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TRANSCRIPT
Modeliranje dinamičnih sistemov z umetnimi nevronskimi mrežami
in sorodnimi metodami
Juš Kocijan
Založba Univerze v Novi Gorici
2015
Modeliranje dinamičnih sistemov z umetnimi nevronskimi mrežami in sorodnimi metodami Juš Kocijan 2. izdaja Recenzenta: prof. dr. Aleš Belič in prof. dr. Igor Škrjanc Jezikovne korekture: prof. dr. Gregor Kocijan Založnik: Založba Univerze v Novi Gorici, Vipavska 13, Rožna dolina, Nova Gorica Leto izida: 2015 ISBN 978-961-6311-92-2 (pdf)
CIP - Kataložni zapis o publikaciji Narodna in univerzitetna knjižnica, Ljubljana 681.5.033:004.032.26(075.8)(0.034.2) KOCIJAN, Juš Modeliranje dinamičnih sistemov z umetnimi nevronskimi mrežami in sorodnimi metodami [Elektronski vir] / Juš Kocijan. - 2. izd. - El. knjiga. - V Novi Gorici : Založba Univerze, 2015 ISBN 978-961-6311-92-2 (pdf) 281954560
To delo je objavljeno pod licenco Creative Commons. Avtor/ji besedila dovoli/jo reproduciranje, distribuiranje, dajanje v najem in priobčitev javnosti samega izvirnega avtorskega dela pod pogoji: priznanje avtorstva in nekomercialno. Celotno besedilo licence je na voljo na spletu, na naslovu http://creativecommons.si/licence Copyright © 2015 Založba Univerze v Novi Gorici. Vse pravice pridržane. Razmnoževanje in fotokopiranje dela v celoti ali po delih brez predhodnega dovoljenja Založbe Univerze v Novi Gorici je prepovedano.
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y = 30[100(x− 0.6)2(x− 0.1)(x− 0.8)− 5e−5x
+ 0.9 sin(20x+π
4)]
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0 0.2 0.4 0.6 0.8 1−150
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50
100
150
200
Rezultati identifikacije (1 skriti nevron)
x
y
prava f.
model
!"#$ %&R( >.<6"*812#" +.,0.+/,86 D .6"- 6.*,868- * 2#,"?
/.- 6"*815
0 0.2 0.4 0.6 0.8 1−150
−100
−50
0
50
100
150
200
Rezultati identifikacije (2 skrita nevrona)
x
yprava f.
model
!"#$ %&T( >.<6"*812#" +.,0.+/,86 D 9*.-$ 6.*,868-$ *
2#,"/.- 6"*815
0 0.2 0.4 0.6 0.8 1−150
−100
−50
0
50
100
150
200
Rezultati identifikacije (5 skritih nevronov)
x
y
prava f.
model
!"#$ %&%S( >.<6"*812#" +.,0.+/,86 2 +./"-" 6.*,86" * 2#,"?
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1
xy
z
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xy
z
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S k r i t in i v o
V h o d i u I z h o d y
@10., !"!KA F-*G, -,90,140= %,(40= :84.605
',605/.& :84.605& f : Rm→R< .0 5* ',+*',+034& &)0/,4,
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y = f(x) =N∑
i=1
wigi(x, γi). !"M$
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:84.605* /&A
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ρ2 )< N,8//&2, :84.605, /10., !"!O$<
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1
2
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Originalna funkcija
y(k)
y(k
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−5
0
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Aproksimacija z veliko baznimi funkcijami
y(k)
y(k
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-# /9:<=
y(k) = a1y(k − 1) + a2y(k − 2)
+ b1u(k − 1) + b2u(k − 2)
+ v(k); /H2K=
u ( k ) y ( k )y ( k )B ( z )
A ( z )
1A ( z )
v ( k )
- 1
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L1)-% H2HD ;!"#$% +%* %+*(), -.%$&%"#. 5 8#581#(!+)
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y(k) = a1y(k − 1) + a2y(k − 2)
+ b1u(k − 1) + b2u(k − 2)
+ v(k) + c1v(k − 1) + c2v(k − 2);
/H2M=
• !"#$% ' )',#$+) 8#0&!(-# /GN=
y(k) = a1[y(k − 1)− v(k − 1)]
+ a2[y(k − 2)− v(k − 2)]
+ b1u(k − 1) + b2u(k − 2)
+ v(k). /H2O=
u ( k ) y ( k )y ( k )B ( z )
A ( z )
C ( z )A ( z )
v ( k )
- 1
- 1
- 1
- 1
L1)-% H2ID ;!"#$% +%* %+*(), -.%$&%"#. ' #$!1# (3 %
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v ( k )
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y(k) = 0.9512y(k − 1) + 0.09754u(k − 1) /H2X=
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H(z) =0.09754z−1
1− 0.9512z−1/H2Y=
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8&!*(+*), "&!+"3-),=D
y(k) = −a1y(k − 1) + b1u(k − 1). /H2ST=
[! .5"%.) # +% !5"# &!0&!5#&*!. /y(k−1), u(k−1)= )+ 8&!4$)-A)* /y(k)= !&*!+! .&!$+#5") 5)0+%1#.> 1%,-# '%8)(! #!+%FB# /H2ST= . %"&)F+) #B1)-)D
y = ΨΨΨθθθ
y(2)y(3)2
2
2
=
−y(1) u(1)−y(2) u(2)2
2
2
2
2
2
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θθθ = [ΨΨΨTΨΨΨ]−1ΨΨΨTy. =>9?>@
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0 10 20 30 40 50 60
8.5
9
9.5
vzorci (Ts=0.01 s)
Stopničasti signal na vhodu
0 10 20 30 40 50 607
7.5
8
8.5
vzorci (Ts=0.01 s)
Odziv sistema
B("!$ A6:;C ?%#"' 1$ 0=&,1"5$0=& 0,-/3/3*&
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G&'&-"(" 3$(& ,/)/6 L"03& "#*-$(" (/ /1/G$ 'F&%1/G$
0 2 4 6 8 10 12 14-2
-1
0
1
2
čas [s]
PNBS vhodni signal v delovni točki
0 20 40 60 80 100 120 140 1600
0.5
1
1.5
krožna frekvenca [rad/s]
Spekter amplitudne gostote
B("!$ A6::C @F&%1" 0"G1$( "1 1)/G&'$ G&0=&=$ $3,("=+%1/G$
0,/!=-$
0"G1$($ 3$-'/5 )/ *"( &,"0$1" 0"G1$( "#*-$1 3/% %/0/="3"
,& ($0=1&0="F ,&%&*1"3" 0"G1$(" HKLMB 0"G1$(" -$#("51"F
%&(2"1 "1 =$!=&' !$- ,&3/1" -$#("51/ 0,/!=-/ $3,("=+%1/
G&0=&=/I #$ !$=/-/ 03& &,-$'"(" 3/-"='/ "1 D/(/ 1$ ,&%($G"
'-/%1&=/1)$ "#*-$(" 1$),-"3/-1/)D" 'F&%1" 0"G1$( "1 +0=-/S
#1& 3/-"=/'6 U=/'"(& '#&-./' 1$) *& 5"3 '/5)/ !$)=" 5"3
'/5 )/ '#&-./' =/3 *&()D$ )/ "%/1="V!$.")$6 @F&%1/3+ 0"S
G1$(+ !" 03& G$ ='&-"(" # -$5+1$(1"!&3 %&%$3& ,-/%1$S
,/=&0= =$!& %$ *& 1)/G&'$ 0-/%1)$ '-/%1&0= /1$!$ %/(&'1"
=&5!" W6O @6 ?%#"' 0"0=/3$ ' %/(&'1" =&5!" # "#(&5/1& /1&S
03/-1& !&3,&1/1=& H%/(&'1$ =&5!$I ,-"!$#+)/ 0("!$ A6:A6
0 2 4 6 8 10 12 14-1.5
-1
-0.5
0
0.5
1
čas [s]
Odziv sistema v delovni točki
0 20 40 60 80 100 120 140 1600
0.5
1
1.5
krožna frekvenca [rad/s]
Spekter amplitudne gostote
B("!$ A6:AC ?%#"' "1 1)/G&'$ G&0=&=$ $3,("=+%1/G$ 0,/!=-$
K-/% $1$(&G1& %"G"=$(1"3 ,-/='&-1"!&3 ($F!& "#'/%/3&
$1$(&G1& V(=-"-$1)/ H0("!$ A6WI "1 1$ =$ 1$5"1 ,-/,-/5"3&
#G"*$1)/ 4-/!'/1. 9:;<6 @ 1$D/3 ,-"3/-+ )/ *"( #$ =& %&*/-
!$- 3/-"(1" &)$5/'$(1"! !" "3$ 4-/!'/151& !$-$!=/-"0="!&
!
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4!.$#)*& %&'$( )*& %$3.$52& $2&5)(&, !#& !'6$ 6$*,! !"7
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496&/
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! @/A@B ! %$ %$-.)1! "6& 24&-! (%,!() 496& %.)3(')6 "&
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3& ) 1! %$ )%$-.)1 )6 6$3),!.&,! 3! &6!#)= #! &'-$%!
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!3) -!+#&;!2$ 3.9*)*&/ F)%&.&6)-.!5 ) 6$3),) $1!5&2 $
9%$.&1!6$ "& (.)3 $-) 2) %&.&6)-.!5 !?/
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D"1!.& '-.9#-9.) 6$3),& 2) %.! !3) -!+#&;!2! %&.&6)-.!5 )*&
6$3),& "),$ %$6)61 &= '&2 ,&?#$ " 6)-$3&6! "& $;) 2)(&7
2) %&.&6)-.$( 3$,$5!6$ ,) (.)3 $'-! %&.&6)-.$(/ G)
6$3), %.$;)'& 1$6$ !"1.&,!= 2) $3(!' $ $3 &6) & 6$3),&/
H &4)6 %.!6).9 I),!6$ ,! )&. ! %&.&6)-.!5 ! 6$3),= #!
1$ 5!6 1$,2 &-& 5 $ $%!'$(&, '!'-)6 $#$,! 3),$( ) -$5#)=
() 3&. &2 ) 1! 1!, %.)(),!#)*& .)3&/ J) !6& 6$3), ()7
,!# .)3= '!;). 1$,2) $%!'92) 3! &6!#$= () 3&. 2) -93! 1$,2
#$6%,)#') ! !6& %$,) ! !5,)= #! ') 6)3 ')1$2 %.!1,!I $
#.&24&2$/
D"1!.&,! '6$ 6)3 6)-$3&6!= #! '$ $%!'& ) ( KALM ! '6$ 2!?
I) $6) !,!/ :.! !3) -!+#&;!2! ,! )&. !? '!'-)6$( ') 6$.&6$
$3,$5!-! ,) "& .)3 6$3),&/ :.($ $;) $ .)3& 6$3),& ,&?#$
3$1!6$ " & &,!"$ ! E$.6&;!2'#) 6&-.!#) &,! %& " !3) -!+#&7
;!2$ )%&.&6)-.!5 )*& 6$3),& 7 E.)#() 5 !6 $3"!($6/
C ! E$.6&;!2'#$ 6&-.!#$ KALM '! ,&?#$ %$6&*&6$ -&#$N H.)7
3 $'- 3)-).6! & -) ! E$.6&;!2'#) 6&-.!#) det( [ΨΨΨ
TΨΨΨ]−1
N)=
#2). '-& ΨΨΨ 6&-.!#& %$3&-#$( >%$3.$1 )2) $%!'& & %$" )2)
" ) &51&6& >@/AOB ! >@/APBB ! N 3!6) "!2& 6&-.!#) %$7
3&-#$(= &.)3! %.)'#$# %.! .)39 ! E$.6&;!2'#) 6&-.!#)= !
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%.$;)'/ H.)3 $'-! 3)-).6! & -) ! E$.6&;!2'#) 6&-.!#) '$
1!,) ( &4)6 %.!6).9N "& %.(! .)3 L/QARS= "& 3.9*! L/LLA@=
"& -.)-2! 4.6501 · 10−7= "& 5)-.-! .)3 1.4037 · 10−10
! -&#$
&%.)2/ F&2()52! %&3); (.)3 $'-! 2) 6)3 3.9*!6 ! -.)-2!6
.)3$6= -$.)2 2) !3) -!+;!.& ! %.$;)' 3.9*)*& .)3&/
T.9*! &5! "& 3$,$5!-)( .)3& 6$3),& 2) )%&.&6)-.!5 &
!3) -!+#&;!2&/ U,)3) & -$= 3& '6$ '! !"1.&,! &%).!$3!5 !
(?$3 ! '!* &,= ,&?#$ 3$,$5!6$ E.)#() 5 ! $3"!( " ()5 6)7
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U,&2) 2) 2) %$'-$%)# +,-.!.& 2& ' '%.)6) ,2!(!6! 9-)I6!
>E.)#() 5 $ &,! 5&'$( $ $# $B= ' #&-).!6! ,&?#$ !",$5!6$
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*,&2) 2)= '&2 6$.& 1!-! -$ $"#$ ( %.!6).2&(! ' 5&'$6 $%&7
"$(& 2& ! 3$($,2 4!.$#$= 3& $?.& ! 1!'-() $ ! E$.6&;!2$/
:$3.$1 $'-! $ *,&2) 29 &23)6$ ( KALM/ X,!#& @/AQ %.!7
#&"92) &6%,!-93 ! ! E&" ! E.)#() 5 ! $3"!( ( "& !6!()6
E.)#() 5 )6 %$3.$529 1.)" *,&2) 2&= #! '6$ *& 3$1!,! '
V$9.!).$($ & &,!"$ >$3"!( 1! 1!, ) &#= 5) 1! ("),! '%)#7
-.&, $ & &,!"$ 1.)" *,&2) 2&B ! " *,&2) 2)6 ' 5&'$( !6
Z&66! *$(!6 $# $6 (),!#$'-! QO ("$.;)(= #! '6$ *& 3$7
1!,! ' '%)#-.&, $ & &,!"$/
100
101
102
10-2
10-1
100
101
Amplitudni odziv
Krožna frekvenca [rad/s]
|G(jw
)|
100
101
102
-400
-200
0
Fazni odziv
Krožna frekvenca [rad/s]
Fa
za
[sto
pin
je]
X,!#& @/AQN [6%,!-93 ! ! E&" ! $3"!( 1.)" *,&2) 2& >5.-7
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%& &!"+)5+ - 0#%&% !")#2+ "#.+; (+<5& .&5+$ 5&%!(#3
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& "+,(&>#/# ) 5/$-2+< FGA; HI7
J.(&6-(- %& # ,+ /+$#/& 0+)/#$'&; 0& $# /#"#51",-)/& %#3
0&.& /+$%+/$'-< 5)+."+0&) , %&.#(&% !& !(&'#/#2+ !&3
2"#'5+ FGAI7 K+ %#0&.+ .+$#; "+,#/ !"- !& #4/- &4(-5- '13
%/#2+ C(0"+; !"- 0"+/ 5# "#,1(0+0#7 E"- 0"+/ 5& 0 (+<5&
,%+/$'+%& , 1 0"#,/-% C(0"-"+/$#%; !"- 6#%#" -0#"+0-)/&
!&-'6#%& 6-% 4&($'- C(0#"7 K+5'/- !& 0&!5- & 2(#.# !"-3
0"+/ 5& 0- !&.&4/- ."12-% %#0&.+% &?#/$#)+/$+ !+"+3
%#0"&) 9%#0&.- "+,'-"$#/-< /+$%+/$'-< 5)+."+0&); %#0&.-
/+$)#6$# !&.&4/& 0-:7
L#0&.+ /+$%+/$'-< 5)+."+0&) ) 4- 0)1 !&%#/- "#'#)+/$#
!"#.&(&6#/#2+ - 0#%+ #/+647 @ !"-%#"1 &?#/$#)+/$+ !+3
"+%#0"&) /#,/+/#2+ !"&?# +; %&0#/#2+ '1%&%; &?#/-%&
!+"+%#0"# 0+5&; .+ 1!&"+4-%& %+0#%+0-6/- %&.#( !"&?# +
-/ %-/-%-,-"+%& 5)+."+0 "+,(-5# %#. -,<&.&% !"&?# + -/
%&.#(+7
M/+64+ %&0#/#2+ !"&?# + ) !"& 0&"1 z # 2(+ -
y(z) =B(z−1)
A(z−1)z−du(z) + n(z), 9H7GB:
5$#" &N A(z−1) 3 -%#/&)+(#? !"#/& /# *1/5?-$# ) !"& 0&"1
z; B(z−1) 3 '0#)#? !"#/& /# *1/5?-$# ) !"& 0&"1 z; y 3 -,3
<&./- -2/+(; u 3 )<&./- -2/+( -/ n 3 %&0/$+;
&,-"&%+ ) 6+ &)/#% !"& 0&"1
y(k) = a1y(k − 1)− a2y(k − 2)− . . .− any(k − n)
+ b1u(k − d− 1) + . . .+ bnu(k − d− n)
+ a1n(k − 1)− a2n(k − 2)− . . .− ann(k − n).
9H7GO:
@#50&" 5- ,+!- 0# #/+64# $#
y(k) = ψψψT (k)θθθ, 9H7GP:
5$#" $#
θθθ = [a1, . . . , an, b1, . . . , bn] 9H7GQ:
)#50&" !+"+%#0"&)7 J?#/& !+"+%#0"&) -,"+61/+%& ) %-3
(1 /+$%+/$'-< 5)+."+0&) 9-,!#($+)+ -/ !&2&$- & !&."&4/&
&!- +/- ) FGAI: !& #/+64-
θθθ = [ΨΨΨTΨΨΨ]−1ΨΨΨTy, 9H7GR:
5$#" $# y )#50&" -,<&./#2+ -2/+(+ -/ $# %+0"-5+ ΨΨΨ # 0+3
)($#/+ -, )#50&"$#) ψψψ ,+ 1 0"#,/# )"#./& 0- #(#%#/0&) )#53
0&"$+ y7
S-(0"-"+/$# $# !"-!&%&6#5; 5+0#"-% (+<5& .& #>#%&; .+
& "#,1(0+0- 6-% 4&($ /#!"- 0"+/ 5-7 E"&?# ; 5- 2+ &!- 1$#
#/+64+ 9H7GB:; (+<5& ,+!-'#%& 01.-
A(z−1)y(z) = B(z−1)z−du(z) +A(z−1)n(z). 9H7GT:
L#0&.+ /+$%+/$'-< 5)+."+0&) .+$# /#!"- 0"+/ 5# "#,1(3
0+0#; 6# $# A(z−1)n(z) = v(z) 4#(- '1%; 0& !&%#/-; 6#
.&4-%& n(z) -, v(z) !"#5 C(0"+
1
A(z−1)7 K& #)#.+ ) !"+5 -
-/ 01.- ) /+'#% !"-%#"1 /- "# ; ,+0& $# !&0"#4/& C(0"-"+3
/$#7 U# # &.(&6-%& ,+ C(0"-"+/$#; %&"+%& C(0"-"+0- , - 0-%
C(0"&% )<&./- -/ -,<&./- -2/+(7
M/+64& 9H7GT: .#(-%& C(0"&% F (z−1) -/ .&4-%&
A(z−1)y(z)
F (z−1)= B(z−1)z−d u(z)
F (z−1)+A(z−1)
F (z−1)n(z).
9H7HA:
V#!"- 0"+/ 5# "#,1(0+0# -.#/0-C5+?-$# .&4-%&; 6# $# /+!+5+
#/+64# 9H7HA: &,-"&%+ /$#/ ,+./$- 6(#/ 4#(- '1%7
@,#%-%&; .+ $# n(z) 4#(- '1%7 @ 0#% !"-%#"1 .&4-%&
/#!"- 0"+/ 5# "#,1(0+0#; 6# )#($+ A(z−1) = F (z−1); 0& !&3
%#/-; 6# C(0"-"+%& , /#,/+/-% -%#/&)+(?#% !"#/& /# *1/53
?-$# /#,/+/#2+ !"&?# +; 5- 2+ -.#/0-C?-"+%&7
@ !"+5 - #)#.+ n(z) /- 4#(- '1% 9$# !+ (+<5& 4(-># 4#(#%1
'1%1 5&0 !+ &)/& &%#$#/#%1 ,
1
A(z−1):7 D+0& 1!&"+4-%&
&4(-5& C(0"+ 9/!"7 W100#"X&"0<&) C(0#":; 5- %&"+ 4-0- 0+3
5'#/; .+ 4& ,+./$- 6(#/ #/+64# 9H7HA: 6-% 4&($ 4#(; 5+" !+
!"#)#"$+%& -0#"+0-)/& , )"#./&0#/$#% "#,1(0+0&)7
E& &%#/$#/#% !& 0&!51 %& -.#/0-C?-"+(- %&.#(# "+,(-63
/-< "#.&) -/ # !& )"#./&0#/$1; 5+0#"#2+ #(#%#/0# &!- 1$#
/+ (#./$# !&2(+)$#; &.(&6-(- ,+ %&.#( ."12#2+ "#.+; !"- 6#3
%#" %& -%#(- /+ )&($& .)# %&>/& 0-; 5- $1 4&%& '# /+!"#$
)"#./&0-(-; .+ 4- -,4"+(- 0- 0&; 5- 4& !"-%#"/#$'+ ,+ /+'
/+%#/7
E"- -,4"+/#% ."12#% "#.1 .&4-%&; 6# /# C(0"-"+%& )<&3
./#2+ -/ -,<&./#2+ -2/+(+; !"#/& /& *1/5?-$& %&.#(+ !"&3
?# +
Gpn(z) =0.0281z−1 + 0.0165z−2
1− 1.6379z−1 + 0.6890z−2z−2. 9H7HG:
K& $# !"#/& /+ *1/5?-$+ ."12#2+ "#.+ ) z !"& 0&"1 0+3
4-(/-%- !&(- -/ *+,/& /#%-/-%+(/& /-6(& 0#" .&.+0/& ,+5+3
/-0)-$& 9.)+ 6+ &)/+ ),&"?+; 5+" $# A7AB #51/.#:7
E& -0#"+0-)/#% !& 0&!51 .&(&6#)+/$+ C(0"+; 5+0#"-% /+$
4- 6-% 4&($ -,!&(/-(- !&2&$ ,+ /#!"- 0"+/ 5& 0; %& # &.3
(&6-(- ,+ W100#"X&"0<&) C(0#" ."12#2+ "#.+ -/ (&%/& *"#53
)#/?& A7APG "+.Y -/ # !& )"#./&0#/$1 &.(&6-(- ,+ %&.#(
34
! "#$%&# $'&()$*+,-$.& %&/,)#0(1+&2, 1$1)&%,
!"#$#% !$ %& '( #% )*(+",$ *!$-)+-% ."-'/(,%
Gpf (z) =0.0590z−1
− 0.0272z−2
1− 1.7558z−1 + 0.7923z−2z−2. 012113
4!( ) 5)6(78(& '%7$!$#% (9:$ );$< :) $5)8 ;):) (9;!%5(&
+( *):%#%:) 9 8!$ -)7$-,$:2 =5%8-$ $5$:$-7$ 8!$ -)7$>
-,% )*(+",$ -%+5$ -,( !%9 $5$'2
"#&1+31 4&5.,4(01)$ %0'&5,
?(+7( $5 *)+7)*'% ( $-7(@'%/(,$& '( *)8$& '%') );$! ,$ :)>
$5 (- '%, :)!%:) +*!$:$-(7( 0!$ & :$7) %& 8<) -( +(#-%5&
6%+ 89)!6$-,%3& ,$ -%,*):$:;-$,A( $5 ( $-7(@'%/(,$2 ?) ,$
*!$+'"+ 8$5,%8-)+7( :) $5% %5( 7" ( 8!$ -)7$-,$ :) $5%&
'( +$ (98%,% $)&#,)$4(0 (- *!( '%7$!$: "*)A7$8%:) +8),%
)*%9)8%-,% (- 9%'5,"6'$ (- +$ -$ 9%-%A%:) 5$ -% $5-$ !$>
9"57%7$& '( ,(< %,$ !%6"-%5-('2 B%7$!( :) $5 +(+7$:% ;):)
(9;!%5(& -( ) 8(+-) 5$ ) (9*)5-,$8%-,% 9%<7$8& %:*%' 7" (
) -%:$-% :) $5%2 C% *!$+'"+ 8$5,%8-)+7( 5)6(:) 8$6 *)>
+7)*')82 4)+7)*'$ ;):) *!('%9%5( 9 !$9"57%7( 8!$ -)7$-,%
(9;!%-(< :) $5)8 (9 $-%6; 0121D3 (- 0121132
"#&1+31 602#&7+, %0'&5,
4!( :$7) ( -%,:%-,A(< '8% !%7)8 -%, ;( ;(5% -%*%'% :)>
$5% $-%'% ;$5$:" A":" EDFG2 C%7) 5%<') "*)!%;5,%:)
-%+5$ -,$ *!$+'"+$H
• I!$ -,% 8!$ -)+7 +(#-%5% -%*%'$ e(t) :)!% ;(7( -(62
J9 5$8(< $5)8 +5(' 12DK (- 12DL 8( (:)& % -%*%'%
"+7!$9% 7$:" *!$+'"+"& 8$- %! ;)5, *!( :) $5"& )>
;5,$-$:" (9 @57!(!%-(< +(#-%5)82
0 2 4 6 8 10 12-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
čas [s]−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.20
10
20
30
40
50
60
70
80
90
napaka e(k)
Porazdelitev napake e(k)
I5('% 12DKH M%*%'% e(k) :$ ) 9(8):% *!$-)+-$ ."-'/(,$
Gpn (- *!)/$+%& "*)!%;5,$-$#% 9% ( $-7(@'%/(,) 05$8% +5('%3&
7$! *)!%9 $5(7$8 +(#-%5% -%*%'$ e(k) 0 $+-% +5('%3
• 4)!%9 $5(7$8 +(#-%5% -%*%'$ e(k):)!% ;(7( +(:$7!(6-%&
'%! 5%<') *)7! (:) -% *) 5%#( $+-(< $5)8 +5(' 12DD
(- 12DK2
• I(#-%5 -%*%'$ e(k) :)!% ;(7( -$) 8(+$- ) *!$7$'5(<
8<) )8 0φeu(τ) = 03 9% τ < 02 4!( );!%7)8%-,"
8 ) *!7( 9%-'( :)!% ;(7( -%*%'% -$) 8(+-% ) 8+$<
8<) )82 I5('( 12DN (- 12DO *!('%9",$7% '!(P-) ')!$5%>
/(,) :$ -%*%') (- 8<) -(: +(#-%5): (- 8( (:)& %
,$ -,"-% 8!$ -)+7 *)8+) :%,<-%2
0 2 4 6 8 10 12-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
čas [s]−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.20
0.02
0.04
0.06
0.08
0.1
0.12
0.14
napaka e(k)
Porazdelitev napake e(k)
I5('% 12DLH M%*%'% e(k) :$ ) 9(8):% *!$-)+-$ ."-'/(,$
Gpf (- *!)/$+%& "*)!%;5,$-$#% 9% ( $-7(@'%/(,) 05$8% +5('%3&
7$! *)!%9 $5(7$8 +(#-%5% -%*%'$ e(k) 0 $+-% +5('%3
-15 -10 -5 0 5 10 15-0.03
-0.02
-0.01
0
0.01
0.02
premik
Križna korelacija med e(k) in u(k)
I5('% 12DNH B!(P-% ')!$5%/(,% :$ -%*%') e(k) (- 8<) -(:
+(#-%5): u(k) 9% *!$-)+-) ."-'/(,) Gpn
-15 -10 -5 0 5 10 15-0.03
-0.02
-0.01
0
0.01
0.02
premik
Križna korelacija med e(k) in u(k)
I5('% 12DOH B!(P-% ')!$5%/(,% :$ -%*%') e(k) (- 8<) -(:
+(#-%5): u(k) 9% *!$-)+-) ."-'/(,) Gpf
• Q87)')8%!(%-6-% ."-'/(,% +(#-%5% -%*%'$ e(k) :)!%
;(7( $57% (:*"592 J9 +5(' 12DR (- 12DS 8( (:)& % (:%
%87)')8%!(%-6-% ."-'/(,% -%,8$6,) 8!$ -)+7 *!%8 *!(
*!$:('" τTF& *!( *!$)+7%5(< *!$:('(< *% ,$ :%,<-%2
U 7) 5%<') +'5$*%:) -% *) );-)+7 0-$ $-%')+73
9 $57% (:*"59):2
4!( 8+$< ):$-,$-(< *!$+'"+(< 5%<') -% *) 5%#( +5(' "#)7)>
8(:)& % );$ *!$-)+-( ."-'/(,( (9*)5-,",$7% *)#),$& 8$- %!
,(< ;)5, (9*)5-,",$ *!$-)+-% ."-'/(,%& );5,$-% 9 ( $-7(@'%>
/(,) @57!(!%-(< +(#-%5)82
B)-+(+7$-/% 8<) -)>(9<) -$#% );-%A%-,%
V) $5 8!$ -)7(:) 7%')& % *!$+'"+(:)& '%') +$ ) 9(8% -%
8<) -$ +(#-%5$& *!( '%7$!(< *)9-%:) ) 9(8 *!)/$+%2 4!(
7$: (:%:) 8$ :)P-)+7(H
• 8!$ -)7$-,$ )*!%8(:) -% *) %7'(<& '( +:) ,(< "*)>
!%;(5( 9% ( $-7(@'%/(,)W
8
!"#$%&'()%*( +%#"(,#%- !%#(.%/#%- 0%0$".12
-15 -10 -5 0 5 10 15-0.5
0
0.5
1
premik
Avtokovariančna funkcija e(k)
!"#$ %&'() *+,-#-+$."$/0/$ 12/#3"4$ 5"6/$!$ /$7$#8 e(k)9$ 7.8/-5/- 12/#3"4- Gpn
-15 -10 -5 0 5 10 15-0.5
0
0.5
1
premik
Avtokovariančna funkcija e(k)
!"#$ %&':) *+,-#-+$."$/0/$ 12/#3"4$ 5"6/$!$ /$7$#8 e(k)9$ 7.8/-5/- 12/#3"4- Gpf
• +.8;/-,8/48 -7.$+"<- /$ 7-;$,#"=> #" 4"= /"5<- 27-?
.$@"!" 9$ ";8/,"A#$3"4-> +8/;$. + "5," ;8!-+/" ,-0#"&
B$#$ +.5,$ +.8;/-,8/4$ 7-+8 +8!"#- +80 - +8!4$+/-5,"
<-;8!$ #-, +.8;/-,8/48 /$ 7-;$,#"=> #" 5<- 4"= C8
27-.$@"!" 9$ ";8/,"A#$3"4-&
!"#" %&%D "/ %&%' 7."#$9248,$ .892!,$,8 +=-;/-?"9=-;/86$
-@/$E$/4$ 9$ -@8 +.5," 7-;$,#-+ 9$ -@8 "9@.$/" 7.8/-5/"
12/#3"4"& !"#$ %&%D 7."#$9248 7."<8.4$+- 0$5-+/"= -;9"+-+
-@8= <-;8!-+ /$ +=-;/" 5"6/$!> 27-.$@!48/ 9$ ";8/,"A#$?
3"4- "/ 0$5-+/" 7-,8# /$7$#8 <8; <8.48/"< -;9"+-< "/ -;?
9"+-< <-;8!$ Gpn ,8. <8; <8.48/"< -;9"+-< "/ -;9"+-<
<-;8!$ Gpf 9$ 7-;$,#8> #" 5<- 4"= 27-.$@"!" 9$ ";8/,"A#$?
3"4-& !"#$ %&%' 7$ 7."#$9248 7."<8.4$+- 0$5-+/"= -;9"+-+
-@8= <-;8!-+ /$ +!$# 7.$+-#-,/"= "<72!9-+ "/ 0$5-+/"
7-,8# /$7$#8 <8; <8.48/"< -;9"+-< "/ -;9"+-< <-;8!$
Gpn ,8. <8; <8.48/"< -;9"+-< "/ -;9"+-< <-;8!$ Gpf
9$ 7-;$,#8> #" 4"= /"5<- 27-.$@"!" 9$ ";8/,"A#$3"4-& F9 5!"#
!$=#- +";"<-> ;$ 5- /$7$#8 7." <-;8!2 Gpf + 7-+7.8042
<$/4E8 #-, 7." <-;8!2 Gpn&
B2;" 9$ 7-;$,#8> #" /"5- @"!" 27-.$@!48/" 9$ ";8/,"A#$3"4->
<-.$4- +8!4$," +5" 7.85#25" "9 7.84E/486$ 7-;7-6!$+4$> + #$.
@" 58 !$=#- 7.87."0$!"> 08 @" "9."5$!" 25,.89/8 ;"$6.$<8&
F9 -;9"+-+ /$ 5!"#$= %&%D "/ %&%' !$=#- +";"<-> ;$ 7.8/-?
5/$ 12/#3"4$> ;-@!48/$ "9 A!,.".$/"= 5"6/$!-+> @-!48 -7"5248
-@/$E$/48 7.-385$&
G-/5"5,8/3$ 1.8#+8/0/86$ -@/$E$/4$
H48<$/48 <8; 1.8#+8/0/"< -;9"+-<> #" 5<- 6$ ;-@"!" 5
0 2 4 6 8 10 12 14-1.5
-1
-0.5
0
0.5
1
čas [s]
Vrednotenje v časovnem prostoru
0 2 4 6 8 10 12 14-0.3
-0.2
-0.1
0
0.1
0.2
čas [s]
Vrednotenje - izhodna napaka
!"#$ %&%D) "6/$! 9$ ";8/,"A#$3"4-
2 3 4 5 6 7 8-1.5
-1
-0.5
0
0.5
1
čas [s]
Vrednotenje v časovnem prostoru
0 2 4 6 8 10 12 14-0.2
-0.1
0
0.1
0.2
0.3
čas [s]
Vrednotenje - izhodna napaka
!"#$ %&%') "6/$! 9$ +.8;/-,8/48
I-2."8.-+- $/$!"9-> "/ 1.8#+8/0/"< -;9"+-< 7.8/-5/8 12/#?
3"48 Gpf 7."#$9248 5!"#$ %&%%& H48<$/42 0$5-+/"= -;9"+-+
<-.$ 5!8;"," ,2;" 248<$/48 1.8#+8/0/86$ -;9"+$ "/ /$.-@8&
J$ 5!"#" %&%% !$=#- +";"<-> ;$ 48 1$9/" #-, 9 /$.$E0$4-0-
1.8#+8/3- $@5-!2,/- 6!8;$/- +8;/- +804"& B- 48 7-5!8;"3$
<.,+86$ 0$5$> #$. 58 248<$ ,2;" 7." <-;8!2> #" "<$ 9$#$?
5/",8+ D&DK 58#2/;8&
!"#$%&'#( )*+*!$(+',- +$./01"2* !'.$%*- 0',*3
+"*&4&* !*(+"0*- &*)*0$ )*+*!$(+',- /)'+*5&'#(
!'.$%*
<"58!/-5, ;"5#.8,/86$ <-;8!$ 9+89/86$ 7.-385$ !$=#- <8;
;.26"< 7.85#25"<- 5 ,8<> ;$ 7-6!8;$<- E,8+"!- 7-!-+ /$
66
! "#$%&# $'&()$*+,-$.& %&/,)#0(1+&2, 1$1)&%,
100
101
102
10-2
10-1
100
101
Amplitudni odziv
Krožna frekvenca [rad/s]
|G(jw
)|
100
101
102
-400
-200
0
Fazni odziv
Krožna frekvenca [rad/s]
Fa
za
[sto
pin
je]
!"#$ %&%%' ()*+#,+ -)*#.*/0/*1$ 23/$4$/5$ 6)*/2+/*
-,/#7"5* Gpf 829:". Gpf ; 62!/$ #)".,!5$< 6)"=*)5$!/" -)*;
#.*/0/" 29:". ; 0)>#$/$ #)".,!5$?
/*1$>"./" )*$!/" 2+"< #" /* +=* 3">" !"@2& A /$4*= 6)"=*),
/"=$=2 /23*/*1$ /$ /*1$>"./" )*$!/" 2+"& ()$. >$#2 +*
=2)$ =29*! + 6)27*+2= ,5*=$>" 62 +>$3"!/2+>"&
>$/9$)9/2 9*."$7"52 923"=2 + #2)*/5*/5*= 9"$12/$!/"@
*!*=*/>2. #2.$)"$/0/* =$>)"#* /$6$# 6$)$=*>)2. BCDE& F$;
#2 923"=2 #.$/>">$>"./2 =*)2 :$,6$/5$ 27*/ 6$)$=*>)2.&
A*05$ 5* #2.$)"$/7$< .*05* +2 9*."$7"5*< =$/54$ 5* :$/*+!5";
.2+> "/ .*!5$./2+> =29*!$&
G2.$)"$/0/2 =$>)"#2 ":)$0,/$=2 : */$032
cov[θθθ − θθθ] = σ2
eE[ΨΨΨTΨΨΨ]−1
, 8%&%H?
#5*) 5* σ2
e .$)"$/7$ /$6$#* =29*!$ BCDE& I$ 6)*/2+/2 -,/#;
7"52 Gpn +2 9*."$7"5* 62+$=*:/"@ 6$)$=*>)2. /$+!*9/5*'
a1 = −1.6379;σ = 0.0117;
a2 = 0.6890;σ = 0.0112;
b1 = 0.0281;σ = 0.0013;
b2 = 0.0165;σ = 0.0017;
:$ 6)*/2+/2 -,/#7"52 Gpf 6$ 923"=2 9*."$7"5*'
a1 = −1.7558;σ = 0.0040;
a2 = 0.7923;σ = 0.0035;
b1 = 0.0590;σ = 0.0014;
b2 = −0.0272;σ = 0.0019.
J$@#2 26$:"=2< 9$ +2 9*."$7"5* 6$)$=*>)2. =29*!$ 9),;
1*1$ )*9$< 923!5*/*1$ ": K!>)")$/"@ +"1/$!2.< . +6!24/*=
=$/54* #2> . 6)"=*), =29*!$< 923!5*/*1$ ": /*K!>)")$/"@
+"1/$!2.& L: .+*@ 6)"#$:$/"@ )*:,!>$>2. .)*9/2>*/5$ !$@#2
,12>2."=2< 9$ =29*!< 923!5*/ ": K!>)")$/"@ +"1/$!2.< /$;
>$/0/*5* 26"+,5* 23/$4$/5* 6)27*+$< #" +=2 1$ "9*/>"K7";
)$!"& M* .*0 629)23/2+>" 2 .)*9/2>*/5, :$ >$ 6)"=*) 5* .
9*!, BNE&
A*=2 BOE< 9$ /* =2)*=2 +#!*6$>" /$ !*12 62!2. . :.*:/*=
6)2+>2), ": !*1* 62!2. . 9"+#)*>/*= 6)2+>2),& P3"0$5/2 +*
/"0!* . /*+#2/0/2+>" . )$./"/" s 6)*+!"#$52 . 62!* /$ /*1$;
>"./" )*$!/" 2+" 2:")2=$ . 3!"Q"/2 #22)9"/$>/*1$ ":@29"40$
. )$./"/" z& L: .+*1$ >*1$ 3" !$@#2 +#!*6$!"< 9$ :.*:/" 6)2;
7*+< #" 1$ "9*/>"K7")$=2< 6)" 6)*962+>$.#"< 9$ 5* 9),1*1$
)*9$< /"=$ /"0*! . #2/0/2+>"& F2 62>)5,5* >,9" +!"#$ %&%%<
6)" #$>*)" !$@#2 ."9"=2< 9$ "=$ $=6!">,9/" 29:". 6)" ."+2;
#"@ -)*#.*/7$@ /$#!2/ ND 9RS9*#$92& L9*/>"K7")$/" =29*!
9),1*1$ )*9$ "=$ /"0!2 . #2/0/2+>" 8s = −37.2989?< :$>2+* /5*12. 29:". 6)" ."+2#"@ -)*#.*/7$@ /$532!5* /* ,5*=$
+ -)*#.*/0/"= 29:".2= 6)27*+$& T* 62/2."=2 62+>26*#
"9*/>"K#$7"5* : 9"+#)*>/"= =29*!2=< #" "=$ /"0!2 6)" :UD<
923"=2 /$+!*9/5" =29*!
Gp(z) =0.0390z−2
1− 1.7144z−1 + 0.7589z−2z−2. 8%&%N?
F$#2 923!5*/" -)*#.*/0/" 29:". 8+!"#$ %&%H? +* :*!2 923)2
,5*=$& V5*=$/5* 0$+2./*1$ 29:".$ 8+!"#$ %&%N? /" >$#2 92;
3)2< .*/9$) 5* 923!5*/ )*:,!>$> :$ .*0"/2 6)$#>"0/"@ ":.*93
.29*/5$ 4* +6)*5*=!5".&
0 2 4 6 8 10 12 14-2
-1
0
1
2
čas [s]
Vrednotenje v časovnem prostoru
!"#$ %&%H' W*! 0$+2./*1$ 62>*#$ 29:".$ 6)27*+$ 80)>#$/$
#)".,!5$? "/ =29*!$ Gp 862!/$ #)".,!5$? /$ .!$# 6)$.2#2;
>/"@ "=6,!:2.
100
101
102
10−2
10−1
100
101
Amplitudni odziv
Kro na frekvenca [rad/s]
|G(jw
)|
100
101
102
−400
−200
0Fazni odziv
Kro na frekvenca [rad/s]
Faza [
sto
pin
je]
!"#$ %&%N' ()*+#,+ -)*#.*/0/*1$ 23/$4$/5$ 6)*/2+/*
-,/#7"5* Gp 829:". Gp ; 62!/$ #)".,!5$< 6)"=*)5$!/" -)*;
#.*/0/" 29:". ; 0)>#$/$ #)".,!5$?
T* 3" Q*!*!" /$>$/0/*54" 2912.2) 2 =29*!,< 6)*9.+*= 2 /5*;
12.*= 23/$4$/5, 6)" ."+2#"@ -)*#.*/7$@< 3" =2)$!" ":.*+>"
929$>/* 62+#,+* 2:")2=$ 62/2.">" 62+>26*# "9*/>"K#$7"5*&
3
!"#$%&'()%*( +%#"(,#%- !%#(.%/#%- 0%0$".12
!"#$%!&! &!#!&'&()# *(+$,! -& ./(0!1&()# *(+$,! +(,(2
'$&$ &!#!&'&()#- 3$ (+%-)&! /0$+%)$* (+ &!*$&! &3$4(%$
./(0!1$5 !#( 3$ #! (6$&! /0$/.7'$&! ./(0!1&-8.5 9(2
+$,$ ,!"8( ./(0!1,3!*( :! 0!:,-'&$ &!*$&$; 8(# )( &!'02
#(%!&3$ 0$4.,!6-3)8-" !,4(0-#*(%; /($&()#!%,3!&3$ 8(*/,$2
8)&-" )-)#$*(%; &!'0#(%!&3$ )-*.,!#(03$%; &!'0#(%!&3$ !,2
4(0-#*(% :! (+80-%!&3$ &!/!8 -#+5 < &!7$* /0-*$0. )*(
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?+$&#->6-0!&- *(+$, 3$ 4,$+$ &! 0$:.,#!#$ %0$+&(#$&3! +(2
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34
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872/.3 >.2,.3 ?'9 #$%#9 @<78302A72B
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737H/<1/7K20H19 $I,O3I.2.9 +<,A. 0Q+.I.B
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y(k) = g(y(k−1), y(k−2), . . . , u(k−1), u(k−2), . . .)+n(k).NJ<UP
!"#$%&'! (#!)! &$ *$%+"+,- "!,%$ *#!*+#+(!.#-/-#+% ($,!01
.2,- 3! %-&$ *#!"!3 %+23!%!4
?R
!"#$%&'()%*( #"+%#"(,#%- !%#(.%/#%- 0%0$".12
!"#$%&'()%*( +,-"#% %.'(#*" /0#')%*" g(·)
y(k|θθθ) = g(ψψψ(k), θθθ), 12345
'*"6 .$(7
ψψψ(k)= ψψψ(u(k − 1), u(k − 2), . . . , y(k − 1), y(k − 2), . . .)8"'$,6 6"96".,6*"8:
θθθ 8"'$,6 +(6(-"$6,83
;6,<="- =(>', 6(?!"=%-, #( !8( +,!+6,<="-(7
• #( %?<%6, 8"'$,6*( 6"96".,6*"8 ψψψ(k) %#
• #( %?<%6, #"=%#"(6#" +6".=%'(8" g(ψψψ)3
3"4,"01,*%
;6% =%#"(6#%> .%.$"-%>: +6% '($"6%> *" .$60'$06( =%#"(6#(:
=(>', .%.$"-: '% *" ,+%.(# 8 ,<=%'%
A(z−1)y(z) =B(z−1)
F (z−1)u(z) +
C(z−1)
D(z−1)v(z), 12325
?(+%@"-, ?"=, "#,.$(8#, ',$
y(k) = ψψψT (k)θθθ. 123A5
B(',6 .-, ,+%.(=% C" 8 +6"*@#*"- +,9=(8*0: ." -"$,!" %!"#D
$%&'()%*" =%#"(6#%> .%.$"-,8 6(?=%'0*"*, +, 0+,6(<=*"#%> 6"D
96".,6*%>3 ;,!,<#, 8"=*( $0!% ?( #"=%#"(6#" .%.$"-" %# -"D
$,!" %-"#0*"-, +, 6(?=%E#%> 0+,6(<=*"#%> 6"96".,6*%> ψψψ: .
'($"6%-% +,#(?,6%-, #"=%#"(6#,.$
y(k) = g(ψψψ(k), θθθ). 123F5
G"96".,6*% ψψψ +6% 6(?=%E#%> #"=%#"(6#%> -,!"=%>7
• -"$,!( ',#E#"9( %-+0=?#"9( ,!?%8( 1(#9=3 #,#=%D
#"(6 &#%$" %-+0=." 6".+,#." D HI G5: '(6 +,-"#%: !(
., 6"96".,6*% .(-, ?('(.#*"#% 8>,!#% .%9#(=% 1u(k −i)5J
• -"$,!( #(*-(#*@%> '8(!6($,8 . +,.+=,@"#%- +,96"@D
',- 1(#9=3 #,#=%#"(6 (0$,6"96"..%8" -,!"= K%$> "L,D
9"#,0. %#+0$ D HMGN5: '(6 +,-"#%: !( ., 6"96".,6*%
?('(.#*"#% 8>,!#% %# %?>,!#% .%9#(=% 1u(k − i), y(k −i)5J
• -"$,!( ? %?>,!#%- +,96"@',- 1(#9=3 #,#=%#"(6 ,0D
$+0$ "66,6 -"$>,! D HOP5: '(6 +,-"#%: !( ., 6"96"D
.,6*% ?('(.#*"#" 86"!#,.$% 8>,!#"9( .%9#(=( %# ,)"#
%?>,!#"9( .%9#(=( 1+6"!%')%* %? +6"$"'=,.$%51u(k − i):y(k − i)5J
• -"$,!( 6(?@%6*"#%> #(*-(#*@%> '8(!6($,8 1(#9=3 #,#D
=%#"(6 (0$,6"96"..%8" (#! -,8%#9 (8"6(9" -,!"= K%$>
"L,9"#,0. %#+0$ D HMGQMN5: '*"6 ., 6"96".,6*% ?(D
'(.#*"#% 8>,!#%: %?>,!#% %# @0-#% .%9#(=% 1u(k − i):y(k − i), ε(k − i) = y(k − i)− y(k − i)5J
• -"$,!( R,L D S"#'%#. 1HRS5: '% %-( ?( 6"96".,6*" ?(D
'(.#*"#" 8>,!#" .%9#(=": ?('(.#*"#" ,)"#" %?>,!#"9(
.%9#(=(: #(+(', #(+,8"!% %# #(+(', .%-0=%6(#" #(D
+,8"!% 1', .-, ?( #(+,8"! 0+,6(<=*(=% ,)"#" %?>,D
!#"9( .%9#(=(51u(k− i), y(k− i), ε(k− i), εu(k− i) =y(k − i)− yu(k − i)5J
• -,C#( *" $0!% +6"!.$(8%$"8 8 +6,.$,60 .$(#*: '*"6 #"D
=%#"(6#% .%.$"- ?(+%@"-, 8 ,<=%'% 7 x(k) = F(x(k −1)u(k − 1)):
• !609% -,C#% 6"96".,6*%3
5"+%#"(,#" 6,"0+%'(2"
H"=%#"(6#" +6".=%'(8" #(*8"E'6($ +6%'(C"-, ',$ 0$"C"#,
8.,$, <(?#%> /0#')%* gk(ψψψ)7
g(ψψψ(k), θθθ) =∑
k
αkgk(ψψψ), 123T5
E"+6(8 <,-, '(.#"*" +6% -,!"=%6(#*0 ? U(0..,8%-% +6,)".%
8%!"=%: !( ., -,9,E" $0!% !609" ,<=%'" +,#(?,6%$"83
V#(# .'(=(6#% +6%-"6 +,#(?,6%$8" ? 0$"C"#, 8.,$, *" #(
+6%-"6 6(?8,* /0#')%*" . I,06%"6,8, 86.$,3 W%+%E#% +6%-"6%
#"=%#"(6#%> +6".=%'(8: +,#(?,6*"#%> ? 0$"C"#, 8.,$,: .,7
• 8(=E#" /0#')%*":
• R ?="+'% 1(#9=3 .+=%#".5:
• XQPWHP HPYGOHZBP QGP[P:
7 8"E#%8,*.'% +"6)"+$6,#:
7 -6"C" 6(!%(=#%> <(?#%> /0#')%* 1(#9=3 6(!%(= <(D
.%. /0#)$%,# #"$K,6'5:
7 !609,:
• QP\B QO]P^ 3
;,9="*-, .% ?(+%." #"'($"6%> ?#(#%> <(?#%> /0#')%*7
• #"86,#.'" -6"C" . .%9-,%!#, ('$%8()%*.', /0#')%*,
gk(ψψψ) = σ(βkψψψ + γk),
'*"6 .$( β @%6%#( %# γ +,=,C(*J
• -6"C" ? 6(!%(=#%-% <(?#%-% /0#')%*(-%
gk(ψψψ) = r(βk(ψψψ − γk));
89
!" #$%&'(& & )*+(,)-./0)1) (+%)(+/2()3 4)4,+5&6
• !"#$ %&!'$
gk(ψψψ) =∑
j
yj(∏
µA(ψψψ));
• (%)*+,-! *!.!
ψψψ(k) = g(ψψψ(k − i), θθθ)
• $- -%/! &*0/!1
7/$&,.) 8/ )89)2& 4,2:.,:2+
2+3(*!3 (*$4-! % 5 5$5,! +,$4-% $67$*% *!/*!5%*3!) %& -+38
7%'3 !-%5,+)-! %.-! #% 7$-+9$3! &% 7%'3 #% ('!#5-$":
• u(k) 8 5,+,$4-+ -!'$-!+*-%5,;
• u(k − i) 8 2<=>;
• u(k − i), y(k − i) 8 2?>@;
• &*0/%1
A% &%'%4+ % *!& %&!'+ %6$*% + B,!)$'% 6+#+5-$,!) )6%*8
9!); 3! #%*$5,-%; &+ 5! 6+)!&+ % C+#!-5%)!/+ ,!%*! +
D-3!/%)+ %7'$#+ 6+ )6703+-! 5$5,! ! 3! ) EFGHI; #$ (%)!; #+#%
$6 )6%*4!-$" )"%&-$" $- $6"%&-$" 5$/-+'%) *!#%-5,*0$*+ %
%&!' 6)!6-!/+ &$-+ $4-!/+ 5$5,! +1 J+&%5,-$ (%/%3 6+
*!#%-5,*0#9$3% 5$5,! + 3!:
n ≥ 2p+ 1, DK1LI
#3!* 3! n *!& &$5#*!,-!/+ %&!'+ $- p *!& %*$/$-+'-!/+ 6)!8
6-!/+ 5$5,! +1 M (*+#5$ 5! $6#+.!; &+ 3! +*5$#&+3 &%)%'3
%&!' +-3B!/+ *!&+; #+* 3! %&)$5-% %& 5$5,! +:
p ≤ n ≤ 2p+ 1. DK1NI
A% 5 % *!& %&!'+ $67*+'$; 5! '%,$ % $67$*! 7+6-$" O0-#9$31
P%.-$" 3! )!4 -+4$-%) $67$*! $- ,% %73!#,$)-$" $- 5073!#8
,$)-$"1 A%-#*!,-$" (*+)$' -$; 5+3 5% )5! 7%'3 6-+-! 7+6-!
O0-#9$3!; #$ 5 % 3$" -+B,!'$; 5(%5%7-! (%'307-% -+,+-4-!
(%-+6%*$,)! -!'$-!+*-%5,$1 2!#+,!*$ -+(%,#$; #$ 7$ 3$" '+"#%
0(%B,!)+'$; 5% EQLH:
• *+&$+'-! 7+6-! O0-#9$3! $67!*! %; #+&+* $ + % %8
(*+)#+ 6 +3"-$ B,!)$'% *!/*!5%*3!) D-(*1 )+'4-!
O0-#9$3! 6+ -+3)!4 K *!/*!5%*3!I;
• 5'! !-+5,! D+-/'1 *$&/!I 7+6-! O0-#9$3! $67!*! %;
#+&+* $ + % %(*+)#+ 6 )!43$ B,!)$'% *!/*!5%*3!)
D-(*1 5$/ %$&-! -!)*%-5#! *!.!I;
• !"#! %&!'! 0(%*+7$ %; #+&+* %75,+3+ )-+(*!3B-3!
"!)*$5,$4-% 6-+-3!1
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%&!'+; %*+ % &%'%4$,$ B! -3!/%)! (+*+ !,*!; #$ 5% 0,!.$
(%5+ !6-$" 7+6-$" O0-#9$3 $- %&)$5-% %& $67*+-! 5,*0#8
,0*! B! -!#+,!*$ &*0/$ (+*+ !,*$1 C% '+"#% 5,%*$ % 5 #+8
,!*% #%'$ 6-+-% $- 05,*!6-% &!,!* $-$5,$4-% +'$ 5,%"+5,$4-%
!,%&% %(,$ $6+9$3!1 P!& &!,!* $-$5,$4-$ $ !,%&+ $ 5!
&%7*% %7-!5!3% R+05582!S,%-%)$ +'/%*$, $; !&,! #% 5%
/*+&$!-,-! !,%&! (*)!/+ *!&+ ) 5('%B-! 4+5%)-% (%,*+8
,-!1 P%.-+ 3! -!5(*%,-+ %(,$ $6+9$3+ (+*+ !,*%) +'$ (+
5(*%,-+ ,+#% $ !-%)+-+ *!#0*6$)-+ $&!-,$T#+9$3+1 A*$,!*$3$
%(,$ $6+9$3! 5% (% -+)+&$ %&)$5-$ %& (%/*!B#+ %&!'+1
;&<2+'+. 5&*+%/
U(,$ $6+9$35#$ #*$,!*$3$ 5% *+6'$4-$1 U&)$5-$ 5% %& -+ !-+
%&!'+ $- %& %(,$ $6+9$35#! !,%&! $- -+*%7!1 A*$,!*$38
5#+ O0-#9$3+; 5 #+,!*% 3! %(,$ $6+9$35#$ #*$,!*$3 6+($5+-; 3!
#)+-,$,+,$)-% $6*+.!-% !*$'% #)+'$,!,! %&!'+1 V% -+)+&$
.!'$ % &%5!4$ 4$ +-3B% )*!&-%5, #*$,!*$35#! O0-#9$3!; #$
3! -+3)!4#*+, O0-#9$3+ (%/*!B#+ %&!'+1 V%/%5,% 0(%*+8
7'3!- (*$ !* 3! #*$,!*$35#+ O0-#9$3+; %&)$5-+ %& #)+&*+,+
(%/*!B#+ %&!'+:
V (θθθ) = E ‖ y(k)− g(ψψψ(k), θθθ) ‖2
= σ2
n + E ‖ g0(ψψψ(k))− g(ψψψ(k), θθθ) ‖2,
DK1WI
#3!* 3! g0(ψψψ(k)) %*$/$-+'-$ 5$5,! $- σ2
n )+*$+-9+ B0 +1
A+&+* %(,$ $6$*+ % %&!' /'!&! -+ #)+&*+, (%/*!B#+; 5%
) 5('%B-! %.-$ ,*$3! )$*$ -+(+#; #$ #)+*$3% #)+'$,!,% %8
&!'+:
• B0 ;
• (*$5,*+-5#%5,; #$ 3% '+"#% (*$#+.! % #%,
V = E ‖ g0(ψψψ(k))− g(ψψψ(k), θθθ(m))) ‖2,
#3!* 3! θθθ(m) )!#,%* %9!-3!-$" )*!&-%5,$ (+*+ !,*%)$67*+-! &%'.$-! m;
• $- )+*$+-9+ )*!&-%5,$ (+*+ !,*%); #$ 3! %&)$5-+ %&
)+*$+-9! B0 + !*$,!) σ2
n
V ≈ σ2
n
dim θθθ
N.
2! %*! % $&!-,$T9$*+,$ 7%'3B!/+ %&!'+; #%, ,% &%(0B4+
)5!7$-+ $-O%* +9$3! ) !*3!-$" )"%&-%8$6"%&-$" (%&+,#$"1
M+*$+-9+ )*!&-%5,$ (+*+ !,*%) $- (*$5,*+-5#%5, 5,+ !&
5!7%3 (%)!6+-+; #%, (*$#+603! 5'$#+ K1F1
7/46+,) 8/ )86+*9& )*+(,)-./0)1+
R'!&! -+ #% ('!#5-%5, (*%7'! + $&!-,$T#+9$3! -!'$-!+*8
-$" &$-+ $4-$" 5$5,! %) 3! ,!.#% (*!&($5+,$ -+,+-4!- (%8
5,%(!#1 M 5('%B-! $ + (%5,%(!# !-+#! #%*+#! #%, 6+
$&!-,$T#+9$3% '$-!+*-$" 5$5,! %); #$ 5 % /+ -+)!&'$ ) (*!38
B-3! (%/'+)301 X+"#% (+ -+B,!3! % -!#+3 -+5)!,%); #$ 7$
(*$(% %/'$ # 04$-#%)$,!3B! 0 &!'01
=>
!"#$%&'()%*( #"+%#"(,#%- !%#(.%/#%- 0%0$".12
V r e d n o s tk r i t e r i j s k ef u n k c i j e
t e v i l op a r a m e t r o v
V a r i a n c aP r i s t r a n s k o s t
!"#$ %&'( )*"#$+ ,-./#$ *$+0/*1$ 0/2 ,*"3.*$43#-3.1- "4
5$*"$46-
'& 78!/1./ 3" 59-24-:"+9-24/ 0/*1/4/ ,-2$.#/& ;+ 41"9
!$9#- 3#!/,$0-< $!" 1/ 3"3./0 +/!- 4/!"4/$*/4 "4 #$:
#=4/ 3- 41/8-5/ >$3-54/ #-43.$4./ ?#-. 3/ ,-8-3.-
@,-*$A!1$ B$*8-4 "+ "2/4."C#$6"1/ !"4/$*4"9 2"4$0">:
4"9 3"3./0-5D&
E& F$1,*/1 ,-3#@3"./ /4-3.$54/ 3.5$*"< .- 1/ /4-3.$54/
3.*@#.@*/ "4 2"0/4+"1/ 0-2/!-5( 4$1,*/1 !"4/$*4" 0-:
2/!" "4 0$194- =./5"!- */8*/3-*1/5< A$+4"9 G@4#6"1 "4
,$*$0/.*-5 +$ -6/41/5$41/&
%& )-"=>"./ C+"#$!4- -+$21/ 2"4$0">4/8$ ,*-6/3$< 3$1
!$9#- "+ 41/8$ 2-A"./ "2/1- +$ "+A"*- */8*/3-*1/5&
H& )-2$.#/ *$+2/!"./ 4$ ."3./ +$ "2/4."C#$6"1- "4 ."3./ +$
5*/24-./41/ 0-2/!$&
I& )-2$.#/ 4-*0$!"+"*$1./ ?*$+!">4/ 5*3./ ,-2$.#-5 .*$4:
3G-*0"*$1./ 5 /4 5/!"#-3.4" *$+*/2D "4 @3*/2"=>"./&
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-25 -20 -15 -10 -5 0 5 10 15 20 25-1
-0.5
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−3 −2 −1 0 1 2 30
10
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100Histogram pogreškov simulacije modela
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−10
12
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0 20 40 60 80 100-2
-1
0
1
2
Vhodni signal za identifikacijo
čas [s]
0 20 40 60 80 100-2
-1
0
1
2
Odziv sistema za identifikacijo
čas [s]
!"#$ %&N( O-,07";#$/"<6#" 6"?0$! "0 2-4"8
0 20 40 60 80 100-2
-1
0
1
2Vhodni signal za vrednotenje
čas [s]
0 20 40 60 80 100-2
-1
0
1
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Odziv sistema za vrednotenje
čas [s]
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5
10
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Porazdelitev za identifikacijo
amplituda signala−1.5 −1 −0.5 0 0.5 1 1.50
5
10
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Porazdelitev za vrednotenje
amplituda signala
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−3−2
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−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.80
20
40
60
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100
120
140Histogram pogreška
pogrešek
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?$ 8-01+,50+30& J? 4!"#0 %&'K: #" /-"#$?C30 5$#, /-"60-B
3$8,: !$E#, C=,5,8"6, /-0203>+30 C306$+30 ,G0E ,1?"8,8&
−100 −50 0 50 100−0.5
0
0.5
1
Tau [s]
Avtokorelacija izhodnega pogreška
!"#$ %&'L) M85,#,-0!$2"3$ "?E,1+0=$ /,=-0>#$
-100 -50 0 50 100-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Tau [s]
Križna korelacija vhodnega signala in izhodnega pogreška
!"#$ %&'D) N-"O+$ #,-0!$2"3$ /,=-0>#$ ? 8E,1+"6 4"=+$!,6
P$> +$60+: #" 30 G"! "!C45-$2"3$ /,45,/#$ "10+5"I#$2"30 +0B
0 20 40 60 80 100-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Odziv sistema na signal za vrednotenje - simulacija
čas [s]
model
sistem
!"#$ %&'K) A-04#C4 #,+4"450+20 8E,1+,B"?E,1+0=$ ,G+$B
>$+3$ +$ 4"=+$!C ?$ 8-01+,50+30
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!
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!$'05'- &'(!% /#+ ) ! *"#-(*89*+'$* )*)(# .+,#6' +#"'% /*
1 $'</! -#$*-#'+-!)( .+*/'=# ! , .+!)(!+13 >' )*)(# #
,*50#6' +#"' 0# ('/! ,+#"-!(#-0# &#$! !(#=#-!3 4 .+* #+1
*"#-(*8/'9*0# .? .+!9#)' ;! ! .+*/'&'$* *"#-(*8/'9*0! )*)7
(# ' ,*50#6' +#"' -' )* 1$*+'-# .+!9#)1% /* !"+'=' +#7
)-*2-* /# *0)/* .+!9#)3
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*- -'2+(!,'-0' ,!"#-0'% ,*"* ! -' )$*/* A3BC3 :' #- (#6'
.+!9#)' 0# /# *0)/' -#,(+'$*&'9*0' ,<!"-# (#/!2*-#3 @+!9#)
* ' (+* ,<!"-# .+#(!/#% *- )*9#+ /*)$*-# DQ1E% ,<!"-# (#/!7
2*-# DQ2E *- ;'&# DQ3E% /* )# & #5'0! , .!)!"* F13 @+#"
#5'-0# 6+# /*)$*-' , .!)!"! F2% /'+ .! #-* "!"'(-!
"*-' */! , .+!9#)13 @+#(!/' /*)$*-# *- ;'&# )(' +#61$*7
+'-' & ',(! '()/* * +#61$'9*0)/* * ,#-(*$*% #"(# /! 0#
,<!"-' (#/!2*-' )' ! #+0#-' & #+*$-*/! .+#(!/' 7 +!7
(' #(+! 3 G&<!"-' ,#$*2*-'% /* 0! #+* !% 0# /*)$!)( 7 .?
DpHE #5'-*9#3 H#+ 0# (*.'$! ,+#"-!)(* .? .!)(',$0#-! -'
*&<!"-* !"(!/ *& .!)!"# F1% * ' ! , #+*(,* /*)$!)(* (1"*
-#/!$*/! +(,#6' 2')'3 4 -'5# .+* #+1 0# ,<!"-' ,#$*7
2*-' , .+!9#) .+#(!/ ;'&# D*- -# ,+#"-!)( .+#(!/' ,<!"-#
(#/!2*-#E3 I!$0 .!"+!;#- !.*) .+!9#)' 0# , JKL3
M*-' *2-* !"#$ .+!9#)' .? -#,(+'$*&'9*0# 0# ;*$ *&.#$0'-
*& +',-!(#=-*< /# *0)/*< #-'2;3 G&.#$0'-* !"#$ ,)#;10#
(1"* "*-' */! #+*$-*/!, *- ,#-(*$!, (#+ "*-' */! <*"+'7
,$*/# *&<!"-#6' .+#(!/'3 @+#".!)(',/# .+* !"#$*+'-01 )!
;*$# *"#'$-! #5'-0# (#/!2*-% -0*<!,' /!-)('-(' 6!)(!(' *-
.!.!$-' (!.-!)( *!-!,3 N!"#$ .? .+!9#)'% !.*)'- ) )*)(#7
! #-'2; DA3OAE *- .!"'(/* *& (';#$# A3O% ) ! -' #)(!
*&,'0'-0' #+*(#, -' -' -#"!)(!.-# .+!9#)1 1.!+';*$*
&' .+*"!;*,'-0# .!"'(/!, &' *"#-(*8/'9*0! !"#$'3 P.!7
+';$0#-* !"#$ ,)#;10# +'&-# -#$*-#'+-!)(*% #" -0* * 0#
,)#/'/!+ (+#;' ! #-*(* * .$*9*(-! +'21-'-! *- &#$! -#$*7
-#'+-! (*(+'9*0)/! /+*,1$0!% /* 0# (*.*2#- #$# #-( ,)#< .?
-#,(+'$*&'9*0)/*< .+!9#)!,Q
x = f(x) + g(x)Q3 + p(x)Q2
c(x, y) = 0
DA3OBE
f(x) =
[q1
A1x3
(Wa1 − x1)q1
A1x3
(Wb1 − x2)
1
A1
(q1 − Cv4(h1 + z)n)
]T
g(x) =
[1
A1x3
(Wa3 − x1)1
A1x3
(Wb3 − x2)1
A1
]T
p H
T 2
T 1 h 1
h 2
Q 4
Q 2
Q 3 Q 1 e
Q 1
R$*/' A3BCQ R<# ' .+!9#)' .? -#,(+'$*&'9*0#
p(x) =
[1
A1x3
(Wa2 − x1)1
A1x3
(Wb2 − x2)1
A1
]T
c(x, y) = x1 + 10y−14− 10−y
+x2
1 + 2 · 10y−pK2
1 + 10pK1−y + 10y−pK2
pH = y DA3OAE
F';#$' A3OQ @'+' #(+* .? .+!9#)' , "#$!,-* (!2/* JKL
|q1|SC3CCA N ?:T3 q1SOU3U $V)
|q2|SC3CA N :'?WT3 q2SC3XX $V)
|q3|SC3CCA N :'T?% q3SOX3U $V)
C3CCCX N :'?WT3 q1eSOU3U $V)
A1SBCY 9 2 q4SAB3Z $V)
A2S[B 9 2 Wa1S3 · 10
−3N
zSOO3X 9 Wb1SC N
pSC3UCY Wa2S7C3CA N
Ka1 = 4.47 · 107 Wb2SC3CA N
Ka2 = 5.62 · 10−11 Wa3 = 3.05 · 103 NKw = 1.00 · 1014 Wb3 = 5.00 · 105 N
∆tSOX ) h1SO[ 9
∆tcSO ) h2SA 9
τpHSOX ) Wa4 = 4.32 · 10−4N
τhSOX ) Wb4 = 5.28 · 10−4N
τvSU ) .?SY3C
θSOC )
R$*/' A3BO .+*/'&10# )* 1$'9*0)/! )<# ! &' .? -#,(+'$*&'7
9*0)/* .+!9#)3 :#$*-#'+-!)( .+!9#)' $'</! .+*/'=# ! (1"*
('/!% "' .+*/'=# ! !"&*,# .+!9#)' -' #-'/ ,<!"-* )*6-'$
, +'&$*2-*< .!"+!20*<3 F'/ .+* #+ 0# ,$'/ .+',!/!(-*< * 7
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T"&*, -'5#6' -#$*-#'+-#6' )*)(# ' -' (' ,<!"-* )*6-'$ .+*7
/'&10# )$*/' A3BA3
G&;+'-* *"#-(*8/'9*0)/* )*6-'$ *- )*6-'$ &' ,+#"-!(#-0# (#+
!"&*,# .+!9#)' -'-0 .+*/'&10# )$*/' A3B[3 H!( $'</! ,*"* !
*& .+* #+0',# )*6-'$!, *& )$*/# A3B[ *- .!+'&"#$*(,# ' .$*7
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4
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Q 3
Q 2
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h 1
p H
2
o u t _ 2
1
o u t _ 1
- l o g 1 0 ( u )
p H
1 / s
h 2 1 / s
h 1
F 1
f i l t e r T v v 2
F 1
f i l t e r T v v 1
F 1
f i l t e r T v v
F 1
f i l t e r T p H
M A T L A B
F u n c t i o n
[ H + ]
1 / s
W b T 1
W b 3
W b 2
W b 1
1 / s
W a T 1
W a 3
W a 2
W a 1M u x
C v 4
C v 1
1 / A 1
1 / A 1
1 / A 1
1 / A 2
u ^ n v 4
s q r t ( u )
1 / u
y ( n ) = C x ( n ) + D u ( n )
x ( n + 1 ) = A x ( n ) + B u ( n )
D i s c r e t e
S t a t e - S p a c e 1
z 4| u |
| u |
3
i n _ 3
2
i n _ 2
1
i n _ 1
Q 3 p H
h 1
Q 2
Q 1
p H p l a n td i s c r e t e d e l a y
t _ v a l
p H _ v a l
u _ v a l
S c o p e u
S c o p e p H
S c o p e h 1
S a t u r a t i o nR a n d o mN u m b e r
Q 2
Q 1
C l o c k
!"#$ %&'() "*+!",# -"*+!$."/-#$ -01*$ 23 ,1456$!"7$."/8
-#19$ 26:.1-$
0 500 1000 1500 2000 2500 3000 3500 40000
2
4
6
8
10
12
14
16
18
20
čas [s]
Q3
[m
l/s]
Vhodni signal
!"#$ %&'') ;0:<," -"9,$! 7$ 26"#$7 ,1!",1$6,:-5" -"-51*$
0 500 1000 1500 2000 2500 3000 3500 40002
3
4
5
6
7
8
9
10
čas [s]
pH
Izhodni signal
!"#$ %&'%) =<7"4 ,1!",1$6,19$ -"-51*$ >Ts = 25 -1#?
0 5000 100000
5
10
15
20
čas [s]
Q3
[m
l/s]
Identifikacijski signal
0 5000 100002
4
6
8
10
čas [s]
pH
Odziv modela
0 5000 100000
5
10
15
20
čas [s]
Q3
[m
l/s]
Signal za vrednotenje
0 5000 100002
4
6
8
10
čas [s]
pH
Odziv modela
!"#$ %&'@) A<1,5"B#$."/-#" -"9,$! ", -"9,$! 7$ 461<,:51,/1
", ,/+,$ :<7"4$
<",$*"C1, ", "*$ D:9$51/E: 2:6$7<1!"514 $*2!"5+<& F: 4
-2!:E,1* 2:*1," 5+<"G <$ :<7"4 7$ "<1,5"B#$."/: ,:-" 41C/:
#:!"C",: ",H:6*$."/1 : <",$*"#" 26:.1-$&
0 5 10 15 200
5
10
15
20
25
30Histogram amplitud identifikacijskega signala
!"#$ %&'() *"+,-./$0 $01!",23 45-367.$ +".6$!$ 8$ "3769
,":#$;"<-
=$#- #-, 4 "!2+,/$,"4670 1/"07/2 4 1/7<>6<70 1-31-.!$4<2
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"8?/$!" 674/-6+#- 0/7@- 8 47A6"4-<+#"0 17/;71,/-6-0 8$
"376,":#$;"<- 0-37!$&
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6$<?-!<>" 0-37!C 4/736-,76 + #/",7/"<+#- D26#;"<- 1/" 6$<9
2.-367<>70 /$807/<2 073 #4$!",7,- 0-37!$ "6 A"0 0$6<9
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B$ +,/2#,2/- +0- "8?/$!" FGHI J07,-3$ + 1-+1!->76"0
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!
!" #$%&'$ %(')*%+,-.%/' 01 0$2.'3-
0 5 10 15 200
10
20
30
40
50
60
70
80
Histogram amplitud signala za vrednotenje
!"#$ %&'() *"+,-./$0 $01!",23 45-367.$ +".6$!$ 8$ 4/79
36-,76:7
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1
2
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+!"#" %&%R !$5#- 1-,/3"0-? #$/ +,$ 1/"#$8$!" L7 +,$,"+,"=6"
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8 674/-6+#- 0/7L-&
0 1000 2000 3000 4000 5000 6000 7000−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Tau [s]
Avtokorelacija izhodnega pogreška
!"#$ %&'N) E4,-#-/7!$B":$ "85-367.$ 1-./7@#$
-6000 -4000 -2000 0 2000 4000 6000-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Tau [s]
Križna korelacija vhodnega signala in izhodnega pogreška
!"#$ %&'O) T/"L6$ #-/7!$B":$ 45-367.$ +".6$!$ "6 "85-9
367.$ 1-./7@#$
0 2000 4000 6000 8000 100002
3
4
5
6
7
8
9
10
čas [s]
Q3
[m
l/s]
Odziv obrezane mreže in modela na signal za vrednotenje
pH
model
!"#$ %&%R) H/7+#2+ #-6+"+,76B7 45-36-9"85-367.$ -<6$9
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čas [s]
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čas [s]
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x(t+ T ) = Ax(t) + Bu(t) + f(ρρρ),
y(t) = Cx(t) + Du(t) + g(ρρρ), &'(BC*
.5$1 "#, x(t) ∈ Rn> u(t) ∈ Rm
!6 "/ A, B, C, D ="#1$7
06/ 2$+!.$ ./6"#,6#6$ %,#1!.$( F$+!6$,16! G=6.8!5! f(·)!6 g(·) "#, /;2!"6! /; 2$.#/15, 1,021"#!+6!- "H1$%$6+5!2.
ρρρ = ρρρ(x(t),u(t)) ∈ Rq> q ≤ m + n> .! !01,I, 6$+!6$,16/
/;2!"6/"# ;!6,%!.$ "!"#$%, /; 65$9/2$9, "#,65, !6 2-/;,
" ./6"#,6#6!%, ∇xρρρ> ∇uρρρ ABCD(
J12/ $6,K3/ !0 &'(BC* +,-./ 1$+,#!26/ 9+$;$ 6, ;$+/26/
#/K./ (x,u) = (x0,u0) 0,H!4$%/ 0 =H/1,3/
δx(t) = x(t)− x0, &'(BB*
δu(t) = u(t)− u0, &'(BL*
./#E
x(t+T ) = A (x0 + δx(t))+ B (u0 + δu(t))+ f(ρρρ). &'(BM*
J1! H1$;H/"#,2.! / +/.,+6! +!6$,16/"#! 2 3+!I!6! ;$+/26$
#/K.$ (x0,u0) "+$;!E
x(t+ T )− x0 + x0 = Ax0 + Aδx(t) + Bu0 + Bδu(t)
+ f0 + fx0δx(t) + fu0δu(t),
&'(BN*
δx(t+ T ) + x0 =(Ax0 + Bu0 + f0
)
+(A+ fx0
)δx(t) +
(B
+ fx0) δu(t),
&'(B'*
δx(t+ T ) = (F0 − x0) +(A+ fx0
)δx(t)
+(B+ fu0
)δu(t), &'(BO*
.5$1 "/ F0 = F(x0,u0)> f0 = f(x0,u0)> fx0 = ∂f∂x
(x0,u0)
!6 fu0 = ∂f∂u
(x0,u0)(
P /;2,5,65$% $6,K3$ &'(BO* H/ K,"= ;/3!%/E
x(t+ T ) =(A+ fx0
)x(t) +
(B+ fu0
)u(t)
&'(BQ*
x(t+ T ) = A(ρρρ)x(t) +B(ρρρ)u(t), &'(BR*
.5$1 "#, A(ρρρ) = (A + fx0) !6 B(ρρρ) = (B + fu0)( J1! !"7
#!- H/9/5!- +,-./ 6, $6,. 6,K!6 ;1=9/ $6,K3/ !0 &'(BC*
0,H!4$%/ ./#E
y(t) = C(ρρρ)x(t) +D(ρρρ)u(t), &'(B)*
.5$1 "#, C(ρρρ) = (C+ gx0) !6 D(ρρρ) = (D+ gu0)>0 gx0 = ∂g
∂x(x0,u0) !6 gu0 = ∂g
∂u(x0,u0)(
F, /H!",6! 6,K!6 +,-./ H1$;"#,2!%/ ;$+/2,65$ "!"#$%, 2
.,#$1! ./+! #/K.!> ;/+/K$6! 0 2$.#/15$% 1,021"#!+6!- "H1$7
%$6+5!2. ρρρ> 0 2$.#/15$% "#,65 x(t) !6 2-/;/% u(t)( !"#$%&'(BC* #,./ +,-./ 0,H!4$%/E
x(t+ T ) = A(ρρρ)x(t) +B(ρρρ)u(t)
y(t) = C(ρρρ)x(t) +D(ρρρ)u(t), &'(LC*
.5$1 "/ %,#1!.$
A(ρρρ) = A+∂f
∂x(x(t),u(t)),
B(ρρρ) = B+∂f
∂u(x(t),u(t)),
C(ρρρ) = C+∂g
∂x(x(t),u(t)),
D(ρρρ) = D+∂g
∂u(x(t),u(t)), &'(LB*
/;2!"6$ /; 2$.#/15, 1,021"#!+6!- "H1$%$6+5!2. ρρρ(
J/9+$5%/ "! H1!%$1 %/;$+!1,65, ;!".1$#6$9, "!"#$%, " -!7
#1/"#6/ +!6$,1!0,8!5/(
0&$1,& 1'2,+$&*)/* ($(%,1*
P,6!%, 6," -!#1/"#6, +!6$,1!0,8!5, 0, 6$+!6$,16! ;!".1$#6!
"!"#$% ABQD> /H!",6 0 $6,K3/
y(k + 1) =y(k)
1 + y2(k)+ u(k)3 &'(LL*
!6 " K,"/% 20/1K$65, T = 1 "( S H1/"#/1= "#,65 +,-./
"!"#$% &'(LL* 0,H!4$%/E
x(k + 1) =x(k)
1 + x2(k)+ u(k)3
y(k) = x(k). &'(LM*
J/6,0/1!#$2 "!"#$%, &'(LM* 2 02$06$% H1/"#/1= 5$
x(t+ T ) =x(t)
1 + x2(t)+ u(t)3
y(t) = x(t), &'(LN*
.,1 6! $.2!2,+$6#$6 "!"#$%> !%, H, $6,. /;0!2 2 #1$6=#7
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x(t+ T ) = (1− x(t)2)/(1 + x(t)2)2x(t) + 3u(t)2u(t)
y(t) = x(t). &'(L'*
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'&
x = w
w = (A+∇f(ρ1)∇xρ)w + (B+∇f(ρ1)∇uρ)u,
E&
x = w
w = (A+∇f(ρ2)∇xρ)w + (B+∇f(ρ2)∇uρ)u,
V&
x = w
w = (A+∇f(ρ3)∇xρ)w + (B+∇f(ρ3)∇uρ)u.
K10@02$ #546"2$8"9$ /0B"10-)
w =3∑
i=1
wiµi(ρ).
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x = w
w =3∑
i=1
[(A+∇f(ρi)∇xρ)w
89
!" #$%&'(% &)*+,-)$(% .%./)+%
+ (B+∇f(ρi)∇uρ)u]µi(ρ).
!"#$%
&'()*+,' -).('/ 0) 1+-2)3
w ≈ w, 4+567+1 87+*+9+ (',:69);62+
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!"##$%& '($)*#& #$ +!,-."/+& '($)*#&0 1!,-."/+& '($)*# .*
'$#'-$2&3*% +!,-."/+* #'(*4*+-.&%,* 5+'(0 %*,3$(6 #,!-!(7
+! +*,& $8 +*$8%&#+&9 #'(*4*+-.&%, $8%&#*+ '($#3$(0 :* .*
%(*8+$#3 +!,-."/+* #'(*4*+-.&%,* % %#!,& 3$/,& 3*;! '($<
#3$(! '$(!=8*-.*+! '$ !"##$%& 5+$(4!-+&7 '$(!=8*-&3%&6
3!,*4" '($)*#" (*/*4$ !"##$% &'$()# 5 >7 ?@AB0 C("<
;!/*D /* .* %9$8 % '($)*# %*,3$( +*$8%&#+&9 #'(*4*+-.&%,
x6 .* 3! '($)*# !"##$%6 /* .* '$(!=8*-&3*% %(*8+$#3& E"+,<
)&.* f(x) =! %#!, %9$8+& %*,3$( x !"##$%!0
F$8*- +! '$8-!;& !"##$%&9 '($)*#$% 5,(!.2* > 4$<
8*-7 $G&/!.+$ &4*+".*4$ +*'!(!4*3(&/+& %*(.*3+$#3+& 4$<
8*- ?HIB0 1!4*#3$ =! 4$8*-&(!+.* G$-. $G&/!.+* $4*.&<
3%* +! +*,& (!=(*8 5'!(!4*3(&=&(!+&97 E"+,)&. # 3*4 +!<
/&+$4 ! &'*$'* 8$'"2/!4$ $'&# +*=+!+*;! #*4! = +*<
#,$+/+$ 4+$J&)$ E"+,)&.0 >(& 3*4 8$'"2/!4$ %*/.$ %*(.*<
3+$#3 E"+,)&.6 =! ,!3*(* 4*+&4$6 8! #* '(& $'&#" #*4!
G$-. %*(.*3+$ '$.!%-.!.$ +'(0 ;-!8,*6 #3!)&$+!(+*6 '*(&$<
8&/+*0
>$8($G+*.* #$ $#+$%* 4$8*-&(!+.! = 4$8*-$4 +! '$8-!;&
!"##$%&9 '($)*#$% $'&#!+* % ?HKB !-& =!/*3+&9 '$;-!%.&9 %
?HIB6 G$-. '$8($G+$ (!=-!;$ +!.8*4$ % ?HI6 HLB0
M*-.! $'$=$(&3&6 8! %9$8 &+ &=9$8 % > 4$8*- +! #&<
;+!-!6 4!(%*/ .* %9$8 % 4$8*- %*,3$( = 5%=$(/*+&4&7 %(*<
8+$#34& +*$8%&#+&9 #'(*4*+-.&%, x6 &=9$8 &= > 4$8*-!
'! %*(.*3+$#3+! '$(!=8*-&3*% &=9$8+* %(*8+$#3& f(x) '(&8!+*4 %9$8+*4 %*,3$(."N 3$ &=9!.! &= O!P*#$%*;! +!/&+!
4$8*-&(!+.! ?HHB6 ,& '(*8#3!%-.! G%*+$ (!=-&,$ $8 8(";&9
+!/&+$%0 >$;-*.4$ #& '$8($G+$#3& 3*;! +!/&+! 4$8*-&(!<
+.!0
Q! '$-."G+& +!G$( N %9$8+&9 %*,3$(.*% xi6 i = 1, . . . , N .*
> 8$-$/*+ = %*,3$(.*4 #(*8+.&9 %(*8+$#3&
m = [m1(x1) . . .mN (xn)]T&+ ,$%!(&!+/+$ 4!3(&,$ K6
K =
K11 . . . K1N
0
0
0
0
0
0
0
0
0
KN1 . . . KNN
, 5A0@7
,.*( #$D
mi(xi) = E[f(xi)] 5A0H7
&+ #$ *-*4*+3& ,$%!(&!+/+* 4!3(&,* Kij 8$-$/*+& ,$3D
Kij = cov(f(xi), f(xj))
= E[(f(xi)−m(xi)(f(xj)−m(xj)], 5A0R7
$G&/!.+$ 8$G-.*+& = +*,$ +$%!'*!,-,$ .",+(*/$ C(xi,xj)D
cov(f(xi), f(xj)) = C(xi,xj). 5A0L7
:* .* '$(!=8*-&3*% +*,* 4+$J&)* #'(*4*+-.&%, !"##$%!6
.* !"##$%! 3"8& '$(!=8*-&3*% ,!3*(* ,$-& '$84+$J&)* *-*<
4*+3$% 3* 4+$J&)*6 ,!( &4*+".*4$ =!93*%! '$ ,$+#*+)&
5!+;-0 )$+#*+)P (*S"&(*4*+370 T$ =! 8*-$%!+.* > 4$<
8*-! '$4*4G+$ -!#3+$#3 8$#*J*4$6 /* #$ *-*4*+3& ,$%!(&<
!+/+* 4!3(&,* ><.! 8$G-.*+& # ,$%!(&!+/+$ E"+,)&.$ ?HIB0
!"#$%#&'&# ()&*+%,#
M(*8+$#3 ,$%!(&!+/+* E"+,)&.* C(xi,xj) &=(!J! ,$(*-!)&.$4*8 %(*8+$#3&4! &=9$8$% f(xi) &+ f(xj) +! '$8-!;& %(*<8+$#3& %9$8+&9 %*,3$(.*% xi &+ xj 0 U$%!(&!+/+! E"+,)&.!
.* -!9,$ ,!3*(* ,$-& $G-&,*6 /* =! '$-."G+& +!G$( N %9$<
8+&9 %*,3$(.*% xi, i = 1, . . . , N 3%$(& +*<+*;!3&%+$ 8*V+&<
3+$ ,$%!(&!+/+$ 4!3(&,$ K0 U$%!(&!+/+* E"+,)&.* #$ -!9,$
#3!)&$+!(+*6 +*#3!)&$+!(+*6 '*(&$8&/+* &3806 '$8($G+*.* #$
+!23*3* % ?HIB0 U$%!(&!+/+* E"+,)&.*6 ,& 8$-$/! $G-&,$ +*<
=+!+* E"+,)&.* f(x)6 +!%!8+$ +* '$=+!4$ %+!'(*.6 -!9,$'! &= =+!+.! $ #'-$2+&9 -!#3+$#3&9 E"+,)&.* f(x) #,-*'!4$+! $G-&,$ ,$%!(&!+/+* E"+,)&.*0
1!.%*/ #* "'$(!G-.! !"##$%! ,$%!(&!+/+! E"+,)&.!6 ,& &=<
(!J! 8%* '$;$#3& -!#3+$#3& '($)*#$%D
• ;-!8,$#36 ,& '$%*6 8! #* G$ &=9$8 '($)*#! = 4!.9+$
#'(*4*4G$ %9$8! (*-!3&%+$ 4!-$ #'(*4*+&- 5,$(*-!<
)&.! .* %*/.! =! 8%! &=9$8!6 /&;!( %9$8! -*J&3! G-&="
#,"'!.76 &+
• #3!)&$+!(+$#36 '(& ,!3*(& .* ,$%!(&!+)! 4*8 8%*4!
%9$8+&4! %*,3$(.*4! $8%&#+! #!4$ $8 +."+* 4*8<
#*G$.+* (!=8!-.* &+ +* 3"8& $8 +."+* !G#$-"3+* -*;*
% '($#3$("0
AR
!"#$%&'()%*( #"+%#"(,#%- .%.$"/01 2 3(4..01%/% 5,0)".%
!"#$%&'" (" !")"*+", -" .&'$'" /.$!#0(1.(0)$ 2.$.($
" *+#3-+3#& !#"40*$5 6#& .(0( (0 -"/$#&$.4$ '07 7/0'$
&28"7"'$ yi = f(xi) &. yj = f(xj)9
Kij = C(xi,xj) = v1 exp
[−
1
2
D∑
d=1
wd(xdi − xd
j )2
]; :;5<=
D (0 7">?&.$ /8"7.0)$ /0-+"#($ x "25 1+0/&>" .0"7/&*.&8
*!#0'0.>(&/- *&*+0'$5 6$#$'0+#& v1 &. wd, d = 1, . . . , D *"
!">(3%." 7">"@>(&/& !$#$'0+#& -"/$#&[email protected] A3.-4&(05 B'0C
.3(0'" (&8 !"#$"%$%&#'$!
DEF, EGH, 7$ !"37$#&'", 7$ *"
+" !$#$'0+#& *&40# .0!$#$'0+#&@.0)$ '"70>$
!
, -& 7">"@$("
"%>&-" .02.$.0 A3.-4&(0 f(x)5 6$#$'0+0# v1 )"/"#& " /0>&C-"*+& -"/$#&$.40, !$#$'0+#& wd !$ "7#$?$(" !"'0'%."*+
!"*$'02.0 -"'!".0.+0 /8"7.0)$ /0-+"#($I /0@(& (0 !$#$C
'0+0# wd, %">( /!>&/.$ (0 *!#0'0'%$ -"'!".0.+0 /0-+"#($
xd.$ /#07."*+ &28"7$5 J$ 7$.$ -"/$#&$.@.$ A3.-4&($ +/"#&
!"2&+&/." 70K.&+." -"/$#&$.@." '$+#&-", '"#$(" %&+& /*&
!$#$'0+#& L$3**"/0 -"/$#&[email protected] A3.-4&(0 /0@(& "7 .&@5
60!"+%,(#*"
M$(>$?0 !#07*+$/&'" 70>"/$.(0 L6 '"70>$ .$ !#&'0#35
N20'&'", 7$ %& #$7& "!&*$>& .0-& *&*+0'
y = f(x) + v, :;5O=
-(0# (0 v %0>& L$3**"/ 13' 2 /$#&$.4" v0, v ∼ N (0, v0)5 M$!"7>$)& N C+&8 /8"7."C&28"7.&8 /2"#40/, +(5 !$#"/ /0-+"#C
(0/ (xi, yi), 2%#$.&8 / '."?&4& D = X,y, ?0>&'" 7">"@&+&
.02.$." /#07."*+ &28"7$ y∗ !#& /#07."*+&8 /8"7.0)$ /0-C
+"#($ x∗
5 N .$7$>(0/$.(3 / -".+0-*+3 L6 '"70>$ N ×D
'$+#&-" X &. N × 1 /0-+"# y "2.$@&'" -"+ ()*+ &*+,!-+,
*$( (&8 3!"#$%>($'" 2$ 3@0.(0 :$.)>5 '$%!*!*.= L6 '"C
70>$5 6"*$'02.& /8"7."P&28"7.& !$# (xi, yi) &2 +0 '."?&40&'0.3(0'" +37& 3@.& /0-+"# "25 3@.$ +"@-$5 6$# (x∗, y∗)"2.$@&'" -"+ !#0&2-3*." "25 +0*+." '."?&4" $>& +37& -"+
'#/'*! 0 +12!3 +15
@.& &28"7& yi, i = 1, . . . , N !#07*+$/>($(" /#07."*+& .$C
->(3@.&8 *!#0'0.>(&/-, &28$($("@&8 &2 L$3**"/0)$ !#"40*$5
6#07!"*+$/&'", 7$ (0 &28"7 *&*+0'$ )>$70- &. 7$ (0 *&*+0'
*+$4&".$#0. +0# 2$ +/"#(0.(0 -"/$#&[email protected] '$+#&-0 K 3!"C
#$%&'" L$3**"/" -"/$#&$.@." A3.-4&(" :;5<= 2 .02.$.&'&
!$#$'0+#& .$ 2$@0+-35
J"%&'"9 y ∼ N (0,K), -(0# *" 0>0'0.+& -"/$#&[email protected] '$C+#&-0 Kij = Σij + v0δij 5 Σij *" 0>0'0.+& -"/$#&[email protected] '$C
+#&-0, 7"%>(0.& * -"/$#&$.@." A3.-4&(" :;5<=, v0δij !$ "!&C
*3(0 /!>&/ 13'$ .$ &28"73 !#"40*$, -(0# (0 δij Q#".04-0#C
(0/ "!0#$+"#5 Q0# *'" !#07!"*+$/&>& %0>& 13', *" .(0)"/0
/#07."*+& -"#0>&#$.0 *$'" *$'0 * *$%"5
!"# $%&' (! )*+","#- ." (! /0")1!(20(" 0!/1*03+" 41!5" , !064
3+16764 06/*(!4- / +"7!1!4 (! 0!3+*080* 27!/6#* 0!/1*0*/- !0"+"
9: 4*.!#;< =6)!1)"1"4!716 .*#*8"(* .63716>;?6(* @36?!1 0!3+*080!A"
27!/6#"B )"1"4!71*/ 7! 0!/1*03+! 41!5!<
!
C*.!# (! 0!)"1"4!7168!0- 3"( ," 0")*/!.*/"0(! )*#!A D6)!1)"1"E
4!71*/ 60 +*/"16"080! F;0+?6(! )*71!>;(!4* 2! 60F*14"?6(* * *>0"E
2"0(; 3637!4" / *>#6+6 /D*.0*G6,D*.06D )*."7+*/- ;)*1">#(!06D )16
4*.!#61"0(;<
Q0# (0 :2$0.-#$+ 10 .02.$.&= &28"7 y∗ 370($.(0.(0 &*+0)$
!#"40*$ -"+ 3@.& &28"7& y, >$8-" 2$!&10'" DFH9 yN+1 =[y
y∗
]∼ N (0,KN+1)5 R-3!." -"/$#&$.@." '$+#&-" KN+1
/0-+"#($ yN+1 >$8-" #$2%&(0'"9
KN+1 =
K
k(x)
[k(x)T
] [k(x)
]
. :;5;=
S$+#&-$ K (0 -"/$#&$.@.$ '$+#&-$ 3@.&8 !"7$+-"/, k(x∗)(0 /0-+"# -"/$#&$.4 '07 3@.&'& &28"7& &. +0*+.&' &28"7"',
k(x∗) !$ $/+"-"/$#&$.4$ +0*+.0)$ &28"7$5
6" T$U0*"/0' .$@&.3 DEEH >$8-" /0#(0+."*+." !"#$270>&C
+0/ /#07."*+& &28"7$ y∗ #$270>&'" .$ 7/$ 70>$9 .$ 70>,
-& 7">"@$ /0#(0+."*+ 3@.&8 &28"7"/ )>070 .$ [email protected] /8"70
:$.)>5 &%$.!*%4 "%$'=9 p(y|X) ∼ N (0,K), &. .$ !")"(.&70> :$.)>5 -+*1!'!+*%4 "%$' =, -& )>070 .$ !#/& 70> &. /8"7
x∗
.$!"/073(0 /0#(0+."*+." !"#$270>&+0/ &28"7$ y∗5 V"#C
'$>." 2$!&*$." (0 &2#$@3. !"#$270>&+/0 &28"7.0 /0#(0+."*+&
"72&/$ y∗ DG;H9
p(y∗|x∗,y,X) =
∫p(y∗|x∗, θθθ,y,X)p(θθθ|y,X)dθθθ. :;5W=
X%&@$(." (0 +$ &.+0)#$> $.$>&+&@." .0&2#$@3.>(&/, &'$'" !$
.$ /">(" 7/0 $>+0#.$+&/& DG;H5 6#/$, %">( !")"*+$, (0 $!#"C
-*&'$4&($ &.+0)#$>$ 2 3!"#$%" .$(%">( /0#(0+.&8 /#07."*+&
.02.$.&8 8&!0#!$#$'0+#"/ θθθMP9
p(y∗|x∗,y,X) ≈ p(y∗|x∗, θθθMP,y,X). :;5Y=
!"#$%&'" +&*+0 /#07."*+& 8&!0#!$#$'0+#"/ θθθMP, !#& -$+0C
#&8 (0 /0#(0+."*+ 3@.&8 &28"7"/ y )>070 .$ /#07."*+& 3@.&8
/8"7"/ X &. -"/$#&$.@." A3.-4&(" C(., .) .$(/0@($5 J"C%&'" (&8 2 '0+"7" .$(/0@(0 !"7"%."*+& :$.)>5 &%5!&(&
4!6#4! ++1 &#' +1 Z S[=5 J$ *0 &2").0'" "!+&'&2$4&(& 2
"'0(&+/$'&, 2$ "!+&'&2$4&(" 3!"#$%&'" >")$#&+0' !"#$2C
70>&+/0 3@.&8 !"7$+-"/ :$.)>5 4+.7&%$.!*%4 4!6#4! ++1 =9
L(θθθ) = log(p(y|X, θθθ))
= −
1
2log(|K|)−
1
2yTK−1
y −
N
2log(2π),
!"#$%
&'() '( θθθ *(&+,) -.)./(+),*0 θθθ = [w1 . . . wD v1 v0]T
12 K
&,*.)1.232. /.+)1&. 4. 532( -,6.+&( D" 7( '( ,-+1/14.8
91'. 14*(6(2. 4 /(+,6, &,2'5:1).21; :).61(2+,* .<1 &.+(),
6)5:, :).61(2+2, /(+,6,%0 '( -,+)(=(2 >( 14).352 ,6*,6,*
-, *?(; ;1-()-.)./(+)1;@
∂L(θθθ)
∂θi= −
1
2?<(6
(K−1
∂K
∂θi
)+
1
2yTK−1
∂K
∂θiK−1
y.
!"##%
A= *?.&(/ &,).&5 ,-+1/14.91'( '( -,+)(=2, 14).352.+1 128
*()4 &,*.)1.232( /.+)1&( K−1
0 &.) '( ).352?&, 4.;+(*2,
4. *(<1&( N "
B)5:. /,C2,?+ 4. .-),&?1/.91', 12+(:).<. !"D% '( 25/(8
)132. 12+(:).91'. 2.6 9(<,+2, -,).46(<1+*1', ;1-()-.)./(8
+),* EFEF /(+,6(0 G#!H%0 6,=<'(2, 4 ,-+1/14.91', *()'(8
+2,?+1 5321; -,6.+&,* !"#$%"
!
!" #$%&&'() *+',-&)
!"#$%$ &' '($ )*+(*"# ,-!,.' /,0'.0$.$)'#.*1 )*2'3$4 #%5
($1-6.,+(* 1.'2(*#'(%' 7$(835 !"##$%&'()&*("+9 :;<=5 >.'?
2(*"#, /,0'.0$.$)'#.*1 ,&@')* 6*# *!,@$%(*4 3' 2$ A@('
0*2$#6' .$-!,%')* ($ n 2'3*15 B$ A@'(%' A0*.$!,)* n−12'3*14 -$ 1.'2(*#'(%' 0$ #,"#'8$4 6, *"#$('5 C*"#*0'6 0*?
(*1,)* n?6.$#4 1"$6,@ - 2.A8,), 0*2$#6, -$ 1.'2(*#'(%'5
D6"#.')(, 0.,)'. %' ($1-6.,+(* 1.'2(*#'(%' - ,-0A&@$(%')
7$(835 ',&%,$"+,$"-* $ .// !"##$%&'()&*("+ 95 E$%1'@%,
0.*!3') #'8$ 0*"#*06$ %' .$@A("6$ -$/#'1(*"#4 "$% )*?
.$)* ($A@,#, n )*2'3*1 :;<=5
.$*'(-/'($01-
F6A0($ 0*.$-2'3,#'1 p(yN+1) %' G$A""*1$H #*.'% %' G$A"?
"*1$ #A2, 0*8*%($ 0*.$-2'3,#'1 p(y∗|x∗,y,X) = p(yN+1)
p(y|X)5
C* 0*'(*"#$1,#1, :IJ4 ;K= 6*# ($0*1'2$( ,-/*2 ","#')$
7J5L9 2*!,)* G$A""*1* 0*.$-2'3,#'1M
p(y∗|x∗,X,y) = N(µ(x∗), σ2(x∗)
)7J5I;9
" ".'2(%* 1.'2(*"#%* µ(x∗) ,( 1$.,$(N* σ2(x∗)M
µ(x∗) = k(x∗)TK−1y 7J5IO9
σ2(x∗) = k(x∗)− k(x∗)TK−1k(x∗) + v0, 7J5IP9
6%'. %' k(x∗) = [C(x1,x∗) . . . C(xN ,x∗)] +' *)'(%'(, N×1
6*1$.,$(@(, 1'6#*. )'2 #'"#(,) ,-/*2*) ,( A@(,), ,-/*2,
#'. k(x∗) = C(x∗,x∗) $1#*6*1$.,$(N$ #'"#('8$ ,-/*2$5Q3A"#.$N,%* *0,"$('8$ 0.,6$-A%' "3,6$ J5I5
*
**
*
y
Y x
U è n e t o è k e
V r e d n o s t y
p r i x = x *
*
**
*
y
Y x
x *
V r e d n o s t y
p r i x = x
i p ( y )
p r i x = x
*
**
*
y
Y x
*
*x *
*
**
*
*
y
Y xx *
F3,6$ J5IM C.,(N,0 )*2'3,.$(%$ - G$A""*1,), 0.*N'",
203-+*+-3$,)1$
GC )*2'3 %' "'"#$13%'( ,- 21'/ 2'3*1M
• ,- 0$.*1 1/*2(*R,-/*2(,/ A@(,/ 0*2$#6*1 7#*@69 D4
6, 0.'2"#$13%$%* *!($&$(%' ('-($('8$ ","#')$4 ,(
• 6*1$.,$(@(' SA(6N,%' C(., .) - -($(,), *-5 *0#,),-,?.$(,), /,0'.0$.$)'#., θθθ4 6, 0*1'2*4 1 6$6&(') .$-?
)'.%A "* 0*2$#6, D5
T'. GC )*2'3 0*#.'!A%' ,(S*.)$N,%* * ('-($(, SA(6N,%, 1
*!3,6, A@(,/ 1/*2*1 ,( ,-/*2*1 #A2, 0* A@'(%A4 %' )*2'3
('0$.$)'#.,@'(5 U,0'.0$.$)'#., ($).'@ 0.'6 6*1$.,$(@('
SA(6N,%' "$)* 0*1'2*4 6$6* "' A@($ ,(S*.)$N,%$ A0*.$!,
-$ ($0*1'2*1$(%'4 (, 0$ 1 (%,/ "0.$13%'($ ,(S*.)$N,%$ *
*0,"*1$(, SA(6N,%,R","#')A5
E$ 1'6#*. k(x∗)TK−11 ,-.$-A -$ ".'2(%* 1.'2(*"# ($0*?
1'2$('8$ ,-/*2$ 7J5IO9 3$/6* 83'2$)* 6*# ($ 1'6#*. A#'+,4
6, 2*3*@$ A#'+,#'1 0*"$)'-(,/ A@(,/ ,-/*2*1 yi 1 y 83'2'
($ .$-2$3%* )'2 A@(,), ,( #'"#(,) 1/*2(,) 1'6#*.%')5 V$
3,('$.($ 6*)!,($N,%$ A@(,/ ,-/*2*1 7$(835 '(+,&! 0!,)( $
*"!9 "' 3$/6* .$-A)' 6*# 83$%'(%' 1 GC )*2'3A 1"'!*1$('
,(S*.)$N,%' * ('-($(') ","#')A 7A@(, 0*2$#6,95 W' 2.A?
8$@' ", 3$/6* ($0*1'2 µ(x∗) 0.'2"#$13%$)* 6*# 3,('$.(*6*)!,($N,%* N %'2.(,/ 7$(835 1,!+,'9 SA(6N,%4 A".'2,&@'(,/
1 A@(,/ #*@6$/M y∗ =∑N
i=1αi C(x∗,xi)5 Q-/*2 ,- ","#')$
%' '( 1-*.'N ,- 2*!3%'(' (*.)$3(' 0*.$-2'3,#1' 7J5I;95
X$%/($ 1$.,$(N$ σ2(x∗) ($0*1'2$(' 0*.$-2'3,#1' ,-/*2$0*)'(, 1'@%' 2&-0&+3, 1 ($0*1'2 ,( ($.*!'5 Y' ", *83'?
2$)* ,-.$- -$ 1$.,$(N*4 1,2,)*4 2$ %' "'"#$13%'( ,- 21'/
2'3*1 :;<=5 2 0.1'8$ 2'3$ k(x∗)4 6, 0.'2"#$13%$ $ 0.,*.,1$.,$(N* GC4 %' *2&#'# ,-.$- k(x∗)TK−1
k(x∗)5 V$ 0.'2?"#$13%$ -)$(%&$(%' $ 0.,*., 1$.,$(N' GC 0., x
∗
-$.$2, A@(,/
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Identifikacija
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čas [s]
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1
2
3
čas [s]
Vrednotenje 2
95% pas negotovosti
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odziv za vrednotenje
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0.3
0.35
0.4
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čas [s]
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