Static Output-Feedback Simultaneous Stabilization of Interval Time-Delay Systems

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1 Sac Oupu-Feedback Sulaneous Sablzaon of Inerval e-delay Syses YUAN-CHANG CHANG SONG-SHYONG CHEN Deparen of Elecrcal Engneerng Lee-Mng Insue of echnology No. - Lee-Juan Road a-shan ape Couny 4305 AIWAN R.O.C. ycchang@oron.ee.nus.edu.w Deparen of Inforaon Neworkng echnology Hsupng Insue of echnology No. Gongye Road Dal Cy achung Couny 480 AIWAN R.O.C. Absrac: hs paper addresses he proble of sulaneous sac oupu-feedback sablzaon of a collecon of nerval e-delay syses. I s shown ha hs proble can be convered no a arx easure assgnen proble. Suffcen condons for guaraneeng he robus sably for consdered syses are derved n er of he arx easures of he syse arces. By usng arx nequales we provde wo cases of obanng a sac oupu-feedback conroller ha can sablze he syse.e. boh P= I and P I cases are consdered where I s a deny arx and P s a coon posve defne arx o guaranee he sably of he overall syse. he suffcen condon wh P= I s forulaed n he fora of lnear arx nequales (LMIs. When P I s consdered he suffcen condon becoes a nonlnear arx nequaly proble and a heursc erave algorh based on he LMI echnque s presened o solve he coupled arx nequales. Fnally an exaple s provded o llusrae he effecveness of our approach. Key-Words: Marx easure; Robus conrol; Oupu feedback; Inerval syses; Lnear Marx Inequaly. Inroducon he sulaneous sablzaon proble was frs nroduced n Saeks and Murray [] and Vdyasagar and Vswanadha []. hs proble consss n answerng he followng quesons: gven plans G K G does here exs a sngle feedback conroller C so ha he conroller can sablze all plans and all correspondng closed-loop syses have sasfacory perforance? One scenaro for hs proble s he relable sablzaon proble where G K G represen G operang n varous odes of falures (e.g. falure of sensor severance of loops sofware breakdown. If falures occur he dynacs of he syse wll ceranly change. hus n such a case a conroller s sough so ha can sablze he syse n all suaons. Anoher applcaon s he desgn of a fxed conroller for a se of lnear plans characerzed by dfferen odes of operaons or for nonlnear plans lnearzed a several regons. Los of researchers have been devoed heselves n solvng he sulaneous sablzaon proble. Blondel and Gevers [0] suded he copuaonal coplexy of sulaneous sablzaon and proved ha he sulaneous sablzaon for hree lnear syses s raonally non-decdable. Fro [5]-[6] s possble o conclude ha hs proble s very dffcul o solve due o s NP-hard naure. Paskoa e al. [3] presened a copuaonal echnque for opal sulaneous sablzaon for lnear syses va lnear sae feedback. Cao and La [7] deal wh sulaneous lnear-quadrac (LQ opal conrol desgn for a se of lnear e-nvaran (LI syses va pecewse consan oupu feedback. Cao and Sun [8] proposed an erave lnear arx nequaly (LMI algorh o seek a sae/oupu feedback conroller for a se of MIMO plans. In [9] a nuercal algorh s nroduced o solve he proble of sulaneous sablzaon of a collecon of MIMO plans va sac 85

2 oupu feedback usng a se of coupled algebrac arx nequales (ARI s. Oher desgn resuls on sulaneously sablzng conroller can also be found n []-[3]. In he above-enoned approaches hese researchers eploy he LQ conrol approach o solve he sulaneous sablzaon proble for a collecon of LI syses whou e delay and unceranes. o our bes knowledge here are no general echnques for solvng he proble of sulaneous sablzaon for a collecon of unceran e-delay syses va sac oupu feedback. For a sngle syse wh e delay and unceranes he robus sably analyss proble s que coplcaed and recenly has been suded va several dfferen echnques. he crera for asypoc sably of such syses can be classfed as delay-ndependen whch are ndependen of he sze of e-delay for exaple [0] or delay-dependen whch nclude nforaon on he sze of delay for exaple [3]. Meanwhle soe dfferen sably crera have also been proposed va he LMI approach [4]-[7]. In hs paper we focused on he proble of sulaneous sablzaon for a collecon of nerval e-delay syses va a sac oupu feedback conroller. I wll be shown ha he consdered proble s solvable f a correspondng arx easure assgnen proble s solvable. he arx easure s wdely appled n he analyss of sably properes of unceran and/or e-delay syses [4 5 4]. Alhough has been wdely eployed n he robusness analyss proble neverheless few has nvesgaed abou he conroller synhess proble. Recenly lnear arx nequales (LMI s have eerged as a powerful forulaon and desgn echnque for a varey of lnear conrol probles [678]. Sofware lke Malab s LMI Conrol oolbox [8] s avalable o solve LMI s probles n a fas and user-frendly anner. In hs paper we shall show ha he arx easure assgnen proble s equvalen o an LMI feasbly proble. hus a conroller solvng he arx easure assgnen proble and hen solvng he sulaneous sablzaon proble for a collecon of nerval e delay syses can be obaned va solvng an LMI proble. he paper s organzed as follows. Secon proposes he proble forulaon and revews he basc properes of he arx easure. In Secon 3 a collecon of nerval e-delay syses are dscussed. In he secon he sably and robusness condons for he consdered syse are also derved. I s shown ha he proble of sac oupu-feedback conroller desgn for a collecon of nerval e-delay syses s solvable f a correspondng arx easure assgnen proble s solvable. Furher he suffcen condon wh P= I s forulaed n he fora of lnear arx nequales (LMIs. When P I s consdered he suffcen condon becoes a nonlnear arx nequaly proble and a heursc erave algorh based on he LMI echnque s presened o solve he coupled arx nequales. Secon 4 shows an llusrave exaple. Fnally conclusons are gven n Secon 5. Noaons: In wha follows O s a zero arx wh an approprae denson I s an deny arx wh an approprae denson M denoes he ranspose of he arx M M * denoes he conjugae ranspose of he arx M M >0 ( M 0 eans ha he arx M s posve defne (sedefne M <0 ( M 0 eans ha he arx M s negave defne (sedefne.. Proble Forulaon and Prelnares Consder a collecon of nerval e-delay syses: x& ( ˆ ( ˆ = Ax + Dx ( h + Bu ( = K p ( y( = Cx ( = K p ( n where x ( R s he sae h s he e-delay of he syse u ( R s he conrol npu and r n y ( R s he conrolled oupu. B R and r n C R are consan arces. ˆ n n A R and ˆ n n D R are arces whose eleens vary n soe prescrbed ranges; e.g.  and Dˆ are such ha ˆ A = [ a] a a a = K p (3 ˆ D = [ d ] d d d where a s he -h eleen of he arx  a and a denoe s low bound and upper bound respecvely d s he -h eleen of he arx Dˆ and d and d denoe s low bound and upper bound respecvely. hose bounds a a d and d are known real values. 86

3 he desgn goal s o fnd a arx F such ha he sac oupu feedback conroller u( = Fy ( = K p (4 ensures all he closed-loop nerval e-delay syses beng asypocally sable. We now nroduce several properes abou arx easure as follows. he arx easure of a consan arx M s defned as ( I+ θm v v ( M l (5 + θ 0 θ where. v s a suable arx nor (see [5]. Lea. [5]: he arx easure has followng properes. (a. v (. s convex;. e. k k v αjmj α j v( M j for all α j= j 0. (6 j= (b. For any nor and any consan arx M M v v( M Re λ ( M v( M M v. (7 (c. Suppose s he j-h eleen of M hen j ( ax Re( jj M = + j (8 j j * ( M = ax ( + / M M (9 λ ( M = ax Re( + j. (0 j 3. Man Resuls Fro ( ( and (4 he collecon of closed-loop syses can be descrbed as: x& ( ( ˆ ( ˆ = A + BFC x + Dx( h = K p y( = Cx ( = K p Denoe A = [ a ] A = [ a ] = K p ( D = [ d ] D = [ d ] = K p ( and le A = ( A + A D = ( D + D = K p (3 M = A A N = D D = K p (4 where A and D are he average arces of A and A and of D and D respecvely. Furherore M and N are he axal bas arces beween A ˆ and A and beween D ˆ and D respecvely. We frs derve a new suffcen condon o ensure he sably of a collecon of unforced e-delay syses whou uncerany x& ( = Ax( + Dx ( h = K p. (5 y ( = Cx ( 3. LMI approach heore 3.: For any > 0 f ( A < D = K p (6 hen he equlbru of a collecon of unforced syse (5 s asypocally sable. Proof: Consder a Lyapunov funcon as V( x( = x ( x( + ( s ( s ds x x -h = K p. he e dervave of V( x ( s V& ( x( = x ( x& ( + x ( x( x ( h x( h = x (( A x ( + Dx ( h + ( Ax ( + Dx ( h x( + x ( x( x( h x( h = x (( A + A+ I x( + x ( Dx ( h + x ( h D x( ( x ( h x ( h Se x ( [ ( ( ] = x x h. We oban A + A + I D V& ( x( = x ( x(. D By Schur copleen f A + A + I+ DD 87

4 A + A + I D hen s negave defne. D Fro (6 we oban ( A + A+ I+ DD ( A + A + ( I + ( DD ( A + + D hs proves A + A + I+ DD and hen A + A + I D < 0. D So V & ( x ( for all x ( 0. hs coplees he proof. heore 3. provdes a sple ehod o verfy he sably of a collecon of unforced e-delay syses (5. In wha follows we shall consder he sably condons of a collecon of unforced e-delay syses wh uncerany. heore 3.: Consder a collecon of nerval e-delay syses x& ( ˆ ( ˆ = Ax + Dx ( h = K p (7 y( = Cx ( where A ˆ and D ˆ are defned n (3. If ( A < M D N D N = K p (8 hen (7 s robusly asypocally sable. Proof: Snce ˆ ˆ ( A + + D = ( A + A + + D + D = ( A + ( A + + D + D D + D < ( A + M + + D + N D + N hs coplees he proof. Wh he above heores we have he followng corollary. Corollary : Suppose ha sac oupu feedback gan F sasfy he followng condons ( A + BFC < M D N D N = K p (9 hen he equlbru of he closed-loop e-delay syse wh nor-bounded unceranes represened as x& ( ( ˆ ( ˆ = A + BFC x + Dx ( h = K p y( = Cx ( (0 s robusly asypocally sable. Corollary reveals ha f he consan conrol gan sasfes (9 hen a collecon of nerval e-delay syses are robusly asypocally sable. For splcy of noaon defne γ = M D N D N = K p. hus (9 becoes ( A + BFC < γ = K p. ( r Defne I( γ { R ( + < γ} F A B FC for = K p. he adssble soluon se s I I( γ II( γ I I Ip( γ p. hen we can have he followng heore. heore 3.4: he adssble soluon se I s convex. Proof: Snce he nersecon of convex ses s convex we only need o prove ha I ( γ s convex for each. Assue F I( γ and F I( γ whch eans ( A + BFC < γ and ( A + BF C < γ. hen o prove ha I ( γ s convex s sae as o prove αf+ ( α F I( γ or equvalenly o prove ( A + B( αf+ ( α F C < γ for all 0 α. Noe ha 88

5 ( A + B ( αf + ( α F C = αa + ( α A + αb F C + ( α B F C ( ( α( A + B FC + ( ( A + BFC ( A + BFC + ( α ( A + BFC = α α < γ hs coplees he proof. Fro he above dscussons s concluded ha he arx easure assgnen proble can be consdered as a convex feasbly proble. hus we now urn our aenon o reduce he arx easure assgnen proble o an LMI feasbly proble. For a arx U defne U as a arx whose coluns for bases of he null bases of U. hen we can have he followng heore whch s he an resul of hs paper. heore 3.5: (. he arx F sasfes ( A + BFC < γ = K p. ( f and only f F sasfes LMIs (A + A γ I + BFC + C F B = K p. (3 (. here exss F sasfes (3 f and only f ( B ( A + A γ I( B = K p (4 and ( C ( A + A γ I( C = K p (5 Proof: We frs prove par (. Fro (9 can be shown ha ( A + BFC < γ = K p are equvalen o * (A + BFC + (A + BFC γ I = K p. (6 whch are equvalen o (A + A γ I + BFC + C F B = K p. hs coplees he proof of par (. For he par ( recall he resul n [9]. Gven a syerc arx n n Ψ R and wo arces U and V boh wh a colun denson n here exss a arx Θ of a copable denson such ha Ψ+ U Θ V+ V Θ U f and only f U Ψ U and V Ψ V. Leng Ψ = (A + A γ I V = B U = C and Θ = F he par ( s obvous. heore 3.3 ells us ha f (4 and (5 hold hen here exss a arx F ha sasfes LMIs (3. In fac such an F also solves (. hs eans ha f (4 and (5 hold hen he adssble soluon se I s no epy. Noe ha a arx F sasfyng LMIs (3 can easly be obaned by usng Malab s LMI Conrol oolbox f I s no epy. he obaned F hen can also solve he consdered proble. Reark: he approach descrbed above can be appled o solve he sulaneous oupu feedback sablzaon proble for a collecon of unceran syses: x& ( = ( A + A x( + Bu ( = K p y( = Cx ( = K p A ρ = K p n where x R s he sae u R s he conrol r npu and y R s he oupu; and A B and C are consan arces of approprae densons. he desgn goal s o fnd a arx F such ha he sac oupu feedback conroller u( = Fy ( = K p can sablze all he closed loop syses n he presence of uncerany A. Snce ( A A s known ha f we can fnd a feedback arx F such ha ( A + BFC < ρ = K p (7 hen all he closed-loop syses are asypocally sable. hs proble can be easly solved va our approach. 3. Ierave LMI approach heore 3.6: For any > 0 f exss a syerc and n n posve defne arx P R such ha he followng nequales are sasfed ( PA < PD = K p (8 hen he equlbru of a collecon of unforced syse (5 s asypocally sable. Proof: Consder a Lyapunov funcon as V( x( = x ( Px( + ( s ( s ds x x -h = K p. he e dervave of V( x ( s V& ( x( = x ( Px& ( + x& ( Px( + x ( x( x ( h x( h 89

6 = x ( P( A x ( + Dx ( h + ( Ax( + Dx( h Px( + x ( x( x( h x( h = x (( A P+ PA+ I x( + x ( PDx ( h + x ( h D Px ( ( x ( h x ( h Se x ( = [ x ( x ( h]. We oban AP + PA + I PD V& ( x( = x ( x(. DP By Schur copleen f AP + PA + I + AP + PA + I PD PDD P hen s DP negave defne. Fro (6 we oban ( AP + PA+ I+ PDDP ( AP + PA + ( I + ( PDDP ( PA + + PD hs proves ( AP + PA + I+ PDDP AP + PA + I+ PDDP and hen AP + PA + I PD < 0. DP So V & ( x ( for all x ( 0. hs coplees he proof. In wha follows we shall consder he sably condons of a collecon of unforced e-delay syses wh uncerany. heore 3.7: If ( PA < P M P D P N D P N = K p (9 hen (7 s robusly asypocally sable. Proof: Snce ˆ ˆ ( PA + + PD = ( PA + P A + + PD + P D = ( PA + ( P A + + PD + PD P D + P D < ( PA + P M + + P D + P N D + P N hs coplees he proof. Wh he above heores we have he followng corollary. Corollary : Suppose ha sac oupu feedback gan F sasfy he followng condons ( PA ( + BFC < P M P D P N D P N = K p (30 hen he equlbru of he closed-loop e-delay syse wh nor-bounded unceranes represened as (0 s robusly asypocally sable. For splcy of noaon defne η = P M P D P N D P N = K p. hus (30 becoes ( PA ( + BFC < η = K p. (3 he arx nequaly (3 leads o nonlnear arx nequaly opzaon a non-convex prograng proble. Non-convexy ples he 90

7 exsence of local na and he nonlnear arx nequaly probles are NP-hard. However we shall reduce he arx easure assgnen proble (3 o a arx nequaly proble fro he followng heore. heore 3.8: he sac oupu feedback gans F sasfy he followng condons ( PA ( + BFC < η = K p. (3 f and only f F sasfy he followng arx nequaly (PA + A P η I + PBFC + C FB P = K p. (33 Proof: Fro (9 can be shown ha ( PA ( + BFC < η = K p are equvalen o P(A + BFC + (P( A + BFC η I = K p whch are also equvalen o (PA + A P η I + PBFC + C FB P = K p. hs coplees he proof. In fac he above arx nequaly (33 proble s generally very dffcul for whch o oban soluons or o deerne feasbly. However f we can derve an erave for for s feasbly we ay consruc an erave algorh based on he LMI echnque [8]. If P s fxed n (33 hen reduces o an LMI proble n he unknowns F he LMI proble s convex and can be solved f a feasble soluon exss. If we sply perurb (33 by β P hen we oban a necessary condon for sac oupu feedback sablzably.e. PA + A P ηi βp + PBFC + C F B P = K p. Consequenly he closed-loop syse arces A + BFC have egenvalues on he lef-hand sde of he lne R ( s = β / n he coplex s-plane. Ierave Lnear Marx Inequaly Algorh: Sep Se = selec S>0. Solve he followng ARE: A P + PA PBB P + S = 0 and se F= P = K p. Sep Solve he followng opzaon proble for P and β. OP: Mnze β subjec o he LMI consrans shown n (34-(35. PA + AP γi βp + PBFC + CFBP (34 P = P > 0 (35 Sep 3 If β 0 P s a feasble soluon. SOP. Sep 4 Solve he followng opzaon proble for P. OP: Mnze race( P subjec o he LMI consrans shown n (34-(35. Sep 5 If F P < δ a predeerned olerance go o Sep 6; else se F= P and =+ hen go o Sep. Sep 6 hs algorh canno ge a feasble soluon. SOP. In Sep s vewed as a generalzed egenvalue nzaon proble. hs sep ensures ha he poles of he global closed-loop syse ove o he lef half-plane gradually. Nuercal experences denoed ha β ay converge slowly n soe cases. he algorh s ernaed when β β s saller han a prescrbed olerance for a fxed nuber of successve eraons. In Sep 3 we se β = 0 and le he algorh connue erang o ake he dfference of F and P as sall as possble f a feasble soluon s obaned and he feedback gan s oo large. he condon (34 guaranees he exsence of a soluon of opzaon proble OP. he soluon P ples ha he sequence race( P s bounded below. If β s fxed for >q and q s a posve consan s no dffcul o fnd ha he soluon sequence race( P s a onoonc decreasng sequence. OP ay be nfeasble due o he effec of nuercal errors n Sep. In such a case one ay se β = β + β for soe sall posve nuber β and solve OP agan. 4. Illusrave Exaples Exaple: Consder wo nerval e-delay syses descrbed by (-(3 wh he sae denson: Syse : 9

8 A = A = D = D = B = C = Syse : A = A = D = D = B = C = he delay es h = and h =. he proble s o fnd F such ha ( A + BFC < γ for = where A = ( A + A D = ( D + D = M = A A N = D D = γ = M D N D N =. We can oban γ = and γ = hen we can easly copue a soluon F fro he followng LMIs usng Malab s LMI Conrol oolbox. (A + A γ I + BFC + C F B (A + A γ I + B FC + C F B A soluon s obaned as: F = I s easy o check ha ( A+ BFC =.3590 whch s less han γ = Slarly ( A + BFC = γ = I hen can be nferred ha he collecon of syses x& ( = ( Aˆ + BFC x( + D ˆ x( h for = are all robusly sable. 5. Conclusons In hs paper fndng an adssble soluon o he arx easure assgnen proble can solve he proble of sulaneously sablzng conroller desgn va sac oupu feedback for a collecon of nerval e-delay syses. We presened an LMI approach o solve he arx easure assgnen proble. I was shown ha he adssble soluon se of he arx easure assgnen proble s convex. I s also shown ha he arx easure assgnen proble s equvalen o an LMI feasbly proble. A necessary and suffcen condon for he exsence of oupu feedback conrollers o he arx easure assgnen proble s obaned. Fnally an llusrave exaple s gven o show he correcness of he proposed approach. Our approach does no need o fnd a coon posve defne arx and he verfcaon of sably s very easy. Sulaon resuls have verfed and confred he effecveness of he new approach n he sulaneous sablzaon of a collecon of nerval e-delay syses. 9

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