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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
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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.
Finally, a brief conclusion is drawn in Section 5.
2. IGC model description
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
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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
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
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)
(1)
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.
Finally, a brief conclusion is drawn in Section 5.
2. IGC model description
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) (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
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
and
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
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
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
(4)
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
(5)
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
(6)
where q is the dynamic pressure; α, β and γ are the angle of
attack, sideslip angle and roll angle;
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
and
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
are the corresponding aerodynamic coefficients;
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
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
and
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
are the external disturbances;
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
and
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
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.
Define the following virtual control moment:
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 ,
(8)
where
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 ,
is the uncertainty in the control effectiveness
matrix, and
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 ,
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 ,
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 ,
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 ,
It is assumed that the angles between LOS and the
interceptor velocity coordinate system are small [1], i.e.,
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 ,
and
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 ,
. Based on equations (1)~(6), we
can obtain the IGC model in the pitch channel:
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 ,
(9)
where
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 ,
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 ,
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
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.
the IGC model in the yaw channel:
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.
(10)
(278~294)14-087.indd 280 2015-07-03 오전 5:06:23
![Page 4: WANG Fei* and LIU Gang**past.ijass.org/On_line/admin/files/13.(278~294)14-087.pdf · 2015-07-15 · independent IGC model for the NSI. Section 3 presents the corresponding IGC design](https://reader035.vdocument.in/reader035/viewer/2022062505/5e8acb2c0c66cd5e17016c3c/html5/thumbnails/4.jpg)
281
WANG Fei Integrated Guidance and Control Design for the Near Space Interceptor
http://ijass.org
where
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.
and the IGC model in the roll channel:
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.
(11)
where
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
;
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
;
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.
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
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 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
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 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:
(12)
Then, the following FTDO is proposed to estimate the
target acceleration :
8
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 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:
(13)
where
8
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 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:
,
8
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 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
8
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 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:
are
the estimated values of
8
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 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
8
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 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:
, respectively;
8
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 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:
,
8
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 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
8
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 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:
are the observer gain coefficients to be determined.
When we define the error variables
8
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 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:
,
8
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 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
8
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 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:
, we can obtain the
following error system:
8
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 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:
(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)
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)
(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)
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)
,
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)
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)
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)
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):
Step 1: Define the first error surface as follows:
(278~294)14-087.indd 281 2015-07-03 오전 5:06:25
![Page 5: WANG Fei* and LIU Gang**past.ijass.org/On_line/admin/files/13.(278~294)14-087.pdf · 2015-07-15 · independent IGC model for the NSI. Section 3 presents the corresponding IGC design](https://reader035.vdocument.in/reader035/viewer/2022062505/5e8acb2c0c66cd5e17016c3c/html5/thumbnails/5.jpg)
DOI: http://dx.doi.org/10.5139/IJASS.2015.16.2.278 282
Int’l J. of Aeronautical & Space Sci. 16(2), 278–294 (2015)
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)
(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)
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)
(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)
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)
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)
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)
(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)
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)
:
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) (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)
Then, the virtual control *4px is constructed as follows:
* *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)
where 4 0p denotes the filter time constant. Then, we can obtain the differentiation of the filtered
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)
and its derivative is:
*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)
Step 2: Denote the second error surface as follows:
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)
Then, the virtual control *4px is constructed as follows:
* *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)
where 4 0p denotes the filter time constant. Then, we can obtain the differentiation of the filtered
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)
and its derivative is:
*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
(22)
and its derivative is:
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)
Then, the virtual control *4px is constructed as follows:
* *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)
where 4 0p denotes the filter time constant. Then, we can obtain the differentiation of the filtered
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)
and its derivative is:
*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
(23)
Then, the 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)
Then, the virtual control *4px is constructed as follows:
* *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)
where 4 0p denotes the filter time constant. Then, we can obtain the differentiation of the filtered
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)
and its derivative is:
*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
is constructed as follows:
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)
Then, the virtual control *4px is constructed as follows:
* *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)
where 4 0p denotes the filter time constant. Then, we can obtain the differentiation of the filtered
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)
and its derivative is:
*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
(24)
where kp3>0. In addition, the following first-order low-pass
filter is proposed to obtain 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)
Then, the virtual control *4px is constructed as follows:
* *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)
where 4 0p denotes the filter time constant. Then, we can obtain the differentiation of the filtered
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)
and its derivative is:
*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
:
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)
Then, the virtual control *4px is constructed as follows:
* *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)
where 4 0p denotes the filter time constant. Then, we can obtain the differentiation of the filtered
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)
and its derivative is:
*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
(25)
where τp4>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)
Then, the virtual control *4px is constructed as follows:
* *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)
where 4 0p denotes the filter time constant. Then, we can obtain the differentiation of the filtered
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)
and its derivative is:
*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
(26)
Step 3: Denote the third error surface as follows:
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)
Then, the virtual control *4px is constructed as follows:
* *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)
where 4 0p denotes the filter time constant. Then, we can obtain the differentiation of the filtered
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)
and its derivative is:
*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
(27)
and its derivative is:
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)
Then, the virtual control *4px is constructed as follows:
* *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)
where 4 0p denotes the filter time constant. Then, we can obtain the differentiation of the filtered
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)
and its derivative is:
*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
(28)
Then, we can construct the following control moment:
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)
Then, the virtual control *4px is constructed as follows:
* *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)
where 4 0p denotes the filter time constant. Then, we can obtain the differentiation of the filtered
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)
and its derivative is:
*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
(29)
where kp4>0, kp5>0 and 0<λp4>0<1.
3.2 Control design in the yaw channel
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)
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, ,
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
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)
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, ,
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
(30)
Then, the following FTDO is proposed to estimate the
target acceleration
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)
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, ,
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
:
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)
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, ,
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
(31)
where
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)
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, ,
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
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
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)
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, ,
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
(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
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)
are the virtual control of the guidance loop and
sideslip angle loop in the yaw channel;
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
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)
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
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283
WANG Fei Integrated Guidance and Control Design for the Near Space Interceptor
http://ijass.org
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
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)
(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.
3.4 Stability analysis
It is assumed that the estimated errors in the FTDO satisfy
the following constraints [6]-[7]:
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)
(34)
where Npi, Nyi and Nri are positive 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)
,
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)
.
Define the following filter errors:
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) (35)
Then, the dynamics of the filter errors satisfy:
13
Then, the dynamics of the filter errors satisfy:
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
x s x s y xs s y k s es s y k s e
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)
From equations (17)~(29) and (32)~(33), we can obtain:
13
Then, the dynamics of the filter errors satisfy:
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
x s x s y xs s y k s es s y k s e
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 ,
(37)
13
Then, the dynamics of the filter errors satisfy:
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
x s x s y xs s y k s es s y k s e
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 ,
(38)
13
Then, the dynamics of the filter errors satisfy:
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
x s x s y xs s y k s es s y k s e
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 ,
(39)
where
13
Then, the dynamics of the filter errors satisfy:
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
x s x s y xs s y k s es s y k s e
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 ,
,
13
Then, the dynamics of the filter errors satisfy:
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
x s x s y xs s y k s es s y k s e
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 ,
. It is assumed
that
13
Then, the dynamics of the filter errors satisfy:
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
x s x s y xs s y k s es s y k s e
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 ,
and
13
Then, the dynamics of the filter errors satisfy:
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
x s x s y xs s y k s es s y k s e
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 ,
.
For the system (9)~(11), consider the following Lyapunov
function:
13
Then, the dynamics of the filter errors satisfy:
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
x s x s y xs s y k s es s y k s e
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 ,
(40)
where
13
Then, the dynamics of the filter errors satisfy:
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
x s x s y xs s y k s es s y k s e
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 ,
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)
3 3 3 4 4 3 3 31( )p p p p p p p ps s s s y k s e
2 2 2 2 2 2 23 3 3 4 3 4 3 310.5( ) 0.5( ) 0.5( | | )p p p p p p p pk s s s s y s e
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 .
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)
3 3 3 4 4 3 3 31( )p p p p p p p ps s s s y k s e
2 2 2 2 2 2 23 3 3 4 3 4 3 310.5( ) 0.5( ) 0.5( | | )p p p p p p p pk s s s s y s e
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 .
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)
3 3 3 4 4 3 3 31( )p p p p p p p ps s s s y k s e
2 2 2 2 2 2 23 3 3 4 3 4 3 310.5( ) 0.5( ) 0.5( | | )p p p p p p p pk s s s s y s e
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 .
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)
3 3 3 4 4 3 3 31( )p p p p p p p ps s s s y k s e
2 2 2 2 2 2 23 3 3 4 3 4 3 310.5( ) 0.5( ) 0.5( | | )p p p p p p p pk s s s s y s e
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 .
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)
3 3 3 4 4 3 3 31( )p p p p p p p ps s s s y k s e
2 2 2 2 2 2 23 3 3 4 3 4 3 310.5( ) 0.5( ) 0.5( | | )p p p p p p p pk s s s s y s e
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 .
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)
3 3 3 4 4 3 3 31( )p p p p p p p ps s s s y k s e
2 2 2 2 2 2 23 3 3 4 3 4 3 310.5( ) 0.5( ) 0.5( | | )p p p p p p p pk s s s s y s e
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 .
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)
3 3 3 4 4 3 3 31( )p p p p p p p ps s s s y k s e
2 2 2 2 2 2 23 3 3 4 3 4 3 310.5( ) 0.5( ) 0.5( | | )p p p p p p p pk s s s s y s e
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 .
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)
3 3 3 4 4 3 3 31( )p p p p p p p ps s s s y k s e
2 2 2 2 2 2 23 3 3 4 3 4 3 310.5( ) 0.5( ) 0.5( | | )p p p p p p p pk s s s s y s e
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 .
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)
3 3 3 4 4 3 3 31( )p p p p p p p ps s s s y k s e
2 2 2 2 2 2 23 3 3 4 3 4 3 310.5( ) 0.5( ) 0.5( | | )p p p p p p p pk s s s s y s e
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 .
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)
3 3 3 4 4 3 3 31( )p p p p p p p ps s s s y k s e
2 2 2 2 2 2 23 3 3 4 3 4 3 310.5( ) 0.5( ) 0.5( | | )p p p p p p p pk s s s s y s e
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 .
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)
3 3 3 4 4 3 3 31( )p p p p p p p ps s s s y k s e
2 2 2 2 2 2 23 3 3 4 3 4 3 310.5( ) 0.5( ) 0.5( | | )p p p p p p p pk s s s s y s e
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 .
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)
3 3 3 4 4 3 3 31( )p p p p p p p ps s s s y k s e
2 2 2 2 2 2 23 3 3 4 3 4 3 310.5( ) 0.5( ) 0.5( | | )p p p p p p p pk s s s s y s e
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)
Based on the same method, we can also obtain:
(278~294)14-087.indd 283 2015-07-03 오전 5:06:30
![Page 7: WANG Fei* and LIU Gang**past.ijass.org/On_line/admin/files/13.(278~294)14-087.pdf · 2015-07-15 · independent IGC model for the NSI. Section 3 presents the corresponding IGC design](https://reader035.vdocument.in/reader035/viewer/2022062505/5e8acb2c0c66cd5e17016c3c/html5/thumbnails/7.jpg)
DOI: http://dx.doi.org/10.5139/IJASS.2015.16.2.278 284
Int’l J. of Aeronautical & Space Sci. 16(2), 278–294 (2015)
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)
3 3 3 4 4 3 3 31( )p p p p p p p ps s s s y k s e
2 2 2 2 2 2 23 3 3 4 3 4 3 310.5( ) 0.5( ) 0.5( | | )p p p p p p p pk s s s s y s e
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.
From equation (37), we have:
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)
3 3 3 4 4 3 3 31( )p p p p p p p ps s s s y k s e
2 2 2 2 2 2 23 3 3 4 3 4 3 310.5( ) 0.5( ) 0.5( | | )p p p p p p p pk s s s s y s e
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 .
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)
3 3 3 4 4 3 3 31( )p p p p p p p ps s s s y k s e
2 2 2 2 2 2 23 3 3 4 3 4 3 310.5( ) 0.5( ) 0.5( | | )p p p p p p p pk s s s s y s e
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
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
(50)
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
(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
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
(52)
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
(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
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
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
(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:
If the design parameters satisfy the following conditions:
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:
(55)
where κ is a positive real number, then we have:
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:
(56)
where
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:
Based on the comparison principle, from equation (56)
we can obtain:
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:
(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.
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 △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
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285
WANG Fei Integrated Guidance and Control Design for the Near Space Interceptor
http://ijass.org
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)
(59)
where Ts is the sampling time.
The following transformation is introduced to avoid the
uneven distribution problem:
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)
(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)
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)
based on the robust least-
squares method in [22] and [23]:
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)
(61)
where
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)
. Furthermore, we can obtain
the following worst-case residual:
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)
(62)
Based on the triangle inequality, we have
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)
(63)
Assume that
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)
(64)
where
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)
.
Then, in the direction of εc, the worst-case residual is:
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) (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
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:
(67)
subject to
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:
and
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:
.
Then, the optimal solution u of the RLSWCA problem can
be given by:
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:
(68)
where
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:
,
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:
,
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.
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:
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
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:
Fig. 2. Structure diagram of a three-channel independent IGC system
(278~294)14-087.indd 285 2015-07-03 오전 5:06:34
![Page 9: WANG Fei* and LIU Gang**past.ijass.org/On_line/admin/files/13.(278~294)14-087.pdf · 2015-07-15 · independent IGC model for the NSI. Section 3 presents the corresponding IGC design](https://reader035.vdocument.in/reader035/viewer/2022062505/5e8acb2c0c66cd5e17016c3c/html5/thumbnails/9.jpg)
DOI: http://dx.doi.org/10.5139/IJASS.2015.16.2.278 286
Int’l J. of Aeronautical & Space Sci. 16(2), 278–294 (2015)
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 ;
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 .
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 ;
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 .
(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 ;
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 .
(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 ;
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 .
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 ;
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 .
.
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
2PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
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
3PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
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
4PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
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
5PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
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
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WANG Fei Integrated Guidance and Control Design for the Near Space Interceptor
http://ijass.org
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
3PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Roll angle
0 1 2 3 4 5 6 7-10
-8
-6
-4
-2
0
2
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Sideslip angle
0 1 2 3 4 5 6 7-5
0
5
10
15
20
25
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Angle of attack
0 1 2 3 4 5 6 7
-20
-10
0
10
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Roll rate
0 1 2 3 4 5 6 7-100
-80
-60
-40
-20
0
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Yaw rate
0 1 2 3 4 5 6 7
-100
0
100
200PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(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
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
3PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Roll angle
0 1 2 3 4 5 6 7-10
-8
-6
-4
-2
0
2
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Sideslip angle
0 1 2 3 4 5 6 7-5
0
5
10
15
20
25
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Angle of attack
0 1 2 3 4 5 6 7
-20
-10
0
10
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Roll rate
0 1 2 3 4 5 6 7-100
-80
-60
-40
-20
0
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Yaw rate
0 1 2 3 4 5 6 7
-100
0
100
200PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(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)
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
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
3PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Roll angle
0 1 2 3 4 5 6 7-10
-8
-6
-4
-2
0
2
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Sideslip angle
0 1 2 3 4 5 6 7-5
0
5
10
15
20
25
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Angle of attack
0 1 2 3 4 5 6 7
-20
-10
0
10
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Roll rate
0 1 2 3 4 5 6 7-100
-80
-60
-40
-20
0
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Yaw rate
0 1 2 3 4 5 6 7
-100
0
100
200PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(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)
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
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
3PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Roll angle
0 1 2 3 4 5 6 7-10
-8
-6
-4
-2
0
2
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Sideslip angle
0 1 2 3 4 5 6 7-5
0
5
10
15
20
25
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Angle of attack
0 1 2 3 4 5 6 7
-20
-10
0
10
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Roll rate
0 1 2 3 4 5 6 7-100
-80
-60
-40
-20
0
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Yaw rate
0 1 2 3 4 5 6 7
-100
0
100
200PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(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)
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
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
3PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Roll angle
0 1 2 3 4 5 6 7-10
-8
-6
-4
-2
0
2
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Sideslip angle
0 1 2 3 4 5 6 7-5
0
5
10
15
20
25
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Angle of attack
0 1 2 3 4 5 6 7
-20
-10
0
10
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Roll rate
0 1 2 3 4 5 6 7-100
-80
-60
-40
-20
0
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Yaw rate
0 1 2 3 4 5 6 7
-100
0
100
200PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(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)
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
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
3PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Roll angle
0 1 2 3 4 5 6 7-10
-8
-6
-4
-2
0
2
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Sideslip angle
0 1 2 3 4 5 6 7-5
0
5
10
15
20
25
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Angle of attack
0 1 2 3 4 5 6 7
-20
-10
0
10
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Roll rate
0 1 2 3 4 5 6 7-100
-80
-60
-40
-20
0
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Yaw rate
0 1 2 3 4 5 6 7
-100
0
100
200PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(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)
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
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
3PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Roll angle
0 1 2 3 4 5 6 7-10
-8
-6
-4
-2
0
2
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Sideslip angle
0 1 2 3 4 5 6 7-5
0
5
10
15
20
25
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Angle of attack
0 1 2 3 4 5 6 7
-20
-10
0
10
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Roll rate
0 1 2 3 4 5 6 7-100
-80
-60
-40
-20
0
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Yaw rate
0 1 2 3 4 5 6 7
-100
0
100
200PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(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)
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
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
3PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Roll angle
0 1 2 3 4 5 6 7-10
-8
-6
-4
-2
0
2
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Sideslip angle
0 1 2 3 4 5 6 7-5
0
5
10
15
20
25
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Angle of attack
0 1 2 3 4 5 6 7
-20
-10
0
10
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Roll rate
0 1 2 3 4 5 6 7-100
-80
-60
-40
-20
0
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Yaw rate
0 1 2 3 4 5 6 7
-100
0
100
200PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(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)
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
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DOI: http://dx.doi.org/10.5139/IJASS.2015.16.2.278 288
Int’l J. of Aeronautical & Space Sci. 16(2), 278–294 (2015)
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
Actual valueEstimated value
(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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Aileron deflection
0 1 2 3 4 5 6 7-5
0
5
10
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Rudder deflection
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Elevator deflection
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
0.8
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(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
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
Actual valueEstimated value
(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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Aileron deflection
0 1 2 3 4 5 6 7-5
0
5
10
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Rudder deflection
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Elevator deflection
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
0.8
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(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 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
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
Actual valueEstimated value
(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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Aileron deflection
0 1 2 3 4 5 6 7-5
0
5
10
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Rudder deflection
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Elevator deflection
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
0.8
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(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 sy
Time (s)
δ sz
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
Actual valueEstimated value
(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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Aileron deflection
0 1 2 3 4 5 6 7-5
0
5
10
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Rudder deflection
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Elevator deflection
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
0.8
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(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 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
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
Actual valueEstimated value
(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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Aileron deflection
0 1 2 3 4 5 6 7-5
0
5
10
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Rudder deflection
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Elevator deflection
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
0.8
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(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 sy
Time (s)
δ sz
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
Actual valueEstimated value
(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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Aileron deflection
0 1 2 3 4 5 6 7-5
0
5
10
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Rudder deflection
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Elevator deflection
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
0.8
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(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 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
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
Actual valueEstimated value
(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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Aileron deflection
0 1 2 3 4 5 6 7-5
0
5
10
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Rudder deflection
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Elevator deflection
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
0.8
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(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 sy
Time (s)
δ sz
(e) Equivalent control surface deflection in yaw direction
Fig. 6. Trajectories of the actual control inputs for Case 1
(278~294)14-087.indd 288 2015-07-03 오전 5:06:39
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WANG Fei Integrated Guidance and Control Design for the Near Space Interceptor
http://ijass.org
27
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) LOS angular rate in vertical plane
0 1 2 3 4 5 6 7
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) LOS angular rate in lateral plane
Fig. 7 Trajectories of the LOS angular rate for Case 2
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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Sideslip angle
0 1 2 3 4 5 6 70
5
10
15
20
25
30
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Angle of attack
0 1 2 3 4 5 6 7-40
-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Roll rate
0 1 2 3 4 5 6 7-150
-100
-50
0
50
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Yaw rate
0 1 2 3 4 5 6 7-100
0
100
200
300
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(f) Pitch rate
Fig. 8 Trajectories of the angle and angular rate for Case 2
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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) LOS angular rate in vertical plane
0 1 2 3 4 5 6 7
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) LOS angular rate in lateral plane
Fig. 7 Trajectories of the LOS angular rate for Case 2
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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Sideslip angle
0 1 2 3 4 5 6 70
5
10
15
20
25
30
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Angle of attack
0 1 2 3 4 5 6 7-40
-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Roll rate
0 1 2 3 4 5 6 7-150
-100
-50
0
50
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Yaw rate
0 1 2 3 4 5 6 7-100
0
100
200
300
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(f) Pitch rate
Fig. 8 Trajectories of the angle and angular rate for Case 2
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
Fig. 7. Trajectories of the LOS angular rate for Case 2
27
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) LOS angular rate in vertical plane
0 1 2 3 4 5 6 7
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) LOS angular rate in lateral plane
Fig. 7 Trajectories of the LOS angular rate for Case 2
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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Sideslip angle
0 1 2 3 4 5 6 70
5
10
15
20
25
30
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Angle of attack
0 1 2 3 4 5 6 7-40
-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Roll rate
0 1 2 3 4 5 6 7-150
-100
-50
0
50
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Yaw rate
0 1 2 3 4 5 6 7-100
0
100
200
300
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(f) Pitch rate
Fig. 8 Trajectories of the angle and angular rate for Case 2
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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) LOS angular rate in vertical plane
0 1 2 3 4 5 6 7
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) LOS angular rate in lateral plane
Fig. 7 Trajectories of the LOS angular rate for Case 2
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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Sideslip angle
0 1 2 3 4 5 6 70
5
10
15
20
25
30
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Angle of attack
0 1 2 3 4 5 6 7-40
-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Roll rate
0 1 2 3 4 5 6 7-150
-100
-50
0
50
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Yaw rate
0 1 2 3 4 5 6 7-100
0
100
200
300
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(f) Pitch rate
Fig. 8 Trajectories of the angle and angular rate for Case 2
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
27
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) LOS angular rate in vertical plane
0 1 2 3 4 5 6 7
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) LOS angular rate in lateral plane
Fig. 7 Trajectories of the LOS angular rate for Case 2
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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Sideslip angle
0 1 2 3 4 5 6 70
5
10
15
20
25
30
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Angle of attack
0 1 2 3 4 5 6 7-40
-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Roll rate
0 1 2 3 4 5 6 7-150
-100
-50
0
50
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Yaw rate
0 1 2 3 4 5 6 7-100
0
100
200
300
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(f) Pitch rate
Fig. 8 Trajectories of the angle and angular rate for Case 2
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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) LOS angular rate in vertical plane
0 1 2 3 4 5 6 7
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) LOS angular rate in lateral plane
Fig. 7 Trajectories of the LOS angular rate for Case 2
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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Sideslip angle
0 1 2 3 4 5 6 70
5
10
15
20
25
30
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Angle of attack
0 1 2 3 4 5 6 7-40
-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Roll rate
0 1 2 3 4 5 6 7-150
-100
-50
0
50
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Yaw rate
0 1 2 3 4 5 6 7-100
0
100
200
300
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(f) Pitch rate
Fig. 8 Trajectories of the angle and angular rate for Case 2
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
27
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) LOS angular rate in vertical plane
0 1 2 3 4 5 6 7
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) LOS angular rate in lateral plane
Fig. 7 Trajectories of the LOS angular rate for Case 2
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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Sideslip angle
0 1 2 3 4 5 6 70
5
10
15
20
25
30
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Angle of attack
0 1 2 3 4 5 6 7-40
-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Roll rate
0 1 2 3 4 5 6 7-150
-100
-50
0
50
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Yaw rate
0 1 2 3 4 5 6 7-100
0
100
200
300
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(f) Pitch rate
Fig. 8 Trajectories of the angle and angular rate for Case 2
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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) LOS angular rate in vertical plane
0 1 2 3 4 5 6 7
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) LOS angular rate in lateral plane
Fig. 7 Trajectories of the LOS angular rate for Case 2
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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Sideslip angle
0 1 2 3 4 5 6 70
5
10
15
20
25
30
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Angle of attack
0 1 2 3 4 5 6 7-40
-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Roll rate
0 1 2 3 4 5 6 7-150
-100
-50
0
50
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Yaw rate
0 1 2 3 4 5 6 7-100
0
100
200
300
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(f) Pitch rate
Fig. 8 Trajectories of the angle and angular rate for Case 2
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. 8. Trajectories of the angle and angular rate for Case 2
(278~294)14-087.indd 289 2015-07-03 오전 5:06:41
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DOI: http://dx.doi.org/10.5139/IJASS.2015.16.2.278 290
Int’l J. of Aeronautical & Space Sci. 16(2), 278–294 (2015)
28
0 1 2 3 4 5 6 7-10
0
10
20
30
40
50
Actual valueEstimated value
(a) Target acceleration in vertical plane
0 1 2 3 4 5 6 7-10
0
10
20
30
40
50
Actual valueEstimated value
(b) Target acceleration in lateral plane
Fig. 9 Actual and estimated values of target acceleration for Case 2
0 1 2 3 4 5 6 7-0.05
0
0.05
0.1
0.15
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Aileron deflection
0 1 2 3 4 5 6 7-5
0
5
10
15
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Rudder deflection
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Elevator deflection
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
0.8
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Equivalent control surface deflection in pitch direction
0 1 2 3 4 5 6 7
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Equivalent control surface deflection in yaw direction
Fig. 10 Trajectories of the actual control inputs
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
Actual valueEstimated value
(a) Target acceleration in vertical plane
0 1 2 3 4 5 6 7-10
0
10
20
30
40
50
Actual valueEstimated value
(b) Target acceleration in lateral plane
Fig. 9 Actual and estimated values of target acceleration for Case 2
0 1 2 3 4 5 6 7-0.05
0
0.05
0.1
0.15
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Aileron deflection
0 1 2 3 4 5 6 7-5
0
5
10
15
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Rudder deflection
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Elevator deflection
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
0.8
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Equivalent control surface deflection in pitch direction
0 1 2 3 4 5 6 7
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Equivalent control surface deflection in yaw direction
Fig. 10 Trajectories of the actual control inputs
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
(a) Target acceleration in vertical plane (b) Target acceleration in lateral plane
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
Actual valueEstimated value
(a) Target acceleration in vertical plane
0 1 2 3 4 5 6 7-10
0
10
20
30
40
50
Actual valueEstimated value
(b) Target acceleration in lateral plane
Fig. 9 Actual and estimated values of target acceleration for Case 2
0 1 2 3 4 5 6 7-0.05
0
0.05
0.1
0.15
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Aileron deflection
0 1 2 3 4 5 6 7-5
0
5
10
15
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Rudder deflection
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Elevator deflection
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
0.8
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Equivalent control surface deflection in pitch direction
0 1 2 3 4 5 6 7
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Equivalent control surface deflection in yaw direction
Fig. 10 Trajectories of the actual control inputs
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
Actual valueEstimated value
(a) Target acceleration in vertical plane
0 1 2 3 4 5 6 7-10
0
10
20
30
40
50
Actual valueEstimated value
(b) Target acceleration in lateral plane
Fig. 9 Actual and estimated values of target acceleration for Case 2
0 1 2 3 4 5 6 7-0.05
0
0.05
0.1
0.15
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Aileron deflection
0 1 2 3 4 5 6 7-5
0
5
10
15
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Rudder deflection
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Elevator deflection
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
0.8
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Equivalent control surface deflection in pitch direction
0 1 2 3 4 5 6 7
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Equivalent control surface deflection in yaw direction
Fig. 10 Trajectories of the actual control inputs
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
(a) Aileron deflection (b) Rudder deflection
28
0 1 2 3 4 5 6 7-10
0
10
20
30
40
50
Actual valueEstimated value
(a) Target acceleration in vertical plane
0 1 2 3 4 5 6 7-10
0
10
20
30
40
50
Actual valueEstimated value
(b) Target acceleration in lateral plane
Fig. 9 Actual and estimated values of target acceleration for Case 2
0 1 2 3 4 5 6 7-0.05
0
0.05
0.1
0.15
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Aileron deflection
0 1 2 3 4 5 6 7-5
0
5
10
15
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Rudder deflection
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Elevator deflection
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
0.8
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Equivalent control surface deflection in pitch direction
0 1 2 3 4 5 6 7
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Equivalent control surface deflection in yaw direction
Fig. 10 Trajectories of the actual control inputs
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
Actual valueEstimated value
(a) Target acceleration in vertical plane
0 1 2 3 4 5 6 7-10
0
10
20
30
40
50
Actual valueEstimated value
(b) Target acceleration in lateral plane
Fig. 9 Actual and estimated values of target acceleration for Case 2
0 1 2 3 4 5 6 7-0.05
0
0.05
0.1
0.15
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Aileron deflection
0 1 2 3 4 5 6 7-5
0
5
10
15
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Rudder deflection
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Elevator deflection
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
0.8
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Equivalent control surface deflection in pitch direction
0 1 2 3 4 5 6 7
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Equivalent control surface deflection in yaw direction
Fig. 10 Trajectories of the actual control inputs
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
(c) Elevator deflection (d) Equivalent control surface deflection in pitch direction
28
0 1 2 3 4 5 6 7-10
0
10
20
30
40
50
Actual valueEstimated value
(a) Target acceleration in vertical plane
0 1 2 3 4 5 6 7-10
0
10
20
30
40
50
Actual valueEstimated value
(b) Target acceleration in lateral plane
Fig. 9 Actual and estimated values of target acceleration for Case 2
0 1 2 3 4 5 6 7-0.05
0
0.05
0.1
0.15
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Aileron deflection
0 1 2 3 4 5 6 7-5
0
5
10
15
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Rudder deflection
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Elevator deflection
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
0.8
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Equivalent control surface deflection in pitch direction
0 1 2 3 4 5 6 7
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Equivalent control surface deflection in yaw direction
Fig. 10 Trajectories of the actual control inputs
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
(e) Equivalent control surface deflection in yaw direction
Fig. 10. Trajectories of the actual control inputs for Case 2
(278~294)14-087.indd 290 2015-07-03 오전 5:06:42
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291
WANG Fei Integrated Guidance and Control Design for the Near Space Interceptor
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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) LOS angular rate in vertical plane
0 1 2 3 4 5 6 7
-0.4
-0.2
0
0.2
0.4
0.6
0.8PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) LOS angular rate in lateral plane
Fig. 11 Trajectories of the LOS angular rate for Case 3
0 1 2 3 4 5 6 7-3
-2
-1
0
1
2
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Roll angle
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
30
40PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Sideslip angle
0 1 2 3 4 5 6 7-40
-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Angle of attack
0 1 2 3 4 5 6 7-40
-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Roll rate
0 1 2 3 4 5 6 7-200
-150
-100
-50
0
50
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Yaw rate
0 1 2 3 4 5 6 7-100
0
100
200
300
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(f) Pitch rate
Fig. 12 Trajectories of the angle and angular rate for Case 3
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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) LOS angular rate in vertical plane
0 1 2 3 4 5 6 7
-0.4
-0.2
0
0.2
0.4
0.6
0.8PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) LOS angular rate in lateral plane
Fig. 11 Trajectories of the LOS angular rate for Case 3
0 1 2 3 4 5 6 7-3
-2
-1
0
1
2
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Roll angle
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
30
40PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Sideslip angle
0 1 2 3 4 5 6 7-40
-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Angle of attack
0 1 2 3 4 5 6 7-40
-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Roll rate
0 1 2 3 4 5 6 7-200
-150
-100
-50
0
50
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Yaw rate
0 1 2 3 4 5 6 7-100
0
100
200
300
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(f) Pitch rate
Fig. 12 Trajectories of the angle and angular rate for Case 3
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
Fig. 11. Trajectories of the LOS angular rate for Case 3
29
0 1 2 3 4 5 6 7-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) LOS angular rate in vertical plane
0 1 2 3 4 5 6 7
-0.4
-0.2
0
0.2
0.4
0.6
0.8PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) LOS angular rate in lateral plane
Fig. 11 Trajectories of the LOS angular rate for Case 3
0 1 2 3 4 5 6 7-3
-2
-1
0
1
2
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Roll angle
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
30
40PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Sideslip angle
0 1 2 3 4 5 6 7-40
-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Angle of attack
0 1 2 3 4 5 6 7-40
-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Roll rate
0 1 2 3 4 5 6 7-200
-150
-100
-50
0
50
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Yaw rate
0 1 2 3 4 5 6 7-100
0
100
200
300
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(f) Pitch rate
Fig. 12 Trajectories of the angle and angular rate for Case 3
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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) LOS angular rate in vertical plane
0 1 2 3 4 5 6 7
-0.4
-0.2
0
0.2
0.4
0.6
0.8PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) LOS angular rate in lateral plane
Fig. 11 Trajectories of the LOS angular rate for Case 3
0 1 2 3 4 5 6 7-3
-2
-1
0
1
2
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Roll angle
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
30
40PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Sideslip angle
0 1 2 3 4 5 6 7-40
-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Angle of attack
0 1 2 3 4 5 6 7-40
-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Roll rate
0 1 2 3 4 5 6 7-200
-150
-100
-50
0
50
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Yaw rate
0 1 2 3 4 5 6 7-100
0
100
200
300
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(f) Pitch rate
Fig. 12 Trajectories of the angle and angular rate for Case 3
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
29
0 1 2 3 4 5 6 7-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) LOS angular rate in vertical plane
0 1 2 3 4 5 6 7
-0.4
-0.2
0
0.2
0.4
0.6
0.8PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) LOS angular rate in lateral plane
Fig. 11 Trajectories of the LOS angular rate for Case 3
0 1 2 3 4 5 6 7-3
-2
-1
0
1
2
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Roll angle
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
30
40PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Sideslip angle
0 1 2 3 4 5 6 7-40
-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Angle of attack
0 1 2 3 4 5 6 7-40
-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Roll rate
0 1 2 3 4 5 6 7-200
-150
-100
-50
0
50
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Yaw rate
0 1 2 3 4 5 6 7-100
0
100
200
300
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(f) Pitch rate
Fig. 12 Trajectories of the angle and angular rate for Case 3
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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) LOS angular rate in vertical plane
0 1 2 3 4 5 6 7
-0.4
-0.2
0
0.2
0.4
0.6
0.8PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) LOS angular rate in lateral plane
Fig. 11 Trajectories of the LOS angular rate for Case 3
0 1 2 3 4 5 6 7-3
-2
-1
0
1
2
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Roll angle
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
30
40PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Sideslip angle
0 1 2 3 4 5 6 7-40
-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Angle of attack
0 1 2 3 4 5 6 7-40
-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Roll rate
0 1 2 3 4 5 6 7-200
-150
-100
-50
0
50
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Yaw rate
0 1 2 3 4 5 6 7-100
0
100
200
300
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(f) Pitch rate
Fig. 12 Trajectories of the angle and angular rate for Case 3
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
29
0 1 2 3 4 5 6 7-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) LOS angular rate in vertical plane
0 1 2 3 4 5 6 7
-0.4
-0.2
0
0.2
0.4
0.6
0.8PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) LOS angular rate in lateral plane
Fig. 11 Trajectories of the LOS angular rate for Case 3
0 1 2 3 4 5 6 7-3
-2
-1
0
1
2
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Roll angle
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
30
40PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Sideslip angle
0 1 2 3 4 5 6 7-40
-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Angle of attack
0 1 2 3 4 5 6 7-40
-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Roll rate
0 1 2 3 4 5 6 7-200
-150
-100
-50
0
50
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Yaw rate
0 1 2 3 4 5 6 7-100
0
100
200
300
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(f) Pitch rate
Fig. 12 Trajectories of the angle and angular rate for Case 3
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
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) LOS angular rate in vertical plane
0 1 2 3 4 5 6 7
-0.4
-0.2
0
0.2
0.4
0.6
0.8PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) LOS angular rate in lateral plane
Fig. 11 Trajectories of the LOS angular rate for Case 3
0 1 2 3 4 5 6 7-3
-2
-1
0
1
2
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Roll angle
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
30
40PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Sideslip angle
0 1 2 3 4 5 6 7-40
-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Angle of attack
0 1 2 3 4 5 6 7-40
-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Roll rate
0 1 2 3 4 5 6 7-200
-150
-100
-50
0
50
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Yaw rate
0 1 2 3 4 5 6 7-100
0
100
200
300
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(f) Pitch rate
Fig. 12 Trajectories of the angle and angular rate for Case 3
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. 12. Trajectories of the angle and angular rate for Case 3
(278~294)14-087.indd 291 2015-07-03 오전 5:06:44
![Page 15: WANG Fei* and LIU Gang**past.ijass.org/On_line/admin/files/13.(278~294)14-087.pdf · 2015-07-15 · independent IGC model for the NSI. Section 3 presents the corresponding IGC design](https://reader035.vdocument.in/reader035/viewer/2022062505/5e8acb2c0c66cd5e17016c3c/html5/thumbnails/15.jpg)
DOI: http://dx.doi.org/10.5139/IJASS.2015.16.2.278 292
Int’l J. of Aeronautical & Space Sci. 16(2), 278–294 (2015)
30
0 1 2 3 4 5 6 7-50
0
50
0 0.50
1020
Actual valueEstimated value
(a) Target acceleration in vertical plane
0 1 2 3 4 5 6 7-50
0
50
0 0.50
1020
Actual valueEstimated value
(b) Target acceleration in lateral plane
Fig. 13 Actual and estimated values of target acceleration for Case 3
0 1 2 3 4 5 6 7
-0.2
-0.1
0
0.1
0.2
0.3
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Aileron deflection
0 1 2 3 4 5 6 7-10
-5
0
5
10
15
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Rudder deflection
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Elevator deflection
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
0.8
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Equivalent control surface deflection in pitch direction
0 1 2 3 4 5 6 7
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Equivalent control surface deflection in yaw direction
Fig. 14 Trajectories of the actual control inputs
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
Actual valueEstimated value
(a) Target acceleration in vertical plane
0 1 2 3 4 5 6 7-50
0
50
0 0.50
1020
Actual valueEstimated value
(b) Target acceleration in lateral plane
Fig. 13 Actual and estimated values of target acceleration for Case 3
0 1 2 3 4 5 6 7
-0.2
-0.1
0
0.1
0.2
0.3
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Aileron deflection
0 1 2 3 4 5 6 7-10
-5
0
5
10
15
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Rudder deflection
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Elevator deflection
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
0.8
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Equivalent control surface deflection in pitch direction
0 1 2 3 4 5 6 7
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Equivalent control surface deflection in yaw direction
Fig. 14 Trajectories of the actual control inputs
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
(a) Target acceleration in vertical plane (b) Target acceleration in lateral plane
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
Actual valueEstimated value
(a) Target acceleration in vertical plane
0 1 2 3 4 5 6 7-50
0
50
0 0.50
1020
Actual valueEstimated value
(b) Target acceleration in lateral plane
Fig. 13 Actual and estimated values of target acceleration for Case 3
0 1 2 3 4 5 6 7
-0.2
-0.1
0
0.1
0.2
0.3
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Aileron deflection
0 1 2 3 4 5 6 7-10
-5
0
5
10
15
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Rudder deflection
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Elevator deflection
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
0.8
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Equivalent control surface deflection in pitch direction
0 1 2 3 4 5 6 7
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Equivalent control surface deflection in yaw direction
Fig. 14 Trajectories of the actual control inputs
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
Actual valueEstimated value
(a) Target acceleration in vertical plane
0 1 2 3 4 5 6 7-50
0
50
0 0.50
1020
Actual valueEstimated value
(b) Target acceleration in lateral plane
Fig. 13 Actual and estimated values of target acceleration for Case 3
0 1 2 3 4 5 6 7
-0.2
-0.1
0
0.1
0.2
0.3
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Aileron deflection
0 1 2 3 4 5 6 7-10
-5
0
5
10
15
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Rudder deflection
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Elevator deflection
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
0.8
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Equivalent control surface deflection in pitch direction
0 1 2 3 4 5 6 7
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Equivalent control surface deflection in yaw direction
Fig. 14 Trajectories of the actual control inputs
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
(a) Aileron deflection (b) Rudder deflection
30
0 1 2 3 4 5 6 7-50
0
50
0 0.50
1020
Actual valueEstimated value
(a) Target acceleration in vertical plane
0 1 2 3 4 5 6 7-50
0
50
0 0.50
1020
Actual valueEstimated value
(b) Target acceleration in lateral plane
Fig. 13 Actual and estimated values of target acceleration for Case 3
0 1 2 3 4 5 6 7
-0.2
-0.1
0
0.1
0.2
0.3
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Aileron deflection
0 1 2 3 4 5 6 7-10
-5
0
5
10
15
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Rudder deflection
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Elevator deflection
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
0.8
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Equivalent control surface deflection in pitch direction
0 1 2 3 4 5 6 7
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Equivalent control surface deflection in yaw direction
Fig. 14 Trajectories of the actual control inputs
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
Actual valueEstimated value
(a) Target acceleration in vertical plane
0 1 2 3 4 5 6 7-50
0
50
0 0.50
1020
Actual valueEstimated value
(b) Target acceleration in lateral plane
Fig. 13 Actual and estimated values of target acceleration for Case 3
0 1 2 3 4 5 6 7
-0.2
-0.1
0
0.1
0.2
0.3
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Aileron deflection
0 1 2 3 4 5 6 7-10
-5
0
5
10
15
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Rudder deflection
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Elevator deflection
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
0.8
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Equivalent control surface deflection in pitch direction
0 1 2 3 4 5 6 7
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Equivalent control surface deflection in yaw direction
Fig. 14 Trajectories of the actual control inputs
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
(c) Elevator deflection (d) Equivalent control surface deflection in pitch direction
30
0 1 2 3 4 5 6 7-50
0
50
0 0.50
1020
Actual valueEstimated value
(a) Target acceleration in vertical plane
0 1 2 3 4 5 6 7-50
0
50
0 0.50
1020
Actual valueEstimated value
(b) Target acceleration in lateral plane
Fig. 13 Actual and estimated values of target acceleration for Case 3
0 1 2 3 4 5 6 7
-0.2
-0.1
0
0.1
0.2
0.3
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(a) Aileron deflection
0 1 2 3 4 5 6 7-10
-5
0
5
10
15
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(b) Rudder deflection
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(c) Elevator deflection
0 1 2 3 4 5 6 7-0.4
-0.2
0
0.2
0.4
0.6
0.8
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(d) Equivalent control surface deflection in pitch direction
0 1 2 3 4 5 6 7
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
PNG+BSM+RLSWCASMG+BSM+RLSWCAIGC+FTDO+RLSWCA
(e) Equivalent control surface deflection in yaw direction
Fig. 14 Trajectories of the actual control inputs
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
(e) Equivalent control surface deflection in yaw direction
Fig. 14. Trajectories of the actual control inputs for Case 2
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293
WANG Fei Integrated Guidance and Control Design for the Near Space Interceptor
http://ijass.org
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.
References
[1] Hou, M., Liang, X. and Duan, G., “Adaptive block
dynamic surface control for integrated missile guidance
and autopilot”, Chinese Journal of Aeronautics, Vol. 26, No. 3,
2013, pp. 741-750.
[2] Ran, M., Wang, Q. and Hou, D., et al., “Backstepping
design of missile guidance and control based on adaptive
fuzzy sliding mode control”, Chinese Journal of Aeronautics,
Vol. 27, No. 3, 2014, pp. 634-642.
[3] Menon, P. and Ohlmeyer, E., “Integrated design of agile
missile guidance and autopilot systems”, Control Engineering
Practice, Vol. 9, No. 10, 2001, pp. 1095-1106.
[4] Menon, P., Sweriduk, G. and Ohlmeyer, E., et al.,
“Integrated guidance and control of moving-mass actuated
kinetic warheads”, AIAA Journal of Guidance, Control and
Dynamics, Vol. 27, No. 1, 2004, pp. 118-126.
[5] Xue, W., Huang, C. and Huang, Y., “Design methods
for the integrated guidance and control system”, Control
Theory & Applications, Vol. 30, No. 12, 2013, pp. 1511-1520.
[in Chinese]
[6] Shtessel, Y. and Tournes, C., “Integrated higher-
order sliding mode guidance and autopilot for dual control
missiles”, AIAA Journal of Guidance, Control and Dynamics,
Vol. 32, No. 1, 2009, pp. 79-94.
[7] Shtessel, Y., Shkolnikov, I. and Levant, A., “Guidance
and control of missile interceptor using second-order sliding
modes”, IEEE Transactions on Aerospace and Electronic
Systems, Vol. 45, No. 1, 2009, pp. 110-124.
[8] Dong, F., Lei, H. and Zhou, C., et al., “Research of
integrated robust high order sliding mode guidance and
control for missiles”, Acta Aeronautica et Astronautica Sinica,
Vol. 34, No. 9, 2013, pp. 2212-2218. [in Chinese]
[9] Vaddi, S., Menon, P. and Ohlmeyer, E., “Numerical
state-dependent Riccati equation approach for missile
integrated guidance control”, AIAA Journal of Guidance,
Control and Dynamics, Vol. 32, No. 2, 2009, pp. 699-703.
[10] Xin, M., Balakrishnan, S. and Ohlmeyer, E., “Integrated
guidance and control of missiles with θ-D method”, IEEE
Transactions on Control Systems Technology, Vol. 14, No. 6,
2006, pp. 981-992.
[11] Yan, H. and Ji, H., “Integrated guidance and control
for dual-control missiles based on small-gain theorem”,
Automatica, Vol. 48, No. 10, 2012, pp. 2686-2692.
[12] Wang, X. and Wang, J., “Partial integrated missile
guidance and control with finite time convergence”, AIAA
Journal of Guidance, Control and Dynamics, Vol. 36, No. 5,
2013, pp. 1399 -1409.
[13] Wang, X. and Wang, J., “Partial integrated guidance
and control for missiles with three-dimensional impact
angle constraints”, AIAA Journal of Guidance, Control and
Dynamics, Vol. 37, No. 2, 2014, pp. 644-657.
[14] Chen, W., “Disturbance observer based control
for nonlinear systems”, IEEE/ASME Transactions on
Mechatronics, Vol. 9, No. 4, 2004, pp. 706-710.
[15] Wei, X. and Guo, L., “Composite disturbance-
observer-based control and H∞ control for complex
continuous models”, International Journal of Robust and
Nonlinear Control, Vol. 20, No. 1, 2004, pp. 106-118.
[16] Yang, J., Su, J. and Li, S., et al., “High-order mismatched
disturbance compensation for motion control systems
via a continuous dynamic sliding-mode approach”, IEEE
(278~294)14-087.indd 293 2015-07-03 오전 5:06:46
![Page 17: WANG Fei* and LIU Gang**past.ijass.org/On_line/admin/files/13.(278~294)14-087.pdf · 2015-07-15 · independent IGC model for the NSI. Section 3 presents the corresponding IGC design](https://reader035.vdocument.in/reader035/viewer/2022062505/5e8acb2c0c66cd5e17016c3c/html5/thumbnails/17.jpg)
DOI: http://dx.doi.org/10.5139/IJASS.2015.16.2.278 294
Int’l J. of Aeronautical & Space Sci. 16(2), 278–294 (2015)
Transactions on Industrial Informatics, Vol. 10, No. 1, 2014,
pp. 604-614.
[17] Chen, W., “Nonlinear disturbance observer-enhanced
dynamic inversion control of missiles”, AIAA Journal of
Guidance, Control, and Dynamics, Vol. 26, No. 1, 2003, pp.
161-166.
[18] Li, S. and Yang, J., “Robust autopilot design for
bank-to-turn missiles using disturbance observers”, IEEE
Transactions on Aerospace and Electronic Systems, Vol. 49,
No. 1, 2013, pp. 558-579.
[19] Yang, J., Li, S. and Sun, C., “Nonlinear-disturbance-
observer-based robust flight control for airbreathing
hypersonic vehicles”, IEEE Transactions on Aerospace and
Electronic Systems, Vol. 49, No. 2, 2013, pp. 1263-1275.
[20] Shtessel, Y., Shkolnikov, I. and Levant, A., “Smooth
second-order sliding modes: missile guidance application”,
Automatica, Vol. 43, No. 8, 2007, pp. 1470-1476.
[21] Johansen, T. and Fossen, T., “Control allocation--a
survey”, Automatica, Vol. 49, No. 5, 2013, pp. 1087-1103.
[22] Ghaoui, L. and Lebret, H., “Robust solutions to least
squares problems with uncertain data”, SIAM Journal on
Matrix Analysis and Applications, Vol. 18, No. 4, 1997, pp.
1035-1064.
[23] Cui, L. and Yang, Y., “Disturbance rejection and robust
least-squares control allocation in flight control system”,
AIAA Journal of Guidance, Control, and Dynamics, Vol. 34,
No. 6, 2011, pp. 1632-1643.
[24] Zhou, D., Sun, S. and Teo, K., “Guidance laws with
finite time convergence”, AIAA Journal of Guidance, Control,
and Dynamics, Vol. 32, No. 6, 2009, pp. 1838-1846.
[25] Zhou, D. and Shao, C., “Dynamics and
autopilot design for endoatmospheric interceptors with dual
control systems”, Aerospace Science and Technology, Vol. 13,
No. 6, 2009, pp. 291-300.
(278~294)14-087.indd 294 2015-07-03 오전 5:06:46