9 24 9 ACTA ELECTRONICA SINICA Vol. 32 No. 9 Se. 24 1,2, 2, 1 (11, 184 ;21, T6G 2V4) :,.,,;,,,,.,. : ; ; ; : TP273 : A : 37222112 (24) 92141427 Identificatio n of No n2uniformly Perio dically Sa mled Multirate Syste ms DING Feng 1,2,CHEN Tong2wen 2,XIAO De2yun 1 (11 Deartment of Automation, Tsinghua University, Beijing 184, China ; 21 Deartment of Electrical and Comuter Engineering, University of Alberta, Edmonton, Alberta, Canada T6 G 2V4) Abstract : For non2uniformly eriodically samled multirate systems,we derive the corresonding lifted state2sace models by using the lifting technique. Furthermore,for multirate systems whose states are measurable,an identification method to estimate the lift2 ed system arameter matrices is given by using the least squares rincile ;for multirate systems whose states are unmeasurable,the lift2 ed multirate systems are decomosed into some subsystems according to the hierarchical identification rincile,and the hierarchical state2sace model identification methods are develoed,which tae causality constraints into consideration. The simulation results indi2 cate that the roosed algorithms are effective. Key words : multirate systems ;state2sace models ;hierarchical identification ;arameter estimation 1 1 S c, H T1 T 1, S T2 T 2. T 1 T 2,T 1 = T 2 = h,,, 1, u ( t), y( t), v ( t),. [1 ].. 1, 1 ( T 1 = T 2 = h) T 1 T 2, (Dual2Rate System) [2 ] ;, ( ), (Multirate System).. [3,4 ]., ( [5 ] ),,,,., [61 ], [11 ], [5,12 ].,,,,,.,,[13,14 ].,,. [15 ].,. [16,17 ].,,. : [182 ], :232921 ; :242426 : (No. 67429)
9 : 1415,. 2,, h., ( 2 ), 2 H T S T, { 1, 2,, }, 3 4, t = T t i ( =,1,2, ), i = 1,2,,, t i = 1 2 i (t = ), T = 1 2 = t. H T [ T, ( 1) T ], u t = T t i ( i = 1,2,, ),, H T u ( t) =, u ( T t 1 ), u ( T t ), T Φ t < T t 1, T t 1 Φ t < T t 2, T t Φ t < ( 1) T., t = T t i ( i = 1,2,, )., { 1, 2,, }. 3 4.,: (1), 3,(2), 4. 4, T,.,.,u y., [5,11,15 ],, u y (),T. :,, { u ( T t i ), y ( T t i ) }, =,1,2,, ( i =,1,2,, ),,,. 3 : (1) (3),(2) (4).,2 S c, S c : gx ( t) = A c x ( t) B c u ( t) y( t) = Cx ( t) Du ( t) v( t), x ( t) R n, u ( t) R r, y ( t) R r, v ( t) R r, A c, B c, C, D. 311 t =, t 1 = t 1 = 1, t 2 = t 1 2,, t = t = T = 1 2, u ( t) [ T t i, T t i 1 ], i =,1,2,,, u ( t) = u ( T t i ), T t i Φ t < T t i 1, i =, 1, 2,,,. 3,u ( T t i ) ( i =,1, 2,, ) y( T). (1), x ( ( 1) T) = e A c T x ( T) ( 1) T e A c ( ( 1) T - r) B c u ( ) d T ) = e A c T x ( T) B c du ( T t i ) = e A c T x ( T) T t i e A ( ( 1) T - r) c i =1 T t i T- t i i =1 = e A c T x ( T) T- t i e A c t dtb c u ( T t i ) t i i =1 = e A c T x ( T) i =1 t i e A c ( T - t) dtb c u ( T t i ) e A c ( T - t i ) t i = : A T x ( T) [ B 1, B 2,, B ] = : A T x ( T) B T A T = e A c T R n n, A ti = e A c t i R n n B i = i B i = e A c ( T - t i ) i e A c t dtb c R n r e A c t dtb c = A T A t i n ( r) B T = [ B 1, B 2,, B ] R = u ( T t 1 ) u ( T t ) x ( ( 1) T) y( T) = A T B 1 B 2 B C D (1) e A c t dtb c u ( T t i ) B i R n r = : u ( ) R r ( x ( T) v( T)
1416 24 x ( ( 1) T) y( T) = A T B T C T D T x ( T) v( T), C T = C R m n, D T = [ D,,, ] R m ( r). 312 4, u ( T t i ) y( T t i ), i =,1,2,,, x ( T t i ) = e At i x ( T) T t i e A( T t i - r) Bu ( ) d T = A ti x ( T) [ B 1, B 2,, B i ] u ( T t 1 ) u ( T t i ) y( T t i ) = Cx ( T t i ) Du ( T t i ) v( T t i ) = CA ti x ( T) [ CB 1, CB 2,, CB i ] Du ( T t i ) v( T t i ) = : C i x( T) [ D 1, D 2,, D i, D ] v( T t i ) u ( T t 1 ) u ( T t i ) u ( T t 1 ) u ( T t i ) u ( T t i ), C i = CA ti, D i = CB i, i = 1,2,,., x ( 1) y( ) = A T B T C T D T x ( ) u ( ) v ( ) n ( r) x ( ) : = x ( T), B T = [ B 1, B 2,, B ] R C D D 1 D C T = C 1 C 2 R ( m) n, D T = C y( ) : = y( T) = v( ) : = v( T) = 4 y( T) y( T t 1 ) y( T t 2 ) y( T t ) v( T) v( T t 1 ) v( T t 2 ) v( T t ) D 1 D 2 ω R ω D D 1 D 2 D D (2) ( m) ( r) R ( m) () R ( m) (),S c. T (No2Pathological), ( C, A c ) ( C, A T ) [2 ].,,,. ( 2), (2).. 411 u ( ), y ( ) x ( ), (2) ( A T, B T, C T, D T ), Z( ) = x ( 1) y( ) ( ) = x ( ) u ( ) Z( ) = T ( ) E( ) (3) R n m, T = A T B T R ( n m) ( n r) C T D T R n r, E( ) = v( ) R n m, T., [1,22 ] (3), : ^( ) = ^( ) L ( ) [ Z T ( ) - T ( ) ^( ) ] L ( ) = P ( ) ( ) / [1 T ( ) P ( ) ( ) ] P ( ) = [ I - L ( ) T ( ) ] P ( ), I, ^( ), ^ T ( ) = ^A T ( ) ^B T ( ) ^C T ( ) ^D T ( ) v ( ) v v = 1 N N ^ =1 [ y ( ) - ^C T x ( ) - ^D T u ( ) ] [ y ( ) - ^C T x ( ) - ^D T u ( ) ] T, N, [ ^A T, ^B T, ^C T, ^D T ] N, [ A T, B T, C T, D T ]., D T,, D T,.,,,,. 412,,, ( )., [182 ]. : (),,,., S c,. : y ( ) = : y ( ) Y( ), Y( ) = Y 1 ( ) Y 2 ( ) Y ( ), Y i ( ) = y ( T t i )
9 : 1417 v ( ) = : u ( ) = : v ( ) V ( ) U 1 ( ) U 2 ( ), V ( ) = V 1 ( ) V 2 ( ) V ( ), U i ( ) = u ( T t i ), V i ( ) = v ( T t i ) U ( ) ( ), x ( ) = M a gx ( ), M R n n = R,(2) n ( r 1) n, a : = [ - a n, - a n,, - a 1 ] T R n, b : b : = [ D, b 1, b 2,, b n ] T r( n 1) R gx ( 1) y ( ) Y( ) A x = M A T M = R n n B x = M B T = : b 1 = A x B x C y D y C Y D Y gx ( ) u ( ) v ( ) V ( ) 1 1 ω 1 - a n - a n - a n - 2 - a 1 b 2 R n ( r), b i R b n 1 ( r) C y = CM = [1,,,,] R 1 n 1 ( r), D y = [ D,,,,] R C Y = D Y = C 1 C 2 M = : C F 1 F 2 R ( ) n, F D 1 D D 1 D 2 ω R ( ) ( r) (5) ω D D 1 D 2 D D M = (4) : C CA T CA n T gx ( 1) = A x gx ( ) B x u ( ) R n n y ( ) = C y gx ( ) D y u ( ) v ( ) = C y gx ( ) Du ( ) v ( ) Y( ) = C Y gx ( ) D Y u ( ) V ( ) (7) (4),, { u ( ), y ( ) } { u ( ), y ( ) },(6) [ A x, B x, D ]gx ( ), gx^ ( t), (7) (6) { C Y, D Y }. [ A x, B x, D ]gx ( ) gx ( ) = gx 1 ( ) gx 2 ( ) R n gx n ( ) ( ) = [ gx T ( ), u T ( n), u T ( n ), u T ( n - 2),, u T ( ) ] T n ( r 1) n R (6) gx 1 ( 1) = gx 2 ( ) b 1 u ( ) gx 2 ( 1) = gx 3 ( ) b 2 u ( ) gx n ( 1) = gx n ( ) b n u ( ) gx n ( 1) = agx ( ) b n u ( ) (8) y ( ) = gx 1 ( ) Du ( ) v ( ) (9) z, zu ( ) = u ( 1), (8) j z n - j ( j = 1,2,, n), gx 1 ( n) = a T gx ( ) b 1 u ( n ) b 2 u ( n - 2) (9) z n b n u ( ) (1) y ( n) = T ( ) v ( n),( ) gx ( ), gx ( ), : P ( ) ^( ) ^( 1) = ^( ) 1 ^ T ( ) P ( ) ^( ) [ y ( n) - ^ T ( ) ^( ) ] (11) P ( ) = P ( ) - P ( ) ^( ) ^ t ( ) P ( ) 1 ^ T ( ) P ( ) ^( ) gx^ ( 1) = ^A x ( ) gx^ ( ) ^B x ( ) u ( ) ( ) C T y (12) [ y ( ) - C y gx^ ( ) - Du ( ) ] (13) ^( ) = [ gx^ T ( ), u T ( n), u T ( n ), ^( ) = u T ( n - 2),, u T ( ) ] T (14) ^a ( ) ^b ( ), ^a ( ) = [ - ^a n ( ), - ^a n ( ),, - ^a 1 ( ) ] T, ^b ( ) = [ ^D ( ), ^b 1 ( ), ^b 2 ( ),, ^b n ( ) ] T ^A x ( ) = 1 1 ω 1 - ^a n ( ) - ^a n ( ) - ^a n - 2 ( ) - ^a 1 ( ),
1418 24 ^B x ( ) = ^b 1 ( ) ^b 2 ( ) ^b n ( ) (15), ^( ), gx^ ( ) gx ( ), [ ^A x ( ), ^B x ( ) ][ A x, B x ], [ ^a i ( ), ^b i ( ) ], ( ) Ε, ( ) =, 2 ( ) <. P () = I,, = 1 6, ^() = ^,^() = 1-6 1 n ( n = dim,1 n 1 n ), gx^ () ^(). 2 [ C Y, D Y ] (7) [ C Y, D Y ],, [ C Y, D Y ]., D Y, D Y 3 : (1) D Y, ; (2) D Y ; (3) D Y i D i, i = 1,2,,, 1 D. 1 D Y, c = [ C Y, D Y ] R ( ) ( n r), c ( ) = gx ( ) u ( ) R n r (7) Y( ) = c c ( ) V ( ), c. : ^ c ( ) = ^ c ( j) ^ T c ( j) ^ c ( j) Y T ( j) (16) gx^ ( ) ^ c ( ) = R n r (17) u ( ) 2 D Y (7) ( ) : Y i ( ) = T di ( ) di V i ( ), i = 1,2,, di = [ F i, D 1, D 2,, D i, D ] T R n ( i 1) r di ( ) = gx ( ) U 1 ( ) U 2 ( ) U i 1 ( ) R n ( i 1) r di ^ di ( ) = ^ di ( ) = ^ di ( j) ^ T di ( j) gx ( ) U 1 ( ) U 2 ( ) U i 1 ( ) ^ di ( j) Y i ( j) (18) R n ( i 1) r, i = 1,2,, (19) Y i ( ) - Y i ( ) - 3 D Y i D i ( i = 1,2,, ) (7) ( ) : i D l U l ( ) - DU i 1 ( ) = F i gx ( ) D i U i ( ) V i ( ) l =1 i D l U l ( ) - DU i 1 ( ) = T ei ( ) ei V i ( ), l =1 i = 1,2,, ei = [ F i, D i ] T R n gx ( ) r, ei ( ) = U i ( ) ei ^ ei ( ) = ^ ei ( ) = ^ ^ ei ( j) ^ T ei ( j) i ei ( j) Y i ( ) - gx^ ( ) U i ( ) l =1 R n r ^D l ( ) U l ( ) - ^D ( ) u ( ) (2) R n r, i = 1,2,, (21) (11) (21) (2). Boot2 stra [21 ],,. 5 1 S c ( s) = 12s 2 1s 1 = 3, 1 = 1s, 2 = 2s, 3 = 3 s,t 1 = 1s, t 2 = 3 s, t 3 = T = 1 2 3 = 6 s. x ( ( 1) T) = 1515 3742 41492 187522 15412 1132763 2162137 4122621 6162568 411219 y ( T) = [,1833] x ( T) v ( T) x ( KT) u ( T 1) u ( T 3), { v ( T) } 2 v = 11 2. ( ) = 1/ ( n ), x ( ( 1) T) = b 11 b 12 b 13 b 21 b 22 b 23 1 - a 2 - a 1 x ( T) y ( T) = [1, ] x ( T) v ( T) u ( T 1) u ( T 3), 1, = ^- / ( x 2 = x T x),, ^.
9 : 1419 5. x ( ( 1) T) = x ( KT) u ( T 1) 1 62482 1139511 13522 15521 13435 1517 198 112815 u ( T 3) 1 ( 2 v = 11 2 ) - a 2 - a 1 b 11 b 12 b 13 b 21 b 22 b 23 ( %) 1 117819 158898 13191 16912 11392 11766 111957 17335 73151195 2 12884 193657 13227 15493 1155 13478 1988 19492 42193566 3 41296 112653 13713 16371 11361 1436 19964 1119 14191779 5 6345 11455 13753 15493 11545 151 19746 11588 311368 1 65989 1142694 13243 15511 12575 1511 1128 11138 419961 2 62623 1139365 13298 15783 12975 15123 19978 112114 1183767 y ( T) = [1, ] x ( T) 3 62482 1139511 13481 15699 1335 1569 1149 11231 1179759 (22) 6653 1137627 13522 15521 13435 1517 198 112815 S c ( s) (22) 6. 1 5,,,. 6, 5 ( 2 v = 11 2 ) S c,. 6, 6 ( 2 v = 11 2 ),,.,,,. : [ 1 ],. [M]. :,1988. [ 2 ] CHEN T,FRANCIS B. Otimal Samled data Control Systems[ M]. London :Sringer2Verlag,1995. [ 3 ] OHSHIMA M,HASHIMOTO I,et al. Multirate multivariable model re2 dictive control and its alication to a semi2commercial olymerization reactor[ A ]. Proceedings of the 1992 American Control Conference [ C]. Chicargo,USA : Evanston,IL. 1992,2. 1576 581. [ 4 ] GUDI R D,SHAH SL,GRAYM R. Multirate state and arameter esti2 mation in an antibiotic fermentation with delayed measurements [J ]. Biotechnology and Bioengineering,1994,44 (11) :1271 278. [ 5 ] LI D,SHAH S L,Chen T,Qi K Z. Alication of dual2rate modeling to CCR octane quality inferential control[j ]. IEEE Trans Control Systems Technology,23,11 (1) :43-51. [ 6 ] CHEN T. On stability robustness of a dual2rate control system [J ]. IEEE Trans Automatic Control,1994,39 (7) :1139 152. [ 7 ] CHEN T,QIU L. H design of general multirate samled2data control systems[j ]. Automatica,1994,3 (7) :1139 152. [ 8 ] QIU L,CHEN T. H 2 otimal design of multirate samled2data systems [J ]. IEEE Trans Automatic Control,1994,39 (12) :256-2511. [ 9 ] QIU L, CHEN T. Multirate samled2data systems : all H subotimal controllers and the minimum entroy controller [J ]. IEEE Trans Aut2 moatic Control,1999,44 (3) :537-55. [ 1 ] TANGIRALA A K,LI D,PATWARDHAN R S,SHAH S L,CHEN T. Rile2free conditions for lifted multirate control systems[j ]. Automati2 ca,21,37 (1) :1637 645. [ 11 ] SHENGJ,CHEN T,SHAH S L. Generalized redictive control for non2 uniformly samled systems [J ]. Journal of Process Control,22,12 : 875-885. [ 12 ] LI D,SHAH SL,CHEN T. Analysis of dual2rate inferential control sys2 tems[j ]. Automatica,22,38 (6) :153 59. [ 13 ] LU W P,FISHER G. Outut estimation with multi2rate samling[j ]. International Journal of Control,1988,48 (1) :149 6. [ 14 ] Lu W P, FISHER G. Least2squares outut estimation with multirate samling[j ]. International Journal of Control,21,74 (7) :68-689. [ 15 ] LI D,SHAH S L,CHEN T. Identification of fast2rate models from mul2 tirate data [J ]. International Journal of Control,21,74 (7) : 68-689. [ 16 ] DING F,CHEN T. Parameter identification and intersamle outut esti2 mation of a Class of Dual2Rate Systems[A]. Proceedings of 43nd IEEE Conference on Decision and Control [ C]. Hawaii,USA : IEEE,Decem2 ber 9212,23. 5555-556. [ 17 ] DING F,CHEN T. Parameter estimation for dual2rate systems with fi2 nite measurement data [J ]. Dynamics of Continuous,Discrete and Im2 ulsive Systems,Series B :Alications &Algorithms,24,11 (1) :11 21. [ 18 ],. [J ].,1999,25 (5) : 647-654. [ 19 ] DING Feng, YANG Jiaben,XU Yongmao. Convergence of hierarchical stochastic gradient identification for transfer function matrix model[j ]. Control Theory and Alication,21,18 (6) :949-953. [ 2 ],,,.
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