This is a repository copy of
Estimation of spatial derivatives and identification of
continuous spatio-temporal dynamical systems
.
White Rose Research Online URL for this paper:
http://eprints.whiterose.ac.uk/74501/
Monograph:
Guo, L.Z., Billings, S.A. and Wei, H.L. (2005) Estimation of spatial derivatives and
identification of continuous spatio-temporal dynamical systems. Research Report. ACSE
Research Report no. 898 . Automatic Control and Systems Engineering, University of
Sheffield
[email protected]
https://eprints.whiterose.ac.uk/
Reuse
Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright
exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy
solely for the purpose of non-commercial research or private study within the limits of fair dealing. The
publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White
Rose Research Online record for this item. Where records identify the publisher as the copyright holder,
users can verify any specific terms of use on the publisher’s website.
Takedown
If you consider content in White Rose Research Online to be in breach of UK law, please notify us by
Estimation of Spatial Derivatives and Identification of
Continuous Spatio-Temporal Dynamical Systems
Guo, L. Z., Billings, S. A. and Wei, H. L.
Department of Automatic Control and Systems Engineering
University of Sheffield
Sheffield, S1 3JD
UK
Æ
Æ ! "# $
! " #
Æ
! "
# #
# #
$ #
# #
% # #
# #
'())) * + &,-, . ' /
0 &,-1 &,2& 34% 5 ())6! " #
## % #
$ # #%
#
% #
# /
# # #
# # ' /
# #
# /
# # %
# 0
/ #
/7Æ $
"
% Æ
# #
# Æ
# # '
8
#'#
'
#
' # % # ' # #
# ' #
$ #
# # '
#
# 9*:
' 5&,-,! 3
# #
#
# 9*:
# ; ( #
# # / # /
# #
6 9*:
# #
/ #
# #
> ? ¼
¼¼
´µ
!@ &!
!
#
#
¼
# @
!%# 3
# &!
)!!?
! (!
!!?
!@ 6!
* #
## # A! ?)&( @ A
$ # !
7 !
% * &,-)% &,-&
;B% &,,&! 7 #
&! # ! /'# #
* %% . &,,(! #
&! @
!? A! #
·½
!?
!C
´·½µ¡
¡
>!?
!C
´·½µ¡
¡
!
¼
!
¼¼
!
´µ
!!
<!
/'# ##
·½
!?
!CA
!C
! 1!
!?
!
¼
!
!!
!
# /'# # 3 1! ? &
# /'# #
# %
A #
D% !
! ? &(
% # 1! "
% # % # # # #
1!
!?
!CA
!
!!C
! =!
#
Æ #
!
!! #
#
!
@ A
=! ##
C&!? !C
A
C& !
C& !!!CC&! 2!
?
CA ?A
$
*
% # #
/'#
# #
2! # &! $
! 4
$ %
# "
$ #
!? !C! -!
# ! !
4 # 2! -!
#
C&! ? C&!CC&! ,!
? !C
A
C& !!CC&!CC&! &)!
? CC&!
C& !?
C& !
C& !! ?&
!
C&!?
C&!CC&!C
!
? !
A
C& !
C& !! &&!
A
C& !
C& !!!
? !
A
C& !
C& !! &(!
A
C& !
C& !!!
?EC&!
℄ &6!
E℄ #
G
?E
℄
E
C&!℄ &<!
$ # # E
C&!℄? ) D
E C&!℄ ? E C&!℄CE C&!℄CE
!℄ &1!
? E
!℄
4
G
# #
G
?CE
℄
½ E
!℄ &=!
#
#
8
& " $ Æ
#
#
"
#
# ;3:; 3 : !
;3:H
( " ' #
## %
#
#
6 "
/
#
$ #
3 Æ
# # #
# #
# # # ' # # ' # # # % # / # ! #
# &,,(!
& ( ! (! 6 ¾ ! ¾ ?
< /# !
!
#:4
# * # ## ?( ( !
:4#
5 # ? ¾ ? ! &2! /# ! # ## !
:4 #
$ # # # ?( ( !
:4 #
!
/ 7
! # # ! # ! # # G # G ! # ! ? ! G " ! &-! ! ? ! G " ! &,!
! " ; I
G
?
G
? 3 '
# # !
!?
! (&!
;
#
?
#
! ?
!C
! # 7 #
#
! C& #
!?
!
G
"
!C
!
G
"
! ((!
$ 7
# #
#
'
#
!?
!
G
"
!C
!
G
"
! (6!
# # #
(6! # ;
# # J H J &,-,!
$ # ! 7
!
' # #
# $ #
#
!?
$& #
!C
$
$& #
&! (<!
#
! #
#
!?
& # )&!
)
!?# ! (=! # # !? %
(&! (2!
Æ % ? &! ( $
&C&! &?) '6$( (-!
3 ' #
! !?# !?# !# &!? ! &! (,! * ()! ! ! !? !? ! G " !? ! 6)! $ # # ! # # ! # # # !? !? ! G " !? ) ! 6&! ) ?( ! # (,!
/ 6)! 6&! # #
! # ! !? ! 6(! ! ) !? ( ! !
' # # #
# 5 K #
* ## # # # K8 K
!?K(
! # # K
#:4#
# # K
!? K ! 66!
;:4
! # # # * %# 5 ½ ! ? # ½ ! ¾ ! ? # ¾ ! ' # # $ $
3 $
$ # # K ½ ¾ !? ½ ! ¾ ! 6<! ( & L ½ ¾
&? &( ( & !
! 6<!
; # # !
' # K
62!
# #
#
/ 7
' #
# 6)! 61!! #
I
# ' #
# *
M
# #
# M
# Æ
# ;3: # M
: # Æ M
6(! 6=! 62!
,! # &! # #
C&!? CC&!CC&!C
! 6-!
?&
!
&&! &(!
; !?!!
¼
! ?
¼
!C
¼
!
!?
!C
!
!! C&!
!
C&!! $
!
#
! #
! 6-!
C&!? C +CC&! <)!
#
# # #!!
!
#
+ Æ 3 <)! C&!
C&! C&! C&!
$ ,! #
% 9 * : '
5 &,-,! 9*: #
# # : : '
N &,--! $
!
!
# # ! 9*:
# "
# # 9*:
! +!# #
$ 9*:
# O
0
$ %
5 # # #
##
?!? !C !+C! <&!
! ? &! #!!
! ? &! #!!
! !
* , ?
?
A
(!
A
#C&!!
9*:
#
.
?.
?&
?
P )
?
& ?/0 ¾ ½ P )
!?/0
¾ ½ ! <6! ? ½ ? <<! /
?& <1!
"& . ?. & & <=! ? P ) ? <2! & ?/0 ¾ P )
!?/0
¾ ! <-! ? ? <,! / ?
?& &' 1)!
&
!1 1&!
1 #
Æ # #
G ? G ½ G ¾ G ? & / / ) & /
) ) &
#
G
C&!?C&! ! G
16!
D
# # !+
# !
! '
/
$ #
!
7%%# J
#
# #
/ 2?2
½
2
! # % 3
# + ? 43! # #
2 / # 5
G 3
# # 3
23
2
G
3 1<!
2
? 2
½
2
! 2 3 2 #
3
G
3 @? @32!
# 2 ?&5 # &(
02
?2
? &
11!
2
# # 4
2 4 # 2 *
# + ?43!
G + ?4
G
# # G
3 ## 1=! $ #
4 % % + #
G
+ ? 62! #
G
+ #
G +
? 62
! # 6! # #
#
Æ # #
2
(1()) ? 1) #
# &,,6! / &
G +
&!?62
&!!
# & ? &(
G +
4
G +
&! ? 62
&!! # #
G +
?
+
&! +
!! : # # &,,6! # #
# # #
/#
#
C&!?!C
A
C&!
C&!!!?C&! !
12!
* # # # # %
! #
!?!
G
!?! !
G
1-!
! ? &! #!!
! ? &! #!!
' #
&!
#!
! #
!?!C7
! 1,!
7 Æ # ) ! 7 & 3 !
# !
!?
!CÆ
!
A
=)!
A
D1,! =)!
!
#
!
G
5
G
& # &?&(¡¡¡
G
? G
¡¡¡ G
!
=&!
G
? &5
G
=&!
G
/
% #
' .&,-= ' Q &,,<! / %
# 5
$# # #
#
" # $
! "
#
!
?
!
C
!
!
! =(!
!
?
!
CÆ
! 8
!
¾@?E)&℄
)!?
)!?64! =6!
)!?
&!?
)!?
C&!
!
! ,,,=1)& ,-6)=)& 61=21)6
!
,,,&<)& &)6&))( (=6)&)6
!
,,,(1)& ==6<&)6 &-&,()1 ¼¼
! 1-2)))< (-<(<)2 =(,2))=
!
66,,2)1 (==<=)- <<61&)=
!
¼
! 2(66-)1 2222,), 66,-6)=
!
!
&-&&6)1 6)(6-), (,)2))=
C&!
!
! 6,,-2C)& ,,,&6)& =&,&&)6
! ()))()& ,26,-)< =&<1))1 ¼¼
! =&-6&)< --2,=)- &6(=&)1
&8 # # & #
* # =(!
?
? )')))=&--Æ ? <) 8 ? )'( +
: &,,2! A ? )')& A ? )')( + ? &5(
;3:# 62'2<62'#
66'2&(&'#
*&! *(!
* # #
/ # &))
# $ #
( #
# 8 # ' # 6
# ' # = #
)')& *1&)!
6 /'# # # 6
# # ##
9*: &! :: : : ;
$ :: &!
!
!
¼
!
!
!
% #
#
Æ ! # /#
9*: #
C&!
!
! ,,,12)& ,2=2&)& <)-(&)6
!
&))21C)) &2)&()( (=(1=)6
!
&)&6(C)) =(=-2)6 &&2-()< ¼¼
! (,2&1)6 22)2-)2 &&<(()<
C&!
!
! <)&<,C)& ,,,)()& =1((2)6
! &,6,-)& ,226()< (,<-<)< ¼¼
! ,1(16)1 &-((=)2 (-),,)<
(8 # # & # #
!
? )')))1-)=
!
C)',,-=2
!
!
)',,,==
!=<!
!
? )')))=&6)1
!
C6,',-2
!C
!
$ # #
# (!
(! =(! #
#
#
?() #
!
? )')))6&
!
C&'))6(
!
&')&,<
!
)',,,=
! =1!
!
?
!
C<)')(<=
!C)'()-=
!
$ #
# #$ %
$ ;# " ;# "
0
50
100
150
200
0
10
20
30
40
50
60
−1
−0.5
0
0.5
1
1.5
t
x
y
1
(x,t)
*&8 &8
!
0
50
100
150
200
0
10
20
30
40
50
60
−6
−4
−2
0
2
4
6
8
t
x
y
2
(x,t)
*(8 &8
0
100
200
0
50
−5
0
5
t
x
dy
1
(x,t)
0
100
200
0
50
−5
0
5
t
x
dy
1
(x,t)
0
100
200
0
50
−50
0
50
t
x
ddy
1
(x,t)
0
100
200
0
50
−20
0
20
t
x
ddy
1
(x,t)
* 68 &8
½
! # 8 #
0
100
200
0
50
−50
0
50
t
x
dy
2
(x,t)
0
100
200
0
50
−20
0
20
40
t
x
dy
2
(x,t)
0
100
200
0
50
−200
0
200
t
x
ddy
2
(x,t)
0
100
200
0
50
−100
0
100
t
x
ddy
2
(x,t)
* <8 &8
! # 8 #
M 8 # !
0
50
100
150
200
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
t
y
1
(10,t)
0
50
100
150
200
−1
−0.5
0
0.5
1
t
y
1
(20,t)
0
50
100
150
200
−1
−0.5
0
0.5
1
t
y
1
(30,t)
0
50
100
150
200
−1
−0.5
0
0.5
1
t
y
1
(40,t)
* 18 &8
0
50
100
150
200
−3
−2
−1
0
1
2
3
t
dy
1
(10,t)
0
50
100
150
200
−1
−0.5
0
0.5
1
1.5
t
dy
1
(20,t)
0
50
100
150
200
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
t
dy
1
(30,t)
0
50
100
150
200
−3
−2
−1
0
1
2
3
t
dy
1
(40,t)
*=8 &8 #
½
!
0
50
100
150
200
−15
−10
−5
0
5
10
15
t
ddy
1
(10,t)
0
50
100
150
200
−10
−5
0
5
10
t
ddy
1
(20,t)
0
50
100
150
200
−10
−5
0
5
10
t
ddy
1
(30,t)
0
50
100
150
200
−10
−5
0
5
10
t
ddy
1
(40,t)
* 28 &8 #
½
0
50
100
150
200
−4
−2
0
2
4
t
y
2
(10,t)
0
50
100
150
200
−6
−4
−2
0
2
4
6
8
t
y
2
(20,t)
0
50
100
150
200
−6
−4
−2
0
2
4
6
8
t
y
2
(30,t)
0
50
100
150
200
−4
−2
0
2
4
6
t
y
2
(40,t)
* -8 &8
!
0
50
100
150
200
−15
−10
−5
0
5
10
15
20
t
dy
2
(10,t)
0
50
100
150
200
−4
−2
0
2
4
6
8
t
dy
2
(20,t)
0
50
100
150
200
−20
−15
−10
−5
0
5
10
15
t
dy
2
(40,t)
0
50
100
150
200
−6
−4
−2
0
2
4
t
dy
2
(30,t)
*,8 &8 #
0
50
100
150
200
−100
−50
0
50
100
t
ddy
2
(10,t)
0
50
100
150
200
−80
−60
−40
−20
0
20
40
t
ddy
2
(20,t)
0
50
100
150
200
−60
−40
−20
0
20
40
t
ddy
2
(30,t)
0
50
100
150
200
−80
−60
−40
−20
0
20
40
60
t
ddy
2
(40,t)
*&)8 &8 #
!
##
9?0
C
!
!99
?0 !99
(
C
!9
C
C(
!9 ==!
* # ==!
0 ? )'& ? & :N
A ? A ? )'&1
# E&)&)℄
A?)')& ; *&&!
/ # &))
# ;3: ?=('2<62' # $
# <
#
8 # '# <
# ' # = 2 #
)'))& < /'# #
# 6 # #
6!
:: : : ; /#
# ::
#
¿
9
! &,162C)) <-<11)& 1&-,,)(
9
! &,(),C)) (6-<1)& 6-)-&)(
9 ! 2)2(6)& (<&6))( 6=66))(
9
! &,,=)C)) &)6&,)( 611-1)(
9
! ,,<2,)& (-11=)( 66<(-)(
9
! ,,2-2)& ,&==6)( (16)2)(
9 !
&)((&C)) -6-&))6 (<<(6)(
9
!
9
! <)6)()6 -,=&<)1 (<<&6)(
9
!9
!9
! &,166)6 <<2=&)1 (<<)-)(
9
!9
!
,1(&2)< &2&,-)< (<6,))(
9
!9
!9
! &)&<-)6 1&,<&)1 (<6-<)(
9
!9
!9
! =<(2=)6 <2(6<)1 (<62,)(
:6/ 6,)=-)( (22-()1 (<62-)(
68 #
* # & ( #
% # ;3:
# ;3: # %
9*: %
Æ # # #
$ #
# # %
#
$ # #
%
# #
#
Æ $
7 #
#
−5
0
5
i
j
y
i,j
(1)
10
20
30
40
5
10
15
20
25
30
35
40
−1
−0.5
0
0.5
1
1.5
i
j
y
i,j
(10)
10
20
30
40
5
10
15
20
25
30
35
40
−1
−0.5
0
0.5
1
i
j
y
i,j
(20)
10
20
30
40
5
10
15
20
25
30
35
40
−1
−0.5
0
0.5
1
i
j
y
i,j
(30)
10
20
30
40
5
10
15
20
25
30
35
40
/# # %
#
$ / 9*:
# # ' '
'
# ' #
#
#
' ()
# % # ;: DN!
!
E&℄ ' ; / ; N J + &,--! $ # J$J9
#
.<, (&12(&-,
E(℄ ' ; / . 0 ; * &,-=!
# . <)3& (61(<<
E6℄ ' ;/ Q RJ &,,<!3
.=)3 = &&)2&&()
E<℄ ; ' ; / 5 0 &,-,! 9
.1) 31 &-26&-,=
E1℄ N &,,(! / 3 B%
E=℄ ' ; / ()))! #
.6& 3& &&&2
E2℄ #':+&,,6!
S "3 B%
E-℄ * ; + 5&,-,! / # 8
. ()3 &6616
E,℄ J ;I&,-,!/ # 8
.6)3&& &6-2&<&1
E&&℄ % 3 + * &,-)! I #
! .<1 2&(2&1
E&(℄ I / &,2&! " # /
3 B%
E&6℄ 0 " * ' %% ; / . 0 &,,(!
$ % & $' & " (
8 D
E&<℄ : + 5 &,,2! 5 # 8 &
.-- &,,((<
E&1℄ ; B%+/ J&,,&!
.=1 12,=&=
E&=℄ ;# " &,22! " T
.&1 3&
6&,6(-E&2℄ % * &,-&! ! $
3 -,- 6==6-&;.
E&-℄ 0 " " &,-1! #
22 6<&61(
E&,℄ . " ' J + / J &,,-! $ #