Kalman ( ) 1) (Kalman filter) ( ) t y 0,, y t x ˆx 3) 10) t x Y [y 0,, y ] ) x ( > ) ˆx (prediction) ) x ( ) ˆx (filtering) )

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1 -- 5 6 2009 3 R.E. Kalman ( ) H 6-1 6-2 6-3 H Rudolf Emil Kalman IBM IEEE Medal of Honor(1974) (1985) c 2011 1/(23)

1 -- 5 -- 6 6--1 2009 3 Kalman ( ) 1) (Kalman filter) ( ) t y 0,, y t x ˆx 3) 10) t x Y [y 0,, y ] ) x ( > ) ˆx (prediction) ) x ( ) ˆx (filtering) ) x ( < ) ˆx (smoothing) 8) 1 [ ] x +1 F x + G w ( ) (6 1) y H x + v, 0 ( ) (6 2) x 0, ( ){w }, {v } E{w } 0, E{v } 0, E{x 0 } x 0, E{[x 0 x 0 ][x 0 x 0 ] T } Σ x0 Kalman 1) Bucy 2) c 2011 2/(23)

E { w v [ T w, v ]} T E{w w T } 0 0 E{v v T } Σ w 0 0 Σ δ, Σ v > 0 v E{x 0 w T } 0, E{x 0 v T } 0 (6 3) {F }, {G }, {H }, x 0, Σ x0, {Σ w }, {Σ v } E{ }, T δ, Σ v > 0 Σ v E{ x ˆx 2 Y } x ˆx E{x Y } x p(x Y )dx (6 4) (Kalman filtering problem) (Kalman filter: KF) 2 a Y x p(x Y ) p(x y, Y 1 ) p(x, y, Y 1 ) p(y, Y 1 ) p(x, y, Y 1 ) p(x, Y 1 ) p(x, Y 1 ) p(y, Y 1 ) p(y x, Y 1 ) p(x, Y 1 ) p(y, Y 1 ) p(y x ) p(x, Y 1 )/p(y 1 ) p(y, Y 1 )/p(y 1 ) p(y x )p(x Y 1 ) p(y Y 1 ) (6 5) {x } x {v i } 1 i0 {v } p(y x, Y 1 ) p(y x ) b Y x +1 ˆx Y x E{x Y } 4),8),9) ξ p (ξ x Y Y )dξ x p(x Y )dx c 2011 3/(23)

p(x +1 Y ) p(x +1, Y ) p(x+1, x, Y )dx p(y ) p(y ) p(x+1, x, Y ) p(x, Y ) dx p(x, Y ) p(y ) p(x +1 x, Y )p(x Y )dx p(x +1 x )p(x Y )dx (6 6) v F x +G w p(x +1 x, Y ) p(x +1 x ) 3 (6 5) (6 6) (6 4) a x 0, w v p(y x ) exp{ 1 2 (y H x ) T Σ 1 v (y H x )} p(x Y 1 ) exp{ 1 2 (x ˆx 1 ) T ˆΣ 1 1 (x ˆx 1 )} ˆx 1 E{x Y 1 }, ˆΣ 1 E{(x ˆx 1 )(x ˆx 1 ) T Y 1 } ˆx 0 1 x 0, ˆΣ 0 1 Σ x0 (6 5) p(y x ) p(x Y 1 ) (x ˆx 1 ) T ˆΣ 1 1 (x ˆx 1 ) + (y H x ) T Σ 1 v (y H x ) (x ˆx 1 ) T ˆΣ 1 1 (x ˆx 1 ) +[(y ŷ 1 ) H (x ˆx 1 )] T Σ 1 v [(y ŷ 1 ) H (x ˆx 1 )] (x ˆx 1 ) T ( ˆΣ 1 1 + HT Σ 1 v H )(x ˆx 1 ) +(y ŷ 1 )Σ 1 v (y ŷ 1 ) (x ˆx 1 ) T H T Σ 1 v (y ŷ 1 ) (y ŷ 1 ) T Σ 1 v H (x ˆx 1 ) [(x ˆx 1 ) ˆΣ H T Σ 1 v (y ŷ 1 )] T ˆΣ 1 [(x ˆx 1 ) ˆΣ H T Σ 1 v (y ŷ 1 )] +(y ŷ 1 ) T (Σ 1 v Σ 1 v H ˆΣ H T Σ 1 v )(y ŷ 1 ) (x ˆx ) T ˆΣ 1 (x ˆx ) + (y ŷ 1 ) T (Σ v + H ˆΣ 1 H T ) 1 (y ŷ 1 ) ˆx ˆx 1 + ˆΣ H T Σ 1 v (y ŷ 1 ), ŷ 1 H ˆx 1 ˆx 1 + ˆΣ 1 H T (Σ v + H ˆΣ 1 H T ) 1 (y ŷ 1 ) (6 7) ˆΣ ( ˆΣ 1 1 + H Σ 1 v H T ) 1 ˆΣ 1 ˆΣ 1 H T (H ˆΣ 1 H T + Σ v ) 1 H ˆΣ 1 (6 8) c 2011 4/(23)

(6 5) p(y Y 1 ) exp{ 1 2 (y ŷ 1 ) T (Σ v + H ˆΣ 1 H T ) 1 (y ŷ 1 )} Y x p(x Y ) p(y x )p(x Y 1 ) p(y Y 1 ) exp{ 1 2 (x ˆx ) T ˆΣ 1 (x ˆx )} (6 9) ˆx, ˆΣ x ( x ) b x w v E{ Y } ˆx +1 F ˆx (6 10) ˆΣ +1 F ˆΣ F T + G Σ w G T (6 11) (6 10) (6 11) Y x +1 p(x +1 Y ) exp{ 1 2 (x +1 ˆx +1 ) T ˆΣ 1 +1 (x +1 ˆx +1 )} (6 12) 4 ( (6 7), (6 8)) ( (6 10), (6 11)) [ ] ˆx ˆx 1 + K (y H ˆx 1 ) ˆx +1 F ˆx ( ) (6 13) K ˆΣ 1 H T (H ˆΣ 1 H T + Σ v ) 1 ( ) (6 14) ˆΣ ˆΣ 1 K H ˆΣ 1 ˆΣ +1 F ˆΣ F T + G Σ w G T ( ) (6 15) ˆx 0 1 x 0, ˆΣ 0 1 Σ x0 ( ) (6 16) ˆΣ y 0,, y E{ x ˆx 2 } t r { ˆΣ } t r { } c 2011 5/(23)

x 0,{w },{v } x 0 ( w v 0) Σ x0, Σ w0,, Σ w, Σ v0,, Σ v y 0,, y ˆx 0 0,, ˆx ˆΣ 0 0,, ˆΣ x N(x ; ˆx, ˆΣ ) N( ) u x +1 F x + D u + G w ( ) (6 17) y H x + v ( ) (6 18) 8),9) [ ] ˆx ˆx 1 + K (y H ˆx 1 ) ˆx +1 F ˆx + D u ( ) (6 19) K ˆΣ 1 H T (H ˆΣ 1 H T + Σ v ) 1 ( ) (6 20) ˆΣ ˆΣ 1 K H ˆΣ 1 ˆΣ +1 F ˆΣ F T + G Σ w G T ( ) (6 21) ˆx 0 1 x 0, ˆΣ 0 1 Σ x0 ( ) (6 22) 9) c 2011 6/(23)

1 -- 5 -- 6 6--2 2009 3, 1 x +1 f (x ) + g (x )w (6 23) y h (x ) + v, 0 (6 24) f ( ), g ( ), h ( ) x 0, {w }, {v } (6 3) x ˆx, ˆx 1 f (x ), h (x ) (Taylar expansion) f (x ) f ( ˆx ) + F [x ˆx ] + (6 25) h (x ) h ( ˆx 1 ) + H [x ˆx 1 ] + (6 26) F f (ξ ) ξ T, H h (ξ ) ξ ˆx ξ T ξ ˆx 1 (6 27) ξ 9) G g ( ˆx ) (6 23) (6 24) x +1 f ( ˆx ) + F [x ˆx ] + G w (6 28) y h ( ˆx 1 ) + H [x ˆx 1 ] + v (6 29) x +1 F x + u + G w (6 30) m H x + v (6 31) u f ( ˆx ) F ˆx m y h ( ˆx 1 ) + H ˆx 1 (6 32) f( ) a, b f(ax + by) a f(x) + b f(y) f( ) c 2011 7/(23)

(D I, u f ( ˆx ) F ˆx ) (6 19) (6 22) (6 23) (6 24) 4),8),9) [ ] ˆx ˆx 1 + K (y h ( ˆx 1 )) ˆx +1 f ( ˆx ) ( ) (6 33) K ˆΣ 1 H T (H ˆΣ 1 H T + Σ v ) 1 ( ) (6 34) ˆΣ ˆΣ 1 K H ˆΣ 1 ˆΣ +1 F ˆΣ F T + G Σ w G T ( ) (6 35) ˆx 0 1 x 0, ˆΣ 0 1 Σ x0 ( ) (6 36) (extended Kalman filter: EKF) (6 23) (6 24) F, G, H ˆx, ˆx 1 ˆΣ K ˆx 0 1 2 (unsented Kalman filter: UKF) ( (unscented transformation : UT)) 5),6),10) x +1 f (x, w ) (6 37) y h (x, v ) (6 38) x R N x y R Ny w R Nw v R Nv (6 3) 10) f : R Nx R Nw R Nx h : R Nx R Nv R Ny 6),10) x +1 f (x ) + g(x )w, y h (x ) + v c 2011 8/(23)

x a [xt wt vt ]T (6 39) 2N a + 1 N a (2N a +1) X a [(Xx ) T (X w ) T (X v ) T ] T N a N x + N w + N v N x, N w, N v [ ] 1. : x a 0 E{x a 0 } [ xt 0 0T 0 T ] T Σ a 0 E{(x a 0 xa 0 )(xa 0 xa 0 )T } Σ x0 0 0 0 Σ w 0 0 0 Σ v w, v, (6 3). 2. For 1, 2,... (a) : X a 1 [ x a 1, ( ) { x a 1 (N a + λ) Σ a 1 i ( ) { x a 1 (N + a + λ) Σ a 1 i } Na i1, } Na i1 ] > 1 x a 1 [ ˆx T 1 1 0T 0 T ] T, Σ a 1 diag{ ˆΣ 1 1, Σ w, Σ v } ( ) (N a + λ) Σ a 1 N a N a (N a + λ) Σ a 1 i i diag{ } λ (b) ( ): c 2011 9/(23)

X x i, 1 f 1 (X x i, 1, Xw i, 1 ), i 0, 1, 2,, 2N a x 1 Σ 1 2N a W (m) i X x i, 1 i0 2N a W (c) i [X x i, 1 x 1][X x i, 1 x 1] T i0 Y i, 1 h (X x i, 1, Xv i, 1 ), i 0, 1, 2,, 2N a ȳ 1 2N a W (m) i Y i, 1 i0 X x i, 1, Y i, 1 X x 1, Y 1 i + 1, W (m) i, W (c) i W (m) 0 λ/(n a + λ), W (c) 0 λ/(n a + λ) + (1 α 2 + β) W (m) i W (c) i 1/{2(N a + λ)}, i 1,..., 2N a, α, β (c) ( ) : Σ y Σ x y 2N a W (c) i [Y i, 1 ȳ 1 ][Y i, 1 ȳ 1 ] T i0 2N a W (c) i [X x i, 1 x 1][Y i, 1 ȳ 1 ] T i0 K Σ x y Σ 1 y ˆx x 1 + K (y ȳ 1 ) ˆΣ Σ 1 K Σ y K T MATLAB chol( ) ˆΣ 1 H T (Σ v + H ˆΣ 1 H T ) 1 E{(x ˆx 1 )(y ŷ 1 ) T Y 1 }E{(y ŷ 1 )(y ŷ 1 ) T Y 1 } 1, ˆΣ 1 H T (H ˆΣ 1 H T + Σ v ) 1 H ˆΣ 1 K (H ˆΣ 1 H T +Σ v )KT K E{(y ŷ 1 )(y ŷ 1 ) T Y 1 }K T c 2011 10/(23)

3 E{x Y } x p(x Y )dx p(x Y ) (particle filter: PF) 5),6) x +1 f (x, w ) (6 40) y h (x, v ) (6 41) x y w v (6 3) f : R Nx R Nw R Nx h : R Nx R Nv R Ny Y [y 1,, y ] x p(x Y ) q x (i) ( ) E{x Y } X [x 0,, x ] E{X Y } E{X Y } X p(x Y )dx p(x Y ) X q(x Y ) q(x Y )dx p(y X )p(x ) X p(y )q(x Y ) q(x Y )dx W X q(x Y )dx (6 42) p(y ) W p(y X )p(x ) q(x Y ) (6 43) (importance weight) q(x Y ) (proposal distribution) p(y ) p(y X )p(x ) q(x Y )dx q(x Y ) E q {W Y } 1 N s N s i1 W (i), W(i) p(y X (i) )p(x(i) ) q(x (i) Y ), X (i) q(x Y ) (6 44) (6 42) (w 0) (Monte Carlo: MC) MC g(x) f(x)π(x) π(x) x (i) π(x) g(x)dx 1 Ns Ns i1 f(x(i) ) c 2011 11/(23)

E{X Y } E q {W X Y } E q {W Y } 1 N s Ns X(i) i1 W(i) 1 Ns j) N s j1 W( (6 45) N s p(x Y ) W (i) δ(x X (i) ) i1 (6 46) N s δ( ) W (i) W (i) W (i) W (i) Ns j) j1 W( p(y X (i) )p(x(i) ) q(x (i) Y ) p(y )p(x (i) Y ) q(x (i) Y ) (6 47) (6 48) X (i) q(x Y ) q(x Y ) q(x X 1, Y )q(x 1 Y 1 ) (6 49) X (i) 1 q(x 1 Y 1 ) x (i) q(x X (i) 1, Y ) X (i) [X (i) 1, x(i) ] q(x Y ) (6 40) (6 41) p(x Y ) p(x 1 Y 1 ), p(y x ) p(x x 1 ) p(x Y ) p(y X, Y 1 )p(x Y 1 ) p(y Y 1 ) p(y X, Y 1 )p(x X 1, Y 1 ) p(x 1 Y 1 ) p(y Y 1 ) p(y x )p(x x 1 ) p(x 1 Y 1 ) p(y Y 1 ) (6 50) (6 49) (6 48) x (i) q(x X (i) 1, Y ) x (i) q(x X (i) 1, Y ) c 2011 12/(23)

W (i) p(y ) p(y x (i) )p(x(i) x(i) 1 )p(x(i) 1 Y 1) p(y Y 1 ) q(x (i) X(i) 1, Y )q(x (i) 1 Y 1) W (i) p(y x (i) )p(x(i) x(i) 1 ) 1 q(x (i) X(i) 1, Y ) (6 51) q(x X (i) 1, Y ) q(x x (i) 1, y ) W (i) W (i) p(y x (i) )p(x(i) x(i) 1 q(x (i) x(i) 1, y ) 1 ), x (i) q(x x (i) 1, y ) (6 52) ) q(x x (i) 1, y ) p(x x (i) 1 ) ( ) ) q(x x (i) 1, y ) N(x ; x, Σ ) (EKF) (UKF) N(x ; x, Σ ) x Σ (6 46) x p(x Y ) N s p(x Y ) W (i) δ(x x (i) ), i1 x(i) q(x x (i) 1, y ) (6 53) x ˆx (6 52) ( g(x)δ(x x (i) )dx g(x (i) )) ˆx E{x Y } N s x p(x Y )dx i1 W (i) x(i) (6 54) P{x (i) x ( j) } W ( j) {x (i) } x ( j) N s W ( j) {x (i) } W(i) 1 1/N s (6 52) 6 1 4 EKF( UKF) 5),6) Bootstrap filter Condensation algorithm P{x (i) x ( j) j) } W( x (i) x (i) x ( j) W ( j) Sampling-Importance Resampling (SIR) c 2011 13/(23)

Draw x (i) -1 from ( j) q ( x -1-2 x, y -1 ) (i) -1 {x -1, N s } ( p(x -1 Y -2 ) ) lielihood (i) p( y -1 x -1 ) Importance weightsw (i) -1 Multiply/Suppress (i) (i) {x -1, w _ -1} ( p(x -1 Y -1 ) ) ^ x -1-1 Draw x (i) from ( j) q ( x x -1, y ) {x (i) -1 s, N } ( p(x Y -1 ) ) lielihood (i) p( x ) y Importance weights w (i) (i) w (i) {x, _ } ( p(x Y ) ) x ^ 6 1. [ ( ) ] 1. ( 0) For i 1,..., N s - p(x 0 ) x (i) 0, EKF ( UKF). 2. For 1, 2,... 2.1) For i 1,..., N s - x (i) 1 x(i) 1, Σ (i) 1 Σ(i) 1, F(i), G(i), H(i), EKF ( UKF) x (i), Σ (i), N(x ; x (i), Σ (i) ). - x (i) ( x (i) N(x ; x (i), Σ (i) ) ) - (W (i) W (i) W (i) W (i) Ns j1 W ( j), W (i) /N s). p(y x (i) )p(x(i) x(i) 1 ) N(x (i) ; x(i), Σ (i) ) 2.2) - x (i) W (i) ˆx. ˆx N s W (i) x(i) i1 2.3) - N s (x (i), Σ(i) ). c 2011 14/(23)

EKF UKF 1) R.E. Kalman : A New Approach to Linear Filtering and Prediction Problem, Trans. ASME, Journal of Basic Engineering, 82D, pp.34-45 (1960). 2) R.E. Kalman and R.S. Bucy : New Results in Linear Filtering and Prediction Theory, Trans. ASME, Journal of Basic Engineering, 83D, pp.95-108 (1961). 3) A. H. Jazwinsi : Stochasic Process and Filtering Theory, Academic press (1970). 4) B. D. O. Anderson and J. B. Moore : Optimal Filtering, Prentice-Hall (1979). 5) R. van der Merwe, N. de Freitas, A. Doucet, and E. Wan : The Unscented Particle Filter, Technical Report CUED/F-INFENG/TR380, Cambridge, England (2000). 6) B. Ristic, S. Arulampalam, and N. Gorden : Beyond the Kalman Filter, Artech House Publishers (2004). 7) :, (1977). 8) :, (1983). 9) :, (2001). 10) :, (2005). c 2011 15/(23)

1 -- 5 -- 6 6--3 H 2009 9 1980 H H H H 1, 2, filter 3) H 6--3--1 H 6 2 x +1 F x + G w y H x + v z L x (6 55) (6 56) (6 57) ν w x +1 F x G w H y H z^ x L z e 6 2 H x R n, y R p z R q w R m, v R p Y l : {y 0, y 1,, y l } x ˆx l, z z l H H error criterion H N 0 z ẑ 2 l J : sup N ( w,v,x 0 0 w 2 + v 2) + (x 0 x 0 ) T Π 1 (x 0 x 0 ) < γ2 (6 58) H J : sup w,v 0 z ẑ l 2 0 ( w 2 + v 2) < γ2 (6 59) c 2011 16/(23)

γ Π x 0 w v w 2 < + v 2 < + F, G, H, L ξ : (w, v ) e : z ẑ l T eξ (z) H (6 59) T eξ (z) H T eξ : sup λ >1 σ ( T eξ (λ) ) sup ω [0,2π) σ ( T eξ (e jω ) ) σ: γ J T eξ 2 H H H ξ e γ H 2 T eξ (z) H 2 T eξ 2 ( 2π Tr[T 0 eξ (e jω ) T eξ (e jω )]dω) 1/2 T eξ (e jω ) T eξ σ ( Teξ ( e jω ) ) ω 6 3 T eξ (z) σ- 6--3--2 H l 1 l 1 H H (6 58) H sup V(w, v, x 0 ; ẑ) < 0 w, v, x 0 N N V(w, v, x 0, ẑ) z ẑ l 2 γ 2 ( w 2 + v 2 ) + (x 0 x 0 ) T Π 1 (x 0 x 0 ) 0 H 0 c 2011 17/(23)

V : min max V(w, v, x 0 ; ẑ) ẑ w,v,x 0 (6 60) minimax estimation problem (w, v, x 0 ) ẑ V (6 56) v y H x v y (6 60) V : min max V(w, y, x 0 ; ẑ) ẑ w,y,x 0 N V(w, y, x 0 ; ẑ) L x ẑ l 2 0 N γ 2 ( w 2 + H x y 2 ) + (x 0 x 0 ) T Π 1 (x 0 x 0 ) 0 (6 61) 1 w x 0 w 0,..., w N, x 0 ẑ Y {y 0,..., y } ẑ y 1 ẑ 1 Y 1 {y 0,..., y 1 } ẑ 1 y 1 y (6 61) 1 V max y 0 min ẑ 0 0 (max y 1 min ẑ 1 1 ( max y N 1 min (max ẑ N 1 N 1 y N min ( max V(w, y, x 0 ; ẑ) )) )) ẑ N N w 0,...,w N,x 0 1 V min ẑ 0 1 max y 0 (min ẑ 1 0 max ( min max y 1 ẑ N 1 N 2 y N 1 (min ẑ N N 1 max y N ( max w 0,...,w N,x 0 V(w, y, x 0 ; ẑ) )) )) H 1 2, 4) 5) c 2011 18/(23)

1 H (6 55) (6 57) F, 0,..., N i H H Riccati difference equation P +1 F P [I + (H T H γ 2 L T L )P ] 1 F T + G G T, P 0 Π (6 62) P P > 0 & γ 2 I L P (I + H T H P ) 1 L T > 0, 0,..., N (6 63) V 0 H (6 58) ˆx +1 F ˆx, ˆx 0 1 x 0 (6 64) ˆx ˆx 1 + K (y H ˆx 1 ) (6 65) ẑ L ˆx K P H T (I + H P H T ) 1 (6 66) (6 67) ii 1 H (6 62) P > 0 & γ 2 I L P L T > 0, 0,..., N (6 68) 1 ˆx +1 F ˆx, ˆx 0 1 x 0 (6 69) ˆx ˆx 1 + K (y H ˆx 1 ) (6 70) ẑ 1 L ˆx 1 K P H T (I + H P H T ) 1 P P (I γ 2 L T L P ) 1 (6 71) (6 72) (6 73) V 0 1 H (6 58) (6 55)-(6 57) H H P 0 0 N + (6 62) P c 2011 19/(23)

2 H (6 55)-(6 57) F, G, H, L F, G, H, L (F, G) (H, F) i H (6 59) H H algebraic Riccati equation P FP[I + (H T H γ 2 L T L)P] 1 F T + GG T (6 74) P P 0 & γ 2 I LP(I + H T HP) 1 L T > 0 (6 75) (6 59) H ˆx +1 F ˆx (6 76) ˆx ˆx 1 + K(y H ˆx 1 ) (6 77) ẑ L ˆx K PH T (I + HPH T ) 1 (6 78) (6 79) ii (6 59) H 1 H (6 74) P P 0 & γ 2 I LPL T > 0 (6 80) 1 H ˆx +1 F ˆx (6 81) ˆx ˆx 1 + K(y H ˆx 1 ) (6 82) ẑ 1 L ˆx 1 K PH T (I + H PH T ) 1 P P(I γ 2 L T LP) 1 (6 83) (6 84) (6 85) 1 H H 1 2 H H H H 3, 6) 7) 2 γ γ (6 62),(6 74) ) γ H c 2011 20/(23)

3 (6 63),(6 68) P, P γ H H γ γ (6 58) (6 59) H γ H H 6--3--3 (w, v) H 8, 9) 3 (6 55) (6 57) x 0 x 0 Π w, v E{w } 0, E{v } 0, E w [ ] v w T l v T l I 0 0 I δ l,, l 0, 1, 2,... E δ l exponential-quadratic error criterion N J EQ E exp γ 2 z ẑ l 2 0 (6 60) 1 H J EQ w, v, x 0 f N f (w, v, x 0 ) c exp ( w 2 + v 2 ) (x 0 x 0 ) T Π 1 (x 0 x 0 ) 0 c J EQ N J EQ exp γ 2 z ẑ l 2 f (w, v, x 0) dx 0 dw dv R p(n+1) R m(n+1) R n 0 exp ( γ 2 V(w, v, x 0 ; ẑ) ) dx 0 dw dv R p(n+1) R m(n+1) R n J EQ. min J EQ ẑ [ c min R p ẑ 0 0 min R p ẑ 1 1 [ [ [ ] ] ] ] min exp(γ 2 V)dx 0 dw dv N dv 1 dv 0 R p ẑ N N R m(n+1) R n c 2011 21/(23)

2 Z(x)/ x x T < 0 Z : R n R { } exp(z(x))dx const exp max Z(x) (const: ) R n x 10) minẑ J EQ { min J EQ const exp max ẑ v 0 min ( ( max ẑ 0 0 v N } min ( max(γ 2 V)) ) ẑ N N w, x 0 (6 86) (6 60) H (6 64) (6 66) J EQ 5) 6--3--4 LMS y ϕ T θ + v, 0, 1, 2,..., N (6 87) θ R n v (y, ϕ ), 0, 1,..., N θ N 1 lim N N N ϕ ϕ T > 0 0 (6 88) θ +1 θ, θ 0 θ (6 89) H sup x 0,v N 0 y ŷ 2 N 0 v 2 + µ 1 θ 0 θ 0 2 < γ2, µ > 0 (6 90) ˆθ 6-5-1 H x θ, z y, F I, G 0, H L ϕ T, x 0 θ 0, Π µi 1 H 1(ii) ˆθ +1 ˆθ + K (y ϕ T ˆθ ), ˆθ 0 θ 0 (6 91) K P ϕ /(1 + ϕ T P ϕ ) P +1 P [I + (1 γ 2 )ϕ ϕ T P ] 1, P 0 µi (6 92) (6 93) c 2011 22/(23)

Q : P 1 (6 93) Q +1 Q + (1 γ 2 )ϕ ϕ T, Q 0 µ 1 I (6 94) (6 68) Q γ 2 ϕ ϕ T > 0, 0, 1,..., N (6 95) (6 94) Q µ 1 I + (1 γ 2 ) 1 i0 ϕ iϕ T i γ < 1 1 γ 2 < 0 (6 88) N 1 Q N γ 1 γ 1 Q µ 1 I ( ) 1 0 < µ < sup ϕ 2 (6 96) Q γ 2 ϕ ϕ T µ 1 I ϕ ϕ T > 0 (6 95) (6 96) µ H γ γ 1 P µi K µϕ ˆθ +1 ˆθ + µϕ (y ϕ T ˆθ ), ˆθ 0 θ 0 (6 97) µ LMS LMS (6 96) H LMS 1) K.M. Nagpal and P.P. Khargonear, Filtering and smoothing in an H setting, IEEE Trans. Automat. Contr., vol.36, no.2, pp.152-166, 1991. 2),, vol.7, no.8, pp.322-331, 1994. 3) B. Hassibi, A.H. Sayed and T. Kailath, Linear estimation in Krein spaces Part I: Theory & Part II: Applications, IEEE Trans. Automat. Contr., vol.41, no.1, pp.18-33, 34-49, 1996. 4),, 9,, 2000. 5) B. Hassibi, A.H. Sayed and T. Kailath, H optimality of the LMS algorithm, IEEE Trans. Signal Processing, vol.44, no.2, pp.267 280, 1996. 6) B. Hassibi, A.H. Sayed and T. Kailath, Indefinite Quadratic Estimation and Control A Unified Approach to H 2 and H Theories, SIAM, 1999. 7) K. Taaba and T. Katayama, Parametrization of discrete-time H filters based on model-matching,, vol.33, no.7, pp.648-655, 1997. 8) J.L. Speyer, C. Fan and N. Banavar, Optimal stochastic estimation with exponential cost criteria, Proc. 31st IEEE Conf. Decision Control, pp.2293-2298, 1992. 9) K. Taaba, Stochastic properties of the H filter, Statistical Methods in Control and Signal Processing (T. Katayama and S. Sugimoto (eds.)), Marcel-Deer, pp.239-256 1997. 10) P. Whittle, Ris-Sensitive Optimal Control, Wiley, 1990. c 2011 23/(23)

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