State Predictive Model Following Control System for the Linear System with Input Time-Delay

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1 245 (28..24) State Predictive Model Following Control System for the Linear System with Input Time-Delay Dazhong Wang, Shigenori Okubo Faculty of Engineering, Yamagata University : (time-delay) (state predictive) (disturbances) (model following control system) : ,Tel.: (238) , sokubo@yz.yamagata-u.ac.jp Tel.: (238) , wdzh68@hotmail.com. 2 3 MFCS 4 5 4] ( 6 ) 5] 8] 2. () (2) ẋ(t) = Ax(t) + B u(t L ) 9] +B 2 u(t L 2 ) + d(t) () y(t) = Cx(t) + d (t) (2)

2 (3),(4). ẋ m (t) = A m x m (t) + B m r m (t) (3) B = e A(L 2 L ) B + B 2 A, B] C, A] y m (t) = C m x m (t) (4) x(t) d(t) R n, u(t); y(t) d (t) R l., x(t) u(t) y(t) d(t) d L, L 2 < = L < 2 CpI A] B L 2 x m (t) R nm r m (t) R lm y m (t) R l 3 pi A. A, B, B 2, C A m, B m, C m ˆx m (t) = x m (t + x(t). L 2 ) x m (t) r m (t). e(t) = y(t) y m (t) (5) + e Am(t τ) B mˆr m (τ)dτ () x(t) = x (t) (t < = ) u(t) = u (t) (t < ) t e(t) (4), (MFCS), rank pi A, B ] = n (9) ] pi A rank = n () C ˆx m (t) = e AL2 x m (t) (3) ˆx m (t) = A mˆx m (t) + B mˆr m (t) (2) 3. ŷ m (t) = C mˆx m (t) (3) z(t) L 2 ẑ(t) = z(t + L 2 ) ˆx(t) ê(t) = ŷ(t) ŷ m (t) (4) ˆx(t) = e AL 2 x(t) e A(t τ) B u(τ + L 2 L )dτ e A(t τ) B 2 u(τ)dτ e A(t τ) ˆd(τ)dτ (6) (7) (2) ˆx(t) ˆx m (t) ˆx(t)=pI A] Bu(t)+pI A] ˆd(t)(5) ˆx m (t) = pi A m ] B mˆr m (t) (6) () (2) (8), (3), (5) (6), ŷ(t) ŷ m (t), ŷ(t) = CpI A] Bu(t) ˆx(t) = Aˆx(t) + Bu(t) + ˆd(t) (7) ŷ(t) = C ˆx(t) + ˆd (t) (8) +CpI A] ˆd(t) + ˆd (t) ŷ m (t) = C m pi A m ] B mˆr m (t) 2

3 D(p)ŷ(t), D m (p)ŷ m (t) D(p)ŷ(t) = N(p)u(t) + ŵ(t) D m (p)ŷ m (t) = N m (p)ˆr m (t) CpI A] B = N(p)/D(p), N(p) = T (p)d m (p)ê(t) = D d (p)d(p)r(p)ŷ(t) +S(p)ŷ(t) T (p)n m (p)ˆr m (t) (2) CadjpI A] B R l l, D(p) = pi A, C m pi A m ] B m = N m (p)/d m (p), N m (p) = C m adjpi A m ]B m R l m l m D m (p) = pi A m. ŵ(t) = CadjpI A] ˆd(t) + D(p) ˆd (t) (7) T (p)d m (p)ê(t) = D d (p)r(p){n(p)u(t) +ŵ(t)} + S(p)ŷ(t) T (p)n m (p)ˆr m (t) = D d (p)r(p)n(p)u(t) + S(p)ŷ(t) T (p)n m (p)ˆr m (t) = {D d (p)r(p)n(p) Q(p)N r }u(t) N(p) N m (p)+q(p)n r u(t) + S(p)ŷ(t) N(p) = diag(p ηi )N r + Ñ(p) N m (p) = diag(p η m i )Nmr + Ñm(p) ri Ñ(p) < η i ri Ñ m (p) < η mi. η i T (p)n m (p)ˆr m (t) (22) Q(p) Q(p) Q(p) = diag(p ρ+n m n+η i ) + Q(p) N(p) p ri Q(p) < ρ + nm n + η i η mi N m (p) ˆN r ê(t) (t ) l l, N r =, 22 ê(t) = ˆd(t), ˆd (t) (22), ˆN r = D d (p) ˆd(t) =, D d (p) ˆd (t) = (8) u(t) u(t) = Nr Q (p){d d (p) D d (p), R(p)N(p) Q(p)N r }u(t) (7) (8) ŵ(t) D d (p)ŵ(t) = (9) (proper), ρ (ρ n d +2n n m η i ) T (p) T (p)d m (p) D d (p)d(p) (i =, 2,, l) R(p), S(p) ρ n d + 2n n m η i (25) T (p)d m (p) = D d (p)d(p)r(p) + S(p) (2), T (p) = ρ, D m (p) = n m, D d (p) = n d, D(p) = n, R(p) = ρ + n m n d n S(p) < = n d + n ê(t) 3 N r Q (p)s(p)ŷ(t) + u m (t) (23) u m (t) = N r Q (p)t (p)n m (p)ˆr m (t) (24) n m η mi n η i (26) u(t) u(t) = H ξ (t) E 2 ŷ(t) H 2 ξ 2 (t) + u m (t)

4 (27) u m (t) = E 3ˆr m (t) + H 3 ξ 3 (t) (28) ˆx m (t) (7) (8) (27) (3) u(t), u m (t) ξ (t), ξ 2 (t), ξ 3 E s ż s (t) = A s z s (t) + d s (t) (35) y s (t) = C s z s (t) + d s (t) (36) ξ (t) = F ξ (t) + G u(t) (29) z s (t) = ˆx T (t), ξ T (t), ξ2 T (t), u T (t)] T, ξ 2 (t) = F 2 ξ 2 (t) + G 2 ŷ(t) (3) y s (t) = ŷ T (t),,, ] T, C s = C,,, ], d s = ξ 3 (t) = F 3 ξ 3 (t) + G 3ˆr m (t) (3) ˆd (t),,, ] I I E s = I, H pi F ] G = Nr Q (p) A B F G {D d (p)r(p)n(p) Q(p)N r } (32) A s = G 2 C F 2 E 2 C H H 2 I E 2 + H 2 pi F 2 ] G 2 = N r Q (p)s(p) (33) E 3 + H 3 pi F 3 ] G 3 = N r Q (p)t (p)n m (p) (34) (27) u(t), z s (t) A s ξ i (t)(i =, 2) ŷ(t), u m (t) (27) u(t) ê(t) (t ) e(t) (t ),, CpI A] B, V (p), 4. pe s A s = N r T (p) l D m (p) l Q(p) V (p) ˆr m (t) ˆd(t) ˆd (t). D d (p), (8) (37) p, ˆd(t), ˆd (t), d(t) d x(t) = x (t)(t < = ), u(t) = d s (t) = u (t)(t < ), ξ i (t) = ξi (t)(t < = )(i =, 2). x m (t) E s (37) 4 ˆd(t) G 2 ˆd (t) E 2 ˆd (t) + u m (t), pe s A s = Q(p) D(p) N r T l (p)dm(p) l D l N(p) (p) CpI A] B = N(p)/D(p) l = U(p) V (p) pe s A s (37) pe s A s (38)

5 p, ranke s = deg pe s A s l = n + 2 (ρ + n m n + η i ) (39) 5.,. i= ] ] z s (t) ẋ m (t) = x m (t) + r m (t), (37) 6 5 ] T (p), D m (p), Q(p), V (p) y m (t) = 2 x m (t), A s, z s (t). ] ] ẋ(t) = x(t) + u(t L ) ] ] L, L 2 + u(t L 2 ) + 2 d(t) ] y(t) = 5 x(t) + d (t) ẋ(t) = Ax(t) + B u(t L ) +B 2 u(t L 2 ) + d(t) ] ] A =, B =, y(t) = Cx(t) + d (t) 3 4 ] ] B 2 =, C = 5 2 ˆx = Aˆx(t) + Bu(t) + ˆd(t) ŷ = C ˆx + ˆd (t) ] ] F i =, G i = (i =, 2, 3) 36 2 () (4) () A, B] C, A] rank rank pi A pi A C, B ] = n ] = n (2) CpI A] B (3) pi A (4) ranke s = deg pe s A s. Fig. Responses of the state predictive model following control system for the linear system with input time-delay d(t) d (t) r m d(t) =.35t. (8 < = t < = 22) 5

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