COMPONENTWISE STABILITY OF 1D AND 2D LINEAR DISCRETE-TIME SINGULAR SYSTEMS

Transcription

1 Proceedgs of te 8t WSEAS Iteratoal Coferece o Automato ad Iformato, Vacouver, Caada, Jue 9-2, COMPONENTWISE STABILITY OF AN 2 LINEAR ISCRETE-TIME SINGULAR SYSTEMS H. Mejed*, N.H. MEJHE** ad A. HMAME * ept GI, ENSA, BP 575, Marraec, moroccco Mejed9@otmal.com. ** épt GII, ENSA, BP 33/S, Agadr, Morocco oramej@otmal.com LESSI épartemet de pysque, Faculté des Sceces, BP. 796, 3, FES, Morroco. Abstract: Ts paper deals wt a specal type of asymptotc (epoetal) stablty amely compoetwse asymptotc (epoetal) stablty for - ad 2- lear dscrete-tme sgular systems. Te ma motvato for tese results s te eed, partcularly felt te evaluato a more detaled maer of te dyamcal beavour of - ad 2- lear dscrete-tme sgular systems. Necessary ad suffcet codtos for compoetwse asymptotc (epoetal) stablty are gve. Key words: - ad 2- dscrete-tme sgular systems; compoetwse asymptotc (epoetal) stablty; sgular Foras-Marces model; raz verse teory. H ( ) R ( a real vector); ( j ) R ( H a real matr); It teror of set δ boudary of set det( M) determat of matr M C ; H matr wt compoet H matr wt compoets H matr wt compoets vector wt compoets vector wt compoets, y vectors R ; y f y,,2,..,; < y f < y,,..,.. Itroducto j sup(j,),,j,2,..,; NOTATIONS j sup( j,),,j,2,..,; j,,j,2,..,, sup(,),,..,; sup(,),,..,; Compoetwse stablty of lear cotuous systems wt or wtout tme delay as bee studed by Hmamed (996), Vocu (984, 987) ad Hmamed ad Bezaoua (997). Te purpose of ts ote s to eted te cocept of Compoetwse asymptotc (epoetal) stablty of - ad 2- lear dscrete-tme systems (see Hmamed, 997) to te compoetwse stablty of of - ad 2- dscrete-tme sgular systems. Te paper s orgazed as follows. Secto 2 deals wt te ma results, gvg ecessary ad suffcet codtos for compoetwse asymptotc (epoetal) stablty of sgular systems. Ts s te eteded to te 2 sgular Foras-Marces Secto 3.

2 Proceedgs of te 8t WSEAS Iteratoal Coferece o Automato ad Iformato, Vacouver, Caada, Jue 9-2, dscrete sgular systems I ts secto, some results about compoetwse asymptotc (epoetal) stablty of sgular systems are establsed. We focus o dscrete-tme sgular systems, wc are descrbed by te mplct form E( ) A(), > () () were R s te state vector, E R, ra(e) q, A R. I certa applcatos, amely electrcal egeerg ad bology, dyamcal systems ave to satsfy some addtoal costrats of te form Ω R (2) were Ω s te set of admssble, states defed by { Ω R / ρ2() () ρ (); ρ (), ρ2() tr } (3) wt lm ρ(), lm ρ2 () (4) Ts s a varat osymmetrcal polyedral set, as s geerally te case practcal stuatos. I certa cases, ρ () ad ρ 2 () tae te form ρ s ( ) sβ (5) wt < β < ad s > for s,2. Te purpose of ts secto s to defe a specal type of asymptotc (epoetal) stablty of (), amely te compoetwse asymptotc (epoetal) stablty caracterzed by (3) ad (4) ((3) ad (5)). Necessary ad suffcet codtos for compoetwse asymptotc (epoetal) stablty of te system () are gve. Frst, recall some mportat propertes of mplct systems tat are assumed trsc te followg aalyss. efto 2.: Te system () s called compoetwse asymptotcally stable wt respect to T T ρ() [ ρ ] T () ρ 2 () (CWAS ρ ) f for every ρ2( ) ρ(), te respose of () satsfes ρ ) () (), (6) 2 ( ρ efto 2.2 Te system () s called compoetwse epoetal asymptotcally (CWEAS) f tere est > ad 2 > suc tat, for every 2, te respose of () satsfes 2 β () β, (7) efto 2.3 [Campbell et al., 976]: Te system () s sad to be regular f det( se A). efto 2.4 [Lews, 987]: Te system () s sad to be mpulse-free f deg det(se A) ra(e) or (se A) s proper. efto 2.5 [Ce, 97] A ratoal fucto G(s) s sad to be proper f G( ) s costat matr, G(s) s sad to be strctly proper f G ( ). For te coveece of te later statemets ts paper, we use te par (E,A) to represet te system (). Teorem 2. [Campbell, 99] Suppose tat (E,A) s regular. Te E ( ) A() f(), s solvable ad te geeral soluto s gve by: ν () A Pq Ê A fˆ() (I P) E Â fˆ( ) (8) were fˆ () ( λe A) f (), Ê ( λe A) E, Â ( λe A) A, A Ê Â, E ÊÂ, P ÊÊ, q R, ν s te de of E ˆ ad λ s a scalar suc tat λ E A s osgular. Te projecto P ad E, A, ν are depedet of λ. Matr Ê s te raz verse of Ê ad te de of a matr s te sze of te largest lpotet bloc ts Jorda caocal form. We ow gve ecessary ad suffcet codtos for compoetwse asymptotc (epoetal) stablty of te system (). Teorem 2.2: Suppose tat (E,A) s regular, a ecessary ad suffcet codto for te system () to be CWAS ρ s ρ ( ) H ρ(), (9) wt H H H T, ρ () [ ρ ] () ρ T 2 () T () H H ad H Ê Â () Proof: From Teorem 2. te soluto of te system () s wrtte te form () Ê Â ÊÊq ÊÂ, te: ( ) ( ) q R ( ) Ê Â() (2) wt te tal codto ÊÊ q, q R. At ts step, we ca use te proof gve Hmamed (997) as te proof remas ucaged.

3 Proceedgs of te 8t WSEAS Iteratoal Coferece o Automato ad Iformato, Vacouver, Caada, Jue 9-2, Remar : From Campbell et al. (976) ad Lews (986), we ow tat te cosstet tal codtos () of system () are defed by ÊÊ ad te, tere always ests a vector q R ÊÊ q (Tarbourec et al. 993). suc ta () t () Remar 2: I te case were matr E s o-sgular, te system () ca be wrtte as a classcal autoomous lear system defed as ( ) E A() (3) Hece, f we apply te prevous result to system (3), te te classcal result of te compoetwse stablty of - lear dscrete-tme systems s obtaed, tat s ρ ( ) H ρ () ( ) ( ) wt E A E A H. ( E A) ( E A) For E I, we obta te result gve [Hmamed, 997]. I te symmetrcal case ρ () ρ2 () ρ(), we ca deduce te followg result. I te symmetrcal case ρ ( ) ρ2() β, we ca deduce te followg result. Corollary 2.3: Te regular system () s CWEAS f ad oly f oe of te followg codtos olds: β H ; wt H Ê Â ( ). Remar 3: We Hmamed (997). j > β ma j j j E I, we obta te result gve by 2- Foras- Marces model I ts secto, we eted te oto of compoetwse asymptotc (epoetal) stablty to mplct 2- Foras- Marces model descrbed by: E (, j ) A(, j ) B(, j) (7) wt te boudary codtos (,) ad (, j) for, j,, (8) were R s te state vector. Corollary 2.: Suppose tat (E,A) s regular, a Assume te tat (E,A) s a regular pecl ad mpulsefree. A smlar dscusso apples f (E,B) s regular. ecessary ad suffcet codto for te system () to be CWAS ρ s Treat j as fed, If te sequece (, j) s cosdered ρ ( ) H ρ() (4) ow, te (7) s a dfferece equato matr H s defed by (). E (, j ) A(, j ) [B(, j)], (9) Proof: By observg tat H H H, te proof for (, j ) wt te terms square bracets ow (see Campbell., 99). follows from Teorem 2.6. Sce (E,A) s a regular pecl, we may apply Teorem 2., we ave Usg te tecques of Hmamed (997), we ca also eted te results of Teorem2.2 to te followg ( ) (, j ) Ê Â ÊÊ q Ê Teorem wc deals wt te compoetwse ( Ê Â) Bˆ(, j) epoetal asymptotc stablty. (2) ν r (I ÊÊ ) (ÊÂ ) Â Bˆ( r, j) Teorem 2.3: Suppose tat (E,A) s regular, te system () (system (2)) s CWEAS f ad oly f oe of te followg codtos olds: ere ν s te de of Ê ) ( β I H ) ; (5) ( ) (, j ) Ê Â ÊÊ q Ê ( Ê Â) Bˆ (, j) ) j j j j ν > β ma ma r j j, j j j,(i ÊÊ ) (ÊÂ ) Â Bˆ ( r, j) j j 2 j 2 2 (6) T T wt [ ] T 2 ad H Ê Â (j).

4 Proceedgs of te 8t WSEAS Iteratoal Coferece o Automato ad Iformato, Vacouver, Caada, Jue 9-2, ( )( ) ( Ê Â Ê Â ÊÊ q Ê Â) Ê ( Ê Â) (I Bˆ (, j) ( Ê ν r Bˆ ) (, j) (I ÊÊ ) (ÊÂ ) Â Bˆ( r, j) ( Ê Â) (, j ) (I ÊÊ ) ν r (ÊÂ ) Â Bˆ (, j) ( Ê ν r Bˆ ) (, j) (I ÊÊ ) (ÊÂ ) Â Bˆ( r, j) ( Ê Â) (, j ) ( Ê Bˆ ) (, j) ÊÊ ) ν (ÊÂ ) Â Bˆ (( Ê Â) (, j) (, j)) ( Ê Â) (, j ) ( Ê Bˆ ) (, j) (I ÊÊ ) ν (ÊÂ ) Â Bˆ (, j) efto 3. : Te regular system (7) s called compoetwse asymptotcally stable wt respect to T T T ρ (, j) [ ρ (, j) ρ (, j ] (CWAS ρ ) f, for every 2 ) ρ2 (,) (,) ρ(,) ρ2 (, j) (, j) ρ(, j) for,j,,2,. (2) Te respose of (7) satsfes ρ2(, j) (, j) ρ(, j) (, j) > (,) (22) were ρ (, j) >, ρ2(, j) > (, j) > (,) lm ρ(, j), lm ρ(, j) ad/or j ad / or j 2 efto 3.2 : Te regular system (7) s called compoetwse epoetal asymptotcally stable (CWEAS) f tere est > ad 2 > suc tat, for every 2β (,) β j j 2γ (, j) γ for,j,,2, (23) te respose of (7) satsfes 2 β γ (, j) β γ < β < < γ <., (, j) > (,) were ad We gve ow ecessary ad suffcet codtos for compoetwse asymptotc (epoetal) stablty of te system (29) or (7). Teorem 3.: Suppose tat (E,A) s regular ad mpulse-free, a ecessary ad suffcet codto for te system (7) to be CWAS ρ s ρ (, j ) H ρ (, j ) H ρ (, j), (, j) > (,) (24) 2 wt (Ê Â) (Ê Â) H, (Ê Bˆ ) (Ê Bˆ ) H, 2 (Ê Â) (Ê Â) (Ê Bˆ ) (Ê Bˆ ) T T T ρ (, j) [ ρ (, j) ρ 2 (, j) ] (25) Proof: Sce system (2) s mpulse free, ts case, ν becomes (Lews, 986). Epress (, j ) by usg (2) wt ν as ( Ê Â) ÊÊ q Ê ( Ê Â) (, j ) Bˆ (, j) (26) te (, j ) Ê (Ê ( Ê Â) Â) ( Ê Â) ÊÊ q Ê Bˆ (, j) Bˆ (, j) (27) From te raz verse teory used [Campbell et al., (976)], we ow tat Ê Â ÂÊ, Ê Â ÂÊ ad Ê Â Â Ê te (, j ) H (, j ) H2 (, j) (28) wt H Ê Â, H Ê 2 Bˆ (29) ad boudary codtos (,) ad, (, j) Ê Êq, q R, j. At ts step, ts follows smlar les to te proof of Teorem 3.3 Hmamed (997). Remar 4: Te boudary values (,) may be tae arbtrary ad te boudary values Ê Ê(, j), j, are arbtrary (Campbell, 99). Te, we ca appled te results gve te last secto, mplyg te estece of a vector q R suc tat: (, j) Ê Êq, j, j By aalogy wt secto 2, we gve some deftos. Remar 5: If E s osgular square matr, te equato (24) s te classcal codto gve Hmamed (997). I te symmetrcal case ρ (, j) ρ2(, j) ρ(, j), we ca deduce te followg result. Corollary 3.. Suppose tat (E,A) s regular ad mpulse-free, a ecessary ad suffcet codto for te system (7) to be CWAS ρ s ρ (, j ) H ρ(, j ) H2 ρ(, j) (3) matrces H ad H2 are defed by (29).

5 Proceedgs of te 8t WSEAS Iteratoal Coferece o Automato ad Iformato, Vacouver, Caada, Jue 9-2, Proof: 2 H2 H2 O observg tat H H H2 ad H, te proof follows from Teorem 3.. Teorem 3.2. Suppose tat (E,A) s regular ad mpulse-free, te system (7) (system (28)) s CWEAS f ad oly f oe of te followg codtos olds: ) βγ (H γ H 2 ; (3) 2 j 2 j >βγ mama[ ( j γ j (j γ j 2]/, j ) (32) 2 j 2 j [ ( j γ j (j γ j ]/ 2 j T wt [ 2 ]. j I te symmetrcal case ρ (, j) ρ2(, j) β γ, we ca deduce te followg result. Corollary 3.3: Suppose tat (E,A) s regular ad mpulse-free, te system (7) s CWEAS f ad oly oe of te followg codtos olds: () βγ ( H γ H2 (33) j 2 () > βγ ( j γ j (34) j matrces H ad H2 are defed by (29). Remar 6. We ca eted te results of ts secto to te Roesser 2 model gve Kaczore (987), Lews ( ) by E E 2,j A A2, j v v (35) E3 E 4 A,j 3 A4, wt te boudary codtos v (, j) ad (,) for,j,, Several tecques may be used to sow tat te mplct Roesser ad mplct FM model are equvalet (Kacsore, 989). Ideed, te Roesser model defe E E3 F, F2 (36) E2 E4 A A2 ad smlar quattes wt respect to A. A3 A4 Te (35) may be wrtte as F (, j) F2 (, j ) A(, j) (37) Cosequetly all te results derved ts secto stll old for te Roesser model (35) o tag to accout te relato (36). Followg te same deas, tose results ca easly be eteded to te followg geeral 2 system model: E(, j ) A (, j) A(, j ) A 2(, j) (38) 3. Cocluso I ts paper, we ave gve a eteso of te cocept of compoetwse asymptotc (epoetal) stablty for sgular ad 2 dscrete lear sgular systems. Necessary ad suffcet codtos for compoetwse asymptotc (epoetal) stablty ave bee gve. Te results for te symmetrcal case ave bee obtaed as a partcular case. Refereces: [] Campbell, S. L., C.. Meyer ad N.J. Rose (976). Applcatos of te raz verse to lear systems of dfferetal equatos wt sgular costat coeffcets. SIAM J. Appl. Mat. 3(3) [2] Campbell, S. L. (99). Commet o 2- escrptor systems. Automatca, vol. 27, o.,pp [3] Ce, C. T (97). Itroducto to lear system Teory. Holt, Reart ad Wsto, New Yor. (984) Lear System Teory ad esg. Holt, Reart ad Wsto, New Yor. [4] Hmamed, A. (996). Compoetwse stablty of cotuous tme delay lear systems. Automatca. 32, 65_653. [5] Hmamed A. (997). Compoetwse stablty of - ad 2- lear dscrete-tme systems, Automatca, vol. 33, o 9, pp, [6] Hmamed, A. ad Bezaoua, A. (997). Compoetwse stablty of lear systems: a osymmetrcal case. It. J. Robust Nolear Cotrol, vol. 7, pp [7] Kaczore, T. (986). Geeral respose formula for two-dmesal lear systems wt varable coeffcets. IEEE Tras. Aurom. Cotrol Ac-3, 278_28. [8] Kaczore, T. (987). Sgular Roesser model ad reducto to ts caocal form. Bullet pols Academy of Sceces, Teccal Sceces, vol.35, pp [9] Kaczore, T. (989). Equvalece of sgular 2- lear models. Bull. Pols Academy Sc., Electr. Electroteccs, 37. [] Lews, F.L. (986). A survey of lear sgular systems. J. Crcut, Syst., Sg. Process., Vol. 5, No, pp [] Lews F.L.(987). Recet wor sgular systems. Proc. It. Symp. Sgular systems, pp.2-24, Atlata, GA. [2] Lews, F.L.(992). A revew of 2- mplct systems. Automatca. Vol. 28. No. 2, pp , 992. [3] Tarbourec, S. ad E.B. Castela. (993). Postvely varat sets for sgular dscrete-tme systems, It. J. Systems SCI., Vol. 24, No. 9,

6 Proceedgs of te 8t WSEAS Iteratoal Coferece o Automato ad Iformato, Vacouver, Caada, Jue 9-2, [4] Vocu, M. (984). Compoetwse asymptotc stablty of lear costat dyamcal systems, IEEE Tras. Autom. Cotrol AC-29, [5] Vocu, M. (987). O te applcato of te flowvarace metod cotrol teory ad desg. I Proc. t IFAC World Cogress, Muc, pp