Minimal order perfect functional observers for singular linear systems
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1 Miimal ode efect fuctioal obseves fo sigula liea systems Tadeusz aczoek Istitute of Cotol Idustial lectoics Wasaw Uivesity of Techology, -66 Waszawa, oszykowa 75, POLAND Abstact. A ew method fo desigig a miimal ode efect fuctioal obseves fo sigula cotiuous-time liea systems is oosed. Necessay sufficiet coditios fo the existece of the miimal ode efect fuctioal obseve ae established. A ocedue fo comutatio of matices of the obseves is deived illustated by a umeical examle. ey-wods: - Desig, existece, miimal ode, efect fuctioal obseve, sigula, liea, system. Itoductio The desig of fuctioal obseves that ecostuct (estimate) diectly a liea fuctio x ( is kow) of the state vecto x of liea systems is a vey imotat oblem. Because the fuctioal obseves do ot eed to ecostuct all state vaiables thei odes ca be sigificatly less tha those of state obseves. This oblem was fist oosed by Luebege i [3]. The desig of obseves fo liea systems has bee cosideed i may aes books [-4,6,-8]. Recetly ew cocets of efect obseves fuctioal obseves fo sigula stad liea systems have bee oosed [5,9]. The cocet of efect obseve has bee exteded fo sigula -D liea systems i [7] the cocet of efect fuctioal obseve fo sigula cotiuous-time liea systems has bee exteded i [8]. The oblem of the ode eductio of fuctioal obseves of liea systems has bee extesively ivestigated by Tsui i [6-8]. I [6] a desig method of miimal ode fuctioal obseves fo stad liea cotiuous-time systems has bee oosed with ue boud of the ode mi (, v +L+ vm) mi (,( v ) + L + ( vm )) fo stictly oe oe obseves, esectively, whee, m, v i ( i,..., ) ae the system (lat) ode, umbe of iuts, umbe of oututs obsevability idexes, esectively. The mai subject of this ae is to eset a ew desig method of miimal ode efect fuctioal obseves fo sigula cotiuoustime liea systems. To the best autho s kowledge the desig of efect fuctioal obseves fo sigula liea systems has ot bee cosideed yet. Necessay sufficiet coditios fo the existece of the efect fuctioal obseves fo sigula liea systems will be established a ocedue fo comutatio of matices of the obseves will be deived. Poblem Fomulatio Let m R be the set of eal matices R R. Coside the sigula cotiuoustime liea system x& Ax + Bu (a) y Cx (b) m whee x, u y ae the semistate, iut outut vectos, esectively m, A R, B, C, det.
2 It is assumed that det[ s A] fo some s C () (the field of comlex umbes) ak C. We ae lookig fo a miimal ode efect fuctioal obseve descibed by the equatios z& Fz + Gu + Hy, z, (3a) R det, q w Lz + My, w, L, M (3b) that ecostuct exactly the give liea fuctio x, is give (4) i.e. w ( t) x( t) fo all t > (5) The oblem ca be stated as follows. Give, A, B, C, fid, F, G, H, L M of (3) such that (5) holds. Solvability coditios fo the oblem will be established a ocedue fo comutatio of the matices of (3) will be deived. 3 Poblem Solutio Let us defie e : z Tx, e (6) Usig (6), () (3) we may wite e& z& Tx& F( e + Tx) + Gu + + HCx TAx TBu (7) Fe + ( FT TA + HC) x + ( G TB) u Fo TA FT + HC G TB (8) fom (7) we obtai e& Fe (9) We choose F so that det[ s F] α () (ozeo scala ideedet of s) Fo examle, we may choose I I whee I k is the, F α k k idetity matix. R () It is easy to show that if the matices F have the caoical fom () the the equatio (9) has the solutio e( t) fo t >. Fom (6) it follows that z Tx if oly if e ( t), t > the fom (b), (3b) (5) we obtai LT + MC () The equatio MC has a solutio M if oly if C ak C ak (3) I this case T the liea fuctio (4) ca be ecostucted by My. I what follows it is assumed that the coditio (3) is ot satisfied. Lemma. If the system () is cotollable obsevable the thee exists a ai of osigula matices P, Q such that I PQ A A B ) m A, PAQ I B, PB B, CQ, [ I ] A, (4) ) m B The oof of Lemma is simila oe to the give i []. Without loss of geeality it is assumed that the matices,a,b C of () have the caoical foms (4). Let C [ I ] (5) ) [ ],, (6) The fom () fo M we obtai [ ] MC LT (7) LT (8)
3 whee, I ) T T [ T ] T (9) Fom (8) it follows that ak mi( q, ) if q the ak. Takig ito accout (5) fom (8) we have I () [ TA FT] [ TA FT] H I () Lemma. Let the matices,a,b,c, F have the caoical foms (4) (), esectively. The the equatios () () ca be witte i the foms whee T A A ) ) ) ) FT () T A T A A3 H T (3) A4 A I, T, A A, A 3 A3 A A I, 4 A4 ) (4) Poof. Takig ito accout (4) I T T fom () we obtai the equality A [ T, T ] FT which is equivalet to A (). Note that I A3 H TA T. I A4 If F have the caoical foms () the the -th ow of the matices T A T A is zeo fom () it follows that the fist ow of T should be zeo sice by assumtio α. Thus ak T to satisfy the coditio ak T ak we have to choose ak + (5) Lemma 3. Let the matices, A, B, C of (), F have the caoical foms (4) (), esectively. The it is always ossible to choose the coefficiet α of F such that the equatio () has uique solutio T fo ay give matices A D T A. Poof. Let ab L a B m q A B R am B amb L be the oecke oduct of the matices A ] m [ a ij R B q. The the equatio () ca be witte i the fom [] St d (6) whee S F I t : A, [ t, t,..., t ], d [ d, d,..., d ] t d ) is the i-th ow of the matix T ( ). i ( i T D (7) The equatio (6) has uique solutio t if oly if the matix S is o-sigula o equivaletly if all its eigevalues ae ozeo. It is easy to show [] that the eigevalues λ i S of i i i i i S ae give by λ λ λ whee λ, λ F A i λ A ae the eigevalues of F, A, esectively. If F has the fom () the det[ I λ F] λ + α.thus, it is always ossible i to choose α so that all λ S ae ozeo. By oecke-caelli theoem the equatio (8) has a solutio L if oly if F
4 T ak T ak (8) The matix T is chose so that the equatio (8) has a solutio L fo give T. Theefoe, the followig theoem has bee oved. Theoem. Let the system () be cotollable obsevable its matices have the caoical foms (4). The thee exists a efect fuctioal obseve (3) of the miimal ode ak + fo () if oly if the coditio (8) is satisfied. If the assumtios of Theoem ae satisfied the a efect fuctioal obseve (3) of the ode fo () ca be comuted by the use of the followig ocedue. Pocedue Ste. Usig the slighty modified method give i [] tasfom the matices,a,b,c to the caoical foms (4). Ste. Usig (6) (7) comute M, [ ] MC. Ste 3. Choose the coefficiet α so that the matix Sis o-sigula solve the equatio () fo a give T A. Ste 4. Choose the matix T such that the coditio (8) is satisfied comute the matices T,T L. Ste 5. Usig (3) G TB (9) comute the matices H G. Ste 6. Usig (3) fid the desied efect fuctioal obseve Remak. If ak the by (3) (9) H G sice T. Hece the equatio (3a) takes the fom z & Fz. 4 xamle Desig a efect fuctioal obseve (3) fo the system () with, A, (3) B, C 3 (3) 4 It is easy to check that the system is cotollable obsevable. I this case we have 5, m, usig Pocedue we obtai Ste. The matices (3) have aleady the caoical foms (4). Ste. Usig (6) (7) we obtai M 3, 4 MC [ ], ak ak fom (5) we have. Ste 3. It is easy to check that fo α the matix S (defied by (7)) is o-sigula. Fo 5 T [ tij ] the equatio () has the fom
5 t4 t5 T T its solutio is T t4 t (3) 5 Ste 4. The eties t 4,t5 of T ae chose so that the coditio ak t4 t5 ak t4 t (33) 5 is satisfied. The coditio (33) holds, fo examle, fo t t. 4 5 I this case the equatio (8) has the fom L its solutio is l L ( l l,l ae abitay) Ste 5. Usig (3), (9) (3) we obtai H G. Ste 6. The desied efect fuctioal obseve has the fom l w l z& z z + 3 y 4 5 Cocludig emaks A ew desig method of miimal ode efect fuctioal obseves fo sigula cotiuoustime liea systems has bee oosed. Necessay sufficiet coditios fo the existece of the miimal ode efect fuctioal obseve have bee established. A ocedue fo comutatio of matices of the efect fuctioal obseve (3) has bee deived illustated by a umeical examle. If the coditio (3) is satisfied the the liea fuctio (4) ca be ecostucted by My the matix M ca be foud fom the equatio MC. With mio modificatios the cosideatios ca be also alied to sigula discete-time liea systems. A oe oblem is a extesio of the cosideatios fo two-dimesioal liea systems [6,7]. Refeeces [] G. Celetao A. Balestio, New techiques fo the desig of obseve, III Tas. Automat. Cot., vol. AC-9, No, 984,. -5. [] R.D. Guta, T.. Faima Hiamoto, A diect ocedue fo the desig of sigle fuctioal obseves, I Tas. Cicuits Syst., vol. CAS-8, No 4, 98,. 94 [3] T.. Faima R.D. Guta, Desig of multi-fuctioal educed ode obseve, It. J. Syst. Sci., vol., 98, [4] T.. Fotma D. Williamso, Desig of low ode obseves fo liea feedback cotol laws, I Tas. Automat. Cot. Vol. AC-7, 97, [5] T. aczoek, Full-ode Pefect Obseves fo Cotiuous-time Liea Systems, Bull. Pol. Acad. Tech. Sci., Vol. 49, No. 4,, [6] T. aczoek, Liea Cotol Systems, vol., Reseach Studies Pess J. Wiley, N.Y [7] T. aczoek, Pefect Obseves fo Sigula -D Foasii-Machesii Models, I Tas. Autom. Cotol, AC- 46, No.,, [8] T. aczoek, Pefect fuctioal obseves of sigula cotiuous-time liea systems, Machie Itelligece Robotic Cotol, Vol. 4, No. / (i ess).
6 [9] T. aczoek, Full-ode efect obseves fo cotiuous-time liea systems, Pomiay Automatyka otola, /,. 3-6 [] T. aczoek, Positive D D Systems, Sige-Velag, Lodo. [] T. aczoek, Vectos Matices i Automatio lektotechics, WNT Wasaw, 998 (i Polish). [] H. imua, Goemetic stuctue of obseves fo liea state feedback cotol laws, I Tas. Autom. Cotol, AC-, 977, [3] D.G. Luebege, Obseves fo multivaiable systems, I Tas. Autom. Cotol, AC-, 966, [4] P. Mudoch, Desig of degeeate obseves, I Tas. Autom. Cotol, AC-9, 974. [5] J. O Reilly, Obseves fo Liea Systems, Academic Pess, New Yok, 983. [6] C.C. Tsui, What is the Miimum Fuctio Obseve Ode, Joual of The Fakli Istitute, Volume: 335, Issue: 4, May, 998, [7] C.C. Tsui, A New Algoithm fo Desig of Multifuctioal Obseves, I Tas. Autom. Cotol, Vol. AC-3, 985, [8] C.C. Tsui, O the ode eductio of liea fuctio obseves, I Tas. Autom. Cotol, Vol. AC-3,986,
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