Identificatio n of No n2uniformly Perio dically Sa mpled Multirate Syste ms

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1 ACTA ELECTRONICA SINICA Vol. 32 No. 9 Se. 24 1,2, 2, 1 (11, 184 ;21, T6G 2V4) :,.,,;,,,,.,. : ; ; ; : TP273 : A : (24) 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 ], : ; : : (No )

2 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)

3 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) , 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) 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 )

4 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 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) = Â 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) â ( ) ^b ( ), â ( ) = [ - â n ( ), - â n ( ),, - â 1 ( ) ] T, ^b ( ) = [ ^D ( ), ^b 1 ( ), ^b 2 ( ),, ^b n ( ) ] T Â x ( ) = 1 1 ω 1 - â n ( ) - â n ( ) - â n - 2 ( ) - â 1 ( ),

5 ^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, ^() = ^,^() = 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 = = 6 s. x ( ( 1) T) = y ( T) = [,1833] x ( T) v ( T) x ( KT) u ( T 1) u ( T 3), { v ( T) } 2 v = ( ) = 1/ ( n ), x ( ( 1) T) = b 11 b 12 b 13 b 21 b 22 b 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),, ^.

6 9 : x ( ( 1) T) = x ( KT) u ( T 1) u ( T 3) 1 ( 2 v = 11 2 ) - a 2 - a 1 b 11 b 12 b 13 b 21 b 22 b 23 ( %) y ( T) = [1, ] x ( T) (22) S c ( s) (22) ,,,. 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, [ 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) : [ 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) : [ 6 ] CHEN T. On stability robustness of a dual2rate control system [J ]. IEEE Trans Automatic Control,1994,39 (7) : [ 7 ] CHEN T,QIU L. H design of general multirate samled2data control systems[j ]. Automatica,1994,3 (7) : [ 8 ] QIU L,CHEN T. H 2 otimal design of multirate samled2data systems [J ]. IEEE Trans Automatic Control,1994,39 (12) : [ 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) : [ 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) : [ 11 ] SHENGJ,CHEN T,SHAH S L. Generalized redictive control for non2 uniformly samled systems [J ]. Journal of Process Control,22,12 : [ 12 ] LI D,SHAH SL,CHEN T. Analysis of dual2rate inferential control sys2 tems[j ]. Automatica,22,38 (6) : [ 13 ] LU W P,FISHER G. Outut estimation with multi2rate samling[j ]. International Journal of Control,1988,48 (1) : [ 14 ] Lu W P, FISHER G. Least2squares outut estimation with multirate samling[j ]. International Journal of Control,21,74 (7) : [ 15 ] LI D,SHAH S L,CHEN T. Identification of fast2rate models from mul2 tirate data [J ]. International Journal of Control,21,74 (7) : [ 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, [ 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) : [ 18 ],. [J ].,1999,25 (5) : [ 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) : [ 2 ],,,.

7 [J ].,22,17 (1) :6. [ 21 ],,. [J ].,24,3. [ 22 ],. [ M]. :, 22. :,1963 3,1984, 4, ( ), 1994,22 6 (Al2 berta),, (,22), 1 7,.,1961 1,1984, , (Calgary),(Alberta). B A Francis Otimal Samled2Data Control Systems ( Sringer, 1995).,,,.,1945 8,197,, CIMS.

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24 5 Vol. 24 No. 5 Cont rol an d Decision 2009 5 May 2009 : 100120920 (2009) 0520738205 1a, 2, 1b (1. a., b., 239012 ; 2., 230039) :,,.,.,. : ; ; ; : TP181 : A Quick algorithm f or computing core attribute