Copyright '6 IFAC Linear Time Delay Systems, Ancona, Italy. 2000 IFAC c: 0 C> Publications www.elsevier.com..locateiifac IDENTIFIABILITY OF LINEAR TIME DELAY SYSTEMS L. Belkoura' Y. Orlov ·· J.P. Richard · • LAIL . Laboratory of control and Indu strial inforrnatic of Lille. Villenew 'e d A scq. Fran ce . lotfi. belkoura rg, un il'-lille l.fr •• CICESE Research Ce nt er. El ec t ro nics and Telecommun icatiun Departm ent . San Diego. M exico Ab s tr act: Thi s paper g iYes sufficie nt co ndition for of lin ea r time In case of comme nsurat e thi s condition reduces to th e cl assica l of weak controllability. Th e proof is provid ed within th e fr amework of dis tribution theory, and gives informa tion con cerning the way to cons tru ct a sufficient rich input law allowing the identifi ca tion . Copyright 2000 IF AC Ke yword s: Time lag Identifia Dis tributions 1. [',TRODUCTIO:\ works ha w been devoted to th e anah'- sis and th e contr ol of lin ea r time (see for instance (Ko lman O\'skii and 1999: Richa rd . 1998) a nd th e numerous references herein ). HO\\' ewL icl entifica tion problem has not receiw d an equiyalent interest. In s pit e of re- ce nt \\'orks Ve rdum Lun el (1997). S. :\aka- giri. et 01.( 1995 ). with a spec tr al a pproach and res trict ed to th e a ut onomous ca se. a nd th e a uthor s in th e only "calar case or in th e single case (Belkoura et al.. 1998: Belko ur a et af.. 2000 ). the qu estion of identifiability is still open for th e nona uton omous case. Thi s remain" a ll obsta cl e for tlw th eor\ ' a nd impleme ntation of ada pt a ti w co ntr o ll ers. for installce. Thi s paper proposes a sufficient co ndition for the identifia- bility of lin ea r time delay \\'ith adequ ate illj)ut. In ti lt' case of systems "'i t h co mmensurate thi s co nditi on turn s to be ra ther sim- ple. since it re du ces to th e classica l. s tructur al prop e rty of wea k co ntr ollability (see for instance (Olbrot. 1972: Senallle. 199-1 )) . 205 2. PROBLE?--I 'Ye sha ll consider a S\'stem gowrn ed th e functi onal differe nti al equ atio n: .d t) = ...10.1 (t ) + AI. T (t - T I ) + .. .+ A ,. .r (t - T l' ) -i- B o u (t ) + B1U (t- hd -1- .. ·+ B[ 1I {t - h[ ). (1 ) "'here 0 < '1 < '2 < .. < '1" 0 < hI < "2 < .. < h i. :r( .) E ]R n. u(.) E !RP . s ubj ect to th e initial co nditi on .1'( 11 ) = ,.::( 11 ). - T r S 11 S O. Th e model used for th e idelltifia bility of unknO\\'I1 pa rameters in (1) is giw n by: --- ...-.. ....... ..-... ....... .. . - Bqu (t- hq). (2) \\'ith initial co nditi on i( l1 ) = :;::( 11 ). -T 11l S 11 S O. t he (.) will stand for th e co rr espo ndin g mode l. Th e identifiability pr oblem of th e coefficient ma tric es alld th e time is s tat ed as fo llO\\' :