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ﾄｸn» ｵ fﾀｯ

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4@&* Design of Multiple Model Controller using SOM Neural Network
Poya Bashivan, Alireza Fatehi
Robust H f Control of an Exerimental Inverted Pendulium using Singular Perturbation
Approach
1
10
Roya Amjadifard, Mohammad T. Hamidi Beheshti, Hamid Khaloozadeh, Kirsten. A. Morris
An Algorithm for Constructing Nonsmooth Lyapunov Functions for Continuous
Nonlinear Time Invariant Systems
Alireza Faraji Armaki, Naser Pariz, Rajab Asgharian
17
AR Order Determination of 3-D ARMA Models Based on Minimum Eigenvalue (MEV)
Criterion and Instrumental Variable Method
25
Mahdiye Sadat Sadabadi, Masoud Shafiee
www.isice.ir
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4@&* Design of Multiple Model Controller using SOM Neural Network
Poya Bashivan, Alireza Fatehi
1
Robust H f Control of an Exerimental Inverted Pendulium using Singular Perturbation
10
Approach
Roya Amjadifard, Mohammad T. Hamidi Beheshti, Hamid Khaloozadeh, Kirsten. A. Morris
An Algorithm for Constructing Nonsmooth Lyapunov Functions for Continuous Nonlinear
Time Invariant Systems
Alireza Faraji Armaki, Naser Pariz, Rajab Asgharian
17
AR Order Determination of 3-D ARMA Models Based on Minimum Eigenvalue (MEV)
Criterion and Instrumental Variable Method
Mahdiye Sadat Sadabadi, Masoud Shafiee
25
Journal of Control, Vol. 3, No. 2, Summer 2009
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.H4 0$ +K #H4 0$ :? >)$ B :( Abstract: In this paper, two methods for estimating the range of two Unmanned Aerial Vehicles
(UAVs) in a vision based aerial refueling is presented and also the sliding mode controller is
designed and simulated to establish and maintain desired relative position between them. There is
no communication between two UAVs, and only vision information from a camera mounted on the
follower UAV, is available to extract range information. Leader length is unknown, so the range
between them is unobservable from the camera's images. In this paper a theoretical method for
range estimation is presented. Using this method, the leader length can be estimated and then the
range would be computable from the images. Unlike other estimation methods based on Kalman
filter, this method shows a good robustness against unknown leader acceleration. In the next section
Kalman algorithm for range estimation is presented, which shows instability when leader
accelerates. Leader acceleration and wind effects are uncertain factors for the control system. In
spite of the simple design of a sliding mode controller in this paper, good robustness against these
uncertainties can be achieved.
Keywords: Aerial Refueling, Own-Ship Maneuvering Algorithm, Kalman Estimator, Sliding
Mode Controller.
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[1] Williamson, W.R,, Abdel Hafez, M.F., Rhee, I.,
Song, E. J., January 2007, "An Instrumentation
System Applied to Formation Flight", IEEE
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[3] Parkash Singh, J., 2005, "Evolution of The Radar
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[5] Betser, A., Vela, P., Tanenbaum, A., December 2004,
"Automatic Tracking of Flying Vehicles Using
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Follower
Leader
45
V e lo c ity (m /s )
[2] Bishop, A. N., Pathirana, P. N., Savkin, A.V.,
December 2007, "Radar Target Tracking Via Robust
Linear Filtering", IEEE Signal Processing Letters,
12.
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30
25
15
0
50
100
150
200
250
300
350
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q! : c # .$ !$) kW ? 0#
.> , > +K @ 7 #H4 !/
. :7 P$.& V $ #&
#H4 P$.&
[9] Stepanyan, V., 2006, "Vision Based Guidance and
Flight Control in Problem of Aerial Tracking,"
Requirements for the Degree of Doctor of Philosophy
in Aerospace Engineering, Virginia.
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[10] Stepanian, V., Hovakimian, N., June 2006, "A
Guidance Law For Visual Tracking of Maneuvering
Target", Proceedings of the 2006 American Control
Conference, Minnesota, USA, 14-16.
<% /s&! ,!
V
r$R Bv4 #Q
Journal of Control, Vol. 3, No. 2, Summer 2009
.$/ ($ . OW :$ : r/ oK
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1388 14 02 ,/ 03 !( 0
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10
&6( 7
09 :!"
[17] Herrnberger, M., October 2004, ”Adaptive control
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Guidance Navigation and Control Conf.
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.> > 9+ 09+ 7 W :$l4 04 ," F$"9 0@ )4 : R) O19 BL* :( Abstract: In this paper, by using dominant gain concept and frequency response, a simple
method is presented to recognize zeros locations resulting of time delay parameters of the
differential-diffrence equations. The concept of dominant gain states that, in a specific frequency
band, the dynamic behavior of a Quasi-Rational Distributed System traces the dynamics of that term
in the model which dominates in its gain with respect to the other term. If this behavior is
nonminimum phase, total function is nonminimum phase in that frequency ranges and that means
Right Half Plane (RHP) zeros exist and if it is minimum phase total function behavior is minimum
phas and that means all of zeros locate in Left Half Plane (LHP) in that frequency ranges. Because
dominant gain is a comprehensive concept then it capable to use for differential-diffrence equation
and by using it, zeros location could be recognized in all of frequency ranges.
Keywords: Diffrential-Diffrence Equations, Dominant Gain Concept, Assymptotic Location of
Zeros, Right Half Plane (RHP) Zeros.
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Journal of Control, Vol. 3, No. 2, Summer 2009
4
Hermite-Biehler
Hurwitz polynomials
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Assymptatic location of zeros theory
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Distributed parameter systems
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.> F!
,@! P1 l1 # ($
.! * B !/ A t B P
oK B7v
> * #! @ R ! #K? E (9) R >71) X !C u 0> ZH 2 OW/ $R
k& P1 P >71 @ Wu$
? P$H
>4 ,! Z !C 0$/ ( B A q6) ? < * .! >"(
45 @ >4) V @ 0> A ![ # @
> : C >"( W 6 0! .!W >"(
> # ,! # $/ K? +K ( (
,!/ p B7v P @ 0> : C q6) >"( :.
* 1 @ X !C W D $/ W A . 7
!C # 0> B $ 4 ,! 0$/ 0/99
@ E @ ![ > [ F$"9 (9) R :KV*
@ ![ > [ :* # >" > ,@$< k& !
>
<
.$/ B U4 K? +K A
Journal of Control, Vol. 3, No. 2, Summer 2009
A c$< $< ,$
A >1W K U4 [ >1W $ * 01
K
1388 14 02 ,/ 03 !( 0
15
1
K i @ ,9 @ )4 : R) O19 BL* :9+ W #*4
/ $' 0 ,$'
4 ," F$"9 " }7 $E .$/ O
B7v #
3
"1
K $4 3 ,!/ p :"W/
!? .!W : 1
K @ 2
.$ !$) { O $H 4 ,"
1.5
t 1
K i ,@$< ,1? :"1 W
# + ,? $4 W K # .
:
+6) > ,!/ , w/
G 1 ( j ) t G 2 ( j ) : ,?
:!/ @ R "1
K ?
"1
K F4 !/ >v 67 $R B$' ! :$ .!/ G 1 ( j ) W K U4 ![
,!/ , 5 4 :"W/ >&< # @ & : >1W
.>
Amplitude ratio
10
10
10
1
0.5
0
0
Z
1
2
>1 cos( E D )@
X
3
Y (rad)
4
5
6
ª¬ A B cos( E D ) º¼ Bs4 :(2) OW/
A B GH C :@ Y
(E D )
OC B7v 0:@K K 4 @ ![ >*4 ( # 4
G 1 ( j ) t G 2 ( j ) , Z
10
Z
@ : 7 l1 :$'4 #u .>K? E
K $4 < 4 ," F$"9 @ ,9 0Y44 #!
X=0.99
X=3
X=1
X=2
2.5
," $ K 4 @ ![ >*4 .!/
.!/ B7v @ 4 , 1 B7v # .$/ B7v >1
W$D > ,!/ rH ) x
K V 1C : 3 OW/
-3 OW/ ! K? .!W G&H >"( ! B $% (b)-3 !/ ? A $% (a)
1
.> ?
0
-1
-2
G1=1/(s+1)
G2=0.25*exp(-5s)/(s2+2s+1)
G=G1+G2
-3
10 -1
10
0
10
Frequency (rad/sec)
10
1
0
Phase (deg.)
-50
-100
,@! : : @ [ >*4 ![ >"( t :(3) OW/
-150
.!/ 4
-200
G1=1/(s+1)
O+ ![ $/ ,! w$; W/ # G2=0.25*exp(-5s)/(s2+s+1)
G=G1+G2
-250
-300 -1
10
0
10
Frequency (rad/sec)
.? >"( 4 ,@! > !/ ![K ! 10
.> Y&S G 1 ( j ) :$ ! $ :(4) OW/
Journal of Control, Vol. 3, No. 2, Summer 2009
1
!/ / 4 ," 1
K ,!D #
!
>*74 [ K @ K? >"( [ qR ![
1388 14 02 ,/ 03 !( 0
1
K i @ ,9 @ )4 : R) O19 BL* :9+ W #*4
16
/ $' 0 ,$'
G 2 ( j ) G 1 ( j ) ," :$ +$'H x
K
0.2
G2=0.25*exp(-5s)/(s2+2s+1)
G1=1/(s+1)
G=G1+G2
# :"1
K ![ B$+ # .!/ U%C
0
: & .
!$) >*74 [ W K @ 4L :"1
K
Im
G 1 ( j ) !/ Y&S @ x ,
>*74 G 2 ( j ) @
-0.2
:> ,!/ , t 7 OW/ >&< #
1
-0.4
0
-0.6
-0.5
Amplitude ratio
10
10
-2
-3
10 -1
10
0.5
Re
1
1.5
> Y&S G 1 ( j ) :$ ! $ :(5) OW/
-1
10
10
0
@K 0 >71 :$ > ZH L :"W/ @
G1(s)=0.25/(s+1)
G=G1+G2
G2=exp(-5s)/(s+1)2
0
10
Frequency (rad/sec)
> G 1 ( j ) Y&S $< ![ : 0>1W
1
10
@ )4 ! C U4 V G1 ( s ) ? < .!
$
.>/ !$) @ )4 ! W K P G ( s ) 0!/
0
“NonDelay @ )4 ! K 0! K @ >&< #
> &< R4 C >&< # 9+ 7 W
-500
C U4 [ n t m , K E 1] or [ n ; m , K
G1=0.25/(s+1)
P4 L G ( s ) U4 }$ :9+ 6 OW/ .!/
G2=exp(-5s)/(s+1)2
G=G1+G2
-1500 -1
10
1]
u > :9+ >" @ : ,@ : QRDS !
-1000
0
10
Frequency (rad/sec)
O
R4 >&< # $/ ,! $R .> ,!/
1
.> L 10
G 2 ( j ) # :"1
K W. ! $ :(7) OW/
.> Y&S G 1 ( j ) L :"1
K
30
25
20
! 1
K i $ > ZH 7 OW/ @ $R
0> ,
>*74 [ @ G 2 ( j ) $ Y&S O& ! Im
Phase (deg.)
:$l4 .$/ :k. QRDS (NDQRDS) Behavior”
15
>*74 G 1 ( j ) @ 4 rad/sec x
K ," U%C4 @ x
10
@ >*74 :)4 !C 0@K $ 7& .> ,
5
,
! K? +K [ @K $ @ > / G 1 ( j )
0
-1
.>
-0.5
0
Re
0.5
1
> Y&S G 1 ( j ) :$ 9+ >*$ :(6) OW/
> 0# x
K :9+ @ :!D !*4 >&< # :!D !C < # K # .!? >
0# :"1
K > #W 0+) >&< ! t
L :"1
K @K $ :!< #.
* .!/7 ? G 2 ( j ) @ G 1 ( j ) # .! @ )4 : :"1 K 7/ $.Q
Journal of Control, Vol. 3, No. 2, Summer 2009
1388 14 02 ,/ 03 !( 0
17
1
K i @ ,9 @ )4 : R) O19 BL* :9+ W #*4
/ $' 0 ,$'
“Delay @ )4 -1@ K [ !/ !$) @K P
“QuasiDelayQRDS (QD @ )4 7/ K 0>&<
OW/ .P :k? F QRDS (DQRDS) Behavior”
:9+ 8 OW/ .$/ :k. QRDS) Behavior”
.> ,!/ , t # G ( s ) :9+ >*$ 11
OW/ # $R .! ,!/ , ,!/ y G ( s )
Amplitude ratio
10
10
10
10
4 rad/sec x
K U%C4 W PS $/ ,!
1
R > K? @K P K G ( s ) @K D "
0
," # *%C4 W@ 1C 0> > 9+ :!*4 !/
$ !< @ ! @K $ ," #. !< 0!/ /! $( -1
-2
.> ,$ @4 G1 ( s ) @K
G1=0.25/(s2+2s+1)
G2=exp(-5s)/(s+1)
G=G1+G2
30
-3
10 -1
10
0
10
Frequncy (rad/sec)
10
1
25
20
Im
0
Phase (deg.)
-50
15
-100
10
-150
5
0
-1
-200
-250
G1=0.25/(s2+2s+1)
G2=exp(-5s)/(s+1)
G=G1+G2
-300 -2
10
-1
0
10
10
Frequency (rad/sec)
10
1
!/ Y&S G 2 ( j ) "1
K W. ! $ :(9) OW/
1
G2=exp(-5s)/(s+1)
2
G1=0.25/(s +2s+1)
G=G1+G2
0.5
-0.5
0
Re
0.5
1
G 2 ( j ) # :"1
K W. 9+ >*$ :(8) OW/
.> Y&S G 1 ( j ) L :"1
K
G 1 ( j ) E G 2 ( j ) }/ ? > y F@L
RC . 47 ? # ) :"1
K "4
B$+ # !K ^94 ) x
K ," U%C4
@ Q ! K !/ !$H > 9+ $.Q
0
Im
.$/ r$1D ND-QRDS m$
-0.5
G 1 ( j ) E G 2 ( j ) :F ,?
-1
-1.5
-1
:W4$+ >v $4 % -0.5
0
0.5
1
1.5
Re
Y&S G 2 ( j ) "1
K W. >1W $ :(10) OW/
.!/
G 1 ( j ) E G 2 ( j ) , Z
@ ![ W K "1
K F4 0!/ ^+
W ($4 0>&< # .
!$) >*74 G 2 ( j )
K : !/ @ )4 B7 : G 2 ( j )
:K #u : 7( ![K ! > @K P S W
.> ,!/ , t 10 9 W/ $ # .$ !$)
> R >&< # 7 :9+ :$l4 ,?! @
# . [ n d m
, K ; 1 ] or [ n E m , K
1]
!$) > > :9+ @ !D @ V QRDS >&<
S :K ,@ ! W K >&< # .>/
Journal of Control, Vol. 3, No. 2, Summer 2009
1388 14 02 ,/ 03 !( 0
1
K i @ ,9 @ )4 : R) O19 BL* :9+ W #*4
18
/ $' 0 ,$'
@ ! @K $ >*74 $/ ,! $H 12 OW/ 30
x
K 0 ," U%C4 !/ U @ x 0 G 2 ( j ) @K $
25
G 2 ( j ) @K >71 @K F!C4 :!C , 4 rad/sec
20
Im
.$ !$)
> > :9+ >" @ : ,@ R >&< #
15
10
13 OW/ .!/ u > :9+ :!D !*4 , 5
0
-1
Im
.> ,!/ , t # G ( s ) : 9+ >*$
-0.5
0
Re
0.5
1
30
.!/ Y&S G 2 ( j ) "1
K W. 9+ >*$ :(11) OW/
25
G 2 ( j ) "1
K F4 ? >&< # 20
4 B$+ # 0!K ^94 ," # *%C4 !/7 Y&S
15
K ![ > Y&S G 1 ( j ) 1
K
10
7S D >/ !$) @ )4 ! W
5
!$) G 2 ( j ) W K U4 ![ G 2 ( j )
0
-1
-0.5
0
Re
0.5
.> ,!/ , 12 OW/ >&< # : $ .!/
1
G 1 ( j ) # :"1
K W. 9+ >*$ :(13) OW/
2
.> Y&S G 2 ( j ) L :"1
K
# W K .
,! 4 ," F$"9 9+
“Retarded-Delay-QRDS (RD-QRDS) K >&<
0! > # :k. # rH > .P Behavior”
Amplitude ratio
7 W :$l4 R4 $4 w$; ^$K :"W/ 10
G1=1/(s+1)
G=G1+G2
G2=0.25exp(-5s)/(s+1)
0
10
-2
10
)4 Bv @ :!C & ! @ )4 6
:K
-4
10 -1
10
$D # :"1
K G 1 ( j ) @ >*74 v @
> ,
@ O*4 @ )4 Bv # > ,!/
0
K # > ,K ^94 # x
K @ )4
-100
.> ,!/ G$ ," U%C4 x
K
:. :"&< 0!/ , w/ L "&< @ S !K ^94 :"1
K "&< # .!K ^94 > #W
Y&S @K P U4 # :"1
K > #W $ :"1
K ,!/ xW >*; # :"1
K !/
xW B$+ >*; #Q $/ Y&S @K P U4 , L
!/ Y&S @K P S U4 ! * !/ !$4 m$;$ #
:"1
K $/ Y&S @K P U4 :"1
K x
Phase (deg.)
K [ “QD-QRDS behavior” K xW >
0
10
Frequeny (rad/sec)
1
10
-200
-300
-400
2
G1=1/(s+1)
G2=0.25exp(-5s)/(s+1)
G=G1+G2
-500 -1
10
0
10
Frequency (rad/sec)
1
10
G1 ( j ) # :"1
K W. ! $ :(12) OW/
.> Y&S G 2 ( j ) L :"1
K
"*; # 4 B$+ $/ Y&S @K P S U4 4L
Journal of Control, Vol. 3, No. 2, Summer 2009
1388 14 02 ,/ 03 !( 0
19
1
K i @ ,9 @ )4 : R) O19 BL* :9+ W #*4
/ $' 0 ,$'
$/ ,! $/ P4 U4 # ! $ ?
W :$l4 $u W&< $ !$) ^+ ,$ 4 ," F$"9
7 W :$l4 > ,K ^94 V V :"1
K
MD L # :"1
K 9+ W $ 9+ 7
U4 K O& # !
t [ !$4 9+
> t !$ > #W $ # # !
Ga (s) K 04 ," F$"9 W&< !
t ,7/
.! F
> > > 9+ $7 :* , ZH4 NDQRDS
.> ZH w$; 14 OW/ ,! [ Amplitude ratio
10
10
10
$O +7 K&B '; (!E ;"$*! -4
& 4&
6,!> $ ) 2
-
1
7 W :$l4 W ($4 !/ y 67 $R
!
MD L # :"1
K 9+ W $ 9+
0
K > #W !K ^94 :"1
K :s4 ? #
10
-1
wR $ # @ Oy .!
t ,7/ !
# I ! > " :$l4 # 1C : !u .!/ !$)
-2
10 -2
10
0
10
Frequency (rad/sec)
10
2
> 9+ 7 W :$l4 4 ," F$"9 .! ,!/ ,[ 15 140 8 BL* I # .P[
0
Phase (deg.)
-100
n (in quasi polynomial)
-200
m (in quasi polynomial)
(14)
N1 Q
N2 Q (in QRDS) (15)
-300
G=G1+G2
-400
G2=(s-1)4exp(-2s)/(0.5s+1)6
G1=10/(0.3s+1)
-500 -2
10
0
10
Frequency (rad/sec)
10
Ga (s) [10 /(0.3s 1)] [(s 1) 4 e 2 s /(0.5s 1) 6 ]
#
2
G a (s) U4 ! $ :(15) OW/
N ( s)
U4
.$/
wR
:.

<
K
K 2W n2 K1W n1
!.
U4 <
E
10(0.5s 1) 6 (1)(0.3s 1)( s 1) 4 e 2 s B$+
(1)(0.3)(1) 4 [10(0.5) 6 ] 1.92 ! 1
6 !m
n
5
#
QD 01
K ,@$< U4 K >K? $4 B$+ Gb ( s ) [10.1 /( 2 s 1)] [5 e s /( s 2 0.002 s 1)]
>1 9+ :!D !*4 B$+ # !/ ! QRDS
!/ N ( s ) (10.1)( s 0.002 s 1) (5)(2 s 1)e
U4
P4 Ga (s) U4 :9+ ? .!/ $($ > > #
.!/ !$H ,! > q% :9+ n (14OW/) $/
2
K
K 2W n2 K1W n1
(5)( 2) [(10.1)(1)
0.99 1 n
s
2 !m 1
100
> q% :9+ n Gb (s ) U4 $/ t 80
$/ , ZH4 NDQRDS U4 K * !/ /!
.$/
60
Im
,! > > 9+ V ( 16 OW/) U4 :9+ P4
40
20
0
-4
-3
-2
-1
0
1
Re
G a (s ) U4 9+ W :(14) OW/
Journal of Control, Vol. 3, No. 2, Summer 2009
1388 14 02 ,/ 03 !( 0
1
K i @ ,9 @ )4 : R) O19 BL* :9+ W #*4
20
/ $' 0 ,$'
# $4 !/ wR O7 :"1 7&R 250
W K $4 4 ," F$"9 @ ,9 !
200
U( O+< B$' "[ C U4 0,1? :![K
150
Im
0!/ @ )4 :< "[ @ W 0C U4
:$l4 R4 >&< "u # .$ :! >&< "u
100
:9+ ,@ #K? :"&< m$ ,$ 9+ 7 W
50
:$D D9+ $$ $D q% ![K ! C U4
0
-20
! @K $ !/ GH :"*; IH B'H
0 Z1 E Z E Z2 01
K ,@ QRDS :"&! : U( @ O+< ![ B$' ![K W K
K v4 >D4 ,$ !/ P1 ! ,! OW4 F$"9 # !/ ? ,@! : > : W
.P 4 ," F$"9 ,@ >*$ : @ QRDS :"&! K ?"u :"&<
F :"k. R "[ q 1
K i K 9+
-10
-5
Re
0
5
10
Gb (s ) U4 9+ W :(16) OW/
@ B$' 0 4 ,!/ wR M<7 $R .! I74
.> :! U( O
-15
:"1
K > O& # 0ZH4 ,7/ #
}$ ! $ .> ,
s4 G2 ( s ) G1 ( s ) U$4 K
# $R > ,!/ , t 17 OW/ Gb (s ) U4
G2 ( s ) U4 0v 1 x
K @ O7 $/ ,! OW/
P S Gb (s ) K Y44 #! ,!/ Y&S G1 ( s ) U4 R ,$ > > 9+ $" ,! $/ @K
.$/ t QDQRDS 0 Gb (s ) U4 K 4 ," F$"9
10
.> ,!/ :! U( @ !( &C # ,!/
5
QRDS K ? "u :"&< -1 !(
10
K- Y"*(* T?
+7 ';
6W Y"*-
0
> A&
n t m , K 1
or
n
m , K
1
iLHP
G1 ( j ) t G2 ( j ) ,
Z
ND
QRDS
10
-5
10
0
10
2
0
K !1
iLHP+ 0 ! Z ! Z gc and
fRHP G 1 ( j ) ! G 2 ( j ) ,
Z gc ! Z ! f
QD
QRDS
G 1 ( j ) G 2 ( j ) ,
D
QRDS
n d m , K !1
or
nm , K
nm ,
K 1
1
iRHP
Z
Journal of Control, Vol. 3, No. 2, Summer 2009
-400
-600
G=G1+G2
G2=5exp(-s)/(s2+0.002s+1)
G1=10.1/(2s+1)
-800
10
0
10
Frequency (rad/sec)
2
Gb (s ) U4 ! $ :(17) OW/
G 1 ( j ) ! G 2 ( j ) ,
iRHP+ 0 Z Z and
gc
f
G 1 ( j ) G 2 ( j ) ,
LHP
Z gc ! Z ! f
-200
Phase (deg.)
G 1 ( j ) G 2 ( j ) ,
n!m ,
RD
QRDS
K&B '; (!E W> X8 -5
6W A!W7 +7
1388 14 02 ,/ 03 !( 0
21
1
K i @ ,9 @ )4 : R) O19 BL* :9+ W #*4
/ $' 0 ,$'
!/ @ )4 B7 !K > *$4 ,! G1 ( s )
:P ,$ ," $) RC x
K Z gc L !( !/ @ )4 : > 47 4 ,! G2 ( s) @ )4 ! U$4 ," !/ 2 16 &* ? i-LHP: Infinite number of LHP zeros
i-LHP + f-RHP : Infinite number of LHP plus
finite number of RHP zeros.
i-RHP: Infinite number of RHP zeros.
i-RHP + f-LHP :Infinite number of RHP plus
finite number of LHP zeros
> #W !/ , ;$4 67 $R !u
!/ @ )4 : U$4 ," @ 4 +) 1
K ,!D 9+ 7 W :$l4 !
s4 U4 K :"1
K
U$4 m$ ," ? !/ !$) [ 0m$ U4 K
MD L # :"1
K 9+ Y : W O& )4 ! U$4 ," @ 1
K ,!D V @ )4 :
!/ "&< # ! K D+ $? t !
,!/ @ )4 : U$4 m$ U4 K !/ @
k& > "1
K 4 4 ," F$"9 $u W&<
W $4 < Y44 #! !/ !$) @K P S
:$l4 @ ,$ "*; # ! K $.
#/ : . ZH4 1
K ,!D U4 :9+
.!/ 9+ 7 W
E 4L* : 0>K? $4 < #
16 &* E 4L* :14 O > ,![ >! 6 &*
>K? $4 4 ," F$"9 ($4 Y44 #! .!/ .> ,!/ ,[ Oy & YR " !/
*
!
:
![K
!
oK
:
G pn ( s) 5e 2 s ( s 1) ! G p ( s) 5e 3s ( s 1)
B$+ # .$/ > c @ ,9 &
P1
Z Z (* T$ $: Q"& Y4 -6
Z Z (* ZT Z? $: V ',
(W&,
!$/ PE4 ,!
: [27] ,!/ y c @ ?
> ,!/ B7v : ID 4 ," F$"9 W ($4 &* Y44 #! .![ > Gc ( s ) 0.2 1 s ,!
&* $4 # .!/ B6( I1 O #
.!/ !$) 17 &* B$+ P1 'H
.! 16 , I1 6
k
1 G p 0 ( s )Gc ( s ) G pn ( s ) Gc ( s ) G p ( s ) Gc ( s )
(17)
5
5e 2 s 5e 3 s
1
1 [
][0.2 ]
s 1 s 1 s 1
s
m$ @ : ( !u 7/ V 17 &* $/ ,! $R
G(s)
l
¦G (s) ¦G
n
n 1
m
(s)eW ms
G1 (s) G2 (s) (16)
m 1
B7 !K > *$4 ,! G1 (s) 0&* # )4 : > 47 4 ,! G2 ( s) ,$ @ )4
#*4 0B9 @ )4 #/ O& !/ 16 2 &*
.!/7 :1 .!W !$4 "@ )4 # !/ @
7 W :$l4 ! F$ :$l4 I$4 [ :! I/
.!/ 2 &* &* # >K? $4 #
4 ," F$"9 @ ,9 < .>1 F O 9+
Op1 9+ W #*4 $ $
4 !/ y 67 $R
17 &* }$ ! $ 18 OW/ . F # $4
, B7 "4 4 &* ! 4L* }$ :!
.> ,!/ P4
,!/ /. :@ BLC ! $( >v @ )4 V
$ U$4 # :! 9+ W 0GH I/
# @ W 9+ Y W :$l4 $ .> K?
.> K? $ GH U( !/ BCCD4
@ )4 #!u O/ 4L* ?!Q O& BL* : *( , ; O< , $
4 !/
s :"@ )4 , U$4 ,!? 16 B$+
6 &* }$ A B7v W ($4 .> ,! p >
0 16 &* > ,!/ ,9 U$4 : t @
Journal of Control, Vol. 3, No. 2, Summer 2009
1388 14 02 ,/ 03 !( 0
1
K i @ ,9 @ )4 : R) O19 BL* :9+ W #*4
22
/ $' 0 ,$'
P S K : U4 @K $< 0@K D $ v O+
2
10
.> ,!/ @K
Amplitude ratio
100
80
-2
10
Im
60
0
10
40
-4
10 -1
10
20
0
-3
-2
-1
Re
0
200
1
100
P4 P1 # @ C< U4 @K D 0YR !/ ; :
.> ,!/
U$4 : @K >71 : $ 21 20 :"W/ Phase (deg.)
,!/ wR P1 'H &* :9+ >*$ :(19) OW/
0
-500 -1
10
P K : U$4 >71 @K PS K : U$4 ,"
K : ? @K PS U$4 @ C< U4 !/ 4 @K
.> > > 9+ $" F$"9 $/ @K P S
: U$4 7 rad/sec 5 ad/sec :"1
K #Q
!$/ 4 ," : @K P S U$4 >71 @K P K
K : U$4 0
P1 @ C< U4 @K D K
F@L .!/ u > :9+ $" :* ? @K P
>71 : D 0'4 w$; : > y .! ,!/ P4 B9 1
K O+$K ?!( B$+ @K
:"1
K @K >71 : D K > "!
> :9+ Y44 #! $/ W4 4 B$+ GH
@ C< U4 :9+ W $ r B$+ u >
.!$/ (22 OW/)
2
10
G=G1+G2
G1=Minimum phase functions
G2=Non minimum phase functions
-300
," 1 rad/sec x
K 4 $/ ,! "W/ # $R
0x
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1
-200
-400
? @K P K : U$4 D U4 0@K D k& 0
10
10
Frequency (rad/sec)
-100
.> ,!/ , 0,!/ wR P1 @ C< $($
:4 @K P S K : U$4 @K P K : U$4
G1=Minimum phase functions
G=G1+G2
G2=Non minimum phase functions
0
1
10
10
Frequency (rad/sec)
2
10
17 &* }$ ! $ :(18) OW/
! G(s) $ $/ ,! OW/ # $R
,!/ U$4 # @K P K : U$4 $ Y&S O&
0
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,!/ u > 9+ V 0@K P K : U$4 @K $<
# !W ! 0/9 rad/sec x
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!
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4 >*; # .> ,!/ > > 9+ V 0@K P
," , 0x
K # !W ! 4 rad/sec x
K
Y&S@K P S K : U$4 @K P K : U$4
: ,@ # !
!@K PK G(s) $ ,!/
&* :9+ 19 OW/ .$/ u > :9+ @
&* $/ ,! OW/ # $R ! ,!/ P4 'H
$/ 0/2223+2/6059 i > > 9+ V O/ 'H
> > 9+ # .> P1 :! :* Journal of Control, Vol. 3, No. 2, Summer 2009
1388 14 02 ,/ 03 !( 0
23
1
K i @ ,9 @ )4 : R) O19 BL* :9+ W #*4
/ $' 0 ,$'
> > 9+ ZH4 P $R #Q
2
10
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W $4 0!
qk< > > :9+ : $?
. $7" C<
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1 C< W $7" : P1 @ C< &*
0
10
Minimum phase Functions
Non minimum phasefunctions
Open loop
-2
10
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ZH4 : #$ / 1
K i @ ,9 &C # -1
0
10
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1
10
Frequency (rad/sec)
10
,!/ y @ C< U4 >71 $ :(20) OW/
< ,!Q B7D @ ! ,$ , 1 c #
$/ ZH $% "7R c # .> ,9 O
500
P ? #*4 $$ $D >71 "[ >*$ W
> > :9+ ZH4 0
!" P P , > @ C< V Y W :! OD4 ? 1 c # lR !u .> $) :
Phase (deg.)
$R .!
>9
P1 :! >"( # 0
-500
BL* $ ,$ !" ,!D @
.? !1 }74 @ )4 : R) O19
Minimum phase functions
Non minimum phase functions
Open loop
-1000 -1
10
ST*
[5] Niculescu, S. I., 2001, “Delay effects on stability: A
robust control approach”, Lecture notes in control
and information science, 269, Berlin: Springer.
[6] Gu, K., Kharitonove, V. L., Chen, J., Stability of time
delay systems, Boston: Birkhäuser, 2003.
[7] Gantmacher, F. R., The theory of matrices, New
York, Chelsea Publishing Company, 1959.
1
30
25
[2] Bellman, R.E., Cooke, K. L., Differential-Difference
Equations, Academic Press, New York, 1963.
[4] Stépán, G., Retarded dynamical systems: Stability
and haracteristic functions, Harlow, UK: Longman,
1989.
10
,!/ y @ C< U4 @K $ :(21) OW/
[1] Pontryagin L.S., 1955, “On the zeros of some
elementary transcendental functions”, Amer. Math.
Soc. Transl. Ser. 2, 1, 95-110.
20
Im
[3] J.K. Hale, Theory of functional differential equations,
Springer, New York, 1997.
0
10
Frequency (rad/sec)
15
10
5
0
-1
-0.5
0
Re
0.5
1
@ C< U4 :9+ W :(22) OW/
< > ZH w$; # $R
[8] Bhattacharyya, S. P., Chapellat, H., Keel, L. H.,
Robust control: The parametric approach,
Englewood Cliffs, NJ: Prentic-Hall, 1995.
E $ P1 :! $4 4 ," F$"9 @ ,9
[9] Hara, T., Sugie, J., 1996, “Stability region for
systems of differential difference equations”, Funk.
Ekvac., 39, 69-89.
.!/ [ :! @ U ,!
: s4
Journal of Control, Vol. 3, No. 2, Summer 2009
, ZH4 $) @ )4 >D+ F! ,!D 1388 14 02 ,/ 03 !( 0
1
K i @ ,9 @ )4 : R) O19 BL* :9+ W #*4
24
/ $' 0 ,$'
[19] Ramanathan, S., Curl R. L., Kravaris, C., 1989,
Dynamics and Control of Quasirational Systems,
AIChE J., 35, 6, 1017-1028.
[20]
Minorsky, N., “Self-excited oscillations in
dynamical systems possessing retarded actions”,
Journal of Applied Mechanics, 9, A65-A71, 1942.
[21] Tsypkin, Ya. Z., 1946, “Stability of systems with
delayed feedback”, Avtomatika I Telemekhanica, 7,
2-3, 107-129.
[22] Ansoff, H. I., Kruhmansel, J. A., 1948, “A general
stability criterion for linear oscillating systems with
constant time lag”, Quareterly of applied
Mathematic, 6, 337-341.
[23] Krall, A. M., Stability Techniques for Linear
Systems, Gordon and Breach, New York, 1967.
[24] Manitius, A., Tran, H., Payre, G. Roy, R., 1987,
“Computation of eigenvalues associated with
functional differential equations”, SIAM J. Sci. Stat.
Comput., 8, 222.
[25] Reda, D., 2006, “Solution operator approximations
for characteristic roots of delay differential
equations”, Applied Numerical Mathematical, 56,
3, 305-317.
[26] Verheyden K., Luzyanina T., Roose D., 2007,
“Efficient computation of characteristic roots of
delay differential equations using LMS methods”,
J. Comput. Appl. Math., 214, 1, 209-226.
[27] Majhi, S., Atherton, D P., 2000, “Obtaining
Controller Parameters for a New Smith Predictor
Using Auto tuning”, Automatica, 36, 11, 1651.
[10] Hara, T., Rinco, M., Morii, T., 1997, “Asymptotic
stability condition for linear diffrential- diffrence
equations with delays”, Dynamic systems Appl., 6,
493-506.
[11] Baptistini, M., Hale, P. J. K., 1997, “On the
stability of some exponential polynomials”, Journal
of Mathematical Analysis and Applications, 205, 1,
259-272.
[12] Cahlon, B., Schmidt, D., 2000, “On stability of
systems of delay differential equations”, Journal of
Computational and Applied Mathematics, 117, 137158.
[13] Malakhovski, E., Mirkin, L., 2006, “On stability of
second-order quasi-polynomials with a single
delay”, Automatica, 42, 1041-1047.
[14] Shirvani, M., Inagaki, M., Shimizu, T., 1993, A
Simplified Model of Distributed Parameter
Systems”, Int. J. Eng., 6, 2, 65-78.
[15] Shirvani, M, Inagaki, M., Shimizu, T., 1995,
“Simplification Study on Dynamic Models of
Distributed Parameter Systems”, AIChE J., 41, 12,
2658-2660.
[16] Shirvani, M, Doustary, M. A., Shahbaz, M., Eksiri
Z., 2004, “Heuristic Process Model Simplification
in Frequency Response Domain”,
I.J.E.
Transactions B: Applications, 17, 1, 19-39.
[17] Ramanathan, S., Curl, R. L., Kravaris, C., 1987.
“Dynamics and control of the cumulative mass
fraction of a particle size distribution”, ACC Proc.,
Minneapolis.
[18] Ramanathan, S., 1988, “Control of Quasirational
Distributed Systems with Examples on the Control
of Cumulative Mass Fraction of Particle Size
Distribution”, Ph.D Thesis, University of Michigan,
Ann Arbor.
Journal of Control, Vol. 3, No. 2, Summer 2009
1388 14 02 ,/ 03 !( 0
B
- D9+ 01388 14 02 ,/ 03 !(
&/? $: (+> !&*! 4 (* ,*") 2
$ ;!< 01 9/ $(
[email protected] 0
,? 0^ !" ,!W 0d! >4 ,. 0:
:$ 1
[email protected] 0
,? 0^ !" ,!W 0d! >4 ,. 2
(1388/7/5 &C ck i4 01388/4/25 &C >K i4)
>"( OD4 $ > *4 RC #!u : ($$
1) K#H4 >&< K'4 $$
P1 :6;?
"4 P1 !
#4 > ,!/ <% :$? &
$ .> K? : @! ,!
V <%
@ WK V $ .!
>
< *4 }C . @ F!WQ > $/ >! r$R *4 RC >
$$
> $* (*4 RC O/) P # : P4 V : :@! 0*4 RC #!u P1 :@!
,!/O+< :E <% 0:@7/ A > ,!/ :@7/ K? $ 0 K $$
B7D
.! !p4
.>$
0P # P4 0K'4 :! 0K'4 O19 BL* 0$$
1 :( Abstract: Quantum trajectory with multiple equilibrium points is analyzed to become globally
stable. The control law is designed such that the quantum system stabilized to one of its wanted
equilibrium point and escape from the other unwanted equilibriums. As a physical example the
global stabilization of one half-spin atom, which is known as a quantum bit (qubit) and has many
applications in quantum computing, is investigated by our control law and simulated. The
simulation result confirms the theoretical desig.
Keywords: Quantum trajectory0 stochastic differential equation0 stochastic stability0 half-spin
atom0 quantum bit (qubit).
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F ! V 0$$
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6 $$
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12
k,! :?@! O :. 0$$
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4 $$
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x .P w/ #
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# @ ,9 ,!/ K'4 :! : #"
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1 : : @! V 0K'4 :! :
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& >&< : ![ >! #
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4
Anti-Linear
5
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9
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P1 0>&< :K
$$
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Pure State
Hermitian (Self-adjoint)
12
Observable
11
Journal of Control0 Vol. 30 No. 20 Summer 2009
1
Hermitian (self-adjoint)
1388 14 02 ,/ 03 !( 0
27
?!u *4 :>&< $$
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!&*! $H 94&*7"$ $: -3
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6 $1K 0:6 1984 #& :
$$
P1 $ . 0$$
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$$
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![
@ B$' U†t U t = U t U†t = I 0W x4 . V :
-# K'4 O19 &* !/ O
:[10] B"/
.$/ dWt dYt tr [L† L]St dt
dU t U tLdA†t (U tSL† )dA t U t(S I)d- t
U t(iH 21 L†L)dt , U0 I , i 1
(2)
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. O< 9
S .!
t
P $$
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10
Rössler
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11
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13
Self-nondemolish: < X U , X t > 0 : U v t
14
Quantum trajectory
Commutator
Anti-Commutator
15
16
Journal of Control0 Vol. 30 No. 20 Summer 2009
1
Annihilation process
Creation process
Number process
4
Amplitude operator
5
Phase operator
6
Photocounter
7
Quantum stochastic calculus
8
Hamiltonian
2
3
1388 14 02 ,/ 03 !( 0
28
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V : ? 0!$? 3< ! X e RC -1
:P/ / 0RC # X & I/ @ .1

¬
. lim sup X t X e F 0 : F 0
X 0 l Xe
tp0
®
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tld
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A t A†t
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Ue
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Ut
X
Xe
H(u t )
Yt
Wt
Xt
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Xe
V $) #" @ 0W P1 V :@!
:E #4: @ W q$& c .> P1
Xt
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:! BE @ :1 . W P1 :! OD4 :
$" 1 !/ $($ R)S :r
q$&
K'4 :P1 : q$& :! c 0,6* .>
Xt
Xe
/$
I$4 0K'4 O19 BL* OW/ q$
X
Oy .> ,!/ , *$4 [22] $p [21] W1 [20]
:! (I) < 7 :! 0(L) < :! -2OW/
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E $ OW/ K'4 O19 &* 0$E # : .$/
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:!.
*$4 @ K'4 :! : 0^$K G*4 @ ,9 :[2022] > K
$ .> K'4 . B7D k. \ ! :(Itô) $ 1
3
Stable in probability
4
Globally stable
5
Asymptotically stable
Journal of Control0 Vol. 30 No. 20 Summer 2009
.! F #4$ 2
Homodyne Detector
1388 14 02 ,/ 03 !( 0
29
?!u *4 :>&< $$
1 : @! $ ;!< 09/ $(
>7 C ,.[ . <Hfree, Se > [0] P/ / P1
1 : @! ,!
V 0W$R B, C 0
:@ > B7 $$
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i <Se,H control >
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†
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†
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14
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@
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: > q$& U4 Bs4 0$$
1
dV(St ) B2 d Se 2C 2 Se d Se C 2 d Se
B2 2C 2 Se
2 Se B 2C
2
2
i[H, S ] dt
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2
e
e
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†
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2
2
(7)
dt
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(8)
†
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C : * V 0 -(1) :}/ !/ < !
: V(X) l f , *(V) l f -(2) X t v X e
K'4 ! X e *4 RC ,.[ 0!$/ 0 X l f
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Bs4 &* F (
*(V) 2 Se i[u tHcontrol, Se ]
L S e Se L L L S e
†
†
(9)
2
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i <Hcontrol , Se >
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†
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Vu$
Bs4 $( B, C 0 >7 C ,.[
:![ >! @
* V(St )
{(B2 2C 2 Se ) i <Se,Hcontrol >
C( Se L L†Se L† + L Se )}2 0
Journal of Control0 Vol. 30 No. 20 Summer 2009
!&*! 4 (,*") -6
: 0K'4 :! E @ ,9 P$) tH # .!/ . Se P1 3 (,) *4 :>&< @ W $ @
rH :$? V(St ) K'4 q$& U4 ! $E # :
P1 O
#$ !
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Journal of Control0 Vol. 30 No. 20 Summer 2009
1388 14 02 ,/ 03 !( 0
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Journal of Control0 Vol. 30 No. 20 Summer 2009
Pauli matrices
1388 14 02 ,/ 03 !( 0
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[1] Krausz, F., Ivanov, M., 2009, “Attosecond physics”,
Review of Modern Physics, 81, 163-234.
[2] Lvovsky, A. I., Raymer, M. G., 2009, “Continuousvariable optical quantum-state tomography”, Review
of Modern Physics, 81, 299-332.
[3] Král, P., Thanopulos, I., Shapiro, M., 2007,
“Colloquium: Coherently controlled adiabatic
passage”, Review of Modern Physics, 79, 53-77.
[4] Hohenester, U., Rekdal, P. K., Borzì, A.,
Schmiedmaye, J., 2007, "Optimal quantum control
of Bose-Einstein condensates in magnetic
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[5] Choi, S., Bigelow, N. P., 2005, "Initial steps
towards quantum control of atomic Bose–Einstein
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Semiclassical Optics, 7, 413–420.
A # .$/ ,! 4OW/ GH & I/ :@
0P1 >&< :K & I/ :@ 0P4 # !
: $R ,$ z B'H ,. >7 > .> ,!/ !
*$W%) (/B& -8
@ $$
:P1 E , 'H $R 0&C # #" @ .!/ MD 0!/ $4$K $ OC4 -,@! O+ @ ,9 P1 >&< #H4 E 0$$
VK
; #CCD ! @ :1 > k}RD :?
P1 >&< #H4 BL* 0O+ # d !1$4 VK
R)S 0K'4 O19 BL* # .![ >! $$
[6] Ketterle, E.W., 2001, "When atoms behave as
waves: Bose-Einstein condensation and the atom
laser", Nobel Lecture, 118-154.
1 @ $$
P1 : &
. <% $E ,$
[7] Dunning, F. B., Mestayer, J. J., Reinhold, C. O.,
Yoshida, S., Burgdörfer, J., 2009, “Engineering
atomic Rydberg states with pulsed electric fields”, J.
Phys. B: At. Mol. Opt. Phys., 42, doi: 2/022001.
P1$4 .> ,!/ p BL* # : :1 &
[8] Fert, A., 2008, “Nobel Lecture: Origin,
development, and future of spintronics”, Review of
Modern Physics, 80, 1517-1530.
[9] Werschnik, J., Gross, E. K. U., 2007, “Quantum
optimal control theory”, Journal of Physics B, 40
pp. 175–211.
[10] Hudson, R. L., Parthasarathy, K. R., 1984,
“Quantum Itô’s formula and stochastic evolutions”,
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301–323.
[11] Belavkin, V. P., 1992, “Quantum Continual
Measurements and a Posteriori Collapse on CCR”,
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[12] Bouten, L., Filtering and Control in Quantum
Optics, PhD Thesis0 University of Nijmegen, 2004.
[13] Ticozzi, F., Viola, L., 2008, “Quantum Markovian
Subsystems: Invariance, Attractivity and Control”,
IEEE Trans. Automatic Control, 3, 9, 2048-2063.
:"/ 0!/ ,/ # $R .!/ > p<
-
V $$
1 : 0K'4 :! E @ ,9
V : ,!
# .P[ >! : @! ,!
$$
B7D (>$
) , $$
P1
V .>K? OD4 4 $ 0$$
: >)
(: :!) r$R A !/ :@7/ >$
,!/
.!/ O+<
:@! MD $4 04CCD4 m$;$ # > ! .>) $$
K'4 O19 BL* :
K'4 :! : 4 > @ ! 0$ # : $
.! *$4 0&C # ,!/
y 2 1: 0$$
-
<% $4 0O+< :! : @ ,9 x
: @ ) [24] U( .>) 03 ! 0,!
>! $$
K'4 O19 BL* : K'4 :!
.> ,![
[14] Van Handel, R., Stockton, J. K., Mabuchi, H., 2005,
“Modeling and feedback control design for quantum
Journal of Control, Vol.3, No.2, Summer 2009
1388 14 02 ,/ 03 !( 0
33
?!u *4 :>&< $$
1 : @! $ ;!< 09/ $(
[20] Kushner, H. J., Stochastic Stability and Control,
New York: Academic Press, 1967.
state preparation”, Journal of Optics B: Quantum
Semiclass Optics, 7, 179-197.
[21] Has’minski, R. Z., Stochastic Stability of
Differential
Equations,
Amsterdam,
the
Netherlands: Sijthoff Noordhoff, 1980.
[15] Van Handel, R., Stockton, J. K., Mabuchi, H., 2005,
“Feedback Control of Quantum State Reduction”,
IEEE Trans. Automatic Control, 50, 6, 768-780.
[22] Mao, X., Stochastic Differential Equations and
Applications, Horwood Publishing Chichester,
1997.
[16] Mirrahimi, M., Van Handel, R., 2007, “Stabilizing
feedback controls for quantum systems”, SIAM
Journal on Control and Optimization, 46, 445–467.
[23] Bertlmann R. A., Krammer, P., 2008, “Bloch
vectors for qudits”, Journal of Physics A, 41, doi:
235303.
[17] Altafini, C., Ticozzi, F., 2005, “Almost global
stochastic feedback stabilization of conditional
quantum dynamics”, quant-ph/0510222v1.
[24] Sharifi, J., Momeni, H. R., “Quantum Stochastic
[18] Tsumura, K., 2008, “Global Stabilization at
Arbitrary Eigenstates of N-dimensional Quantum
Spin Systems via Continuous Feedback”, American
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Stability”, quan-ph/0907.3452v1.
[19] Breuer, H. P., Petruccione, F. The Theory of Open
Quantum Systems, Oxford University Press,
Clarendon, 2006.
Journal of Control0 Vol. 30 No. 20 Summer 2009
1388 14 02 ,/ 03 !( 0
Journal of Control
Vol. 3, No. 2, pp. 1-9, Summer 2009
Design of Multiple Model Controller using SOM Neural Network
Poya Bashivan1, Alireza Fatehi2
Advanced Process Automation and Control (APAC) Research Group, Department of Control Engineering,
K. N. Toosi University of Technology, Tehran, Iran
1
[email protected], [email protected]
(Manuscript received: Feb. 20 2009, accepted Aug. 15 2009)
Abstract: A multiple-model adaptive controller is developed using the Self-Organizing Map (SOM)
neural network. The considered controller which we name it as Multiple Controller via SOM
(MCSOM) is evaluated on the pH neutralization plant. In MCSOM multiple models are identified
using an SOM to cluster the model space. An improved switching algorithm based on excitation level
of plant has also been suggested for systems with noisy environments. Identification of pH plant using
SOM is discussed and performance of the multiple-model controller is compared to the Self Tuning
Regulator (STR).
Keywords: Multiple models, Adaptive control, Self-organizing map, pH neutralization plant.
.? !&$4 (SOM) ,!@ $) 7' W7/ @ ,9 ?!u :"&! C7R4 ,!
V &C # :6;?
![K @ &! : 0P MCSOM +6) $% SOM V
?!u ,!
[ $
k ,!
:! SOM 7' W7/ :?W ?!u :"&! 0MCSOM ,!
.? @ pH !C :@)
L $ : :"1 : P1 $($ VD4 R K $7" P$.& V #Q .!? rH "&! :K
V ?!u ,!
W K? MD $ SOM 7' W7/ pH ![K / .> ,!/ "
.> ,!/ 1C (STR) PE4 $) $4L$? ,!
pH PE4 ![K 0,!@ $) 7' W7/ 0C7R4 0?!u :"&! :( 1- Introduction
There are many industrial processes which their
nonlinear behavior cannot be modeled and controlled
by a single mathematical model at least in their full
operating range. Various solutions for controlling
these systems have been suggested over past
decades. Robust and adaptive control is two major
approaches toward solving this problem. But these
techniques can become quite restrictive in many
applications. A more recent approach is the concept
of multiple models along with a switching algorithm
[14] which has been an area of interest in control
theory in order to simplify both the modeling and
controller design. Many global controller designs
with the aid of multiple models have been reported
on different applications [9, 5]. Narendra et al. [15]
suggested an adaptive MM structure with switching
based on a performance function. The key idea to
this approach is the ability to approximate the
behavior of nonlinear processes within a predefined
neighborhood of operating point with a relatively
Corresponding Author: Poya Bashivan
simple linear model with a desired accuracy. By
repeating this job in different key operating points of
the nonlinear process, a bank of linear models can be
created with each model corresponding to one of the
operating points. An algorithm for switching
between these models then should be used to find
and select the best approximation of the nonlinear
process from this bank as fast and as accurate as
possible.
Selecting the proper bank of models is a key
subject in control using multiple models. In many
previous works on multiple model control, the bank
of models is created by dividing the range of
variations of all parameters of the assumed model
structure and place a model for each combination of
parameters [15, 3]. This method becomes inefficient
when working with a nonlinear system with a high
degree of nonlinearity. The proposed algorithm
which is used here is an effort to solve this problem
and introduce a method to create the model bank for
any nonlinear system in a straight forward manner.
Journal of Control, Iranian Society of Instrument & Control EngineersK.N. Toosi University of Technology
2
Design of Multiple Model Controller using SOM Neural
P. Bashivan, A. Fatehi
In this paper, a multiple model adaptive strategy
with pole placement controllers is considered.
Models have the same structure but their parameter
values are different for each model. To identify the
bank of model, the multiple modeling using selforganizing map neural network (MMSOM) [6] is
applied to the input/output data of the plant.
Kohonen’s self-organizing map (SOM) neural
networks [12] is used in this algorithm to
automatically assign parameters of models in the
model bank based on the clustering of identification
data from the RLS method. SOM have previously
been used in a number of local modeling applications
to divide the operating region of systems into local
regions. In [6], at first, model of the system is
identified using the linear identification methods and
then the identification data are fed into the SOM
network as training data. In this way we actually
divide the parameter space of system with respect to
the model structure in order to find the suitable
models. Another method was used in [4], in which
the SOM network was used to divide the state space
of the system directly from the input/output data of
the system.
will then be searched for in each cycle. For this
purpose, a performance index is defined based on
estimation errors of models. The performance index
is given by:
The quality of the multiple controller depends not
only on the bank of models but also on how to select
the best model. An improved switching algorithm is
used to find the best representing model from the
generated bank of models. Parameters of the best
model are then used in pole placement algorithm to
generate control signal in each cycle. By including
an adaptive model beside the fixed model in the
bank, the multiple model controller will be able to
control new operating areas of system with a
performance at least as good as the conventional
adaptive controller.
where G is an arbitrary constant. This function
prevents fast switches between models and decreases
the unnecessary switches. But in the presence of
measurement noise in the system this condition is not
sufficient. Hence, another complementary condition
is introduced in this paper to properly eliminate the
remaining undesired switches and reduce chattering
in the system response.
The paper is organized as follows. The general
structure of multiple model control strategy together
with some modification on it is described in the next
section. A brief description of SOM and its
application in MMSOM on generation of the bank of
models is presented in section 3. In section 4 the total
structure of the multiple model controller by SOM
(MCSOM) is introduced. Simulation results from
implementing the described control strategy on a
simulated pH neutralization process are presented in
section 5 and the paper is concluded in the final
section.
2- Control Based on Multiple Models
A general block diagram of the closed loop system
is shown in Fig.1. In this approach the understudy
nonlinear system will be approximated by a set of
linear models which will form a bank of models. The
model which best suits the actual system’s behavior
Journal of Control, Vol. 3, No. 2, Summer 2009
M
J i (t ) D ei2 (t ) E ¦ e O k ei2 (t k ) (1)
k 1
D , E ! 0,0 O d 1 are three weighting constants and
M t 1 determines the range of effective past data.
Relative values of D and E weights the current and
previous estimation errors of models and O is used
as a forgetting factor for the past errors. In this
manner the model corresponding to the lowest
J i will be the best model describes it at the time t. In
order to avoid fast and unnecessary switches, a
hysteresis function is added to the switching
condition. The switch occurs only if the performance
index for the in-loop model ( J inloop ) and the new
best model ( J min ) satisfy the following condition:
J min G J inloop
(2)
The proposed complementary condition is based on
the excitation level of the process. The basic idea is
to prevent switching between models if the process is
in a steady state where it is not excited properly by
the control signal. The level of excitation of the
process is measured using the method proposed by
Hugglund et al. [11] using a high pass filter. A
predefined threshold is used to indicate the required
level of excitation in order to allow a switch to a new
model. This threshold is selected according to the
existing noise characteristics. The excitation
condition allows switching between models only if
the following condition is satisfied:
yhp (t ) ! w0
(3)
where w0 is the desired switching threshold
defined on the filtered output of the plant. Care must
be taken when introducing this condition to the
switching algorithm as high values for this threshold
brings unnecessary delays into the switching
algorithm and therefore can make the closed loop
system to oscillate and even become unstable.
1388 14 02,/ 03 !( 0
Design of Multiple Model Controller using SOM Neural
P. Bashivan, A. Fatehi
3
the vector of controller parameters, e j is the error
between actual output of system and output of
models and w is the sensitivity vector used in model
reference method. Let Tmin ! 0 be the minimum
time between each successive switching of models.
There exists a Ts ! 0 such that if Tmin (0, Ts ) , then
all the signals in the overall system, as well as the
performance indices ^J j (t )` , are uniformly bounded.
In the following lemma it is shown that in case of
applying the excitation condition (3) for switching
between models, the stability condition of theorem 1
can still be proved while Tmin remains less than Ts .
Fig.1 – General block diagram of the Multiple Model
Structure
Note 1: A band-pass filter H f is used to filter out
the low and high frequency components from the
identification data used in the adaptive estimator
block [16] in which the poles of the filter are chosen
3 to 5 times faster than the dominant desired closed
loop poles of the control system. This filter is
necessary in order to map linear models into the
nonlinear behaviors of the actual system. Also, a
high-pass filter H hp is used to impose excitation
condition on the switching states. H hp was designed
by try and error such that the closed loop system
remains stable and also the resulting switching
condition only permits switching when enough
variation exists in the system’s output (i.e. enough
excitation in the system’s input).
Stability of the multiple model controller with
switching algorithm based on performance indices
similar to (1) and controller designed based on model
reference adaptive controller (MRAC) method was
previously studied in [15]. Theorem 1 summarizes
the stability properties of this configuration as
suggested in [15].
Theorem 1: Consider the switching and tuning
system described above with N1 fixed models and
N 2 t 1 free-running adaptive models, where the
latter are assumed to satisfy the identification
conditions below:
pˆ j ,T j L
pˆ ,T Lf L2
f
j
1 wT w
H T (t ) o 0
L2
the high pass filter H hp can be selected such that
y hp (Tmin ) ! w0
and the closed loop system remains
uniformly bounded.
Proof: Suppose that output of the plant diverges
and moves along
y (t ) CeOt
(4)
(5)
where O ! 0 . It will be shown that for every
supposed trajectory like (5), parameters of the
excitation condition can be found such that the time
delay imposed by excitation condition before
permitting a switch to a new model becomes less
than the Tmin . Therefore the condition of theorem 1
satisfies despite the excitation condition.
Suppose that the output of system changes along
the trajectory stated in (5) and the high pass filter
used for detecting the excitation in system is as
below:
Ghp ( s )
s
s J
(6)
Therefore, the filtered output of the plant is
obtained by:
Yhp ( s )
j
ej
Lemma 1: Assume the switching algorithm as
described in theorem 1 together with the excitation
condition of equation (3) for an observable plant. For
any assumed variation of system output and
Tmin selected according to theorem 1, parameters of
U (s)
Cs
( s J )( s O )
(7)
The time response of which will be as:
j
where p̂ j is the vector of model parameters, T j is
Journal of Control, Vol. 3, No. 2, Summer 2009
1388 14 02 ,/ 03 !( 0
4
Design of Multiple Model Controller using SOM Neural
P. Bashivan, A. Fatehi
yhp (t )
a1
a2
a1e J t a2eOt
another scalar-valued learning-rate factor and the
parameter V (t ) defines the width of the function.
CJ
O J
CO
O J
(8)
y hp (t )
Define t 0 as the moment in which
reaches
w0 . w0 can be computed from (9) while desired
values are selected for t 0 and J :
w0
a1e
J t0
a2e
O t0
(9)
Therefore, by selecting t 0 less than Tmin , the
closed loop stability of the control system is
.guaranteed.
Q.E.D.
3- Multiple Modeling by Self-organizing
Map
Kohonen [12] developed the SOM with the ability
to transform an input signal of arbitrary dimension
into a lower dimensional discrete representation
preserving topological neighborhoods. The SOM
network has an input vector \ k with an arbitrary
high dimension k. Each node in the network has a
reference vector (RV) wi ,k with the same dimension
as the input vector. Training of SOM is started by
introducing the index of the closest reference vector
of the nodes to the input vector:
i
arg min j \ k w j ,k
(10)
Then RV’s of this node and its neighbors up to a
certain geometric distance are updated as follows:
wi , k (t 1) wi , k (t ) hci (t )>\ k (t ) wi , k (t )@
(11)
where hci (t ) is the neighborhood function. For
convergence
it
is
necessary
that
hci (t ) o 0 when t o f . A typical choice in terms of
the Gaussian function is[12]:
§ rc ri 2
hci (t ) D (t ). exp¨ ¨ 2V 2 (t )
©
·
¸
¸
¹
(12)
where rc 2 and ri 2 are the location vectors
of nodes c and i, respectively, in the array. D (t ) is
Journal of Control, Vol. 3, No. 2, Summer 2009
Both D (t ) and V (t ) are some monotonically
decreasing functions of time. In this way, the trained
SOM will have more nodes (i.e. close RV's) in the
regions where more input vectors existed.
Cho et al. [4] trained the SOM directly with
input/output data and then the neuron weights of the
trained SOM were converted to ARX model
parameter vectors using the least square method.
Here we use the MMSOM algorithm proposed by
Fatehi et al. [6] for identification of nonlinear plant.
In
MMSOM,
an
input
vector
of
\ mn [a0 ,..., an1 , b0 ,..., bnd0 1 ] is considered as
the input to the SOM. This input vector is the
identification parameters of an instantaneous model
of the plant, which identified using some online
identification like recursive least square (RLS)
algorithm. Therefore, the reference vector of the i th
node wi ,k represents the parameters for the i th
model in the bank of models. After training the SOM
neural network, models parameters approximate the
statistical distribution of the input data [7].
The second approach will be more efficient in the
sense that different operating regions with similar
characteristics and model parameters will not be
considered as different operating regions by the
SOM and therefore less neurons is placed in the
boundary regions of clusters compared to the first
approach.
4- Structure of Multiple Model
Controller by SOM (MCSOM)
The structure of multiple model controller by SOM
(MCSOM) is as depicted in figure 1 in which the
bank of models is constructed using MMSOM of
section 3. Controllers for each of the models can be
any kind of indirect adaptive controller like MRAC
for continuous models or STR for discrete ones.
If MRAC is employed to design multiple
controllers, stability of the control system can be
guaranteed. For this purpose, first it should be noted
that MCSOM without the excitation condition of
section 2 has exactly the same structure as the
controller assumed in by Narendra et. al [15]. The
main difference between the MCSOM and the
controller structure of [15] is in the way the bank of
models is created. In MCSOM, SOM neural network
is used but in [15] the bank of models is created by
dividing the space of model parameters into a
1388 14 02,/ 03 !( 0
Design of Multiple Model Controller using SOM Neural
P. Bashivan, A. Fatehi
selecting the R polynomial by the following form
number of equal subspaces and assigning each a
model. Since the bank of models created via the
SOM network also satisfies the conditions of
theorem 1 and the method with which the model
bank was created does not violate theorem 1, the
multiple model controller by SOM without the
excitation condition stabilize the closed loop control
system. Lemma 1 also proofs the stability under the
excitation condition of equation (3). Therefore,
stability of the whole control system of MCSOM is
guaranteed for MRAC controller.
R
The continuous stirred tank neutralizer has three
inlet streams: base, acid and buffer. Acetic acid of
concentration Ca flows in at a rate of f a and reacts
with sodium hydroxide of concentration Cb which
flows into the tank at a rate of f b . Buffer flow rate is
f c and is considered to control the tank level through
a PID controller. The liquid exit the tank with the
flow rate of f o . Nominal operating conditions of the
simulated pH system are summarized in Table 1.
(13)
The nonlinear model of this pH neutralization plant
is presented in [10]. The model is assumed to be
carried out under the assumptions of perfect mixing,
constant temperature and complete solubility of the
ions involved. Complete details about the chemical
reactions and the exact mathematical model can be
found in [10]. The objective is to control the pH
value of the outlet stream by controlling the base
flow.
where y is the output, u is the input, v is a
disturbance and q is the forward shift operator. It is
assumed that the A polynomial is monic and also A
and B polynomials are relatively prime. A general
controller with the following structure is assumed
Ru(t ) Tuc (t ) Sy(t )
(18)
The MCSOM algorithm is demonstrated on a
model of pH neutralization process. This process is a
highly nonlinear plant which is widely used in
industries. A schematic diagram of the pH plant is
shown in Fig. 2.
In STR an ARX model is used to describe the
dynamic of the process:
B(q 1 )u (t ) v(t )
Rc(q 1)
5- Simulation Results
In the following section on application of MCSOM
we used self-tuning regulator (STR) as the adaptive
controller. In STR the controller is design based on
pole placement method. Pole placement controller
[1] is a simple and also a practically useful
controller. The idea is to determine a controller with
predefined closed loop poles. A brief description of
this controller is presented here.
A(q 1 ) y (t )
5
(14)
The controller has two degrees of freedom. The
closed loop characteristic polynomial is
AR BS
Ac
(15)
The R and S polynomial can be solved from the
diophantine equation (15) as the minimum degree
solution when the closed loop poles are known. The
desired closed loop response from the command
signal to system output is described as
Am ym (t )
Bmuc (t )
(16)
which is a design parameter and is selected such
that the closed loop response has suitable speed and
characteristics. The T polynomial is then found from
the following condition:
BT
Bm
(17)
The R and S polynomials can be designed to
integrate different characteristics into the controller.
For example an integral action can be added by
putting one root of the R polynomial in 1 and
Journal of Control, Vol. 3, No. 2, Summer 2009
Fig.2 – Schematic Diagram of pH Plant
Table 1. Operating conditions of simulated pH plant
Operating Parameter
Parameter Value
Acid concentration ( Ca )
0.001 mol/lit
Acid flow rate (
fa )
0.3 mlit/sec
1388 14 02 ,/ 03 !( 0
6
Design of Multiple Model Controller using SOM Neural
P. Bashivan, A. Fatehi
Base concentration ( C b )
0.001 mol/lit
Cross sectional area of tank
70 cm2
acid dissociation constant ( K a )
1.75e-5
Water dissociation constant ( K w )
the value of derivative along the full operating
region. Selected operating points are shown in Fig.4.
Fig. 5 demonstrates the total input signal applied to
the base flow pump.
10
X: 19
Y: 9.839
9
-14
1e
X: 16.5
Y: 8.003
Tank level (h)
17 cm
PH steady state
8
X: 14
Y: 6.4
6
X: 11.5
Y: 5.096
5-1- Identification of pH Process
Fig. 3 illustrates the block diagram of identification
of the pH process model. Here we have used
MMSOM to extract the statistical features of
identified parameters produced by the RLS method.
7
5
X: 10.25
Y: 3.864
4
10
11
12
13
14
15
16
17
18
19
20
21
q
Fig.4 – Operating points on steady state diagram of input
flow/pH
20
19
18
17
q (mlit/sec)
16
15
14
13
12
11
10
0
1
2
3
4
time (sec)
5
6
7
8
4
x 10
Fig.5 – Applied input signal for pH neutralization plant
identification
Fig.3 – Block Diagram of identification and model
generation process
In the first step plant is excited by a suitable input
sequence of enough persistently excitation (PE)
order. A random binary signal (RBS) pattern was
used as the identification input of the pH plant. The
RBS signal was constructed such that the plant
reaches the steady state in about 20 percent of
toggles between high and low limits so that both low
and high frequencies are excited. The input pattern
was biased to identify the plant around 5 different
operating points. The titration curve of the under
study pH plant is shown in Fig.4. As the control
variable is the base flow the curve has a positive
derivative. Operating points were selected based on
Journal of Control, Vol. 3, No. 2, Summer 2009
The RLS method was applied to estimate the
model parameters. A first order ARX model was
used as the model structure. A forgetting factor of
0.98 was used to discard the old data and dealing
with the problem of time-varying parameters.
In the next step, data from the RLS estimation was
given to the SOM network. Input vectors are the
estimated parameters of the ARX model which are
two in our case. A two dimensional SOM network
was then used to cluster the estimation parameter
into some clusters. SOM distributes its RVs across
the input space according to the statistical properties
of the input data. Therefore, relative number of input
data in each region acts as a weight for the number of
required models in each region. This means that
more identification data in a specific operating
region of system forces SOM to place more models
in that region on the cost of less models in other
regions. Equal identification time intervals were used
for each operation region to give equal weights to
1388 14 02,/ 03 !( 0
Design of Multiple Model Controller using SOM Neural
P. Bashivan, A. Fatehi
each region of pH plant.
Fig.6 shows the graphical representation of Umatrix [13] of the trained SOM. Areas with lighter
color code indicate the closeness of adjacent neurons
and those with darker color code indicates higher
distance between adjacent neurons. By utilizing this
graphical representation we can distinguish clusters
of neurons. Each cluster represents one operating
region of the plant behavior. Clusters are
distinguished with areas with lighter color codes and
their boundaries with lines of neurons with darker
color codes.
Although the pH system was excited around 5
operating points, only three different regions can be
distinguished from the U-matrix of trained SOM.
This is because of closeness of model parameters in
some of the operating points and also the continuity
of SOM lattice. Similar results have been obtained
by Galan et al. [8] using the gap metric method
which confirms the results from utilizing SOM on
model generation.
Here the SOM network functions as a nonlinear
map to automatically assign proper models to
different operating areas of the nonlinear system
based on the identification data of the nonlinear
system.
U-matrix
0.309
0.163
0.0169
Fig.6 U-Matrix of trained network
5-2 Controller design and implementation
In this section, the proposed control strategy is
evaluated using the nonlinear model of pH
neutralization process. Results have been compared
with a self tuning regulator (STR).
A total of 26 models are used in order to control
the pH plant in this paper. The bank of models
consists of 25 fixed models generated by the SOM
Journal of Control, Vol. 3, No. 2, Summer 2009
7
and one free running adaptive model to give the
control system the flexibility of working in
unexplored regions. In selecting the dimensions of
the SOM network, i.e. the number of fixed models, a
compromise should be made between the required
computational load and the accuracy of models.
Weighting constants of the performance index are
selected as D , E 1, O 0.65 , M = 30 and the
Hysteresis constant is G 0.8 . Pole placement
controller with integral action is employed as the
closed loop controller. The desired closed loop pole
is placed at 0.95.
Note 2: If the applied control technique does not
include an integral action, there will be bias in
tracking and steady state errors will be inevitable due
to slight model mismatches.
Closed loop response of pH neutralization plant for
big setpoint steps is shown in Fig.7. The result
illustrates that the MCSOM controller have faster
and more stable response comparing to the STR.
Some observations from the simulations are given
below:
1) The MCSOM improve the transient response of
the closed loop system compare to the STR. This
improvement is the consequence of switching to
more appropriate models during the transient
response. Switching in MCSOM skips the transient
time of adaptation needed in the STR. The switching
delay in MCSOM which is a result of the hysteresis
and the excitation condition can be adjusted using
the design parameters and are much lower than the
transient adaptation time in STR.
2) The STR suffer from oscillations as the output
approaches the high gain area of around pH = 8.5
due to the delay caused by adaptation process.
Unlike the STR, the MCSOM avoided the oscillation
by switching to a more appropriate fixed model from
the model bank.
Fig.8 shows the results from [3]. A multiple-model
PID controller was used to control a similar pH
neutralization plant. Performance of the switching
controller was compared to a multiple model
interpolation (MMI) controller which is a tuning
method which estimates the model parameters
whenever poor performance of the current controller
is detected. Results show that although the switching
controller has improved the response in case of
stability and speed, but the switching controller still
experiences very different performances around
different setpoints. For example around the high-gain
region of pH=8, oscillations are observed while
around the lower-gain area of pH=6, we have an
over-damped response.
1388 14 02 ,/ 03 !( 0
8
Design of Multiple Model Controller using SOM Neural
P. Bashivan, A. Fatehi
PH vs Command
9
PH
8
7
Uc
STR
MCSOM
6
5
2000
3000
4000
5000
6000
7000
6000
7000
8000
Control Signal
q (mlit/sec)
20
An improved switching algorithm based on the
excitation level of the process is suggested when
multiple model controller is applied to the process.
Simulation studies indicated better performance of
the multiple-model controller when this condition is
applied to the switching algorithm.
15
STR
MCSOM
10
2000
3000
4000
5000
8000
Index of Inloop Model
PH vs Command
20
9
15
8
PH
Model Index
25
10
7
Uc
Without Exc Cond
With Exc Cond
6
5
5
2000
3000
4000
5000
time (sec)
6000
7000
8000
2000
3000
4000
5000
6000
7000
8000
Control Signal
Fig.7 – Closed loop response of MCSOM and STR
controllers in presence of measurement noise
q (mlit/sec)
25
Without Exc Cond
With Exc Cond
20
15
10
5
2000
3000
4000
5000
6000
7000
8000
Index of Inloop Model
25
Without Exc Cond
With Exc Cond
20
15
10
5
2000
3000
4000
5000
time (sec)
6000
7000
8000
Fig.9 - Effect of adding the new excitation condition to the
switching algorithm on the closed loop response in
presence of measurement noise.
High Pass Filtered Input Signal
4
3
Excitation Threshold
Fig.8 – Results from [3].switching controller (—), MMI
re-tuning controller (- - -), setpoint and true values (…).
2
1
Fig.9 illustrates the effect of the excitation
condition on the number of unnecessary switches. A
threshold of 0.9 is considered for the excitation
condition as illustrated in Fig. 10. During the steady
state, the plant behavior does not change. However,
due to the effect of noise and similarity of the
behavior of some of the models, there might be some
unnecessary switches between the models. Using
excitation condition on the switches decreases the
unnecessary switches between the models and
smoothes the output signal.
0
-1
-2
Excitation Threshold
-3
-4
2000
3000
4000
5000
6000
7000
8000
Fig.10 - High pass filtered input signal and the excitation
condition
References
6- Conclusion
In this paper, a multiple model pole placement
control strategy via SOM which divides the
operating region of plant into sub-regions is
presented. The model set design problem in multiple
model controller is solved by using SOM. SOM
clusters the instantaneous models into some models
which stores in the bank of model. Simulation results
indicated superior performance compared to the STR
controller.
Journal of Control, Vol. 3, No. 2, Summer 2009
[1] Astrom, K. J. and Wittenmark, B., Adaptive Control,
2nd ed. Addison-Wesley, NY, 1995.
[2] Bashivan, P., Fatehi, A., December 2008, “Multiplemodel control of pH neutralization plant using the
SOM neural networks”, IEEE Conference &
Exhibition on Control, Communication and
Automation (INDICON), Kanpur, India, 11-13.
[3] Böling, J. M., Seborg D. E., Hespanha J. P., 2007,
“Multi-model adaptive control of a simulated pH
neutralization process”, Control Engineering
Practice, 15, 663-672.
1388 14 02,/ 03 !( 0
Design of Multiple Model Controller using SOM Neural
P. Bashivan, A. Fatehi
9
[4] Cho, J., Principe, J. C., Erdogmus, D., and Motter, M.
A., 2007 “Quasi-Sliding Mode Control Strategy
Based
on
Multiple-Linear
Models”,
Neurocomputing, 70, 960-974.
[5] Dougherty, D., Cooper, D., 2003, “A Practical
Multiple
Model
Adaptive
Strategy
for
Multivariable Model Predictive Control”, Control
Engineering Practice, 11, 649-664.
[6] Fatehi, A., Abe, K., Aug/Sept 1999, “Plant
identification by SOM neural networks”, The
European
Control
Conference
(ECC’99),
Karlsruhe, Germany, F190.
[7] Fatehi, A., Abe, A., October 2007, “Statistical
properties of multiple modeling by self-organizing
map (MMSOM),” The Mediterranean Journal of
Measurement and Control, 3, 4.
[8] Galán, O., Romagnoli, J. A., Palazoglu, A., Arkun,
Y., 2003, “Gap Metric Concept and Implications
for Multilinear Model-Based Controller Design”,
Ind. Eng. Chem. Res, 42, 2189-2197.
[9] Galán, O., Romagnoli, J. A., Palazoglu, A., 2004,
“Real-time implementation of multi-linear modelbased control strategies”, an application to a benchscale pH neutralization reactor. Journal of Process
Control, 14, 571–579.
[10] Hall, R. C., Seborg, D. E., “Modeling and SelfTuning Control of a Multivariable pH
Neutralization Process, Part I: Modeling and
Multiloop Control. In: Proc. ACC. Pitts-burgh. PA.,
1822-1828.
[11] Hugglund, T., Astrom, K. J., 2000, “Supervisory of
Adaptive Algorithms”, Automatica, 36, 8, 11711180.
[12] Kohonen, T., Self-Organizing Maps, Springer,
Berlin, 1995.
[13] Kraaijveld, M. A., Mao, J., Jain, A. K., 1992, “A
Nonlinear Projection Method Based on Kohonen’s
Topologypreserving Maps”, Proc. 11th IAPR Int.
Conf. on Pattern Recognition.
[14] Murray-Smith, R., Johnsen, T.A., 1997, “Multiple
Model Approaches to Modeling and Control”,
Taylor & Francis Inc, London.
[15] Narendra, K. S., Balakrishnan, J., 1997, “Adaptive
Control Using Multiple Models”, IEEE Trans. on
Automatic Control, 42, 2, 171-187.
[16] Peymani F., E., Fatehi, A., Khaki Sedigh, A.,
September 2008, "Automatic Learning in Multiple
Model
Adaptive
Control",
International
Conference on Control, UKACC, Manchester, UK.
Journal of Control, Vol. 3, No. 2, Summer 2009
1388 14 02 ,/ 03 !( 0
Journal of Control
Vol. 3, No. 2, pp. 10-16, Summer 2009
Robust H f Control of an Exerimental Inverted Pendulum using
Singular Perturbation Approach
R. Amjadifard1, M. T. H. Beheshti2, H. Khaloozadeh3, K. A. Morris4
1
Department of Computer Engineering, Faculty of Engineering, Tarbiat Moallem University, Tehran, Iran,
[email protected]
2
Department of Electrical Engineering, Faculty of Engineering, Tarbiat Modares University, Tehran, Iran,
[email protected]
3
Faculty of Electrical and Computer Engineering, K. N. Toosi University of Technology, Tehran, Iran,
[email protected]
4
Department of Applied Mathematics, University of Waterloo, Waterloo, Canada
[email protected]
(Manuscript received: Feb. 23 2009, accepted Jul. 25 2009)
Abstract: In this paper, robust H f control of an experimental system is considered. This system
consists of a pendulum free to rotate 360 degrees that is attached to a cart. The cart can move in one
dimension. The linearized model of the system is used and transformed to a linear diagonal form. The
system is separated into slow and fast subsystems. We consider the fast dynamics as disturbance and
this is used to design a H f controller for a system with lower order than the original system.
It is shown via a theorem that there is a state feedback controller such that the closed loop system will
be stable. Experimental results indicate that the performance is superior to the full-order LQR
controller previously used. Material presented at 16th IFAC world congress.
Keywords: H-infinity control, Singular Perturbation Methods, Robust performance.
> d$W* ! V 0P1 # .$/ K? E .@[ P1 V H f FC &C # :6;?
:R FK P1 ,!/ R) ! . !* V >
< $4 :? .> O' :? V > ( 360 t)u
,!
! ,!/ K? E cS $* !4 :"W .? VW94 !4 !
:"1 @ x ,!/ O!74
V!K ,!
> ,!/ , V @ ,9 .> ,!/ <% + P1 @ ( 1 : H f
,!/ P1 @ " P1 # W !! .@[ A .@ ! 1 C< P1 $( &<
.> ,!? <% .@[ P1 # : 7 > LQR O
74 ,!
.FC W 0#W4 .9/[ :"/ 0 H f :( 1- Introduction
Van der Schaft [1] indicated that in control of
nonlinear systems, if the H f control problem for the
linearized system is solvable, then one obtains a local
solution to the nonlinear H f control problem. One
problem with H f designs is that the order of the
controller is at least the order of the plant, and larger
if, as is common, weights are included in the design.
Corresponding Author: Roya Amjadifard
An approach to reduced order controller design
based on the idea that one can consider the fast
dynamics of a system as disturbances is first
introduced by Khalil [2] and then is discussed by
Yazdanpanah et al. and Yazdanpanah and Karimi
[3]-[4], in which the authors introduced a new
algorithm for the problem of robust regulation for
linear singularly perturbed systems via treating the
fast modes of system as uncertainty using the small
gain theorem. Then the authors in [5] extended the
method introduced in [3] to a class of nonlinear
Journal of Control, Iranian Society of Instrument & Control EngineersK.N. Toosi University of Technology
Robust H f Control of an Experimental Inverted Pendulum using Singular Perturbation Approach
11
R. Amjadifard, M. T. H. Beheshti, H. Khaloozadeh, K. A. Morris
affine systems. Also, in [6] and [7] robust stability
and disturbance attenuation for a class of linear
singularly perturbed systems has been considered. In
[8], problem of disturbance attenuation via H f
approach for nonlinear singularly perturbed systems
has been solved by considering the related HJI
(Hamilton-Jacobi-Isaacs) inequality, defining a
reduced Hamiltonian system, fast H -independent
PDE (Partial Differential Equation), and then
constructing the H f composite controller. And in
[9] the writers have shown that if the reduced order
system associated with the original system is
stabilizable or has uncertainties matched with the
input, then the closed loop reduced-order system has
the same property.
In the present paper, the fast stable part of the
system is considered as uncertainty and then the
controller is designed for the remaining part of the
system. The remaining slow subsystem has an order
less than the original one. The only information must
be known, is the H f norm of the fast subsystem.
The part of the system regarded as uncertainty is not
entirely arbitrary since the small gain theorem must
hold.
Most systems have a lower gain in high
frequencies than in the low frequencies and so this
approach has wide applicability. No other dynamical
information is required. This is advantageous since
in general the high frequency aspect of a model is
not well determined. With this idea, one can use the
H f method to design a robust controller using the
slow subsystem as the nominal plant.
The proposed method is applied to a flexible joint
robot manipulator ([5]) and the simulation results
showed the desired behavior of system.
In this work, the approach to an unstable system is
extended. The stabilization of an inverted pendulumcart is considered. First, the nonlinear part of the
system is eliminated, since it is stable and small.
Then the linearized model is transformed to Jordan
canonical form and the slow and fast modes are
separated. The stability of the controlled system is
proved through a theorem [5] and it has been verified
on an experimental apparatus. The performance is
shown to be superior to a linear quadratic regulator
previously implemented by Landry et al. [10].
force F (t ) can be applied to the cart in the x
direction. In Table 1 there is a complete list of
notation.
The equations of motion for the system are (which
is mentioned, e.g. by Landry et al., [10])
( M m) x Hx mlT cos T mlT 2 sin T F (t ),
mlx cos T 4 ml 2T mgl sin T 0.
(1)
3
Parameter values for the apparatus that is made by
Quanser Consulting Inc. [11] are given in Table 2.
Based on previous experiments, a value H 8 for
the friction parameter was used.
Using the state variables
X
( x1 , x2 , x3 , x4 ) ( x, x,T ,T)T
(2)
equations (1) can be written in first-order form as
X f ( X )
ª
º
x2
«
»
2
«
»
4F (t ) 4H x2 4mlx sin x3 3mg sin x3 cos x3
4
«
»
«
»
2
4(
M
m
)
3
m
cos
x
«
»
3
«
»
x4
«
»
«
»
2
« (M m) g sin x3 ( F (t ) H x2 )cos x3 mlx4 sin x3 cos x3 »
«
»
«
»
l ( 4 (M m) m cos2 x3 )
«¬
»¼
3
(3)
The force F (t ) on the cart is due to a voltage V(t)
applied to a motor:
F (t ) DV (t ) Ex (t ) .
(4)
2- System Definition
A pendulum is attached to the side of a cart by
means of a pivot that allows the pendulum to swing
in the xy-plane over 360 degrees. (See Fig. 1.) A
Journal of Control, Vol. 3, No. 2, Summer 2009
Fig. 1. Inverted pendulum system
1388 14 02 ,/ 03 !( 0
Robust H f Control of an Experimental Inverted Pendulum using Singular Perturbation Approach
12
R. Amjadifard, M. T. H. Beheshti, H. Khaloozadeh, K. A. Morris
Table 1. Notation
x(t )
Displacement of the centre of mass of the cart
from point O
T (t )
Angle the pendulum makes with the top vertical
M
Mass of the cart
m
Mass of the pendulum
L
Length of the pendulum
l
Distance from the pivot to the centre of mass of
the pendulum
P
Pivot point of the pendulum
F (t )
Force applied to the cart
A
ª0
«
«0
«0
«
«0
¬«
b
0
º
ª
4D
»
«
« m 4M »
»
«
0
« 3D »
»
«
«¬ l ( m 4M ) »¼
1
4(H E )
m 4M
0
3(H E )
l (m 4M )
0
3mg
m 4M
0
3(m M ) g
l (m 4M )
0º
»
0»
1»,
»
0»
¼»
(6)
It is well-known (e.g. as indicated by [10]) that for
the uncontrolled system (V(t)=0), the cart-pendulum
at rest in any upright position (x,0,nS, 0) is at an
unstable equilibrium point.
3- H f Controller design
The second term is due to electrical resistance in
the motor. The physical constants are
D
Km K g
Rd
, E
D 2 R.
y
The voltage V (t ) can be varied and is used to
control the system. A system of differential
equations can be written in a simpler form using the
normal form method. In general, the normal form
method is a series of nonlinear coordinate
transformations in order to eliminate or simplify the
equation nonlinearities. Although the transformations
are nonlinear functions of the state variables, they are
found by solving a sequence of linear equations [12].
In this paper using the Taylor expansion of f (X)
about the equilibrium point (upright position of
pendulum) we obtain the linearized model of the
system (3). Then, using the idea of application of the
normal form method ([13]-[14]), we apply a
similarity transformation to convert the linearized
model to a diagonal form.
The model for the controlled system linearized
about the upright position is then
X
AX bV (t )
where
A similarity transformation X Ty is used (see
also, [5] and [14]), where T contains the system
eigenvectors, to transform A into Jordan canonical
form. Equation (5) becomes
(5)
Jw BV (t )
Table 2. Parameter values of the inverted pendulum
Parameter
wM
Value
Description
0.360 Kg
Weight mass
0.455
Kg+ wM
Mass of the cart
0.210 Kg
Mass of the pendulum
0.61 m
Unknown
0.00767
V/(rad/sec)
Length of the pendulum
Acceleration due to
gravity
Viscous friction
Motor torque and back
emf constant
Kg
3.7
Gearbox ratio
R
d
2.6
M
m
L
g
H
Km
9.8 m/s
:
0.00635 m
where J
Motor
resistance
armature
Motor pinion diameter
T 1 AT is a diagonal matrix of system
eigenvalues and B
T 1b .
We can recognize the slow and fast dynamics of
the system equation (7), which is in a diagonal form,
and decompose it into two subsystems as
X 1
X
2
Journal of Control, Vol. 3, No. 2, Summer 2009
(7)
/11 X 1 B1u
/ 22 X 2 B2u
(8)
1388 14 02,/ 03 !( 0
Robust H f Control of an Experimental Inverted Pendulum using Singular Perturbation Approach
13
R. Amjadifard, M. T. H. Beheshti, H. Khaloozadeh, K. A. Morris
where /11
diag{0,5,4.74}, and / 22
18.33 .
The vector B is a permutation of elements of B1
and B 2 . Here X 1 indicates the slow dynamics of
system, and X 2 the fast dynamics of system. Also,
u indicates the input control V (t ) .
If we consider the slow subsystem as the nominal
system [5] with the controlled output Z
Z
ªI º
ª0 º
C1 X 1 D12 u : « » X 1 « »u
¬0 ¼
¬I ¼
and the measured output Y
Y
C 2 X 1 D21 X 2
0
0
D21
B1 º
D12 »» .
0 ¼»
dynamics will be considered as uncertainty
'.
This means rewriting the system (8) is needed so
that the fast dynamics appear as disturbance to the
nominal system.
A state transformation [ X 1 , X 2 ]T M [ X 1 , X 2 ]T is
applied to the system, where M -1 has the structure
M 1
M 12 º
.
M 22 »¼
/X B u
B
0 º
ª/
M 1 « 11
»M
0
/
22 ¼
¬
ªB º ªB º
M 1 « 1 » « 1 » .
¬ B2 ¼ ¬ B2 ¼
Note that the coefficient of X 1 in the fast subsystem is zero. Also, the equations for the slow subsystem, or nominal block, become
ª /11
«
P ~ « C1
« C2
¬
/12
D11
D21
B1 º
»
D12 »,
0 »¼
Z
C1 X 1 D11 X 2 D12 u ,
Y
C 2 X 1 D21 X 2
C1
1
C1 M 11
, C2
1
C 2 M 22
,
D11
1
1
M 12 M 22
C1 M 11
,
D21
1
1
D21 C 2 M 11
M 12 M 22
.
'( s ) ( sI / 22 ) 1 B2 , '
where J 1
f
d J 1,
0.301 .
Since
' ( s ) C 2 ( sI / 22 ) 1 B2
C 2 ( sI / 22 ) 1 B2 , the
H f -norm of the uncertainty block is J 1 = 0.301.
The H f controller design problem for the system
slow sub-system. The H f controller will be
designed here via state feedback (or fullinformation). The next step is to determine
J 2 min J , where J 2 indicates the H f -norm of
the controlled slow sub-system, and a corresponding
controller that achieves this.(9)
The transformation M must be chosen so
By trial and error, a suitable
transformation was found:
J 1 .J 2 1 .
where
/
(9)
shown in Fig. 2, will lead to a H f controller for the
The equations of system (8) after transformation
are (see Fig. 2)
X
/ 22 X 2 B2u
In Fig. 2, Z is the input to the uncertainty block.
The fast dynamics are exponentially stable.
Indicating the transfer function by
As mentioned earlier the stable subsystem with fast
ª M 11
« 0
¬
X2
where
(where C2 is chosen to be I and C1 , C 2 , D12 , D21
are all matrices with proper dimensions), then the
nominal system with full information and with no
disturbance can be written as
ª/11
P ~ «« C1
¬« C 2
uncertainty block, is
ª /11
«
¬ 0
/12 º
»,
/ 22 ¼
The new equation for the fast sub-system, or
Journal of Control, Vol. 3, No. 2, Summer 2009
M
ª I3
«0
¬ 3u1
M 12 º
, M 12
1 »¼
0.0095[1 1 1]T .
It is straightforward to verify that the slow subsystem is stablizable and detectable. As indicated by
1388 14 02 ,/ 03 !( 0
Robust H f Control of an Experimental Inverted Pendulum using Singular Perturbation Approach
14
R. Amjadifard, M. T. H. Beheshti, H. Khaloozadeh, K. A. Morris
Doyle et al. [15], it then follows that J 2 is the
smallest value of J such that the eigenvalues of the
Hamiltonian matrix
H
Proof: The state space realization of the original
closed loop system with controller (10) can be
written as
ª X º ª / B K
1
« 1 » « 11
0
«¬ X 2 »¼ ¬
ªX º
N « 1 » h.o.t.
¬X2 ¼
ª /11
J 2 /12 /T12 B1 B1T º
«
»
T
/T11
¬« C1 C1
¼»
are not on the imaginary axis.
X 2
/ 22 X 2 B2 u
ªX º
Z « 1»
¬u¼
X2
Nominal
X 1
/11 X 1 /12 X 2 B1u
Y
u
H f controller
/12 º ª X 1 º
» « » h.o.t.
/ 22 ¼ ¬ X 2 ¼
(10)
where h.o.t. denotes higher order terms in the
Taylor expansion. It is obvious that the eigenvalues
of N contain the eigenvalues of matrices / 22 and
/11 B1 K . As assumption, /22 is a stability matrix;
also, based on Theorem 1, the controller (10) is a
stabilizing controller for the nominal system P ,
thus /11 B1 K is a stability matrix. It follows from
the small gain theorem that the feedback connection
of two input-output stable systems will be inputoutput stable provided the J 1 .J 1 or J J 1 1 , in
which J 1 is the Hf norm of uncertainty block, and
J
Fig. 2. Block diagram of system with fast dynamics as
uncertainty
Theorem 1 [15]: Under the standard assumptions
of stabilizability-detectability of [16], for a
given J ! 0 , there is an internal stabilizing controller
such that TzX
2
f
d J , if and only if X f is a
positive semi-definite solution of algebraic Riccati
equation
T
T
/11
X f X f /11 X f (J 2 /12 /12
B1 B1T ) X f
C1T C1
0
and the matrix / 11 ( B1 B1T J 2 / 12 /T12 ) X f is a
stability matrix. Then the related controller will be in
the form
u (t )
is the H f norm of nominal closed loop system
with controller that is greater than J 2 determined
from the Hamiltonian matrix. Thus, the linearized
approximation of the whole dynamics is
asymptotically stable, and therefore, using
Lyapunov's linearization theorem, the original
system will be locally asymptotically stable.
B1T X f X 1 (t )
KX 1 (t ) , K
(10)
The following theorem guarantees the stability of
the closed loop system.
4- Experimental Results
The H f controller of equation (10) is applied to
the pendulum system. Only the position of the cart
and the pendulum angle can be measured. An
observer is required to obtain x and T . In order to
compare to the results shown by Landry et al. ([10]),
the same Luenberger observer was used. The same
linear-quadratic regulator (LQR) used in [10] was
used as a comparison for the H f state-feedback
controller designed using the slow-fast approach in
this paper (or for simplicity, the 'slow-fast
controller').
The controlled pendulum angle and the cart
position are shown in Fig. 3. The equilibrium x 1
(cart position) is arbitrary, as can be seen from
equation (3).
Theorem 2 [5]: For the original system (3) and
1
with J 2 J J 1 , there is a state feedback
controller as (10) such that the closed loop system
will be locally asymptotically stable.
Journal of Control, Vol. 3, No. 2, Summer 2009
In Fig. 4, the input controller signals, produced by
the slow-fast and the LQR controllers, are shown.
Although the performance of the two controlled
systems is similar, the slow-fast controller achieves
1388 14 02,/ 03 !( 0
Robust H f Control of an Experimental Inverted Pendulum using Singular Perturbation Approach
15
R. Amjadifard, M. T. H. Beheshti, H. Khaloozadeh, K. A. Morris
this performance with a smaller controller signal and
also, with a system of lower degree.
The response of the controlled pendulum system to
a disturbing “tap” on the pendulum controller was
investigated for each controller. This “tap” was a
disturbing impulse which was applied to the
pendulum controlled by both controllers, similarly.
Fig. 5 shows the angular position of pendulum and
the cart position under this disturbance.
In Fig. 6, the behavior of the two controlled
systems with a time delay of 0.035 seconds in the
controller output is shown.
The performance of the slow-fast controller is
superior to that of the LQR controller for both the
disturbed and delayed systems.
Fig. 3. System behavior via the two controllers without any
additional disturbance or noise. (The equilibrium
cart position, is arbitrary)
x1 , i.e.
Fig. 5. The behavior of pendulum system with an
additional disturbance on pendulum via the two
controllers. (The equilibrium x1 , i.e. cart position, is
arbitrary)
Fig. 4. The input controller signal, produced by the two
controllers in a condition without any additional
disturbance or noise.
Journal of Control, Vol. 3, No. 2, Summer 2009
1388 14 02 ,/ 03 !( 0
Robust H f Control of an Experimental Inverted Pendulum using Singular Perturbation Approach
16
R. Amjadifard, M. T. H. Beheshti, H. Khaloozadeh, K. A. Morris
[2] Khalil, H. K., Nonlinear Systems, (2nd Ed.), chapter
10, 433-435. Prentice Hall, New Jersey, 1996.
[3] Yazdanpanah, M. J., Patel, R. V. and Khorasani, K.,
1997, “Robust Regulation of a Flexible-Link
Manipulator Based on a New Modeling Approach,”
Proceeding of the 36th Conf. on Decision and
Contr., pp1321-1326, USA.
[4] Yazdanpanah, M. J., Karimi, H. R., 2002, “The
Design of H f Controller for Robust Regulation of
Singularly Perturbed Systems,” AmirKabir Journal,
No. 49, pp. 9-23.
[5] Amjadifard, R., Beheshti, M.T.H., Yazdanpanah, M.
J., Momeni, H. R., 2004, ”Robust Regulation of a
Nonlinear Flexible Joint Robot Manipulator using
Slow-Fast Decomposition Approach,” Journal of
Engineering Faculty- Ferdowsi university of
mashhad, 16th year, 1.
[6] Karimi, H. R., Yazdanpanah, M. J., 2001, “Robust
stability disturbance attenuation for a class of
uncertain singularly perturbed systems,” Int. J. of
Control, Automation and system, 3, 3, 164-169.
[7] Karimi, H. R., Yazdanpanah, M. J., Patel, R. V. and
Khorasani, K., 2006, “Modeling and Control of
Linear Two-Time Scale Systems: Applied to Single
Link Flexible Manipulator,” Journal of Intelligent
& Robotic Systems, 45, 3, 235-265.
Fig. 6. Pendulum system behavior with a transport delay of
0.035 sec. in the controller output. (The equilibrium
i.e. cart position, is arbitrary)
x1 ,
5- Conclusion
In this paper robust stabilization of an experimental
pendulum system using slow-fast decomposition
approach is considered. Considering the fast
dynamics as norm-bounded uncertainty, a H f
controller for the reduced order system (slow
subsystem) was designed. It was shown through a
theorem that the closed- loop system would be
stable. The resulting controller was implemented.
Experimental results indicate that the performance is
superior to the full-order LQR controller previously
used, i.e. the slow-fast controller achieves the
performance with a smaller controller signal and
also, with a system of lower degree.
References
[1] Van der Schaft, A. J., 1992, “ L2 -Gain Analysis of
Nonlinear Systems and Nonlinear State Feedback
Hf
control,” IEEE Trans. on Automatic Control,
37, 6, 770-784.
Journal of Control, Vol. 3, No. 2, Summer 2009
[8] Fridman E., 2001, “State Feedback H f Control of
Nonlinear
Singularly
Perturbed
Systems,”
International Journal of Robust and Nonlinear
Control, 11, 12.
[9] Corless, M., Garofalo, F., Glielmo, L., 2007, “Robust
Stabilization of Singularly Perturbed Nonlinear
Systems”, International Journal of Robust and
Nonlinear Control, 3, 2, 105-114.
[10] Landry, M., Campbell, S. A., Morris, K. A. Aguilar,
C., 2005, “Dynamics of an Inverted Pendulum with
Delayed Feedback,” SIAM Jour. on Applied
Dynamical Systems, 4, 2, 333-351.
[11] Quanser Consulting Inc. IP-02 Self- Erecting,
“Linear Motion Inverted Pendulum”, Quanser
Consulting Inc, 1996.
[12] Kahn, P. B. and Zarmi, Y., Nonlinear Dynamics:
Exploration Through Normal Forms, John Wiley &
Sons. 1998.
[13] Khajepour, A., Golnaraghi, M. F., Morris, K. A.,
1997, “Application of Centre Manifold Theory to
Regulation of a Flexible Beam,” ASME Journal of
Vibration and Acoustics, 119, 158-165.
[14] Khajepour, A., 2000, “Nonlinear Controller Design
for Asymmetric Actuators”, Journal of Vibration
and Control, 6, 1-23.
[15] Doyle, J. C., Glover, K., Khargonekar, P. P.,
Francis, B. A., 1989, “State Space Solutions to
Standard H 2 and H f control problems,” IEEE
Trans. on Automatic Control, 34, 831-846.
[16] Green, M., Limebeer, D. J. N., Linear Robust
Control, Chapter 8, Prentice Hall, 1995.
1388 14 02,/ 03 !( 0
Journal of Control
Vol. 3, No. 2, pp. 17-24, Summer 2009
An Algorithm for Constructing Nonsmooth Lyapunov Functions
for Continuous Nonlinear Time Invariant Systems
1
Alireza Faraji Armaki1, Naser Pariz2, Rajab Asgharian3
PhD student, Electrical Engineering Department, Ferdowsi University of Mashhad, [email protected]
2
Associate professor, Electrical Engineering Department, Ferdowsi University of Mashhad, [email protected]
3
Professor, Electrical Engineering Department, Ferdowsi University of Mashhad,
[email protected]
(Manuscript received: Jun. 22 2009, accepted Aug. 28 2009)
Abstract: This paper presents an algorithm based on the Generalized Lyapunov Theorem (GLT) for
constructing nonsmooth Lyapunov Function (LF) for nonlinear time invariant continuous dynamical
systems which can be differentiable almost every-where. A new method is firstly defined that a
neighborhood of the equilibrium point (origin) is partitioned into several regions by means of the
coordinate hyperplans (axes) and system state equations (nullclines); hence, the number of regions is
a function of number of system states. Then, this method selects a LF in each region by original
nonlinear model of system, based on the several proposed analytical Notes. These Notes select LF’s
and solve continuity problem of them on the boundaries of regions in more cases. The existing
methods that use piecewise model of system in each region for constructing piecewise LF are
approximate and computational, but, the defined method is completely exact and analytic. The
different steps of this method are proposed by means of a non-iterative algorithm for constructing a
nonsmooth continuous Generalized Lyapunov Function (GLF) in whole neighborhood of the origin.
The ability of this algorithm is demonstrated via a few examples for constructing LF and analyzing
system stability.
Keywords: Stability analysis, Continuous nonlinear dynamical systems, Generalized Lyapunov
theorem, Nonsmooth continuous Lyapunov functions.
S W :"1 : $ q$& U$4 >) >"( 0K P*4 q$& d $.& &C # :6;?
0[ I$4 $/ K* !!( / ! .$/ p 0!/ k }C ) !$4 @ s $ R)
,!/ :! P1C4 <$ !*4 # 0? P1C4 < #!u P1 >&< BL* $D I$4 (!7) *4 RC .1
0!
! q$& U4 V < P1 R)S + ! @ ,9 c # x .> >&< :s !*4 @ *4
$($ :"/ q6) .!/ D4 W !u 7 <$ :@ : "[ $ $ q$& U$4 # rH 6
c # 0!1 47D 7C4 !
,9 P1 : W4 ! @ < : W4 q$& U$4 >) >"( ,!/ K* c d !7 .1 O
: $ $ K P*4 q$& U4 V >) O< .> D4 , !u "1 :! OD4 q$& U$4 >) >"( [ $4 ,!/ ,!$/ W4 S P$.& V Y&
.> ,!/
$ $ K P*4 q$& U4 0K P*4 q$& 0$ R)S W P10:! OD4 :( 1- Introduction
Lyapunov theorem is used for system stability
analysis, which is an important issue in nonlinear
dynamical systems theory. A main advantage of this
theorem is reduction of system stability analysis with
several dimensional equations, to the study of a LF
with one-dimensional equation. There is no a
systematic approach to choose LF for any nonlinear
system, and the choice of LF is not unique.
Corresponding Author: Alireza Faraji Armaki
Several nonsmooth Lyapunov stability theorems
are defined in the articles. These theorems can be
classified in two main categories. The first category
determines the generalized derivative of a
nonsmooth LF on its nonsmooth surfaces, via
differential inclusion or similar approaches. In these
theorems, the important step is to verify the
generalized derivative of a nonsmooth LF on its
nonsmooth surfaces. For example, [5] defined a
nonsmooth Lyapunov stability theorem for a class of
Journal of Control, Iranian Society of Instrument & Control EngineersK.N. Toosi University of Technology
18
An Algorithm for Constructing Nonsmooth Lyapunov Functions for Continuous Nonlinear Time Invariant
Alireza Faraji Armaki, Naser Pariz, Rajab Asgharian
nonsmooth Lipschitz continuous LF’s using
Filipov’s differential inclusion and Clarke’s
generalized gradient. Based on the latter paper, [6]
constructed the LF’s for several complicated
systems.
systems. It approximates a PM by a switching fuzzy
model in each quadrant. The stability is analyzed via
a derived Piecewise Quadratic (PWQ) LF for each
region. The parameters of quadratic matrix are
solved by Linear Matrix Inequalities (LMI).
The second category of nonsmooth Lyapunov
stability theorems does not determine the generalized
derivative of a nonsmooth LF on its nonsmooth
surfaces. These theorems analyze system stability
without calculation of gradient vector to the system
solutions. Several of these theorems are mentioned
below.
Johansson and Rantzer defined a method for time
invariant nonlinear systems with Piecewise Affine
(PWA) dynamic model [3], [4]. In this method
around the origin is divided into some polyhedral
cells with pair-wise disjoint interior, then, a PWQ LF
is computed in each of them. The search for a PWQ
LF is formulated as a convex optimization problem
in terms of LMI.
[1] proved a GLT for nonlinear dynamical systems,
in which, LF can be discontinuous except for the
origin, so, all regularity assumptions are removed for
the system dynamics and LF’s. Our algorithm in this
paper is proposed using this GLT.
[7] proved a version of the Lyapunov's theorem for
time invariant systems of ordinary differential
equations, whose right hand side is continuous, but
not Lipschitz continuous, in general. For such
systems, stability cannot be characterized in general
by means of smooth LF’s.
[8] defined another theorem for constructing weak
GLF for time invariant continuous systems. In
nonsmooth Lyapunov stability theorems, the LF’s
can be nonsmooth except for the origin. Therefore,
based on these theorems, nonsmooth LF’s can be
constructed for both continuous and discontinuous
systems.
A number of articles have dealt with the continuity
type for LF, for example; [2] proved for nonlinear
systems, which are at least continuous, that the
existence of a continuous LF does not imply the
existence of a locally Lipschitz continuous LF, and
also the existence of a Lipschitz continuous LF
doesn’t imply the existence of continuously
differentiable LF.
The nonlinear systems can be analyzed by
partitioning the state space into several divisions. By
this method, firstly, in each division a Piecewise
Model (PM) of the original nonlinear system is
selected, and using it, a LF is constructed in each
region. After that, the constructed LF’s under special
conditions are combined, and a piecewise LF for the
PM of the whole system is obtained. This method
should finally prove this obtained piecewise LF is
useful for stability analysis of original nonlinear
system.
Various applications of this method have been
reported in the literature; for example; [9] obtained a
switching LF for a class of nonlinear continuous
Journal of Control, Vol. 3, No. 2, Summer 2009
[10] defined a construction method of PWQ LF for
a simplified Piecewise Linear (PWL) model of the
original nonlinear system. This method divided
around the origin into a lot of simplices, then,
computes a PWQ LF in each division by means of
the variable gradient method.
[11] proposed an algorithm for constructing a PWA
LF for nonlinear continuous time invariant ordinary
differential equations in a family of simplices by
linear programming.
[12] considered a parametric PWL model of
nonlinear system, over a simplicial partitions in an
area around the equilibrium point. It constructed a
PWL LF using linear programming methods.
[13] computed global LF for nonlinear systems by
means of radial basis functions.
All PM methods in above, have these
disadvantages;
they
are
approximate
and
computational, also, the result of the system analysis
depends on the state space partitioning. To obtain
sufficient resolution in the analysis, it is often
necessary to refine an initial partition. Such
refinements can be targeted towards increasing the
accuracy of the model, or towards increasing the
flexibility of the LF computations.
This paper describes a non-iterative algorithm,
which is introduced for constructing nonsmooth
continuous GLF for nonlinear time invariant
continuous systems that can be differentiable almost
every-where. The proposed algorithm is based on a
GLT in [1].
This algorithm has three main stages. In the first
stage, it defines a method in accordance to PM
method for dividing neighborhood of the origin into
several regions by means of coordinate hyperplans
(axes) and state equations (nullclines), therefore, in
this method the number of regions is a function of
number of system states.
1388 14 02,/ 03 !( 0
An Algorithm for Constructing Nonsmooth Lyapunov Functions for Continuous Nonlinear Time Invariant
Alireza Faraji Armaki, Naser Pariz, Rajab Asgharian
In the second stage, this method constructs a LF in
each region by means of the original nonlinear model
of system and several analytical Notes. Unlike the
existing PM methods, that use an approximate model
of nonlinear system, this method is completely exact.
Also, the existing PM methods are computational,
but this method is analytic.
In the final stage, it combines selected LF’s and
constructs a nonsmooth continuous GLF with a
condensed formula based on a proved theorem.
The restrictions of this algorithm are; original
selection of LF’s for regions, and then, continuity of
LF’s on boundaries of regions, hence, the Notes are
proposed to solve these restrictions in more cases.
In the next section, the mathematical framework
for this paper is given. A new method for
partitioning the neighborhood of the origin into
several regions is presented in section 3. Section 4
explains construction of GLF. A proposed algorithm
for obtaining GLF is defined in section 5. The
capability of this algorithm is illustrated, when
employed on two examples, in section 6.
2- Mathematical framework
Consider a nonlinear time invariant continuous
dynamical system (1), which can be differentiable
almost every-where.
x f (x(t)), t t0, x(t)DR , f : DoR , f (0) 0D (1)
n
n
where D is an open set and x : T R o D is said to be
a solution to (1) on the time interval T , providing
x(t ) satisfies (1) for all t T . f is such that the
solution x(t ) is well defined on T [0, f) , that is,
assume, for every y D there is a unique solution
x(t ) of (1) on T , such that x(0) y , and all the
solutions x(t ) , t t 0 are continuous functions of the
initial conditions x0 x(0) D [1]. GLF is lower semicontinuous and differentiable almost every-where.
Two definitions and a theorem are recalled from [1],
below.
Definition 1 [1]: A function V : D o R is lower
semi-continuous on D , if for every sequence
then
^xn `fn 0 D such that, lim nof xn x ,
V ( x ) d lim inf nof V ( xn ) .
Definition 2 [1]: A lower semi-continuous,
positive definite function V (x) , which is continuous
at the origin, and satisfies V ( x(t )) d V ( x(W )) for
Journal of Control, Vol. 3, No. 2, Summer 2009
19
all t t W t 0 is called a GLF.
Theorem 1 [1]: Consider the nonlinear dynamical
system (1) and let, x(t ) , t t 0 , denotes the solution
to (1). Assume that, there exists a lower semicontinuous, positive-definite function V : D o R such
that V (x) is continuous at the origin and
V ( x(t )) d V ( x(W )) for all t t W t 0 . Then the zero
solution x(t ) { 0 is Lyapunov stable.
3. Partition method (definition of region)
Let, the coordinate hyperplans (axes) xi 0 , and
nullclines x i f i 0 , i ^1,2,..., n`, partition an
open set D R n in a neighborhood of the origin into
several regions, where each region is denoted
by R j , j ^1,2,..., m`. Obviously, a common boundary
of two neighboring regions is a coordinate hyperplan
or a nullcline. If a nullcline is along a coordinate
hyperplan, then the coordinate hyperplan is
considered. The common vertex of all regions is the
origin.
Each
region
has
common
n
boundaries S ji Rj Rji , which are the coordinate
hyperplans or nullclines with its neighboring
regions R ji .
4. Construction of GLF
For constructing GLF for (1), a smooth LF is
selected in each region; hence, each LF is nonincreasing within its corresponding region.
Moreover, if all neighboring LF’s be equal on their
common boundary, therefore, the condition
V ( x(t )) d V ( x(W )) for all t t W t 0 is satisfied, so, one
can use theorem 1 for constructing GLF.
Assume, v j (x ) in R j satisfies (2),(3).
j , v j ( x) : R j D R n o R, v j (0) 0
(2)
x Bj B Rj : (vj (x) ! 0 for x z 0) andvj (x) d 0 (3)
x Bj
B Rj : (vj (x) ! 0 for x z 0) andvj (x) 0 (3’)
Where B is an open set in neighborhood of the
origin and j 0Bj B D .
x S jk B : v j ( x) vk ( x)
(4)
If (4) is satisfied on all common boundaries of
regions, then all neighboring LF’s are continuous on
1388 14 02,/ 03 !( 0
20
An Algorithm for Constructing Nonsmooth Lyapunov Functions for Continuous Nonlinear Time Invariant
Alireza Faraji Armaki, Naser Pariz, Rajab Asgharian
their common boundary S jk , in B .
proper
Definition 3: A proposed parametric LF is a timeinvariant smooth function v j (x ) , in R j , that satisfy
(2) and (3).
At first, v j (x ) is often selected for more regions as
v j ( x) D hl (x) ,D R , where,
hl , l 1,2
is a time-
invariant smooth function in R j .
Definition 4: A proper LF is a proposed parametric
LF which satisfies (4) at some common boundaries
of its region.
Definition 5: A special LF is a proper LF which
satisfies (4) on all common boundaries of its region,
and its parameters are identified.
Definition 6: An orthant is the n-dimensional
generalization of the two dimensional quadrant and
three dimensional octant.
To construct GLF, the following steps must be
carried out: firstly, proposed LF’s are chosen for
more regions. Secondly, using these proposed LF’s,
proper LF’s are constructed. Special LF’s are
obtained by means of the proper LF’s, and finally, a
GLF is defined using the special LF’s. Each kind of
boundaries, which are the coordinate hyperplans,
nullclines or both of them, will provided different
relationships for LF’s. An algorithm is proposed for
constructing GLF.
LF’s
for
Divide a neighborhood of the origin into several
regions by the coordinate hyperplans and nullclines.
Select the lowest order of all LF’s equal together to
satisfy (4) on the coordinate hyperplans, else, the
continuity of GLF is not provided on them.
Select proper LF for regions, which are on either
side of the nullclines that aren’t along the coordinate
hyperplans by the next Note.
Note 1: Let, a nullcline S jk : xi
boundary of R j and Rk
0 , be the common
and it isn’t along the
coordinate hyperplans. Let, f i1 ( x ) and f i 2 ( x) be
proposed LF’s in R j and Rk , respectively, such
that, f i1 ( x) f i 2 ( x) f i ( x) xi
(5)
then, v j ( x) D jk f i1 ( x) and vk ( x) D jk f i 2 ( x) , D jk R are
Journal of Control, Vol. 3, No. 2, Summer 2009
because,
fi (x) 0
so, x S jk B fi1(x) fi 2 (x) 0 f i1 ( x)
thus, v j ( x) vk ( x) and holds (4) true.
f i 2 ( x) ,
The previous step offers n LF’s for a region whose
all boundaries are nullclines which are not along the
coordinate hyperplans. For such a region, compare
the offered LF’s with LF’s of its neighboring
regions, and then, for this region select a LF equal to
one of the LF’s of its neighboring regions.
Select proposed LF for a region whose boundaries
are only coordinate hyperplans, by the next Note.
Note 2: Suppose, R j is a region whose boundaries
are the coordinate hyperplans S ji : xi
0 , this
region is an orthant. Let, LF’s of all neighboring
regions of R j , v ji , are selected by the previous
is a LF for
steps. Therefore, each v ji ( x)
x
0
i
S ji : xi
0 in B .
To satisfy (4), v ( x)
j
x
i
0
must be satisfied
v ji ( x)
xi 0
for all S ji : xi 0 in B . It imply that, (6) can satisfy
(4) on all common boundaries of
R j by adding
some statements with each v ( x) or a selection of
ji
x 0
i
appropriate parameters for LF’s in the next step.
v j ( x)
¦b
ji
v ji ( x)
i 1
Please, trace each step with its corresponding step
in examples, to illustrate algorithm.
Rk ,
fi1(x) fi2 (x) fi (x) xi and xS jk B xi
n
5. Proposed algorithm
and
Rj
xi 0
(6)
, b ji R (when, n 2 , if bj1 bj 2 1, then v ( x)
j
x
1
, v j ( x)
x2 0
v j 2 ( x)
x2 0
0
v j1 ( x )
,
x1 0
, hence, v j ( x) v j1 ( x) v j 2 ( x)
x 0
x
1
2
0
satisfies (4) in R j .) Moreover, if (6) satisfies (3) in
R j , then, (6) is a proper LF on this orthant.
If the following Note is satisfied on all coordinate
hyperplans, then parameters of LF’s and special
LF’s are specified
Note 3: Let, R j QJ
and
Rk QK
be two
neighboring regions, where Q J and QK denote
orthants and S jk : xi 0 is their common boundary.
All LF’s in Q J and QK are already selected in
previous steps.
Assume, d jk (x) d j (x) dk (x)
(7)
1388 14 02,/ 03 !( 0
An Algorithm for Constructing Nonsmooth Lyapunov Functions for Continuous Nonlinear Time Invariant
Alireza Faraji Armaki, Naser Pariz, Rajab Asgharian
Such that, d j (x) vj (x) , dk (x) vk (x) .
x 0
x 0
i
i
If a selection of appropriate parameters for vj (x)
and vk ( x) satisfies d jk ( x) 0 , then these proper LF’s
are continuous on S jk : xi 0 , and holds (4) true.
Else if, the lowest order of vj (x) and vk ( x) is deleted
in d jk (x) by a selection of appropriate parameters for
them, then, the value of d jk (x) is smaller than the
value of each LF in neighborhood of the origin (note
that, the lowest order of all LF’s for system must be
equal).
Therefore, by adding d jk (x) with all LF’s in QK
(or d jk (x) with all LF’s in QJ ), new LF’s in Q J and
QK are constructed, as the new constructed v j (x) and
vk (x) are continuous on S jk : x i
0 , and holds (4)
true.
21
and also, if it satisfies (4) on all common boundaries
of the regions in B , then, (8) is a GLF for (1), and
the origin is Lyapunov stable. Moreover, If all
special LF’s satisfy (3’) within their regions, then,
origin is asymptotically stable.
Proof:
* Since, V (x) satisfies (2), j , v j (0) 0 V (0) 0 ,
therefore, V (x) is continuous at the origin.
* Since, V (x) satisfies (3),
x B j B : V ( x) v j ( x) ! 0 for x z 0 , hence, V (x) is
a positive-definite function, also, x B j , v j ( x) d 0 ,
but, x S jk ^0`, V (x) is non-differentiable in
general, therefore, Vf (x) isn’t often defined on the
$
boundaries. Thus, x B j B : V f ( x) v j ( x) d 0 , i.e.
V (x ) is differentiable and non-increasing within all
regions in B .
If this Note is repeated on all coordinate
hyperplans, then, parameters of LF’s and all special
LF’s may be identified.
V ( x(t )) d V ( x(W )) for all t tW t 0 for any x0 B .
If the special LF’s for all regions are identified,
then, construct a nonsmooth continuous GLF for
system (1) by following Note.
* Since, V (x) has a zero minimum value in B , for
every sequence ^xn `fn 0 B , so, limnof xn 0. Hence,
Note 4: A nonsmooth continuous function is
constructed by combination of the special LF’s,
v j ( x), j ^1,..., m`
m
V ( x)
¦ v ( x)< ( x)
j
< j ( x)
j
j 1
1 x B j
®
¯0 x B j
(8)
where < j ( x) is a characteristic function.
(9) defines derivative of (8) almost every where.
V f ( x )
m
¦ v ( x)< ( x)
j
j
a.e.
(9)
j 1
According to theorem 2, V (x) in (8) is a GLF, and
the origin is (asymptotically) stable.
Theorem 2: Consider the nonlinear dynamical
$
system (1). Let, B D be an open set, 0 B where is
$
the interior of B and B j be the interior of B j ,
0 B j R j . Suppose, D is divided by the coordinate
hyperplans and nullclines of system into several
regions R j .
If V (x) in (8) which is constructed by the special
LF’s, satisfies (2) and (3) within all regions in B ,
Journal of Control, Vol. 3, No. 2, Summer 2009
* Since, x S jk B : V (x) v j (x) vk (x) in (4), thus,
infnof V ( xn )
exists and lim inf nof V ( xn ) t V (0) 0 , i.e.
V (x) is a lower semi-continuous function.
* According to theorem 1, since, function V : D o R
is lower semi-continuous, positive-definite and
continuous at the origin, and moreover,
V (x(t)) d V (x(W )) for all t t W t 0 , (8) is a GLF for (1) in
B and x(t ) { 0 is Lyapunov stable.
* Moreover, If each special LF’s satisfies (3’)
$
within its region x B j B : V f ( x) v j ( x) 0 , it means
that V (x) is decreasing within all regions in B ,
therefore, V f ( x) 0 a.e. along the system solutions in
B.
* In addition, the special LF’s satisfy (4) on all
common boundaries, x S jk B : V (x) v j (x) vk (x) ,
so, V (x(t)) V (x(W )) for all t t W t 0 for any x0 B .
* V (x) in (8) has a zero minimum value in B , if
t o fV (x(t)) o0 , that it means the origin is
asymptotically stable.
(18)
1388 14 02,/ 03 !( 0
22
An Algorithm for Constructing Nonsmooth Lyapunov Functions for Continuous Nonlinear Time Invariant
Alireza Faraji Armaki, Naser Pariz, Rajab Asgharian
6- Examples
In this section, the GLF is constructed for two
systems. The stability of the origin in these examples
is approved by simulation with MATLAB software.
These examples demonstrate ability of the proposed
algorithm for system stability analysis.
Similarly, since, this function satisfies (3) in R3 ,
it’s a proper LF for this region.
Since, system is two dimensional; the continuity of
LF’s on all axes is provided in the previous step.
Thus, D 23 , D 56 R , the special LF’s are specified.
By assumption, D 23 D 56 1 ,
v1 ( x) x1 4 x2 , v2 ( x ) 4 x2 sin 2 ( x1 ) ,
v3 ( x) 3x1 , v4 ( x) 3x1 4 x2 ,
Example 1:
x1 f1 ( x) x2 x1 tan(x22 )
®
2
¯ x2 f 2 ( x) (sgn(x1 ) 2) x1 4sat( x2 ) sin ( x1 )
0.2 d sat ( x) d 0.2
6
V ( x)
Figure 1 shows a neighborhood of the origin for
this continuous system. The simulation in figure 2
shows that the system is stable. The proposed
algorithm is implemented for constructing GLF.
The neighborhood of the origin is divided by the
axes and nullclines into 6 regions.
In the second quadrant, on x 2
4 x2 sin 2 ( x1 )
v2 ( x) D 23 f 21 ( x)
v3 ( x) D 23 f 22 ( x)
D 23 (4 x2 sin 2 ( x1 ))
,
3D 23 x1 .
v5 ( x) D 56 f 21 ( x)
D 56 (4 x2 sin 2 ( x1 ))
,
D 56 x1 .
No region exists with nullcline boundaries.
Consider the first and third quadrants of this two
dimensional system:
In the first quadrant,
v2 ( x ) x
1
0
2
0
0
6
¦v (x)< (x) 0
j
j
3D 23 x1 , v5 ( x) x
1
a.e.
j 1
It is a GLF for the system and the origin is
asymptotically stable.

°° x1
®
° x 2
°¯
x2
f1
3x1 x1 f2
x1
x2
1 x1 x2
1 x12
Two functions f1 (.) and f 2 (.) are continuous, but,
f1 (.) is non-differentiable on the axes. Two figures
3, 4 show the neighborhood of the origin for this
stable system and its phase plan, respectively. The
proposed algorithm is used for constructing GLF for
the system.
The neighborhood of the origin is divided by the
axes and nullclines into 8 regions.
In the first quadrant, x2 0 x1 x2 x1 x22
0
4D 56 x2 ,
so, v4 ( x) 3D 23 x1 4D 56 x2 , satisfies (4).
Journal of Control, Vol. 3, No. 2, Summer 2009
D12 ( x1 )
,
D12 ( x2 x1 x22 ) .
v2 ( x) D 12 f 22 ( x)
In the first quadrant, x1
v2 ( x) D 23 f11 ( x)
v3 ( x) D 23 f12 ( x)
In the third quadrant,
2
V f ( x)
v1 ( x) D12 f 21 ( x)
D 56 x1 ,
so, v1 ( x) D 56 x1 4D 23 x2 , and holds (4). After
checking, we find that, (3), is satisfied by it in R1 ,
hence, it’s a proper LF.
v3 ( x) x
( x )< j ( x )
0,
x1 , therefore,
4D 23 x2 , v6 ( x) x
j
j 1
Example 2:
4 x2 sin 2 ( x1 )
v6 ( x) D 56 f 22 ( x)
¦v
x1 .
0,
3 x1 , therefore,
In the fourth quadrant, on x 2
4 x2 sin 2 ( x1 ), v6 ( x )
v5 ( x )
0 2 x1
x 2 2 x13
2D 23 x1
,
D 23 ( x2 2 x13 ) .
In the third quadrant, x 2
v5 ( x) D 56 f 21 ( x)
0 x1
x2 x1 x22
D 56 ( x1 ) ,
1388 14 02,/ 03 !( 0
An Algorithm for Constructing Nonsmooth Lyapunov Functions for Continuous Nonlinear Time Invariant
Alireza Faraji Armaki, Naser Pariz, Rajab Asgharian
v6 ( x) D 56 f 22 ( x)
23
D 56 ( x2 x1 x22 ) .
In the fourth quadrant, x1 0 x1 x2 2 x13
v7 ( x) D 78 f11 ( x)
v8 ( x) D 78 f12 ( x)
0.5D 78 ( x2 2 x13 ) ,
D 78 x1 .
The previous step offered two LF for R2 , if
v1(x) v2 (x) 2D23x1 D12x1, then, (4) is satisfied on their
common boundary.
Since,
v4 ( x )
the
v3 ( x) x
1
2
system
0
v5 ( x) x
2
0
is two dimensional,
D 23 x2 D 56 x1 satisfies (4)
Figure 2: Simulation of example 1
1 .
on x and x Moreover, after checking, we get
that, (3), is satisfied by it in R4 , hence, it’s a proper
LF for this region.
For continuity of LF’s on x1 , x2 ,
2D 23 x1 , d 8 ( x) D 78 x1 , if,
d1 ( x)
D 23
0.5D 78 d18 ( x)
0.
d 6 ( x) D 56 x2 , d 7 ( x) 0.5D 78 x2 ,
if, D 56 0.5D 78 d 67 ( x) 0 .
Therefore, 0.5D78 D56 D23 , by assumption, D 23 1 ,
the special LF’s of the regions are identified.
v1 ( x )
v2 ( x )
v8 ( x)
v4 ( x )
x2 x1 , v5 ( x)
v7 ( x)
x2 2 x13 .
8
V ( x)
¦v
j
2 x1 , v3 ( x )
Figure 3: The regions for example 2
x2 2 x13 ,
x1 , v6 ( x) x2 x1 x22
( x)< j ( x) , V f ( x)
8
¦ v
j
( x) < j ( x) 0 a.e.
j 1
j 1
V ( x) is a GLF for this system and the origin is
asymptotically stable.
Figure 4: Simulation of example 2
7. Conclusion
Figure 1: The regions for example 1
Journal of Control, Vol. 3, No. 2, Summer 2009
In this paper, a non-iterative algorithm was
proposed for constructing Generalized Lyapunov
Function for nonlinear time invariant system which
can be differentiable almost every-where, such that,
the system solutions be well defined. The proposed
algorithm was based on the Generalized Lyapunov
theorem, hence, it didn’t require calculation of the
generalized derivative of nonsmooth LF’s on their
nonsmooth surfaces.
1388 14 02,/ 03 !( 0
24
An Algorithm for Constructing Nonsmooth Lyapunov Functions for Continuous Nonlinear Time Invariant
Alireza Faraji Armaki, Naser Pariz, Rajab Asgharian
Unlike the methods that for constructing piecewise
LF, used approximate piecewise model of system in
each region, the defined method used original
nonlinear model of system, hence, this method was
exact. Furthermore, these other methods are
computational and more detailed analysis comes to
the cost of increased computations, but, this method
was analytic.
The steps of algorithm were defined by means of
several proposed Notes, which select LF with
attention to kind of boundaries of each region.
According to the algorithm, a GLF for the whole
system was constructed by a condensed formula. The
capability of the algorithm was demonstrated by
successful construction of GLF’s for two nonsmooth
examples.
The main restrictions of this algorithm were
original selection of LF’s for regions, and then,
continuity problem of LF’s on their common
boundaries. The Notes are proposed to solve these
restrictions in many cases.
[7]
Bacciotti, A., 2002, “Stability in the continuous
case,” Journal of Mathematical Analysis and
Applications, 270, 488–498.
[8]
Bacciotti, A., Rosier, L., Lyapunov functions
stability in control theory, (2nd ed.), Berlin:
Springer, 2005.
[9]
Ohtake, H., Tanaka, K., Wang, H., 2002, “A
construction method of switching Lyapunov
function for nonlinear systems,” Proceeding of the
IEEE conference on Fuzzy Systems, 221-226.
[10] Nakamura A., Hamada, N., 1988, “A construction
method of Lyapunov functions for piecewise
Linear systems,” Proceeding of the IEEE
symposium on Circuits and Systems, 3, 2217-2220.
[11] Marinoasson, S.F., 2002, “Lyapunov function
construction for ordinary differential equations with
linear programming,” Dynamical Systems, 17, 2,
137-150.
[12] Juliaan, P., Guivant J., Desages, A., 1999, “A
parameterization of piecewise linear Lyapunov
functions via linear programming,” International
Journal of Control, 72, 7/8, 702 -715.
[13] Giesl, P., Construction of Global Lyapunov
Functions Using Radial Basis Functions, Berlin:
Springer, 2007.
In the next researches, one can suggest these
subjects; can this algorithm obtain GLF for every
stable continuous system? Is there a systematic
approach for selection of LF’s and continuity of
them on the boundaries?
References
[1]
Chellaboina, V., Leonessa, A., Haddad, W.M.,
1999, “Generalized Lyapunov and invariant set
theorems for nonlinear dynamical systems,”
Systems & Control Letters, 38, 289-295.
[2]
Bacciotti, A., Rosier, L., 2000, “Regularity of
Liapunov functions for stable systems,” Systems &
Control Letters, 41, 265-270.
[3]
Johansson, M., Rantzer, A., 1997, “On the
Computation of Piecewise Quadratic Lyapunov
Functions,” Proceeding of the 36th IEEE
Conference on Decision & Control, 4, 3515-3520.
[4]
Johansson, M., Rantzer, A., 1998, “Computation of
Piecewise Quadratic Lyapunov Functions for
Hybrid Systems,” IEEE Transaction on Automatic
Control, 43, 4, 555-559.
[5]
Shevitze, D., Paden, B., 1994, “Lyapunov stability
theory of nonsmooth systems”, IEEE Transaction
on Automatic Control, 39, 9, 1910-1914.
[6]
Wu, Q., Sepehri, N., 2001, “On Lyapunov’s
stability analysis of non-smooth systems with
applications to control engineering”, International
journal of non-linear mechanics, 36, 1153-1161.
Journal of Control, Vol. 3, No. 2, Summer 2009
1388 14 02,/ 03 !( 0
Journal of Control
Vol. 3, No. 2, pp. 25-32, Summer 2009
AR Order Determination of 3-D ARMA Models Based on
Minimum Eigenvalue (MEV) Criterion and Instrumental
Variable Method
Mahdiye Sadat Sadabadi, Masoud Shafiee
Electrical Engineering Department, Amirkabir University of Technology, Tehran, Iran
[email protected], [email protected]
(Manuscript received: Aug. 20 2009, accepted Oct. 10 2009)
Abstract: Model order determination, the first step of system identification, plays a dominant role in
modeling any dynamic system. In this paper, a new method for AR order determination of 3-D
ARMA models is proposed. The proposed method is based on minimum eigenvalue (MEV) criterion
and instrumental variable (IV) approach. The model is assumed to be causal, stable, linear, and spatial
shift-invariant with quarter space (QS) region of support. Numerical simulations are presented to
confirm the theoretical results.
Keywords: Three-Dimensional ARMA Model, AR Order, Model Order Determination, Minimum
Eigenvalue Criterion and Instrumental Variable.
.!
9 W :P1 :@1&! 1 tC 0P1 / ![K < #& $ ! 74 #*4 :6;?
s c (MEV) , !C O!< * d :!* ARMA ! AR tH 74 #*4 : :!!( c 0&C #
.!/ (QS) D9+ U 7 < >9/ ks4 R) 0! 0 0! > [ oK .> ,!/ wR W
.> ,!/ ,[ 0&C ,!/ wR :$l4 A !p4 >"( :! ::@7/ 0"
.W
s , !C O!< * 0! 74 #*4 0AR 74 0:!* ARMA ! :( Nomenclatures:
AIC:
AR:
Akaike Information Criterion
Autoregreesive
ARMA: Autoregreesive Moving Average
CRA:
Column Ratio Array
Instrumental
Variable
IV:
LRA:
MA:
MDL:
MEV:
1-D:
QS:
ROS:
RRA:
SVD:
3-D:
2-D:
Layer Ratio Array
Moving Average
Minimum Description Length
Minimum Eigenvalue
One-Dimensional
Quarter Space
Region of Support
Row Ratio Array
Singular Value Decomposition
Three-Dimensional
Two-Dimensional
1- Introduction
Recently, there has been considerable interest in
three-dimensional (3-D) systems by 3-D autoCorresponding Author: Mahdiye Sadat Sadabadi
regressive (AR) models and 3-D autoregressive
moving-average (ARMA) models. These models are
used in several areas such as modeling, system
identification, spectral analysis, etc [1]-[6]. In most
cases, the model order is assumed to be known.
However, in most realistic situations, the model
order is not known and must be estimated.
Obviously, selecting the model order is an important
first step towards the goal of system modeling.
Model order determination of 3-D ARMA models is
a difficult task.
During the last three decades, several new methods
and algorithms have been proposed for one
dimensional and two-dimensional (2-D) model order
selection, but 3-D model order selection has not
received so much attention.
Generally, the existing order determination
methods can be divided into two categories, namely,
information theoretic criterion methods and linear
algebraic methods [7]-[10]. Information criterion
methods, e.g., Akaike information (AIC) criterion
and minimum description length (MDL) criterion,
are evaluated by minimizing an expression that
Journal of Control, Iranian Society of Instrument & Control EngineersK.N. Toosi University of Technology
26
AR Order Determination of 3-D ARMA Models
Based on Minimum Eigenvalue (MEV) Criterion and Instrumental Variable Method
Mahdiye Sadat Sadabadi, Masoud Shafiee
depends on the prediction error variance and free
parameter number. The value of order that yields the
lowest value of selected criterion is chosen as the
best estimate of the true model order [9]. In applying
AIC or MDL criterion, one usually has to estimate
the parameters corresponding to all possible model
structures. Therefore, these methods are very time
consuming [9]. The minimum eigenvalue (MEV)
criterion is based on MDL criterion. It permits the
choice of the true order with high accuracy and
without any parameter estimation [9].
The linear algebraic methods are based on
determinant and rank testing algorithms, SVD-based
methods, etc. In these methods, the order of the
system is usually determined using the rank of
special matrices. References [8], [11] are examples
of this class.
In this paper, a new technique for AR order
selection of 3-D ARMA models is proposed. The
proposed method consists of the minimum
eigenvalue (MEV) criterion and instrumental
variable (IV) approach (with delayed observations).
Usually, the techniques based on the computation of
an information criterion need the parameter
estimation of all the possible models including the
true order. On the contrary, this method only needs
the computation of matrix eigenvalues.
Based on the authors’ knowledge, threedimensional ARMA model order determination has
not been studied as much as one-dimensional and
two-dimensional case. However, some references
([1-2]) have proposed some methods for 3-D AR
model order determination. These methods cannot be
used for 3-D ARMA model order determination.
The model considered here is assumed to be causal,
stable, linear, and spatial shift invariant with quarter
space (QS) region of support. The paper is organized
as follows: The problem formulation and the basic
algorithm are presented in section 2. Section 3
provides numerical simulations in order to illustrate
the effectiveness of the proposed method. Section 4
concludes the paper.
2- 3-D AR Order Determination of a 3-D
ARMA Model
2-1- Preliminaries
Consider a 3-D causal, stable, linear, and spatial
shift invariant ARMA model defined by
Journal of Control, Vol.3, No.2, Summer 2009
p1*
p3*
p*2
¦¦ ¦
ai1,i2 ,i3 yt1 i1,t2 i2 ,t3 i3
i1 0 i2 0 i3 0
q3*
q1* q*2
¦ ¦ ¦ b j1, j2 , j3 et1 j1,t2 j2 ,t3 j (1)
j1 0 j2 0 j3 0
a0,0,0
1
where ( p1* , p2* , p3* ) and (q1* , q2* , q3* ) are the AR
order and the MA order of a 3-D ARMA model,
respectively. The following conditions are assumed
to hold.
Assumption 1: et1 ,t 2 ,t 3 is a white noise with zeromean and variance V e2 .
Assumption 2: The true AR model order is
( p1* , p2* , p3* ) such that
p1*
max{i1 ;
ai1 ,i2 ,i3 z 0}
p2*
max{i2 ;
ai1,i2 ,i3 z 0}
p3*
max{i3 ;
ai1,i2 ,i3 z 0}
Assumption 3: The 3-D ARMA model in (1) is
stable.
Note that the stability analysis of 3-D models is
much more difficult than one-dimensional case. One
of the most important reasons is that
multidimensional systems have infinite poles. As a
result, the convention methods and theorems for 1-D
stability
analysis
cannot
be
used
for
multidimensional case. For more information in
multidimensional stability conditions, one can refer
to [12]-[13].
Since the true orders ( p1* , p2* , p3* ) and (q1* , q2* , q3* )
are unknown, the general case of (1) with
( p1* , p2* , p3* ; q1* , q2* , q3* ) replaced by unknown orders
( p1 , p2 , p3 ; q1 , q2 , q3 ) is considered.
2-2- Algorithm for AR Order Determination
Assuming the data length is N1 u N 2 u N 3 (that is
0, 1, ..., N1 1 ,
t2 0, 1, ..., N 2 1 ,
and
t3 0, 1, ..., N3 1 ), the equation (1) can be
rewritten in a matrix form as follows:
t1
1388 14 02 ,/ 03 !( 0
AR Order Determination of 3-D ARMA Models
Based on Minimum Eigenvalue (MEV) Criterion and Instrumental Variable Method
Mahdiye Sadat Sadabadi, Masoud Shafiee
YT
W
(2)
In the above equation, Y is an output data matrix
with
dimension
( N1 N 2 N 3 ) u ( p1 1)( p2 1)( p3 1) , vector T is a
( p1 1)( p2 1)( p3 1) u 1 parameter vector, and W
(3a)
[T 0 T1 T p1 ]T
[T i1,0 T i1 ,1 T i1, p2 ]T
T i1
[ai1 ,i2 ,0 ai1,i2 ,1 ai1,i2 , p3 ]T
T i1 ,i2
[W0 W1 W N1 1 ]T
W
[Wt1,0 Wt1,1 Wt1 , N 2 1 ]T
Wt1
[ wt1,t2 ,0 wt1,t2 ,1 wt1,t2 , N3 1 ]T
Wt1 ,t2
q1
wt1 ,t2 ,t3
q2
0
0
ª y t1,t 2 ,0
º
«
»
y
y
0
« t ,t ,1
»
t1,t 2 ,0
Y t1,t 2 « 1 2
»
«
»
« y t ,t ,N 1 y t ,t ,N 2 y t ,t ,N 1 p »
1 2 3
1 2 3
3¼
¬ 1 2 3
t 2 0,1, , N 2 1
(5c)
is an ( N1 N 2 N 3 ) u 1 input data vector.
T
27
(3b)
Note that matrices O and Oc in (5a) and (5b) are
zero matrices with dimensions N 2 u ( p2 1) and
N 3 u ( p3 1) , respectively. An instrumental variable
(IV) matrix can be defined as
O
O º
ª Z0
«
»
Z
Z
O »
1
0
«
« »
«
»
¬« Z N1 1 Z N1 2 Z N1 1 k1 ¼»
(3c)
Z
(4a)
(4b)
(4c)
q3
¦ ¦ ¦ bi ,i ,i
e
1 2 3 t1 i1 ,t2 i2 ,t3 (6a)
Oc
Oc
º
ª Z t1 ,0
»
«
Z t1,0 Oc
« Z t1,1
»
Z t1 «
»
»
«
« Z t , N 1 Z t , N 2 Z t , N 1 k »
1 2
1 2
2¼
¬ 1 2
t1 0,1, , N1 1
(6b)
i1 0 i2 0 i3 0
0
ª zt1,t2 ,0
«
zt ,t ,0
« zt ,t ,1
1 2
Zt1,t2 « 1 2
«
« zt ,t , N 1 zt ,t , N 2
1 2 3
¬ 1 2 3
t1 0,1, , N1 1
and
Y
O
O
ª Y0
º
«
»
Y0
O
« Y1
»
« »
«
»
Y
Y
Y
«¬ N1 1 N1 2
N1 1 p1 »¼
t2
º
»
0
»
»
»
zt1,t2 , N3 1 k3 »
¼
0
0,1, , N 2 1
(5a)
(6c)
Oc
Oc
ª Yt1 ,0
º
«
»
c
Y
Y
O
« t1,1
»
t1 ,0
«
»
«
»
«Yt , N 1 Yt , N 2 Yt , N 1 p »
1 2
1 2
2¼
¬ 1 2
Yt1
t1
0,1, , N1 1
where zt1 , t 2 , t 3 , O, and Oc are an IV sequence and
zero matrices with dimensions N 2 u ( k 2 1) and
(5b)
N 3 u (k3 1) , respectively. Several choices of
zt1 ,t 2 ,t 3 are possible as long as IV sequence is
uncorrelated with the noise part wt1 ,t 2 ,t 3 and fully
correlated with the observed part yt1 ,t 2 ,t 3 [14], [15].
In this paper, the instrument zt1 ,t 2 ,t 3 is formed
Journal of Control, Vol.3, No.2, Summer 2009
1388 14 02 ,/ 03 !( 0
28
AR Order Determination of 3-D ARMA Models
Based on Minimum Eigenvalue (MEV) Criterion and Instrumental Variable Method
Mahdiye Sadat Sadabadi, Masoud Shafiee
from the delayed observed data yt1 l1,t2 l2 ,t3 l3 with
l1 ! q1 , l 2 ! q 2 , and l3 ! q3 . Premultiplying (2) by
1
1
Z T and considering V
Z TW ,
N1 N 2 N 3
N1 N 2 N 3
the following equation is obtained:
1
ZTY T
N1 N 2 N 3
V
(7)
V is an asymptotically Gaussian distribution with
D
zero mean [15]. If
is defined as
1
T
D
Z Y , the equation (7) can be
N1 N 2 N 3
rewritten as
DT
V
(8)
In which, the dimensions of
respectively are
D
and
V
(k1 1)(k2 1)(k3 1) u ( p1 1)( p2 1) ( p3 1) and
( k1 1)( k2 1)( k3 1) u 1 . Now the matrix R is
defined as
R
(9)
DT D
Note that R is a symmetric and positive semidefinite matrix. The method proposed in this paper
permits the choice of the AR order of 3-D ARMA
models in (1) with high accuracy and without any
parameter estimation. This method uses both 3-D
MDL criterion and the minimum eigenvalue of
matrix R .
In the 3-D case, the MDL order determination
criterion appears as follows [2]
J MDL ( p1, p2 , p3 )
log( f (V )) K
1
( p1 1)( p2 1)( p3 1) log((k1 1)(k2 1)( k3 1))
2
K
(10)
where f (V ) is the probability density function of
V
such
that
V
[v0,0,0 ... v0,0,k3 ... vk1 ,k2 ,0 ... vk1,k2 ,k3 ]T .
vt1 , t 2 ,t 3 is zero-mean white Gaussian noise,
Journal of Control, Vol.3, No.2, Summer 2009
Since
f (V )
1
( k1 1)( k2 1)( k3 1)
2
(2SV 2 )
1
(2SV
2
( k1 1)( k2 1)( k3 1)
2
)
exp (
exp (
1
2V 2
1
2V 2
VT
(11)
T
T RT )
where V 2 is the variance of vt1 , t 2 ,t 3 . Replacing
f (V ) by (11) results in
MDL ( p1 , p2 , p3 , T )
(k1 1)(k2 1)(k3 1)
log(2SV 2 )
2
1
T T RT K
2V 2
(12)
For fixed-order ( p1 , p2 , p3 ) and constraining T to
have unit Euclidean norm, the choice of T that
minimizes (12) is found to be the eigenvector
associated with the minimum eigenvalue (Omin ) of
R [2]. In other words
1
T
T min
RT min
( k1 1)( k2 1)( k3 1)
1
V TV | V 2
( k1 1)(k2 1)( k3 1)
(13)
Therefore,
1
Omin
( k1 1)(k 2 1)(k3 1)
V2
Substituting and dropping all the terms not
depending on p1 , p2 , p3 or T ,
J MDL ( p1 , p2 , p3 )
(k1 1)(k2 1)(k3 1)
log(Omin )
2
K
(14)
The term T in the argument of MDL has been
dropped since it has been incorporated into the Omin
term. Multiplying both sides of the above equation
2
by
, and combining terms
( k1 1)(k 2 1)(k3 1)
lead to
1388 14 02 ,/ 03 !( 0
AR Order Determination of 3-D ARMA Models
Based on Minimum Eigenvalue (MEV) Criterion and Instrumental Variable Method
Mahdiye Sadat Sadabadi, Masoud Shafiee
2
J MDL ( p1 , p2 , p3 )
(k1 1)(k2 1)(k3 1)
log(Omin (k1 1)(k2 1)(k3 1)
( p1 1)( p2 1)( p3 1)
( k1 1)( k2 1)( k3 1)
(15)
J
vertical plan of J ( p1 , p2 , p3 ) array by the previous
one; i.e.
CRA(i1 , i2 , i3 )
)
Since log(.) is a monotonically increasing function,
a different criterion can be formed that contains the
same information as J MDL ( p1 , p2 , p3 ) , and
combining terms. Therefore, using a combination of
MDL criterion and instrumental variable (IV)
method, the minimum eigenvalue criterion for AR
order selection of an ARMA model is as follows:
29
J (i1 , i2 , i3 ) / J (i1 , i2 1, i3 )
3) Layer Ratio Array (LRA): by dividing each
layer of J ( p1 , p2 , p3 ) array by the previous one;
i.e.
LRA(i1 , i2 , i3 )
J (i1 , i2 , i3 ) / J (i1 , i2 , i3 1)
An estimate of p1* is set equal to the row number
( p1 ) that contains the minimum value in the row
Omin
[((k1 1)(k2 1)(k3 1))
( p1 1)( p2 1)( p3 1) (16)
( k1 1)( k2 1)( k3 1)
]
ratio array. The number of columns ( p2 ) which have
the minimum value in the column ratio array will be
the estimate of p2* . Finally, the number of layers
where Omin is the minimum eigenvalue of positive
semi-definite matrix R .
( p3 ) which have the minimum value in the layer
From the above equation, it can be seen that when
k1 o f or/and k2 o f or / and k3 o f the
From the proposed results, the following algorithm
for AR order estimation of a 3-D ARMA model is
suggested.
( p1 1)( p2 1)( p3 1)
( k1 1)( k 2 1)( k3 1)
[((k1 1)(k 2 1)(k3 1))
] part of
(16) is approximately one and AR model selection is
asymptotically simplified by examining the
minimum eigenvalue of R for different values of
p1 , p2 , p3 .
Note that if p1 , p2 , p3 are chosen such that
p1 t p1* , p2 t p2* ,and p3 t p3* , Omin will be
small compared with case p1 p1* or p2 p2* or
p3 p3* . Because if
p1 p1*
or
p2 p2*
or
p3 p3* , the model dose not have enough
parameters to fit the signal very well.
Consequently, the procedure for model order
selection consists of computing J ( p1 , p2 , p3 ) for
different orders and selecting the triplet which
correspond to the corner where Omin drops very
quickly [2]. In practice, several corners can be found
instead of a single corner. In order to select the
correct corners, three arrays are constructed as
follows:
1) Row Ratio Array (RRA): by dividing each
horizontal plan of
J ( p1 , p2 , p3 ) array by the
previous one; i.e.
RRA(i1 , i2 , i3 )
J (i1 , i2 , i3 ) / J (i1 1, i2 , i3 )
ratio array will be the estimate of p3* .
Step 1) Fix the AR order ( p1, p2 , p3 ) over the set
[ p1min , p1max ] u [ p2 min , p2 max ] u [ p3 min , p3 max ]
S
, and suppose that is the true order.
Step 2) Compute the matrix R
determine its eigenvalues.
by (9) and
Step 3) Evaluate (16) for all values of p1 , p2 , and
p3 .
Step 4) Construct the row, column, and layer ratio
arrays.
Step 5) Choose the minimum value of the row,
column, and layer ratio arrays.
Step 6) An estimate of the true value of p1* is set
equal to the row number that contains the minimum
value of the row ratio array. The number of columns
having the minimum value in the column ratio array
will be the estimate of the true value of p 2* . Finally,
the number of layers which have the minimum value
in the layer ratio array will be the estimate of p3* .
3- Numerical Simulations
In this section, two numerical examples are
presented to provide verification of the theoretical
2) Column Ratio Array (CRA): by dividing each
Journal of Control, Vol.3, No.2, Summer 2009
1388 14 02 ,/ 03 !( 0
30
AR Order Determination of 3-D ARMA Models
Based on Minimum Eigenvalue (MEV) Criterion and Instrumental Variable Method
Mahdiye Sadat Sadabadi, Masoud Shafiee
results. In these examples, input sequence et1 ,t 2 ,t3 is
a Gaussian white noise with zero-mean and variance
one: N (0,1) . The data length is N1 u N 2 u N 3 :
N1
30 , N 2
30 , N 3
1.3 yt1 1,t2 ,t3 0.3825 yt1 2,t2 ,t3
1.27 yt1,t2 1,t3 1.6510 yt1 1,t2 1,t3
0.4858 yt1 2,t2 1,t3 0.39 yt1,t2 2,t3
30 .
0.5070 yt1 1,t2 2,t3 0.1492 yt1 2,t2 2,t3
Example 1: The true model is given by
yt1,t2 ,t3
yt1,t2 ,t3
1.13 yt1,t2 ,t3 1 1.4696 yt1 1,t2 ,t3 1
0.9 yt1 1,t2 ,t3 0.88 yt1,t2 1,t3 0.95 yt1,t2 ,t3 1
0.792 yt1 1,t2 1,t3 0.855 yt1 1,t2 ,t3 1
0.4322 yt1 2,t2 ,t3 1 1.4251 yt1,t2 1,t3 1
1.8656 yt1 1,t2 1,t3 1 0.5489 yt1 2,t2 1,t3 1
0.4407 yt1,t2 2,t3 1 0.5729 yt1 1,t2 2,t3 1
0.836 yt1 ,t2 1,t3 1 0.7524 yt1 1,t2 1,t3 1
0.1686 yt1 2,t2 2,t3 1 0.3120 yt1,t2 ,t3 2
et1,t2 ,t3 0.8 et1 1,t2 1,t3 1
(17)
0.4056 yt1 1,t2 ,t3 2 0.1193 yt1 2,t2 ,t3 2
0.3962 yt1,t2 1,t3 2 0.5151 yt1 1,t2 1,t3 2
This model is a 3-D stable model of AR order
(1,1,1) and MA order (1,1,1).
0.1516 yt1 2,t2 1,t3 2 0.1217 yt1,t2 2,t3 2
The data for order determination is collected from
the above model in (17). Using these data and the
algorithm proposed in the end of section (2-2), the
AR order of the ARMA model in (17) can be
determined.
et1,t2 ,t3 1.2 et1 2,t2 2,t3 2
In this example, the fixed-order interval set is
( p1 , p 2 , p 3 ) [0,3] u [0,3] u [0,3] .
Using
row,
column, and layer ratio arrays, the AR order of the
ARMA model in the above example is estimated.
Results are shown in tables 1-3. As can be seen from
the tables, the minimum value of the row, column,
and layer ratio arrays has occurred in p1 1 ,
p2 1 , and p3 1 . Therefore, based on the
proposed algorithm, the true AR order is
( p1* , p2* , p3* )
(1,1,1) .
The example is simulated 100 times and it is
checked how often this method choose the correct
AR order. The results obtained with the proposed
method are displayed in table 4. Note, the symbol *
is used in tables to identify the true order of model.
Example 2: In the second example, a 3-D ARMA
model is considered as follows:
0.1582 yt1 1,t2 2,t3 2 0.0465 yt1 2,t2 2,t3 2
(18)
The same procedure as in Example 1 was followed.
The simulation for AR order determination is
performed 100 times using the proposed method.
The results are displayed in table 5. In this example,
the
fixed-order
interval
set
is
( p1 , p2 , p3 ) [1,5] u [1,5] u [1,5] .
4- Conclusion
In this paper, an effective approach for AR order
determination of 3-D causal, stable and shiftinvariant ARMA models with quarter-plane ROS
was proposed. The proposed method is based on the
minimum eigenvalue (MEV) criterion and the
instrumental variable method that is computationally
more efficient than some methods such as AIC and
MDL criteria. In spite of 3-D AIC criterion and
MDL criterion, this method permits choice of the
true order with high accuracy and without any
parameter estimation. Numerical examples were
given that illustrated the good performance of the
results that can be obtained with this approach.
References
[1] Aksasse, B., Stitou, Y., Berthoumieu, Y., Najim, M.,
2006, “3-D AR model order selection via rank test
procedure”, IEEE Trans. Signal Processing, 54,
2672-2677.
[2] Aksasse, B., Stitou, Y., Berthoumieu, Y., Najim, M.,
2005, “Minimum eigenvalue based 3-D AR model
Journal of Control, Vol.3, No.2, Summer 2009
1388 14 02 ,/ 03 !( 0
AR Order Determination of 3-D ARMA Models
Based on Minimum Eigenvalue (MEV) Criterion and Instrumental Variable Method
Mahdiye Sadat Sadabadi, Masoud Shafiee
31
order selection”, 13th Workshop on Statistical
Signal Processing, 431-436.
[3] Digalakis, V. V., Ingle, V., Manolakis, D. G., 1993,
“Three-dimensional linear prediction and its
application
to
digital
angiography”,
Multidimensional Systems and Signal Processing,
4, 4, 307-329.
[4]
Kokaram, A. C., Morris, R. D., Fitzgerald, W.,
Rayner, P. J. W., 1995, “Interpolation of missing
data in image sequences”, IEEE Trans. Image
Process, 4, 1509-1519.
[5] Kwan, H. K., Lui, Y. C., 1989, “Lattice predictive
modeling of 3-D random fields with application to
interframe predictive coding of picture sequences”,
International Journal of Electronics, 66, 489-505.
[6] Szummer, M., Picard, R.W., 1996, “Temporal
texture modeling”, IEEE International Conference
on Image Processing (ICIP), 3, 823-826,
Lausanne, Switzerland.
[7] Rital, S., Meziane, A., Rziza, M., Aboutajdine, D.,
2002,
“Two-dimensional
non-Gaussian
autoregressive model order determination”, IEEE
Signal Processing Letter, 9, 426-42.
[8] Aksasse, B., Radouane, L., 1999, “A rank test based
approach to order estimation-Part I: 2-D AR
models application”, IEEE Trans. Signal
Processing, 47, 2069-2072.
[9] Aksasse, B., Radouane, L., 1999, “Two-dimensional
autoregressive (2-D AR) model order estimation”,
IEEE Trans. Signal Processing, 47, 2072-2077.
[10] Sadabadi, M. S., Shafiee, M., Karrari, M., 2008, “A
new technique for order determination of twodimensional ARMA models”, SYSTEMS SCIENCE
Journal, 34, 2, 49-53.
[11] Sadabadi, M. S., Shafiee, M., Karrari, M., October
2007, “Determination of the two-dimensional
ARMA model order using rank test based
approach”, Proceedings of the 16th IEEE
International Conference on Control and
Applications (CCA), Singapore, 1156-1160.
[12] Bose, N. K., Multidimensional Systems-Theory and
Applications, Kluwer Academic Publishers,
Second Edition, 2003.
[13] Mastorakis, N. E., Gonos, I. F., Swamy, M. N. S.,
2003, “Stability of multidimensional systems using
genetic algorithms,” IEEE Trans. Circuits and
Systems I: Fundamental Theory and Applications,
50, 7, 962–965.
[14] Xiao, C. B., Zhang, X. D., Li, Y. D., 1996, “A new
method for AR order determination of an ARMA
process”, IEEE Trans. Signal Processing, 44,
2900-2903.
[15] Söderström, T., Stoica, P., 1981, “Comparison of
some instrumental variable methods-consistency
and accuracy aspects”, Automatica, 17, 101-115.
Journal of Control, Vol.3, No.2, Summer 2009
1388 14 02 ,/ 03 !( 0
AR Order Determination of 3-D ARMA Models
Based on Minimum Eigenvalue (MEV) Criterion and Instrumental Variable Method
Mahdiye Sadat Sadabadi, Masoud Shafiee
32
Table 1: Row Ratio Array (Example 1)
p1
2
*1
p3
p2 / p3
p2
3
p3
p2 / p3
0
1
2
3
0
1
0.21
0.26
0.31
0.5
0.54
0.58
0.43
0.92
2
0.57
0.84
0.54
0.38
3
0.56
0.72
0.45
10
e-7
p2
p3
p2 / p3
0
1
2
3
0
1
0.43
0.81
0.84
0.93
0.66
0.57
0.69
0.45
2
1.05
0.59
3
1
0.59
15
e-7
21
e-7
22
e-7
1.23
p2
0
1
2
3
0
1
0.84
0.93
0.93
0.91
0.99
0.68
2
0.75
0.94
0.88
0.79
16
e-7
1.16
3
0.91
17
e-7
1.16
1.31
Table 2: Column Ratio Array (Example 1)
p2
2
*1
p3
p1 / p3
p1
3
p3
p1 / p3
0
1
2
3
0
1
0.35
0.44
0.37
0.58
0.48
0.52
0.23
0.50
2
0.82
0.65
0.45
0.33
3
0.92
0.64
0.31
67
e-8
p1
p3
p1 / p3
0
1
2
3
0
1
0.38
0.81
0.42
0.71
0.51
0.48
0.92
0.38
2
1.05
0.45
3
0.84
0.46
13
e-6
16
e-7
19
e-7
1.37
p1
0
1
2
3
0
1
0.78
0.77
0.83
0.72
0.82
0.67
2
0.73
0.73
0.93
0.83
27
e-7
1.20
3
0.89
13
e-7
1.23
1.36
Table 3: Layer Ratio Array (Example 1)
p3
*1
2
p2
p1 / p2
p1
p1 / p2
3
p2
p2
p1 / p2
0
1
2
3
0
1
2
3
0
1
2
3
0
1
0.23
0.24
0.27
0.29
0.71
0.86
0.85
0.50
0.9
0.91
0.45
0.40
0.37
0.93
0.83
0.56
0.52
0
1
1.02
0.34
0
1
0.55
0.82
0.80
0.62
2
0.66
0.52
0.22
0.22
2
0.73
0.51
0.85
0.62
0.93
0.73
0.51
0.28
42
e-8
3
0.79
0.38
18
e-7
1.24
2
3
14
e-7
13
e-7
20
e-7
1.20
3
0.67
15
e-7
1.23
1.36
p1
p1
Table 4: 3-D AR Order estimation results from 100 simulation runs (Example 1)
AR Order
%
*(1,1,1)
81
(2,1,1)
(1,2,1)
(1,1,2)
(3,1,1)
16
0
0
3
Table 5: 3-D AR Order estimation results from 100 simulation runs (Example 2)
AR Order
%
*(2,2,2)
88
(2,1,2)
(2,2,3)
(2,3,2)
(3,2,2)
(2,3,3)
(3,2,3)
(3,3,2)
1
0
2
8
0
0
1
Journal of Control, Vol.3, No.2, Summer 2009
1388 14 02 ,/ 03 !( 0
Journal of Control
A Joint Publication of the Iranian Society of Instrument and Control
Engineers and the K.N. Toosi University of Technology
Vol. 3, No. 2, Summer 2009
Persian Part
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English Part
Design of Multiple Model Controller Using SOM Neural Network
Poya Bashivan, Alireza Fatehi
Robust H f control of an Exerimental Inverted Pendulium Using Singular Perterbation
Approach
1
10
Roya Amjadifard, Mohammad T. Hamidi Beheshti, Hamid Khaloozadeh, Kirsten. A. Morris
An Algorithm for Constructing Nonsmooth Lyapunov Functions for Continuous
Nonlinear Time Invariant Systems
Alireza Faraji Armaki, Naser Pariz, Rajab Asgharian
18
AR Order Determination of 3-D ARMA Models Based on Minimum Eigenvalue (MEV)
Criterion and Instrumental Variable Method
26
Mahdiye Sadat Sadabadi, Masoud Shafiee
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