Ta . p ch´ı Tin ho . c v`a Diˆe ` u khiˆe ’ n ho . c, T.21, S.3 (2005), 244—247 NH ˆ A . N DA . NG M ` U CHU ˆ O ˜ I HAMMERSTEIN B ˆ A . C HAI TR ˆ A ` N THI . HO ` ANG OANH 1 , D ˆ O ` NG S ˜ I THI ˆ EN CH ˆ AU 2 1 Tru . `o . ng Da . i ho . c Cˆong nghiˆe . p, Tp Hˆo ` Ch´ı Minh 2 Tru . `o . ng Da . i ho . c B´an cˆong Tˆon D´u . c Th˘a ´ ng, Tp Hˆo ` Ch´ı Minh Abstract. In this paper, a method of blind identification of second order Hammerstein series is considered. This method is developed on the combination of stochastic approximation and Tixonop method. T´om t˘a ´ t. Du . . a trˆen su . . kˆe ´ t ho . . p gi˜u . a hai phu . o . ng ph´ap chı ’ nh h´oa Tixonop v`a l´y thuyˆe ´ t xˆa ´ p xı ’ ngˆa ˜ u nhiˆen, b`ai b´ao dˆe ` cˆa . p dˆe ´ n mˆo . t phu . o . ng ph´ap nhˆa . n da . ng chuˆo ˜ i Hammerstein bˆa . c hai. GI ´ O . I THI ˆ E . U C´ac mˆo h`ınh Hammerstein du . o . . c ´u . ng du . ng dˆe ’ mˆo ta ’ hˆe . thˆo ´ ng phi tuyˆe ´ n d˜a v`a dang du . o . . c ´u . ng du . ng nhiˆe ` u trong c´ac qu´a tr`ınh sinh ho . c, h´oa ho . c, viˆe ˜ n thˆong, diˆe ` u khiˆe ’ n v`a xu . ’ l´y t´ın hiˆe . u [1, 2, 3]. Ta x´et chuˆo ˜ i Hammerstein bˆa . c hai sau dˆay: y(n) = k + 1  k=k − 1 h k (n)x(n − k) + k + 2  k=k − 2 h kk (n)x 2 (n − k); (1) h 1 (0) = 1; k 1 , k 2 l`a bˆa . c cu ’ a hˆe . thˆo ´ ng; x(n) l`a t´ın hiˆe . u dˆa ` u v`ao d`u . ng c´o trung b`ınh b˘a ` ng khˆong da . ng Gauss. B`ai to´an nhˆa . n da . ng m`u du . o . . c d˘a . t ra l`a du . . a trˆen c´ac thˆong tin dˆa ` u ra y(n) v`a dˆa ` u v`ao x(n) , h˜ay x´ac di . nh c´ac gi´a tri . h k (n) v`a h kk (n) . Du . a y(n) = X T (n)h(n), o . ’ dˆay ta k´y hiˆe . u: h(n) = (h k − 1 h k + 1 . . . h k − 2 k − 2 h k + 2 k + 2 ) T , X(n) = (x(n − k − 1 ) x(n − k + 1 ) . . . x 2 (n − k − 2 ) x 2 (n − k + 2 )) T . (2) X´et mˆo h`ınh: ˆy(n) = k + 1  k=k − 1 ˆ h k (n)x(n − k) + k + 2  k=k − 2 ˆ h kk (n)x 2 (n − k), (3) ˆy(n) = X T (n) ˆ h(n), (4) o . ’ dˆay ta k´y hiˆe . u ˆ h(n) = ( ˆ h k − 1 ˆ h k + 1 . . . ˆ h k − 2 k − 2 . . . ˆ h k + 2 k + 2 ) T . Lˆa . p tiˆeu chuˆa ’ n d´anh gi´a tˆo ´ i u . u: NH ˆ A . N DA . NG M ` U CHU ˆ O ˜ I HAMMERSTEIN B ˆ A . C HAI 245 E{e 2 (n)} → min ˆ h , o . ’ dˆay sai sˆo ´ c´o da . ng: e(n) = y(n) − ˆy(n) = X T (n)(h(n) − ˆ h(n)). (6) T`u . diˆe ` u kiˆe . n tˆo ´ i thiˆe ’ u h´oa theo tiˆeu chuˆa ’ n d´anh gi´a (5) ta thu du . o . . c: ˆ h k (n + 1) = γ ˆ h k (n) + F k ( ˆ h k (n) ˆ h k (n − 1)) − η(n)E{e(n)x(n − k)}, k = k − 1 , , 0, 1, 2, , k + 1 , 0 < γ  1, 0 < F k << 1, (7) ˆ h kk (n + 1) = γ ˆ h kk (n) + F kk ( ˆ h kk (n) − ˆ h kk (n − 1)) − η(n)E{e(n)x 2 (n − k)}, k = k − 2 , , 0, 1, 2, , k + 2 . (8) Trong (7) v`a (8) c´ac bu . ´o . c l˘a . p η(n) du . o . . c cho . n sao cho tho ’ a m˜an c´ac diˆe ` u kiˆe . n hˆo . i tu . cu ’ a Robbin—Monro [4] du . . a theo l´y thuyˆe ´ t xˆa ´ p xı ’ ngˆa ˜ u nhiˆen: 0 < η(n) → 0 khi n → ∞; η(n − 1) − η(n) η(n) → 0 khi n → ∞, (9) ∞  n=1 η(n) = ∞; ∞  n=1 η p (n) < ∞; p ≥ 2. (10) C´o thˆe ’ du . a c´ac thuˆa . t to´an (7) v`a (8) vˆe ` da . ng kh´ac sau dˆay: ˆ h k (n + 1) =  I − η(n) x(n − k)x T (n − k) x(n − k) 2 2  ˆ h k (n) + F k ( ˆ h k (n) − ˆ h k (n − 1)) + η(n)y(n) x(n − k) x(n − k) 2 2 , ˆ h kk (n + 1) =  I − η(n) x(n − k)x T (n − k) x(n − k) 2 2  ˆ h kk (n) + F k ( ˆ h kk (n) − ˆ h kk (n − 1)) + η(n)y(n) x(n − k) x(n − k) 2 2 . Ta x´et hˆe . thˆo ´ ng dˆo . ng phi tuyˆe ´ n c´o dˆa ` u ra du . o . . c mˆo ta ’ bo . ’ i phu . o . ng tr`ınh: y(n) = N a  i=0 a i (n)x(n − i) + N d  i=0 N d  i=0 d ii (n)x 2 (n − i) + N b  i=1 b i (n)y(n − i) + N c  i=1 N c  i=1 c ii (n)y 2 (n − i). (11) 246 TR ˆ A ` N THI . HO ` ANG OANH, D ˆ O ` NG S ˜ I THI ˆ EN CH ˆ AU Ta c´o h`am quan s´at z(n) : z(n) = y(n) + v(n), (12) o . ’ dˆay v(n) l`a nhiˆe ˜ u quan s´at da . ng Gauss E{v(n)} = 0, E{v 2 (n)} = σ 2 v < ∞, (13) E{.} l`a k`y vo . ng to´an ho . c. Diˆe ` u kh´ac biˆe . t o . ’ dˆay v´o . i c´ac t`ai liˆe . u d˜a cˆong bˆo ´ [1, 2, 3] ta gia ’ thiˆe ´ t c´ac thˆong sˆo ´ a i (n), d ii (n), b i (n), c ii (n) biˆe ´ n dˆo . ng theo th`o . i gian. O . ’ dˆay, b`ai to´an nhˆa . n da . ng th´ıch nghi khˆong d`u . ng du . o . . c du . . a v`ao c´ac quan s´at dˆa ` u ra z(n) v`a dˆa ` u v`ao x(n) dˆe ’ d´anh gi´a c´ac thˆong sˆo ´ hˆe . thˆo ´ ng dˆo . ng ˆa i (n), ˆ d ii (n), ˆ b i (n), ˆc ii (n). Gia ’ thiˆe ´ t r˘a ` ng ta c´o thˆe ’ du . a hˆe . thˆo ´ ng dˆo . ng phi tuyˆe ´ n (11) vˆe ` da . ng vecto . sau dˆay: z(n, θ) = φ T (n)θ(n) + v(n) = θ T (n)φ(n) + v(n). (14) B˘a ` ng c´ach du . a v`ao vecto . thˆong sˆo ´ θ(n) θ(n) = (a i (n) (i = 0, , N a ); d ii (n) (i = 0, , N d ); b i (n) (i = 0, , N b ); c ii (n) (i = 0, , N c )) T , v`a vecto . quan s´at φ(n) φ(n) = (x(n − i) (i = 0, , N a ); x 2 (n − i) (i = 0, , N d ); y(n − i) (i = 0, , N b ); y 2 (n − i) (i = 0, , N c )) T . Dˆo ` ng ´y v´o . i c´ac t´ac gia ’ Erik Weyer v`a M.C. Campi [5] ta du . a v`ao c´ac tiˆeu chuˆa ’ n d´anh gi´a tˆo ´ i u . u: V ( ˆ θ) = E{ε 2 (n, ˆ θ)}, (15) o . ’ dˆay ε(n, θ) = z(n) − ˆy(n, ˆ θ), (16) ˆy(n, ˆ θ) = φ T (n) ˆ θ(n − 1). (17) Su . ’ du . ng phu . o . ng ph´ap chuˆa ’ n b`ınh phu . o . ng tˆo ´ i thiˆe ’ u du . . a theo ´y tu . o . ’ ng cu ’ a Alimed 2003 d˜a du . o . . c ca ’ i biˆen, ta thu du . o . . c: ˆ θ(n + 1) = ˆ θ(n) + F (n)( ˆ θ(n) − ˆ θ(n − 1)) + µ(n)ε(n, ˆ θ) φ(n) φ(n) 2 2 . (18) T`u . diˆe ` u kiˆe . n tˆo ´ i thiˆe ’ u h´oa (17), ta thu du . o . . c l`o . i gia ’ i d´anh gi´a tˆo ´ i u . u vecto . θ : θ opt = R −1 f, o . ’ dˆay R = E{φ(n)φ T (n)}; f = E{φ(n)z(n)}. (19) Du . . a theo ´y tu . o . ’ ng cu ’ a c´ac t´ac gia ’ Erik Weyer v`a M.C. Campi ta lˆa . p tiˆeu chuˆa ’ n tˆo ´ i thiˆe ’ u h´oa phiˆe ´ m h`am Tixonop: NH ˆ A . N DA . NG M ` U CHU ˆ O ˜ I HAMMERSTEIN B ˆ A . C HAI 247 V N ( ˆ θ) = lim N→∞ 1 2N N  n=1 ε 2 (n, ˆ θ) + α 2  ˆ θ 2 → min ˆ θ , (20) N > N a + N b + N c + N d , 0 < α. Thˆong sˆo ´ b´e Tixonop c´o thˆe ’ cho . n: α opt = min{σ 2 v , 1 n ). (21) T`u . diˆe ` u kiˆe . n tˆo ´ i thiˆe ’ u h´oa tiˆeu chuˆa ’ n d´anh gi´a tˆo ´ i u . u (12) kˆe ´ t ho . . p v´o . i c´ach lu . . a cho . n thˆong sˆo ´ b´e Tixonop theo (21) ta thu du . o . . c d´anh gi´a b`ınh phu . o . ng tˆo ´ i thiˆe ’ u: ˆ θ Nα = R −1 Nα f N , (22) o . ’ dˆay R Nα = lim N→∞ 1 N N  n=1 φ(n)φ T (n) + α opt I, (23) I l`a ma trˆa . n do . n vi . N × N, f N = lim N→∞ 1 N N  n=1 φ(n)z(n). (24) Kˆe ´ t luˆa . n. Viˆe . c t`ım mˆo h`ınh gˆa ` n d´ung xˆa ´ p xı ’ c´ac hˆe . phi tuyˆe ´ n l`a mˆo . t vˆa ´ n dˆe ` rˆa ´ t quan tro . ng du . o . . c nhiˆe ` u nh`a nghiˆen c´u . u quan tˆam. Du . . a theo ´y tu . o . ’ ng cu ’ a c´ac t´ac gia ’ Erik Weyer v`a M.C. Campi c`ung v´o . i viˆe . c su . ’ du . ng phiˆe ´ m h`am chı ’ nh h´oa Tixonop ch´ung ta du . a ra du . o . . c c´ac thuˆa . t to´an tˆo ´ i u . u bˆe ` n v˜u . ng dˆe ’ nhˆa . n da . ng chuˆo ˜ i Hammerstein bˆa . c hai. C´ac kˆe ´ t qua ’ thu du . o . . c s˜e du . o . . c ´ap du . ng trong qu´a tr`ınh cˆong nghˆe . h´oa ho . c, sinh ho . c v`a viˆe ˜ n thˆong. T ` AI LI ˆ E . U THAM KHA ’ O [1] P. Koukoulas, N. Kalouptsidis, Blind identification of second order Hammerstein series, Signal Processing 83 (2003) 213—234. [2] N. Kalouptsidis, P. Koukoulas, Blind identification of Bilinear Systems, IEEE Transac- tions on Signal Processing 51 (2) (2003) 484—499. [3] Jean-Marc Le Caillec, Rene Garello, Time series nonlinearity modeling: A Giannakis formula type approach, Signal Processing 83 (2003) 1759—1788. [4] H. Robbins, S. Monro, Stochastic approximation method, Ann Math, Startist 22 (1951) 400—407. [5] Erik Weyer, M. C. Campi, Non-asymptotic confidence ellipsoids for the least-square esti- mate, Automatica 38 (2002) 1539—1547. Nhˆa . n b`ai ng`ay 1 - 6 - 2005 . (2005), 244—247 NH ˆ A . N DA . NG M ` U CHU ˆ O ˜ I HAMMERSTEIN B ˆ A . C HAI TR ˆ A ` N THI . HO ` ANG OANH 1 , D ˆ O ` NG S ˜ I THI ˆ EN CH ˆ AU 2 1 Tru . `o . ng Da . i. Tixonop method. T´om t˘a ´ t. Du . . a trˆen su . . kˆe ´ t ho . . p gi˜u . a hai phu . o . ng ph´ap chı ’ nh h´oa Tixonop v`a l´y thuyˆe ´ t xˆa ´ p xı ’ ngˆa ˜ u nhiˆen,