194 8 DING Feng Syste dentfcaton Part H Coupled dentfcaton concept and ethods IEEE Transactons on Autoatc Control C-LS

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1 FIR CAR CARMA CARAR CARARMA OEMA OEAR TP273 A fdng@ jangnan edu cn Lorentz transforaton specal relatvty J 3

2 194 8 DING Feng Syste dentfcaton Part H Coupled dentfcaton concept and ethods IEEE Transactons on Autoatc Control C-LS 2 1 SLS 2 2 C-LS 2 3 C-SG 3 PC-SG 3 1 SG 3 2 SSG 3 3 PC-SG 4 PC-LS 4 1 LS 4 2 SLS 4 3 PC-SLS 4 4 PC-LS

3 Journal of Nanjng Unversty of Inforaton Scence and Technology Natural Scence Edton θ j θ j j j t 6 j t j t 1 j y t = T t T t θ + v t 3 coupled dentfcaton concept y t = T t T t θ + v t =1 2 5 t 3 2 j j 5 j θ R n y t = φ T t u t y t φ t R n y j t = φ T y t = φ T j t + v j t 7 t θ + v t 1 y t = θ T φ t + v t y t = φ T t + v t = v t y t =φ T t φ T t [ ] + v t = θ y t = 1 2 r 2 u j t j = 1 2 r y t y t = φ T t θ + v t = θ R n r u j t y t φ t R n y 1 t φ T 1 t y 2 t y t = φ T R 2 t Φ t = R n y t = φ T t θ + v t 2 y v t t φ T t = v 1 t v 2 t v t = R 1 2 v t θ j θ j j 10 φ t j φ j t y t = Φ t θ + v t 12 j ultvarable 2 y j t = T t T j t θ j + v j t j 4 + v t 6 11 lnear regresson odel ult-

4 196 8 DING Feng Syste dentfcaton Part H Coupled dentfcaton concept and ethods varate lnear regresson odel 5 13 batch process k = t d 24 h k = 1 1 t = 0 k = A z y t =- ψ t T L 2 2 t = 0 tera- tve learnng control syste 13 k + 1 t A z y 24k + t = B z u 24k + t + v 24k + t u k + 1 t u k t k = e k t + 1 t = lt y d t t y d 24k + t A z y k t = B z u k t + v k t 13 y d 24 + t = y d t t 0 t < 24 y d t v k t d u k t y k t z - 1 z - 1 y t = y t - 1 zy t = y t j A z B z z - 1 α j θ j A z = 1 + a 1 z a 2 z a na z - n a φ t j B z = b 1 z b 2 z b nb z - n b - ψ t j φ T t α θ u k t t = lt l 0 1 L - 1 y j t =- ψ j t φ T t α + v j t 16 θj y k t t = lt l d d + 1 d + L - 1 y d t t = lt l d d + 1 d + L - 1 y t =- ψ t φ T t [ α θ ] + v t = e k t = y d t y k t - ψ t α + φ T t θ + v t = - ψ t α + θ T φ t + v t = L L k k 0 1 ψ 1 t 2 ψ 2 t d G z = B z A z ψ t = R n θ = θ 1 θ 2 θ R n 1 ψ t d = 1 17 u k + 1 t = Q z u k t + R z e k t + 1 Q z R z z - 1 y 1 t - ψ 1 t α θ T 1 φ t v 1 t y Q- 14 k = t - ψ 2 t α = θ T 2 φ t v + 2 t y t - ψ t α θ T φ t v t + v t 15

5 Journal of Nanjng Unversty of Inforaton Scence and Technology Natural Scence Edton y t = - ψ t α + θ T φ t + v t 2 C-LS y t + ψ t α = θ T φ t + v t 1 n 1 n nput-output representaton I I n n X 2 = trxx T α θ coupled dentfcaton ethod 17 partally coupled dentfcaton ethod y t = Φ t θ + v t 19 y t = y 1 t y 2 t y t T R Φ t R n θ R n v t = v 1 t v 2 t v t T R 6 7 t 0 y t = 0 Φ t = 0 9 v t = ^θ t θ t J θ = y j - Φ j θ J θ = t y j - Φ j θ 2 j = 1 ^θ t = ^θ t L t y t - Φ t ^θ t L t R n Partally coupled stochastc gradent dentfcaton ethods for nonunforly sapled systes 13 R 0 = 1 L t = ΦT t R t R t = R t Φ t 2 24 θ SG ^θ t = ^θ t L t y t - Φ t ^θ t - 1 ^θ 0 = 1 n /p 0 20 L t = ΦT t R t 21 R t = R t Φ t 2 R 0 = 1 22

6 198 8 DING Feng Syste dentfcaton Part H Coupled dentfcaton concept and ethods L t = P t Φ T t 19 θ RLS y t = T t θ + v t = ^θ t = ^θ t L t y t - Φ t ^θ t - 1 θ ^θ 0 = 1 n /p 0 23 R n L t = P t Φ T t 24 P -1 t =P -1 t -1 +Φ T t Φ t P 0 =p 0 I n P t R n n n > L t = P t - 1 Φ T t I + Φ t P t - 1 Φ T t P t = I n -L t Φ t P t -1 P 0 =p 0 I n 29 ^θ t ^θ t ^θ 0 = 1 n /p 0 p 0 = 10 6 RLS j ^θ t j ^θ t RLS ^θ t SLS I + Φ t P t - 1 Φ T t - 1 R ^θ t = ^θ t P t t y t - RLS P - 1 t = P - 1 t t T t P 0 = p 0 I n = SLS 1 RLS Table 1 The densons of the varables n the RLS algorth ^θ t =^θ t -1 + L t y t - T t ^θ t -1 y t R ^θ 0 = 1 n /p 0 35 θ R n L t = P t t = P t - 1 t 1 + ^θ t R n Φ t R n L t R n P t R n n SLS SLS T t R 1 n Φ t T 1 t T 2 t Φ t = R n T t θ θ 30 RLS SLS ^θ t =^θ t -1 +P t t y t - T t ^θ t -1 A + BC -1 = A -1 - A -1 B I + CA -1 B -1 CA ^θ 0 = 1 n /p 0 31 ^θ t = ^θ t L t y t - Φ t ^θ t - 1 P - 1 t = P - 1 t t T t ^θ 0 = 1 n /p 0 27 P 0 = p 0 I n = P t R n n T t ^θ t - 1 ^θ 0 = 1 n /p 0 T t P t - 1 t - 1 P t = I n - L t T t P t P 0 = p 0 I n = ^θ t 1 Fg 1 The schenatc dagra of the subsyste least squares algorth

7 Journal of Nanjng Unversty of Inforaton Scence and Technology Natural Scence Edton = ^θ t θ Coupled Least Squares dentfcaton algorth θ SLS θ ^θ t =^θ -1 t + L t θ L t = P t t ^θ t = ^θ 1 t + ^θ2 t + + ^θ t P - 1 R n t = P t + t T t = ^θ t ^θ1 t =^θ t -1 + L 1 t y 1 t - 1 t ^θ T t -1 θ ^θ 0 = 1 n /p 0 49 ^θ t L 1 t = P 1 t 1 t 50 P t = P - 1 t t T 1 t ^θ t P 0 = p 0 I n 51 ^θ t - 1 SLS C-LS ^θ t - 1 C-SLS ^θ t =^θ -1 t + L t y t - T t ^θ -1 t 52 ^θ t = ^θ t P t t y t - P - 1 T t ^θ t - 1 ^θ 0 = 1 n /p 0 t = P - 1 t t T t P 0 = p 0 I n = ^θ t = ^θ 1 t + ^θ2 t + + ^θ t 41 ^θ t =^θ t -1 + L t y t - T t ^θ t -1 ^θ 0 = 1 n /p 0 42 L t = P t t = P t - 1 t 1 + T t P t - 1 t - 1 P t = I n - L t T t P t - 1 P 0 = p 0 I n = ^θ t = ^θ 1 t + ^θ2 t + + ^θ t 45 C-LS y t - T t ^θ -1 t 46 ^θ t 1 t = 1 ^θ 0 = 1 n /p 0 P 0 = p 0 I n p 0 = L t =P -1 t t / 1 + T t P -1 t t 53 P t = I - L t T t P - 1 t = ^θ1 t =^θ t -1 + L 1 t y 1 t - 1 t ^θ T t -1 ^θ 0 = 1 n /p 0 55 L 1 t = P t t / 1 + T 1 t P t t P 1 t = I - L 1 t T 1 t P t P 0 = p 0 I n 57 ^θ t P t ^θ t R n L t R n P t R n n t C-LS y t Φ t T t 2 2 C-LS R 1 n Φ t 3 56 L 1 t 57 P 1 t 55-1 t ^θ 1 t ^θ - 1 t t = t - 1 θ 10 L t 54 P t ^θ t Jacob Gauss-Sedel 5 t 1 2 ^θ - 1 t 33 ^θ t - 1 = RLS ^θ t = 1 ^θ 1 t - 1 I + Φ t P t - 1 Φ T t C-LS

8 200 8 DING Feng Syste dentfcaton Part H Coupled dentfcaton concept and ethods C-LS L t = P t - 1 t / 1 + RLS T t P t - 1 t P t = I - L t T t P t RLS P 0 = p 0 I n = ^θ t = ^θ t 24 P t = P t 33 RLS ^θ 0 = 1 n /p P 0 = p 0 I n ^θ 0 = 1 n /p 0 P 0 = p 0 I n = RLS 63 3 ^θ t = ^θ t ^θ t P t 2 3 C-SG ^θ1 t =^θ t -1 + L 1 t y 1 t - 1 t ^θ T t -1 C-SLS ^θ 0 = 1 n /p 0 58 C-SSG L 1 t = P 1 t t / 1 + T 1 t P 1 t t P 1 t = I - L 1 t T 1 t P 1 t - 1 P 1 0 = p 0 I n ^θ t = ^θ - 1 t + L t y t - T t ^θ - 1 t t ^θ t = ^θ t -1 + r t y t - T t ^θ t -1 ^θ 0 = 1 n /p 0 64 r t = r t t 2 r 0 = 1 = Fg 2 2 The scheatc dagra of the coupled least squares algorth Fg 3 3 The scheatc dagra of the paraeter estaton coupled least squares algorth

9 Journal of Nanjng Unversty of Inforaton Scence and Technology Natural Scence Edton ^θ t = ^θ 1 t + ^θ2 t + + ^θ t r 1 0 = ^θ 0 λ 1 C-SSG t = ^θ -1 t + t r t y t - T t ^θ -1 t r t = λr t t 2 r 0 = 1 t ^θ t = ^θ t -1 + r t y t - T = t ^θ t -1 ^θ 0 = 1 n /p 0 67 r t = λr t t 2 r 0 = 1 0 λ 1 = ^θ t = ^θ 1 t + ^θ2 t + + ^θ t 69 convergence ndex C-SSG x t + 1 = Ax t + Bu t t ^θ t =^θ t -1 + r t y y t = Cx t t - T t ^θ t -1 { 81 x t R n u t =u 1 t u 2 t ^θ 0 = 1 n /p < 1 70 r t = r t t 2 r 0 = 1 = ^θ t = ^θ 1 t + ^θ2 t + + ^θ t y t = G z u t G z = C zi - A - 1 B = C adj zi - AB det zi - A = r t = r - 1 t + t 2 = ^θ1 t =^θ t t r 1 t y 1 t - 1 t ^θ T t Q r 1 t = r t t 2 1 z Q 2 z Q n z - n Q R r r 0 = 1 76 α z C-SG C-SG 82 y t = Q z u t α z an odel ^θ t r t G z = Q z r t λ α z C-SG ^θ1 t =^θ t t r 1 t y 1 t - 1 t ^θ T t -1 ^θ 0 = 1 n /p 0 77 r 1 t = λr 1 t t 2 0 λ 1 3 PC-SG u r t T R r y t = y 1 t y 2 t y t T R A B C 81 G z R r z - n C adj zi - AB r t = trp - 1 t z - n det zi - A = Q z α z C-SG α z n ^θ t = ^θ - 1 t + t r t y t - T t ^θ - 1 t G z Q z z - 1 ^θ 0 = 1 n /p 0 73 α z = z - n det zi - A= α 1 z α 2 z α n z - n α R Q z = z - n C adj zi - AB = subodel α z y t = Q z u t v t R CAR ARX CAR-lke

10 202 8 DING Feng Syste dentfcaton Part H Coupled dentfcaton concept and ethods ARX-lke 37 α z y t = Q z u t + v t v t = v 1 t v 2 t v t T R z - 1 α z Φ ^t = ^t T t -1 + R t y t - Φ t ^t Q z R t = R t Φ t 2 R 0 = 1 88 y t + α 1 y t α 2 y t α n y t - n = Q 1 u t -1 + Q 2 u t Q n u t - n + v t α 2 y t y t - 2 y t - n = αn u t - 1 u t - 2 Q 1 Q 2 Q n + v t 83 u t - n θ φ t ψ t θ T =Q 1 Q 2 Q n R nr α 1 u t - 1 α 2 u t - 2 α = R n φ t = R nr α u t - n n ψ t = y t - 1 y t - 2 y t - n R n 83 ARX-lke y t + ψ t α = θ T φ t + v t α 1 84 t J 3 = y t - Φ t Φ t ψ t R n ψ t ψ 1 t ψ 2 t ψ t = R n ψ t Φ t Φ t =- ψ t I φ T t = - α ψ 1 t φ T t ψ 2 t 0 φ T t 0 - ψ 1 t 0 0 φ T t 3 1 SG t = - ψt t 84 α R n φ t Rn + nr 90 θ R nr 89 Φ t 86 α θ 84 θ R nr θ y 1 t y 2 t = α θ 1 y t α = [ colθ ] = θ 2 R n + nr - ψ 1 t φ T t 0 0 α - ψ 2 t 0 φ T t 0 θ 1 v 1 t v 2 t θ θ 2 + Φ t =- ψ t I φ T t R n + nr 85 - ψ 1 t 0 0 φ T t v t 84 θ y t = Φ t + v t 86 y t =- ψ t φ T t α ^α t ^θ t ^t α θ θ R n + nr 89 Φ t n r nr SSG + v t

11 Journal of Nanjng Unversty of Inforaton Scence and Technology Natural Scence Edton = T t [ α θ ] + v t = J 4 α θ = y t - T t α 2 { [ θ ] = 1 2 } α α θ 3 3 PC-SG SSG 91 α ^α t ^θ t = ^α t - 1 [ ^θ t - 1 ] { } t R t y ^α t - 1 t T t - 1 ^θ t R t = R t t 2 R 0 = 1 = SSG α t θ = 1 2 α R t y ^α t - 1 t T t - 1 ^θ t - 1 α ^θ 0 = 1 nr /p ^α t SSG R t = R t t 2 R 0 = 1 ^α t ^θ t = ^α t - 1 [ ^θ t - 1 ] { } t R t y ^α t - 1 t T t - 1 ^θ t - 1 ^α 0 = 1 n /p 0 ^θ 0 = 1 nr /p 0 R t = R t t 2 R 0 = 1 94 = R 1 t = R 1 t t 2 R 1 0 = φ t = u T t - 1 u T t - 2 u T t - n T ψ t = y t - 1 y t - 2 y t - n ^θ t θ ψ 1 t ^α t α ψ 2 t ^α t α ψ t = t = - ψt t [ φ t ] 106 ψ ^α t = ^α t 1 t + ^α 2 t + + ^α t R n ^α t - 1 SSG ^α t - 1 PC-SSG ^α t ^θ t = ^α t - 1 [ ^θ t - 1 ] { } t R t y ^α t - 1 t T t - 1 ^θ t - 1 ^α 0 = 1 n /p 0 ^θ 0 = 1 nr /p 0 R t = R t t 2 R 0 = 1 97 = ^α t = ^α 1 t + ^α 2 t + + ^α t 99 R n α Partally Coupled Stochastc Gradent PC-SG 13 ^α t ^θ t = ^α t - 1 [ ^θ t - 1 ] { } = 2 3 ^α t ^θ t = ^α t - 1 [ ^θ t - 1 ] { } 101 t R t y t T t - 1 ^α t - 1 ^θ t - 1 ^α 0 = 1 n /p 0 ^θ1 0 = 1 nr /p ^α t ^θ t α θ 1 θ 2 θ ^α t = ^α t ^θ 1 t ^θ 2 t ^θ t 4 SG PC-SSG PC-SG α 2 n = 5 5

12 204 8 DING Feng Syste dentfcaton Part H Coupled dentfcaton concept and ethods Fg 4 4 The scheatc dagra of the subsyste stochastc gradent algorth Fg 5 5 The scheatc dagra of the partally coupled stochastc gradent algorth 5 5 t 2 PC-SG R 1 t = R t 100 ^α t ^θ t Table 2 The coparson of coputatonal effcency 5 t 1 2 PC-SG SG 3 n + nr + = n + nr = ^α t ^θ t = PC-SSG 3 n + nr + + n = 460 n + nr n = 470 PC-SG 3 n + nr + = n + nr = 450 λ λ R t = λr t Φ t 2 R 0 = 1 flop 32 0 λ SG = R t = λr t φ t 2 R 0 = flop PC-SSG = 930 flop PC-SSG = 905 flop PC-SG ^α t ^θ t = t = ^α 0 = 1 n /p 0 ^θ 0 = 1 n0 /p 0 R 0 = 1 = 1 2 p 0 = u t y t 104 ^α 1 t ^θ 1 t = λ λ SG λ PC-SG PC-LS φ t 105 ψ t 106 ψ t =

13 Journal of Nanjng Unversty of Inforaton Scence and Technology Natural Scence Edton ^α t L t =P t -1 Φ T t I +Φ t P t -1 Φ T t P t = I n + nr - L t Φ t P t - 1 P 0 = p 0 I n + nr 111 ^t t P t n + nr n + nr 5 RLS L t = P t t = P t P t -1 t 1 + T t P t -1 t P t = I n + nr - L t T t P t SLS 86 J α θ = t j = 1 { } y j - T j α θ 2 = 1 2 α θ 4 3 PC-SLS SLS ^α t - 1 SLS ^α t ^θ t = ^α t - 1 ^θ t { } ^α t - 1 P t φ t y t - T t ^θ t P -1 t =P -1 t -1 + t T t = α ^α t ^α t ^θ t = ^α t - 1 [ ^θ t - 1 ] { } P t t y t - T t ^α t - 1 ^θ t α α ^α t = ^α 1 t + ^α 2 t + + ^α t 115 ^θ t = SLS Fg 6 The flowchart of coputng the paraeter estates ^α t and ^θ t = 1 2 ^α t [ ^θ t ] = ^α t - 1 [ ^θ t - 1 ] + 38 P 4 1 LS t φ t y t - T t ^α t - { [ ^θ t - 1 ] } P - 1 t = P - 1 t t T t J = t y j - Φ j 2 P 0 = p 0 I n + nr = j = 1 ^α t = ^α 1 t + ^α 2 t + + ^α t ^t = ^t -1 + L t y t - Φ t ^t ^α 0 = 1 n /p 0 ^θ 0 = 1 nr /p ^α t ^θ t = ^α t - 1 [ ^θ t - 1 ] { } L t y t - T t ^α t - 1 ^θ t - 1 P 0 = p 0 I n + nr = 1 2 ^α t = ^α 1 t + ^α 2 t + + ^α t ^α 0 = 1 n /p 0 ^θ 0 = 1 nr /p ^α t - 1 PC-SLS

14 206 8 DING Feng Syste dentfcaton Part H Coupled dentfcaton concept and ethods ^α t ^θ t = P - 1 ^α t - 1 [ ^θ t - 1 ] { } ^α t - 1 P t t y t - T t ^θ t t = P - 1 t t T t P 0 = p 0 I n + nr = 1 2 ^α t = ^α 1 t + ^α 2 t + + ^α t ^α 0 = 1 n /p 0 ^θ 0 = 1 nr /p 0 ^α t ^θ t = ^α t - 1 [ ^θ t - 1 ] { } ^α t - 1 L t y t - T t ^θ t - 1 L t = P t t = P t -1 t 1 + T t P t -1 t P t = I n + nr - L t T t P t - 1 P 0 = p 0 I n + nr = 1 2 ^α t = ^α 1 t + ^α 2 t + + ^α t 128 ^α 0 = 1 n /p 0 ^θ 0 = 1 nr /p PC-SLS P t R nr nr 4 4 PC-LS R n + nr n + nr ^α t = ^α t ^θ 1 t C-LS ^θ 2 t ^θ t α θ 1 θ 2 θ Partally Coupled Least Squares dentfcaton algorthpc- PC-LS LS 38 1 t = 1 ^α 0 = 1 n /p 0 ^θ 0 = 1 nr /p 0 ^α 1 t ^θ 1 t = P - 1 ^α t - 1 [ ^θ 1 t - 1 ] { } P 1 t 1 t y 1 t - T 1 t ^α t - 1 ^θ 1 t t = P - 1 t t T 1 t P 0 = p 0 I n + nr ^α 0 = 1 n /p 0 ^α t ^θ t = ^α - 1 t [ ^θ t - 1 ] + { } 131 P t t y t - T t ^α - 1 t ^θ t P - 1 t = P t + t T t ^θ 0 = 1 nr /p 0 = PC-LS ^α 1 t ^θ 1 t = ^α t - 1 [ ^θ 1 t - 1 ] { } L 1 t y 1 t - T 1 t ^α t - 1 ^θ 1 t P t t L 1 t = 1 + T 1 t P t t 135 P 1 t = I - L 1 t T 1 t P t - 1 P 0 = p 0 I n + nr ^α 0 = 1 n /p 0 ^α t ^θ t = ^α - 1 t [ ^θ t - 1 ] + { } L t y t - T t ^α - 1 t ^θ t P - 1 t t L t = 1 + T t P - 1 t - 1 t 138 P t = I - L t T t P - 1 t ^θ 0 = 1 nr /p 0 = t = u T t - 1 u T t - 2 u T t - n T 140 ψ t = y t - 1 y t - 2 y t - n 141 ψ 1 t ψ 2 t ψ t = t = ψ t - ψt t t 142 PC-LS P t P 0 = p 0 I n + nr = 1 2 p 0 = u t y t 140 φ t 141 ψ t 142 ψ t ψ t = 1 2 φ t L 1 t 136 P 1 t 134 ^α 1 t ^θ 1 t 4 = L t 139 P t 137 ^α t ^θ t 5 t PC-LS ^α t

15 Journal of Nanjng Unversty of Inforaton Scence and Technology Natural Scence Edton P t n + nr n + nr α z y t = Φ t θ + C - 1 z D z v t 144 RLS P t n + nr n + nr P t n 2 r n 2 r P t α z = n = r = 5 A z = I + A 1 z A 2 z A na z - n a A R n 2 r = n 2 r = y t = Φ t α z θ + C - 1 z D z v t y t = A - 1 z Φ t θ + C - 1 z D z v t 147 C z D z C z D z ultvarable ovng average pseudo-lnear syste y t = Φ t θ + d z v t y t = Φ t θ + D z v t c z = 1 + c 1 z c 2 z c nc z - n c c R d z = 1 + d 1 z d 2 z d nd z - n d d R C z = I + C 1 z C 2 z C nc z - n c C R D z = I + D 1 z D 2 z D nd z - n d D R c d C R D R n c n d t 0 y t = 0 Φ t = 0 v t = y t α z A z A z y t = Φ t θ + C - 1 z D z v t ultvarable autoregressve pseudo-lnear syste y t = Φ t θ + C - 1 z D z v t 143 y t = y 1 t y 2 t y t T R y t = Φ t θ + 1 c z v t 150 Φ t R n v t = v 1 t v 2 t v t T R y t = Φ t θ + C - 1 z v t 151 θ R n C z D z z - 1 z - 1 x t = x t ultvarable autoregressve ovng average pseudo-lnear syste y t = Φ t θ + d z v t c z y t = Φ t θ + D z v t c z

16 208 8 DING Feng Syste dentfcaton Part H Coupled dentfcaton concept and ethods y t = Φ t θ + C - 1 z d z v t 154 y t = Φ t θ + C - 1 z D z v t α z y t = Q z u t + C - 1 z d z v t 163 α z y t = Q z u t + C - 1 z D z v t CARARMA CARARMA-lke ARARMAX 5 ARARMAX-lke α z y t = Q z u t + C - 1 z D z v t 156 C z D z ARARMAX-lke 1 y t = Q z α z u t + C - 1 z D z v t 165 CARMA CARMA-lke ARMAX ARMAX-lke -lke CARARMA-lke ARARMAXlke α z y t = Q z u t + d z v t c z 161 α z y t = Q z u t + D z v t c z CARARMA CARARMA-lke ARARMAX 7 C z D z 165 α z y t = Q z u t + d z v t ultvarable OElke syste α z y t = Q z u t + D z v t y t = Q z u t + v t α z CARARlke ARARX-lke 2 ultvarable OEMA-lke syste α z y t = Q z u t + 1 c z v t 159 y t = Q z u t + d z v t α z 167 α z y t = Q z u t + C - 1 z v t 160 y t = Q z u t + D z v t α z ultvarable OEAR-lke syste y t = Q z α z u t + 1 c z v t 169

17 Journal of Nanjng Unversty of Inforaton Scence and Technology Natural Scence Edton y t = Q z α z u t + C - 1 A z y t = B z u t + d z v t 177 z v t A z y t = B z u t + D z v t ultvarable e- quaton error autoregressve syste e ultvarable EEAR syste CARAR ARARX 4 A z y t = B z u t + 1 ultvarable OEARMA-lke syste e ulltvarable Box-Jenkns-lke syste c z v t 179 y t = Q z d z u t + v t α z c z 171 A z y t = B z u t + C - 1 z v t ultvarable CARARMA syste y t = Q z D z u t + v t α z c z 172 A z y t = B z u t + d z v t c z 181 y t = Q z α z u t + C - 1 z d z v t 173 y t = Q z α z u t + C - 1 z D z v t A z y t = B z u t + C - 1 z D z v t OEARMA Box- 5 4 Jenkns y t = A - 1 z B z u t + C - 1 z D z v t 185 CARARMA ζ z y t = A -1 z B z u t + C -1 z D z v t 186 CARARMA-lke ARARMAX ARARMAX-lke A z y t = B z u t + C - 1 z D z v t 175 B z B z = B 1 z B 2 z B nb z - n b B R r C z D z C z D z ultvarable equaton error syste CAR ARX A z y t = B z u t + v t ultvarable 188 equaton error ovng average syste e ultvarable EEMA syste CARMA ARMAX A z y t = B z u t + D z v t c z 182 A z y t = B z u t + C - 1 z d z v t 183 F z y t = A -1 z B z u t + C -1 z D z v t 187 ζ z = 1 + ζ 1 z ζ 2 z ζ n z - n ζ R F z = I + F 1 z F 2 z F nf z - n f F R 1 ultvarable output error syste y t = A - 1 z B z u t + v t 188

18 210 8 DING Feng Syste dentfcaton Part H Coupled dentfcaton concept and ethods 2 ultvarable Snha Kwong Autoatca OEMA syste y t = A - 1 z B z u t + d z v t y t = A - 1 z B z u t + D z v t ultvarable References OEAR syste y t = A - 1 z B z u t + 1 c z v t J DING Feng Martngale hyperconvergence theore and convergence of the forgettng factor least squares algorthj Control Theory and Applcatons y t = A - 1 z B z u t + C - 1 z v t ultvarable OEARMA syste Box-Jenkns J y t = A - 1 z B z u t + d z v t c z 193 DING Feng YANG Jaben Rearks on the artngale hyperconvergence theore and convergence analyss of the forgettng factor least squares algorths J Control y t = A - 1 z B z u t + D z Theory and Applcatons v t c z J y t = A - 1 z B z u t + C - 1 DING Feng YANG Huzhong JI Zhcheng Te-varyng z d z v t 195 syste dentfcaton ethods and convergence theores J Journal of Southern Yangtze Unversty Natural Scence Edton y t = A - 1 z B z u t + C - 1 z D z v t J DING Feng Syste dentfcaton Part A Introducton to the dentfcaton J Journal of Nanjng Unversty of Inforaton Scence & Technology Natural Scence Edton J DING Feng Syste dentfcaton Part B Basc odels for syste descrpton J Journal of Nanjng Unversty of Inforaton Scence & Technology Natural Scence E- dton J DING Feng Syste dentfcaton Part C Identfcaton accuracy and basc probles J Journal of Nanjng Unversty of Inforaton Scence & Technology Natural Sc- ence Edton J DING Feng Syste dentfcaton Part D Auxlary odel dentfcaton dea and ethods J Journal of Nanjng Unversty of Inforaton Scence & Technology Natural Scence Edton J

19 Journal of Nanjng Unversty of Inforaton Scence and Technology Natural Scence Edton puter Modellng / DING Feng Syste dentfcaton Part E Iteratve search 21Zhang Z N Dng F Lu X G Herarchcal gradent based prncple and dentfcaton ethods J Journal of Nanjng Unversty of Inforaton Scence & Technology Nat- output error ovng average systesj Coputers & teratve paraeter estaton algorth for ultvarable ural Scence Edton Matheatcs wth Applcatons J 22Wang D Q Dng F Input-output data flterng based recursve least squares paraeter estaton for CARARMA DING Feng Syste dentfcaton Part F Mult-nnovaton dentfcaton theory and ethodsj Journal of 999 systes J Dgtal Sgnal Processng Nanjng Unversty of Inforaton Scence & Technology 23Dng F Lu X P Lu G Identfcaton ethods for Haersten nonlnear systesj Dgtal Sgnal Process J ng Natural Scence Edton M DING Feng Syste dentfcaton Part G Herarchcal dentfcaton prncple and ethods J Journal of Nanjng Unversty of Inforaton Scence & Technology Nat- M Bejng Tsnghua Unversty Press 1988 FANG ChongzhXIAO Deyun Process Identfcaton ural Scence Edton J J DING Feng YANG Jaben Herarchcal dentfcaton of large scale systes J Acta Autoatca Snca DING Feng Stochastc gradent estaton algorth for ultple-nput systes based on the output estaton 26Dng F Yang J B Xu Y M Convergence of herarchcal J Journal of Nanjng Unversty of Inforaton Scence stochastc gradent dentfcaton for transfer functon atrx odels J Control Theory and Applcatons 2001 & Technology Natural Scence Edton J 27Dng F Chen T Perforance analyss of ult-nnovaton gradent type dentfcaton ethods J Autoatca DING FengXIAO DeyunDING Tao Mult-nnovaton stochastc gradent dentfcaton ethodsj Control 28Dng F Lu X P Lu G Mult-nnovaton least squares - Theory and Applcaton dentfcaton for syste odelng J IEEE Transactons 13Dng F Lu G Lu X P Partally coupled stochastc gradent dentfcaton ethods for non-unforly sapled on SystesMan and Cybernetcs Part B Cybernetcs systesj IEEE Transactons on Autoatc Control 29Dng F Several ult-nnovaton dentfcaton ethods J Dgtal Sgnal Processng Brstow D A Tharayl M Alleyne A G A survey of tera- 30Wang D Q Dng F Perforance analyss of the auxlary tve learnng controlj IEEE Control Ssystes Magazne ton algorth for output error systes J Dgtal Sgnal 15Dng F Chen T Herarchcal gradent-based dentfca- Processng odels based ult-nnovaton stochastc gradent estaton of ultvarable dscrete-te systes J Autoat- 31Lu Y J Yu LDng F Mult-nnovaton extended stochastc gradent algorth and ts perforance analyss 16Dng F Chen T Herarchcal least squares dentfcaton J Crcuts Systes and Sgnal Processng ca ethods for ultvarable systes J IEEE Transactons on Autoatc Control Golub G H Van Loan C F Matrx CoputatonsM 3rd 17Wang L Y Dng F Lu X P Consstency of HLS estaton algorths for MIMO ARX-lke systes J Appled 33Dng F Lu Y J Lu X G et al Coupled least squares - Ed Baltore MD Johns Hopkns Unv Press 1996 Matheatcs and Coputaton dentfcaton for ultvarable systesj IET Control 18 Theory Subtted J DING Feng XIAO Deyun Herarchcal dentfcaton of J state space odels for ultvarable systes J Control DING Feng DING Tao YANG Jaben et al Convergence and Decson of forgettng gradent estaton algorth for te-varyng 19Dng F Chen T Herarchcal dentfcaton of lfted statespace odels for general dual-rate systesj IEEE paraetersj Acta Autoatca Snca Transactons on Crcuts and Systes I Regular Papers 35Dng J Sh YWang H G et al A odfed stochastc gradent based paraeter estaton algorth for dual- 20Han H Q Xe L Dng F et al Herarchcal least squares rate sapled-data systes J Dgtal Sgnal Processng based teratve dentfcaton for ultvarable systes wth ovng average noses J Matheatcal and Co- 36Chen J Dng F Modfed stochastc gradent algorths

20 212 8 DING Feng Syste dentfcaton Part H Coupled dentfcaton concept and ethods wth fast convergence rates J Journal of Vbraton and for dscrete-te ultple-nput ultple-output systes Control C The th Chnese Control and Decson Con- 37Lu Y J Sheng J Dng R F Convergence of stochastc ference 2012 CCDC Tayuan Chna gradent algorth for ultvarable ARX-lke systes 39L J H Dng R F Yang Y Iteratve paraeter dentfca- J Coputers & Matheatcs wth Applcatons 2010 ton ethods for nonlnear functons J Appled Matheatcal Modellng Lu Y J Dng R Partally coupled estaton algorth Syste dentfcaton Part H Coupled dentfcaton concept and ethods DING Feng School of Internet of Thngs Engneerng Jangnan Unversty Wux Control Scence and Engneerng Research Center Jangnan Unversty Wux Key Laboratory of Advanced Process Control for Lght Industry Mnstry of Educaton Jangnan Unversty Wux Abstract Coupled dentfcaton s an portant branch of syste dentfcaton and s a new dentfcaton concept whch s used anly to study dentfcaton probles of lnear and nonlnear ultvarable systes wth coplex structures and paraeter couplng The auxlary odel dentfcaton dea the ult-nnovaton dentfcaton theory the herarchcal dentfcaton prncple and the coupled dentfcaton concept are new dentfcaton research deas concepts and ethods and can be used to study dentfcaton probles of systes wth unknown process varables to prove the convergence rates and accuraces of dentfcaton ethods to solve dentfcaton probles of largescale ultvarable systes wth coplex structures and of ultvarable systes wth paraeter couplng reducng the coputatonal load of the dentfcaton algorths Ths paper ntroduces the coupled dentfcaton concept of ultvarable systes dscusses the full coupled least squares dentfcaton ethods the full coupled stochastc gradent dentfcaton ethods the partally coupled stochastc gradent dentfcaton ethods the partally coupled least squares dentfcaton ethods etc for ultvarable systes Fnally we show that the coupled dentfcaton ethods can be appled to ultvarable systes wth colored noses lst soe odel structures of soe ultvarable systes and ndcate that the coupled dentfcaton concept can cobne the auxlary odel dentfcaton dea the ult-nnovaton dentfcaton theory herarchcal dentfcaton prncple the teratve search prncple the gradent teraton the least squares teraton the Newton teraton to study dentfcaton probles of lnear and nonlnear ultvarable systes wth colored noses Key words teratve dentfcaton recursve dentfcaton paraeter estaton FIR odel equaton error odel CAR odel CARMA odel CARAR odel CARARMA odel output error odel OEMA odel OEAR odel auxlary odel dentfcaton ult-nnovaton dentfcaton herarchcal dentfcaton coupled dentfcaton