《系统辨识基础》-sysppt54

= 0§=Zeá"
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Z9C §Z 9C"Z JD( '§ ê Φ;—'"XJ¤^ 9C÷vü^§ Kk N →∞ θIV → θ w.p.1 (7) θIV 9Cê O"§ J9C§ ÷vü^§ê OθIV ± — O"
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2. 9C J ¤J 9CAe(k)'§ u(k)Φ 0 A~^ J{" {1S“9C{ w(n) w(1) u(n + 1) w(n + 1) w(2) u(n + 2) Z= . . . w(n + N 1) w(N ) u(n + N ) {w(k)} nX §=nX .
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3. 9C4í{ X5 § .
5.7 9C{
A(z 1 )y(k) = B(z 1 )u(k) + e(k) N *ê O
(16)
5.8 ' 5.9
{ {
T T θ = [WN ΦN ]1 WN YN
(17) (18)
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5.7 9C{ 5.8 ' 5.9 { {
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5.7 9C{ 5.8 ' 5.9 { {
{3X |^\9C§9C u(2n) Z= u(2n + N 1)
u(0) . . . u(N 1)
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5.8 '
{
5.7 9C{ 5.8 ' 5.9 { {
'– {(Correlation–Least Square§COR–LS)^ü{ Oê . ê E{"k^'{EX ó Aê§( X ê .§, 2A^ {ò ê .[¤ê ." b X \{u(k)}{y(k)} { -‘S " * \ê §± O\& g'êRu (τ ) \&m p'êRuy (τ ) R (τ ) = lim 1 N 1 u(k τ )u(k) u N →∞ N k=0 (23) N 1 R (τ ) = lim 1 uy u(k τ )y(k) N
y(k)r
'" e
5.7 9C{ 5.8 ' 5.9 { {
u(1) u(2) . . . u(N )
(8)
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A(z 1 )w(k) = B(z 1 )u(k) A(z )B(z ) Xê LS“{)"
1 1
(9) w(k)Q±
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(11)
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\&u(k)±Y-y&§…D(e(k)'§K9C÷ vü^"
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{4Tally n XJD(e(k)w¤e
MA .
1
5.7 9C{
e(k) = D(z )ε(k) ε(k)" '‘D(§… D(z 1 ) = 1 + d1 z 1 + + dm z m K9C A w(k) = y(k D), 9C |^ò D≥m
5.7 9C{ 5.8 ' 5.9 { {
N →∞ k=0
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XJêN k§'ê O^êO"|^ù * ê§O l g'êRu (τ )l p' êRuy (τ ) R (τ ) = 1 N 1 u(k τ )u(k) u N k=0 (24) N 1 R (τ ) = 1 uy u(k τ )y(k) N
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T PN = [WN ΦN ]1
#O*êyN +1 §K PN +1 = =
WN T wN +1
1 PN T 1
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ΦN T +1 N
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1 wN +1 T +1 N
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4í{Xe θN +1 = θN + GN +1 [yN +1 T +1 θN ] N GN +1 PN +1 T = N yN 1 yN n uN 1 uN n PN wN +1 = 1 + T +1 PN wN +1 N (19)
5.7 9C{ 5.8 ' 5.9 { {
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3'– {§2|^óAS 5 OX ê ." X ê . gn ‘ §
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y(k) = a1 y(k1) an y(kn)+b0 u(k)+b1 u(k1)+ +bn u(kn)+e(k) (26) b \ & {u(k)} ‘ D ( S {e(k)} ' " ü > ±u(k τ )§ "§ Ruy (τ ) = a1 Ruy (τ 1) an Ruy (τ n)+b0 Ru (τ )+b1 Ru (τ 1)+ +bn Ru (τ n) (27)
θIV = (Z T Φ)1 Z T Y θIV = (Z T Φ)1 Z T (Φθ + e) = θ + (Z T Φ)1 Z T e OθIV ìC ^
(6)
5.9
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(1) lim (2) lim
1 T N Z Φ N →∞
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1 T NZ e N →∞
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5.7 9C{
1. g ¤{ 9C{óC{(Instrumental Variable, IV)"éu5 A(z 1 )y(k) = B(z 1 )u(k) + e(k) l\ =\ {u§ W =Z Z W é "8Iê J = eT W e K\ O θW LS = (ΦT Z Z T Φ)1 ΦT Z Z T Y (4) (3)
(12)
5.8 ' 5.9
{ {
(13) (14)
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\9C§=
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y(D) u(n) u(0) . . . . Z= . . y(n D + N 2) y(N D 1) u(n + N 1) u(N 1) (15) òD ≥D( .MA g" y(n D 1)
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ê§X
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k^ {o θ§2òθ“\(9)§ w(k)"2(8) E9C §|^(5) θ 9C OθIV §, 2òθIV “\ (9)2g w(k)"Xd4í O9Cê§ ÷ E(J" {2gA{ ù{¤J 9Cw(k)9C Z /{ §9 .êθ O{{k¤ " θ(k) = (1 α)θ(k 1) + αθ(k d) (10) α 0.01 0.1§d 0 10§θ(k)k¤ ê O" .(1) D(e(k)u(k)'§…u(k)±Y-y&§¤ 9C÷vü^"
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5.7 9C{
¤
Ruy (τ ) = T (τ )θ + e(τ ) (28) "…
5.8 ' 5.9
{ {
e(τ )L'ê^éA
O“O
¤E¤
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(τ ) = [Ruy (τ 1) Ruy (τ n) Ru (τ ) Ru (τ n)]T θ = [a1 a2 an b1 bn ]T -τ = 1, 2, , N §K §
k=0
5.7 9C{ 5.8 ' 5.9 { {
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§m
'XdWiener–Hopf§5§=
N 1
Ruy (τ ) =
i=0
g(i)Ru (i τ )
(25)
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X \ ê § X J ò e Ru (τ ) |“ê§"dd§ Ruy (τ ), τ = 1, 2, , p.Kd§(25) )óA G = [g(1) g(1) g(p)]T ù0 L '{"
5.7 9C{ 5.8 ' 5.9 { {
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θCORLS = (ΦT ΦN )1 ΦT Ruy,N N N e±xD(&–‘ &X XA^ OXê . ê"
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\&§?
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1. xD(\& \&xD(§g'ê Ru (τ ) = ù§Wiener–Hopf§{z Ruy (τ ) = g(τ )Ru (0) = σ 2 g(τ ) ù §§(27) (32) σ2 0 τ =0 τ =0 (31)
5.7 9C{ 5.8 ' 5.9 { {
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(20)
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PN wN +1 T +1 N = I PN T 1 + N +1 PN wN +1
(21)
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9
.
T wN = xN 1 xN n uN 1 uN n T xN = wN θN
#-V>f&Egza‘§U á 5XE:6 ‘
15 E
5.8 ' 5.9
{ {
{
5.7 9C{ 5.8 ' { {
5.7 9C{
5.9
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I“‰E>ó§gz Emailgaoxia73@163.com IY|Y> ¤k
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. (1)
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5.8 ' 5.9
{ {
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1 N ΦT Z §N "K
O (5)
5.7 9C{ 5.8 ' { {
θIV = (Z T Φ)1 Z T Y θIV = (Z T Φ)1 Z T (Φθ + e) = θ + (Z T Φ)1 Z T e
(6)
5.9
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1 N ΦT Z §N "K
O (5)
5.7 9C{ 5.8 ' { {
θIV = (Z T Φ)1 Z T Y θIV = (Z T Φ)1 Z T (Φθ + e) = θ + (Z T Φ)1 Z T e OθIV ìC ^
(6)
5.9
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(1) lim (2) lim
1 T N Z Φ N →∞
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1 T NZ e N →∞
= 0§=Zeá"
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1 N ΦT Z §N "K
O (5)
5.7 9C{ 5.8 ' { {
5.7 9C{ 5.8 ' 5.9 { {
(22)
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P0 = α2 I, θ0 = 0,
α"
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x0 = x1 = = xn = 0 éa"
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":éP0
U?50*100:^4í
§2=\IV4í§K
J
"
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15 E
5.7 9C{ 1. g ¤{ 2. 9C J
{
5.7 9C{ 5.8 ' 5.9 { {
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3. 9C4í{ 5.8 ' 5.9 { {
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Ruy,N = ΦN θ + EN
(29)
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Ruy,N = [Ruy (1) Ruy (2) Ruy (N )]T EN = [e(1) e(2) e(N )]T T (1) T ΦN = (2) T (N ) ù^ { '¨ { OθCORLS (30)