An Improved Stabilization Method for Linear Time-Delay Systems

Transcription

1 IEEE RASACIOS O AUOMAIC COROL, VOL. 47, O. 11, OVEMBER 191 An Improved Sablzaon Mehod for Lnear me-delay Sysems Emla Frdman and Ur Shaked Absrac In hs noe, we combne a new approach for lnear medelay sysems based on a descrpor represenaon wh a recen resul on boundng of cross producs of vecors. A delay-dependen creron for deermnng he sably of sysems wh me-varyng delays s obaned. hs creron s used o derve an effcen sablzng sae-feedback desgn mehod for sysems wh parameer uncerany, of eher he polyopc or he norm-bounded ypes. Index erms Delay-dependen sably, lnear marx nequaly (LMI), sablzaon, me-delay sysems, me-varyng delay. I. IRODUCIO he problem of reducng he conservasm enaled n applyng fne-dmensonal echnques o asses he sably of lnear sysems wh me delay has araced much aenon n he pas few years [1] [6]. All hese echnques provde suffcen condons only for he asympoc sably of hese sysems and hey enal a consderable conservasm whch sems from wo man sources. he frs cause for conservasm s he model ransformaon used o descrbe he sysem whch makes more amenable for analyss [7], [8] and he second reason for conservasm s he boundng mehod used o derve he bounds on weghed cross producs of he sae and s delayed verson whle ryng o secure a negave value o he dervave of he correspondng Lyapunov Krasovsk funconal. he search for he mos approprae model ransformaon has led o four man approaches [9] [11]. he mos recen one [9], he one ha s based on a descrpor represenaon of he sysem, whch s equvalen o he orgnal sysem, mnmzes he overdesgn ha sems from he model ransformaon source of conservasm [11]. he conservasm ha sems from he boundng of he cross erms has also been sgnfcanly reduced n he pas few years. An mporan resul for mprovng he sandard boundng echnque of, e.g., [], has been proposed n [1]. Indeed, combnng he laer wh he descrpor model ransformaon lead n [1] and [11] o an effcen delay-dependen sably creron ha was also used n synhess for sablzaon and opmal performance. Only recenly, an mprovemen of he boundng echnque has been proposed [1]. he laer generalzes he one n [1] and he resulng crera ha are obaned n [1] are, herefore, more effcen han hose found n [1]. I s he purpose of hs noe o combne he boundng mehod of [1] wh he descrpor model ransformaon of [9] and [11] n order o derve a mos effcen sably creron for sysems wh me-varyng delays. hs creron s hen appled o solve he problem of robus sablzng he sysem n presence of eher norm-bounded or polyopc unceranes by means of sae-feedback conrol. he resulng creron s appled o an example aken from [1], and s superory o he resuls of he laer s demonsraed. oaon: hroughou hs noe, he superscrp sands for marx ransposon, R n denoes he n-dmensonal Eucldean space, R nm s he se of all n m real marces, and he noaon >, for Manuscrp receved Ocober 15, 1; revsed February 8, and June 8,. Recommended by Assocae Edor L. andolf. hs work was suppored by he C&M Maus Char a el Avv Unversy. he auhors are wh he Deparmen of Elecrcal Engneerng-Sysems, el-avv Unversy, el-avv 69978, Israel. Dgal Objec Idenfer 1.119/AC R nn, means ha s symmerc and posve defne. he space of vecor funcons ha are square negrable over [ 1) s denoed by L. II. A EW SABILIZAIO MEHOD We consder he followng lnear sysem wh me-varyng delays: _x() = A x( ()) Bu(); x() =(); [h; ] = (1) x() R n s he sysem sae, u() R q s he conrol npu,, A and B are consan n n marces, s a connuously dfferenable nal funcon, and h s an upper-bound on he medelays, =1;. For smplcy only, we ook wo delays 1 and. he resuls of hs secon can be easly appled o he case of mulple delays 1;...; m. he marces of he sysem are no exacly known. Denong we assume ha =[A A1 A B ] = f j j ; for some f j 1; he verces of he polyope are descrbed by j =[A (j) A (j) 1 A (j) B (j) ] : f j =1 () In Secon III, we exend our resuls o he case he uncerany n he sysem parameers obeys he norm-bounded model [17]. As n [11], we consder wo dfferen cases for me-varyng delays () are dfferenable funcons, sasfyng for all : () h ; _ () d < 1; =1; : () () are connuous funcons, sasfyng for all, () h, =1;. oe ha n he pas, he Razumkhn s approach was he only one ha was o cope wh Case I) of fasly varyng delays. he Krasovsk approach for hs case was nroduced recenly n [11]. We seek a conrol law ha wll asympocally sablze he sysem. u() =Kx() (4) A. Sably Issue In hs secon, we consder B =. Represenng (1) n an equvalen descrpor form [9] or _x() =y() /$17. IEEE = y() A x() = E _x() = _x() I = = A I x() A A (5a) y(s)ds (5b) () y(s)ds (5c) ()

2 19 IEEE RASACIOS O AUOMAIC COROL, VOL. 47, O. 11, OVEMBER wh x() = colfx(); y()g, E = dagfi;g, he followng Lyapunov Krasovsk funconal s appled: V () =x ()E x() V V (6) = 1 ; 1 > E = E V = V = h y (s)r y(s)dsd x ( )S x( )d: () (7a-e) he followng resul s obaned for Case I). Lemma 1: Under Case I), (1), wh B =, s asympocally sable f here exs n n marces < 1 ; ; ; S ; Y 1 ; Y ; Z 1 ; Z ;Z, and R >, =1; ha sasfy he followng lnear marx nequales (LMIs): and R Y = < 9 Y1 A 1 A Y S 1(1 d 1) S (1 d ) Z ; =1; (8a,b) Y =[Y 1 Y ] Z = 9= I A I roof: oe ha h Z Y Z1 Z Z ; =1; I A I S h R Y x ()E x() =x () 1 x() : (9a-c) and, hence, dfferenang he frs erm of (6) wh respec o gves d x ()E x() =x () 1 _x() =x () _x() : (1) d Subsung (5) no (1), we oban (11), as shown a he boom of he page, = 1 I A A I = = I I S h R () = 1 x () A y(s)ds: (1) Snce, by [1], for any a R n, b R n, R nn, R R nn, Y R nn, Z R nn, he followng holds: b a a R Y a b Y Z b R Y (1) Y Z we apply he laer on he expresson we have prevously obaned for. From (1), akng = =, R = R, Z = Z, A Y = Y, a = y(s) and b = x(), we oban, for =1;, (14) found a he boom of he page. Subsung he laer and (1) no (11), we oban ha dv () () 1 () d he frs equaon shown a he boom of he nex page holds, and () = colfx(); y(); x( 1); x( )g. Snce 1 = he LMIs n (8) lead o _ V <, whle V and, hus, (1) wh B = s asympocally sable [5], [15]. Choosng n Lemma 1 S! and Y = [ A ] we oban he followng resul for he case B. Corollary 1: Under Case II), (1), wh B =, s asympocally sable f here exs n n marces < 1; ; ; Z 1; Z ; Z and R >, =1; ha sasfy he followng LMIs: 9 1 < and R [ A ] Z ; =1; dv () x () x() d (1 d )x ( )S x( ) h y ( )R y( )d (11) () = = h [ y (s) x ()] y (s)r y(s)ds y (s)r y(s)ds R Y [ A ] Y Z A y (s)(y [ A ] )x()ds y(s) x() ds x() Z x()ds _x (s)(y [ A ] )x()ds x() Z x() y (s)r y(s)ds x ()(Y [ A ] )x() x ( )(Y [ A ] )x() h x() Z x(): (14)

3 IEEE RASACIOS O AUOMAIC COROL, VOL. 47, O. 11, OVEMBER 19 Z = Z 1 Z Z I 91 = = A I h Z ; =1; I = A I : hr Remark 1: I follows from (8a) ha he dagonal elemens S (1 d ), = 1, are negave and, hus, S >, snce by assumpon d < 1. Remark : A queson may arse as o wheher he sandard Lyapunov creron can be resored when leng h go o. akng R = I, Z = 1=,!1, Y =[ A ] and <S! we oban 9= I I A I A I Y Y = ( = A) ( = A ) 1 ( = A ) For = I;! and = 1 > he requremen ha 9 < becomes 1( A )( A )1 < ; 1 > : (15) = = I follows from (15) ha f he sysem wh h = s asympocally sable, hen here exss 1 > ha solves (15) and, hus, (8a),(b) possess a soluon for small enough h>. he laer can be readly used o verfy he sably of (1) over he uncerany polyope () [1]: = f jj ; for some f j 1; he verces of he polyope are descrbed by j =[A (j) A (j) 1 A (j) ] f j =1 by solvng he LMI smulaneously for all he verces, applyng he same 1,, S, Y 1, Y, and R, =1,. : In he sequel, wll be mporan o deermne he condons for achevng H1 norm of (1) less han 1, u s he npu vecor and he conrolled oupu s gven by z() =Lx() L1x( 1) Lx( ): (16) Smlarly o he dervaon of he bounded real lemma (BRL) n [11], we oban he followng. Lemma : Under Case I) he H1 norm of (1) and (16) s less han one f here exs n n marces <1; ; ; S ; Y 1; Y ; Z 1; Z ;Z and R, =1, ha sasfy (17), as shown a he boom of he page, and (8b), 9 s gven by (9c). roof: Addng he erm z ()z() w() w() o dv ()=d n (11) and subsung for z() from (16), he resul follows from he argumens used o derve Lemma 1 he las column and row blocks n (17) are obaned by applyng he sandard Schur s formula [1]. B. Sae-Feedback Sablzaon he resuls of Lemma 1 can also be used o verfy he sably of he closed loop obaned by applyng (4) o (1) (wh B 6= ) f we replace A n (8a) by A BK and verfy ha he resulng nequaly s feasble over he polyope defned n () by solvng he LMI smulaneously for all he verces, applyng he same 1,, S, Y 1, Y, and R, =1,. he problem wh (8a) s ha s lnear n s varables, only when he sae-feedback gan K s gven. In order o fnd K, consder he nverse of. I s obvous from he requremen of <1, and he fac ha n (8)( ) mus be negave defne, ha s nonsngular. Defnng 1 =Q = Q 1 Q Q 1=dagfQ; Ig (18a) (18b) we mulply (8a) by 1 and 1, on he lef and on he rgh, respecvely, and (8b), on he lef and on he rgh, by dagfr 1 ; Q g and dagfr 1 ; Qg, respecvely. Applyng Schur formula o he emergng quadrac erm n Q, denong S = S 1, Z 1 Z Z = Z = Q Z Z Q and R = R 1, =1, and choosng [Y 1 Y ] = " A [ ], " R nn s a dagonal marx, we oban, smlarly o [14], he followng. heorem 1: he conrol law of (4) asympocally sablzes (1) for all he delays ha belong o Case I) and for all he sysem parameers ha resde n he uncerany polyope, f for some dagonal marces 1 = h Z I (Y [ A ] ) Y [ I ] Y1 A Y A A1 S1(1 d1) S(1 d) 9 Y1 Y [ L ] B A1 A I q S1(1 d1) L1 S(1 d) L I r < (17)

4 194 IEEE RASACIOS O AUOMAIC COROL, VOL. 47, O. 11, OVEMBER " 1; " R nn, here exs: <Q 1; Q (j) 1 ; S ; R > ; Z (j) j R nn, =1,, j =1,, and Y R qn ha sasfy he LMIs shown n (19) a he boom of he page, 4 (j) = Q (j) Q (j) Q 1(A (j) h Z (j) he sae-feedback gan s hen gven by " A (j) ) Y B (j) ;j =1; ;...;: K = YQ 1 1 : () he prevous resul represens a delay-dependen suffcen condon for he conroller of (4) o guaranee, for Case I), sably over he enre uncerany polyope. he correspondng delay-ndependen resul s obaned, sll for Case I), by subsung " =, R = I n and Z = n (19) and akng he lm ends o nfny. he las wo row and column blocks of (19a) wll dsappear due o R! 1. Consderng, sll n he delay-ndependen case, he more general conrol law we replace A (j) u() = n (19) by A (j) = K x( ) (1) B (j) K and oban he followng. Corollary : In Case I), he conrol law of (1) asympocally sablzes (1) ndependenly of he delay lenghs, for all he sysem parameers ha resde n he uncerany polyope, f here exs:q 1 >, S 1, S, Q (j), Q (j) R nn and Y R qn, =; 1; ha sasfy he equaons shown a he boom of he page 4 (j) g = Q (j) Q (j) Q 1A (j) Y B (j) ; j =1; ;...;: (a,b) he sae-feedback gans are hen gven by K = Y Q 1 1 K = Y S 1 ; =1; : () Remark : In Case I), he conrol law of (1) canno be readly ncorporaed n he resul of heorem 1 because of he quadrac erm ha wll emerge n (19b) when A (j) s replaced by A (j) B (j) K. One can, however, solve he desgn problem wh he feedback law of (1) by applyng he mehod of [14] whch convers he problem of dealng wh delayed componens n he npu o one wh he conrol law of (4) by addng, n seres o he npu, smple lnear componens. he ransference of hese componens s almos I and he augmened sysem ha resuls can be readly solved usng he LMIs of heorem 1. he delay-dependen resul for Case II) s obaned by deleng V n (6) and choosng " = I n, =1,. he heory hen develops along he lnes ha led o heorem 1. hus, he resul for Case II) s he followng. Corollary : he conrol law of (4) asympocally sablzes (1) for all he delays ha belong o Case II) and for all he sysem parameers ha resde n he uncerany polyope, f here exs: Q 1 >, Q (j), Q (j) Q (j) h Z (j) 1 4 (j) Q 1 Q 1 Q (j) Q (j) h Z (j) A (j) 1 (I n " 1 ) S 1 A (j) (I n " ) S (1 d 1 ) S 1 (1 d ) S S 1 S h 1Q (j) h Q (j) h 1 Q (j) h Q (j) h 1 R 1 < (19a) h R R R " A (j) Z (j) 1 Z (j) Z (j) ; =1; (19b) Q Q Q (j) Q (j) 4 (j) g Q 1 Q 1 S 1 B (j) Y 1 S B (j) Y A (j) 1 A (j) (1 d 1 ) S 1 (1 d ) S S 1 S <

5 IEEE RASACIOS O AUOMAIC COROL, VOL. 47, O. 11, OVEMBER 195 Q (j), R 1, R R nn and Y R qn ha sasfy he LMIs shown n he equaon a he boom of he page, and (19b), " = I, and ^4 (j) = Q (j) Q (j) Q 1 ( = A (j) ) he sae-feedback gan s hen gven by (). h Z (j) Y B (j) ; j =1; ;...;: III. SABILIZAIO OF SYSEMS WIH ORM-BOUDED UCERAIIES he resuls of Secon II were derved for he case he unknown parameers of (1) le n a gven polyope. An alernave way of dealng wh unceran sysems s o assume ha he devaon of he sysem parameers from her nomnal values s norm bounded [17]. In our case, consder he sysem _x() = (A H1()E )x( ()) = (B H1()E )u() x(s) =(s)s (4) x() and u() are defned n Secon II and he me delays are defned n (). he marces A, =,1,, B, H and E, =;...; are consan marces of approprae dmensons. he marx 1() s a me-varyng marx of unceran parameers sasfyng 1 ()1() I 8 : (5) We consder also, for a gven posve scalar ^", he followng augmened sysem: _() = = (s) =(s) s z() =^"E () wh he performance ndex A ( ()) Bu()^" 1 Hw() J(w) = 1 ^"E ( )^"E u() (6a,b) (z z w w)d (7) w L s an exogenous sgnal. I has been explcly proved n [17], n he case whou delays, ha he exsence of a soluon o he Rcca equaons or LMIs ha are obaned when solvng he H1 sae-feedback conrol problem for he augmened sysem (6) wh he ndex (7), whou delays, guaranees he sably of (4), under he same feedback law, for all 1() ha sasfy (5). he proof follows, n fac, from he small gan heorem [16] whch can also be appled o our case of rearded sysems. he sysem (4) can be wren as _x() = A x( )Bu() = ^" 1 H1^"[ E E 1 E E ] colfx() x( 1 ) x( ) u()g: hs sysem can be looked a as (6), 1 of (5) s he feedback gan from z of (6b) o w n (6a). Consder he closed-loop sysem G ha s obaned from (6a),(b) by applyng he sae-feedback conroller. I follows from he exsence of a soluon o he above H1 sae-feedback conrol problem, ha G s asympocally sable and ha he H1 norm of he ransference of G from w o z s less han 1. Applyng, herefore, he feedback gan 1(); whch sasfes (5), around G, follows from he small gan heorem [16] ha he resulng closed-loop sysem wll reman asympocally sable. Snce he laer closed-loop sysem s dencal o he closed-loop obaned from (4) by applyng he same sae-feedback conroller (n he sense ha x() ()), hs conroller also sablzes (4). In order o apply he aforemenoned argumen o (4), one should use a BRL creron ha wll guaranee a H1-norm less han one o he closed-loop sysem obaned from (6). An effcen delay-dependen BRL has recenly been derved n [11]. he laer s based however on he boundng echnque of [1]. In order o benef from he new mehod n [1], we derve he followng resul from Lemma, applyng he same ransformaon ha was used n dervng heorem 1 and akng = ^". heorem : In case A, (4) s sablzed va he conrol law of (4) for all 1() ha sasfy (5), f for some dagonal " 1 ; " R nn and a scalar < ^, here exs <Q 1;Q 1; S ; R > ; Z j ; R nn, =1,, j =1,, and Y R qn ha sasfy he LMIs, shown n he equaon a he boom of he nex page. 4 g = Q Q Q 1 (A " A ) h Z Y B : (8a-c) he sae-feedback gan s hen gven by K = YQ 1 1 : (9) Remark 4: he delay-ndependen verson n Case I) s obaned by solvng (8a) and (8c), " =, Z =and he eghh and he nnh row and column blocks are omed. In Case II), he correspondng delay-dependen resul s obaned by solvng he LMIs of heorem for " = I, =1;, n (8a) he fourh, ffh, sxh, and sevenh row and column blocks are deleed. Remark 5: he resuls of heorems 1 and apply he unng parameers " 1 and ". he queson arses how o fnd he opmal combnaon of hese parameers. One way o address he unng ssue s o choose for a cos funcon he parameer mn ha s obaned whle solvng he feasbly problem usng Malab s LMI oolbox [18]. hs scalar parameer s posve n cases he combnaon of he unng parameers s one ha does no allow a feasble soluon o he se of LMIs consdered. Applyng a numercal opmzaon algorhm, such as he program fmnsearch n he opmzaon oolbox of Malab Q (j) Q (j) h Z (j) 1 ^4 (j) h 1 Q (j) h Q (j) Q (j) Q (j) h (j) Z h 1 Q (j) h Q (j) h 1R1 h R <

6 196 IEEE RASACIOS O AUOMAIC COROL, VOL. 47, O. 11, OVEMBER [19], o he above cos funcon, a locally convergen soluon o he problem s obaned. If he resulng mnmum value of he cos funcon s negave, he unng parameers ha solve he problem are found. he laer opmzaon procedure s me consumng. Our experence shows ha akng " 1 = " ="I, " s a scalar, a one-dmensonal search for " s easly performed. he cos funcon, or he bound on delay ha sll manans sably, exhb hen a convex behavor wh respec o " and a clear opmum value of he laer s obaned. In he examples we solved, he sngle unng parameer " acheved resuls ha are que close o hose obaned by he fmnsearch program. IV. EXAMLES We demonsrae he applcably of he above heory by solvng he second example n [1] for a sysem wh norm-bounded uncerany and he hrd example from [], we neglec unceranes. Example 1: he problem n [1] s one a sae-feedback conrol s sough ha sablzed (4) for one delay wh A = 1 :5 A1 = 1 B = 1 H =:I E =I: For he case of d =, he maxmum bound h for whch he sysem s sablzed by a sae-feedback was found n [1], afer 99 eraons, o be.45. Applyng he resul of heorem, a maxmum bound of h =:5865 s obaned usng " =:757I and =:8. he correspondng feedback-gan marx s K = [:155 4:4417 ]. Usng he mehod of [11] (an mproved verson of [1]), whch s based on he boundng mehod of [1], a maxmum value of h =:55 was acheved wh = :;" = : and K = [:9 5:8656 ]. I s noed ha he compuaonal complexy of he soluons n [1] and [11] and he presen mehod s he same. Comparng o [1], he eraons requred here almos compensae he ncrease n he dmenson of he LMIs ha s caused by usng he descrpor approach. In Case II) (fasly varyng delays), he correspondng resuls are: h = :496, K = [:4 5:168 ] and = :8 by Remark and h = :489, K = [:884 1:8558 ] and =:1 by [11]. I follows from hs ha he heory of [11] provdes sablzaon resuls ha are superor o hose obaned n [1]. hs s rue n spe of he fac ha he former apples he old boundng mehod of [1] and ha handles me-delays ha can vary very fas. he resuls ha are obaned usng he heory of he presen noe surpass hose found by he mehods of [11]. he combnaon of he descrpor approach and he new boundng mehod of [1] s shown o be superor o all oher soluons ha were proposed n he leraure. Example []: We address he problem of fndng a sae-feedback sablzng conroller for (4) wh one delay and whou unceranes (H = E =), A = A1 = :9 B = 1 : Applyng he mehod of [, Cor..], was found ha, for _, he sysem s sablzable for all < 1. For, say, = :999 a mnmum Q Q h Z 1 4 g Q1 Q1 h1q Q Q h Z ^H A1(I n "1) S1 A(I n ") S h1q ^I (1 d1) S1 (1 d) S S1 S h1r1 hq Q1E Y E hq S1E SE < hr ^I R R " A Z 1 Z Z ; =1;

7 IEEE RASACIOS O AUOMAIC COROL, VOL. 47, O. 11, OVEMBER 197 value of = 1:88 resuls for K = [ : ]. By [11], he sysem s sablzable for h 1:48:. By heorem 1, he correspondng value s h = 1:51 for " = :59 and K = [58:1 94:95 ]. V. COCLUSIO he problem of fndng a sae-feedback conroller ha asympocally sablzes a lnear me-delay sysem wh eher polyopc or norm-bounded uncerany has been solved. A delay-dependen soluon has been derved usng a specal Lyapunov Krasovsk funconal. he resul s based on a suffcen condon and hus enals an overdesgn. hs overdesgn s consderably reduced due o he fac ha s based on he descrpor represenaon and snce apples a new boundng mehod. REFERECES [1] S. Boyd, L. El Ghaou, E. Feron, and V. Balakrshnan, Lnear Marx Inequaly n Sysems and Conrol heory. hladelpha, A: SIAM, [] X. L and C. de Souza, Crera for robus sably and sablzaon of unceran lnear sysems wh sae delay, Auomaca, vol., pp , [] V. Kolmanovsk and J.-. Rchard, Sably of some lnear sysems wh delays, IEEE rans. Auoma. Conr., vol. 44, pp , May [4] V. Kolmanovsk, S. I. culescu, and J.. Rchard, On he Lapunov- Krasovsk funconals for sably analyss of lnear delay sysems, In. J. Conrol, vol. 7, pp , [5] V. Kolmanovsk and A. Myshks, Appled heory of Funconal Dfferenal Equaons. orwell, MA: Kluwer, [6] M. Mahmoud, Robus Conrol and Flerng for me-delay Sysems. ew York: Marcel Decker,. [7] V. Kharonov and D. Melchor-Agular, On delay-dependen sably condons, Sys. Conrol Le., vol. 4, pp ,. [8] K. Gu and S.-I. culescu, Addonal dynamcs n ransformed medelay sysems, IEEE rans. Auoma. Conr., vol. 45, pp ,. [9] E. Frdman, ew Lyapunov-Krasovsk funconals for sably of lnear rearded and neural ype sysems, Sys. Conrol Le., vol. 4, pp. 9 19, 1. [1] E. Frdman and U. Shaked, A descrpor sysem approach o H conrol of lnear me-delay sysems, IEEE rans. Auoma. Conr., vol. 44, Feb.. [11], Delay dependen sably and H conrol l: Consan and mevaryng delays, In. J. Conrol, o be publshed. [1]. ark, A delay-dependen sably creron for sysems wh unceran me-nvaran delays, IEEE rans. Auoma. Conr., vol. 44, pp , Apr [1] Y. S. Moon,. ark, W. H. Kwon, and Y. S. Lee, Delay-dependen robus sablzaon of unceran sae-delayed sysems, In. J. Conrol, vol. 74, pp , 1. [14] E. Frdman and U. Shaked, ew bounded real lemma represenaons for me-delay sysems and her applcaons, IEEE rans. Auoma. Conr., vol. 46, pp , Dec. 1. [15] E. Frdman, Sably of lnear descrpor sysems wh delay: A Lyapunov-based approach, J. Mah. Anal. Appl., vol. 7, no. 1, pp. 4 44,. [16] K. Zhou, J. C. Doyle, and K. Glover, Robus and Opmal Conrol. Upper Saddle Rver, J: rence-hall, [17].. Khargonekar, I. R. eersen, and K. Zhou, Robus sablzaon of unceran lnear sysem: Quadrac sablzably and H conrol heory, IEEE rans. Auoma. Conr., vol. 5, pp , Mar [18]. Gahne, A. emrovsk, A. J. Laub, and M. Chlal, LMI Conrol oolbox for Use wh Malab. ack, MA: MahWorks, [19]. Coleman, M. Branch, and A. Grace, Opmzaon oolbox for Use wh Malab. ack, MA: MahWorks, Ulmae erodcy of Orbs for Mn Max Sysems Ypng Cheng and Da-Zhong Zheng Absrac he ulmae perodcy heorem s an mporan resul n mn max sysems heory. I was frs proved by Olsder and erennes n her unpublshed work. In hs noe, we presen a new proof. hs proof s also based on wo mporan heorems: he exsence of cycle me for any mn max funcon and he ussbaum Sne heorem. However, wo dfferen echnques, pure mn max funcon and condonal redundancy, are used o oban wo mporan nermedae resuls. he purpose of hs noe s o provde a smple alernae proof o he ulmae perodcy heorem. Index erms Dscree-even sysems, mn max funcons, ulmae perodcy. I. IRODUCIO Mn max funcons (e.g., [1] and []) arse n modelng he dynamc behavor of dscree-even sysems wh maxmum and mnmum consrans, such as dgal crcus, compuer neworks, manufacurng plans, ec. Mahemacally, a mn max funcon F n :! n s bul from erms of he form x a, 1 n and a, by applcaon of fnely many max and mn operaons n each componen. In such a model, f we denoe he me of he kh occurrence of even by x(k), hen x(k 1)=F (x(k)). Mn max funcons are homogeneous 8x n ; 8h ;F(x h) =F (x) h monoonc wh respec o he usual produc orderng on 8x; y n ;x y ) F (x) F (y) and nonexpansve n he sup norm 8x; y n ; kf (x) F (y)k kx yk: In hs noe, we sudy a mn max funcon as a dynamcal sysem, and we are concerned wh he behavor of he orbs of a mn max sysem F, namely he sequences x(), x(1), and x(),..., x() n and x(k 1) = F (x(k)). herefore, we shall be usng funcon and sysem nerchangeably n he sequel, dependng on he conex. In sudyng he behavor of mn max funcons, one s emped o fnd ou wheher all or some mn max funcons exhb he followng properes. ropery C(F ): he lm (F; ) =lmk!1f k ()=k exss. I s o be noed ha f for some he lm (F; ) exss, hen for all hs lm exss and s ndependen of, because F s nonexpansve n he sup norm. hs lm s called cycle me of F, and we wll denoe by (F ) n he sequel. ropery I(F ): F has a cycle me wh dencal coordnaes,.e., here s a such ha (F )=(;...;). Manuscrp receved ovember 1, ; revsed Ocober 9, 1 and February 18,. Recommended by Assocae Edor R. S. Sreenvas. hs work was suppored n par by he aonal Scence Foundaon of Chna under Gran 6741, n par by he Key Fundamenal Research Foundaon of Chnese Mnsry of Scence and echnology under Gran 97117, and n par by he Research Foundaon of snghua Unversy, Bejng, Chna. he auhors are wh he Deparmen of Auomaon, snghua Unversy, Bejng 184, Chna (e-mal: ypng99@mals.snghua.edu.cn). Dgal Objec Idenfer 1.119/AC n /$17. IEEE