i 1 1 1.1... 1 1.1.1... 2 1.1.2... 7 1.2... 9 1.3... 1 1.4... 12 1.4.1 s... 12 1.4.2... 12 1.5... 15 1.5.1... 15 1.5.2... 18 2 21 2.1... 21 2.2... 23

2 III Copyright c 2 Kazunobu Yoshida. All rights reserved.

i 1 1 1.1... 1 1.1.1... 2 1.1.2... 7 1.2... 9 1.3... 1 1.4... 12 1.4.1 s... 12 1.4.2... 12 1.5... 15 1.5.1... 15 1.5.2... 18 2 21 2.1... 21 2.2... 23 2.2.1... 23 2.2.2... 25 2.2.3... 28 2.3... 32 2.4 -... 34 2.5... 35 2.5.1... 35 2.5.2... 37 2.5.3... 39 3 41 3.1... 41 3.1.1... 41 3.1.2... 43 3.2... 46

ii 3.2.1... 47 3.2.2... 48 3.2.3... 51

1 1 1.1 ẋ(t) =Ax(t)+bu(t) (1.1) y(t) =cx(t)+du(t) (1.2) (1.1) (1.2) x(t)(n 1) : u(t)(1 1) : y(t)(1 1) : A(n n), b(n 1), c(1 n), d(1 1) u y d = m l b, c B(n m), C(l n)

2 1 x C P u 1.1: 1.1.1 1.1 C u x Cẋ = u (1.3) ẋ = 1 C u (1.4) x y = x (1.5) C θ θ 1 R 1.2: 1.2 θ 1 θ Newton θ 1 C θ 1 = 1 R (θ θ 1 ) (1.6)

1.1. 3 C R x = θ 1,u= θ,y= θ 1 (1.7) ẋ = 1 CR x + 1 CR u (1.8) y = x (1.9) RC R C v i i vo 1.3: RC Ri + v o = v i (1.1) v o = 1 idt (1.11) C (1.11) v o = 1 C i (1.12) i = C v o (1.13) (1.1) v o = 1 RC (v i v o ) (1.14) x = v o, u = v i (1.15) ẋ = 1 RC x + 1 RC u (1.16) y = x (1.17)

4 1 x u m 1.4: 1.4 m u mẍ = u (1.18) x 1 = x, x 2 =ẋ, y = x (1.19) 1 ẋ = x + 1 u (1.2) m y = 1 x (1.21) x = x1 x 2 (1.22) x k m u 1.5: 1

1.1. 5 1.5 mẍ + kx = u (1.23) ẍ = k m x + 1 m u (1.24) x k x 1 = x, x 2 =ẋ, y = x (1.25) ẋ = 1 k m x + 1 u (1.26) m y = 1 x (1.27) RLC R L C v i i vo 1.6: RLC L i + Ri + v o = v i (1.28) v o = 1 idt (1.29) C (1.29) v o = 1 C i (1.3)

6 1 i = C v o (1.31) (1.28) LC v o + RC v o + v o = v i (1.32) x 1 = v o, x 2 = v o, u = v i (1.33) ẋ 1 = x 2 ẋ 2 = 1 LC x 1 R C x 2 + 1 LC u (1.34) 1 ẋ = 1 LC R x + 1 u (1.35) L LC y 1 y 2 A B k 1 k 2 m 1 m 2 u 1.7: 1.7 A y 1 A m 1 ÿ 1 k 1 y 1 k 2 (y 1 y 2 ) = (1.36) A ÿ 1 = k 1 + k 2 m 1 y 1 + k 2 m 1 y 2 (1.37)

1.1. 7 B m 2 ÿ 2 k 2 (y 2 y 1 )+u = (1.38) B ÿ 2 = k 2 m 2 y 1 k 2 m 2 y 2 + 1 m 2 u (1.39) x 1 = y 1, x 2 = y 2,x 3 =ẏ 1,x 4 =ẏ 2 (1.4) 1 1 ẋ = k 1 + k 2 k 2 x + m 1 m 1 k 2 k 2 m 2 m 2 y = 1 1 1 m 2 u (1.41) x (1.42) 1.1.2 (1.1) (1.2) h q a C 1.8:

8 1 1.8 q a a C v = 2gh (1.43) g h Cḣ = q fav (1.44) f 1 q q = q h q = fa 2gh (1.45) h q h = h + x, q = q + u (1.46) h,q x, u (1.46) (1.44) x/h h + x = h (1 + x h ) 1/2 h (1 + x 2h ) (1.47) ẋ = 1 CR x + 1 C u (1.48) 1 R = fa g 2h (1.49) x y = x (1.5) 1.9 θ J θ mgl sin θ + u = (1.51)

1.2. 9 u θ l 1.9: mg J u J = ml 2 θ = g l sin θ + 1 ml 2 u (1.52) θ sin θ θ θ = g l θ + 1 ml 2 u (1.53) x 1 = θ, x 2 = θ, y = θ (1.54) ẋ = 1 g x + 1 u (1.55) l ml 2 y = 1 x (1.56) m 1.2 ẋ(t) = Ax(t)+bu(t) y(t) = cx(t) (1.57) sx(s) x() = AX(s) +bu(s) Y (s) = cx(s) (1.57) (1.58)

1 1 x() = G(s) = Y (s) U(s) = c(si A) 1 b (1.59) G(s) U(s) Y (s) (si A) 1 = adj (si A) si A (1.6) G(s) G(s) = cadj (si A)b si A (1.61) G(s) = si A = (1.62) G(s) = cadj (si A)b = (1.63) p12 8.1 R 1 C 1,θ 1 θ R2 C 2,θ 2 1.1: 1.3 1. 1.1 1 2 C 1 θ 1 C 2 θ 2 R 1 R 2 θ x 1 = θ 1,x 2 = θ 2

1.3. 11 h 1 q 1 C 1 a 1 h 2 C 2 a 2 1.11: m mg l θ u 1.12: 2. 1.11 q 3. 1.12 x 1 = θ x 2 = θ 4. 1.7 k 1 k 2 c 1 c 2 5. p134 8.1

12 1 1.4 1.4.1 s ẋ(t) = Ax(t)+bu(t) y(t) = cx(t) sx(s) x() = AX(s) +bu(s) Y (s) = cx(s) (1.64) (1.65) (si A)X(s) =x() + bu(s) s X(s) =(si A) 1 x() + (si A) 1 bu(s) (1.66) Y (s) =cx(s) =c(si A) 1 x() + c(si A) 1 bu(s) (1.67) 1.4.2 ẋ(t) =Ax(t) (1.68) e At := I + At + A 2 t2 t3 + A3 + (1.69) 2! 3! e At (1.68) x(t) =e At x() (1.7) t ẋ(t) =Ae At x() = Ax(t) (1.71) (1.68) e At

1.4. 13 1. de At dt 2. e = I = Ae At 3. e At e Aτ = e A(t+τ) 4. (e At ) 1 = e At y(t) ẋ(t) =Ax(t)+bu(t) (1.72) x(t) =e At x() + t y(t) =cx(t) =ce At x() + (1.73) (1.72) e A(t τ) bu(τ )dτ (1.73) t ce A(t τ) bu(τ )dτ (1.74) x(t) =e At v(t) (1.75) ẋ(t) = Ae At v(t)+e At v(t) = Ax(t)+bu(t) e At v(t) =bu(t) v(t) =e At bu(t) dv(t) =e At bu(t)dt t dv(t) = t v(t) v() = e Aτ bu(τ )dτ t e Aτ bu(τ )dτ

14 1 v(t) =v() + t x(t) =e At {v() + e Aτ bu(τ )dτ t } e Aτ bu(τ )dτ t = v() = x() t } x(t) = e {x() At + e Aτ bu(τ )dτ s = e At x() + t X(s) =(si A) 1 x() + (si A) 1 bu(s) e A(t τ) bu(τ)dτ (1.76) e At = L 1 (si A) 1 (1.77) e At RC ẋ = 1 RC x + 1 RC u (1.78) RC =1, x() = 1, u(t) =1,t i) s A = 1, b =1 (si A) 1 =(s +1) 1 = 1 s +1 s X(s) = 1 s +1 + 1 s +1 1 s = 1 s +1 + 1 s 1 s +1 = 1 s 2 s +1 (1.79)

1.5. 15 x(t) =1 2e t (1.8) ii) e At = L 1 (si A) 1 =L 1 1 = e 1 s +1 x(t) = e t ( 1) + t e (t τ) dτ t = e t + e t e τ dτ = e t + e t (e t 1) = 1 2e t (1.81) p124 8.2 p134 8.2, 8.3 1.5 1.5.1 ẋ(t) =ax(t) (1.82) x(t) =e at x() (1.83) a (1.82) a< (1.84)

16 1 x(t) x(t) x(t) x() x() x() t t t a<( ) a =( ()) a>() 1.13: n ẋ(t) =Ax(t) (1.85) x(t) =e At x() (1.86) A λ 1 λ n A si A = s n + α 1 s n 1 + + α n 1 s + α n = (1.87) A T (n n) T =v 1, v 2,, v n (1.88) T 1 AT = A = T ΛT 1 e At λ 1 λ 2... λ n = Λ (1.89) e At = e T ΛT 1t = Te Λt T 1 (1.9)

1.5. 17 e Λt = e λ 1t e λ 2t... e λnt (1.91) (1.83) x(t) e λ1t e λnt x 1 (t) =f 11 e λ 1t + f 12 e λ 2t + + f 1n e λnt f 11 f 1n x() x() lim x(t) = (1.92) t lim t eλ it =, i =1 n (1.93) λ i = a i + jb i, j = 1 (1.93) Re λ i = a i <, i =1 n (1.94) e (a i+jb i )t = e a it e jb it = e a it (cos b i t + j sin b i t) A J 1 = e J1t = e λ 1t Jordan λ 1 1 λ 1 1 λ 1 1 t t 2 /2! 1 t 1

18 1 (1.92) t k e λ it lim t k! Re λ i < = (1.95) A λ i Re λ i 1.5.2 1868 Maxwell 1875 Routh Strum (1829) Routh 1895 Stodola Hurwitz Hermite (1856) 1914 Lienard Chipart Hurwitz n α s n + α 1 s n 1 + + α n 1 s + α n =, α > (1.96) s i,i=1 n Re s i <, i =1 n (1.97) Routh

1.5. 19 1.1: Routh α α 2 α 4 α 6 1 α 1 α 3 α 5 α 7 2 b 1 = α 1α 2 α α 3 b 2 = α 1α 4 α α 5 b 3 = α 1α 6 α α 7 α 1 α 1 α 1 3 c 1 = b 1α 3 α 1 b 2 c 2 = b 1α 5 α 1 b 3 b 1 b 1. n ω 1 1.1 α > α n > (1.98) α >, α 1 >,, ω 1 > (1.99) Re (s i ) <, i =1 n (1.1) α 1 α 3 α 5 α 7 α α 2 α 4 α 6 α 1 α 3 α 5 H(n n) = α α 2 α 4 α 1 α 3........ α n (1.11)

2 1 H i H i α 1 α 3 H 1 = α 1, H 2 =, (1.12) α α 2 1.2 α > α n > (1.13) H 1 > H n > (1.14) Re (s i ) <, i =1 n (1.15) 1.3 α > α n > (1.16) H i > i n (1.17) H i > i n (1.18) Re (s i ) <, i =1 n (1.19)

21 2 2.1 x 2 x e 2 e 1 x 1 2.1: 1 x t 2 t 1 2.2: 2

22 2 e 1 = 1, e 2 = x = e 1 x 1 + e 2 x 2 = x = t 1 z 1 + t 2 z 2 = 1 e 1 e 2 x 1 x 2 t 1 t 2 z 1 z 2 x t 1, t 2 (2.1) = Ix (2.2) = Tz (2.3) T T (2.4) t 1, t 2 x = Tz T (2.5) n x, z : n T : n n ẋ(t) = Ax(t)+bu(t) y(t) = cx(t) x(t) =Tz(t) (2.6) ẋ(t) =T ż(t) (2.6) T ż(t) = AT z(t)+bu(t) y(t) = ct z(t) ż(t) = Ãz(t)+ bu(t) y(t) = cz(t) (2.7) (2.8) Ã= T 1 AT, b = T 1 b, c = ct (2.9)

2.2. 23 1. si à = T 1 (si A)T = T 1 T si A = si A (2.1) 2. c(si Ã) 1 b = ct T 1 (si A)T 1 T 1 b = ct T 1 (si A) 1 TT 1 b = c(si A) 1 b (2.11) 2.2 R.E.Kalman(1959) u(t) x u x 1 x 1 u x 2 x 2 2.3: 2.2.1 x() x f t f > u(t), t t f x(t f )=x f b Ab A n 1 b (2.12)

24 2 y x 1 x 1 y x 2 x 2 2.4: rank b Ab A n 1 b = n (2.13) u x 1 C 1 x 2 C 2 2.5: u, x 1,x 2 C 1 ẋ 1 = 1 x 1 + u R 1 C 2 ẋ 2 = 1 R 1 x 1 1 R 2 x 2 (2.14) ẋ = a11 a 21 a 22 b1 x + u (2.15)

2.2. 25 U c = b Ab b1 a 11 b 1 = a 21 b 1 (2.16) rank U c =2 x 1 C 1 u x 2 C 2 2.6: C 1 ẋ 1 = 1 R 1 x 1 C 2 ẋ 2 = 1 R 1 x 1 1 R 2 x 2 + u (2.17) a11 ẋ = x + u a 21 a 22 b 2 (2.18) U c = b Ab = b 2 a 22 b 2 (2.19) rank U c =1 2.2.2 ẋ = Ax + bu (2.2) (2.2) rank b Ab A n 1 b = n x() = x x(t f )= u(t)

26 2 t = x t = t f 2.7: (2.2) t x(t) =e At x + e A(t τ) bu(τ)dτ tf = e At f x + e A(t f τ) bu(τ)dτ e At f x = tf e Aτ bu(τ )dτ (p159) e Aτ = Iφ (τ)+aφ 1 (τ)+ + A n 1 φ n 1 (τ) x = = tf {bφ (τ)+abφ 1 (τ)+ + A n 1 bφ n 1 (τ)}u(τ )dτ b Ab... A n 1 b tf φ (τ) φ 1 (τ). φ n 1 (τ) u(τ)dτ x rank b Ab... A n 1 b = n

2.2. 27 W t = t e Aτ bb T e AT τ dτ y T W t y = y = y T W t y = = t t τ = y T b = τ τ = y T Ab = y T A 2 b =. y T A n 1 b = y T e Aτ bb T e AT τ ydτ (y T e Aτ b) 2 dτ y T e Aτ b y T b Ab... A n 1 b = u(t) = b T e AT t W 1 t f x W 1 t f x(t f ) = e At f x + e At f tf = e At f x + e At f ( x )= e Aτ b( b T e AT τ )dτ W 1 t f x

28 2 2.2.3 t f t t f y(t) u(t) x() ẋ(t) =Ax(t) y(t) =cx(t) rank c ca ca 2. ca n 1 = n y(t) =cx 1 (t) =ce At x 1 () y(t) =cx 2 (t) =ce At x 2 () =ce At (x 1 () x 2 ()) z t = cz = t t = caz = ca 2 z =. ca n 1 z =

2.2. 29 c ca. ca n 1 rank c ca z =, z. ca n 1 <n y x 1 x 1 y x 2 x 2 2.8: a11 A =, c = 1 a 21 a 22 x 1 y = cx = 1 = x 2 x 2 c 1 U o = = ca a 21 a 22

3 2 rank U =2 a11 A =, c = a 21 a 22 1 U o = a 11 rank U o =1 1 x u m 2.9: mẍ = u y = x x 1 = x, x 2 =ẋ 1 ẋ = x + u 1/m y = 1 x U o = 1 1 rank U o =2 y =ẋ = 1 U o = 1 rank U o =1

2.2. 31 x k m u 2.1: mẍ + kx = u x 1 = x, x 2 =ẋ 1 ẋ = x + k/m 1/m y = x = 1 U o = 1 1 x rank U o =2 y =ẋ = 1 U o = 1 k/m rank U o =2 m =1,c=2,k=1, y = x +ẋ 1 ẋ = x + 1 2 1 u u

32 2 k x c m u 2.11: y = 1 1 x 1 1 U o = 1 1 rank U o =1 2.3 A λ i,i=1 n v i,i=1 n Av i = λ i v i x(t) =Tx(t), T = v 1 v 2 v n (2.21) ż(t) =Ãz(t)+ bu(t) (2.22) y(t) = cx(t) (2.23) β 1 Ã = T 1 AT =, b = T 1 β 2 b =. λ 1 λ 2... λ n c = ct = θ 1 θ 2 θ n β n

2.3. 33 ż i (t) =λ i z i (t)+β i u(t), i =1 n (2.24) y(t) =θ 1 z 1 (t)+θ 2 z 2 (t)+ + θ n z n (t) (2.25) sz 1 = λ 1 Z 1 + β 1 U U + 1 β 1 + s Z 1 λ 1 2.12: Z 1 + 1 β 1 + s θ 1 λ 1 Z 2 + 1 β 2 + s θ 2 U λ 2 + + + Y Z n + 1 β n + s θ n λ n 2.13:

34 2 Ã, b, c : z i (t) : (2.24) (2.25) G(s) = c(si A) 1 b = c(si Ã) 1 b = θ 1β 1 s λ 1 + θ 2β 2 s λ 2 + + θ nβ n s λ n (2.26) n =2 G(s) = θ 1 θ 2 s 1 1 λ1 λ 2 1 β1 β 2 = θ 1β 1 s λ 1 + θ 2β 2 s λ 2 β i,i=1 n θ i,i=1 n 2.4 - S 1 S 2 u y S 3 S 4 2.14:

2.5. 35 S 1 : S 2 : S 3 : S 4 : G(s) =c(si A) 1 b = i I β i θ i s λ i I β i,θ i i 2.5 2.5.1 G(s) = Y (s) U(s) = h 3s 3 + h 2 s + h 1 s 3 + a 3 s 2 + a 2 s + a 1 (2.27) G (s) = Y (s) U(s) = 1 s 3 + a 3 s 2 + a 2 s + a 1 (2.28) Y Y Y (s) =h 1 Y (s)+h 2 sy (s)+h 3 s 2 Y (s) y(t) =h 1 y (t)+h 2 ẏ (t)+h 3 ÿ (t) (2.29) (2.28) (s 3 + a 3 s 2 + a 2 s + a 1 )Y (s) =U(s) t y (3) (t)+a 3ÿ (t)+a 2 ẏ (t)+a 1 y (t) =u(t) y (3) (t) = a 1y (t) a 2 ẏ (t) a 3 ÿ (t)+u(t) (2.3) z 1 = y, z 2 =ẏ, z 3 =ÿ (2.31)

36 2 (2.29) (2.3) (2.31) ż 1 = z 2 ż 2 = z 3 ż 3 = a 1 z 1 a 2 z 2 a 3 z 3 + u y = h 1 z 1 + h 2 z 2 + h 3 z 3 Ã b ż 1 1 z 1 ż 2 = 1 z 2 + u ż 3 a 1 a 2 a 3 z 3 1 c y = h 1 h 2 h 3 z 1 z 2 z 3 Ã, b, c sz 1 = Z 2 sz 2 = Z 3 sz 3 = a 1 Z 1 a 2 Z 2 a 3 Z 3 + U Y = h 1 Z 1 + h 2 Z 2 + h 3 Z 3 n G(s) = h ns n 1 + + h 2 s + h 1 s n + a n s n 1 + + a 2 s + a 1 Ã = 1.... 1..................... 1 a 1 a 2... a n

2.5. 37 h 3 h 2 U + Z 1 3 Z 1 2 Z 1 1 + h 1 + s s s + Y a 3 a 2 a 1 2.15: b =., c = 1 h 1 h 2 h n 2.5.2 Ã, b, c G(s) = h ns n 1 + + h 2 s + h 1 s n + a n s n 1 + + a 2 s + a 1 Ã = 1.... 1..................... 1 a 1 a 2... a n b =., c = h 1 h 2 h n 1

38 2 A, b, c Ã, b, c T n =3 Ã = T 1 AT, b = T 1 b, T = t 1 t 2 t 3 AT = T Ã, b = T b = T 1 = t 3 (2.32) A t 1 t 2 t 3 = t 1 t 2 t 3 1 1 = a 1 a 2 a 3 a 1 t 3 t 1 a 2 t 3 t 2 a 3 t 3 At 1 = a 1 t 3, At 2 = t 1 a 2 t 3, At 3 = t 2 a 3 t 3 (2.33) (2.32) (2.33) t 3 = b t 2 = At 3 + a 3 t 3 = Ab + a 3 b t 1 = At 2 + a 2 t 3 = A 2 b + a 3 Ab + a 2 b T = = A 2 b + a 3 Ab + a 2 b Ab+ a 3 b b a 2 a 3 1 b Ab A 2 b a 3 1 1 n T = U c W = b Ab A n 1 b a 2 a 3 a 4 a n 1 a 3 a 4 a n 1. a n 1. a n 1 1

2.5. 39 2.5.3 G(s) = Y (s) U(s) = h 3s 2 + h 2 s + h 1 s 3 + a 3 s 2 + a 2 s + a 1 s 3 Y (s) = a 3 s 1 Y (s) a 2 s 2 Y (s) a 1 s 3 Y (s) +h 3 s 1 U(s)+h 2 s 2 U(s)+h 1 s 3 U(s) h 3 h 2 U(s) + 1 + 1 1 h 1 + + + Y (s) s Z s s 1 Z 2 Z 3 a 3 a 2 a 1 2.16: ż 1 = a 1 z 3 + h 1 u ż 2 = z 1 a 2 z 3 + h 2 u ż 3 = z 2 a 3 z 3 + h 3 u y = z 3 ż 1 ż 2 ż 3 = y = 1 a 1 1 a 2 1 a 3 z 1 z 2 z 1 z 2 z 3 + h 1 h 2 h 3 u z 3

4 2 n... a 1 1... a 2 1. h 1.... Ã =...., b h 2 =............ h n 1 a n c = 1 A, b, c Ã, b, c 1. 2. 3. G(s) = θ 1β 1 s λ 1 + θ 2β 2 s λ 2 + + θ nβ n s λ n

41 3 3.1 ẋ(t) = Ax(t)+bu(t), A(n n), b(n 1) (3.1) y(t) = cx(t), c(1 n) (3.2) x : n u : y : x() x 3.1.1 u = f 1 x 1 f 2 x 2 f n x n = fx, f = f 1 f 2 f n (3.3) ẋ =(A bf)x (3.4) u = fx A A bf f A bf ż =(Ã b f)z

42 3 Ã = 1.... 1..................... 1 a 1 a 2... a n b =., f = ɛ 1 ɛ 2 ɛ n 1 Ã b f = 1.... 1..................... 1 (a 1 + ɛ 1 ) (a 2 + ɛ 2 )... (a n + ɛ n ) (3.5) si (Ã b f) = s n +(a n + ɛ n )s n 1 + +(a 2 + ɛ 2 )s +(a 1 + ɛ 1 )(3.6) ɛ i µ 1,µ 2,..., µ n (s µ 1 )(s µ 2 ) (s µ n )=s n + d n s n 1 + + d 2 s + d 1 (3.7) ɛ i = d i a i,i=1 n T T 1 (A bf)t = T 1 AT T 1 bft = Ã b f

3.1. 43 f = ft, 1 f = ft f = d 1 a 1 d 2 a 2 d n a n T 1 3.1.2 ẋ = Ax + bu, A(n n), b(n n) (3.8) J = {x(t) T Qx(t)+u(t) 2 }dt Q Q = Q T Q (Q, A) 196 R.E.Kalman u (t) u (t) = f x(t) f = b T P P (n n) A T P + PA+ Q Pbb T P = J min J = x() T Px()

44 3 DP x(t) u J = x(t) T Px(t) u { û t τ<t+ t u(τ) = u (τ) τ t + t û x(t + t) x(t) u u 3.1: û u(τ) J(û) ={x(t) T Qx(t)+û 2 } t + x(t + t) T Px(t + t) x(t + t) x(t) + x(t) =x(t) +{Ax(t) +bû} t t 2 J(û) = {x T (t)qx(t)+û 2 } t + x(t) T Px(t) +x(t) T (A T P + PA)x(t) t +2ûb T Px(t) t (3.9) J(û) û J(û) û = 2û +2b T Px(t) =

3.1. 45 û = u (t) = b T Px(t) (3.1) (3.9) J(u ) = x(t) T Px(t) = x(t) T (Q + Pbb T P )x(t) t + x(t) T Px(t) +x(t) T (A T P + PA)x(t) t 2x(t) T Pbb T Px(t) t x(t) T (A T P + PA+ Q Pbb T P )x(t) = A T P + PA+ Q Pbb T P = (3.11) (3.1) P (3.11) J = x(t) T Px(t) > P 3.1 x u m 3.2: m =1, x 1 = x, x 2 =ẋ 1 ẋ = x + u 1 J = (x 2 1 + u 2 )dt min 1 1 A =, b =, Q = 1

46 3 A T P + PA+ Q Pbb T P = p11 p 12 p11 p 12 1 + + 1 p 12 p 22 p 12 p 22 p11 p 12 p11 p 12 = p 12 p 22 1 p 12 p 22 p11 + + p 11 p 12 p 12 1 p 2 12 p 11 p 12 p 22 2p 12 p 2 22 = 1 1 p 2 12 = p 12 =1 (P ) 1 p 2 12 p 12 p 22 p 12 p 22 p 2 = 22 2p 12 p 2 22 = p 22 = 2 (p 22 > 2 ) p 11 p 12 p 22 = p 11 = 2 2 1 P = 1 2 f = b T P = 1 2 1 = 1 2 1 2 u = f x(t) 3.2 u = fx y(= cx) x x

3.2. 47 3.2.1 1. y(t) u(t) n =3 ż 1 = a 1 z 3 + h 1 u ż 2 = z 1 a 2 z 3 + h 2 u ż 3 = z 2 a 3 z 3 + h 3 u (3.12) y = z 3 (3.13) (3.12) (3.13) z 2 = ẏ + a 3 y h 3 u z 1 = ÿ + a 3 ẏ h 3 u + a 2 y h 2 u 2. y(t) u(t) = ẋ = Ax, x() (3.14) (3.14) x(t) =e At x() y(t) =cx(t) =ce At x() (3.15) (3.15) e AT t c T t 1 t1 ( t1 ) e AT t c T y(t)dt = e AT t c T ce At dt x() (3.16) D(t 1 ):= t1 e AT t c T ce At dt (c, A) D(t 1 ), t 1 > D(t 1 ) 1 (3.16) t1 D(t 1 ) 1 e AT t c T y(t)dt = x()

48 3 3. x u y ˆx ŷ ẋ = Ax + bu (3.17) ˆx = Aˆx + bu (3.18) e = ˆx x (3.17) (3.18) ė = A(ˆx x) =Ae e(t) =e At e() ˆx(t) =x(t)+e At (ˆx() x()) x() ˆx() = x() ˆx(t) =x(t) x() e At (ˆx() x()) 3.2.2 ẋ = Ax + bu (3.19) y = cx (3.2)

3.2. 49 u y k + ŷ 3.3: (3.19) (3.2) ˆx = Aˆx + bu ŷ = cˆx ˆx = Aˆx + bu + k(y ŷ) = Aˆx + bu + kc(x ˆx) = (A kc)ˆx + bu + ky (3.21) e = ˆx x (3.19) (3.21) ė =(A kc)e A kc (c, A) k A kc (A kc) T = A T c T k T à = A T, b = c T, f = k T A kc à b f (Ã, b) 3.2 (Ã, b) =(c, A)

5 3 x k m u 3.4: y = x, m =1, k =1 ( 4, 4) mẍ + kx = u ẍ + x = u x 1 = x, x 2 =ẋ 1 ẋ = x + u 1 1 y = 1 x ˆx =(A kc)ˆx + bu + ky k (A kc) ( 4, 4) 1 k1 k 1 1 A kc = 1 = 1 1 k 2 k 2 s + k 1 1 si (A kc) = 1+k 2 s = s2 + k 1 s +1+k 2 (s +4) 2 = s 2 +8s +16 k 1 =8, k 2 =15

3.2. 51 3.2.3 ẋ = Ax + Bu, u : m A(n n) B(n m) y = Cx, y : r C(r n) x R n y R r n r G.Gopinath(1971) rank C = r C T = (n n) D D((n r) n) x(t) =Tx(t) (3.22) x(t) =TAT 1 x(t)+tbu(t) (3.23) y(t) =CT 1 x(t) (3.24) I r T = I r C D = C CT 1 = I r x 1 (t) x 2 (t) y(t) =. x r (t) x y(t) x(t) =, z : n r z(t)

52 3 A11 A 12 B1 TAT 1 = A 21 A 22, TB = B 2 A 11 (r r), A 12 (r (n r)), A 21 ((n r) r), A 22 ((n r) (n r)) B 1 (r m), B 2 ((n r) m) (3.23) ẏ(t) =A 11 y(t)+a 12 z(t)+b 1 u(t) ż(t) =A 21 y(t)+a 22 z(t)+b 2 u(t) ż(t) =A 22 z(t)+b 2 u(t)+a 21 y(t) (3.25) A 12 z(t) =ẏ(t) A 11 y(t) B 1 u(t) (3.26) (3.25) z(t) (3.26) z(t) n r (C, A) (A 12, A 22 ) x(t) ẑ(t) =A 22 ẑ(t)+b 2 u(t)+a 21 y(t) A 12 z(t) A 12 ẑ(t) G((n r) r) ẑ(t) = A 22 ẑ(t)+b 2 u(t)+a 21 y + G(A 12 z A 12 ẑ(t)) = A 22 ẑ(t)+b 2 u(t)+a 21 y(t)+ga 12 (z(t) ẑ(t)) = (A 22 GA 12 )ẑ(t)+b 2 u(t)+a 21 y(t) +G(ẏ(t) A 11 y(t) B 1 u(t)) (3.27) e(t) =ẑ(t) z(t) (3.25) (3.27) ė(t) = A 22 e(t) GA 12 e(t) = (A 22 GA 12 )e(t) (A 12, A 22 ) G (A 22 GA 12 )

3.2. 53 (A 22 GA 12 ) ẑ(t) z(t) (3.27) ẏ(t) ẏ(t) z(t) w = z Gy(t) (3.28) (3.27) ŵ(t) = ẑ(t) Gẏ(t) =(A 22 GA 12 )ẑ(t)+b 2 u(t)+a 21 y(t) GA 11 y(t) GB 1 u(t) =(A 22 GA 12 )(ẑ(t) Gy(t)) + (A 22 GA 12 )Gy(t) +B 2 u(t)+a 21 y(t) GA 11 y(t) GB 1 u(t) =(A 22 GA 12 )ŵ(t)+(b 2 GB 1 )u(t) +{(A 22 GA 12 )G + A 21 GA 11 }y(t) w(t) (3.28) z(t) =w(t)+gy(t) (3.22) x(t) =T 1 x(t) =T 1 y(t) z(t) T 1 = H 1 H 2 = T 1 y(t) w(t)+gy(t) x(t) =H 1 y(t)+h 2 (w(t)+gy(t)) = (H 1 + H 2 G)y(t)+H 2 w(t) x(t) ˆx(t) =(H 1 + H 2 G)y(t)+H 2 ŵ(t)

1 197. 2 H.Kwakernaak and R.Sivan: Linear optimal control systems, Wiley- Interscience, 1972. 3 1982. 4 1994. 5 W.M.Wonham: Linear multivariable control: a geometric approach, Springer-Verlag, 1979. 6 F.R.Gantmacher: The theory of matrices Vol.2, Chelsea Publishing Company, 1959. 2 3 1 c