MPC.dvi

1997 1 Copyright c 1997 by Manabu Kano All rights reserved 1

1 3 11 3 12 3 121 3 122 5 13 6 14 6 15 7 2 9 21 9 22 10 221 10 222 11 23 12 24 13 25 14 26 15 27 17 2

1 11 (MPC; Model Predictive Control) 12 MPC ( ) (FIR; Finite Impulse Response) (FSR; Finite Step Response) (TF; Transfer Function) ARX(Auto-Regressive exogenous input) (SS; State Space) 121 MPC Fig11 {h i } {h i } s +1 3

u y h 1 h 2 h s Fig 11 h i = 0 for i>s (11) T d h i = 0 for i =1, 2,,T d (12) t u(t) t + j y M (t + j) = h j u(t) (13) y M (t + j) = = h k u(t + j k) s h k u(t + j k) (14) y M (t) = s h k u(t k) (15) d s y M (t) = h k d k u(t) = dg M (d)u(t) (16) d x(t 1) = dx(t) (17) 4

u y a 1 a 2 a s Fig 12 122 Fig12 1 {a i } {a i } s a i = a s for i s (18) T d a i = 0 for i =1, 2,,T d (19) t u(t) t + j y M (t + j) = a j u(t) (110) y M (t + j) = a k u(t + j k) (111) y M (t) = a k u(t k) (112) d y M (t) = a k d k u(t) = dg M (d) u(t) (113) G M (d) 5

13 Sec12 y P (t + j) y P (t + j) = y M (t + j) (114) t y P (t + j) = y M (t + j)+y(t) y M (t) (115) y(t) y M (t) 14 r y P y P (t +1) = r(t + 1) (116) MPC Eq(116) (Reference Trajectory) y R (t + j) = α j y(t)+(1 α j )r(t + j) for j>0 (117) α α =0 α =1 y(t) α 6

α y(t) 1 Eq(117) y P (t +1) = y R (t + 1) (118) 15 1 1 (1 ) y M (t) = dg M (d)u(t) (119) y P (t +1) = y M (t +1)+y(t) y M (t) = (1 d)g M (d)u(t)+y(t) (120) y R (t +1) = αy(t)+(1 α)r(t + 1) (121) Eq(118) y P (t +1) = y R (t + 1) (122) u(t) u(t) = 1 α {r(t +1) y(t)} (123) (1 d)g M (d) Fig13 Eq(123) 1 d Fig13 1 r 1 α d u 1 y (1 d)g M (d) Fig 13 1 7

Eq(120) 1 Eq(120) 1 Eq(123) y M (t) u(t) u(t) = 1 α (1 αd)g M (d) {r(t +1) y(t)+y M (t)} (124) Fig14 IMC(Internal Model Control) r 1 α d u 1 y (1 αd)g M (d) dg M (d) y M Fig 14 1 (IMC ) 8

2 21 Chap1 1 MPC 1 MPC MPC 3 ( ) Fig21 MPC t y(t) y(t) r y R (t + j) y P (t + j) L P [L, L + P 1] M u(t),u(t +1),,u(t + M 1) u(t) t +1 t +1 y(t +1) t +1 [1,L+ P 1] (Prediction Horizon) () [L, L + P 1] (Coincidence Horizon) M (Control Horizon) r y R y(t) y P u(t) Fig 21 MPC 9

22 Sec12 [L, L + P 1] 221 y M (t + j) = s h k u(t + j k) (21) t y M (t + j) = j s h k u(t + j k)+ h k u(t + j k) (22) k=j+1 t M u(t),u(t +1),,u(t + M 1) t + M u(t + M 1) u(t + i) = u(t + M 1) for i M 1 (23) y M (t + j) = + j M+1 h k u(t + M 1) + h j M+2 u(t + M 2) + + h j 1 u(t +1)+h j u(t) s j h j+k u(t k) (24) MPC [L, L + P 1] L M+1 h L h L 1 h L M+2 h k y M (t + L) L M+2 u(t) y M (t + L +1) = h L+1 h L h L M+3 h k u(t +1) y M (t + L + P 1) L+P M u(t + M 1) h L+P 1 h L+P 2 h L+P M+1 h L+1 h L+2 h s 1 h s h L+2 h L+3 h s 0 + h L+P h L+P +1 0 0 u(t 1) u(t 2) u(t s + L) h k (25) t y M (t) = s h k u(t k) (26) 10

y M (t + L) y M (t) y M (t + L +1) = y M (t) y M (t + L + P 1) y M (t) L M+1 h L h L 1 h L M+2 L M+2 h L+1 h L h L M+3 + L+P M h L+P 1 h L+P 2 h L+P M+1 h k h k h k h L+1 h 1 h L+2 h 2 h s h s L h s h L+2 h 1 h L+3 h 2 0 h s L h s + h L+P h 1 h L+P +1 h 2 0 h s L h s u(t) u(t +1) u(t + M 1) u(t 1) u(t 2) u(t s) (27) f,o Y M = Y M0 + H f u f + H o u o (28) 222 y M (t + j) = a k u(t + j k) (29) t y M (t + j) = = j a k u(t + j k)+ a k u(t + j k) k=j+1 j 1 a j k u(t + k)+ a j+k u(t k) (210) k=0 t + M u(t + M 1) u(t + i) = 0 for i M (211) y M (t + j) = M 1 k=0 a j k u(t + k)+ a j+k u(t k) (212) 11

y M (t + j) t y M (t) = a k u(t k) (213) Eq(212) y M (t + j) y M (t) M 1 s 1 y M (t + j) = y M (t)+ a j k u(t + k)+ (a j+k a k ) u(t k) (214) k=0 y M (t + j) MPC [L, L + P 1] y M (t + L) y M (t) y M (t + L +1) = y M (t) y M (t + L + P 1) y M (t) a L a L 1 a L M+1 u(t) a L+1 a L a L M+2 u(t +1) + a L+P 1 a L+P 2 a L+P M u(t + M 1) a L+1 a 1 a L+2 a 2 a L+s 1 a s 1 u(t 1) a L+2 a 1 a L+3 a 2 a L+s a s 1 u(t 2) + (215) a L+P a 1 a L+P +1 a 2 a L+s+P 1 a s 1 u(t s +1) Y M = Y M0 + A f u f + A o u o (216) 23 Sec13 y P (t + j) = y M (t + j)+y(t) y M (t) (217) [L, L + P 1] Y P = Y M + Y Y M0 (218) Y P = y P (t + L) y P (t + L +1) y P (t + L + P 1) (219) 12

Y = y(t) y(t) y(t) (220) MPC Sec14 L y R (t + j) = α j L+1 y(t)+(1 α j L+1 )r(t + j) for j L (221) [L, L + P 1] Y R = αy +(I α)r (222) Y R = r = α = y R (t + L) y R (t + L +1) y R (t + L + P 1) r(t + L) r(t + L +1) r(t + L + P 1) α 0 α 2 0 α P (223) (224) (225) 24 1 (Sec15) () (P >1) () MPC J = Y P Y R 2 (226) J {u(t + j)}(j =0, 1,,M 1) Eqs(28),(218) Y P = Y + H f u f + H o u o (227) 13

J J = (Y + H f u f + H o u o Y R ) T (Y + H f u f + H o u o Y R ) (228) u f J J = u f H T f H f J u f = 0 (229) J u(t) J u(t +1) J u(t + M 1) = 2H T f H f u f +2H T f (Y + H o u o Y R ) = 0 (230) u f = (H T f H f ) 1 H T f (Y R Y H o u o ) (231) Eqs(216),(218) Y P = Y + A f u f + A o u o (232) u f A T f A f u f = (A T f A f ) 1 A T f (Y R Y A o u o ) (233) 25 MPC J J = Y P Y R 2 Q + u f 2 R (234) Y P Y R 2 Q = (Y P Y R ) T Q(Y P Y R ) (235) u f 2 R = u T f R u f (236) 14

Q, R Y R r 1 () 2 R R Eq(234) Eq(232) u f A T f QA f +R u f = (A T f QA f + R) 1 A T f Q(Y R Y A o u o ) (237) J = Y P Y R 2 Q + u f 2 R + u f u ss 2 S (238) 1,2 3 u ss u ss MPC S Q,R (i, j) { 0 for i = j = M 1 s ij (239) = 0 for other i, j 2 MPC 26 MPC (over-parameterization) 15

Output Step Fig 22 2 1 Sec12 s s MPC G(s) = 2 10s +1 (240) 1 20 Fig22 5 20 20 s ARX u f = (A T f A f ) 1 A T f (Y R Y A o u o ) (241) A f L, P, M s Y + A o u o u(t 1) ARX s 16

27 MPC L, M, P, α, Q, R, S L =1, S = 0 1 PT s T s P>20 2 M 3 5 3 α α 1 0 (a) Q SISO Q (b) R R R 4 α =0 R = 0 P M 17