Int. J. Computers & Electrical Engineering, vol. 30, no. 1, pp , AN OBSERVER DESIGN PROCEDURE FOR A CLASS OF NONLINEAR TIME-DELAY SYSTEMS

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1 In. J. Compues & Elecical Engineeing, vol. 3, no., pp. 6-7, 3. AN OBSERVER DESIGN PROCEDURE FOR A CLASS OF NONLINEAR TIME-DELAY SYSTEMS * ** H. Tinh *, M. Aleen ** an S. Nahavani * School o Engineeing an Technology, Deakin Univesiy, Geelong, 37, Ausalia. Depamen o Elecical an Eleconic Engineeing, The Univesiy o Melboune, Pakville, Vicoia, 3, Ausalia. Absac: This pape consies a class o unceain, nonlinea ieenial sae elaye conol sysems an pesens a euce-oe obseve esign poceue o asympoically esimae any veco sae uncionals. The meho popose involves ecomposiion o he elaye poion o he sysem ino wo pas: a mache an mismache pa. Povie ha he ank o he mismache pa is less han he numbe o he oupus, a euceoe linea uncional obseve, wih any pescibe sabiliy magin, can be consuce by using a simple poceue. A numeical example is given o illusae he new esign poceue an is eaues. Key Wos: Time-Delay Sysems, Linea Funcional Obseves, Reuce-Oe Sae Obseves, Unceain Sysems.. Inoucion The sae esimaion poblem o ime-elay sysems has been he subjec o consieable eseach aciviies ove he yeas. This is ue o he ac ha ime elays ae inheen in many eal physical sysems, such as mechanical an chemical pocesses, powe an wae isibuion newoks, ai polluion sysems, economeic sysems, ec. Seveal esign poceues have been popose o esign asympoic sae obseves o ime-elay sysems (see o example [] o a bie suvey an he eeences cie heein). In mos cases, epoe esign mehos ([]-[3]) involve he compuaion o he eigenvalues o he ime-elay sysems. Ohe esign mehos eihe assume he socalle maching-coniion on he elaye sae maix o he esuls ae applicable o sysems wih a small ime elay ([4]-[5]). In [6], a inie-oe memoyless sae obseve o ime-elay sysems is popose. As commene in [7], he obseve in [6], howeve, oes no povie acking o inpu signals ha o no convege o zeo. On he ohe han, he use o linea uncional obseves o asympoically geneae conol signals an/o o esimae a subse o saes has pove o be a pacical alenaive o oupu eeback conol. A numbe o esign poceues have been epoe in he lieaue [8]-[] o non-elay sysems. Recenly, he poblem o esigning low-oe linea uncional sae obseves o ime-elay sysems was is suie an a simple obseve esign poceue, which involves solving only a se o linea algebaic equaions, was popose in [3]. So a, epoe obseve esign mehos []-[3] have no eal wih ime-elay sysems ha may conain unceainies an/o nonlineaiies. The esign meho popose in his pape is capable o ealing wih a class o unceain an/o nonlinea imeelay sysems. The poceue oes no involve he compuaion o he eigenvalues o he elaye sysem elaive o some igh hal-plane, as in [-3,6]. Also, unlike ha in [4-5], hee is no esicion impose on he size o he ime elay an he elaye em o he sysem is no assume o saisy maching coniions. Insea, he elaye em is ecompose ino wo poions: a mache an mismache poion. The mismache poion may conain any unceainies an/o nonlineaiies an is eae as an unknown inpu (o isubance) o he sysem [4]. Subjec o he well-known coniion egaing he numbe o unknown inpus an oupus [4], an unknowninpu-ee euce-oe ime-elay sysem can be eive. Then a euce-oe linea uncional obseve, wih any pescibe sabiliy magin, can be sysemaically consuce base on he euce-oe sysem moel.. Sysem Descipion an Peliminaies Consie a class o ime-elay sysems moelle by he ollowing equaions x () = Ax () + Bu () + (, x ( τ )); (a) x () = g (); τ (b) y () = Cx () (c) τ is a posiive eal numbe epesening he ime elay in he sae an g() is a coninuous uncion on he n m ineval [ τ,]. The vecos x() R, u () R an y () R ae he sae, inpu an oupu, especively. n n Maices A R n m, B R n an C R ae consan. Wihou loss o genealiy, i is assume ha C has ull n ank, i.e. ank( C) =. The elaye uncion (, x ( τ )) R may be consue o epesen he unceainies an/o nonlineaiies o he sysem. I is assume ha (, x( τ )) can be ecompose ino a mache poion an a mismache poion, as ollows 6

2 In. J. Compues & Elecical Engineeing, vol. 3, no., pp. 6-7, 3. (, x ( τ )) = LCx ( τ) + Dη(, x ( τ)) () L R n n q, D R q, an η(, x ( τ)) R. Fuhemoe, D is assume o have a ull column ank, q. The em Dη (, x( τ )) can be use o escibe aiive isubance as well as many ieen kins o moelling unceainies (ee o [5]-[6] o uhe eails). In his pape, i will be eae as an unknown inpu o he sysem (). Now, he poblem o be aesse in his pape is ha o esigning a euce-oe obseve o imension p o geneae any equie l veco sae uncionals o he om x () = Fx() (3) l n F R. This ype o obseve can be use o iecly ealise a ull sae eeback conol law an/o o esimae any subse o he sae veco x() by leing maix F in equaion (3) o compise hose ows o I n ha coespon o he sae vaiables o be esimae. 3. Reuce-oe Time-Delay Moel In his secion, a euce-oe ime-elay moel ee o he unknown inpu is eive o sysem (). The euce-oe moel is hen use o eive a euce-oe linea uncional obseve wih any pescibe sabiliy magin, making use o he Unknown Inpu Obseves heoy [4]. Le us now consie he ollowing ansomaion x() = T x() (4a) n n T R is a non-singula maix eine as T = [ N D] (4b) an N R n ( n q) is any abiaily ull column ank maix. Subsiue () in () an hen use (4a) on he esulan, he ollowing equivalen sysem is obaine x () = Ax () + Bu () + LCx ( τ) + Dη(, x ( τ)); (5a) x () = g (); τ (5b) y () = Cx () (5c) x () = F x() (5) x () x (), () n q x R, () q x R, = x () A A A = T AT =, A A B B = T B =, B L C = CT = [ CN CD] = C C, L = T L =, D = T D = L I an F = FT = F F q (5e) Fom (5), he ollowing unknown-inpu-ee sysem is obaine x () = A x () + A x () + Bu() + LC x ( τ) + LC x ( τ); (6a) x () = g (); τ (6b) y () = C C x () (6c) x () = F F x() (6) Assuming ha q an maix C = CD has ull column ank q, hen hee exiss a non-singula maix U R > 6

3 In. J. Compues & Elecical Engineeing, vol. 3, no., pp. 6-7, 3. U U = C Q an U = (7) U gives an Q R is any abiaily ull column ank maix, ( q ) U R q ( q) an U R. Using (7) ino (6c) Uy () = UCx () + x() (o x () = Uy () UCx () ) (8a) U y() = U C x () (8b) Subsiuing equaion (8a) ino (6) an ae some manipulaions, he ollowing euce-oe ime-elay sysem is obaine x () = Ax () + Bu() + E y() + E y( τ) + L ( I CU ) C x ( τ); (9a) x () = g (); τ (9b) y () = C x() (9c) x () = F x () + E y() (9) 3 3 A = A AUC, E = A U, E = LC U, C = C U, F = F FU C, E = FU an ( q) y () = Uy () R (9e) Consie maix L ( I C U ) C in equaion (9a) an ecall om (7) ha UC = I q. Then i is clea ha ank{( I C U )} = ( q). Accoingly, hee always exiss a ull column ank maix K R ( n q) ( q) such ha L ( I C U ) = KU () U (SVD). Theeoe has been eine in (7). Noe ha maix K can be easily eive by using singula value ecomposiion L ( I C U ) C x ( τ ) can now be expesse as L( I CU) Cx( τ ) = KUCx( τ ) = KCx ( τ) = Ky ( τ) = KU y ( τ) () As a esul, he ollowing (n-q)-h euce-oe ime-elay sysem is obaine x () = Ax () + Bu() + E y() + E y( τ ); (a) 4 x () = g (); τ (b) y () = C x() (c) x () = F x () + E y() () 3 E = ( E + KU ) R 4 ( n q). Base on he above euce-oe moel, a linea uncional obseve o an/o a euce-oe Luenbege ype obseve o x () can now be easily synhesise. This is consiee in he nex secion. 4. Obseve Design This secion pesens wo ypes o obseves. One is a p-h euce-oe linea uncional obseve o asympoically esimae he veco sae uncionals x (). The ohe is a (n-q)-h euce-oe Luenbege ype obseve o esimae he enie sae veco x(). 4. Linea Funcional Obseve x () 63

4 In. J. Compues & Elecical Engineeing, vol. 3, no., pp. 6-7, 3. Le us now ecompose, abiaily, maix F as ollows F = KT + WC (3) l p K R p ( n q) T R l ( q) W R.,, Subsiuing equaion (3) ino () gives x () = F x () + E y() = KT x () + W y() + E y() 3 3 = K z () + ( WU + E) y () (4a) 3 p z() = Tx () R (4b) Consie he ollowing ieenial-elay equaion z () = Ez () + TBu () + TEy () + T ( E + KU) y ( τ ) + Gy (); (5) wih iniial coniions z () = h (), τ (6) p p E R ( ), G R p q an h() is a coninuous uncion on he inicae close ineval. i Deine an eo e () as e () = z () Tx () (7) Taking he eivaive o (7) gives E{() z Tx ()} ( GC T A ET ) x () = Ee() + ( GC T A + ET ) x () (8) e () = z () Tx () = + + I is clea ha i GC T A ET (9) + = hen equaion (8) becomes e () = Ee (). (a) wih iniial coniion e () = h () Tg (); τ. (b) The above evelopmen implies ha a ynamic elay sysem (5) can be consuce o asympoically geneae linea uncions o he sae (3), povie ha he ollowing coniions hol E = sable GC T A + ET = () F = KT + WC () 64

5 In. J. Compues & Elecical Engineeing, vol. 3, no., pp. 6-7, 3. Une he assumpion ha he pai ( A, C ) is obsevable an maix E is chosen o be sable, maices T, W, K, G can be solve using he poceues epoe in []-[] (ee o [] o a simple an MATLAB base poceue). The esign o a linea muli-uncional obseve is hus complee. 4. A Reuce-Oe Luenbege Obseve Povie ha he pai ( A, C ) is obsevable o eecable, a (n-q)-h Luenbege ype obseve can be eive o esimae he sae veco x() o sysem (). Le us inouce he ollowing ynamic ime-elay sysem v () = ( A LC ) v () + Bu () + ( E+ LU ) y( ) + ( E + KU) y( τ ); (3a) v () = m (); τ (3b) Deine an eo e () as e () = v () x() (4) Taking he eivaive o (4) an subsiuing equaions (3a) an (a) ino he esulan gives e () = v () x () = ( A LC ) e( ) (5) Povie ha maix we have ( n q) ( q) L R is chosen such ha A LC ( ) is sable, v () x ˆ () as. As a esul () ˆ v x ˆ( ) = Tx () = T Uy () UCv () (6) This complees he esign o (n-q)-h euce-oe obseve o esimae he enie sae o sysem (). (Noe ha q he unknown inpus η(, x ( τ)) R may also be esimae base on he euce-oe obseve (3). This can be one in a simila way o he wok epoe in [4]). Remak : The esuls o his pape ae applicable o a class o ime-elay sysems he ank o he mismache poion q is less han he numbe o oupus (i.e. q < ). This assumpion is less esicive han he maching coniion assumpion commonly mae in he lieaue (see [4] an he eeences cie heein). One o he novelies o his pape is he eucion o he ime-elay sysem () ino a lowe-oe sysem () o which available obseve esign heoy can be use. Remak : Noe ha o he case (, x( τ )) = A x( τ ) is a linea uncion [-7,3], he ecomposiion () can be easily peome by some simple maix manipulaions, as shown in he Appenix A. I is also shown in he Appenix ha o he case he numbe o oupus is moe han hal o he numbe o he saes (i.e. >.5n), he coniion q < is always saisie. 5. Numeical Example In his secion, a numeical example is given o illusae he esign poceue an is eaues. Consie he ollowing sysem x( ) sin( x ( )) x( τ ) +.5 x3( ) x ( ) = x ( ) + u ( ) + x( ) + x3( ) 3 x ( ) + x ( ) 3 y() = x(), τ, Fo illusaive pupose, le us now esign a euce-oe obseve o esimae he sae x (), ie. [ ] x () = Fx () = x () The nonlinea uncion can be ecompose ino he om (), 65

6 In. J. Compues & Elecical Engineeing, vol. 3, no., pp. 6-7, 3..5 L =, 3 D = an η(, x ( τ)) = sin{ x()} x( τ) R Base on he poceue given in Secion 3 o he pape, a euce-oe sysem o he om () is easily eive, =, A B =, E =, E 4 3 =, [ ] F = [ ] an [ ] C =, E 3 =. (Noe ha in eiving he above euce-oe moel, he maices N an Q eine in (4b) an (7), especively, ae chosen abiaily as: N = an = ). The pai ( A, C ) is obsevable. Base on he evelopmen in Secion 4 o his pape, an by choosing maix E as E = 5, he ollowing is-oe obseve o he sae x () is easily eive [ ] [ ] [ ] xˆ () = z() + 5 y() z ( ) = 5 z ( ) + u ( ) + y ( ) y ( ), The ollowing simulaion suy was caie ou wih he conol inpu signal is as shown in igue () an he elay is in he nonlinea uncion ( sin( x ( )) x( τ )) τ =. Figue () shows he simulae esponses o x () an xˆ (). Figue (3) shows he eo sae x () = x () xˆ (). The iniial coniions o he sysem an obseve wee aken o be x () = an z () = 5, [,]. Figues ()-(3) clealy show ha he sae esimaion eo conveges o zeo. Exensive compue simulaions have been conuce o ieen nonlinea uncions an ieen values o ime-elay, τ,. In all o he ese cases, he sae esimaion eo convege o zeo. 6. Conclusion In his pape a euce-oe linea uncional obseve esign poceue o a class o unceain an/o nonlinea ieenial sae elaye conol sysems has been popose. The meho involves he ecomposiion o he unceainies an/o nonlinea uncion ino wo pas: a mache an mismache pa. Povie ha he ank o he mismache pa is less han ha o he numbe o he oupus, a euce-oe linea ime-elay moel can be eive. Base on he eive moel, linea uncional obseve an/o euce-oe Luenbege sae obseve, wih any pescibe sabiliy magin, can be consuce by using a simple an sysemaic poceue. A numeical example has been given o illusae he new esign poceue an is main eaues. Reeences [] A.E. Peason an Y.A. Fiagbezi, An obseve o ime lag sysems, IEEE Tans. Auoma. Con., 34 (8), 989, [] K.P.M. Bha an H.N. Koivo, An obseve heoy o ime-elay sysems, IEEE Tans. Auoma. Con.,, 976, [3] D. Salamon, Obseves an ualiy beween obsevaion an sae eeback o ime elay sysems, IEEE Tans. Auoma. Con., 5, 98, [4] G.A. Hewe an G.J. Nazao, Obseve heoy o elaye ieenial equaions, In. J. Conol, 8 (), 973, -7. [5] A. Tonambe, Simple obseve-base conol law o ime lag sysems, In. J. Sysems Sci., 3 (9), 99, [6] J. Leyva-Ramos an A.E. Peason, An asympoic moal obseve o linea auonomous ime lag sysems, IEEE Tans. Auoma. Con., 4 (7), 995, [7] H. Tinh an M. Aleen, Commens on An asympoic moal obseve o linea auonomous ime lag sysems, IEEE Tans. Auoma. Con., 4, 997, u () 66

7 In. J. Compues & Elecical Engineeing, vol. 3, no., pp. 6-7, 3. [8] F.W. Faiman an R.D. Gupa, Design o muli-uncional euce-oe obseve, In. J. Sysems Science,, 98, [9] J. O'Reilly, Obseves o Linea Sysems (Acaemic Pess, Lonon, 983). [] C.C. Tsui, 986, On he oe eucion o linea uncional obseves, IEEE Tans. Auoma. Con., 3, 986, [] M. Aleen an H. Tinh, Reuce-oe linea uncional obseve o linea sysems, IEE Poceeings, Pa D, Conol Theoy an Applicaions, 46, 999, [] L. Lin, M. Aleen an H. Tinh, A ecenalise obseve scheme o ineconnece sysems wih highoe ineconnecions, Fouh Inenaional Coneence on Conol, Auomaion, Roboic an Vision, Singapoe, vol. 3, pp , 996. [3] H. Tinh, Linea uncional sae obseve o ime-elay sysems, In. J. Conol, 7(8), 999, [4] M. Hou an P.C. Mulle, Design o obseves o linea sysems wih unknown inpus, IEEE Tans. Auoma. Con., 37, 99, [5] M. Hou an P.C. Mulle, Faul eecion an isolaion obseves, In. J. Conol, 6(5), 994, [6] J. Chen, R.J. Paon an H.Y. Zhang, Design o unknown inpu obseves an obus aul eecion iles, In. J. Conol, 63(), 996, Appenix A: Maix Decomposiion C n n Une he assumpion ha ank( C) =, we can consuc a non-singula maix M as M = R, H ( n ) n H R. Then C an A can be ansome ino Cnew = CM [ I ] = (A) A = MA M = L C + A,, new new new new (A) Fom (A)-(A), n n n new can be chosen o compise he is columns o A new,. Maix A new, R is hen L R easily obaine wih is is o Anew,. Accoingly, columns being all zeos an is las (n ) columns being he las ( n ) columns A can be ecompose ino A = LC + A = n ; L R (A3) A = M A M ; A n n R (A4) L M L new, new I easy o see, om he above evelopmen, ha he ank o Thus, hee exiss a ull column ank maix n q D R such ha A saisies he coniion ank( A ) ( n ) q. Ax ( τ ) = DVx ( τ) = Dw ( τ ) (A5) w ( τ) = Vx ( τ) R q is eine as an unknown inpu. I is also clea om above ha o saisy he coniion q< equies ( n ) < (i.e. >.5n). This means ha o he case he numbe o oupus is moe han hal o he numbe o he saes hen he coniion q < is always saisie. 67

8 In. J. Compues & Elecical Engineeing, vol. 3, no., pp. 6-7, Figue (): Inpu signal u () F Figue (): Responses o x () (soli line) an xˆ () (ashe line) Figue (3): Response o he eo sae x () 68