Applied Mathematics, 2011, 2, 565-574
doi:10.4236/am.2011.25075 Published Online May 2011 (http://www.SciRP.org/journal/am)
Pulse-Width Pulse-Frequency Based Optimal Controller
Design for Kinetic Kill Vehicle Attitude Tracking Control
Xingyuan Xu, Yuanli Cai
Department of Automatic Contro l, Xi’an Jiao Tong University, Xi’an, China
Received December 30, 2010; revised March 22, 2011; accepted March 25, 2011
Abstract
The attitude control problem of the kinetic kill vehicle is studied in this work. A new mathematical model of
the kinetic kill vehicle is proposed, the linear quadratic regulator technique is used to design the optimal atti-
tude controller, and the pulse-width pulse-frequency modulator is used to shape the continuous control
command to pulse or on-off signals to meet the requirements of the reaction thrusters. The methods to select
the appropriate parameters of pulse-width pulse-frequency are presented in detail. Numerical simulations
show that the performance of the LQR/PWPF approach can achieve good control performance such as
pseudo-linear operation, high accuracy, and fast enough tracking speed.
Keywords: Attitude Control, Kinetic Kill Vehicle, Linear Quadratic, Pulse-Width Pulse-Frequency
1. Introduction
It is known that the earliest interceptor required a very
large and powerful propulsion system in combination
with a very high powered warhead to successfully de-
stroy an incoming reentry vehicle. With the development
of small-size propulsion systems, reliable and accurate
sensors and guidance systems, and rapid servo-mecha-
nism techniques, the kinetic kill vehicle (KKV) is now
becoming both technically and economically feasible.
KKV adopts the way of direct collision to destroy target,
which makes the promise of a low-cost defense system
more reasonable. A KKV should include a sensor that
identifies and tracks the target, the guidance and control
system computes the course changes for the interceptor,
and a propulsion system supplies the forces to maneuver
the vehicle [1-3]. Each KKV requires six to eight thrust-
ers to maneuver its pitching angle, yawing angle, and
rolling angle. A procedure for finding the optimal thruster
configuration with desired control effectiveness was pro-
posed by [4].
The reaction thruster attitude control system (RTACS)
has been widely studied. Aventine et al. [5] studied the
attitude dynamics for rigid spacecraft. The firing logic of
reaction thrusters and the methods of tracking attack an-
gle are presented by He et al. [6], but the coupling be-
tween three channels have not been taken into account in
their work. Reference [7] employed switching strategy in
combination with PID controller applying to the ducted
rocket system. Luo et al. [8] proposed an optimization
approach combining primer vector theory and evolution-
ary algorithms for fuel-optimal non-linear impulsive
rendezvous, which is designed to seek the optimal num-
ber of impulses as well as the optimal impulse vectors.
Reference [9] presented an adaptive attitude control law
based on neural network for a rigid spacecraft without
requiring angular velocity measurements, but in the
course of this research, we found that control perform-
ance indices deteriorated obviously without tracking an-
gular velocity. Reference [10] studied the coupled dy-
namics of spacecraft, and introduced angle decoupling
conditions of attitude control problem.
This work adopts pulse or on-off reaction thruster for
attitude maneuver. Pulse-width pulse-frequency (PWPF)
modulator is used to translate continuous control com-
mands to on-off signals due to its advantages over other
types of pulse modulators such as bang-bang controller
which has excessive energy consumption [11-13]. The
PWPF modulator is composed of a Schmidt trigger, a
first-order-filter, and a feedback loop. The design of
PWPF requires iterative tuning of the filter and the
Schmidt trigger. References [13-16] presented detailed
methods to select the appropriate parameters of PWPF,
but the relationship between signal attenuation and
PWPF parameters has not been taken into account. The
fuel decides the lifetime of the KKV, therefore, reduced

X. Y. XU ET AL. 566
energy consumption is highly required, therefore, i.e.,
decrease the number of firings.
In this work, the PWPF is combined with the linear
quadratic regulator (LQR) technique for the KKV atti-
tude control system. The paper is divided into 5 sections.
The nonlinear model of the KKV is presented in Section
2; Section 3 presents a detail description of the LQR/
PWPF controller design which includes the description
of the tuning methods for the parameters of the PWPF
modulator; Section 4 presents the result of the numerical
simulation for the RTACS; Conclusions are presented in
Section 5 based on the obtained results.
2. Mathematical Mode l of the KKV
This section will describe the mathematical model of the
attitude motion, including kinematics and dynamics equ-
ations of the KKV. In general, the response, energy con-
sumption, effective torque and fault tolerance of the con-
trol system are affected by the thruster configuration [4].
The basic thruster configuration of the KKV in this work
is illustrated in Figure 1. There are 6 thrusters parallel to
the 11
plane of the vehicle. These thrusters are
installed in the tail of the KKV. Positive pitching mo-
ment is acquired by turning on thruster number 1 and 2,
and negative pitching moment is acquired by turning on
thruster number 4 and 5. Positive yawing moment is ac-
quired by turning on thruster number 3, and negative
yawing moment is acquired by turning on thruster num-
ber 6. Positive rolling moment is acquired by turning on
thruster number 2 and 5, and negative rolling moment is
acquired by turning on thruster number 1 and 4. The
control commands are given in 3 channels: pitching,
yawing, and rolling. Obviously, there happens control
cross coupling phenomena. Smaller effective torque
compared to the control command is produced when 3
channel commands are given simultaneously.
OY OZ
2.1. The Rotational Dynamic Equation of the
KKV
In this work, the attitude of the KKV is defined by the
body reference frame with respect to the
ground reference frame, 222
is obtained though
translation of the ground reference frame and O is the
center of gravity. Figure 2 presents the relationship of
the body reference frame 11 and the ground ref-
erence frame 222
. For 111 OX , 1 axis
align with the longitudinal symmetric axis of the body,
1 axis and 1 axis are perpendicular to 1
OX
axis, and completes a three-axis right-hand orthogonal
system. For 222
, 2 axis is parallel to the
plane of the ground, in the direction of the launch direc-
tion; 2 axis is perpendicular to the ground; 2
axis is perpendicular to 2 and 2
OY , also completes
a three-axis right-hand orthogonal system.
111
OXY Z
OXY Z
OXY Z
OX
1
OXY Z
OZ
OXY Z
Y ZOX
OY
OY OZ
OX
The motions of the KKV are composed of transla-
tional and rotational motions. Their independent charac-
teristics allow one to deal with the two motions sepa-
rately. For attitude control, only rotational equations are
required. The rotational motions of a rigid spacecraft in
space are described by Euler’s Equation (1)
d
dtt



HH
H
M  (1)
here


,, 
111
xyz is the angular velocity of the
body frame relative to the ground frame, and the projec-
tion of
T
 

in the body reference frame are 11 ,
x
y

 and
1
z

respectively;


H
J

is the moment of mo-
mentum, and J is the inertia matrix. Suppose the KKV in
this work is an axis-symmetric vehicle, thus the product
of inertia in J is zero. 1
x
J
, 1
y
J
and 1
z
J
are the mo-
ment of inertia relativing to body axes. M represents the
control torques used for controlling the attitude. 1
x
M
,
1
y
M
and 1
z
M
are the control torques about the body
axes which is provided by the thrusters.
The moment of momentum can be expressed as
1
Y
1
Z
#
Figure 1. Configuration of attitude control thruste r s.
1
X
1
Y
1
Z
2
X
2
Y
2
Z



Figure 2. Relationship of the body reference frame and the
ground reference frame.

X. Y. XU ET AL.567
11
1
xx
y







11
111
11 11
00
00
00
xx
yyy
zz zz
JJ
JJ
JJ





 




HJ
 (2)
1
1
1
1
1
1
d
d
d
d
d
d
x
x
y
y
z
z
Jt
J
tt
Jt












H


(3)



1111
1111
1111
zyzy
xyxz
yxyx
JJ
JJ
JJ







 




H
 (4)
The control torques about the body axes are
11
11
11
xx
yy
zz
M
TR
M
TL
M
TL







M






(5)
where R is the radius of the body; L is the distance from
the location of thruster to the center of mass; 1
x
T,
and
1
y
T
1
z
T are the thruster force in different directions.
According to (1-5), the rotational dynamic equation of
KKV can be expressed as



111111
1
111111
1
111111
1
xyzyzx
x
yzxzxy
y
zxyxyz
z
J
JTRJ
J
JTLJ
J
JTL




 



 



 





J


0
(6)
2.2. The Rotational Kinematic Equation of the
KKV
According to the relationship of the body reference
frame and the ground reference frame, the rotational an-
gular velocity of KKV in body reference frame can be
deduced as follow
 
1
1
1
00
00
0
sin
cos cossin
cos sincos
x
yxz x
z
LL L



 



 
 
 
 

 
 
 
 
 



















1sin 0
0coscos sin
0cossincos




 
















(7)
where ,

 and

are pitching angle, yawing angle
and rolling angle respectively, then we obtain


11
11
11 1
sin cos
cossin cos
cossin tan
yz
yz
xy z
 
 

 




 






(8)
3. Design of LQR/PWPF Controller
In this section, the LQR/PWPF controller design is de-
scribed. LQR is a method for designing optimal control-
ler. However, the control signals from LQR controller
are of continuous type, the PWPF is used to modulate
continuous control commands to discrete signals.
3.1. LQR Controller
According to (6) and (8), the rotational dynamic and ki-
nematic equation groups of KKV have the characteristic
of strong coupling and nonlinearity. Only linearized rea-
sonably, RTACS can use linear system theory. Suppose
the attitude angle and angular velocity of KKV are all
small, we can consider RTACS as a three-axis self-gov-
erned second-order linear model, and it is linearized and
decoupled in the following section.
Pitching channel

1
11 1
z
z
zzz
TL Jku








z
(9)
Yawing channel

1
11 1
y
y
yyy
TL Jku









y
(10)
Rolling channel

1
11 1
x
x
xxx
TL Jku









x
(11)

In (9-11),
x
u,
y
u and
z
u are the thruster control
signal in three different channel;
x
k,
y
k and
z
k are
coefficients of
x
u,
y
u and
z
u.
The three channels can be expressed by a second-order
system


 
12
2
tt
tkt







xx
xu

(12)

X. Y. XU ET AL. 568


1
2
x
t
x
t




y
(13)
The problem has the following state-space representa-
tion





xAxBu
y
CxDu (14)
For the controller design, it is necessary to check the
limitations and constraints imposed by the plant. The
matrix [A B] must be controllable and [A C] must be
detectable. If both conditions are satisfied for the attitude
model, the next step is to design a controller which
achieves the system performance required. Stable, fast
regulating angle is required. It is desired to accomplish
the regulation to maintain the KKV in the required atti-
tude.
Suppose the initial and final values of the system are
given in the follow


10 10
20 20
x
tx
x
tx





(15)
By considering the state-space (14) and (15), the op-
timal control problem is formulated as a two point
boundary value problem. In order to reduce energy con-
sumption, we formulate the following cost function
 
 
0
T
TT
1
2
1d
2
f
ff
t
t
tt
ttt ttt





Je Fe
eQe uRut


(16)
where is the error of the expectation output

et


t

y
and the reality output.
 
tt

eyyt (17)




1
2
x
t
t
x
t







y (18)
The first term in (16) corresponds to the energy of the
final error, the second term corresponds to the energy of
the instantaneous error and the third term corresponds to
the energy of the control signal. The weighting matrices
F, Q and R are defined the trade-off between regulation
performance and control efforts [17]. Q is assumed at
positive definite, and R is assumed at semi-positive defi-
nite. Weighting matrices should be selected according to
specific control requirements. Fast regulating angle re-
quires large control variable, and large control variable
means small R; Saving fuel requires small control vari-
able, while small control variable means large R. In ad-
dition, the thruster is not a simple linear links which has
dead zone and saturation, therefore a larger matrix R
relative to matrix Q is necessary, thus the thruster save
more fuel and work more in the linear region. Smaller
control variables inevitably lead to lengthen regulation
time and unable to meet intercept performance indices.
So it is necessary to find a compromise when choosing R.
The control signal


tu is obtained by minimizing J in
the optimization problem [18].
For infinity time fixed-length tracking system, the ap-
proximate optimal control to minimize the cost function
(16) is shown below [19]
 
1T
ˆ
ˆˆ
()tttt

 

uRBPx

g
(19)
where is a symmetry positive constant matrix. is
obtained by solving the following Riccatti equation
ˆ
Pˆ P
T1TT
ˆˆˆ ˆ


0PAAPPBRBPCQC 
g
(20)
Constant concomitance vector satisfies the fol-
lowing equation
ˆ
g

1
1T TT
ˆ
ˆt






gPBRBA CQy (21)
Thus, from (14) and (19), the approximate optimal
closed-loop track system is
 
1T 1T
ˆˆ
tt


 


xABRBPxBRB (22)
The solution that satisfies the initial condition is the
approximate optimal


*tx.
3.2. Pulse-Width Pulse-Frequency Modulator
Equation (19) shows the optimal attitude control com-
mands for the KKV, however, it is of continuous type.
The PWPF can modulate the continuous control com-
mand to on-off signals which meet the requirement of
thrusters. In order to ensure the modulation is in a qua-
si-linear mode, we must modulate the width of the pulse
proportionally to the magnitude of the continuous control
command. In PWPF, the distance between the pulses is
also modulated. Compared with other methods of modu-
lation, the PWPF modulator has several advantages such
as close-to linear operation and high accuracy, which
provide scopes for the advanced control [14].
The basic structure of PWPF is shown in Figure 3.
The PWPF modulator is composed of a Schmidt trigger
which is simply an on-off relay with dead zone and hys-
teresis, a first-order-filter, and a negative feedback loop
[15]. When a positive input to the Schmidt trigger is
greater than on, the trigger output is m; While the
input falls below off , the trigger output is 0, and this
response is also reflected for negative inputs. The error
U U
U


et is the difference between the Schmidt trigger out

X. Y. XU ET AL.569
1sT
K
m
m
m
u
off
u
on
u
Figure 3. Pulse-width pulse-fr eque ncy (PWPF) modulator.
put m and the system input .The error is fed into
the first-order-filter whose output is fed to the
Schmidt trigger. The parameters of interest are the first-
order-filter coefficients m
U

rt

ut
K
and U, and the Schmidt
trigger parameters and off which defines the
hysteresis as (in this paper, is nor-
malized, = 1).
m
on
U
on
 U
off
UhU m
U
m
U
3.2.1. Static Characteristics of PWPF
If the input


rt is constant, e.g., E, the PWPF modu-
lator output on-off pulse sequence having a nearly linear
duty cycle with input amplitude. It is worth to note that
the modulator is independent of the system in which it is
used [6]. The static characteristics indicate how the
modulator will perform in most cases. Choosing appro-
priate static parameters of the PWPF is the first step of
attitude control design.
1) On-time
The relationship between


ut and E can be repre-
sented by
  

/
0101e
m
tT
m
utuK Eu  

(23)
For (23), as ,
t

1
m
utK E 

(24)
The time from on to off
U is defined as the relay
on-time or pulse width, denoted by . can be
solved from (23)
U
on
Ton T

ln 11
on mmo
h
TT KEU
n

 




(25)
2) Off-time
The off-time is defined as the time from 0 to on U.
According to (23), the off-time denoted by can be
solved
off
T

ln 1
off mmon
h
TT
K
EU h

 




(26)
3) Modulator frequency
The frequency of the PWPF modulator is defined as
the inverse of the period and is given by the following
equation
1
on off
fTT  (27)
4) Duty cycle
The duty cycle of PWPF is the ratio of on-time to the
period and is given by
on
DCf T

 (28)
5) Saturation input and dead band
The maximum input
s
E for pseudo-linear operation
can be solved by equating the maximum value of the
first-order-filter output to ,

1
ms
KEoff U
s
E can
be determined below
1
s
off m
EUK

 (29)
The dead band of the modulator is defined as the
minimum input d to the modulator such that .
can be determined below
E0
on
T
d
E
don
EUK
m

(30)
From (30), we can see that increasing of m
K
can re-
duce the size of dead band and it is reasonable to keep
m to ensure that U is an upper bound of the
dead band.
1Kon
6) Minimum pulse-width
The effective dead band of the modulator is defined
as the minimum input to the modulator when on .
Substituting (30) into (26), we obtain an expression for
the minimum on-time, which is defined as the mini-
mum pulse-width. The minimum pulse-width is usually
denoted by relay operational constrains and is shown
as
0T
ln 1m
mmm
hT
h
T
K
K

 

 (31)
In addition, the normalized hysteresis width is defined
as


ms d
ahKEE , and the normalized input is de-
fined as




dsd
x
EEEE .
Suppose 5
m
K

, 0.05
m
T

, on and 0.8u0.5h

,
then d0.16E

and s1.06E

. The curve of static
characteristics of PWPF verse input is shown in Figure 4.
From Figure 4, it can be seen that when ds EEE

,
is proportional to the input, and T is inversely
on
Toff
y(t) r(t) u(t)
e
(
t
)

X. Y. XU ET AL. 570
proportional to the input. The duty cycle keep a good
linear relationship with input. Furthermore, when
d, relay work in dead zone, and the output is zero;
When
EE
s
EE, relay work in saturation zone, and the
output is 1. The on-off frequency varies with the input,
the maximum frequency occurs at

2
sd
EE ap-
proximately.
3.2.2. Selection of PWPF Parameters
The aim of this section is to recommend a general me-
thod to select PWPF parameters, which will be used later
for design the PWPF modulator in this work. The design
is done by comparing performance indices for different
parameter settings.
(a)
(b)
(c)
Figure 4. The curve of static characteristics of PWPF (a)
Ton and Toff, (b) On-off frequency, (c) Duty cycle.
1) and U
on off
Simulations with a constant input E = 0.5 are per-
formed to study the influence of parameters on U and
off on the on-off frequency. Figure 5 illustrates the
on-off frequency verse on and
U
UUoff on
UU. It indicates
that for 0.6
off on
UU, the on-off frequency increases
much faster than it does for 0.6
off on
UU, and for
0.3
on
U

the on-off frequency increases much faster
than it does for . To avoid excessive on-off
frequency, on and
0.3
on
U
0.3U0.6
off on
UU are sug-
gested in PWPF modulator designs.
2) m
K
A large number of simulations indicate that m
K
is an
important parameter which affects performance indices
of PWPF. Selection of m
K
is presented in detail below.
Suppose 0.05
m
T

， and , the in-
fluence of m
0.8
on u0.h5
K
is discussed in two cases: 1
m
K

and
5
m
K

. When 1
m
K

, we obtain 0.8
d
E

and
1.3
s
E

; When m5K

, we obtain d and 0.E16
1.06
s
E

. The curve of the on-off frequency verse input
is shown in Figure 6. The maximum frequency is 9 and
50 respectively. The random signal and its corresponding
discrete one after PWPF are shown in Figure 7. From
Figure 6 and Figure 7, it can be seen that the smaller the
m
K
, the smaller the maximum frequency which means
m
K
is as small as possible.
PWPF translates continuous signal to discrete one,
however, the process bring about changes of the signal
energy. In order to study how the parameters of PWPF
influence the signal energy, integral continuous signal
and its corresponding discrete one respectively. Suppose
the input signal is sinusoidal signal. The integral of si-
nusoidal signal and its discrete one after PWPF are
shown in Figure 8. From Figure 8, it can be concluded
that the larger m
K
lead to smaller energy loss. In this
sense, the larger m
K
is preferred.
According to above discussion, there should be a tra-
deoff when choosing m
K
. Simulations with varying
m
K
show that 16
m
K

 should be maintained.
Figure 5. The on-off frequency verse Uon and Uoff /Uon.

X. Y. XU ET AL.571
(a)
(b)
Figure 6. On-off frequency curve of PWPF (a) Km = 1, (b)
Km = 5.
Figure 7. The random signal and its corresponding discrete
signal after PWPF when Km = 1 and Km = 5.
(a)
(b)
Figure 8. The integral of sinusoidal signal and its discrete
signal after PWPF (a) Km = 1, (b) Km = 5.
3)
m
indicates a phase lag of an input. If is too
small, there is little phase lag for all input frequencies,
and bring about excessive energy consumption; If the
time constant is too large, there will be no firing numbers
for the modulator to react to the high-frequency input
[14].
T
m
Tm T
Generally, better control performance requires larger
control command, i.e., requires more energy consump-
tion, but the fuel is limited in KKV, and the number of
firings gives an indication of the life-time of the thrusters.
In the case of meeting the control requirements, we must
minimize the number of firings, i.e., minimize the firing
frequency. This tradeoff has to be kept in mind during
the design of the PWPF. In order to reduce the energy
consumption, an iterative searching of the tradeoff can be
carried out, and all the involved requirements should be
satisfied. The preferred range of parameters for PWPF is
recommended in Table 1.
(Km = 5)
Table 1. The preferred range of PWPF parameters.
Parameter Recommended setting
Modulator gain, Km 1 ≤ Km ≤ 6
On-threshold, Uon Uon > 0.3
Off-threshold, Uoff Uoff < 0.6 Uon
Time constant, Tm 0.02 < Tm < 0.2
(Km = 1)

X. Y. XU ET AL. 572
,


2

1
.



4. Numerical Results The optimal control law is


*
12
14.145 2.5214.145utx xy  1
,.
(37)
According to above analysis, the PWPF parameters for
the particular problem in this work are presented in Ta-
ble 2. Table 3 shows the dynamic and structure parame-
ters of the KKV in this work.
3) Rolling channel
The matrixes of the yawing channel are shown as fol-
lows
010010
,,
00200 01k
  

 
  
   0AB CD
4.1. Design of Optimal Control Law
1) Pitching channel
The matrixes of the pitching channel are shown as
follows
010 010
,,
00150 01k
  

  
 0AB CD.
Select 100 0
01.5







Q, R = 0.5, according to (19-21),
we obtain
12.5328 0.0354
0.0354 0.0044







P (38)
Select , R = 0.5, according to (19-21),
we obtain
1000
01.3



Q1
1
12.5328 1.5
0.0354
yy
y
2









g (39)
11.8080 0.0471
0.0471 0.0056



P (32) The optimal control law is


*
12
14.16 1.7614.16utx xy 1 (40)
1
1
11.8080 1.3
0.0471
yy
y






g (33)
4.2. Numerical Result
The optimal control law is Suppose the initial values of the three attitude angles and
angle velocity are all zero, the final values are all 1 rad,
the tracking profile are shown in the following. Figure 9
presents the tracking profile when the input is step signal;
Figure 10 presents the tracking profile when the input is
square signal. Figures 11-13 present the control com-
mands of the three channels when the final values are 1
rad, 0.5 rad, 0.1 rad respectively.

*
12
14.13 1.6814.13utx xy (34)
2) Yawing channel
The matrixes of the yawing channel are shown as fol-
lows
010 010
,,,
0075 01k
  

  
   0AB CD
Select , R = 0.5, according to (19-21),
we obtain
100 0
03



QTable 2. The selected PWPF parameters.
Parameter Recommended setting
Km 5
Uon 0.8
Uoff 0.3
Tm 0.05
17.8565 0.0943
0.0943 0.0168



P (35)
12
1
17.8565 3
0.0943
yy
y






g (36)
Table 3. Dynamic and structure parameters of attitude control system.
Pitching channel Yawing channel Rolling channel
Arm of force (m) 1 1 0.2
Moment (Nm) 600 300 60
Moment of inertia (kgm2) 4 4 0.3
Thruster force (N) 600 300 300

X. Y. XU ET AL. 573
Figure 9. Attitude tracking profile when the input is step
signal.
Figure 10. Attitude tracking profile when the input is
square signal.
Figure 11. Control commands when inputs are 1 rad.
Figure 12. Control commands when inputs are 0. 5 rad.
Figure 13. Control commands when inputs are 0. 1 rad.
From the simulation results, it can be seen that there
are very good tracking performance of the three attitude
angles. When tracking large attitude angle, the thrusters
need a high on-off frequency; when tracking small atti-
tude angle, the thrusters reduce the number of firings.
Thus, we conclude that for terminal guidance, better pre-
cision in midcourse control will reduce the number of
firings in terminal phase. It is worth to note that the
pitching and rolling channels share four attitude control
thrusters, the two channels are coupled stronger than the
yawing channel.
5. Conclusions
This work presents the design process of LQR/PWPF
controller, which can meet the requirements of RTACS
for KKV. Firstly, a mathematical model of RTACS for
KKV is deduced. Then, the design methods of LQR and

X. Y. XU ET AL. 574
PWPF are introduced in detail. The LQR technique is an
efficient way to achieve stability; it allows to select the
level of input signal, and to restrict the control com-
mands till acceptable performance is obtained. The
guidelines to select the weighting matrices of the cost
function for the optimal controller are also presented and
discussed in this work. The influence of PWPF modula-
tor parameters on the control performance is studied. The
analysis of the PWPF modulator presents an effective
method to tune its parameters. The tuning ranges of some
parameters are presented, and a set of approximate opti-
mal parameters for the PWPF is also obtained in this
work. Finally, simulation results demonstrate the feasi-
bility of the LQR/PWPF controller for RTACS, and rea-
sonable tracking accuracy in attitude is achieved. It is
worth to note that the LQR/PWPF controller can stabi-
lize the system for all initial states. The simulation re-
sults also show that improving the control precision in
midcourse will reduce the on-off frequency of thrusters
in terminal phase, and this is what we hope to achieve.
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