CONTROL ROUTH ARRAY AND ITS APPLICATIONS

Transcription

1 3 Asa Joual of Cool, Vol 5, No, pp 3-4, Mach 3 CONTROL ROUTH ARRAY AND ITS APPLICATIONS Dazha Cheg ad TJTa Bef Pape ABSTRACT I hs pape he Rouh sably ceo [6] has bee developed o cool Rouh aay Soe foulas fo calculag he aay ae povded Cea popees ae vesgaed Usg, he poble of sablzao of blea syses va cosa cools s solved The by coveg he pobles, he ehod of cool Rouh aay s also used o solve soe ohe sably elaed pobles, such as he sablzao of cool syses, he sably of ucea Huwz ax ad he sably of eval aces Seveal algohs have bee developed o povde uecal soluos KeyWods: Rouh sably ceo, cool ouh aay, feedbac sablzao, eval aces, gowh algoh I PRELIMINARY The Rouh aay of he polyoal q(λ) s defed as Cosde a blea syse x = Ax+ u N x, x R () Ths syse evolves as a accuae appoxao o ueous dyacal pocesses egeeg, ecoocs, bology, ecology, ec Fo sace, he dyacs of he pedao-pey eaco descbed by he Loa-Volea equao [6] ay be cosdeed as a sae feedbac fo of () We sa wh he sablzao of syse () wh cosa cools Whe = ad = [7] povded ecessay ad suffce codos I hs pape, we povde a uecal ehod o solve hs poble Ou ew appoach s based o Rouh sably ceo Fo a easy efeece, we ce as he followg Defo [6] Cosde a polyoal q( λ) = q λ + q λ + + q () whee q q q q q q a a a a a a a a a q q q q q q q q a =, a = a q a q a q a q a =, a = a a 3 a a 3 a = a q q a a (3) (4) Mauscp eceved Sepebe 8, ; evsed Augus 3, ; acceped Jauuay 3, 3 Dazha Cheg s wh Laboaoy of Syse ad Cool, Isue of Syses Scece, Chese Acadey of Sceces, Bejg 8, PRCha TJ Ta s wh Dep of Syses Scece ad Mah, Washgo Uvesy, MO, USA Ths eseach s suppoed paly by NSF G598377, G99838 ad Naoal Key Pojec of Cha Theoe [6] The ecessay ad suffce codo fo q(λ) o be Huwz s ha all elees he fs colu of s Rouh aay (3) have he sae sg, ad ha oe vashes Whe he polyoal q(λ) s he chaacesc polyoal of a ax, he q = I s easy o see ha whe q > he deoaos (4) ca be eoved as

2 D Cheg ad TJ Ta: Cool Rouh Aay ad Is Applcaos 33 p = q, p = q 3 p = q q q q, p = q q q q p = p q p q, p = p q p q p = p p 3 p p (5) Fd a effce algoh o solve he polyoal equales They wll be dscussed hs seco To beg wh, we cosde he degees of cool Rouh aay I copue codg, hs foao s equed fo eoy aagee ec We gve a Lea, whch s well ow ad easly vefable The, a odfed Rouh ceo ca be saed as: Coollay 3 The polyoal q(λ) () wh q > s Huwz f ad oly f he elees of he fs colu of Rouh aay, as ( p, p,, p ) T (6), ae posve, whee he elees p j ae calculaed by (5) Poof I fac, s easy o chec ha F+ + = + ( ), = p a a whee a = p = q ad F, =,,, s he Fboacc sequece (,,, 3, ) Fo he above equaly s obvous ha a >, ples p >, Covesely, f p >,, ca be poved by aheacal duco ha a > The cocluso follows edaely The advaage of (5) s ha whe he ees of he ax ae polyoals of cool u The, he odfed Rouh aay, defed by (5), eas polyoals I wll be called he cool Rouh aay, whch s used o solve he sablzao poble of blea syse () We wll show ha seveal sablzao pobles ca be coveed o he sablzao poble of blea syse () Hece, he cool Rouh aay ay be used o povde uecal soluos o soe ohe sably ad sablzao pobles The pape s ogazed as follows: I seco, he sucue ad soe popees of he cool Rouh aay ae vesgaed Seco 3 dscusses he poble of solvg Rouh polyoal equales Seco 4 uses he echque developed o solve soe ohe sablzao pobles Paculaly, he sulaeous sablzao of a se of lea syses s cosdeed The he sably of he eval aces s vesgaed Seco 5 Two algohs ae poposed seco 6 o ge sably bouday of a sable ax II CONTROL ROUTH ARRAY To use cool Rouh aay appoach o solve he sably ad sablzao pobles wo hgs have o be doe: Fd foulas fo he polyoals (6) Lea Fboacc sequece, {,,,3, }, sasfes he dffeece equao F = F + F (7) + + Usg he al codo F = F =, he soluo of (7) s F = 5 5 Poposo Fo syse () he degee of p, as a polyoal of cool u = (u,, u ), s of possble hghes degee deg( p( u)) = 5 5 (8) (9) Poof Deoe by D = deg(p (u)) Noe ha deg( ps+ ( u)) = deg( ps( u)) +, s, s easy o ge he followg dffeece equao: D+ = D+ + D + D =, D = 3 Expess () a ax fo as D Befly, se D = + D+ D D X =, A=, b= D + The, X = AX + b Usg hs equao ecusvely, we have X A X ( A I) ( A I) b () () = + () Now we have oly o fgue ou A Noe ha A ca be

3 34 Asa Joual of Cool, Vol 5, No, Mach 3 expessed va Fboacc sequece as A F = F F Usg (7) ad he aheacal duco, s easy o ge ha A F F = F F+ The follows ha D F F = D+ F F+ 3 Paculaly, F F + F F+ D = F + 3F + F = F + 3F + + Usg (8), he cocluso follows fo a saghfowad copuao Nex, we povde soe foulas o calculae he cool Rouh aay Fs, we cosde he plaa case A saghfowad copuao shows he followg: Poposo 3 Le A, N, =,,, be axes Deoe by ( ) P( λ, u) = de λi ( A+ u N ) : = λ + P( u) λ+ P ( u) The P ( u) = ( ( A) + u ( N )), P ( u) = de( A) + ( ( A) ( N ) ( AN )) u < j + ( ( N ) ( N ) ( N N )) u u de( N) u j j j + (3) Le I = {,,, }, J = {,, }, ad I J be he se of appgs fo J o I The = (,, ) I J eas Z ad j, j =,, Lae o, o ephasze we also use I, I 3 ec fo J ={,}, J = {,,3} ec especvely Fo oaoal ease, se N = A, u =, ad defe a se, Σ, of + aces as Σ= { A, N, N,, N } = { N, N, N,, N } We deoe he -h colu of a ax M by M Le = (,,, ) I J The we ca cosuc a ax Σ fo Σ as ( N N N ) Σ = We also deoe u = u u u The we have he followg lea: Lea 4 Le A, N, =,, be axes, I = (,,, ), J = (,,, ) The de( A+ u N ) = de( Σ ) u (4) J I Poof Usg aheacal duco Whe = s val Assue (4) s ue fo, he fo +, splg fs colu o + colus wh espec o N, =,, We ge + deeas Expadg each deea alog fs colu ad usg duco assupo fo each oe caefully, he esul follows The oaos ad esul of Lea 4 ae deosaed he followg exaple Exaple 5 Le = 3, = The 3 3 = 7, ad J I = {,,,,, } J I = ( + ) = = {(), (), (), (),, ()} The, seg A as he -h colu of A ec, we have Σ = A, Σ = ( A A N ), Σ = ( A A N ), Σ = Σ = u u u u u u u u u ( AN A),, N; 3 =, =, = =,, = Hece de( A+ u N + u N ) = de( A) + de( Σ ) u + de( Σ ) u 3 3 de( 4 ) de( ) + Σ u + + N u To buld he cool Rouh aay, he oly dffcul pa s o ge he fs wo ows of I ohe wods, f we ow he coeffces of ( λi A+ un ) de ( )

4 D Cheg ad TJ Ta: Cool Rouh Aay ad Is Applcaos 35 he cosucg of Rouh polyoal aay becoes a saghfowad copuao We eed soe oe oaos: Gve Z + Le We deoe by S = { s = ( s,, s ) Z s < s < < s } I fac, S s a se of odeed subses of E = {,,, } cosuced by selecg dffee elees fo E ad odeg by he heed ode Le s = (s,, s ) S A o of A M, deoed by A (s), cosss of all s -h,, s -h ows ad s -h,, s -h colus elees of A e, A a a a = as s a s s a s s ss ss ss ( s) Slaly, we deoe ( s) ( A N N ) ( s) ( s) ( s) Σ =,,, Now we ae eady o pese ou a esul Theoe 6 Fo syse () he fs wo ows of s cool Rouh polyoal aay q ( u) q 4( u) q ( u) q ( u) q ( u) 3 5 ae deeed by q u u j ( s) j( ) = ( ) { de[( Σ ) ] }; j j s S I j =,,, (5) Poof I s well ow ha fo a ax A he chaacesc equao, λ + q λ has he coeff ces as q = A j = j j ( ) de( ( s) );,,, j s S Usg ad Lea 4, he esul follows by a caeful calculao III SOLVING ROUTH POLYNOMIAL INEQUALITIES I hs seco we povde a algoh fo solvg Rouh polyoal equales Oe way s o solve he ul-vaable polyoal equales decly We efe o [] fo soe elaed wos Ou pupose s o avod solvg he equales decly The algoh s based o he followg wo obvous facs: F: Syse () s sablzable by cosa cools, f oe of he followg axes s Huwz: A, ±N, =,, F: Syse () s sablzable by cosa cools, f ad oly f, hee exss λ, =,,, λ =, such ha λ A+ λ ( ± N ) (6) s Huwz Eve hough (6) s obvous, s advaage s ha we ow have oly o seach possble cool ove a co pac se: λ, =,, ad λ Noe ha sce (6) N ca be ehe posve o egave, o es we have o seach he possble soluo es fo ±N, =,, Fo sgle pu case, () becoes x = Ax+ unx We popose he followg algoh: Fs, we chec whehe oe of A, N, N s Huwz If yes, we ae doe If o, we use he followg algoh: Algoh 3 (Chec A, N:) Sep Calculae he cool Rouh aay fo A + un o ge he Rouh polyoal equales Sep Choose a sep legh < δ << Fd a ege M > such ha δ M < δ (M + ) Sep 3 Se δ u =, =,, M + δ Se = (Fs e =, ad lae wll be e-se) Fd j such ha p j ( u ) s he salles oe, e, p ( u ) = p ( u ) j If p j ( u ) >, we ae doe Ohewse, chec he values of hs p j o fd he salles such ha p j us o be posve If such does exs, go o Sep 5 Ohewse go o he ex Sep Sep 4 Fd aohe j such ha p j ( ) s he salles oe If p j ( ) > we ae doe Ohewse, sa fo ad go bac o Sep 3 Sep 5 If δ >, se δ = δ Sa wh = ad go bac Sep 3 fo oppose deco seachg Ohewse, ed he algoh The advaage of hs algoh s ha: bascally a each sep (each ) we have oly o chec he value of oe

5 36 Asa Joual of Cool, Vol 5, No, Mach 3 polyoal I s depede of he deso of Rea The cool u he algoh s chose by usg (6) u follows by choosg δ + δ λ =, λ = Whe = M,,,,, M, λ goes fo o The egave s ae eplaced by egave δ A oue has bee ceaed o ge he Rouh polyoals by usg (5) ad o es he exsece of he cool whch sablzes syse () The followg exaple s obaed by choosg axes ado Exaple 3 Le A, S be as: A= 3 3, S = 3 3 (7) 3 Boh ±S ae o Huwz The algoh shows ha o posve u ca ae A + us sable Bu o egave deco he Rouh polyoals of A us ae p = 39u 37 3 p = 9966u 937u + 865u p = 358u + 488u 97u The fs suable cool, evealed by he poga, s u = 766, whch sablzes A + us Fd a sable covex cobao of wo usable axes s useful soe ohe pobles, eg, swched syses [5] ec Nex, we cosde wo pu case Recall (6), we ca choose λ ad λ as depede paaees So he oly dffeece bewee oe ad wo pu cases s he seachg ego Fo wo pu case he seachg ego s R = {( λ, λ) R λ, λ, λ+ λ < } (8) We popose he followg couous oue, whch ca save seach e: Selec hee decos as D = (, ), D = (, ), D 3 = (, ) Sa fo he wegh cee of R, P = (/3, /3) Se < δ << as sep legh The go o δd oce, δd wce, δd 3 hee es, δd aga fo fou es,, ad so o The we have a agula spal whch flls he whole ego of R Oce such seachg oue s se, Algoh 3 eas avalable We gve aohe uecal exaple Exaple 33 Le A be as Exaple 3, M ad N be as he follows 5 M = 3 3, N = 4 (9) The algoh shows ha boh ±M ad ±N ae o Huwz, ad A + υ(±m) s o sablzable Moeove, A + υ(±m) + un s o sablzable Bu fo A υm un he coespodg Huwz polyoals becoe: p = υ p = 4 + 6υ+ 59υ 38u 3uυ 93υ p = u+ 865υ + 86u uυ+ 567υ u 3376u υ uυ 845υ p3 = 3744 u 36υ 348u 3594uυ υ 3966u 9 95u υ+ 776uυ υ The seachg pocess foud he fs pa of avalable cool s u = , υ = The A un υm becoes A un υm = whch s Huwz As fo >, Algoh 3 eas avalable Bu as = case, we have o fd a seachg oue whch fll he paaee ego R = {( λ,, λ ) R λ, =,, ; λ < } To ae he algoh oe effce, we should educe he possble dscouy ad epea Keepg oue couous ca educe he ube of shfg fo esg oe Rouh polyoal o aohe IV STATE OR OUTPUT FEEDBACK STABILIZATION I hs seco we wll cove he sablzao poble of ohe ds of syses o he poble of blea syses The he pevous ehod ca be used Sce he esuls hs seco ae followed by soe eleeay copuao, we sp all he poofs

6 D Cheg ad TJ Ta: Cool Rouh Aay ad Is Applcaos 37 Fs we cosde lea syse Le B be a ax Deoe a se of aces, B j, =,,, j =,,, as j ( B ) s, = Bsj, = whee subscpos aached o a ax, ae used o dcae he elees of he ax Hece, he -h colu of B j s he j-h colu of B, ad he ohe colus of B j ae zeo Cosde a lea syse x = Ax+ Bu, x R, u R () The sae feedbac sablzao poble ca be coveed o he sablzao of he blea syse j x = Ax+ u jb x j= () by cosa cools Le {u j, =,,, j =,,,} be a se of cools, whch sablze () Cosuc a ax F as F j = u j, he u = Fx sablzes () Ths appoach ca be used fo he poble of sulaeous sablzao of a se of syses [,5] Exaple 4 Cosde he followg se of syses wh =, =, d d d a a b x = x u, d I d d + d a a b () I s easy o wo ou he Rouh polyoal equales as d d d d a + b u + a + b u < d d d d d d d d d d d d aa aa + ( ab ab ) u + ( ab ab ) u >, d I (3) Ths se of Rouh polyoal equales becoes a se of lea equales Solvg s saghfowad The sulaeous oupu feedbac sablzao ca be solved slaly Exaple 4 [5] Cosde he followg se of syses: A =, A =, A3 = ; 3 Bj =, j =,,3; C = C = (), C3 = ( ) (4) I s easy o wo ou he Rouh polyoal equales fo he blea syses, deduced by he oupu feedbac syses as 3+ u < u < u < So hey cao be sablzed by cosa ga oupu feedbac V STABILITY OF INTERVAL MATRICES As aohe applcao of Rouh polyoal aay, we cosde he poble of he sably of eval aces Fs, we cosde oe paaee case: Gve A, N, whee A s Huwz Fd he lages eval (a, b), such ha { Au ( ) = A+ un u ( a, b)} (5) ae all sable Ths poble has bee dscussed by seveal papes [,3] We gve a uecal ehod o solve Algoh 5 Sep Fd he Rouh polyoal se (6) fo A + un Sep Choose a sep legh < δ << Fd he lages M > such ha δ M < Sep 3 Se δ u =, δ fd salles such ha p ( u ): = p ( u ) ; + + Sep 4 Choose a sep legh < δ << u + u Fd he lages M > such ha δ M < u + u Se u = u + * δ, fd salles such ha p ( u ): = p ( u ) + + Se b = u ; Sep 5 Replace N by N, use he sae pocedue o fd u ad se a = u Rea I hs algoh we seach u wce The ea-

7 38 Asa Joual of Cool, Vol 5, No, Mach 3 so s he fs seach u + u ay be lage So we eed secod seach The sep legh of he secod seach ay be chose accodg o he legh u + u I Algoh 3 we do eed hs because he sablzables of A + un ad N + A ae he sae The s u easy o see ha he sablzably of A + un s o sesve o lage u Exaple 5 Cosde he sably of A + un, whee 4 3 A+ un = u The algoh povded he Rouh polyoals as p = 56u+ 89 p = u + u u+ p = 7956u + 7u 3355u u + Se sep legh as a fs seach, ad ( u ) a secod seach The esul s: Fo fs seach we have u = 888, u + = 8585 The fo secod seach we have u = 84579, u + = 8468 Replace N by N A fs seach we have u = , u + = A secod seach we have u = 3563, u + = So he uecal aswe s [ 3563, 84579] The poble of sably of eval aces ca be saed as [4]: Gve wo Huwz aces: P Q, Le I[P, Q] be he se of eval aces, ad V[P, Q] be he se of veex aces, e, I[ P, Q] = { A p a q }, j j j VPQ [, ] = { A IPQ [, ] a = p o a = q} j j j j A codo gve [4], whch s supposedly ecessay ad suffce fo he sably of I[P, Q], as f all veex aces V[P, Q] ae sable A coue-exaple has bee gve [] We show ha he cla [4] s coec whe = Theoe 53 Whe = ad 6 veex aces V[P, Q] ae sable, he all aces I[P, Q] ae sable Poof I fac, wha we have o pove s ha he followg ax s sable: λ p + ( λ) q λ p + ( λ ) q A( λ) = (6) λ p + ( λ) q λ p + ( λ ) q whee λ j,, j =, I s equvale o showg ha ( A( λ)) < de( A( λ)) > (7) l Deoe by V s = (υ j ) V[P, Q], he veex ax, such a way as f l = p he υ = p, f l = q he υ = q ec Fo sace, V qp qq q p = q q pp ec Now sce V pp ad V pp qq ae sable, we ow ha he aces ae egave, e, Hece p q + p < + p < λ ( p + p ) + ( λ )( q + p ) Slaly, sce pp qq = λ p + ( λ ) q + p < (8) V ad V qq ae sable we have qq λ p + ( λ ) q + q < (9) Mulplyg (8) by λ, ad (9) by ( λ ) ad sug up yeld ( A( λ)) = λ p + ( λ ) q + λ p + ( λ ) q < Fs equaly of (7) s poved Now cosde he secod equaly Fs of all, we show ha λ p + ( λ) q υ de υ υ λ p + ( λ) q υ = de λυ + ( λ) υ υ p υ q υ = λ de + ( λ) de > υ υ υ υ (3) Nex we fx υ = λ p + ( λ q ) Usg he sae c o υ, υ ad υ shows ha de(a(λ)) > We esae he sably poble of eval aces a slghly dffee fo: Gve a Huwz ax A ad a peubao bouday ax W, Fd lages such ha

8 D Cheg ad TJ Ta: Cool Rouh Aay ad Is Applcaos 39 A+ A, <, Aj Wj (3) s Huwz If all he elees of W ae, becoes a sadad obusess poble of a Huwz ax [] We call W he wegh ax We popose a ew algoh, aely, Rouh posve polyoal algoh We defe Defo 54 A pa (c, p(u)) s called a posve polyoal fo f c R, ad p(u) s a polyoal wh he popey ha p() = ; All he coeffces of p(u) ae oegave Gve a polyoal q(u) = cu The pa (c, p(u)) s sad o be q(u) s posve (polyoal) fo f c = q() pu ( ) = c u Noe ha f u R he = (,, ) Z + ad = 3 Le P be he se of posve polyoal fos, e, P = {(c, p(u))} We defe wo opeaos: : P P as (c, p (u)) (c, p (u)) = (c c, p (u) + p (u)), ad : P P as (c, p (u)) (c, p (u)) = (c c, c p (u) + c p (u) + p (u)p (u)) Iuvely, wha we dd s o elage he o- cosa es of a polyoal so ha he eo ca be bouded by he value of posve polyoals Tha s, f (c 3, p 3 (x)) = (c, p (x)) (c, p (x)), (c 4, p 4 (x)) = (c, p (x)) (c, p (x)), ad x <, he c3 p3() q() x ± q() x c3 + p3() c4 p4( ) q( x) q( x) c4 + p4( ) We call he followg algoh he Rouh posve polyoal algoh, whch poduces a se of equales They wll be called Rouh posve polyoal equales The basc dea fo hs algoh s o educe ul-vaable polyoals o sgle-vaable polyoals by elagg he eo Algoh 55 (Rouh posve polyoal algoh) Sep Cosuc he fs wo ows of he Rouh polyoal aay fo A + un by usg (5) Sep Cove elees he fs wo ows o he posve polyoal fos as P( u) = ( c, p ( u)), =,, Sep 3 Replace u P (u) by o geeae P (); Sep 4 Usg P (), ad o cosuc Rouh posve polyoal aay as a = P (), a = P 3 () a P P P P 3 =, a = P P 4 P P 5 a = a P 3 a P, a = a P 5 a3 P a = a a a a,, 3,, 3, We call a = (c, p ()), =,,, Rouh posve polyoals ad call he followg Rouh posve polyoal equales c p () >, =,,, The ex poposo follows fo he defo ad cosuco Poposo 56 Rouh posve polyoal equales (3), has soluo >, whch assues (3) o be Huwz Exaple 57 [] The followg exaple has bee used [] o copae he esul of [] wh [3] 3 A = Usg above algoh he Rouh posve polyoal equales ae 3 > > The soluo s < 3968 e, A < Hee he o s defed as ax a j Copae hs esul wh he esul obaed [] The esul [] s A < 389 Sce we use dffee o, s easy o chec ha ehe oe ca cove he ohe I s obvous ha he soluo obaed by Rouh posve polyoal algoh s oly a cosevave esao To ge eal bouday we popose he followg gowh algoh Sla dea ca be foud [8] We eed soe ew coceps ad oaos Le W j be a ax wh ees as

9 4 Asa Joual of Cool, Vol 5, No, Mach 3 j ( W ) s, ( s, ) (, j) = Ws, s =,ad = j Defo 58 Gve a adus >, a po V, whch s a ax, s sad o be a veex of he cube C (A), f V = A+ ± W j= A se S j ± as j j s ± ( s) ( j) j S = { A ± W + µ W, µ } s called he ±j-face especvely The uo of he faces s called he suface of he cube, deoed by S 3 Gve a sep d = a po H s called a ode o j S ±, f H = A± W + c dw, c Z [, ] j j s j s s ( s ) ( j) 4 Fo a sable ax, P, usg Algoh 55, a sable adus ρ > s obaed The ay eal ube, < ρ s sad o be a feasble value of P Algoh 59 (gowh algoh) Sep Fd he al adus ρ() va Algoh 55; Sep Afe -h eao assue we ge a adus ρ(), wh sable cube C() Fd salles p such ( ) ha s a feasble value of all veces of ρ p C() Sep 3 If such p does exs, he sably adus ρ = ρ() Teae he algoh If p =, se ρ( + ) = ρ() Go bac o sep ρ( ) If p >, se sep d = Go o ex sep p Sep 4 Fo all odal pos wh espec o sep d fd ρ( ) he salles q such ha s feasble + p q fo all odal pos Sep 5 If q =, se ρ( + ) = ρ()( + ) Go o p sep ρ( ) If q >, se sep d = ad eplace p by p p + q + q Go o Sep 4 Theoe 5 The sequece ρ(), =,,, poduced by Algoh 59 coveges o he sably adus ρ Poof I fac, we have oly o pove hee hgs: Fs of all, a each eao ew ρ() ca be geeaed I pogag laguage: Sep 4 - Sep 5 s o a dead-loop To see hs, we cla ha afe a fe e q us be zeo The a ew ρ() s obaed Noe ha ρ() j S ± of Cube C() < ρ So a each po, p, o he suface hee exss a posve feasble ube p Sce he value of a polyoal depeds o s coeffces couously ad he fs colu elees of he Rouh posve polyoal aay ae oooe fucos (hs s a ey po), s easy o see ha p couously depeds o p Sce S s copac ad p > s a couous fuco of p Thee exss a posve ube >, whch s he u value of p, such ha p, p S ρ( ) Now a wos case whe he sep d = <, p ( ) he s feasble fo all odal pos e, q = ρ p We ae ou of he loop: Sep 4 Sep 5 Secodly, we have o show ha ρ() ρ as Sce ρ() < ρ s oooously ceasg, he l, deoed by ρ *, exss Now assue ρ* = l ρ( ) < ρ The we cosde he suface S ρ of he cube C * ρ * Le ρ * > be he lages coo feasble value of all pos o S ρ By couy, hee exss a posve δ < * ρ * /4 ad s sall eough such ha whe ρ * δ < < ρ * he lages coo feasble value,, o he suface S, sasfes: > ρ* / Now le be lage eough such ha ρ() > ρ * δ Recall he seleco of p Sep of he algoh, oe ρ( ) sees ha > p, whee s he lages coo fesble value o S ρ The we have ρ( ) ρ( + ) = ρ( ) + > ρ( ) + p / > ρ * δ + ρ* /4 ρ * whch s a coadco Fo a eval aces I[P, Q], se he ddle ax M = (P + Q)/ ad N = (Q P)/ The he eval aces ae Huwz, ff, M s sable ad hee exss > such ha ρ() >, whee ρ() s geeaed by usg Algoh 59 o M + un Noe ha based o Theoe 53 whe = we eed o chec odes So he Algoh ca be splfed

10 D Cheg ad TJ Ta: Cool Rouh Aay ad Is Applcaos 4 VI ILLUSTRATIVE EXAMPLES Exaple 6 (Couao of Exaple 57) Recall ha Exaple 57 we have 3 A = Usg coe gowh algoh, we have he followg esul: ρ[] = 3776, ρ[] = 37847, ρ[] = 4847, ρ[3] = 443, ρ[4] = 44, ρ[ ] = 444, 5 The cocluso s he lages sable adus s 444 I has bee poed by [4] ha he sablzao ego of A ca be A < 54 No oe ca cove he ohe Exaple 6 Cosde 6 λ 6 λ M = 5 ; N λ = 6 λ 6 λ, 6 λ λ =,, 3, 4, 5 The algoh yelds he followg esuls: Table The sable adus fo dffee λ Case(λ) ρ[] Ieao No ρ VII CONCLUSION Based o a ld odfcao of Rouh aay, he cocep of cool Rouh aay was oduced he pape The calculag foulas ad soe popees wee peseed I was used o solve uecally he sablzao of blea syses usg cosa cools The was show ha he sulaeous sablzao of a se of lea cool syses ad he sably of eval aces ec ca be solved by coveg he o blea cool syses Fally, wo eave algohs wee poposed fo solvg he poble of sably of eval aces ad soe elaed pobles The advaage of hs ew appoach s ha ca povde pecse soluos Bu fo hghe ode syses he copug coplex peve s plee REFERENCES Adeso, BDO, NK Bose, ad EI Juy, Oupu Feedbac Sablzao ad Relaed Pobles- Soluo va Decso Mehods, IEEE Tas Auoa Co, Vol, No, pp (975) Bash, BR ad CV Hollo, Coue-exaple o a Rece Resul o he Sably of Ieval Maces by S Balas, I J Co, Vol 39, No 5, pp 3-4 (984) 3 Bese, D ad W Haddad, Robusess Sably ad Pefoace Aalyss fo Lea Dyac Syses, IEEE Tas Auoa Co, Vol 34, pp Balas, S, A Necessay ad Suffce Codo fo he Sably of Ieval Maces, I J Co, Vol 37, No 4, pp 77-7 (983) 5 Boussad, JR ad CS McLea, A Algoh fo Sulaeous Sablzao Usg Decealzed Cosa Ga Oupu Feedbac, IEEE Tas Auoa Co, Vol 38, No 3, pp (993) 6 Cas, JL, Nolea Syse Theoy, Acadec Pess, Ic Ld Olado (985) 7 Chabou, R, G Salle ad JCVvalda, Sablzao of Nolea Syses: A Blea Appoach, Mah Co Sgal Sys, Vol 6, pp 4-46 (993) 8 Esla, M, Se by Se Geeao of Robus Sable Maces, IEEE Tas Auoa Co, Vol 38, No 4, pp (993) 9 Gacoa, G ad JBeussou, Pole Assge fo Ucea Syses a Specfed Ds by Sae Feedbac, IEEE Tas Auoa Co, Vol 4, No, pp 84-9 (995) Gog, C ad S Thopso, Sably Mag Evaluao fo Ucea Lea Syses, IEEE Tas Auoa Co, Vol 39, No 3, pp (994) Gudes, AN, ad MG Kabul, Sulaeous Sablzao of Lea Syses Ude Sable Addve o Feedbac Peubaos, IEEE Tas Auoa Co, Vol 4, No, pp (995) Lee, JH, WH Kwo ad JW Lee, Quadac Sably ad Sablzao of Lea Syses wh Fobeus No-bouded Uceaes, IEEE Tas Auoa Co Vol 4, No 3, pp (996) 3 Ooba, T ad Y Fuahash, Sably Robusess fo Lea Space Models -a Lyapuov Mappg Appoach, Sys Co Le, Vol 9, pp 9-96 (997) 4 Su, J, Coes o Sably Mag Evaluao fo Ucea Lea Syses, IEEE Tas Auo-

11 4 Asa Joual of Cool, Vol 5, No, Mach 3 a Co, Vol 39, No, pp (994) 5 Wcs, MA ad P Pelees, Cosuco of Pecewse Lyapuov Fucos fo Sablzg Swched Syses, Poc 33d IEEE Cof Decs Co, pp (994) 6 Wlles, JL, Sably Theoy of Dyacal Syses, Joh Wley & Sos, Ic New Yo (97) 7 Zhou, K ad PP Khagoea, Sably Robusess Bouds fo Lea Sae-space Models wh Sucued Uceay, IEEE Tas Auoa Co, Vol 3, No 6, pp 6-64 (987)