In the articlc proposed an effective method estim ating transfer function model o f controlled plant includ ing dead-time delay, based on slochatstic tim e series o f inpul-output sign
Trang 1V N U Journal o f Scicnce, M athem atics - Physics 24 (2008) 101-109
An application o f random process for controlled object
identification w ith traffic delay problem
V u Tien Viet*
D e p a rim e n t o f M a th em a tics, M echanics, Inform atics, C o lleg e o f Science, V N Ư
334 N guyen Trai, H anoi, Vietnam
R eceived 23 January 2007; received in revised form 20 March 2008
A b stra c t In the articlc proposed an effective method estim ating transfer function model o f
controlled plant includ ing dead-time delay, based on slochatstic tim e series o f inpul-output
signals T he model slm cture is modified w ith parameters optim ized until the m odel error
b ccom cs ”w h itc-n o isc” series that w ith inough srnal auto-corrclation function.
1 Propose
T he R eal sign als w h ic h occur in the control process alw ays im lp y influen ces o f m an y random factors, so th e D irective O b ject Identification Problem is often related to random process
M athem atically, th e C ontrolled O bjcct Iden tification p rob lem is th e pro blem th a t p red icts the
trend o f R andom P roccss: y { t ) — f { t ^ ĩ i ) -ị- v{ t ) , w here t - tim e; u - vector o f non -ran do m input variab les; / ( i , ĩi) - regressive fun ctio n th at reflects th e trend o f no n-rand om p ro cess or is th e m odel o f
the id en tificatio n p ro b lem ; v { t ) - random error
T h e T h eory o f P rediction and Identification has been stu d ied and developed w ith th ou san ds o f scien tific w o rk s m ade p u b lic since last century We can find th e fun d am en tal results o f studies o f
statistics and p red ictio n in [1,2], o f kinetics system identification in detail in [3,4
To use lin ear alg eb ra m etho ds, wci often try to changc th e regressive m odels into linear com
b in a tio n fo n n s o f coefficients: f { t ^ u ) — 5 w h ere Ci - param eters, - given
co m po nent fu nction s B y u sin g th is m odel, th e P aram eter Id en tificatio n P roblem can b e solved easily
H ow ever, th is m odel is n o t used to solvs th e analysis and syn th esise p ro b lem o f system s and w e have
to tran sfo rm th is m odel in to th e fo m i o f sets o f state equations (sets o f C auchy d ifferen tial equations)
or tra n sfe r fun ction fom i T here is a close, easy to exchange relation b etw een set o f state equations and tran sfer fu nctio n T h e tra n s fe r fu n ctio n ’s m odel o f controlled o bject is often in th e follo w in g form;
W { C , s) = ^ ^ ^ ^
w h ere Ó' - co m p lex nu m b er, r ^ 0 - th e dead tim e delay; r a ^ n - degree o f n um erato rs and denom inators;
c = { r , 6o, 6i , bjrv^ ao, a i , a-n} - vector o f param eters to be d eterm ined.
In th e classic w o rk s o f identification, all th e authors con centrated on dev elo p in g identification
m ethods b ased on pu re p o ly n o m ial fraction m odels w ith o u t the dead tim e d elay com ponents (i.e set Coưesponding author E -m ail: vutienvict.56@i]mail.com
101
Trang 2r — 0) In fact, exists r ^ 0, w e n o rm a lly t i y to use approxm iate p o ly n o m ial fractio n m od els w ith
h is h e r degrees o f m , n to in c re ase th e m o d el accuracy W ith th is ap proach, th e object id en tificatio n problem w ith o u t dead tim e d e la y is co n sid ered to b e co m p letely solved in th e o ry [1,4
In fact, how ever, a p p ly in g th e p u re p o ly n o m ial fractio n m eth od s to th e ob jects w ith dead tim e delay is relu ctant and in e ffe ctiv e in c o n tro llin g te ch n o lo g y p ro cesses such as energy, m etallurgy, becau se m ost o f th e o b je cts o b v io u sly h av e th e dead tim e delay To have th e n ece ssary m odel accuracy,
w e norm ally increase th e d eg ree o f p o ly n o m ial fractio n to a g reat value, and therefore m ak in g th e s}Tithesise pro b lem o f sy stem s m o re co m p lex , ev en lose its essence
D isregarding th e c h a ra c te ristic s o f dead tim e d elay o f an o b ject is one o f th e reasons th a t leads to
a great n u m b er o f research d ire c tio n s o f c o ntro l th e o ry im p ractically developed, even caused a ’’crisis”
in th e previous cen tu ry [5] To ac c u ra te ly re fle c t th e controlled object, w e have to co n sid er dead tim e
d elay as an existing p a ra m e te r in c lu d ed in th e m o del W hereas, clearly, th e m o d el is non -Iincar for the param eters In th is case, cla ssic m e th o d s are e ith e r ineffective o r inapp licab le
B ecause o f the ab o v e reaso n s, in o rd er to increase th e ap plicability, w e recom m end a controlled object identification m e th o d b a se d on u s in g d irc c tly m o del ( 1) alo n g w ith the dead tim e d elay r and other param eters T he fo llo w in g m eth o d is b ased on co n sid erin g th e tim e respo nse o f the object as a random data series
2 Estim ation o f the object m odel from output response data series
Suppose the c o n tro lled o b je ct has w e ig h t fiintion w { t ) w ith cfFect in pu t u{t)~ p red eterm ined , output response is m e asu re d : y { t ) — x { t ) + v { t ) , w h ere v { t ) - ad ditive n o ise (figure 1).
102 Vu Tien Viet / VNƯ Journal o f Science, Mathematics - Physics 24 (2008) Ỉ0 Ỉ-1 0 9
y[ - ^
Fig 1 Linear control system under random effect
W ithout loss o f g en era lity , w e re stric t v { t ) ^ u { t ) b e in g th e n o n -in terco irelatio n scalars, w here
v{t ) is W hite N oise, u { t ) is step p u lse:
^ ' [ 1 w h e n Í > 0
F orm erly [3,4], so as to solv e th e moQol estim atio n problem , w e b ased on p o p u la r relation
betw een output resp o n se x { t ) o f o b je c t and in p u t signal u{t )\
x { t ) = [ w { ^ u { t - Ẹ ) d ^
Jo
(3)
w here w { t ) is th e w e ig h t fu n ctio n
From (3), w e e sta b lis h th e P ro b lem o f D efin in g w e ig h t fiintion w { i ) upon least square condition:
d t m in (4)
\ 2
[ (y{t) - [
Jo \ Jo
Trang 3and then, define the tra n s fe r function i y ( 6) from th e w e ig h t fo n tio n w { t )
If w c p aram eterize the funtion w { t ) in th e fom i o f lin ear c o m b in a tio n , the (4) problem w ill
b cco m c lin ear to coefficien ts and can be solved easily H ow ever, in th is w a y it is com plcx to sclcct
co m p o n en t fu n ctio n s and causes the problem biiigcn and th e re fo re m ::kcs th e problem illconditioncd
To avoid th is d raw b ack , w e rccom m cnd d ircc tly u s in g th e m o d e l o f tran sfer function in fom i (.1) and salve th e m o d el estim ation problem b ased on th e in v erse l.a p la c e transform ation Indeed, if
w e c o n sid e r argu m en ts in th e L.aplace imatỉc dom ain, w e obtain:
Vu Turn Viet / m ư Journal o f Science, M athem atics - Physics 24 (2008) Ỉ 0Ỉ - Ỉ 09 103
— rs
X ( C , 6-) = VK(C, s ) U { s ) ^ ^0 + + + ^ (5 ^
1 -Ị-' + + cijiS
w h ere x { s ) L { x { t ) } , v y ( c s ) ^ L { w { t ) } , u { s ) = L{ u { t ) } - , s - com plcx variable; L {.}
roo
- n o tatio n fo r L aplacc tran sform ation; G{ s ) = L { g { t ) } = / g{ t ) e~' ' ^dt - Laplace transfom iation,
Jo from any g{t ) fun ctio n in real dom ain (g{t) — ^ 0) into c o rrcsp o n d in ơ im aac G{ s ) in com plex
dom ain
A cco rdin g to th e Inverse Laplacc tran sform ation , in [6 ] w e in fcrcd a sim p le form ula to computti
tim e response x { t ) from its L aplacc im aac:
1 rgijoo
w h e re g ' co n v erg in g ab scissa o f Laplace integral ( if o bjects are sta b le , w e can select ^ > 0 sm all enough, for in stance Q — 0.0 1 ); = —1 , P ( C ,o j i = R e { X ( C ,c j -\' j o c ) } - the real part o f
X { C , c u j o c )
S clect the u p p e r lim it (u^m) o f th e integral w h ic h is b ig en ougli, th e n trasfo rm into approxim ate sum fo n n , w c obtain:
x { C , t) = — V P ( C , LUr) COs{uJrt)
7T ^
r -^1
w h ere M “ the n u m b e r o f d iscrete po in ts in frcq u cn cy range: UJ = 0 ^
COM-From here, w e obtain square error betw een o u tp u t resp o n se an d real data:
f T
ơ ‘^ { C ) = / [y{t) — x { C , t ) ] ' ^ d t
Jo
m m c 0
w h ere T - th e am o u n t o f tim e to observe the random o u tp u t d ata series o f real objects.
R egard in g th e d iscrete po in ts o f tim e series, w e o b tain m in im iz a tio n objective function:
w here N - th e n u m b e r o f d iscrete p oints in an interval o f ob served tim e: Í = 0 ~ T
O bjective fu n c tio n ’s v alu e a ^ ( C ) is d eterm in ed a fte r a c o m p u tin g p ro cess in the follow ing order: c — > W { C , g - \ - j i o ) — ^ X (C , Ơ + jc j) — ^ P{ C ^ u j ) — > x { C ^ t ) — ^ (J^(C ) Therefore,
a ^ ( C ) is a co m p u tab le function and is continuous an d d iffere n tiab le every'w here O n the other hand, is obvious n o n -lin e a r fu n ctio n s to param eters and esp ecially h av e th e co m p le x c left (ravine) characteristic
W ith these ch ara cteristics, th e m o st effective m eth o d s to solve th e m in im iz e d problem (8 ) is to apply
”c le ft-o v c f’ o p tim izatio n algo rithm [7,8
Trang 4T h e so lu tio n to th e ( 8 ) pro blem w ith the selected structu re ( m , n , q) o f m odel (1) give u s an
op tim al estim atio n a;(C*, t) w ith ĩj{ti) series, and to s e th s r w ith th e optim al tran sfer function W^(C* s)
respectively
3 D eterm ine the optim al estim ation model
A s above, w ith each selected struture ( m n , q), an optim al so lutio n is ou tput response m odel a;(C*, t) and th e W { C * , s) op tim al tran sfer fu nction, respectively D ep en d in g on the selected (m , n , q) com b in atio n , T h e re are infinite structures o f th e m odel So, th e facin g pro b lem is to find a (rn, n , q) stru tu re so th a t th e co ư e sp o n d in g solution to th e pro b lem (8 ) b rin g s out th e response x (C * , t), w h ich
is the p ro p e r estim atio n fo r th e y{ti) tim e series
A c c o rd in g to [1,2], th e m odel is cosidered as a p ro p e r estim ation if th e obtained erro r series
b etw ee n th e g iv en m odel and tim e series becom o a radom d istrib u tio n range in the fon n o f "w hite
n o ise” A ssu m e a a sig nifican ce level, the m odel is consid ered to be accurate if G oưelation co efticien ts value ri o f e rro r series satisfy th e follow ed condition;
w h ere U a - is th e lim ited v alue obeying th e norm al d istrib u tio n rules, N - th e n u m b e r o f d ata sets o f
series
O n th e o th e r hand, th e (1) m ode] is fractio nal, so if w e increase th e (m , n ) degrees its accuracy
w ill increase as a resu lt P articularly, q is the n o n static degree o f m odel, it dep ends on th e b eh av io u r o f
ou tpu t resp o n se an d is equal to the degree o f th e asym ptote o f o utpu t response (J = 0 if the asym ptote
o f o u tp u t resp o n se is h o rizo n tal asym ptote <7 = 1 if the asy m p tote o f o utp ut response is oblique
asym ptote, q > I i f output has no asym ptote In fact, Ọ < 1 in m o st eases.
To d efin e th e g lobal optim al estim ation m o d el, th e step s to solve th e identificatio n prob lem arc
as follow s:
1 S ele ct th e degree o f q and fix it from th e o utpu t re sp o n se ’s behaviour
2 E x p lo rativ e ly select values o f d en o m in ato r’s degree n, and v alu es o f n o m in ato r’s degree
m = n — I sim ultaneously.
3 F o r each selected { m, n, q) structure w e solve th e (8 ) estim atio n m odel problem
4 W ith th e respective s) and x { c * , t) o btained, w e d etcm iin e error scries and check
th e cond itio n u p o n m o del suitability
5 I f th e co nd itio n is satisfied, th e obtained m odel is op tim al and th e respective W { C * , s)
tra n s fe r iu n c tio n is th e solu tion to th e identificatio n problem
6 I f th e m o del is n o t suitable, w e w ill select other m o dels w ith m and n ’s degree g rad ually increased and re o e a t from step 3
4 E x a m p le
S u p p o se th e ou tp ut sig nal o f an im plem en t controlled ob ject is the step p u lse in form (2 ) A t
th a t tim e, from th e output, th e m easured response signal in form o f tim e series is as follows:
104 Vu Tien Viet / VNU Jo u ’Tial o f Science, Mathematics - Physics 24 (2008) I0Ỉ-109
ti 0 0 1 0 0 0.2054 0.40 08 0.5962 0.7 9 1 6 0.9 87 0 1.1824 1.3778 1.5732 1.7686
Vi - 0 0 0 2 3 0.0 06 0 0.2 5 4 0 0.6000 1.0641 1.3125 1.4700 1.4126 1.2500 1.2191
Trang 5Vu Tien Viet / VNƯ Journal o f Science, Mathematics - Physics 24 (2008) Ỉ 0 Ỉ - Ỉ 0 9 105
tr 3.9181 4.1 1 3 5 4.3 08 9 4.5043 4.6 9 9 7 4.8951 Õ.0905 5.2859 5.4 8 1 3 5.6 76 7
in 0.9Õ00 0.9 9 0 0 0.9560 0.9960 0.9539 1.0400 0.9680 1.0510 0.9 9 0 0 1.0211
The G raph o f the tim e series y {t ) obtained from cx periem cnt is show n in figure 2
! ị ị \'•
-> ị
[ i
: ị
Fig 2 The curv'e o f output response data scries o f dircclivc object.
We id e n tify the tra n sfe r function o f objcct b ased on m odel (5) b y solving th e (8 ) problem w ith
d iftcrcn t stru ctu res T he im age o f input signal step p ulse; u { s ) = 1 / s T he b e h a v io u r o f the d ata
series above is co ư c sp o n d in g to th e nonstatic degree w h ere <7 = 0 H ence, w e on ly have to selcct the
suitable degree o f n u m erato r and denom inator ( m , n ) o f tran sfer fu nctio n (5 ), H av in g selected the
( r n ,n ) structure, w e solve the (8 ) problem by the ”cleft-o v er” algorithm [7,8'
o ái p
-vis.
a
0^
t>,7.
6.0
b.
t
o 4
0^
C.Í G.o ủ:
Fig 3 The error series of models
Trang 6Strutnres o f tested models'.
1 T h e first struture, w e choose; 771 = 0, n — 1 The rcsp ectivc o ptim al m odel is:
e - 0 3 6 4
T h e e ư o r series b etw een the output respo nse X i( c * , t) and the m easured d ata is on flizurc 3-a.
T he root-m ean-sq uarc E ư o r o f the m odel is ã = 0, 1429 B y u sin g th is m odel, th e estim atio n IS
o b v io u sly in c o ư e c t b ecau se th e condition \ri\ ^ >/]V is d e a r ly no t satisfied
2 T he second struture, w c choose: m = 0, n = 2 T he optim al m odel is:
W o(C * s) = - _g - 0,108.,
’ ’ l + a i 5 + a 2 s 2 ' l + 0 , 2 3 9 s + 0, 1 2 3 s 2'
T h e erro r series b etw een th e output response X 2 {C*^ t) and the m easured data is on figure 3-b
T h e root-m can-sq uare E rro r o f th e m odel is Ỡ “ 0, 0682 T h is m odel briniĩs out the in c o ư e c t estim ation
because the condition \ri\ ^ U 2 / \ / 7 ĩ is still not satisfied
3 W ith th e third strutu re, w e choose; m — 1, n — 3 T he optim al m odel is:
, 6 o ( l + ?ii6') 0 , 9 8 5 ( 1 + 1,1666')
(1 -I- a i s + a26'2) ( l + a s s ) ■ (1 + 0, 3 6 5 s + 0, 0986-^)(l + 0, 6836') ■
T h e erro r series b etw een th e output response ,X3(C*, t) and th e m easured d ata is on figure 3-c.
T h e roo t-m can -sq uarc E rro r o f th e m odel is Ỡ = 0, 0342 T h is m odel bring s out the n early correct
e s tim a tio n , th e c o n d itio n u p o n m o d el s u ita b ility |r,;| ^ U ẹ / \ Í N is n e a rly sa tis fie d
4 T he fourth stru tu re, w e choose: m = 2, n = 4 T he op tim al m odel is;
(1 + a i s + a 2 S^) {ỉ + a a s + 045^)
1, 001(1 + 0, 7 1 6 s + 0, 37s2) - 0,183
“ (1 + 0, 2 0 2 s + 0, 0 8 4 s 2 ) ( l + 0, 5 4 5 s + 0, 414s2)
T h e e ư o r series b etw ee n the output response a;4( c * , t) and the m easured d ata is on figure 3-d
T h e root-m ean-sq uare E ư o r o f th e m odel is o' — 0, 0212 T h is m o del b rings out the c o ư c c t estim ation
sin ce th e con dition up on m odel su ita b ility Ir^l ^ U i y / \ / N is co m p letely satisfied.
W h ile increasing th e degrees o f m ,n o f th e m odel, th e e ư o r series is alm o st n on-decrcasing
In th e o p tim izin g p rocess, th e old coefticients are alm o st in variable, w hereas th e ad ded co eftlcicn ts -
a risin g w h ile in creasin g th e degree o f m odel - are alw ays forced to zero b y the algorithm T he optim al
so lu tio n is n early at a stand still A ccording to th e se resu lts, th e tra n sfe r fun ctio n m odel W 4{C*^ s)
h as e rro r series w h ich is sim ilar to ’’w hite n o is e ” , and sim u ltan e o u sly yields T he root-m ean-square
E rro r m inim um T herefore, w e can considered it as th e global op tim al m od el o f object T he respective
o u tp u t resp on se o f th e m o del is show n on figure 2
Ỉ0 6 Vu Tien Viet / VNƯ Journal o f Science, M athematics - Physics 24 (2008) Ỉ 0 Ỉ - Ỉ 0 9
5 C onclusion
1 The objects w ith dead tim e d elay are p o p u la r class o f objects in in du strial control, its tra n s fe r fu n ctio n has n o n -lin ear pro perty for p aram eters, therefo re classic identification m ethods have
low eftectiv eness
Trang 72 R eco m m en d in g u sin g th e tran sfer fun ction w ith dead tim e delay as th e basic m odel and by
u sing the inverse L aplace transfo m iation w c obtain th e ou tp ut response o f the m odel On th is basis
w c solve the objcct id en tification problem in the fo n n o f th e tim e series estim ation problem based on
m easured random d a ta o f controlled objcct
3 T he rccom m cn dcd m ethod in this report en ab le us to solve the directive objcct id en tification problem licncrally and effectively u n d er th e random noise
4 The o p tim al m odel o f objcct d ctcm iincd b y the estim ation m ethod fo r non -linear random
m odel ensu res th e su ita b ility according to p ro b ab ility and s ta tis tic ’s p o in t o f view
References
|1] G Christian, A Monibrt, Times Deries and Dynamic Models, Cambridge University Press 1997.
[2] Nguyen Van IIuu Nguyen fluu Du, Statislic Analysis and Forecast, Hanoi National University' Press, 2003 (in Viet
namese).
13] P Eykluiil', System Identification Wiley 1974.
[4] P I''ykh<iir, Trench Ẵ: Progress in System Identification Pcrgamon Press, 1981.
[5] A.A Krasovski, Summarization o f Development History' and Situation of the Control Theory, Automation and ỉndĩistry Jour, Moscow, No 6-7 (1999) 1 (in Russian).
[6] Nguyen Van Manh, A Frcqucncy Method Calculating the Output Response o f a Line Automatic Control System without
Use of Il-lunction Table, hiformatic and Control Jour, Hanoi, No 1 (1995) 30 (in Vietnamese).
|7J Nguyen Van Manh, The Aliinc projection mclliod for solving non-lincar optimization problems, Proceeding o f NCST
of VictNam, No 2 (1992) 53.
[H] Nguyen Van Mdnh, M ethods Optimization o f Control Svslem fo r Unceriaini Processes Disscrtiilion of Science Doctor Moscow Power Energy Institute, 1999, (in Russian).
Pu Tien lĩet / VNU Journal o f Science, Mathematics - Physics 24 (200H) Ỉ 0Ĩ - Ỉ 09 107
Trang 8108 Vu Tien Viet / VNU Journal o f Science, Mathematics - Physics 24 (2008) 101-109
APPENDIX TESTED MODELS
Model 1: f v ^ ( c \ s ) = ^°—
1 + a^s The obtaúicd optimal coefficients: Òq = 1.065 Uj = 0.167 r = 0.364
Coưesponding output response Xj(c\t) computed by (6) on Oie basis o f image function
0 0 1 0 0 - 0 0 0 2 3 0 2 0 5 4 0 0 0 6 0
0 4 0 0 8 0 0 4 3 0 0 5 9 6 2 - 0 2 0 0 3
0 7 9 1 6 0 0 8 0 7 0 9 8 7 0 0 2 7 3 0
1 1 8 2 4 0 4 1 2 4 1 3 7 7 8 0 3 4 9 9
1 5 7 3 2 0 1 8 5 2 1 7 6 8 6 0 1 5 4 3
1 9 6 4 0 0 0 1 5 0 2 1 5 9 4 0 0 1 7 8
2 3 5 4 8 - 0 0 5 2 7 2 5 5 0 2 - 0 0 3 5 2
2 7 4 5 7 - 0 0 7 5 4 2 9 4 1 1 - 0 0 3 4 1
3 1 3 6 5 - 0 1 2 4 6 3 3 3 1 9 - 0 0 7 5 4
3 5 2 7 3 - 0 1 0 5 1 3 7 2 2 7 - 0 0 9 1 2
3 9 1 8 1 - 0 1 1 5 2 4 1 1 3 5 - 0 0 7 5 1
4 3 0 8 9 - 0 1 0 9 1 4 5 0 4 3 - 0 0 6 9 1
4 6 9 9 7 - 0 1 1 1 3 4 8 9 5 1 - 0 0 2 5 2
5 0 9 0 5 - 0 0 9 7 3 5 2 8 5 9 - 0 0 1 4 3
5 4 8 1 3 - 0 0 7 5 2 5 6 7 6 7 - 0 0 4 4 0
Model 2: ỈVyiC , s )
-1 + ^ -1^ + « 2*^
x e
The obtaine(toptimal coefficients; bọ = 1.033 ứị - 0.123 r = 0 1 0 8
Coưesponding output response X2(C*,l) computed by (6) on the basis o f image function
X , ( C , s ) = ^ , ( C , s ) / s ,
Tlie eưor series = X2 ( C \ t i ) - y ^ ( i = l, ,3 0 ) o fth e sccond model is:
0 0 1 0 0 - 0 0 0 2 3 0 2 0 5 4 - 0 0 3 1 4
0 4 0 0 8 - 0 0 3 0 5 0 5 9 6 2 - 0 0 4 1 8
0 7 9 1 6 0 0 7 9 6 0 9 8 7 0 0 0 8 1 5
1 1 8 2 4 0 1 2 0 3 1 3 7 7 8 0 0 6 1 3
1 5 7 3 2 - 0 0 2 2 5 1 7 6 8 6 c 0 6 0 5
1 9 6 4 0 0 0 3 0 6 2 1 5 9 4 0 1 1 2 1
2 3 5 4 8 0 0 7 9 7 2 5 5 0 2 0 0 9 7 6
2 7 4 5 7 0 0 3 2 6 2 9 4 1 1 0 0 3 7 5
3 1 3 6 5 - 0 0 8 7 5 3 3 3 1 9 - 0 0 6 3 2
3 5 2 7 3 - 0 1 0 5 3 3 7 2 2 7 - 0 0 9 1 6
3 9 1 8 1 - 0 1 0 7 7 4 1 1 3 5 - 0 0 5 6 3
4 3 C 8 9 - 0 0 7 9 3 4 5 0 4 3 - 0 0 3 1 3
4 6 9 9 7 - 0 0 6 9 6 4 8 9 5 1 0 0 1 6 7
5 0 9 0 5 - 0 0 5 7 8 5 2 8 5 9 0 0 2 1 6
5 4 8 1 3 - 0 0 4 2 9 5 6 7 6 7 - 0 0 1 4 3
Trang 9Vu Tien Viet / VNƯ Journal o f Science, M athematics - Physics 24 (2008) Ỉ 0Ỉ - Ỉ 0 9 109
—rs
Model 3; ll';(6''\,s) = - - : X e
(] + Oj.s + cir^s )(1 + a^ s )
The obtained optimal cocfficicnts: = 1.033, òj = 1.166, a ^ = 0 365, 0,2 = 0.098, a3 = 0.683,
r 0 2 2 6
Corresponding output response X3(C*,t) computed by (6) on Oic basis of image function
The error series ôy^ = — Ị ,3 0 ) o f Ihc lliừd model is:
0 0 1 0 0 - 0 0 0 2 3 0 2 0 5 4 0 0 0 6 0
0 4 0 0 8 0 0 5 2 2 0 5 9 6 2 - 0 0 5 0 6
0 7 9 1 6 - 3 5 E - 0 4 0 9 8 7 0 - 0 0 1 1 3
1 1 8 2 4 0 0 5 1 2 1 3 7 7 8 0 0 1 8 6
1 5 7 3 2 - 0 0 5 6 6 1 7 6 8 6 0 0 1 5 2
1 9 6 4 0 - 0 0 3 5 0 2 1 5 9 4 0 0 3 1 3
2 3 5 4 8 - 0 0 0 1 5 2 5 5 0 2 0 0 3 3 5
2 7 4 5 7 - 9 8 E - 0 4 2 9 4 1 1 0 0 3 9 6
3 1 3 6 5 - 0 0 5 2 6 3 3 3 1 9 - 0 0 0 4 7
3 5 2 7 3 - 0 0 3 4 8 ' 3 7 2 2 7 - 0 0 1 9 7
3 9 1 8 1 - 0 0 4 2 0 4 1 1 3 5 - 1 7 E - 0 4
4 3 0 8 9 - 0 0 3 2 6 4 5 0 4 3 0 0 0 8 6
4 6 9 9 7 - 0 0 3 2 6 4 8 9 5 1 0 0 5 4 0
5 0 9 0 5 - 0 0 1 7 8 5 2 8 5 9 0 0 6 5 4
5 4 8 1 3 0 0 0 4 5 5 6 7 6 7 0 0 3 5 6
W , { C \ s ) — 0^(1 + bịS + ^2'"’'^) 2 \ / i _ _.2x
(1 + Cí^ố' + + CLz^i +
The oblaincd optimal cocfficicnls: òy = 1.001 = 0.716 = 0.202 = 0.084 = 0.545
a, = 0.414 T 0.183
Coưesponding output response X3(C*,l) compuícd by (6) on tlie basis o f image function
Thc eưor scries ỗy^ = x A C \ t J - 2/^ ĩ 1, ,30) o f the fourlli model is;
0 0 1 0 0 - 0 0 0 2 3 0 2 0 5 4 - 0 0 0 3 0
0 4 0 0 8 0 0 2 5 5 0 5 9 5 2 - 0 0 3 3 5
0 7 9 1 6 0 0 1 8 0 0 9 8 7 0 - 0 0 1 8 2
1 1 8 2 4 0 0 2 9 5 1 3 7 7 8 0 0 0 5 7
1 5 7 3 2 - 0 0 4 9 8 1 7 6 8 6 0 0 3 4 4
1 9 6 4 0 - 0 0 2 0 1 2 1 5 9 4 0 0 2 9 1
2 3 5 4 8 - 0 0 2 1 5 2 5 5 0 2 0 0 0 5 5
2 7 4 5 7 - 0 0 2 3 0 2 9 4 1 1 0 0 3 4 1
3 1 3 6 5 - 0 0 3 8 6 3 3 3 1 9 0 0 2 4 4
3 5 2 7 3 6 9 E - 0 4 3 7 2 2 7 0 0 1 3 2
3 9 1 8 1 - 0 0 1 7 9 4 1 1 3 5 0 0 1 2 7
4 3 0 8 9 - 0 0 3 0 1 4 5 0 4 3 0 0 0 3 3
4 6 9 9 7 - 0 0 4 3 0 4 8 9 5 1 0 0 4 0 7
5 0 9 0 5 - 0 0 3 2 4 5 2 8 5 9 0 0 5 0 0
5 4 8 1 3 - 0 0 1 1 3 5 6 7 6 7 0 0 1 9 7