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Developments and Applications of Adaptive Cerebellar Model Articulation
Controller
Chih-Min Lin林 志 民
Yuan Ze University, IEEE Fellow , IET Fellow
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Content1. Introduction of Cerebellar Model Articulation Controller (CMAC)2. Missile Guidance Law Design Using CMAC3. Linear Piezoelectric Ceramic Motor (LPCM) Control Using Adaptive CMAC4. Recurrent CMAC Control for Unknown Nonlinear Systems
Referred Papers
5. RCMAC Fault Accommodation Control of a Biped Robot
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3
The Cerebellar Cortex
A model of the cerebellar cortex (1969 Marr )
1. Introduction of Cerebellar Model Articulation Controller (CMAC)
GranuleCell
Layer
Mossy FiberFeedback from Limbs
Mossy Fiber Inputfrom Higher Centers
Selection of ActiveParallel Fibers
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-
+
+
+
+
+
+
+
PurkinjeCells
StellateCells
BasketCells
AdjustWeights
ClimbingFiber Input
Output
AdjustableWeight Synapses
Summation ofSynaptic Influence
1. Information is stored in overlap layers
2. Quickly recall of the stored information
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4
1w
2w
3w4w
Rnw
kw
Rna
ka
1a2a
3a4a
5a 5w
LearningAlgorithm
oy
dy
Input Space Q
Output
Weight MemorySpace W
Association Memory Space A
Commands fromHigher Level
Feedback fromSensors
AaAbBaBbCc
Ef
Hh
Sum ofSelected Weights
Referenceof Output
Receptive-FieldSpace T
Original Cerebellar Model Articulation Controller
The basic concept of an original CMAC (Albus 1971).
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5
Mapping : Transforms the input vector into an association memory selection vector .
AQ qAqa )(
Tnk Raaaa )](,),(,),(),([)( 21 qqqqqa
Mapping : Each location of A corresponds to a receptive-field (binary receptive-field).
TA
Mapping :WT T
nk Rwwww ],,,,,[ 21 w
Output computation :oy
k
n
kk
To way
R
1
)()()( qwqaq
(1.1)
(1.2)
(1.3)
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6
1 Variable q
A B
HG
FE
DC
a
h
g
f
e
d
c
b
1
1
2
2 3
3
4
5
5
2 Variable q
Layer 1
Layer 1 Layer 2Layer 2Layer 3
State (2,2)
Layer 3
Layer 4
Layer 4
4
Bb
Cc
Ee
Gg
Layer 1
Layer 2
Layer 3
Layer 4
The schematic representation of a 2-D CMAC. (binary receptive-fields )
1 Variable q
A B
HG
FE
DC
a
h
g
f
e
d
c
b
1
1
2
2 3
3
4
4
5
5
2 Variable q
Layer 1
Layer 1 Layer 2Layer 2Layer 3
State (2,2)
Layer 3
Layer 4
Layer 4
Bb
Cc Ee Gg
Non-differentiable receptive-fieldsNon-smooth function approximationNo stability analysis
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7
A B
IHG
FED
C
a
i
h
g
f
e
d
c
b
71
1
2
2 3
3
4
4
6
5
65
Layer 1
Layer 1 Layer 2Layer 2Layer 3
Gg
State (3,3)7
Layer 3
J K L
l
k
j
8
9
8 9Layer 4
Layer 4
Jj 1 Variable q
2 Variable q
Bb
Ee
The schematic diagram of a general 2-D CMAC. General Cerebellar Model Articulation Controller
Differentiable receptive-fieldsSmooth function approximationStability analysis
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8
Gaussian receptive-field basis function :
2
2)(exp)(ik
ikiiik v
mqq
Multidimensional receptive-field function:
n
i ik
ikin
iiikkkk v
mqqb1
2
2
1
)(exp)(),,( vmq
(1.4)
(1.5)
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9
The output of CMAC
The receptive-field Bc with including a Gaussian receptive-field basis function.
Rn
kkkkk
To bwy
1),,(),,( vmqvmqw
Ca
A
B
C
cb
aBc
Ba
Aa
AcAb
Bb
Cb
Cc
1 Variable q
2 Variable q
(1.6)
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10
1q
nq
Input Space Q
Outputoy
Receptive-Field Space T
Weight Memory Space W
Association Memory Space A
k1
nk
kb kw
CMAC Neural Network
Layer 1
Layer 2 Layer 3 Layer 4
1x
2x
2jy 3ky
4oy
Good generalization capabilityLess computationBetter approximation ability
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11
Formulation of Missile-Target Engagement
IZ
IX
IY
MXMYMZ
my
mx
m
ty
tx t
tm
mm
Target
Missile
Ground tracker
mRtR
xaycazca
The 3-D missile-target pursuit diagram.
2. Missile Guidance Law Design Using CMAC
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12
The motion of the missile in the inertial frame.
(2.1)mmvmmczcmmcycmmmmczcmmmcycm
vmmczcmmcycmxm
mmcmmmczc
mmcmmmcycmmxm
mmcmmmczc
mmcmmmcycmmxm
vgvava
vava
gaaaza
aaya
aax
/cos/cos/sin
)cos/(sin)cos/(cos
coscoscossinsin)cossinsinsin(cos
)coscossinsin(sinsincos)sinsincossin(cos
)sincoscossin(sincoscos
Target
Missile
LYLZ
LX
LOSGround tracker
mR
pR1e
2e
P
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13
m
m
m
tmtmtmtmtm
tmtm
zyx
ee
)cos()sin()sin()cos()sin(0)cos()sin(
2
1
The missile position in the LOS frame.
(2.2)
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14
The tracking error dynamic equation
)(),(),(
)(),(),(),(),(
),(),(
2221
1211
2
1
2
1
ttt
ttGtGtGtG
tFtF
ee
uxGxF
uxxxx
xx
(2.4)
Define
Tzcyc
T
Tmmmmmmmm
T
aauu
zyxzyx
xxxxxxxx
],[],[
],,,,,,,[
],,,,,,,[
21
87654321
u
x
(2.3)
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15
The CMAC control system.The control law:
CCMACuuu (2.5)
CMACe
dtd
AdaptationLaws
21 ˆ,ˆ ww
21,
CMACu
CompensationController
CuuTarget
Maneuvertt , Calculation of the
Tracking Errormm ,
MissileManeuver
Limiter
Adaptive CMAC Control System
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16
A feedback linearization control law
]),()[,(1 eKeKxFxGu pvT tt
Substituting (2.6) into (2.4), yields 0eKeKe pv
(2.6)
(2.7)
The minimum approximation error TjjjCMACjj uu ),(
*wq
CMAC control system )()(ˆ)()ˆ,( quqwquwquu C
TCCMACT
Error equation])[,( TCCMACpv t uuuxGeKeKe
(2.9)
(2.10)
(2.8)
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),(),(00
),(),(00
),(),,(),(
2221
121121
tGtG
tGtGttt mmm
xx
xxxGxGxG
whereTeeee ],,,[ 2211 E
22
11
001000000010
vp
vp
kk
kk
The error dynamics in the state-space form
])[,(
])[,(
])[,(
2222
1111
TCCMACm
TCCMACm
TCCMACm
uuut
uuut
t
xG
xGE
uuuxGEE
(2.11)
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Theorem 2.1: If the control law is designed as (2.5), in which the adaptation law of CMAC
and the compensation control is designed as
jmjT
wjjj PGEww ~ˆ
then the stability of the guidance system can be guaranteed.
)sgn( mjT
jCju PGE
(2.12)
(2.13)
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222
111
~~2
1~~2
121)( wwwwPEE T
w
T
w
TtV
The Lyapunov equation RPP T
Taking the derivative of the Lyapunov function.
222222
111111
22
2
11
1
~][
~][
~~1~~121)(
T
m
T
Cm
T
T
m
T
Cm
T
T
w
T
w
T
uu
tV
wPGEPGEwPGEPGE
wwwwREE
Proof: A Lyapunov function is defined as
(2.14)
(2.15)
(2.16)
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20
Choosing (2.12), then (2.16) can be simplified
||
||21)(
2222
1111
m
T
Cm
T
m
T
Cm
TT
u
utV
PGEPGE
PGEPGEREE
Setting (2.13), then (2.17) can be rewritten as
021
)|(|)|(|21)(
222111
REE
PGEPGEREE
T
m
T
m
TTtV
(2.17)
(2.18)
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21
Numerical SimulationsThe target motion model.
ttvtzt
tttyt
vttzt
tttzttyt
tttzttyt
vga
vagaz
aay
aax
/)cos(
)cos/(cos
sinsincos
cossinsin
For scenarios 1 and 2:for the first 2.5 sec until interception
vty ga 5 vtz ga vty ga 5 vtz ga 5
For scenarios 3: for the first 2.5 sec until interception
vty ga 0vty ga 5.0
vtz ga
vtz ga
(2.19)
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The feedback linearization guidance law ]),()[,(1 eKeKxFxGu pvT tt
where ,140014
vK
490049
pK
Adaptive CMAC-based guidance law
The design parameters are set as follows:
,
16830083588000016830083588
R ,1521 ww 01.021
(2. 20)
(2.21)
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23
Engagement scenario 1 with feedback linearization guidance law.
0 1 2 3 4 5 6 7-2
0
2
4
0 1 2 3 4 5 6 7-4
-2
0
2
1e
2e
(sec) Time
(sec) Time
0 1 2 3 4 5 6 7-300
-200
-100
0
100
200
0 1 2 3 4 5 6 7-100
0
100
200
300
)(m
/sec
2
yca)
(m/s
ec
2zca
(sec) Time
(sec) Time
0
1000
2000
3000
0
2000
4000
60000
200
400
600
800
1000
1200
(m)
z
(m)x(m)
y
MD=4.4539m
y trajectorMissile
jectoryTarget trapointIntercept
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Engagement scenario 2 with feedback linearization guidance law.
0 1 2 3 4 5 6 7-2
0
2
4
6
0 1 2 3 4 5 6 7-0.5
0
0.5
1
1.5
2
1e
2e
(sec) Time
(sec) Time
0 1 2 3 4 5 6 7-400
-200
0
200
0 1 2 3 4 5 6 7-400
-200
0
200
400
)(m
/sec
2
yca)
(m/s
ec
2zca
(sec) Time
(sec) Time
0
2000
4000
6000
0
100
200300
4000
1000
2000
3000
4000
(m)
z
(m)x(m)
y
MD=3.758m
y trajectorMissile
jectoryTarget tra
pointIntercept
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Engagement scenario 3 with feedback linearization guidance law.
0 1 2 3 4 5 6 7 8 9-2
0
2
4
6
0 1 2 3 4 5 6 7 8 9-20
-10
0
10
20
1e
2e
(sec) Time
(sec) Time
0 1 2 3 4 5 6 7 8 9-400
-200
0
200
400
0 1 2 3 4 5 6 7 8 9-400
-200
0
200
400
)(m
/sec
2
yca)
(m/s
ec
2zca
(sec) Time
(sec) Time
0
2000
4000
6000
0
2000
4000
60000
2000
4000
6000
8000
(m)
z
(m)x(m)
y
MD=1.8434m
y trajectorMissile
jectoryTarget tra
pointIntercept
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26
Engagement scenario 1 with adaptive CMAC-based guidance law.
0 1 2 3 4 5 6 7-2
0
2
4
0 1 2 3 4 5 6 7-4
-2
0
2
1e
2e
(sec) Time
(sec) Time
0 1 2 3 4 5 6 7-200
-100
0
100
200
0 1 2 3 4 5 6 7-200
-100
0
100
200
)(m
/sec
2
yca)
(m/s
ec
2zca
(sec) Time
(sec) Time
0
1000
2000
3000
0
2000
4000
60000
200
400
600
800
1000
1200
(m)
z
(m)x(m)
y
MD=0.5737m
y trajectorMissile
jectoryTarget trapointIntercept
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27
Engagement scenario 2 with adaptive CMAC-based guidance law.
0 1 2 3 4 5 6 7-4
-2
0
2
4
6
0 1 2 3 4 5 6 7-2
0
2
4
1e
2e
(sec) Time
(sec) Time
0 1 2 3 4 5 6 7-400
-200
0
200
400
0 1 2 3 4 5 6 7-400
-200
0
200
400
)(m
/sec
2
yca)
(m/s
ec
2zca
(sec) Time
(sec) Time
0
2000
4000
6000
0
100
200300
4000
1000
2000
3000
4000
(m)
z
(m)x(m)
y
MD=1.5612m
y trajectorMissile
jectoryTarget tra
pointIntercept
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28
Engagement scenario 3 with adaptive CMAC-based guidance law.
0 1 2 3 4 5 6 7 8 9-4
-2
0
2
4
6
0 1 2 3 4 5 6 7 8 9-15
-10
-5
0
5
1e
2e
(sec) Time
(sec) Time
0 1 2 3 4 5 6 7 8 9-400
-200
0
200
400
0 1 2 3 4 5 6 7 8 9-400
-200
0
200
400
)(m
/sec
2
yca)
(m/s
ec
2zca
(sec) Time
(sec) Time
0
2000
4000
6000
0
2000
4000
60000
2000
4000
6000
8000
(m)
z
(m)x(m)
y
y trajectorMissile
jectoryTarget tra
pointIntercept
MD=0.2781m
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0.27811.56120.5737Adaptive CMAC-based Guidance
Law
1.84343.7584.4539Feedback
LinearizationGuidance Law
Scenario 3Scenario 2Scenario 1Scenario
GuidanceLaw
Comparison of Miss-distance
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Summary
CMAC guidance law can achieve satisfactory performance for different engagement scenarios.
CMAC guidance law performs better than the feedback linearization guidance law.
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3. Linear Piezoelectric Ceramic Motor (LPCM) Control Using Adaptive CMAC
Structure of LPCM
Unknown dynamic equation);()();();()( txdtutxgtxftx (3.1)
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Adaptive CMAC Control System
The adaptive CMAC control law
CCMAC uuu (3.2)
CMAC
dtd
AdaptationLaws
ikik vm ˆ,ˆ,ŵ
CMACu
CompensatedControl
Cu
u x
Linear Piezoelectic Ceramic Motor Drive System
*x
dx+
cvmw ,,,
+ +e
LC ResonantInverter
Linear PiezoelectricCeramic Motor
PerformaceIndex
ReferenceModel
r
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Tracking errorxxe d
Performance index dekeketr t 012 )()(
Ideal control law]);();([);( 12
1 ekektxdtxfxtxgu dT
Substituting (3.5) into (3.1), then0)( 12 ekeketr
(3.3)
(3.4)
(3.5)
(3.6)
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CT
CCMAC uuuu wvmwq ˆ)ˆ,ˆ,ˆ,(
Adaptive CMAC control
Error equation])ˆ,ˆ,ˆ,()[;()( CCMACT uuutxgtr vmwq
(3.7)
(3.8)A minimum approximation error
TTCMACT uuu***** ),,,( wvmwq (3.9)
According to (3.9), (3.8) can be rewritten as
]~)[;(
])ˆ()[;()( *
CT
CT
utxg
utxgtr
w
ww
(3.10)
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Theorem 3.1: The adaptive CMAC control law is designed as (3.8), in which the adaptation law
then the stability of the control system can be guaranteed.
with bound estimation algorithm given in
and the compensated control is designed as);()(ˆ txgtrww
)];()(sgn[ˆ txgtruC
|);()(|ˆ txgtrc
(3.11)
(3.12)
(3.13)
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Proof: A Lyapunov function is chosen as 22 ~
21~~
21)(
21)(
cT
wtrtV ww
ˆ~1ˆ~1]~)[;()()(c
T
wC
T utxgtrtV www
Substituting (3.11)-(3.13) into (3.15), gives
0|);()(||)|(
ˆ]ˆ[1);()(|||);()(|
~~1);()();()()(
txgtr
utxgtrtxgtr
utxgtrtxgtrtV
cC
cC
(3.14)
(3.15)
(3.16)
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According to the gradient descent method,
kCMAC
wk
wkwk buV
wVbtxgtrw
ˆ
);()(̂
The adaptation laws of means and variances
21
1
)()(2ˆ);()(
ˆ
ik
ikik
n
kkm
ik
ik
ik
kn
k k
CMAC
CMACmik
vmqbwtxgtr
mb
bu
uVm
R
R
(3.17)
(3.18)
(3.19)
R
R
n
k ik
ikikkv
n
k ik
ik
ik
k
k
CMAC
CMACvik
vmqbwtxgtr
vb
bu
uVv
13
21
)()(2ˆ);()(
ˆ
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)];(sgn[)(ˆ txgtrww
)];(sgn[)](sgn[ˆ txgtruC
|)(|ˆ trc
21 )(
)(2ˆ)];(sgn[)(ˆik
ikik
n
kkmik v
mqbwtxgtrmR
Rn
k ik
ikikkvik v
mqbwtxgtrv1
3
2
)()(2ˆ)];(sgn[)(ˆ
The adaptation laws can be reconstructed as (3.20)
(3.21)
(3.22)
(3.23)
(3.24)
The in the tuning algorithms can be reorganized as in practical applications.
);( txg)];(sgn[|);(| txgtxg
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dx
ParallelI/O
EncorderInterface
and Timer
D/AConverter
Servo Control Card
Personal Computer
DigitalOscilloscope
CW,CCW
u
x
LinearScale
Linear PiezoelectricCeramic Motor Moving Table
LC ResonantDrivingCircuit
x
Control Computer andServo Control Card
LC ResonantDriving System
DigitalOscilloscope
Linear Scale
Linear PiezoelectricCeramic Motor
MovingTable
The PC-based experimental control system.Experimental Results
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40
Block diagram of PI position control system.
eV
u x
dx LC ResonantInverter+
Linear Piezoelectric CeramicMotor Drive System
PK
dtd
sKI
SK Linear Piezoelectric
Ceramic MotorReference
Model
*x
+ ++
x
The parameters of the PI position control system are chosen as follows:
,20SK ,1PK 25IK
36.7313.17s36.73
2 2222
sss nnn
The reference model for the periodic step command:
(3.25)
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41
Experimental results of PI position control for LPCM due to periodic step command.
Reference Model
Table Position
Tracking Error
Control Effort
Start
0 cm
4.5 cm
2 sec
Reference Model
Table Position
Start
0 cm
4.5 cm
2 sec
Start
0 cm
2 sec
0.5 cm Tracking Error
Start
0 cm
2 sec
0.5 cm
Start
0 V
2 sec
5 V Control Effort
Start
0 V
2 sec
5 V
(a)
(b)
(c)
(d)
(e)
(f)
No load caseLoad case
-
42
Reference Model
Table Position
Start
0 cm
4.5 cm
2 sec
-4.5 cm
Reference Model
Table Position
Start
0 cm
4.5 cm
2 sec
-4.5 cm
Tracking Error
Start
0 cm
2 sec
0.5 cm Tracking Error
Start
0 cm
2 sec
0.5 cm
Control Effort
Start
0 V
2 sec
5 V Control Effort
Start
0 V
2 sec
5 V
(a)
(b)
(c)
(d)
(e)
(f)
Experimental results of PI position control for LPCM due to sinusoidal command.
-
43
The adaptive CMAC control Parameters
,04.0w 01.0c,02.0 vm
20ikv]35,25,15,5,5,15,25,35[
],,,,, ,,[ 87654321
iiiiiiii mmmmmmmm
The initial values of the parameters are chosen as
,251 k ,102 k
,4 ,5En 422 Rn
]/)(exp[)( 22 ikikiiik vmqq
The receptive-field basis functions are chosen as
,42Bn
-
44
Reference Model
Table Position
Start
0 cm
4.5 cm
2 sec
Reference Model
Table Position
Start
0 cm
4.5 cm
2 sec
Tracking Error
Start
0 cm
2 sec
0.5 cm Tracking Error
Start
0 cm
2 sec
0.5 cm
Control Effort
Start
0 V
2 sec
5 V Control Effort
Start
0 V
2 sec
5 V
(a)
(b)
(c)
(d)
(e)
(f)
Experimental results of robust CMAC control for LPCM due to periodic step command.
-
45
Reference Model
Table Position
Start
0 cm
4.5 cm
2 sec
-4.5 cm
Reference Model
Table Position
Start
0 cm
4.5 cm
2 sec
-4.5 cm
Tracking Error
Start
0 cm
2 sec
0.5 cm Tracking Error
Start
0 cm
2 sec
0.5 cm
Control Effort
Start
0 V
2 sec
5 V Control Effort
Start
0 V
2 sec
5 V
(a)
(b)
(c)
(d)
(e)
(f)
Experimental results of robust CMAC control for LPCM due to sinusoidal command.
-
46
Summary
The successful development of adaptive CMAC control system.
The successful application of adaptive CMAC control for an LPCM.
-
47
4. Recurrent CMAC Control for Unknown Uncertain Nonlinear Systems
Problem FormulationThe nth-order nonlinear dynamic system
xytdtugfx n )()()()()( xx
The tracking error vector is defined asTneee ],,,[ )1( E
(4.1)
(4.2)The ideal control law
])()([)(
1 )( EKxx
TndI xtdfg
u (4.3)
-
48
The error dynamics0)1(1
)( ekeke nnn
Recurrent CMACArchitecture of a recurrent CMAC
(4.4)
1q
nq
Input Space Q
Receptive-Field Space T
Output
oy
Weight Memory Space W
Association Memory Space A
1z
1z
1rw
rnw
kwkb
1rq
nrq
k1
nk
-
49
The inputs of every block are represented as
nTnrrrr qqq ],,,[ 21 q
)1( Nyorr wqq (4.5)The receptive-field basis function
2
2)(exp)(ik
ikririik v
mqq (4.6)
The RCMAC is utilized to estimate the perfect control law, so that
TorrRCMAC yu wwvmwq ),,,,( (4.7)
-
50
Recurrent CMAC Control SystemControl law
CRCMAC uuu
Recurrent CMAC
riikikk wvmw ˆ,ˆ,ˆ,ˆ
RCMACu
CompensatedController
u x +
E
AdaptiveEstimation Law
̂
Cu
dx
e
Adaptive Recurrent CMAC
Plantxytdtugfx n ),()()()()( xx
Adaptive Laws
TrackingError Vector
E
++
(4.8)
-
51
Theorem 4.1: The adaptive law of the recurrent CMAC is designed as
and the compensated controller is designed as
with the adaptive estimation law given in
where and are positive constants, then the stability of the control system can be guaranteed.
mT
w PBEw ̂
)sgn(ˆ mT
Cu PBE
||ˆ mT
c PBE
w c
(4.9)
(4.10)
(4.11)
-
52
On-line parameter training algorithm
),,,(
ˆ),,,(ˆ
rkkrkRCMAC
w
k
RCMAC
RCMACwrkkrkm
Twk
bu
Vw
uu
Vbw
wvmq
wvmqPBE
(4.11)
-
53
The adaptive laws of means, variances and recurrent weights:
2)(2ˆˆ
ik
ikrikkm
Tmik v
mqbwm PBE
3
2)(2ˆˆik
ikrikkm
Tvik v
mqbwv PBE
)1()(2ˆ
ˆˆ
2
Nuv
qmbw
wq
qb
bu
uVw
RCMACik
riikkkm
Tr
ri
ri
ri
ik
ik
k
k
RCMAC
RCMACrri
PBE
(4.13)
(4.14)
(4.12)
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54
where and .
If is unknown:
rT
w PBEw ̂
)sgn(ˆ rT
Cu PBE
||ˆ rT
c PBE
2)(2ˆˆ
ik
ikrikkr
Tmik v
mqbwm PBE
3
2)(2ˆˆik
ikrikkr
Tvik v
mqbwv PBE
)1()(2ˆˆ 2
Nuv
qmbww RCMACik
riikkkr
Trri PBE
jj nT
r ]1,,0,0[ B
(4.16)
(4.17)
(4.15)
(4.19)
(4.20)
(4.18)
)(xg
|);(| txg
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55
Illustrative Examples
Example 4.1: Consider the Duffing forced oscillation system
)cos(121.0 3 tuxxx
)(10
0010
2
1
2
1 dugfxx
xx
It can be rewritten as
(4.21)
(4.22)
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56
Time (sec)
State response
Time (sec)
State response
Phase-plane portrait
2x 2x2x 2x
1x 1x
1x(a)
(b)
(c)
Simulated results of the Duffing forced oscillation system (without control)
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57
Phase-plane portrait
1x
1dx
2x
2dx
Time (sec)
Time (sec)
Control effort u
Tacking error e
Time (sec)
Time (sec)
State response
State response
1x2x 2x 2x 2x
1x 1x
u
e
(a)
(b)
(c)
(e)
(d)
Simulated results of RCMAC control for the Duffingforced oscillation system.
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58
Example 4.2: The delta wing of the wing rock motion control
uqcqqcqqcqcqccq 3543210 ||||
)(10
0010
2
1
2
1 dugfxx
xx
The state equation
The aerodynamic parameters of the delta wing for a 25-deg angle of attack are chosen as
,00 c ,01859521.01 c ,015162375.02 c,06245153.03 c ,00954708.04 c
02145291.05 c
(4.23)
(4.24)
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59
Z
80
x
Wing
Axis of rotation
d
d
Axis of rotation
Wing
U
(a)
(b)
(c)
Two initial conditions:
a small initial condition
a large initial condition
deg6)0(1 xsecdeg/3)0(2 x
deg30)0(1 x
secdeg/10)0(2 x
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60
The reference model is defined as
,25.6w
,]6,9[ TK
05.0c,75.0 vm ,01.0r
,29990
R ,
15530
P
The parameters are selected as
2
1
2
16.164.0
10
d
d
d
dxx
xx
(4.25)
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61
Phase-plane portrait
c)(d
egre
e/se
2xc)
(deg
ree/
se2x
c)(d
egre
e/se
2xc)
(deg
ree/
se2x
(degree)1x (degree)1x (degree)1x (degree)1x
State response
Time (sec)
State response
Time (sec) Time (sec)
Time (sec)
State response
State response
(deg
ree)
1x(d
egre
e)1x
c)(d
egre
e/se
2xc)
(deg
ree/
se2x
c)(d
egre
e/se
2xc)
(deg
ree/
se2x
(deg
ree)
1x(d
egre
e)1x
Phase-plane portrait
(a)
(b)
(c) (f)
(e)
(d)
Simulated results of the wing rock motion system (without control)
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62
Initial condition
Intelligent hybrid control
RNN adaptive control
Intelligent hybrid control
RNN adaptive control
Intelligent hybrid control
RNN adaptive control
1x
1dx
(RNN adaptive control)
1x (Intelligent hybrid control)
2x
2dx
(RNN adaptive control)
2x (Intelligent hybrid control)
(degree)1x (degree)1x
c)(d
egre
e/se
2xc)
(deg
ree/
se2x
)c
(deg
ree/
se2
u)
c(d
egre
e/se
2u
Phase-plane portrait
State response
Time (sec)
State response
Time (sec)
(deg
ree)
1x(d
egre
e)1x
c)(d
egre
e/se
2xc)
(deg
ree/
se2x
Time (sec)
Control effort u
Tacking error e
(deg
ree)
e(de
gree
)e
(a)
(b)
(c)
(e)
(d)Time (sec)
Simulated results of RCMAC control and RNN control for small initial condition.
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63
Initial condition
Intelligent hybrid control
RNN adaptive control
Intelligent hybrid control
RNN adaptive control
Intelligent hybrid control
RNN adaptive control
State response
1x
1dx
(RNN adaptive control)
1x (Intelligent hybrid control)
2x
2dx
(RNN adaptive control)
2x (Intelligent hybrid control)
c)(d
egre
e/se
2xc)
(deg
ree/
se2x
(degree)1x (degree)1x
)c
(deg
ree/
se2
u)
c(d
egre
e/se
2u
Phase-plane portrait
Time (sec)
State response
Time (sec)
(deg
ree)
1x(d
egre
e)1x
c)(d
egre
e/se
2xc)
(deg
ree/
se2x
Time (sec)
Control effort u
Tacking error e
(deg
ree)
e(de
gree
)e
(a)
(b)
(c)
(e)
(d)Time (sec)
Simulated results of RCMAC control and RNN control for large initial condition.
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64
Summary
An RCMAC control scheme has been proposed for a class of nonlinear dynamical system.
RCMAC is introduced which has both the merits of RNN and conventional CMAC.
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65
Input Space
Receptive -FieldSpace
Weight MemorySpace
Association MemorySpace
Recurrent Unit
k1
nk
Output Space
kow
-
kpw anp
1p k1
kna
k
ikr Tikr T
Ono
1o
sI
sA
sR
sW
sO Input Space
Receptive -FieldSpace
Weight MemorySpace
Association MemorySpace
Recurrent Unit
k1
nk
Output Space
kow
-
kpw anp
1p k1
kna
k
ikr Tikr Tikr TTikr T
Ono
1o
sI
sA
sR
sW
sO
Structures of MIMO RCMAC)()()( Ttrtptp ikikirik
dn
kkkp
Tpp wo
1Φw
(5.1)
(5.2)
6. RCMAC Fault Accommodation Control of a Biped Robot
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66
4l
1
4m
x
1a 1l
1m1q
2
2q 2a
2l2m
3a3
3m3q
5a
4a 4q
5q
5l 5
6
b
6q
6m
6a6l
4
5m
4l
1
4m
x
1a 1l
1m1q
2
2q 2a
2l2m
3a3
3m3q
5a
4a 4q
5q
5l 5
6
b
6q
6m
6a6l
4
5m
yy
),() ()(),()( t01 qqfqgqqqCτqMq tt
• The unknown fault-occurrence time
,1 ,0
) (0
00 ttif
ttiftt
• A biped robot is subjected to nonlinear faults
(5.3)
(5.4)
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67
In the absence of a fault )(),()(1 qgqqqCτqMq
Computed torque controller
)(),(2)( 20 qgqqqCeKeKqqMτ dqqe dwhere the tracking error vector
Error dynamics 0 eKeKe 22
Fault occurs:
Robust fault-accommodation controller
(5.5)
(5.6)
(5.7)
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68
RCMAC-based fault accommodation control
RCMAC-Based Fault-Tolerant Control System
_
Nonlinear Estimation Model
Computed TorqueController
rvcw ,,, Adaptive Laws
RCMAC
Biped Robot
dq],[ qq
],[ dd qq 0τ
tf̂rτ
+
_
+_
qq
ζ
+],[ ee
tf̂
τ
ω
)(qM
rvcW ˆ,ˆ,ˆ,ˆ
RCMAC-Based Fault-Tolerant Control System
_
Nonlinear Estimation Model
Computed TorqueController
rvcw ,,, Adaptive Laws
RCMAC
Biped Robot
dq],[ qq
],[ dd qq 0τ
tf̂rτ
+
_
+_
q
q
ζ
+],[ ee
tf̂
τ
ω
)(qM
rvcW ˆ,ˆ,ˆ,ˆ
Accommodation controller
),(ˆ)( t qqfqMτ r
Fault-accommodation control law
rτττ 0
t1 ˆ)(),()()( fqgqqqCτqMωqω c
RCMACTo estimate the nonlinear fault
(5.8)
(5.9)
(5.10)
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69
Simulation Results The joint angle of each link
CMAC RCMAC
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70
Simulation Results The fault function and the output
CMAC RCMAC
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71
RCMAC can achieve favorable accommodation control for the faults of a biped robot.
Summary
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72
Chih-Min Lin and Ya-Fu Peng, “Adaptive CMAC-based supervisory control for uncertain nonlinear systems,” IEEE Trans. System, Man, and Cybernetics Part B, Vol. 34, No. 2, pp. 1248-1260, 2004.
Chih-Min Lin, Ya-Fu Peng, and Chun-Fei Hsu, “Robust cerebellarmodel articulation controller design for unknown nonlinear systems,” IEEE Transactions on Circuits and Systems-II, Vol. 51, No. 7, pp. 354-358, 2004.
Chih-Min Lin and Ya-Fu Peng, “Missile guidance law design using adaptive cerebellar model articulation controller,” IEEE Trans. Neural Networks, Vol. 16, No. 3, pp. 636-644, 2005.
Chih-Min Lin and Chiu-Hsiung Chen, “Robust fault-tolerant control for biped robot using recurrent cerebellar model articulation controller,” IEEE Trans. Systems, Man, and Cybernetics, Part B, Vol. 37, No. 1, pp. 110-123, 2007.
Referred Papers
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73
Thank You for Attention