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Application of Type-1 and Type-2 Fuzzy CMAC to AutomaticLanding System
Journal: Transactions on Industrial Electronics
Manuscript ID: 10-1315-TIE
Manuscript Type: Regular paper
Manuscript Subject: Control and Signal Processing
Keywords: Intelligent control, Control systems, Fuzzy control
Are any of authors IEEEMember?:
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Transactions on Industrial Electronics
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Application of Type-1 and Type-2 Fuzzy CMAC to Automatic Landing System
AbstractIntelligent control scheme that utilizes
adaptive fuzzy cerebellar model articulation controller
(FCMAC) to aircraft automatic landing system is proposed
in this paper. The control scheme uses CMAC and type-1
and type-2 fuzzy systems. Current flight control law is
adopted in the controller design. Lyapunov stability theory is
applied to obtain adaptive learning rule and to guarantee
stability of the control system. Performance on tracking
desired landing path and environment adaptive capability are
demonstrated through software simulations. The proposed
intelligent controllers can guide the aircraft to a safe landing
in severe turbulence condition and act as an experienced
pilot.
I. INTRODUCTIONOn March 23, 2009 a FedEx cargo plane crashed and burst
into flame as it landed at Narita Airport, Tokyo's main
international airport, killing the American pilot and copilot.
The plane bounced twice on the runway and veered to left as
it turned on its side before bursting into flames. The fire
destroyed the aircraft [1]. Investigators believe wind shear,
or a sudden gust of wind, may have been a factor. An
accident survey of 1,300 aircraft accidents from 1950
through 2008 classified the causes into several categories asshown in Table 1. Weather was a main contributing factor of
the causes. The percentage of weather related to total
accidents was 22%. Conventional automatic landing systems
(ALS) can provide a smooth landing, which is essential to
the comfort of passengers. However, these systems work
only within a specified operational safety envelope. When
the conditions, such as turbulence or wind shear, are beyond
the envelope, they often cannot be used. The ALS relies on
the Instrument Landing System (ILS) to guide the aircraft
into the proper altitude, position, and approach angle during
the landing phase. The atmospheric disturbances affect not
only fly qualities of an airplane but also flight safety. Most
conventional control laws generated by the ALS are basedon the gain scheduling method [2]. Control parameters are
preset for different flight conditions within a specified safety
envelope, which is relatively defined by the Federal Aviation
Administration (FAA) regulations [3]. Environmental
conditions considered in the determination of dispersion
limits are: headwinds up to 25 knots, tailwinds up to 10 knots,
crosswinds up to 15 knots, moderate turbulence, and wind
shear of 8 knots per 100 feet from 200 feet to touchdown.
When the conditions, such as turbulence, are beyond the
envelope, the ALS often is disabled. It is therefore desirable
to develop an intelligent ALS that expands the operational
envelope to include more safe responses under a wider range
of conditions.
In recent years, modern control techniques and intelligent
concepts such as neural networks, fuzzy systems, genetic
algorithms, and mixed intelligent schemes have been applied
to various scientific and engineering researches [4]-[7].
There are also obvious achievements in flight control
domain [8]-[17]. In 1975, the cercbellar model articulation
controller (CMAC) was first introduced by J.S. Albus. It is
neural nwtworks inspired by the cercbellar [18]. It imitates
the structure of human cerebellum, which is a kind of
associative memory neural network. Unlike the
back-propagation based neural network which is using the
global weight updating rule, CMAC is distinguished by the
constant local weight updating rule. CMAC not only
combines the advantages of rapid convergence speed and
low computation but also can be realized easily by hardware.
With these attractive characters, the CMAC can be used to
approximate a wide variety of nonlinear functions and was
widely applied in real time automatic control, signal
processing, image coding and pattern recognition [19]-[22].
Currently, many approaches have been introduced to
improve the performance of CMAC. In 2000, a fuzzy
CMAC was developed by Zhang and Qian [23]. The
characteristics of fuzzy system are human-like reasoning and
expert knowledge. Therefore, fuzzy CMAC includes thelearning ability of neural networks and the advantages of
fuzzy system. Fuzzy logic was first proposed by Zadeh [24].
The main idea of fuzzy set is to imitate human experiences to
fuzzy rules applied in fuzzy inference system. And type-2
fuzzy logic [25] is based on the extension principle proposed
by Zadeh. A fuzzy system that uses type-2 fuzzy sets and
type-2 fuzzy logic and inference is called a type-2 fuzzy
system. In contrast, a fuzzy system uses traditional fuzzy sets,
logic, and inference is called type-1 fuzzy system. In
traditional fuzzy system models, the structure is
characterized by using type-1 fuzzy sets. Type-1 fuzzy sets,
defined on a universe of discourse, map an element of the
universe of discourse onto a crisp number in the unit interval[0, 1]. However, type-2 fuzzy can translate the linguistic and
numerical uncertainty from original data into fuzzy rule
uncertainty, while the type-1 cannot.
Another improvement effect is to develop adaptive
learning rule for the CMAC. In 2004, C.M. Lin proposed
adaptive method of CMAC for linear piezoelectric ceramic
motor [26]. After four years, he designed a robust adaptive
CMAC system for BLDC motors [27], which adds an
improved adaptive method into CMAC and obtains better
result. In this paper, we propose an intelligent aircraft
automatic landing control system that uses type-1 fuzzy
CMAC and type-2 fuzzy CMAC with adaptive learning rule
to improve the performance of conventional ALS and guide
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the aircraft to a safe landing. Some researchers have applied
intelligent concepts to the problem of landing control
[10]-[17], but these intelligent concepts are not adaptive to
severe disturbance environment. This paper presents
adaptive learning rule to the control scheme. This approach
can overcome those problems in [10]-[17] and make thecontroller more robust and adaptive to various wind
disturbance conditions.
Table 1. Causes of fatal accidents by decade (%), 1950
through 2008 [1]
Cause 50s 60s 70s80s
90s00s
Pilot Error 40 32 24 25 27 25Pilot Error
(weather related)11 18 14 17 21 17
Pilot Error(mechanical related)
7 5 4 2 4 3
Total Pilot Error 58 57 42 44 53 45Other Human Error 0 8 9 6 8 9
Weather 16 10 13 15 9 8Mechanical Failure 21 20 23 21 21 28
Sabotage 5 5 11 13 10 9Other Cause 0 2 2 1 0 1
II. SYSTEM DESCRIPTIONAt the aircraft landing phase, the pilot descends from the
cruise altitude to an altitude of approximately 1200 ft above
the ground. The pilot then positions the aircraft so that the
aircraft is on a heading towards the runway centerline. When
the aircraft approaches the outer airport marker, which is
about 4 nautical miles from the runway, the glide path signal
is intercepted, as shown in Fig. 1 [28]. As the airplanedescends along the glide path, its pitch, attitude, and speed
must be controlled. The descent rate is about 10 ft/sec and
the pitch angle is between -5 to +5 degrees. Finally, as the
airplane descends 20 to 70 feet above the ground, the glide
path control system is disengaged and a flare maneuver is
executed. The vertical descent rate is decreased to 2 ft/sec so
that the landing gear may be able to dissipate the energy of
the impact at landing. The pitch angle of the airplane is then
adjusted, between 0 to 5 degrees for most aircraft, which
allows a soft touchdown on the runway surface.
~
~
0 ft
50 ft
touchdown
1200 ftglide path
flare ath
Altitude
Fig. 1. Glide path and flare path
A simplified model of a commercial aircraft that moves
only in the longitudinal and vertical plane is used in the
simulations for implementation ease [12]. The motion
equations of the aircraft are given as follows:
( ) ( )
( ) cos( )0180
u X u u X w w X qu g w g q
g X XE E T T
(1)
TTEEqgwgu MMqMwwMuuMq )()( (2)
( ) ( ) ( )0180
( ) sin( )0180
w Z u u Z w w Z U qu g w g q
g Z ZE E T T
(3)
q (4)
0180
Uwh (5)
where u is the aircraft longitudinal velocity (ft/sec), w is
the aircraft vertical velocity (ft/sec), q is the pitch rate
(rate/sec), is the pitch angle (deg), h is the aircraft
altitude (ft),E is the incremental elevator angle (deg), T
is the throttle setting (ft/sec), o is the flight path angle
(-3deg), and g is the gravity (32.2 ft/sec2). The
parameters ii ZX , and iM are the stability and control
derivatives.
To make the ALS more intelligent, reliable wind profiles
are necessary. Two spectral turbulence forms models by von
Karman and Dryden are mostly used for aircraft response
studies. In this study the Dryden form [12] was used for its
demonstration ease. Fig. 2 shows a turbulence profile with a
wind speed of 30 ft/sec at 510 ft altitude.
Fig. 2. Turbulence profile
III. CONTROL SCHEMEPID controller is a simplified structure of an aircraft
landing controller as shown in Fig. 3. Its inputs consist of
altitude and altitude rate commands along with aircraft
altitude and altitude rate. Via aircraft landing controller we
can obtain the pitch commandc . Then, the pitch autopilot
is controlled by pitch command. Detail descriptions can be
found in [12]. In order to enable aircraft to land more steady
when an aircraft arrives to the flare path, a constant pitch
0 5 10 15 20 25 30 35 40 45-40
-35
-30
-25
-20
-15
-10
-5
0
5
10Wind Gust velocity components:Longitudinal (Solid) & Vertical (Dashed)
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angle will be added to the controller. In general, the PID
controller is simple and effective but there are some
drawbacks such as apparent overshoot and sensitive to
external noise and disturbance. When severe turbulence is
encountered the PID controller may not be able to guide the
aircraft to land safely. With CMAC compensator theproposed controller can overcome these disadvantages [17].
It uses a traditional PID controller to stabilize the system and
train the CMAC to provide precise control. The gains of PID
controller are adjusted based on experiences, what it
provides are tolerable solutions, not desired solutions. The
CMAC can effectively meliorate these conditions.
s
wh1
pitchuppitchup
ch
ch
h
ch
hk
hk
applied only during flare
typical value : 4pitchup
1.0hw
25.0hk
3.0h
k
Fig. 3. PID controller
The overall control scheme is described in Fig. 4, in which
the control signal U is the sum of the PID controller output
and the fuzzy CMAC output. The inputs for the fuzzy
CMAC and PID controller are: altitude, altitude command,altitude rate, and altitude rate command. The PID controller
provides tolerable solutions. In each time step k, the fuzzy
CMAC involves a recall process and a learning process. In
the recall process, it uses the desired system output of the
next time step and the actual system output as the address to
generate the control signal fuzzyCMACU . In the learning process,
the control signal of the pitch autopilot, U, is treated as a
desired output. It is used to modify the weights of fuzzy
CMAC stored at location which is addressed by the actual
system output and the system output of the next time step.
The output of the fuzzy CMAC is the compensation for pitch
command. When the wind turbulence is too strong, the ALS
can not control the aircraft to land safely. In here we use
fuzzy CMAC control scheme to improve the ability of
turbulence resistance of the ALS.
Fig. 4. The fuzzy CMAC control scheme
A. Type-1 Fuzzy CMAC
The structure of fuzzy CMAC is shown in Fig. 5. fuzzy
CMAC is a kind of associative memory network. Not only it
has faster self learning rate than normal neural network by
quantities with a few adjustments of memory weights, but
also it has good local generalization ability. The function offuzzy CMAC is similar to a look-up table, and the output of
CMAC is figured from a linear combination of weights
which are stored in memory. The concept of fuzzy CMAC is
to store data (knowledge) into overlapped storage
hypercubes (remembering region) in an associative manner
such that the stored data can easily be recalled. Two kinds of
operations are included in the fuzzy CMAC, one is
calculating the output result and the other is learning and
adjusting the weight. The output of fuzzy CMAC can be
obtained by the mapping process XSCWY as
follows.
1x
2x
nx
dY
Fig. 5. Conventional fuzzy CMAC structure
Step 1. Quantization (XS):
X is an n-dimensional input vector. For the givenT
nxxxX ]...[ 21 ,T
nsssS ]...[ 21 represents the
quantization vector of X. It is specified the corresponding
state of each input variable before the fuzzification.
Step 2. Associative Mapping segment (SC):
It is to fuzzify the quantization vector which is quantized
from x . Fuzzy CMAC uses the fuzzification method of the
fuzzy theorem as its addressing scheme. After the inputvector is being fuzzified, the input state values are
transformed to firing strength, which is based on
corresponding membership functions.
Step 3. Memory Weight Mapping (CW):
After fuzzifying block regions, thethi rules firing strength
in fuzzy CMAC could be computed as:
)()(...*)(*)()( 11
2211 ij
n
injnjjj xcxcxcxcxC
(6)
where ( )ij i
c x is theth
j membership function of theth
i
input vector and nis the number of total states. The asterisk
* denotes a fuzzy t-norm operator. And there are several
kinds of t-norms such as the max, min and product operators.
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We choose the product inference method as the t-norm
operator because it is easy to implement.
Step 4. Output generation with memory weight learning
(WY):
Due to partial proportional fuzzy rules and existent overlap
situation, more than one fuzzy rules are fired simultaneously.The consequences of multi-rules are merged by a
defuzzification process. The defuzzification approach we
applied is to sum assigned weights of the activated fuzzy
rules on their firing strengths, denoted as )(xCj . The output
of network is,
))(/)((11
xCxCwy i
N
ijj
N
j (7)
The work on learning of fuzzy CMAC is to update the
memory weight according to the error between the desired
output and the actual output. The weight update rule for
fuzzy CMAC is as follows:
)(/)()(1
)1()( xCxCyym
ww iN
ijd
ij
ij
(8)
where is the learning rate, mis the size of floor (called
generalization),ydis the desired output.
The conventional fuzzy CMAC weight updating rule uses
fixed learning rate, which might make the learning process
updating too slow or too fast. In order to improve the
shortcoming of conventional fuzzy CMAC on the learning
process, an adaptive learning rate is introduced. Here, we use
the discrete-time Lyapunov function to define the adaptive
learning rate . Let the tracking errore(t)be
)()( tyyte d (9)
where tis the time index. A discrete-type Lyapunov function
can be expressed as
)(2
1 2teV (10)
Thus, the change in the Lyapunov function is obtained by
)()1(2
1)()1(
22tetetVtVV (11)
The error difference can be represented by [16]
)()(
)( tWW
tete
(12)
Using gradient descent method, we have
)(
)()(
tw
tV
mtw
j
j
(13)
Since
))(/)())((()(
)()(
)(
)(xCxCtyy
tw
tete
tw
tV Td
jj
(14)
Thus
NjtyyxCxCm
tw djj ,.....,2,1)),()((/)(()(
(15)
( ) ( ( ) / ( )1
( ) ( ( ) / ( )2 2( ) ( ( ))
( ) ( ( ) / ( )
w t C x C xi
w t C x C xW t y y t
dm
w t C x C x
N N
( ( ) / ( )( ( ))C x C x y y t dm
(16)
From (7) to (9) we have
)(/)()(
),(/)(
)(
)(xCxC
W
texCxC
tw
te Tj
j
(17)
From (10) to (17) we have
1 12 2( 1) ( ) ( 1) ( ) ( 1) ( )2 2
1 1( ) 2 ( ) ( ) ( ) ( ) ( )
2 2
V e t e t e t e t e t e t
e t e t e t e t e t e t
( )( ( ) / ( )) ( )
1 ( )( ) ( ( ) / ( )) ( )
2
e tC x C x e t
W m
e te t C x C x e t
W m
(18)
Since )(/)(
)(
xCxCW
te T
, we have
( ) / ( ) ( ) / ( ) ( )
1( ) ( ) / ( ) ( ) / ( ) ( )
2
2( )1 ( ) ( )
( ) 2 ( ) ( )2 22 ( ( )) ( ( ))
2 2( ) ( )1 2( ) 2
22 ( ( )) (
TV C x C x C x C x e t m
Te t C x C x C x C x e t m
C xTC x C x
e t e t e t m mC x C x
C x C x
e tm mC x C
2( ))x
(19)
Let 0))((
)(2
2
2
xC
xC
m
then 0V , i.e., select the
learning rate as
0)(
))((2
2
2
xC
xCm (20)
V becomes negative definite. This implies that 0)( te
for t . The convergence of the adaptive type-1 fuzzyCMAC learning process is then guaranteed. The aircraft
landing control system is locally asymptotically stable.
B. Type-2 Fuzzy CMAC
The type-2 fuzzy theorem is utilized into CMAC structure
in order to promote more accurate resolution than
conventional fuzzy CMAC. The mapping procedure of
type-2 fuzzy CMAC is similar as type-1 fuzzy CMAC. The
diagram structure of type-2 fuzzy CMAC is shown in Fig. 6.
Each phase of mapping is described as follows. TheXis an
n-dimensional input space, as shown in Fig 7. Type-2 fuzzy
CMAC uses the interval type-2 fuzzification method of the
fuzzy theorem as its addressing scheme. After the input
vector to the interval type-2 fuzzy set is being fuzzified, the
input state values are transformed to upper firing strength
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and lower firing strength, which is based on corresponding
interval type-2 membership functions. We choose the
product inference method as the t-norm operator. The jth
rules upper firing strength jc and lower firing strength
firing strengthj
c in type-2 fuzzy CMAC could becomputed as:
)()(......*)(*)()(1
21 21 ij
n
i
njjjj
xcxcxcxcxcin
(21)
)()(......*)(*)()(1
2121
ij
n
i
njjj
jxcxcxcxcxc
in
(22)
1S
2
S
Fig. 6. Diagram of type-2 fuzzy CMAC in 3-D
1x
2x
nx
dY
Fig. 7. Architecture of type-2 fuzzy CMAC network
The type-reduced set of the type-2 fuzzy CMAC using the
center of sets type reduction :
11 1[ , ]
1 11[ , ] ....[ , ] [ , ]cos
1.... 1/[ , ]
1
...c c c
N NMy y y w w w w w wl r
nj jc w
jM NMc c c n
jc
j
(23)
It is an interval type-1 set determined by its left and right end
points ly and ry , which can be written as follows [29]:
N
Rj
jR
j
j
N
Rj
jjR
j
jj
N
j
j
N
j
jj
r
cc
wcwc
c
wc
y
11
11
1
1 (24)
N
Lj
jL
j
j
N
Lj
jjL
j
jj
N
j
j
N
j
jj
l
cc
wcwc
c
wc
y
11
11
1
1 (25)
w and w are the corresponding weights of c and c ,
respectively. L and R can be obtained as follows:
Step 1.Assume that the pre-computed jw are arranged in
ascending order, i.e., Nwww ...21
Step 2.Compute ry by initially setting 2/)(jjj ccc
for Nj ,....2,1 and let rr yy
Step 3.Find )11( NRR such that 1 RrR wyw
Step 4. Compute ry withjj
cc for Rj and jj cc
for Rj and let rr yy
Step 5. If rr yy then go to step 6. If rr yy then stop
and set rr yy
Step 6. Set rr yy and return to Step 3.
The procedure for computing ly is very similar to the one
just given for ry . In Step 3, find )11( NLL such
that1 Ll
Lwyw . Additionally, in Step 2 compute ly
initially setting 2/)(
jjj
ccc
for Nj ,....2,1
and inStep 4 compute ly with
jjcc for Lj and jj cc
for Lj . The defuzzified output is simply the average ofyr
andylas
lr yyy (26)
The work on learning of type-2 fuzzy CMAC is to update the
memory weight according to the error between the desired
output and the actual output. The learning rules for type-2
fuzzy CMAC is as follows:
N
j
jjd
i
j
i
j xcxcyym
ww1
1)1()( )(/)()(
(27)
N
j
jjdi
ji
j xcxcyym
ww1
2)1()( )(/)()( (28)
where 1 and 2 are the learning rates, m is the size of
floor (called generalization) , dy is the desired output.
The conventional type-2 fuzzy CMAC weight updating
rule uses fixed learning rate, which might make the learning
process too slow or too fast. In order to improve the
shortcoming of conventional type-2 fuzzy CMAC, adaptive
learning rule is introduced. Here, we use the discrete-time
Lyapunov function to define the adaptive learning rate .
The tracking error e(t) is given in (9). A discrete-type
Lyapunov function can be expressed in (10). The change in
the Lyapunov function is given in (11). The error difference
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can be represented by (12). Using gradient descent method,
we have
N
j
jjdj xcxcyym
tw1
1 )(/)()()(
(29)
N
j
jjdjxcxcyy
mtw
1
2 )(/)()()( (30)
Since
( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( )
( ) ( ) / ( )1
V t V t e t e t e t
w t e t w t w t j j j
Ny y c x c xjd j
j
(31)
( ) ( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( )
( ) ( ) / ( )1
V t V t e t e t ie t wjw t e t w t w t j j j
N
y y c x c xj jdj
(32)
Thus
Njxcxcyym
twN
j
jjdj ,.....,2,1,)(/)()()(1
1
(33)
Njxcxcyym
twN
j
jjdj ,.....,2,1,)(/)()()(1
2
(34)
( ) / ( )11
( )1
( ) / ( )2( )2 1( ) ( ( ))1
( )
( ) / ( )
1
1 ( ( ) / ( ))( ( )1
Nc x c xj
jw t
N
c x c xw t jW t y y t j dm
w tNN
c x c xN jj
NC x c x y y t j dm j
(35)
( ) / ( )11
( )1( ) / ( )( ) 22 2( ) ( ( ))1
( )
( ) / ( )
1
2 ( ( ) / ( ))( ( )1
Nc x c xj
j
w t Nc x c xw t j
W t y y t j dm
w tNN
c x c xN jj
NC x c x y y t j dm j
(36)
From (21, (22), (27), and (28) we have
N
j
jT
N
j
jj
j
xcxCW
te
xcxctw
te
1
1
)(/)()(
)(/)()(
)(
(37)
N
j
j
T
N
j
jj
j
xcxCW
te
xcxctw
te
1
1
)(/)()(
)(/)()(
)(
(38)
From (9) to (12) and (29) to (38), with respect to W , we
have
1 12 2( 1) ( ) ( 1) ( ) ( 1) ( )2 2
1 1( ) 2 ( ) ( ) ( ) ( ) ( )
2 2
( ) 1( ( )/ ( )) ( )
1
1 ( ) 1( ) ( ( ) / ( )) ( )2 1
V e t e t e t e t e t e t
e t e t e t e t e t e t
Ne tC x c x e t j
mW j
Ne te t C x c x e t j
mW j
(39)
Since
N
j
jT xcxC
W
te
1
)(/)()(
, we have
1( ( ) / ( )) ( ( ) / ( )) ( )
1 1
1 1( ) ( ( ) / ( )) ( ( ) / ( )) ( )2 1 1
2( )1 ( ) ( )1 1( ) 2 ( ) ( )2 2 2( ( )) ( ( ))
1 1
N NTV C x c x C x c x e t j j
mj j
N NTe t C x c x C x c x e t j j
mj j
C xTC x C xe t e t e t
N Nm mc x c xj j
j j
2 2( ) ( )1 21 1( ) 22 2 2( ( )) ( ( ))
1 1
C x C xe t
N Nm mc x c xj j
j j
(40)
Let 0
))((
)(2
2
1
2
1
N
j
j xc
xC
m
then 0V , i.e., we can
select the learning rate 1 in the following range
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0)(
))((2
12
2
1
xC
xcmN
j
j
(41)
From (9) to (12) and (29) to (38), with respect to W , wehave
1 12 2( 1) ( ) ( 1) ( ) ( 1) ( )2 2
1 1( ) 2 ( ) ( ) ( ) ( ) ( )
2 2
( ) 2 ( ( ) / ( )) ( )
1
1 ( ) 2( ) ( ( ) / ( )) ( )2 1
V e t e t e t e t e t e t
e t e t e t e t e t e t
Ne tC x c x e t jW m j
Ne te t C x c x e t jW m j
(42)
Since
N
j
j
T xcxCW
te
1
)(/)()(
, we have
2( ( ) / ( )) ( ( ) / ( )) ( )
1 1
1 2( ) ( ( ) / ( )) ( ( ) / ( )) ( )2 1 1
2( )1 ( ) ( )2 2( ) 2 ( ) ( )2 2 2( ( )) ( ( ))
1 1
N NTV C x c x C x c x e t j jmj j
N NTe t C x c x C x c x e t j jmj j
C xTC x C xe t e t e t
N Nm mc x c xj j
j j
2 2( ) ( )1 22 2( ) 22 2 2( ( )) ( ( ))
1 1
C x C x
e tN Nm m
c x c xj jj j
(43)
Let 0
))((
)(22
1
2
2
N
j
jxc
xC
m
then 0V , i.e., we can
select the learning rate 2 in the following range
0)(
))((2
22
2
1
xC
xcmN
j
j
(44)
V becomes negative definite. This implies that 0)( te
for t . The convergence of the adaptive type-2 fuzzyCMAC learning process is then guaranteed. The aircraft
landing control system is locally asymptotically stable.
The four inputs of the aircraft are altitude, altitude rate,
altitude command, and altitude rate command and also are
the inputs for the PID controller and type-2 fuzzy CMAC.
The input of pitch autopilot U, which is the control signal of
aircraft model,is the summation of the PID controller output
PIDU and the type-2 fuzzy CMAC output fuzzyCMACU . The
conventional PID controller is used to stabilize the aircraft
and to help type-2 fuzzy CMAC in learning process, then the
type-2 fuzzy CMAC improves the performance of the
intelligent controller in severe wind disturbance condition.
At each sampling instant k, the function of type-2 fuzzyCMAC includes two phases, which are recall process and
learning process. First, type-2 fuzzy CMAC will utilize
)1( kYd and )(kY to address the corresponding memory
weights in order to generate an output fuzzyCMACU in the
recall process, where the )(kY is the output of the dynamic
aircraft model at sampling instant k and the )1( kYd represents the desired dynamic aircraft model output at the
next time step, as shown in Fig. 8. fuzzyCMACU in the recall
process is taken to be an calculation of the demanded control
signal U. And then it is integrated with the output of PID
controller PIDU to form the demanded control signal U. In
the learning process, as shown in Fig. 9, Uobtained in the
recall process is regarded as the desired output. The error we
get from fuzzyCMACUU is used to update the corresponding
memory weights stored at location )(kY and )1( kY . The
error will converge after several iterations, then the type-2
fuzzy CMAC network can compensate for the PID
controller.
type-2 fuzzy CMACU
Fig. 8. The control process of type-2 fuzzy CMAC
( 1)Y k
( )Y k
U
type-2 fuzzy CMACU
Fig. 9. The learning process of type-2 fuzzy CMAC
IV. SIMULATIONSThe aircraft starts the initial states of the ALS as follows:
the flight height is 500 ft, the horizontal position before
touching the ground is 9240 ft, the flight angle is -3 degrees,
the speed of the aircraft is 234.7 ft/sec. Successful
touchdown landing conditions are defined as follows:
-3 )(Th ft/sec 0, 200 )(Tx ft/sec 270,
-300 )(Tx ft 1000, -10 )(T degree 5,
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where T is the time at touchdown, )(Th is vertical speed of
the aircraft at touchdown, )(Tx is the horizontal position at
touchdown, )(Tx is the horizontal speed, )(T is the pitch
angle at touchdown.
Table 2 shows the results from using PID controller withdifferent turbulence speeds [15]. The conventional controller
with original control gains can only successfully guide an
aircraft flying through wind speeds of 0 ft/sec to 30 ft/sec.
The situations at turbulence 30 ft/sec are that the pitch angle
is -0.17 degrees, vertical speed is -2.19 ft/sec, horizontal
velocity is 234.7 ft/sec, and horizontal position at touchdown
is 844 ft as show in Fig. 10 to Fig. 12 . If the wind speed is
higher than 30 ft/sec, the ALS will be unable to guide an
aircraft to land safely.
Table2.Result from using conventional PIDWind speed
(ft/sec)
Landing
point (ft)
Aircraft vertical
speed (ft/sec)
Pitch angle
(degree)0 797 -2.83 -1.41
10 910 -2.55 -0.85
20 809 -2.38 -0.59
30 844 -2.19 -0.17
40 1020 -1.72 0.44
0 5 10 15 20 25 30 35 40 45-5
-4
-3
-2
-1
0
1
2
Time (sec.)
de
g.
Pitch (Solid) & Pitch Command (Dashed)
Fig. 10. Aircraft pitch and pitch command at turbulence 30
ft/sec
0 5 10 15 20 25 30 35 40 45-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
Time (sec.)
ft.
/sec.
Vertical Velocity (Solid) & Vertical V elocity Command (Dashed)
Fig. 11. Aircraft vertical velocity and command at turbulence
30 ft/sec
5 10 15 20 25 30 35 40 450
50
100
150
200
250
300
350
400
450
500
Time (sec.)
ft.
Altit ude (Solid) & Altit ude Command (Dashed)
Fig. 12. Aircraft altitude and command at turbulence 30
ft/sec
Table 3 shows the results from using type-1 fuzzy CMAC
with different turbulence speeds. This controller can guidethe aircraft to land safely through wind speed at 10 ft/sec to
90 ft/sec.
Table 3.Result from using type-1 fuzzy CMACWindspeed
(ft/sec)
Landingpoint (ft)
Aircraftvertical
speed (ft/sec
Pitch angle(degree)
10 656.1882 -1.5376 -0.878720 574.0509 -1.8942 -0.409430 855.6644 -2.5421 -0.182240 656.1882 -2.2989 0.062550 515.3814 -2.2081 0.676560 421.5102 -1.2063 2.22
70 773.5271 -1.6393 1.365980 468.4458 -2.5756 1.255990 808.7288 -2.4467 2.2739
Table 4 shows the results from using adaptive type-1 fuzzy
CMAC with adaptive learning rate. Simulations are done by
using original control parameters of pitch autopilot. The
learning rate is2
2
)(
))((
xC
xCm and the number of blocks m
is 10. This controller can successfully guide the aircraft
flying through wind speeds to 100 ft/sec as shown in Fig. 13
to Fig. 15.
Table4.Results from using adaptive type-1 fuzzy CMACWindspeed
(ft/sec)
Landingpoint (ft)
Aircraft verticalspeed (ft/sec
Pitch angle(degree)
10 843.9305 -2.5073 -0.90420 879.1322 -2.2857 -0.547130 69.4933 -2.8413 -0.094140 773.5271 -2.0937 0.104450 738.3255 -2.4378 0.255760 656.1882 -2.7763 0.86870 550.5831 -2.362 1.506280 480.1797 -2.0533 1.945790 433.2441 -1.7998 2.3901
100 620.9865 -2.7555 2.3027
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Fig. 13. Aircraft pitch and pitch command at turbulence 100
ft/sec
Fig. 14. Aircraft vertical velocity and command atturbulence 100 ft/sec
Fig. 15. Aircraft altitude and command at turbulence 100
ft/sec
Table 5 shows the results from using type-2 fuzzy CMAC.
Table 6 shows the results from using adaptive type-2 fuzzy
CMAC. This controller can successfully guide the aircraft
flying through wind speeds to 165 ft/sec as shown in Fig. 16
to Fig. 18.
Table 5. The results from using type-2 fuzzy CMAC
Table 6. The results from using adaptive type-2 fuzzy
CMAC
0 5 10 15 20 25 30 35 40 45-10
-5
0
5
10
15
20
25
30
Time (sec.)
deg.
Pitch (Solid) & Pitch Command (Dashed)
Fig. 16. Aircraft pitch and pitch command at turbulence 165
ft/sec
0 5 10 15 20 25 30 35 40 45-80
-60
-40
-20
0
20
40
Time (sec.)
ft.
/sec.
Vertical Velocity (Solid) & Vertical Velocity Command (Dashed)
Fig. 17. Aircraft vertical velocity and command at
turbulence 165 ft/sec
Wind speed(ft/sec)
Landingpoint (ft)
Aircraft verticalSpeed (ft/sec)
Pitch angle(degree)
30 703.1238 -2.2958 0.0532
50 667.9221 -2.1930 0.191470 644.4543 -2.4343 0.7688
90 562.3170 -2.0297 1.8769110 691.3899 -1.8194 2.8465
130 609.2526 -1.7036 3.6364
Wind speed(ft/sec)
Landingpoint (ft)
Aircraft verticalspeed (ft/sec)
Pitch angle(degree)
10 867.3983 -2.6675 -0.977130 996.4712 -2.473 -0.5183
50 796.9949 -1.9923 0.7827
70 550.5831 -1.9903 1.569290 163.3645 -2.5925 2.7191
110 268.9696 -2.6133 3.5923
130 480.1797 -2.1648 3.8796150 351.1069 -2.9028 3.3518
165 961.2695 -1.4312 4.8714
5 10 15 20 25 30 35 40 450
50
100
150
200
250
300
350
400
450
500
Time (sec.)
ft.
Altitude (Solid) & Altitude Command (Dashed)
0 5 10 15 20 25 30 35 40 45-30
-25
-20
-15
-10
-5
0
5
Time (sec.)
ft.
/sec.
Vertical Velocity (Solid) & Vertical Velocity Command (Dashed)
0 5 10 15 20 25 30 35 40 45-10
-5
0
5
10
15
Time (sec.)
deg.
Pitch (Solid) & Pitch Command (Dashed)
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5 10 15 20 25 30 35 40 450
50
100
150
200
250
300
350
400
450
500
Time (sec.)
ft.
Altitude (Solid) & Altitude Command (Dashed)
Fig. 18. Aircraft altitude and command at turbulence 165
ft/sec
V. CONCLUSIONSThe purpose of this paper is to investigate the use of
intelligent control techniques in the automatic landing
system, which includes the conventional fuzzy CMAC and
the fuzzy CMAC with adaptive learning rule. Tracking
performance and environment adaptive capability are
demonstrated through software simulations. The type-2
fuzzy CMAC has better disturbance adaptive capability than
conventional PID type controller and type-1 fuzzy CMAC, it
can tolerate the turbulence strength to 130 ft/sec. The
performance of the adaptive type-2 fuzzy CMAC is more
robust than type-2 fuzzy CMAC. The adaptive type-2 fuzzy
CMAC can guide the aircraft safely under the turbulence
strength up to 165 ft/sec. Stability of the proposed control
system is guaranteed by using gradient descent method andthe Lyapunov theory.
ACKNOWLEDGEMENT
This work is supported by the National Science Council,
Taiwan, ROC under Grant NSC 97-2221-E- 019-025.
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