 4 40.5( )p p p py y M (51)


2 2 2 22 2 2 2 3 3 2

2 2 2 23 3 3 3 4 4 3

2 24 4 4 5 4 4 5

1 2 2 23 3 3 3 3 3

1 2 2 24 4 4 4 4 4

(1.5 ) 0.5 0.5 0.5

(1.5 ) 0.5 0.5 0.5

(0.5 1.5 ) 0.5 0.5

0.5( )

0.5( )

y y y y y y y

y y y y y y y

y y y y y y y

y y y y y y

y y y y y y

s s k s s y N

s s k s s y N

s s k k s N k

y y y y M

y y y y M

(52)

2 2 2 23 3 3 3 4 4 3

2 24 4 4 5 4 4 5

1 2 2 24 4 4 4 4 4

(1.5 ) 0.5 0.5 0.5

(0.5 1.5 ) 0.5 0.5

0.5( )

r r r r r r r

r r r r r r r

r r r r r r


(53)






(51)

Based on the same method, from equations (38) and (39),

we can obtain:

15

44 4 4 4 4 5 4 4 41[ | | sgn( ) ]p


4 12 2 24 4 4 41 5 40.5( | | ) | | p


2 2 24 4 4 41 5 4 40.5( | | ) | |(1 | |)p p p p p p pk s s e k s s

2 2 2 2 24 4 4 41 5 4 40.5( | | ) (0.5 0.5 )p p p p p p pk s s e k s s

2 24 5 4 4 5(0.5 1.5 ) 0.5 0.5p p p p pk k s N k (49)

1 *3 3 3 3 3 3( )p p p p p py y y y x 1 2 2 * 2

3 3 3 30.5( | | )p p p py y x

1 2 2 23 3 3 30.5( )p p p py y M (50)

1 *4 4 4 4 4 4( )p p p p p py y y y x 1 2 2 * 2

4 4 4 40.5( | | )p p p py y x

1 2 2 24 4 4 40.5( )p p p py y M (51)


2 2 2 22 2 2 2 3 3 2

2 2 2 23 3 3 3 4 4 3

2 24 4 4 5 4 4 5

1 2 2 23 3 3 3 3 3

1 2 2 24 4 4 4 4 4

(1.5 ) 0.5 0.5 0.5

(1.5 ) 0.5 0.5 0.5

(0.5 1.5 ) 0.5 0.5

0.5( )

0.5( )

y y y y y y y

y y y y y y y

y y y y y y y

y y y y y y

y y y y y y

s s k s s y N

s s k s s y N

s s k k s N k

y y y y M

y y y y M

(52)

2 2 2 23 3 3 3 4 4 3

2 24 4 4 5 4 4 5

1 2 2 24 4 4 4 4 4

(1.5 ) 0.5 0.5 0.5

(0.5 1.5 ) 0.5 0.5

0.5( )

r r r r r r r

r r r r r r r

r r r r r r


(53)






(52)

15

44 4 4 4 4 5 4 4 41[ | | sgn( ) ]p


4 12 2 24 4 4 41 5 40.5( | | ) | | p


2 2 24 4 4 41 5 4 40.5( | | ) | |(1 | |)p p p p p p pk s s e k s s

2 2 2 2 24 4 4 41 5 4 40.5( | | ) (0.5 0.5 )p p p p p p pk s s e k s s

2 24 5 4 4 5(0.5 1.5 ) 0.5 0.5p p p p pk k s N k (49)

1 *3 3 3 3 3 3( )p p p p p py y y y x 1 2 2 * 2

3 3 3 30.5( | | )p p p py y x

1 2 2 23 3 3 30.5( )p p p py y M (50)

1 *4 4 4 4 4 4( )p p p p p py y y y x 1 2 2 * 2

4 4 4 40.5( | | )p p p py y x

1 2 2 24 4 4 40.5( )p p p py y M (51)


2 2 2 22 2 2 2 3 3 2

2 2 2 23 3 3 3 4 4 3

2 24 4 4 5 4 4 5

1 2 2 23 3 3 3 3 3

1 2 2 24 4 4 4 4 4

(1.5 ) 0.5 0.5 0.5

(1.5 ) 0.5 0.5 0.5

(0.5 1.5 ) 0.5 0.5

0.5( )

0.5( )

y y y y y y y

y y y y y y y

y y y y y y y

y y y y y y

y y y y y y

s s k s s y N

s s k s s y N

s s k k s N k

y y y y M

y y y y M

(52)

2 2 2 23 3 3 3 4 4 3

2 24 4 4 5 4 4 5

1 2 2 24 4 4 4 4 4

(1.5 ) 0.5 0.5 0.5

(0.5 1.5 ) 0.5 0.5

0.5( )

r r r r r r r

r r r r r r r

r r r r r r


(53)






(53)

By taking the time derivative of V(t) along the system

(9)~(11), we have:

15

44 4 4 4 4 5 4 4 41[ | | sgn( ) ]p


4 12 2 24 4 4 41 5 40.5( | | ) | | p


2 2 24 4 4 41 5 4 40.5( | | ) | |(1 | |)p p p p p p pk s s e k s s

2 2 2 2 24 4 4 41 5 4 40.5( | | ) (0.5 0.5 )p p p p p p pk s s e k s s

2 24 5 4 4 5(0.5 1.5 ) 0.5 0.5p p p p pk k s N k (49)

1 *3 3 3 3 3 3( )p p p p p py y y y x 1 2 2 * 2

3 3 3 30.5( | | )p p p py y x

1 2 2 23 3 3 30.5( )p p p py y M (50)

1 *4 4 4 4 4 4( )p p p p p py y y y x 1 2 2 * 2

4 4 4 40.5( | | )p p p py y x

1 2 2 24 4 4 40.5( )p p p py y M (51)


2 2 2 22 2 2 2 3 3 2

2 2 2 23 3 3 3 4 4 3

2 24 4 4 5 4 4 5

1 2 2 23 3 3 3 3 3

1 2 2 24 4 4 4 4 4

(1.5 ) 0.5 0.5 0.5

(1.5 ) 0.5 0.5 0.5

(0.5 1.5 ) 0.5 0.5

0.5( )

0.5( )

y y y y y y y

y y y y y y y

y y y y y y y

y y y y y y

y y y y y y

s s k s s y N

s s k s s y N

s s k k s N k

y y y y M

y y y y M

(52)

2 2 2 23 3 3 3 4 4 3

2 24 4 4 5 4 4 5

1 2 2 24 4 4 4 4 4

(1.5 ) 0.5 0.5 0.5

(0.5 1.5 ) 0.5 0.5

0.5( )

r r r r r r r

r r r r r r r

r r r r r r


(53)






15

44 4 4 4 4 5 4 4 41[ | | sgn( ) ]p


4 12 2 24 4 4 41 5 40.5( | | ) | | p


2 2 24 4 4 41 5 4 40.5( | | ) | |(1 | |)p p p p p p pk s s e k s s

2 2 2 2 24 4 4 41 5 4 40.5( | | ) (0.5 0.5 )p p p p p p pk s s e k s s

2 24 5 4 4 5(0.5 1.5 ) 0.5 0.5p p p p pk k s N k (49)

1 *3 3 3 3 3 3( )p p p p p py y y y x 1 2 2 * 2

3 3 3 30.5( | | )p p p py y x

1 2 2 23 3 3 30.5( )p p p py y M (50)

1 *4 4 4 4 4 4( )p p p p p py y y y x 1 2 2 * 2

4 4 4 40.5( | | )p p p py y x

1 2 2 24 4 4 40.5( )p p p py y M (51)


2 2 2 22 2 2 2 3 3 2

2 2 2 23 3 3 3 4 4 3

2 24 4 4 5 4 4 5

1 2 2 23 3 3 3 3 3

1 2 2 24 4 4 4 4 4

(1.5 ) 0.5 0.5 0.5

(1.5 ) 0.5 0.5 0.5

(0.5 1.5 ) 0.5 0.5

0.5( )

0.5( )

y y y y y y y

y y y y y y y

y y y y y y y

y y y y y y

y y y y y y

s s k s s y N

s s k s s y N

s s k k s N k

y y y y M

y y y y M

(52)

2 2 2 23 3 3 3 4 4 3

2 24 4 4 5 4 4 5

1 2 2 24 4 4 4 4 4

(1.5 ) 0.5 0.5 0.5

(0.5 1.5 ) 0.5 0.5

0.5( )

r r r r r r r

r r r r r r r

r r r r r r


(53)






(54)

16

2 2 2 1 22 2 3 3 4 5 4 3 3(1.5 ) (2 ) (1 1.5 ) (1 )y y y y y y y y yk s k s k k s y

1 2 2 2 2 2 24 4 2 3 4 3 4 5(1 ) 0.5 0.5 0.5 0.5 0.5 0.5y y y y y y y yy N N N M M k

2 2 1 2 2 2 23 3 4 5 4 4 4 3 4 4 5(1.5 ) (1 1.5 ) (1 ) 0.5 0.5 0.5 0.5r r r r r r r r r r rk s k k s y N N M k

(54)

If the design parameters satisfy the following conditions:

2 3 4 5

1 13 4 2

3 4 5

1 13 4 3

14 4 5

1.5 0.5 , 2 0.5 , 1.5 1 0.5

1 0.5 , 1 0.5 , 1.5 0.5

2 0.5 , 1.5 1 0.5

1 0.5 , 1 0.5 , 1.5 0.5

1 0.5 , 1.5 1 0.5

p p p p

p p y

y y y

y y r

r r r

k k k k

kk k k

k

k k

(55)

where is a positive real number, then we have:

V V C (56)

where 2 2 2 2 2 2 22 3 4 3 4 5 2 30.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5p p p p p p y yC N N N M M k N N

2 2 2 2 2 24 3 4 5 3 4 4 50.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5y y y y r r r rN M M k N N M k .

Based on the comparison principle, from equation (56) we can obtain:

[ (0) ]( )tV C e CV t

(57)

If the parameter is selected large enough, the parameter /C can be made arbitrarily small.

Therefore, the error surfaces and filter errors are all uniformly ultimately bounded, and we can obtain

the small miss distance.

3.5 Robust least-squares weighted control allocation

In this section, we proposed a novel RLSWCA algorithm to distribute the virtual control command

among the individual actuators (aerodynamic fins and reaction jets), while also taking into account the

uncertainty in the control effectiveness matrix.

The elements of the uncertain terms cB are all usually small bounded. Therefore, we assume

that the uncertainty in the control effectiveness matrix satisfies the following constraint:


16




(54)


2 3 4 5

1 13 4 2

3 4 5

1 13 4 3

14 4 5

1.5 0.5 , 2 0.5 , 1.5 1 0.5

1 0.5 , 1 0.5 , 1.5 0.5

2 0.5 , 1.5 1 0.5

1 0.5 , 1 0.5 , 1.5 0.5

1 0.5 , 1.5 1 0.5

p p p p

p p y

y y y

y y r

r r r

k k k k

kk k k

k

k k

(55)


V V C (56)




[ (0) ]( )tV C e CV t

(57)










(55)

where κ is a positive real number, then we have:

16




(54)


2 3 4 5

1 13 4 2

3 4 5

1 13 4 3

14 4 5

1.5 0.5 , 2 0.5 , 1.5 1 0.5

1 0.5 , 1 0.5 , 1.5 0.5

2 0.5 , 1.5 1 0.5

1 0.5 , 1 0.5 , 1.5 0.5

1 0.5 , 1.5 1 0.5

p p p p

p p y

y y y

y y r

r r r

k k k k

kk k k

k

k k

(55)


V V C (56)




[ (0) ]( )tV C e CV t

(57)










(56)

where

16




(54)


2 3 4 5

1 13 4 2

3 4 5

1 13 4 3

14 4 5

1.5 0.5 , 2 0.5 , 1.5 1 0.5

1 0.5 , 1 0.5 , 1.5 0.5

2 0.5 , 1.5 1 0.5

1 0.5 , 1 0.5 , 1.5 0.5

1 0.5 , 1.5 1 0.5

p p p p

p p y

y y y

y y r

r r r

k k k k

kk k k

k

k k

(55)


V V C (56)




[ (0) ]( )tV C e CV t

(57)










Based on the comparison principle, from equation (56)

we can obtain:

16




(54)


2 3 4 5

1 13 4 2

3 4 5

1 13 4 3

14 4 5

1.5 0.5 , 2 0.5 , 1.5 1 0.5

1 0.5 , 1 0.5 , 1.5 0.5

2 0.5 , 1.5 1 0.5

1 0.5 , 1 0.5 , 1.5 0.5

1 0.5 , 1.5 1 0.5

p p p p

p p y

y y y

y y r

r r r

k k k k

kk k k

k

k k

(55)


V V C (56)




[ (0) ]( )tV C e CV t

(57)










(57)

If the parameter κ is selected large enough, the parameter

C/κ can be made arbitrarily small. Therefore, the error

surfaces and filter errors are all uniformly ultimately

bounded, and we can obtain the small miss distance.


In this section, we proposed a novel RLSWCA algorithm

to distribute the virtual control command among the

individual actuators (aerodynamic fins and reaction jets),

while also taking into account the uncertainty in the control

effectiveness matrix.

The elements of the uncertain terms △Bc are all usually

small bounded. Therefore, we assume that the uncertainty

in the control effectiveness matrix satisfies the following

constraint:

17

|| ||c B (58)

where is known constant. Generally, the individual actuators constraints are given as follows:

min

max

( ) ( ) ( ), ( )( ) max{ , ( ) }( ) min{ , ( ) }

s s

s s

t t t tt t T Tt t T T

u u u u u uu u u uu u u u

(59)

where sT is the sampling time.

The following transformation is introduced to avoid the uneven distribution problem:

1uu W u (60)

where 0u W is the transformation matrix.

For the control allocation problem (8) and (58)~(60), we can obtain the optimal input u based on

the robust least-squares method in [22] and [23]:

1 1 || ||( ) ( ) ( )arg min max

cu uct t t

J

BW u u W uu (61)

where ||( ) ||c c c uJ B B W u v . Furthermore, we can obtain the following worst-case residual:

|| ||( ) max ||( ) ||

cc c u

Bu B B W u v (62)

Based on the triangle inequality, we have

|| ||( ) max (|| || || ||)

cc u c u

Bu B W u v B W u

|| |||| || max || ||

cc u c u

BB W u v B W u (63)

Assume that

c u B W|| ||

Tc

uu

(64)

where || || , if

any unit norm vector,otherwise

c u

c u c uc

B W u v

B W u v B W u v.

Then, in the direction of c , the worst-case residual is:

( ) || || || ||c u u B W u v u (65)

(58)

where η is known constant. Generally, the individual

actuators constraints are given as follows:

(278~294)14-087.indd 284 2015-07-03 오전 5:06:32

285


http://ijass.org

17

|| ||c B (58)


min

max

( ) ( ) ( ), ( )( ) max{ , ( ) }( ) min{ , ( ) }

s s

s s



(59)



1uu W u (60)




1 1 || ||( ) ( ) ( )arg min max

cu uct t t

J

BW u u W uu (61)


|| ||( ) max ||( ) ||

cc c u

Bu B B W u v (62)


|| ||( ) max (|| || || ||)

cc u c u

Bu B W u v B W u

|| |||| || max || ||

cc u c u

BB W u v B W u (63)

Assume that

c u B W|| ||

Tc

uu

(64)

where || || , if


c u

c u c uc

B W u v

B W u v B W u v.


( ) || || || ||c u u B W u v u (65)

(59)

where Ts is the sampling time.

The following transformation is introduced to avoid the

uneven distribution problem:

17

|| ||c B (58)


min

max

( ) ( ) ( ), ( )( ) max{ , ( ) }( ) min{ , ( ) }

s s

s s



(59)



1uu W u (60)




1 1 || ||( ) ( ) ( )arg min max

cu uct t t

J

BW u u W uu (61)


|| ||( ) max ||( ) ||

cc c u

Bu B B W u v (62)


|| ||( ) max (|| || || ||)

cc u c u

Bu B W u v B W u

|| |||| || max || ||

cc u c u

BB W u v B W u (63)

Assume that

c u B W|| ||

Tc

uu

(64)

where || || , if


c u

c u c uc

B W u v

B W u v B W u v.


( ) || || || ||c u u B W u v u (65)

(60)

where Wu>0 is the transformation matrix.

For the control allocation problem (8) and (58)~(60), we

can obtain the optimal input

17

|| ||c B (58)


min

max

( ) ( ) ( ), ( )( ) max{ , ( ) }( ) min{ , ( ) }

s s

s s



(59)



1uu W u (60)




1 1 || ||( ) ( ) ( )arg min max

cu uct t t

J

BW u u W uu (61)


|| ||( ) max ||( ) ||

cc c u

Bu B B W u v (62)


|| ||( ) max (|| || || ||)

cc u c u

Bu B W u v B W u

|| |||| || max || ||

cc u c u

BB W u v B W u (63)

Assume that

c u B W|| ||

Tc

uu

(64)

where || || , if


c u

c u c uc

B W u v

B W u v B W u v.


( ) || || || ||c u u B W u v u (65)

based on the robust least-

squares method in [22] and [23]:

17

|| ||c B (58)


min

max

( ) ( ) ( ), ( )( ) max{ , ( ) }( ) min{ , ( ) }

s s

s s



(59)



1uu W u (60)




1 1 || ||( ) ( ) ( )arg min max

cu uct t t

J

BW u u W uu (61)


|| ||( ) max ||( ) ||

cc c u

Bu B B W u v (62)


|| ||( ) max (|| || || ||)

cc u c u

Bu B W u v B W u

|| |||| || max || ||

cc u c u

BB W u v B W u (63)

Assume that

c u B W|| ||

Tc

uu

(64)

where || || , if


c u

c u c uc

B W u v

B W u v B W u v.


( ) || || || ||c u u B W u v u (65)

(61)

where

17

|| ||c B (58)


min

max

( ) ( ) ( ), ( )( ) max{ , ( ) }( ) min{ , ( ) }

s s

s s



(59)



1uu W u (60)




1 1 || ||( ) ( ) ( )arg min max

cu uct t t

J

BW u u W uu (61)


|| ||( ) max ||( ) ||

cc c u

Bu B B W u v (62)


|| ||( ) max (|| || || ||)

cc u c u

Bu B W u v B W u

|| |||| || max || ||

cc u c u

BB W u v B W u (63)

Assume that

c u B W|| ||

Tc

uu

(64)

where || || , if


c u

c u c uc

B W u v

B W u v B W u v.


( ) || || || ||c u u B W u v u (65)

. Furthermore, we can obtain

the following worst-case residual:

17

|| ||c B (58)


min

max

( ) ( ) ( ), ( )( ) max{ , ( ) }( ) min{ , ( ) }

s s

s s



(59)



1uu W u (60)




1 1 || ||( ) ( ) ( )arg min max

cu uct t t

J

BW u u W uu (61)


|| ||( ) max ||( ) ||

cc c u

Bu B B W u v (62)


|| ||( ) max (|| || || ||)

cc u c u

Bu B W u v B W u

|| |||| || max || ||

cc u c u

BB W u v B W u (63)

Assume that

c u B W|| ||

Tc

uu

(64)

where || || , if


c u

c u c uc

B W u v

B W u v B W u v.


( ) || || || ||c u u B W u v u (65)

(62)


17

|| ||c B (58)


min

max

( ) ( ) ( ), ( )( ) max{ , ( ) }( ) min{ , ( ) }

s s

s s



(59)



1uu W u (60)




1 1 || ||( ) ( ) ( )arg min max

cu uct t t

J

BW u u W uu (61)


|| ||( ) max ||( ) ||

cc c u

Bu B B W u v (62)


|| ||( ) max (|| || || ||)

cc u c u

Bu B W u v B W u

|| |||| || max || ||

cc u c u

BB W u v B W u (63)

Assume that

c u B W|| ||

Tc

uu

(64)

where || || , if


c u

c u c uc

B W u v

B W u v B W u v.


( ) || || || ||c u u B W u v u (65)

(63)

Assume that

17

|| ||c B (58)


min

max

( ) ( ) ( ), ( )( ) max{ , ( ) }( ) min{ , ( ) }

s s

s s



(59)



1uu W u (60)




1 1 || ||( ) ( ) ( )arg min max

cu uct t t

J

BW u u W uu (61)


|| ||( ) max ||( ) ||

cc c u

Bu B B W u v (62)


|| ||( ) max (|| || || ||)

cc u c u

Bu B W u v B W u

|| |||| || max || ||

cc u c u

BB W u v B W u (63)

Assume that

c u B W|| ||

Tc

uu

(64)

where || || , if


c u

c u c uc

B W u v

B W u v B W u v.


( ) || || || ||c u u B W u v u (65)

(64)

where

17

|| ||c B (58)


min

max

( ) ( ) ( ), ( )( ) max{ , ( ) }( ) min{ , ( ) }

s s

s s



(59)



1uu W u (60)




1 1 || ||( ) ( ) ( )arg min max

cu uct t t

J

BW u u W uu (61)


|| ||( ) max ||( ) ||

cc c u

Bu B B W u v (62)


|| ||( ) max (|| || || ||)

cc u c u

Bu B W u v B W u

|| |||| || max || ||

cc u c u

BB W u v B W u (63)

Assume that

c u B W|| ||

Tc

uu

(64)

where || || , if


c u

c u c uc

B W u v

B W u v B W u v.


( ) || || || ||c u u B W u v u (65)

.

Then, in the direction of εc, the worst-case residual is:

17

|| ||c B (58)


min

max

( ) ( ) ( ), ( )( ) max{ , ( ) }( ) min{ , ( ) }

s s

s s



(59)



1uu W u (60)




1 1 || ||( ) ( ) ( )arg min max

cu uct t t

J

BW u u W uu (61)


|| ||( ) max ||( ) ||

cc c u

Bu B B W u v (62)


|| ||( ) max (|| || || ||)

cc u c u

Bu B W u v B W u

|| |||| || max || ||

cc u c u

BB W u v B W u (63)

Assume that

c u B W|| ||

Tc

uu

(64)

where || || , if


c u

c u c uc

B W u v

B W u v B W u v.


( ) || || || ||c u u B W u v u (65) (65)

To this end, the worst-case residual (65) also satisfies the

following constraint:

18

To this end, the worst-case residual (65) also satisfies the following constraint:

|| || || ||c u e B W u v u (66)

where e is the upper bound of the residual to be minimized.

Based on the method [23], the RLSWCA problem can be rewritten as a second order cone program

(SOCP) problem:

, ,min

e ee

u (67)

subject to || ||c u e e B W u v , || || e u and 1 1( ) ( ) ( )u ut t t W u u W u .

Then, the optimal solution u of the RLSWCA problem can be given by:

1

1

( ) if 0

( ) else

T Tu u c c u u c

T Tu u c c u u c

W I W B B W W B vu

W W B B W W B v (68)

where 2 2 2( ) ( ) 0e e e e s , 1 2 1 2|| || || ||u us W u W u , e and e are the optimal

solutions of the minimization problem (67). The proof procedure can be easily obtained from [23].

Based on the above analysis, the structure diagram of a three-channel independent IGC system for

the NSI is shown in Fig. 2.

Fig. 2 Structure diagram of a three-channel independent IGC system

4. Simulation results

Some simulation results are presented to illustrate the effectiveness of the proposed three-channel

independent IGC algorithm and RLSWCA scheme.

The parameters of the FTDOs (13)~(16) and (31) are selected as follows:

(66)

where λe is the upper bound of the residual to be minimized.

Based on the method [23], the RLSWCA problem can be

rewritten as a second order cone program (SOCP) problem:

18


|| || || ||c u e B W u v u (66)



(SOCP) problem:

, ,min

e ee

u (67)



1

1

( ) if 0

( ) else

T Tu u c c u u c

T Tu u c c u u c

W I W B B W W B vu











(67)

subject to

18


|| || || ||c u e B W u v u (66)



(SOCP) problem:

, ,min

e ee

u (67)



1

1

( ) if 0

( ) else

T Tu u c c u u c

T Tu u c c u u c

W I W B B W W B vu











and

18


|| || || ||c u e B W u v u (66)



(SOCP) problem:

, ,min

e ee

u (67)



1

1

( ) if 0

( ) else

T Tu u c c u u c

T Tu u c c u u c

W I W B B W W B vu











.

Then, the optimal solution u of the RLSWCA problem can

be given by:

18


|| || || ||c u e B W u v u (66)



(SOCP) problem:

, ,min

e ee

u (67)



1

1

( ) if 0

( ) else

T Tu u c c u u c

T Tu u c c u u c

W I W B B W W B vu











(68)

where

18


|| || || ||c u e B W u v u (66)



(SOCP) problem:

, ,min

e ee

u (67)



1

1

( ) if 0

( ) else

T Tu u c c u u c

T Tu u c c u u c

W I W B B W W B vu











,

18


|| || || ||c u e B W u v u (66)



(SOCP) problem:

, ,min

e ee

u (67)



1

1

( ) if 0

( ) else

T Tu u c c u u c

T Tu u c c u u c

W I W B B W W B vu











,

λe and τe are the optimal solutions of the minimization

problem (67). The proof procedure can be easily obtained

from [23].

Based on the above analysis, the structure diagram of a

three-channel independent IGC system for the NSI is shown

in Fig. 2.


Some simulation results are presented to illustrate the

effectiveness of the proposed three-channel independent

IGC algorithm and RLSWCA scheme.

The parameters of the FTDOs (13)~(16) and (31) are

selected as follows:

19

20 20p , 30 40 2p p , 1 1.5pi , 2 1.1pi , 0.5piq , 8pip , {2, 3, 4}i ;

20 20y , 30 40 2y y , 1 1.5yi , 2 1.1yi , 0.5yiq , 8yip , {2, 3, 4}i ;

0 2ri , 1 1.5ri , 2 1.1ri , 0.5riq , 8rip ， {3, 4}i ;

2 2 3 100p y rL L L , 3 3 10p yL L , 4 4 4 50p y rL L L .

The parameters of the proposed three-channel independent IGC algorithm (17)~(29) and (32)-(33)

are selected as follows:

2 2 2.5p yk k , 3 3 3 10p y rk k k , 4 4 4 40p y rk k k ,

5 5 5 5p y rk k k , 3 3 0.1p y , 4 4 4 0.025p y r ,

4 4 4 0.6p y r , 2 2 3 0p d y d r dx x x .

The proposed three-channel independent IGC algorithm incorporating the RLSWCA method

(denoted by IGC+FTDO+RLSWCA) is firstly applied to the NSI guidance and control system.

Furthermore, we also studied the proportional navigation guidance (PNG) law and sliding mode

guidance (SMG) law combined with the traditional backstepping sliding mode control law and

RLSWCA method (denoted by PNG+BSM+RLSWCA and SMG+BSM+RLSWCA, respectively) for

the simulation comparison. This was in line with traditional guidance and control separation design.

The PNG law and SMG law are given as follows:

4m pa N rq , 4m pa N rq (69)

4 sgn( )m sa N rq q , 4 sgn( )m sa N rq q (70)

where 3.2pN , 4.5sN and 200 .

In this paper, we select the following six interception conditions to illustrate the effectiveness of the

above three guidance and control schemes:

Case 1 and Case 4: No maneuvering target, 20 sm/t ta a ;

Case 2 and Case 5: Step maneuvering target, 24 sm0 /t ta a ;

Case 3 and Case 6: Sinusoidal maneuvering target, 240sin( ) m s/t ta a t .

18


|| || || ||c u e B W u v u (66)



(SOCP) problem:

, ,min

e ee

u (67)



1

1

( ) if 0

( ) else

T Tu u c c u u c

T Tu u c c u u c

W I W B B W W B vu











Fig. 2. Structure diagram of a three-channel independent IGC system

(278~294)14-087.indd 285 2015-07-03 오전 5:06:34



The parameters of the proposed three-channel

independent IGC algorithm (17)~(29) and (32)-(33) are

selected as follows:

19

20 20p , 30 40 2p p , 1 1.5pi , 2 1.1pi , 0.5piq , 8pip , {2, 3, 4}i ;

20 20y , 30 40 2y y , 1 1.5yi , 2 1.1yi , 0.5yiq , 8yip , {2, 3, 4}i ;

0 2ri , 1 1.5ri , 2 1.1ri , 0.5riq , 8rip ， {3, 4}i ;





5 5 5 5p y rk k k , 3 3 0.1p y , 4 4 4 0.025p y r ,

4 4 4 0.6p y r , 2 2 3 0p d y d r dx x x .
















The proposed three-channel independent IGC

algorithm incorporating the RLSWCA method (denoted by

IGC+FTDO+RLSWCA) is firstly applied to the NSI guidance

and control system. Furthermore, we also studied the

proportional navigation guidance (PNG) law and sliding

mode guidance (SMG) law combined with the traditional

backstepping sliding mode control law and RLSWCA method

(denoted by PNG+BSM+RLSWCA and SMG+BSM+RLSWCA,

respectively) for the simulation comparison. This was in line

with traditional guidance and control separation design. The

PNG law and SMG law are given as follows:

19

20 20p , 30 40 2p p , 1 1.5pi , 2 1.1pi , 0.5piq , 8pip , {2, 3, 4}i ;

20 20y , 30 40 2y y , 1 1.5yi , 2 1.1yi , 0.5yiq , 8yip , {2, 3, 4}i ;

0 2ri , 1 1.5ri , 2 1.1ri , 0.5riq , 8rip ， {3, 4}i ;





5 5 5 5p y rk k k , 3 3 0.1p y , 4 4 4 0.025p y r ,

4 4 4 0.6p y r , 2 2 3 0p d y d r dx x x .
















(69)

19

20 20p , 30 40 2p p , 1 1.5pi , 2 1.1pi , 0.5piq , 8pip , {2, 3, 4}i ;

20 20y , 30 40 2y y , 1 1.5yi , 2 1.1yi , 0.5yiq , 8yip , {2, 3, 4}i ;

0 2ri , 1 1.5ri , 2 1.1ri , 0.5riq , 8rip ， {3, 4}i ;





5 5 5 5p y rk k k , 3 3 0.1p y , 4 4 4 0.025p y r ,

4 4 4 0.6p y r , 2 2 3 0p d y d r dx x x .
















(70)

where

19

20 20p , 30 40 2p p , 1 1.5pi , 2 1.1pi , 0.5piq , 8pip , {2, 3, 4}i ;

20 20y , 30 40 2y y , 1 1.5yi , 2 1.1yi , 0.5yiq , 8yip , {2, 3, 4}i ;

0 2ri , 1 1.5ri , 2 1.1ri , 0.5riq , 8rip ， {3, 4}i ;





5 5 5 5p y rk k k , 3 3 0.1p y , 4 4 4 0.025p y r ,

4 4 4 0.6p y r , 2 2 3 0p d y d r dx x x .
















and

19

20 20p , 30 40 2p p , 1 1.5pi , 2 1.1pi , 0.5piq , 8pip , {2, 3, 4}i ;

20 20y , 30 40 2y y , 1 1.5yi , 2 1.1yi , 0.5yiq , 8yip , {2, 3, 4}i ;

0 2ri , 1 1.5ri , 2 1.1ri , 0.5riq , 8rip ， {3, 4}i ;





5 5 5 5p y rk k k , 3 3 0.1p y , 4 4 4 0.025p y r ,

4 4 4 0.6p y r , 2 2 3 0p d y d r dx x x .
















.

In this paper, we select the following six interception

conditions to illustrate the effectiveness of the above three

guidance and control schemes:

Case 1 and Case 4: No maneuvering target, atε=atβ=0m/s2;

Case 2 and Case 5: Step maneuvering target, atε=atβ=40m/s2;

Case 3 and Case 6: Sinusoidal maneuvering target,

atε=atβ=40sin(t)m/s2.

The initial conditions for Cases 1~3 are give as follows:

The initial positions of the interceptor are xm0=ym0=zm0=0m;

the velocities of the interceptor and target are Vm0=1500m/s

and Vt0=1700m/s, respectively; the initial flight-path angle

and heading angle of the interceptor and target are θm0=26o,

ψvm0=-5o, θt0=-10o and ψvt0=120o; the initial relative distance is

r0=1.8×104; the azimuth angle and elevation angle are qε0=20o

and qβ0=-36o, respectively. The initial conditions for Cases

4~6 are also give as follows: The initial positions of the target

are xt0=15204.5m, tt0=6840.4m and zt0=11046.7m; the other

conditions are the same as Cases 1~3.

In the simulation process, we select the external

disturbance of the control system loop as dr=dβ=dα=0.2sin(t)

and dωx=dωy

=dωz=sin(t), respectively.

Figure 3 shows the response curves of the LOS angular rate

for Case 1 based on the above three guidance and control

schemes, and we can see that the proposed three-channel

independent IGC algorithm can make the LOS angular rate

converge to the steady state value more quickly; i.e., the

small miss distance can be obtained. Fig. 4 and Fig. 6 show

the curves of the interceptor attitude and control inputs.

The actual and estimated value responses of the target

acceleration are shown in Fig. 5, and we can observe that

the proposed FTDO can provide a satisfactory estimation of

performance. The simulation curves for Case 2 and Case 3

are shown in Figs. 7~14, the simulation curves for Case 4~6

and the analysis processes are similar, which are omitted

here for brevity.

Define the control energy function as follows:

20

The initial conditions for Cases 1~3 are give as follows: The initial positions of the interceptor are

0 0 0 0mm m mx y z ; the velocities of the interceptor and target are 0 1500m/smV and

0 1700m/stV , respectively; the initial flight-path angle and heading angle of the interceptor and

target are 0 26m , 0 5vm , 0 10t and 0 120vt ; the initial relative distance is

40 1.8 10 mr ; the azimuth angle and elevation angle are 0 20q

and 0 36q ,

respectively. The initial conditions for Cases 4~6 are also give as follows: The initial positions of the

target are 0 152 m04.5tx , 0 68 m40.4ty and 0 110 m46.7tz ; the other conditions are the

same as Cases 1~3.

In the simulation process, we select the external disturbance of the control system loop as d

0.2sin( )d d t and sin( )x y z

d d d t , respectively.

Figure 3 shows the response curves of the LOS angular rate for Case 1 based on the above three

guidance and control schemes, and we can see that the proposed three-channel independent IGC

algorithm can make the LOS angular rate converge to the steady state value more quickly; i.e., the

small miss distance can be obtained. Fig. 4 and Fig. 6 show the curves of the interceptor attitude and

control inputs. The actual and estimated value responses of the target acceleration are shown in Fig. 5,

and we can observe that the proposed FTDO can provide a satisfactory estimation of performance.

The simulation curves for Case 2 and Case 3 are shown in Figs. 7~14, the simulation curves for Case

4~6 and the analysis processes are similar, which are omitted here for brevity.

Define the control energy function as follows: 2

00.5 || ||

t

u cJ dt B u . Table 1 shows the miss

distances, interception time and control energy of the three guidance and control schemes for Cases

1~6. The proposed IGC algorithm required more control energy to obtain the small miss distance.

Compared with PNG+BSM+RLSWCA and SMG+BSM+RLSWCA, it can be observed from Table 1

that the miss distance and interception time for the proposed IGC algorithm are minimal, since this

method considers the effects of the target maneuver and external distance in the control system loop,

and the couplings between the guidance loop and control loop.

. Table 1 shows the miss distances,

interception time and control energy of the three guidance

and control schemes for Cases 1~6. The proposed IGC

Table 1. Simulation results of the three guidance and control schemes

21

Table 1. Simulation results of the three guidance and control schemes

Case Design method Miss distance Interception time Control energy

1PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA

0.4139 m 0.2652 m 0.1282 m

6.318 s 6.318 s 6.318 s

1.361×106

2.268×106

2.513×107


1.2580 m 1.0120 m 0.1928 m

6.321 s 6.319 s 6.318 s

3.216×107

2.582×107

4.069×107


1.2360 m 1.1970 m 0.2110 m

6.320 s 6.318 s 6.318 s

2.848×107

1.347×107

3.755×107


0.4640 m 0.2948 m 0.1344 m

7.020 s 7.020 s 7.020 s

1.195×106

1.902×106

2.468×107


1.1140 m 0.9619 m 0.1968 m

7.023 s 7.022 s 7.020 s

3.496×107

2.283×107

4.375×107

PNG+BSM+RLSWCA 0.9931 m 7.022 s 1.180×107

6 SMG+BSM+RLSWCAIGC+FTDO+RLSWCA

1.3220 m 0.1481 m

7.021 s 7.020 s

1.260×107

4.235×107

Remark 1. In general, the knowledge and experience of experts (including control engineers and

operators) are employed to determine the design parameters. For the design parameters of the pitch

channel, the FTDO parameters are often selected as 20 20p , 30 40 2p p , 1 1.5pi ,

2 1.1pi , 0.5piq and 8pip , {2, 3, 4}i ; the filter time constants are often selected as

3 [0.01,0.2]p and 4 [0.01,0.2]p , and then the controller parameters are often selected as

2 [2,5]pk , 3 31/p pk , 4 41/p pk , 5 [1,10]pk and 4 (0,1)p . The design parameters of

the yaw channel and roll channel are similar.

5. Conclusion

In this paper, a novel three-channel independent IGC design method is proposed for the NSI, based

on the backstepping sliding mode and FTDO, which also take into account the couplings of the

guidance and control system and those between the different channels of interceptor. In the proposed

method, the FTDO is firstly constructed to estimate the target acceleration and the disturbance of the

control system loop, and the IGC algorithm based on the backstepping sliding mode is given to obtain

the desired control moment. Moreover, the stability analysis based on the Lyapunov theory shows that

(278~294)14-087.indd 286 2015-07-03 오전 5:06:35

287


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25

0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05

0.1

PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA

(a) LOS angular rate in vertical plane

0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05

0.1PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA

(b) LOS angular rate in lateral plane

Fig. 3 Trajectories of the LOS angular rate for Case 1

0 1 2 3 4 5 6 7

-1

0

1

2


(a) Roll angle

0 1 2 3 4 5 6 7-10

-8

-6

-4

-2

0

2


(b) Sideslip angle

0 1 2 3 4 5 6 7-5

0

5

10

15

20

25


(c) Angle of attack

0 1 2 3 4 5 6 7

-20

-10

0

10


(d) Roll rate

0 1 2 3 4 5 6 7-100

-80

-60

-40

-20

0

20


(e) Yaw rate

0 1 2 3 4 5 6 7

-100

0

100


(f) Pitch rate

Fig. 4 Trajectories of the angle and angular rate for Case 1

Time (s)

dqɛ

(deg

/s)

Time (s)

dqβ

(deg

/s)

Time (s)

γ (d

eg)

Time (s)

β (d

eg)

Time (s)

α (d

eg)

Time (s) ω

x (d

eg/s

)

Time (s)

ωy

(deg

/s)

Time (s)

ωz

(deg

/s)

25

0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05

0.1



0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05




0 1 2 3 4 5 6 7

-1

0

1

2


(a) Roll angle

0 1 2 3 4 5 6 7-10

-8

-6

-4

-2

0

2


(b) Sideslip angle

0 1 2 3 4 5 6 7-5

0

5

10

15

20

25


(c) Angle of attack

0 1 2 3 4 5 6 7

-20

-10

0

10


(d) Roll rate

0 1 2 3 4 5 6 7-100

-80

-60

-40

-20

0

20


(e) Yaw rate

0 1 2 3 4 5 6 7

-100

0

100


(f) Pitch rate


Time (s)

dqɛ

(deg

/s)

Time (s)

dqβ

(deg

/s)

Time (s)

γ (d

eg)

Time (s)

β (d

eg)

Time (s)

α (d

eg)

Time (s)

ωx

(deg

/s)

Time (s)

ωy

(deg

/s)

Time (s)

ωz

(deg

/s)

(a) LOS angular rate in vertical plane (b) LOS angular rate in lateral plan

Fig. 3. Trajectories of the LOS angular rate for Case 1

25

0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05

0.1



0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05




0 1 2 3 4 5 6 7

-1

0

1

2


(a) Roll angle

0 1 2 3 4 5 6 7-10

-8

-6

-4

-2

0

2


(b) Sideslip angle

0 1 2 3 4 5 6 7-5

0

5

10

15

20

25


(c) Angle of attack

0 1 2 3 4 5 6 7

-20

-10

0

10


(d) Roll rate

0 1 2 3 4 5 6 7-100

-80

-60

-40

-20

0

20


(e) Yaw rate

0 1 2 3 4 5 6 7

-100

0

100


(f) Pitch rate


Time (s)

dqɛ

(deg

/s)

Time (s)

dqβ

(deg

/s)

Time (s)

γ (d

eg)

Time (s)

β (d

eg)

Time (s)

α (d

eg)

Time (s)

ωx

(deg

/s)

Time (s) ω

y (d

eg/s

)

Time (s)

ωz

(deg

/s)

25

0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05

0.1



0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05




0 1 2 3 4 5 6 7

-1

0

1

2


(a) Roll angle

0 1 2 3 4 5 6 7-10

-8

-6

-4

-2

0

2


(b) Sideslip angle

0 1 2 3 4 5 6 7-5

0

5

10

15

20

25


(c) Angle of attack

0 1 2 3 4 5 6 7

-20

-10

0

10


(d) Roll rate

0 1 2 3 4 5 6 7-100

-80

-60

-40

-20

0

20


(e) Yaw rate

0 1 2 3 4 5 6 7

-100

0

100


(f) Pitch rate


Time (s)

dqɛ

(deg

/s)

Time (s)

dqβ

(deg

/s)

Time (s)

γ (d

eg)

Time (s)

β (d

eg)

Time (s)

α (d

eg)

Time (s)

ωx

(deg

/s)

Time (s)

ωy

(deg

/s)

Time (s)

ωz

(deg

/s)

(a) Roll angle (b) Sideslip angle

25

0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05

0.1



0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05




0 1 2 3 4 5 6 7

-1

0

1

2


(a) Roll angle

0 1 2 3 4 5 6 7-10

-8

-6

-4

-2

0

2


(b) Sideslip angle

0 1 2 3 4 5 6 7-5

0

5

10

15

20

25


(c) Angle of attack

0 1 2 3 4 5 6 7

-20

-10

0

10


(d) Roll rate

0 1 2 3 4 5 6 7-100

-80

-60

-40

-20

0

20


(e) Yaw rate

0 1 2 3 4 5 6 7

-100

0

100


(f) Pitch rate


Time (s)

dqɛ

(deg

/s)

Time (s)

dqβ

(deg

/s)

Time (s)

γ (d

eg)

Time (s)

β (d

eg)

Time (s)

α (d

eg)

Time (s)

ωx

(deg

/s)

Time (s)

ωy

(deg

/s)

Time (s)

ωz

(deg

/s)

25

0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05

0.1



0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05




0 1 2 3 4 5 6 7

-1

0

1

2


(a) Roll angle

0 1 2 3 4 5 6 7-10

-8

-6

-4

-2

0

2


(b) Sideslip angle

0 1 2 3 4 5 6 7-5

0

5

10

15

20

25


(c) Angle of attack

0 1 2 3 4 5 6 7

-20

-10

0

10


(d) Roll rate

0 1 2 3 4 5 6 7-100

-80

-60

-40

-20

0

20


(e) Yaw rate

0 1 2 3 4 5 6 7

-100

0

100


(f) Pitch rate


Time (s)

dqɛ

(deg

/s)

Time (s)

dqβ

(deg

/s)

Time (s)

γ (d

eg)

Time (s)

β (d

eg)

Time (s)

α (d

eg)

Time (s)

ωx

(deg

/s)

Time (s)

ωy

(deg

/s)

Time (s)

ωz

(deg

/s)

(c) Angle of attack (d) Roll rate

25

0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05

0.1



0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05




0 1 2 3 4 5 6 7

-1

0

1

2


(a) Roll angle

0 1 2 3 4 5 6 7-10

-8

-6

-4

-2

0

2


(b) Sideslip angle

0 1 2 3 4 5 6 7-5

0

5

10

15

20

25


(c) Angle of attack

0 1 2 3 4 5 6 7

-20

-10

0

10


(d) Roll rate

0 1 2 3 4 5 6 7-100

-80

-60

-40

-20

0

20


(e) Yaw rate

0 1 2 3 4 5 6 7

-100

0

100


(f) Pitch rate


Time (s)

dqɛ

(deg

/s)

Time (s)

dqβ

(deg

/s)

Time (s)

γ (d

eg)

Time (s)

β (d

eg)

Time (s)

α (d

eg)

Time (s)

ωx

(deg

/s)

Time (s)

ωy

(deg

/s)

Time (s)

ωz

(deg

/s)

25

0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05

0.1



0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05




0 1 2 3 4 5 6 7

-1

0

1

2


(a) Roll angle

0 1 2 3 4 5 6 7-10

-8

-6

-4

-2

0

2


(b) Sideslip angle

0 1 2 3 4 5 6 7-5

0

5

10

15

20

25


(c) Angle of attack

0 1 2 3 4 5 6 7

-20

-10

0

10


(d) Roll rate

0 1 2 3 4 5 6 7-100

-80

-60

-40

-20

0

20


(e) Yaw rate

0 1 2 3 4 5 6 7

-100

0

100


(f) Pitch rate


Time (s)

dqɛ

(deg

/s)

Time (s)

dqβ

(deg

/s)

Time (s)

γ (d

eg)

Time (s)

β (d

eg)

Time (s)

α (d

eg)

Time (s)

ωx

(deg

/s)

Time (s)

ωy

(deg

/s)

Time (s)

ωz

(deg

/s)

(e) Yaw rate (f ) Pitch rate

Fig. 4. Trajectories of the angle and angular rate for Case 1

(278~294)14-087.indd 287 2015-07-03 오전 5:06:37



26

0 1 2 3 4 5 6 7-1

-0.5

0

0.5

1

1.5

Actual valueEstimated value

(a) Target acceleration in vertical plane

0 1 2 3 4 5 6 7-0.5

0

0.5

1

1.5

2


(b) Target acceleration in lateral plane

Fig. 5 Actual and estimated values of target acceleration for Case 1

0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05

0.1

0.15


(a) Aileron deflection

0 1 2 3 4 5 6 7-5

0

5

10


(b) Rudder deflection

0 1 2 3 4 5 6 7-30

-20

-10

0

10

20


(c) Elevator deflection

0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6

0.8


(d) Equivalent control surface deflection in pitch direction

0 1 2 3 4 5 6 7-0.4

-0.3

-0.2

-0.1

0

0.1


(e) Equivalent control surface deflection in yaw direction

Fig. 6 Trajectories of the actual control inputs

for Case 1

Time (s)

a tɛ

(m/s

2 )

Time (s)

a tβ

(m/s

2 )

Time (s)

δ x (

deg)

Time (s)

δ y (

deg)

Time (s)

δz

(deg

)

Time (s) δ s

y

Time (s)

δ sz

26

0 1 2 3 4 5 6 7-1

-0.5

0

0.5

1

1.5



0 1 2 3 4 5 6 7-0.5

0

0.5

1

1.5

2




0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05

0.1

0.15



0 1 2 3 4 5 6 7-5

0

5

10



0 1 2 3 4 5 6 7-30

-20

-10

0

10

20



0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6

0.8



0 1 2 3 4 5 6 7-0.4

-0.3

-0.2

-0.1

0

0.1




for Case 1

Time (s)

a tɛ

(m/s

2 )

Time (s)

a tβ

(m/s

2 )

Time (s)

δ x (

deg)

Time (s)

δ y (

deg)

Time (s)

δz

(deg

)

Time (s)

δ sy

Time (s)

δ sz

(a) Target acceleration in vertical plane (b) Target acceleration in lateral plane

Fig. 5. Actual and estimated values of target acceleration for Case 1

26

0 1 2 3 4 5 6 7-1

-0.5

0

0.5

1

1.5



0 1 2 3 4 5 6 7-0.5

0

0.5

1

1.5

2




0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05

0.1

0.15



0 1 2 3 4 5 6 7-5

0

5

10



0 1 2 3 4 5 6 7-30

-20

-10

0

10

20



0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6

0.8



0 1 2 3 4 5 6 7-0.4

-0.3

-0.2

-0.1

0

0.1




for Case 1

Time (s)

a tɛ

(m/s

2 )

Time (s)

a tβ

(m/s

2 )

Time (s)

δ x (

deg)

Time (s)

δ y (

deg)

Time (s)

δz

(deg

)

Time (s)

δ sy

Time (s)

δ sz

26

0 1 2 3 4 5 6 7-1

-0.5

0

0.5

1

1.5



0 1 2 3 4 5 6 7-0.5

0

0.5

1

1.5

2




0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05

0.1

0.15



0 1 2 3 4 5 6 7-5

0

5

10



0 1 2 3 4 5 6 7-30

-20

-10

0

10

20



0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6

0.8



0 1 2 3 4 5 6 7-0.4

-0.3

-0.2

-0.1

0

0.1




for Case 1

Time (s)

a tɛ

(m/s

2 )

Time (s)

a tβ

(m/s

2 )

Time (s)

δ x (

deg)

Time (s)

δ y (

deg)

Time (s)

δz

(deg

)

Time (s)

δ sy

Time (s)

δ sz

(a) Aileron deflection (b) Rudder deflection

26

0 1 2 3 4 5 6 7-1

-0.5

0

0.5

1

1.5



0 1 2 3 4 5 6 7-0.5

0

0.5

1

1.5

2




0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05

0.1

0.15



0 1 2 3 4 5 6 7-5

0

5

10



0 1 2 3 4 5 6 7-30

-20

-10

0

10

20



0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6

0.8



0 1 2 3 4 5 6 7-0.4

-0.3

-0.2

-0.1

0

0.1




for Case 1

Time (s)

a tɛ

(m/s

2 )

Time (s)

a tβ

(m/s

2 )

Time (s)

δ x (

deg)

Time (s)

δ y (

deg)

Time (s)

δz

(deg

)

Time (s)

δ sy

Time (s)

δ sz

26

0 1 2 3 4 5 6 7-1

-0.5

0

0.5

1

1.5



0 1 2 3 4 5 6 7-0.5

0

0.5

1

1.5

2




0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05

0.1

0.15



0 1 2 3 4 5 6 7-5

0

5

10



0 1 2 3 4 5 6 7-30

-20

-10

0

10

20



0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6

0.8



0 1 2 3 4 5 6 7-0.4

-0.3

-0.2

-0.1

0

0.1




for Case 1

Time (s)

a tɛ

(m/s

2 )

Time (s)

a tβ

(m/s

2 )

Time (s)

δ x (

deg)

Time (s)

δ y (

deg)

Time (s)

δz

(deg

)

Time (s)

δ sy

Time (s)

δ sz

(c) Elevator deflection (d) Equivalent control surface deflection in pitch direction

26

0 1 2 3 4 5 6 7-1

-0.5

0

0.5

1

1.5



0 1 2 3 4 5 6 7-0.5

0

0.5

1

1.5

2




0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05

0.1

0.15



0 1 2 3 4 5 6 7-5

0

5

10



0 1 2 3 4 5 6 7-30

-20

-10

0

10

20



0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6

0.8



0 1 2 3 4 5 6 7-0.4

-0.3

-0.2

-0.1

0

0.1




for Case 1

Time (s)

a tɛ

(m/s

2 )

Time (s)

a tβ

(m/s

2 )

Time (s)

δ x (

deg)

Time (s)

δ y (

deg)

Time (s)

δz

(deg

)

Time (s)

δ sy

Time (s)

δ sz


Fig. 6. Trajectories of the actual control inputs for Case 1

(278~294)14-087.indd 288 2015-07-03 오전 5:06:39

289


http://ijass.org

27

0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6



0 1 2 3 4 5 6 7

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1




0 1 2 3 4 5 6 7-2

-1

0

1

2

3 PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA

(a) Roll angle

0 1 2 3 4 5 6 7-25

-20

-15

-10

-5

0

5


(b) Sideslip angle

0 1 2 3 4 5 6 70

5

10

15

20

25

30


(c) Angle of attack

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20


(d) Roll rate

0 1 2 3 4 5 6 7-150

-100

-50

0

50


(e) Yaw rate

0 1 2 3 4 5 6 7-100

0

100

200

300


(f) Pitch rate


Time (s)

dqɛ

(deg

/s)

Time (s)

dqβ

(deg

/s)

Time (s)

γ (d

eg)

Time (s)

β (d

eg)

Time (s)

α (d

eg)

Time (s) ω

x (d

eg/s

)

Time (s)

ωy

(deg

/s)

Time (s)

ωz

(deg

/s)

27

0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6



0 1 2 3 4 5 6 7

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1




0 1 2 3 4 5 6 7-2

-1

0

1

2


(a) Roll angle

0 1 2 3 4 5 6 7-25

-20

-15

-10

-5

0

5


(b) Sideslip angle

0 1 2 3 4 5 6 70

5

10

15

20

25

30


(c) Angle of attack

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20


(d) Roll rate

0 1 2 3 4 5 6 7-150

-100

-50

0

50


(e) Yaw rate

0 1 2 3 4 5 6 7-100

0

100

200

300


(f) Pitch rate


Time (s)

dqɛ

(deg

/s)

Time (s)

dqβ

(deg

/s)

Time (s)

γ (d

eg)

Time (s)

β (d

eg)

Time (s)

α (d

eg)

Time (s)

ωx

(deg

/s)

Time (s)

ωy

(deg

/s)

Time (s)

ωz

(deg

/s)

(a) LOS angular rate in vertical plane (b) LOS angular rate in lateral plane


27

0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6



0 1 2 3 4 5 6 7

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1




0 1 2 3 4 5 6 7-2

-1

0

1

2


(a) Roll angle

0 1 2 3 4 5 6 7-25

-20

-15

-10

-5

0

5


(b) Sideslip angle

0 1 2 3 4 5 6 70

5

10

15

20

25

30


(c) Angle of attack

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20


(d) Roll rate

0 1 2 3 4 5 6 7-150

-100

-50

0

50


(e) Yaw rate

0 1 2 3 4 5 6 7-100

0

100

200

300


(f) Pitch rate


Time (s)

dqɛ

(deg

/s)

Time (s)

dqβ

(deg

/s)

Time (s)

γ (d

eg)

Time (s)

β (d

eg)

Time (s)

α (d

eg)

Time (s)

ωx

(deg

/s)

Time (s)

ωy

(deg

/s)

Time (s)

ωz

(deg

/s)

27

0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6



0 1 2 3 4 5 6 7

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1




0 1 2 3 4 5 6 7-2

-1

0

1

2


(a) Roll angle

0 1 2 3 4 5 6 7-25

-20

-15

-10

-5

0

5


(b) Sideslip angle

0 1 2 3 4 5 6 70

5

10

15

20

25

30


(c) Angle of attack

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20


(d) Roll rate

0 1 2 3 4 5 6 7-150

-100

-50

0

50


(e) Yaw rate

0 1 2 3 4 5 6 7-100

0

100

200

300


(f) Pitch rate


Time (s)

dqɛ

(deg

/s)

Time (s)

dqβ

(deg

/s)

Time (s)

γ (d

eg)

Time (s)

β (d

eg)

Time (s)

α (d

eg)

Time (s)

ωx

(deg

/s)

Time (s)

ωy

(deg

/s)

Time (s)

ωz

(deg

/s)


27

0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6



0 1 2 3 4 5 6 7

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1




0 1 2 3 4 5 6 7-2

-1

0

1

2


(a) Roll angle

0 1 2 3 4 5 6 7-25

-20

-15

-10

-5

0

5


(b) Sideslip angle

0 1 2 3 4 5 6 70

5

10

15

20

25

30


(c) Angle of attack

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20


(d) Roll rate

0 1 2 3 4 5 6 7-150

-100

-50

0

50


(e) Yaw rate

0 1 2 3 4 5 6 7-100

0

100

200

300


(f) Pitch rate


Time (s)

dqɛ

(deg

/s)

Time (s)

dqβ

(deg

/s)

Time (s)

γ (d

eg)

Time (s)

β (d

eg)

Time (s)

α (d

eg)

Time (s)

ωx

(deg

/s)

Time (s)

ωy

(deg

/s)

Time (s)

ωz

(deg

/s)

27

0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6



0 1 2 3 4 5 6 7

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1




0 1 2 3 4 5 6 7-2

-1

0

1

2


(a) Roll angle

0 1 2 3 4 5 6 7-25

-20

-15

-10

-5

0

5


(b) Sideslip angle

0 1 2 3 4 5 6 70

5

10

15

20

25

30


(c) Angle of attack

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20


(d) Roll rate

0 1 2 3 4 5 6 7-150

-100

-50

0

50


(e) Yaw rate

0 1 2 3 4 5 6 7-100

0

100

200

300


(f) Pitch rate


Time (s)

dqɛ

(deg

/s)

Time (s)

dqβ

(deg

/s)

Time (s)

γ (d

eg)

Time (s)

β (d

eg)

Time (s)

α (d

eg)

Time (s)

ωx

(deg

/s)

Time (s)

ωy

(deg

/s)

Time (s)

ωz

(deg

/s)


27

0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6



0 1 2 3 4 5 6 7

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1




0 1 2 3 4 5 6 7-2

-1

0

1

2


(a) Roll angle

0 1 2 3 4 5 6 7-25

-20

-15

-10

-5

0

5


(b) Sideslip angle

0 1 2 3 4 5 6 70

5

10

15

20

25

30


(c) Angle of attack

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20


(d) Roll rate

0 1 2 3 4 5 6 7-150

-100

-50

0

50


(e) Yaw rate

0 1 2 3 4 5 6 7-100

0

100

200

300


(f) Pitch rate


Time (s)

dqɛ

(deg

/s)

Time (s)

dqβ

(deg

/s)

Time (s)

γ (d

eg)

Time (s)

β (d

eg)

Time (s)

α (d

eg)

Time (s)

ωx

(deg

/s)

Time (s)

ωy

(deg

/s)

Time (s)

ωz

(deg

/s)

27

0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6



0 1 2 3 4 5 6 7

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1




0 1 2 3 4 5 6 7-2

-1

0

1

2


(a) Roll angle

0 1 2 3 4 5 6 7-25

-20

-15

-10

-5

0

5


(b) Sideslip angle

0 1 2 3 4 5 6 70

5

10

15

20

25

30


(c) Angle of attack

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20


(d) Roll rate

0 1 2 3 4 5 6 7-150

-100

-50

0

50


(e) Yaw rate

0 1 2 3 4 5 6 7-100

0

100

200

300


(f) Pitch rate


Time (s)

dqɛ

(deg

/s)

Time (s)

dqβ

(deg

/s)

Time (s)

γ (d

eg)

Time (s)

β (d

eg)

Time (s)

α (d

eg)

Time (s)

ωx

(deg

/s)

Time (s)

ωy

(deg

/s)

Time (s)

ωz

(deg

/s)



(278~294)14-087.indd 289 2015-07-03 오전 5:06:41



28

0 1 2 3 4 5 6 7-10

0

10

20

30

40

50



0 1 2 3 4 5 6 7-10

0

10

20

30

40

50




0 1 2 3 4 5 6 7-0.05

0

0.05

0.1

0.15



0 1 2 3 4 5 6 7-5

0

5

10

15



0 1 2 3 4 5 6 7-30

-20

-10

0

10

20



0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6

0.8



0 1 2 3 4 5 6 7

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1




for Case 2

Time (s)

a tɛ

(m/s

2 )

Time (s)

a tβ

(m/s

2 )

Time (s)

δ x (

deg)

Time (s)

δ y (

deg)

Time (s)

δz

(deg

) Time (s)

δ sy

Time (s)

δ sz

28

0 1 2 3 4 5 6 7-10

0

10

20

30

40

50



0 1 2 3 4 5 6 7-10

0

10

20

30

40

50




0 1 2 3 4 5 6 7-0.05

0

0.05

0.1

0.15



0 1 2 3 4 5 6 7-5

0

5

10

15



0 1 2 3 4 5 6 7-30

-20

-10

0

10

20



0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6

0.8



0 1 2 3 4 5 6 7

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1




for Case 2

Time (s)

a tɛ

(m/s

2 )

Time (s)

a tβ

(m/s

2 )

Time (s)

δ x (

deg)

Time (s)

δ y (

deg)

Time (s)

δz

(deg

)

Time (s)

δ sy

Time (s)

δ sz


Fig. 9. Actual and estimated values of target acceleration for Case 2

28

0 1 2 3 4 5 6 7-10

0

10

20

30

40

50



0 1 2 3 4 5 6 7-10

0

10

20

30

40

50




0 1 2 3 4 5 6 7-0.05

0

0.05

0.1

0.15



0 1 2 3 4 5 6 7-5

0

5

10

15



0 1 2 3 4 5 6 7-30

-20

-10

0

10

20



0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6

0.8



0 1 2 3 4 5 6 7

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1




for Case 2

Time (s)

a tɛ

(m/s

2 )

Time (s)

a tβ

(m/s

2 )

Time (s)

δ x (

deg)

Time (s)

δ y (

deg)

Time (s)

δz

(deg

)

Time (s)

δ sy

Time (s)

δ sz

28

0 1 2 3 4 5 6 7-10

0

10

20

30

40

50



0 1 2 3 4 5 6 7-10

0

10

20

30

40

50




0 1 2 3 4 5 6 7-0.05

0

0.05

0.1

0.15



0 1 2 3 4 5 6 7-5

0

5

10

15



0 1 2 3 4 5 6 7-30

-20

-10

0

10

20



0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6

0.8



0 1 2 3 4 5 6 7

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1




for Case 2

Time (s)

a tɛ

(m/s

2 )

Time (s)

a tβ

(m/s

2 )

Time (s)

δ x (

deg)

Time (s)

δ y (

deg)

Time (s)

δz

(deg

)

Time (s)

δ sy

Time (s)

δ sz


28

0 1 2 3 4 5 6 7-10

0

10

20

30

40

50



0 1 2 3 4 5 6 7-10

0

10

20

30

40

50




0 1 2 3 4 5 6 7-0.05

0

0.05

0.1

0.15



0 1 2 3 4 5 6 7-5

0

5

10

15



0 1 2 3 4 5 6 7-30

-20

-10

0

10

20



0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6

0.8



0 1 2 3 4 5 6 7

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1




for Case 2

Time (s)

a tɛ

(m/s

2 )

Time (s)

a tβ

(m/s

2 )

Time (s)

δ x (

deg)

Time (s)

δ y (

deg)

Time (s)

δz

(deg

)

Time (s)

δ sy

Time (s)

δ sz

28

0 1 2 3 4 5 6 7-10

0

10

20

30

40

50



0 1 2 3 4 5 6 7-10

0

10

20

30

40

50




0 1 2 3 4 5 6 7-0.05

0

0.05

0.1

0.15



0 1 2 3 4 5 6 7-5

0

5

10

15



0 1 2 3 4 5 6 7-30

-20

-10

0

10

20



0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6

0.8



0 1 2 3 4 5 6 7

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1




for Case 2

Time (s)

a tɛ

(m/s

2 )

Time (s)

a tβ

(m/s

2 )

Time (s)

δ x (

deg)

Time (s)

δ y (

deg)

Time (s)

δz

(deg

)

Time (s)

δ sy

Time (s)

δ sz


28

0 1 2 3 4 5 6 7-10

0

10

20

30

40

50



0 1 2 3 4 5 6 7-10

0

10

20

30

40

50




0 1 2 3 4 5 6 7-0.05

0

0.05

0.1

0.15



0 1 2 3 4 5 6 7-5

0

5

10

15



0 1 2 3 4 5 6 7-30

-20

-10

0

10

20



0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6

0.8



0 1 2 3 4 5 6 7

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1




for Case 2

Time (s)

a tɛ

(m/s

2 )

Time (s)

a tβ

(m/s

2 )

Time (s)

δ x (

deg)

Time (s)

δ y (

deg)

Time (s)

δz

(deg

)

Time (s)

δ sy

Time (s)

δ sz



(278~294)14-087.indd 290 2015-07-03 오전 5:06:42

291


http://ijass.org

29

0 1 2 3 4 5 6 7-0.8

-0.6

-0.4

-0.2

0

0.2

0.4



0 1 2 3 4 5 6 7

-0.4

-0.2

0

0.2

0.4

0.6




0 1 2 3 4 5 6 7-3

-2

-1

0

1

2


(a) Roll angle

0 1 2 3 4 5 6 7-30

-20

-10

0

10

20

30


(b) Sideslip angle

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20


(c) Angle of attack

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20


(d) Roll rate

0 1 2 3 4 5 6 7-200

-150

-100

-50

0

50


(e) Yaw rate

0 1 2 3 4 5 6 7-100

0

100

200

300


(f) Pitch rate


Time (s)

dqɛ

(deg

/s)

Time (s)

dqβ

(deg

/s)

Time (s)

γ (d

eg)

Time (s)

β (d

eg)

Time (s)

α (d

eg)

Time (s) ω

x (d

eg/s

)

Time (s)

ωy

(deg

/s)

Time (s)

ωz

(deg

/s)

29

0 1 2 3 4 5 6 7-0.8

-0.6

-0.4

-0.2

0

0.2

0.4



0 1 2 3 4 5 6 7

-0.4

-0.2

0

0.2

0.4

0.6




0 1 2 3 4 5 6 7-3

-2

-1

0

1

2


(a) Roll angle

0 1 2 3 4 5 6 7-30

-20

-10

0

10

20

30


(b) Sideslip angle

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20


(c) Angle of attack

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20


(d) Roll rate

0 1 2 3 4 5 6 7-200

-150

-100

-50

0

50


(e) Yaw rate

0 1 2 3 4 5 6 7-100

0

100

200

300


(f) Pitch rate


Time (s)

dqɛ

(deg

/s)

Time (s)

dqβ

(deg

/s)

Time (s)

γ (d

eg)

Time (s)

β (d

eg)

Time (s)

α (d

eg)

Time (s)

ωx

(deg

/s)

Time (s)

ωy

(deg

/s)

Time (s)

ωz

(deg

/s)

(a) LOS angular rate in vertical plane (b) LOS angular rate in lateral plane


29

0 1 2 3 4 5 6 7-0.8

-0.6

-0.4

-0.2

0

0.2

0.4



0 1 2 3 4 5 6 7

-0.4

-0.2

0

0.2

0.4

0.6




0 1 2 3 4 5 6 7-3

-2

-1

0

1

2


(a) Roll angle

0 1 2 3 4 5 6 7-30

-20

-10

0

10

20

30


(b) Sideslip angle

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20


(c) Angle of attack

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20


(d) Roll rate

0 1 2 3 4 5 6 7-200

-150

-100

-50

0

50


(e) Yaw rate

0 1 2 3 4 5 6 7-100

0

100

200

300


(f) Pitch rate


Time (s)

dqɛ

(deg

/s)

Time (s)

dqβ

(deg

/s)

Time (s)

γ (d

eg)

Time (s)

β (d

eg)

Time (s)

α (d

eg)

Time (s)

ωx

(deg

/s)

Time (s)

ωy

(deg

/s)

Time (s)

ωz

(deg

/s)

29

0 1 2 3 4 5 6 7-0.8

-0.6

-0.4

-0.2

0

0.2

0.4



0 1 2 3 4 5 6 7

-0.4

-0.2

0

0.2

0.4

0.6




0 1 2 3 4 5 6 7-3

-2

-1

0

1

2


(a) Roll angle

0 1 2 3 4 5 6 7-30

-20

-10

0

10

20

30


(b) Sideslip angle

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20


(c) Angle of attack

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20


(d) Roll rate

0 1 2 3 4 5 6 7-200

-150

-100

-50

0

50


(e) Yaw rate

0 1 2 3 4 5 6 7-100

0

100

200

300


(f) Pitch rate


Time (s)

dqɛ

(deg

/s)

Time (s)

dqβ

(deg

/s)

Time (s)

γ (d

eg)

Time (s)

β (d

eg)

Time (s)

α (d

eg)

Time (s)

ωx

(deg

/s)

Time (s)

ωy

(deg

/s)

Time (s)

ωz

(deg

/s)


29

0 1 2 3 4 5 6 7-0.8

-0.6

-0.4

-0.2

0

0.2

0.4



0 1 2 3 4 5 6 7

-0.4

-0.2

0

0.2

0.4

0.6




0 1 2 3 4 5 6 7-3

-2

-1

0

1

2


(a) Roll angle

0 1 2 3 4 5 6 7-30

-20

-10

0

10

20

30


(b) Sideslip angle

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20


(c) Angle of attack

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20


(d) Roll rate

0 1 2 3 4 5 6 7-200

-150

-100

-50

0

50


(e) Yaw rate

0 1 2 3 4 5 6 7-100

0

100

200

300


(f) Pitch rate


Time (s)

dqɛ

(deg

/s)

Time (s)

dqβ

(deg

/s)

Time (s)

γ (d

eg)

Time (s)

β (d

eg)

Time (s)

α (d

eg)

Time (s)

ωx

(deg

/s)

Time (s)

ωy

(deg

/s)

Time (s)

ωz

(deg

/s)

29

0 1 2 3 4 5 6 7-0.8

-0.6

-0.4

-0.2

0

0.2

0.4



0 1 2 3 4 5 6 7

-0.4

-0.2

0

0.2

0.4

0.6




0 1 2 3 4 5 6 7-3

-2

-1

0

1

2


(a) Roll angle

0 1 2 3 4 5 6 7-30

-20

-10

0

10

20

30


(b) Sideslip angle

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20


(c) Angle of attack

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20


(d) Roll rate

0 1 2 3 4 5 6 7-200

-150

-100

-50

0

50


(e) Yaw rate

0 1 2 3 4 5 6 7-100

0

100

200

300


(f) Pitch rate


Time (s)

dqɛ

(deg

/s)

Time (s)

dqβ

(deg

/s)

Time (s)

γ (d

eg)

Time (s)

β (d

eg)

Time (s)

α (d

eg)

Time (s)

ωx

(deg

/s)

Time (s)

ωy

(deg

/s)

Time (s)

ωz

(deg

/s)


29

0 1 2 3 4 5 6 7-0.8

-0.6

-0.4

-0.2

0

0.2

0.4



0 1 2 3 4 5 6 7

-0.4

-0.2

0

0.2

0.4

0.6




0 1 2 3 4 5 6 7-3

-2

-1

0

1

2


(a) Roll angle

0 1 2 3 4 5 6 7-30

-20

-10

0

10

20

30


(b) Sideslip angle

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20


(c) Angle of attack

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20


(d) Roll rate

0 1 2 3 4 5 6 7-200

-150

-100

-50

0

50


(e) Yaw rate

0 1 2 3 4 5 6 7-100

0

100

200

300


(f) Pitch rate


Time (s)

dqɛ

(deg

/s)

Time (s)

dqβ

(deg

/s)

Time (s)

γ (d

eg)

Time (s)

β (d

eg)

Time (s)

α (d

eg)

Time (s)

ωx

(deg

/s)

Time (s)

ωy

(deg

/s)

Time (s)

ωz

(deg

/s)

29

0 1 2 3 4 5 6 7-0.8

-0.6

-0.4

-0.2

0

0.2

0.4



0 1 2 3 4 5 6 7

-0.4

-0.2

0

0.2

0.4

0.6




0 1 2 3 4 5 6 7-3

-2

-1

0

1

2


(a) Roll angle

0 1 2 3 4 5 6 7-30

-20

-10

0

10

20

30


(b) Sideslip angle

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20


(c) Angle of attack

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20


(d) Roll rate

0 1 2 3 4 5 6 7-200

-150

-100

-50

0

50


(e) Yaw rate

0 1 2 3 4 5 6 7-100

0

100

200

300


(f) Pitch rate


Time (s)

dqɛ

(deg

/s)

Time (s)

dqβ

(deg

/s)

Time (s)

γ (d

eg)

Time (s)

β (d

eg)

Time (s)

α (d

eg)

Time (s)

ωx

(deg

/s)

Time (s)

ωy

(deg

/s)

Time (s)

ωz

(deg

/s)



(278~294)14-087.indd 291 2015-07-03 오전 5:06:44



30

0 1 2 3 4 5 6 7-50

0

50

0 0.50

1020



0 1 2 3 4 5 6 7-50

0

50

0 0.50

1020




0 1 2 3 4 5 6 7

-0.2

-0.1

0

0.1

0.2

0.3



0 1 2 3 4 5 6 7-10

-5

0

5

10

15

20



0 1 2 3 4 5 6 7-30

-20

-10

0

10

20



0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6

0.8



0 1 2 3 4 5 6 7

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1




for Case 3

Time (s)

a tɛ

(m/s

2 )

Time (s)

a tβ

(m/s

2 )

Time (s)

δ x (

deg)

Time (s)

δ y (

deg)

Time (s)

δz

(deg

)

Time (s) δ s

y

Time (s)

δ sz

30

0 1 2 3 4 5 6 7-50

0

50

0 0.50

1020



0 1 2 3 4 5 6 7-50

0

50

0 0.50

1020




0 1 2 3 4 5 6 7

-0.2

-0.1

0

0.1

0.2

0.3



0 1 2 3 4 5 6 7-10

-5

0

5

10

15

20



0 1 2 3 4 5 6 7-30

-20

-10

0

10

20



0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6

0.8



0 1 2 3 4 5 6 7

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1




for Case 3

Time (s)

a tɛ

(m/s

2 )

Time (s)

a tβ

(m/s

2 )

Time (s)

δ x (

deg)

Time (s)

δ y (

deg)

Time (s)

δz

(deg

)

Time (s)

δ sy

Time (s)

δ sz


Fig 13. Actual and estimated values of target acceleration for Case 2

30

0 1 2 3 4 5 6 7-50

0

50

0 0.50

1020



0 1 2 3 4 5 6 7-50

0

50

0 0.50

1020




0 1 2 3 4 5 6 7

-0.2

-0.1

0

0.1

0.2

0.3



0 1 2 3 4 5 6 7-10

-5

0

5

10

15

20



0 1 2 3 4 5 6 7-30

-20

-10

0

10

20



0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6

0.8



0 1 2 3 4 5 6 7

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1




for Case 3

Time (s)

a tɛ

(m/s

2 )

Time (s)

a tβ

(m/s

2 )

Time (s)

δ x (

deg)

Time (s)

δ y (

deg)

Time (s)

δz

(deg

)

Time (s)

δ sy

Time (s)

δ sz

30

0 1 2 3 4 5 6 7-50

0

50

0 0.50

1020



0 1 2 3 4 5 6 7-50

0

50

0 0.50

1020




0 1 2 3 4 5 6 7

-0.2

-0.1

0

0.1

0.2

0.3



0 1 2 3 4 5 6 7-10

-5

0

5

10

15

20



0 1 2 3 4 5 6 7-30

-20

-10

0

10

20



0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6

0.8



0 1 2 3 4 5 6 7

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1




for Case 3

Time (s)

a tɛ

(m/s

2 )

Time (s)

a tβ

(m/s

2 )

Time (s)

δ x (

deg)

Time (s)

δ y (

deg)

Time (s)

δz

(deg

)

Time (s)

δ sy

Time (s)

δ sz


30

0 1 2 3 4 5 6 7-50

0

50

0 0.50

1020



0 1 2 3 4 5 6 7-50

0

50

0 0.50

1020




0 1 2 3 4 5 6 7

-0.2

-0.1

0

0.1

0.2

0.3



0 1 2 3 4 5 6 7-10

-5

0

5

10

15

20



0 1 2 3 4 5 6 7-30

-20

-10

0

10

20



0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6

0.8



0 1 2 3 4 5 6 7

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1




for Case 3

Time (s)

a tɛ

(m/s

2 )

Time (s)

a tβ

(m/s

2 )

Time (s)

δ x (

deg)

Time (s)

δ y (

deg)

Time (s)

δz

(deg

)

Time (s)

δ sy

Time (s)

δ sz

30

0 1 2 3 4 5 6 7-50

0

50

0 0.50

1020



0 1 2 3 4 5 6 7-50

0

50

0 0.50

1020




0 1 2 3 4 5 6 7

-0.2

-0.1

0

0.1

0.2

0.3



0 1 2 3 4 5 6 7-10

-5

0

5

10

15

20



0 1 2 3 4 5 6 7-30

-20

-10

0

10

20



0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6

0.8



0 1 2 3 4 5 6 7

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1




for Case 3

Time (s)

a tɛ

(m/s

2 )

Time (s)

a tβ

(m/s

2 )

Time (s)

δ x (

deg)

Time (s)

δ y (

deg)

Time (s)

δz

(deg

)

Time (s)

δ sy

Time (s)

δ sz


30

0 1 2 3 4 5 6 7-50

0

50

0 0.50

1020



0 1 2 3 4 5 6 7-50

0

50

0 0.50

1020




0 1 2 3 4 5 6 7

-0.2

-0.1

0

0.1

0.2

0.3



0 1 2 3 4 5 6 7-10

-5

0

5

10

15

20



0 1 2 3 4 5 6 7-30

-20

-10

0

10

20



0 1 2 3 4 5 6 7-0.4

-0.2

0

0.2

0.4

0.6

0.8



0 1 2 3 4 5 6 7

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1




for Case 3

Time (s)

a tɛ

(m/s

2 )

Time (s)

a tβ

(m/s

2 )

Time (s)

δ x (

deg)

Time (s)

δ y (

deg)

Time (s)

δz

(deg

)

Time (s)

δ sy

Time (s)

δ sz



(278~294)14-087.indd 292 2015-07-03 오전 5:06:45

293


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algorithm required more control energy to obtain the small

miss distance. Compared with PNG+BSM+RLSWCA and

SMG+BSM+RLSWCA, it can be observed from Table 1 that

the miss distance and interception time for the proposed

IGC algorithm are minimal, since this method considers

the effects of the target maneuver and external distance

in the control system loop, and the couplings between the

guidance loop and control loop.

Remark 1. In general, the knowledge and experience

of experts (including control engineers and operators)

are employed to determine the design parameters. For

the design parameters of the pitch channel, the FTDO

parameters are often selected as λp20=20, λp30=λp40=2, λpi1=1.5,

λpi2=1.1, qpi=0.5 and ppi=8, i∈{2, 3, 4}; the filter time constants

are often selected as τp3∈{0.01, 0.2} and τp4∈{0.01, 0.2}, and

then the controller parameters are often selected as kp2∈{2,

5}, kp3=1/τp3, kp4=1/τp4, kp5∈{1, 10} and λp4∈(0, 1). The design

parameters of the yaw channel and roll channel are similar.

5. Conclusion

In this paper, a novel three-channel independent IGC

design method is proposed for the NSI, based on the

backstepping sliding mode and FTDO, which also take into

account the couplings of the guidance and control system

and those between the different channels of interceptor.

In the proposed method, the FTDO is firstly constructed to

estimate the target acceleration and the disturbance of the

control system loop, and the IGC algorithm based on the

backstepping sliding mode is given to obtain the desired

control moment. Moreover, the stability analysis based on

the Lyapunov theory shows that the proposed IGC scheme

can make the LOS angular rate to converge into a small

neighborhood of zero. Finally, simulation results also

illustrate the effectiveness of the proposed IGC algorithm.

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