() 218 3 2 / 218 1/18 218 2/18 2 http://stlab.ssi.ist.hokudai.ac.jp/yuhyama/lecture/digital/ 2 (Continuous-Time System) t R, Controller u(t) Con nuous-time System y(t) or x(t) (Discrete-Time System) k Z k =..., 2, 1,, 1, 2,... ( ) Controller u(k) Discrete-Time System y(k) or x(k) (Sampled-Data System) Sampler Controller u*(k) Holder u(t) Con nuous-time System y(t) or x(t) y*(k) or x*(k) Sampler 218 3/18 218 4/18
2 2 () () AD/DA sampling rate () bit AD/DA bit 2 ( ) () ( ) ( ) ( ) (...) 218 5/18 218 6/18 z- (1) z- z- z- z- z- z- {x(k)}: x(),x(1),x(2),... ()z-: {x(k)} z- X(z) = x(k)z k Z [x(k)] = X(z) [] z-: X(z) = x(k)z k k= z- 218 7/18 218 8/18
z- (2) z- (1) z- z- z- z z {x(k)} z ( ) z-: x(k) =Z 1 [X(z)] = 1 X(z)z k 1 dz 2πj U U X(z) ( ) z-: X(z) z 1 z- z- z- : a, b Z [ax(k)by(k)] = ax(z)by (z) Z {ax(k)by(k)} = {ax(k)by(k)}z k = ax(z)by (z) : x(k) =(k <) s z s X(z) =Z [x(k s)] z s X(z) = x(k)z k s = x(k s)z k = Z [x(k s)] k = X(z) =x() x(1)z 1 x(2)z 2 218 9/18 218 1/18 z- (2) z- (3) z- z- z- 1: {x 1 (k)} {x 2 (k)} k y(k) = x 1 (m)x 2 (k m) m= z- Y (z) =X 1 (z)x 2 (z) z- z- z- 2: y(k) =x 1 (k)x 2 (k) z- Y (z) = 1 X 1 (z 1 )X 2 (z/z 1 )z1 1 2πj dz 1 U z = e jωt, z 1 = e jω1t Y (e jωt )= T 2π π/t π/t X 1 (e jωt )X 2 (e j(ω ω1)t )dω 1 k Y (z) = x 1 (m)x 2 (k m)z k m= = x 1 (m)x 2 (n)z (mn) = x 1 (m)z m x 2 (n)z n m= n= m= n= = X 1 (z)x 2 (z) { 1 Y (z) = 2πj U 1 X 1 (z 1 ) 2πj U 1 X 1 (z 1 )X 2 (z/z 1 )z1 1 2πj dz 1 U X(z 1 )z k 1 1 dz 1 } x 2 (k)z k = x 2 (k)(z/z 1 ) k z 1 1 dz 1 = 218 11/18 218 12/18
z- (4) z- (1) z- z- z- : {x(k)} k x x =[(1 z 1 )X(z)] z=1 x(k) =(k <) lim (1 z 1 z 1 )X(z) = lim z 1 x(k)(z k z k 1 ) n = lim lim (x(k) x(k 1))z k z 1 n = lim x(n) n () z- z- z- x(t) (t ) T x (t) = x(kt)δ(t kt) δ( ) Dirac δ x (t) ( ) x(t) t x*(t) t 218 13/18 218 14/18 z- (2) z- (3) z- z- z- x (t) X (s) = {x(kt)} z- X(z) = x(kt)e kst x(kt)z k z- z- z- z- X(s) =L [x(t)] = N(s)/D(s) p i (i =1, 2,...,m) X (s) =L[x(t)δ T (t)] = 2πj = m Res[X(p)/(1 e (s p)t )] p=p i i=1 m 1 lim (p p i )X(p) p p i 1 e = m (s p)t i=1 i=1 N(p i ) D (p i ) 1 1 e (s pi)t z = e st z- X(e st )=X (s) δ T (t) 1 L[δ T ]=1/(1 e st ) X(z) =Z [x(z)] = m i=1 N(p i ) D (p i ) 1 1 e pit z 1 218 15/18 218 16/18
z- (1) f(t) F (s) {f(k)} F (z) δ(t) 1 {1,,,...} 1 1 z 1(t) {1, 1, 1,...} s z 1 1 Tz t s 2 kt (z 1) 2 e at 1 e akt z s a z e at te at 1 (s a) 2 kte akt Te at z (z e at ) 2 s z(z cos ωt) cos ωt s 2 ω 2 cos kωt z 2 2z cos ωt 1 ω z sin ωt sin ωt s 2 ω 2 sin kωt z 2 2z cos ωt 1 e at s a cos ωt (s a) 2 ω 2 e akt z(z e at cos ωt) cos kωt e at sin ωt ω (s a) 2 ω 2 e akt sin kωt z 2 2e at z cos ωt e 2aT e at z sin ωt z 2 2e at z cos ωt e 2aT z- z- z- : x(t) ( X(ω) =F[x(t)]) x(nt )= 1 2π = 1 2π = T 2π X(ω)e jωnt dω (2m1)π/T m= (2m 1)π/T π/t π/t ˆX(ω) = 1 T ˆX(ω )e jω nt dω m= X(ω)e jωnt dω ( X ω 2πm ) T ˆX(ω) 2π/T (*) x(nt ) ( ) 218 17/18 218 18/18 (2) (2) z- z- z- ˆX(ω) = n= x(nt )e jωnt = X (jω) ˆX ( ˆX(ω) =F[x (t)]) ( ): x(t) ω s = π/t X(ω) = ( ω ω s ) z- z- z- x(t) =T h(τ)x (t τ)dτ h(t) =F 1 [H(ω)] = 1 2π ωs ω s e jωt dω = 1 T sin(ω s t) ω s t ( ): {x(kt)} ω s ˆX(ω) x(t) = k= x(kt) sin(ω s(t kt)) ω s (t kt) X(ω) =H(ω) T ˆX(ω) { 1 ( ω <ω s ) H(ω) = ( ω ω s ) 218 19/18 218 2/18
(3) (4) z- z- z- T =.5sin(πt) sin(3πt) 1 z- z- z- ˆX(ω) = 1 T m= ( X ω 2πm ) T ω s ω s = ( ).5 -.5 { 3!s {!s!s 3!s -1 -.5.5 1 1.5 2 2.5 3 Aliasing 218 21/18 218 22/18 (1) u (t) y (t) u(t) u*(t) y(t) y*(t) G(s) u (t) = u(kt)δ(t kt), y (t) = y(kt)δ(t kt) G(s) g(t) Y (s) =G(s)U (s) =G(s)L [u (t)] nt n y(nt )= g(nt τ)u (τ)dτ = g((n k)t )u(kt) y (t) = u(kt) g((n k)t )δ(t nt ) n=k Y (s) =L [y (t)] = U (s)g (s) G (s) G(s) g(t) G(s) G (s) 218 23/18 218 24/18
(2) (1) Y D (z) =Z [y(t)], U D (z) =Z [u(z)] Y D (e st )=Y (s), G (s) z G D (z) = U D (e st )=U (s) g(kt)z k z : Y D (z) =G D (z)u D (z) G D (z) G D (e st ) G(s) G (s) = z = e st z-g(s) G D (z) : r*(t) H(s) : (1/s) T [sec] (e st /s) 2 1 ((1 e st )/s) : H(s) = 1 e st s c(t) 218 25/18 218 26/18 (2) (3) z- u*(t) y(t) y*(t) H(s) G(s) HG(z) = Y (z) U(z) = Z [L 1 [(1 e st )(G(s)/s)] ] G(s)/s f(t) [ ] HG(z) =Z (f(kt) f((k 1)T ))δ(t kt) =(1 z 1 )Z [f (t)] =(1 z 1 )Z [L 1 G(s)/s] ] : G(s) = 1 1as H(s)G(s) H(s)G(s) = 1 e st s ( 1 1 1as =(1 e st ) s a ) 1as z- z e st z 1 ( ) HG(z) =(1 z 1 1 ) 1 z 1 1 = 1 e T/a 1 e T/a z 1 z e T/a G(s) HG(z) 218 27/18 218 28/18
(1) : z y(t) = a n 1 y(t 1) a y(t n)b n u(t) b u(t n) z Y (z) = a n 1 z 1 Y (z) a z n Y (z)b n U(z) b z n U(z) G(z): ( ) Y (z) U(z) = G(z) = b n b z n 1a n 1 z 1 a z n z n a n 1 z n 1 a = b n z n b = () z z n z Y (z) U(z) = G(z) = b n z n b z n a n 1 z n 1 a ( ) ( ) ( ) < ( ) (b n =) 218 29/18 218 3/18 (2) G(z) z 1 () G(z) =h() h(1)z 1 h(2)z 2 {h(t)} y(t) = t h(τ)u(t τ) τ= t < u(t) =, y(t) = G(z) U(z) z 1 218 31/18 218 32/18
(1) (2) : y(t) = a n 1 y(t 1) a y(t n)b n u(t) b u(t n) : u (t) Y (z) =b n U(z)z 1 ( a n 1 Y (z)b n 1 U(z) z 1 ( a n 2 Y (z)b n 2 U(z) z 1 ( a Y (z)b U(z)) )) b z 1 b 1 a a 1 z 1 b n{1 a n{1 z 1 b n y (t) 218 33/18 218 34/18 (3) (4) b n z n b z n a n 1 z n 1 a = u (t) b { a b n z 1 b n (b n 1 a n 1 b n )z n 1 (b a b n ) z n a n 1 z n 1 a b 1 { a 1b n x 1 (t) x 2 (t) x n{1 (t) x n (t) z 1 z 1 b n{1 { a n{1b n b n y (t) x 1 (k),...,x n (k) (, observable canonical form): : x =(x 1,...,x n ) a. x(k 1)= 1.. a1.... x(k) 1 a n 1 y(k) =( 1)x(k)b n u(k) c i = b i a i b n c c 1... c n 1 u(k) a a 1 a n{1 G(z) = b n z n b c n 1 z n 1 c z n a n 1 z n 1 = b n a z n a n 1 z n 1 a 218 35/18 218 36/18
(1) (2) (1 1 ) A: n n, b: n, c: n, d: n x(t 1)=Ax(t)bu(t) y(t) =cx(t)du(t) ( 1) A, b, c, d t ( 2) x : x = Tx x (k 1)=Tx(k 1)= T (Ax(k)bu(k)) = TAT 1 x (k)tbu(k) y(k) =ct 1 x (k)du(k) : x = Tx x (k 1)=A x (k)b u(k) y(k) =c x (k)d u(k) A = TAT 1, c = ct 1, d = d b = Tb 218 37/18 218 38/18 (1) (2) : x() = x, : u(),u(1),u(2),... k 1 x(k) =A k x A k τ 1 bu(τ) τ= k 1 y(k) =ca k x ca k τ 1 bu(τ)du(k) τ= ( 1) {h(k)} : k y(k) = h(τ)u(k τ) τ= { d (k =) x =, h(k) = ca k 1 b (k =1, 2,...) ( 2) ẋ = Ax bu, y = cx du x(t) = exp(at)x t exp(aτ)bu(t τ)dτ exp(at) A k 218 39/18 218 4/18
(1) (2) z- X(z) zx(z) =AX(z)bU(z), Y(z) =cx(z)du(z) Y (z) ={c(zi n A) 1 b d}u(z) : (A, b, c, d) G(z) = Y (z) U(z) = c(zi n A) 1 b d G 11 (z) C 1m (z) Y (z) =G(z)U(z), G(z) = G l1 (x) C lm (x) x(k 1)=Ax(k)Bu(k) y(k) =Cx(k)Du(k) G(z) =C(zI n A) 1 B D 218 41/18 218 42/18 (1) (2) : ẋ = A c x B c u y = C c x D c u T T x((k 1)T )=exp(a c T )x(kt) exp(a c τ)dτb c u(kt) x(kt) x(k) : x(k 1)=A d x(k)b d u(k) y(k) =C d x(k)d d u(k) A d = exp(a c T ), B d = C d = C c, D d = D c T exp(a c τ)dτb c 218 43/18 218 44/18
G(z) = G(s) = (41,42 ) (36,48 ) ẋ = A c x B c u y = C c x D c u Cc(sI Ac) 1 Bc Dc G(s) = Y (s) U(s) A d = e AcT, T B d = e Acτ B c dτ, C d = C c, D d = D c (44 ) HG(z) (27, 28 ) x(k 1)=A d x(k)b d u(k) y(k) =C d x(k)d d u(k) Cd(zI Ad) 1 Bd Dd G(z) = Y (z) U(z) G(z) =C(zI A) 1 B D : G (z) =B (zi A ) 1 C D m l : x(k 1)=Ax(k)Bu(k) y(k) =Cx(k)Du(k) (l m ) x (k 1)=A x (k)c u (k) y (k) =B x (k)d u (k) 1 1 218 45/18 218 46/18 (1) { x(k 1)=Ax Bu y = Cx Du { z(k 1)=A z C u y = B x D u x = Tx = z =(T ) 1 z = { x(k 1)= Ā x Bu y = C x Du Ā = TAT 1, B = TB C = CT 1, D = D { z(k 1)= Ā z C u y = B x D u Ā = TAT 1, B = TB C = CT 1, D = D : (): 1....... x(k 1)= 1 x(k). u(k) a a 1 a n 1 1 y(k) =(c, c 1,...,c n 1 )x(k)b n u(k) c i = b i a i b n G(z) = b n z n b c n 1 z n 1 c z n a n 1 z n 1 = b n a z n a n 1 z n 1 a 218 47/18 218 48/18
(2) : y (t) b { a b n x 1 (t) z 1 b 1 { a 1b n a a 1 x 2 (t) z 1 b n{1 { a n{1b n a n{1 x n (t) z 1 b n u (t) 218 49/18 218 5/18 ( ) [ x1 (k 1) A1 A = 2 x 2 (k 1) A 3 ]( ) ( x1 (k) b1 x 2 (k) x 2 ( ) [ x1 (k 1) A1 = x 2 (k 1) A 2 A 3 y(k) = ( c 1 ) ( ) x 1 (k) x 2 (k) ) u(k) ]( ) x1 (k) bu(k) x 2 (k) du(k) x 2 : x f : x s () = () ( ) ( ) 218 51/18 218 52/18
(1) : u(k) y(), y(1), y(2),...,y(n) x() x(n) N x f N 1 x(n) =A N x() A N j 1 Bu(j) (=x f ) j= u(n 1) [B ABA 2 B A N 1 B]. = x f A N x u() x N x f rank [B ABA 2 B A N 1 B]=n 218 53/18 218 54/18 (2) (1) Cayley-Hamilton N n rank [B ABA 2 B A N 1 B]=rank [B ABA 2 B A n 1 B] : rank [B ABA 2 B A n 1 B]=n [B ABA 2 B A n 1 B] : : dim{span [B ABA 2 B A n 1 B] ImA} = dim ImA Cx() = y() du() CAx() = y(1) du(1) CBu() 1 CA 2 x() = y(2) du(2) CA 1 j Bu(j) j=. N 2 CA N 1 x() = y(n 1) du(n 1) CA N j 2 Bu(j) j= 218 55/18 218 56/18
(2) (3) y() du() y(1) du(1) CBu() C CA 1 y(2) du(2) CA CA 2 1 j Bu(j) x() = j=.. CA N 1 N 2 y(n 1) du(n 1) CA N j 2 Bu(j) j= Cayley-Hamilton : (n = ) C CA rank CA 2 = n.. CA n 1 218 57/18 218 58/18 (1) (2) 1 1 x(k 1)=Ax(k)bu(k), y(k) =cx(k)du(k) x (k) =T 1 x(k) A det(λi n A) =λ n a n 1 λ n 1 a 1 λ a () a 1 a 2 a n 1 1 a 2 a 3 1 T 1 =.... a n 1 1 1 1 U 1 c U c =[baba 2 b A n 1 b] 218 59/18 218 6/18
(3) (4) (): T 1 1 =[t 1 t 2 t n ] n i 1 a ji A j b A n i b (i =1, 2,...,n 1) t i = j= b (i = n) At 1 = a b = a t n (Cayley-Hamilton ) At 2 = t 1 a 1 b = t 1 a 1 t n. At n = t n 1 a n 1 t n 1....... A[t 1 t n ]=[t 1 t n ] 1 a a 1 a n 1 [t 1 t n ]( 1) = b 1 T 1 AT1 1........ = 1, T 1b =. a a 1 a n 1 1 c ct 1 1 218 61/18 218 62/18 (1) (2) 1 1 x(k 1)=Ax(k)bu(k), y(k) =cx(k)du(k) x (k) =T 2 x(k) A det(λi n A) =λ n a n 1 λ n 1 a 1 λ a : a 1 a 2 a n 1 1 a 2 a 3 1 T 2 =.... a n 1 1 1 U o U o =[c (ca) (ca 2 ) (ca n 1 ) ] 218 63/18 218 64/18
(3) (4) () T 2 =[t 1 t n ] n i 1 a ji ca j ca n i (i =1, 2,...,n 1) t i = j= c (i = n) t 1 A = a c = a t n t 2 A = t 1 a 1 c = t 1 a 1 t n. a. [t 1 t n ] A = 1.. a1.... [t 1 t n ] 1 a n 1 ( 1)[t 1 t n ] = c a. T 2 AT2 1 = 1.. a1......, ct 1 2 =( 1) 1 a n 1 t n A = t n 1 a n 1 t n b T 2 b 218 65/18 218 66/18 (1) (2) z(k) =Tx(k) x(k 1)=Ax(k)Bu(k) y(k) =Cx(k)Du(k) z(k 1)=Ãz(k) Bu(k) y(k) = Cz(k)Du(k) 2 2 () [ B Ã B Ã n 1 B] =T [BAB,A n 1 B] C CÃ. CÃn 1 = C CA. CA n 1 T 1 218 67/18 218 68/18
rank U c = s<n (n s) n T 2 T 2 U c = T 2 [ B AB A n 1 B ] = T =[T 1,T 2 ] T 1 z = Tx [ ] [ ] A1 A z(k 1)= 2 B1 z(k) u(k) A 3 T 2 B = T 2 AU c = T 2 A = A 3 T 2 T 2 x(k) () rank U o = s<n P s n T 1 = PU o T =[T 1,T 2 ] T 2 z = Tx [ ] A1 z(k 1)= z(k)bu(k) A 2 A 3 y(k) = [ C 1 ] z(k)du(k) T 1 x(k) 218 69/18 218 7/18 (1) ( ) ( ) ( ) ( ) ( ) () : [ A1 A x(k 1)= 2 A 3 ] [ B1 x(k) G(z) = [ ] ( [ A1 A C 1 C 2 zi 2 A 3 = C 1 (zi A 1 ) 1 B 1 D ] u(k) ]) 1 [ B1 ] D 218 71/18 218 72/18
(2) T [ ] ẋ = [ ω ω ] x T [ cos ωt sin ωt x(k 1)= sin ωt cos ωt y(k) = ( 1 ) x(k) ωt = nπ (n =1, 2,...) [ ] ±1 x(k 1)= x(k) 1 ±1 ω ( ) u, y = ( 1 ) x 1 ] x(k) 1 ω ( ) 1 cos ωt u(k) sin ωt ( ) 1 1 u(k), y(k) = ( 1 ) x(k) 218 73/18 218 74/18 (1) x(k 1)=f(x(k)) ( ) δ (> ) ɛ (> ) x() <ɛ x(k) <δ(k =, 1,...) x(k 1)=f(x(k)) x() x(k) (k ) 218 75/18 218 76/18
(2) ( /) (1) x(k 1)=f(x(k)) ( ) x() x(t) (t >) x(k 1)=f(x(k)) ( ) x(k) (k ) x(k 1)=Ax(k) ( /) ( /) ( )( ) LS, ( ) LAS, ( ) GS, GAS : A : x(k 1)=Ax(k) λ 1 A = T 1... T λ n : z = Tx : λ 1 x(k 1) = Ax(k) = z(k 1) = TAT 1 z(k) =... z(k) λ n 218 77/18 218 78/18 ( /) (2) - () (1) : λ k 1 z(k) =... z() λ k n : x(k 1)=f(x(k),u(k)) T 1 z() z() z(k) : λ i 1 (i =1,...,n) z(k) : λ i < 1 (i =1,...,n) ( ) A 1 ( 1 ) A 1 ( ) x(k) β( x(),k)χ max k <k u(k ) - (Input to State Stable; ) β( x(),k) x() k β(,k)=, β( x(),k) (k ) χ( ), χ() = : x(k 1)=Ax(k)Bu(k) 218 79/18 218 8/18
- () (2) - () (3) : x(k 1)=Ax(k)Bu(k) u(k) = A 1 A 1 max λ i (A) = ρ(a) <c<1 i : A = σ max (A) = ρ(a A) : Ax A x A Ã = T 1 AT ( ) x(k) Lc k x() L(1 c) 1 B max k <k u(k ) β( x(),k)=lc k x(), χ(d) =L(1 c) 1 B d A 1 A k m T T 1 k m ρ(a) k <Lc k (k>1) m () 218 81/18 218 82/18 (1) (2) ( ) : Y (z) U(z) = G(z) BIBO BIBO : BIBO (Bounded-Input Bounded-Output Stable) ( ): x(k 1)=Ax(k)Bu(x), y(k) =Cx(k)Du(k) A B (b i ), C (c i ) λ 1, u(i) =(i =1, 2,...), u() = 1 y(k) =CA k 1 Bu() = n λ k i c i b i i=1 k λ 1 λ 1 1 y(k) 1 y(k) BIBO 218 83/18 218 84/18
(3) Schur-Cohn-Jury (1) (, A ) 1 a......... 1,. 1.. a1.... a a 1 a n 1 1 a n 1 z n a n 1 z n 1 a 1 z a A ( 1 ) / : a > f(z) =a n z n a 1 z a (a > ) Schur-Cohn-Jury (1): f j (z) =a (j) n j zn j a (j) 1 z a(j), j =, 1,...,n f (z) =f(z) BIBO 1 f j1 (z) =a (j) f j(z) a (j) n j f j (z) f j (z) =a (j) zn j a (j) n j 1 z a(j) n j 218 85/18 218 86/18 Schur-Cohn-Jury (2) (1) Schur-Cohn-Jury (2): a (1) < a (j) >, j =2,...,n : w = z 1 z 1, z = 1w 1 w z w G ( ) 1w = 1 w G(z) = (z μ 1) (z μ m ) (z λ 1 ) (z λ n ) (μ 1 1)w (μ 1 1) 1 w (λ 1 1)w (λ 1 1) 1 w (μ m 1)w (μ m 1) 1 w (λ n 1)w (λ n 1) 1 w = (1 w)n m {(μ 1 1)w (μ 1 1)} {(μ m 1)w (μ m 1)} {(λ 1 1)w (λ 1 1)} {(λ n 1)w (λ n 1)} = P w (w)/q w (w) 218 87/18 218 88/18
(2) P w (w), Q w (w) w Q w (w) = () Q w (w) n z = 1 w =(z 1)/(z 1) z =exp(st ) 1.5 1.5.5 1 D z (z) =z 2 z/21/2 f (z) =D z (z) f 1 (z) =(1/2)(z 2 z/21/2) (1/2z 2 z/2 1) = z/4 3/4 f 2 (z) =( 3/4)( z/4 3/4) ( 1/4)( 3z/4 1/4) = 1/2 3/4 <, 1/2 > z =(1w)/(1 w) ( ) 1w D w (w) =D z = (1 w)2 (1 w)(1 w)/2(1 w) 2 /2 1 w (1 w) 2 D w (w) =.5 D w (w) =w 2 w 2= 1 D w (w) 218 89/18 218 9/18 : x(k 1)=Ax(x)Bu(k), y(k) =Cx(k) FB FB FB FB () : u(k) =Fx(k)Gv(k) PID PID x(k 1)=(A BF)x(k)BGv(k) A A BF 218 91/18 218 92/18
(1) (2) FB FB PID : 1........ x(k 1)= 1 x(k). u(k) a a 1 a n 1 1 u(k) =Fx(k)Gv(k) =(f,...,f n 1 )x(k)gv(k) 1....... x(k 1)= 1 x(k). v(k) f a f 1 a 1 f n 1 a n 1 G FB FB PID : z n a n 1 z n 1 a 1 z a = z n (a n 1 f n 1 )z n 1 (a 1 f 1 )z (a f )= F =(f,...,f n 1 ) (=) = (Pole assignment) 218 93/18 218 94/18 (3) (1) FB FB PID : u(k) =Fx(k), F = (,...,, 1)G 1 c P (A) G c P (A) P (z) =z n β n 1 z n 1 β 1 z β A : P (A) =A n β n 1 A n 1 β 1 A β I FB FB PID U(z) E(z) G (z) Y(z) G(z) = Y (z) U(z) = G (z) 1G (z), E(z) U(z) = 1 1G (z) C(z) U(z) E(z) C(z) G P (z) Y(z) G(z) = Y (z) U(z) = C(z)G P (z) 1C(z)G P (z), E(z) U(z) = 1 1C(z)G P (z) G (z) =C(z)G P (z) 218 95/18 218 96/18
(2) (3) FB FB PID 2 U(z) E(z) C 1 (z) C 2 (z) G(z) = Y (z) U(z) = C 1 (z)g P (z) 1C 1 (z)c 2 (z)g P (z), E(z) U(z) = 1 1C 1 (z)c 2 (z)g P (z) G P (z) Y(z) : E(z) G (z) =C 1 (z)c 2 (z)g P (z) [] E(z) U(z) = 1 1G (z) FB FB PID : G (z) = N(z) D(z) G(z) = Y (z) U(z) = N(z) N(z)D(z) E(z) U(z) = D(z) N(z)D(z) 218 97/18 218 98/18 FB FB PID U(z) E(z) C(z) G P (z) Y(z) e(k) =Z 1 [E(z)] = Z 1 [U(z)/(1 G (z)) G (z) =C(z)G P (z) e(k) k : u(k) =1() : u(k) =kt( ) : u(k) =k 2 T 2 /2 N(z), D(z) G (z) = N(z) D(z) FB FB PID G (z)/(1 G (z)) e( ) = lim(1 z 1 1 ) z 1 1G (z) 1 1 z 1 = lim z 1 1 1G (z) = D(1) D(1) N(1) D(1) = D(z) z 1 G (z) z 1 1 1 G (z) = b m z m b 1 z b (z 1)(a n 1 z n 1 a 1 z a ) 218 99/18 218 1 / 18
PID FB FB PID e( ) = lim(1 z 1 1 ) z 1 1G (z) Tz (z 1) 2 = T N(1) lim D(z) z 1 z 1 D(z)/(z 1) (z 1) D(z) z 1 2 G (z) z 1 2 2 G (z) = b m z m b 1 z b (z 1) 2 (a n 2 z n 2 a 1 z a ) FB FB PID K P U(z) E(z) TK Y(z) I (z 1) G P (z) 2(z 1) K D (z 1) Tz PID, P (), I ( ), D () () PID PID 218 11 / 18 218 12 / 18 (1) (2) FB FB PID : G(e jωt ) ( π/t ω π/t) N: 1 Z: P : G (z) N = P Z : N = P FB FB PID G (z) / P 218 13 / 18 218 14 / 18
(3) (1) FB FB PID U(z) E(z) K G P (z) Y(z) G P (z) N 1 1/K G P (z) 1/K G P (z) FB FB PID G(z) =N(z)/D(z) cos kωt Y (z) = N(z) D(z) z(z cos ωt) z 2 2z cos ωt 1 = zp(z) D(z) zp(z)/d(z): zq(z)/(z 2 2z cos ωt 1): zq(z) z 2 2z cos ωt 1 q(z) =k 1 z k q(z) =k 1 z k =(z cos ωt)g(z) p(z)(z2 2z cos ωt 1) D(z) z = e jωt q(e jωt )=k 1 cos ωt k jk 1 sin ωt = j sin ωt G(e jωt ) 218 15 / 18 218 16 / 18 (2) FB FB PID k 1 cos ωt k = sin ωtim[g(e jωt )] k 1 =Re[G(e jωt )] q(z) =Re[G(e jωt )](z cos ωt) Im[G(e jωt )] sin ωt [ ] Z 1 zq(z) z 2 2z cos ωt 1 =Re[G(e jωt )] cos kωt Im[G(e jωt )] sin kωt FB FB PID ω G(e jωt ) arg[g(e jωt )] G(e jωt ) π <arg[g(e jωt )] < arg[g(e jωt )] < arg[g(e jωt )] <π arg[g(e jωt )] : z = e jωt = G(e jωt ) cos(kωt ϕ) ϕ =arg[g(e jωt )] 218 17 / 18 218 18 / 18