Stability of time-varying linear system

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1 KNWS 39 Sbiliy of im-vrying linr sysm An Szyd Absrc: In his ppr w considr sufficin condiions for h ponnil sbiliy of linr im-vrying sysms wih coninuous nd discr im Sbiliy gurning uppr bounds for diffrn msurs of prmr vriions r drivd Kywords: im-vrying linr sysm ponnil sbiliy, sbiliy of linr sysm sympoic sbiliy, Lpunov ponn Srszczni: W ryul rozwżn są wruni wysrczjąc n sponncjlną sbilność dl zminnych w czsi ułdów liniowych zrówno ciągłych j i dysrnych Prznown są różn miry zpwnijąc sbilność orz wyprowdzon zosło górn ogrniczni n zminność prmrów zpwnijąc sbilność (Sbilność zminnych w czsi ułdów liniowych) Słow luczow: zminn w czsi ułdy liniow, sponncjln sbilność, sbilność ułdów liniowych, sbilność sympoyczn, wyłdni Lpunow INTRODUCTION Sbiliy nd sbiliy condiions r on of h mos imporn problm in sysm projcing procss Ensuring h sbiliy for sysms is min issu h dcids h corrc cion During h consrucion, subsqun ss nd simulion i is ncssry o chc whhr h s objc is sbl nd/or w nd o dfin h sings for which his sysm is sbl In drmining sbiliy, usful nd quic o ssss r h sbiliy condiions For sionry sysms sbiliy condiions r wll nown Thr r mny mhods dscribd in h lirur for such sysms h drmin condiions for h sysm sbiliy For compl or nonsionry sysms whr s dpnds on swiching signl h sbiliy condiions r no so obvious nd problm of nsuring sbiliy or drmining whhr h sysm is sbl or no, is no n sy nd rivil s In his ppr w sudy problms concrning ponnil sbiliy of linr im-vrying sysm of h form: nd (, () ( ) ) ), n () If h funcion in () is picwis consn hn sysm () is clld swichd linr sysm If ll of h mrics r Hurwiz, hn i is possibl o nsur h sbiliy of h ssocid swichd sysm by swiching sufficinly slowly bwn h sympoiclly sbl consiun ppropri im invrin sysms This mns h insbiliy riss in () s rsul of rpid swiching bwn hs vcor filds Givn his bsic fc, nurl nd obvious mhod o nsur h sbiliy of () is o somhow consrin h r which swiching s plc Th bsic id of consrining h swiching r hs pprd in mny sudis on im vrying sysms ovr h ps numbr of dcds [, 8, 3] On of h bs nown nd mos informiv of hs sudis ws givn by Chrls Dsor in 99 in his sudy of slowly vrying sysms [4] Th bsic problm considrd by Dsor ws o find condiions on h swiching r h would nsur h sbiliy of n unforcd sysm of h form (), whr is mri vlud coninuous funcion such h for som h condiion R A is sisfid Thr r wo y poins o mphsiz hr; firsly, h sbiliy of h im-vrying sysm cn b nsurd by suibly consrining h r of vriion of, nd scondly h consrin on driviv of is drmind by Lypunov funcion ssocid wih h sysm I is wll-nown h if, for ch, ( ), ll ignvlus of, ()) r lying in h propr opn lf hlf compl pln (in opn uni circ, hn h sysm (), (()) is no ncssrily ponnilly sbl (s g ) Eponnil sbiliy is scurd if, ddiionlly, h prmr vriion of,, ()) is "slow nough", s [, ] Howvr, hs r quliiv rsuls In [8] quniiv rsuls r drivd for coninuous im This mn uppr bounds for h ignvlus nd for h r of chng of which nsur ponnils sbiliy of () r drmind Rsuls of his ind r prsnd in Scion I sms h similr rsuls for discr im r unnown In his ppr w prsn such rsuls In Scion prvious chivmns nd drivr sbiliy condiions in coninuous im sysms r prsnd, imporn horm ws in nw, diffrnly wy provd, hr r lso mpls of coninuous im sysms shown In Scion 3 similr condiions for discr im sysms r drivd nd illusrd wih numricl mpl in Scion 4 Scion 5 includs conclusion CONTINUOUS TIME SYSTEM Considr h homognous linr im-vrying nn diffrnil quion (), wih PC( R, R ), whr nn PC( R, R ) dnos h s of picwis coninuous rl n by n mri funcions on R, L, b h usul innr produc on n R, h ssocid norm

2 4 KNWS nd B inducd opror norm for ny linr opror B L( R n, R n ) Dno by (, h rnsiion mri of () nd by X, fundmnl mri W will ssum h is boundd i hr is consn M such h M for ll (3) Dfiniion Th sysm () is sid o b ponnilly sbl if hr is C, such h s, s C for ll s (4) I pprs h ponnil sbiliy cn b chrcrizd in rms of Bohl ponns Dfiniion Th Bohl ponn (A) of h sysm () is givn by ln(, lim sup (5) s s I cn b shown [3] h for h Bohl ponn w hv h following formul (A) s inf : M s, s M () Morovr h Bohl ponn is uppr smi-coninuous wh is crucil in mos prurbion qusions nd sysm () is ponnilly sbl if nd only if If is consn mri i is wll-nown h () is ponnilly sbl iff h rl prs of h ignvlus of A r lying in h opn lf hlf pln For im-vrying sysm vn if hy r nlyic nd priodic, ponnil sbiliy dos nihr imply A nor dos for som > h condiion R C (7) R A for ll (8) gurn ponnil sbiliy Empl L [7] cos sin 5 cos sin (9) sin cos sin cos Thn A for ll nd i cn b sily vrifid h fundmnl mri is givn by cos sin X cos sin 3 3 Thus () is no ponnilly sbl Empl L [] cos sin () cos sin 5 5 sin cos 5 5 () cos sin Thn A, 3 for ll nd i cn b sily vrifid h fundmnl mri is givn by X () whr cos 3sin cos 3sin cos 3sin cos 3sin 3cos sin 3 cos sin 3 cos sin 3cos sin nd consqunly () is ponnilly sbl Th sysm prsnd in Empl is in som sns "oo fs" in ordr h condiion R A for ll implis ponnil sbiliy Vrious ssumpions on h prmr vriion of r nown, such h if is sufficinly smll hn nyon of h following condiions gurns ponnil sbiliy of (): A ( for ll [] (3) A A for ll, [] (4) sup A A h (5) As consqunc of h following Thorm, (5) implis ponnils sbiliy if is smll nough (5) is lss rsriciv hn similr condiion in [9], Lmm 3: lim sup A A for ll h () h Th disdvng of (3) - (5) is h hy r quliiv condiions in h sns h mus b smll nough W cn improv h rsuls nd giv quniiv bounds nn Thorm Suppos PC( R, R sisfis for som, M nd ll M (7) A sc : R s Thn h sysm () is ponnilly sbl if on of h following condiions holds ru for ll : (i) 4m (ii) ) is picwis diffrnibl nd 4n A ( 4n4 n- M Proof: W will us h following imporn inquliy du o []: n M ( ) for ll, nd for ll,m For fid () cn b rwrin in h form (8) ( A ) (9) nd for ) is soluion is givn by )( ) s A ( ) ( ) ds ()

3 KNWS 4 Hnc by (8) s ) ds () n for ll M, whr Muliplying his inquliy by Gronwll's Lmn yilds Thus ( ) nd pplying p ) ds () p ) ds (3) Now w prov h smns (i ) nd (ii) (i): Sinc ) M for ll s,, (3) implis for,m nd som h>: p M ) ( p M 4M h (4) Th funcion f :,M R dfind s f ( ) M M h is coninuous nd f(m)=-h Thus hr iss,m such h f (ii): Considr which solvs Lypunov quion A ( ) ( ) T s A s ds (5) T A ( I () nd sisfis for som c, c c ) Th driviv of ) is givn by I c I (7) ( ) ( ) R A ( T s A ( A T A s ( ds Now w show h (8) V (,, (9) is Lypunov funcion of () Is im driviv long ny soluion is d V (, d I, (3) W hv o show h R ( I (3) for ll Applying Coppl's inquliy (8) o (8) nd (5) yilds 4( n) M R ( (3) nd hus (3) holds if for som, 4( n) (33) M Dfin funcion g :, R, by 4( n) g ( ) M I is sily vrifid, h g() n chivs is minimum on, n vrifis (33) nd nds h proof 3 DISCRETE TIME SYSTEM For sysm () w dfin (, I nd This, s ) ) for s, s N (34) W will considr sysm () undr h ssumpions h hr iss consn M> such h ) M for ll N (35) W hv h following dfiniion Dfiniion 3 Th sysm () is sid o b ponnilly sbl if hr is C,, such h s, s C for ll s (3) I pprs h ponnil sbiliy cn b chrcrizd in rms of Bohl ponns Dfiniion 4 Th Bohl ponn (A) of h sysm () is givn by s s lim sup (, (37) Morovr for h Bohl ponn w hv svrl lrniv formuls (s []) limsup ( = inf sup ( = sn N sn s = inf r : M r s, s M rr (38) nd sysm () is ponnilly sbl if nd only if For discr im-vrying sysms similrly s for coninuou ponnil sbiliy dos nihr imply nor dos h condiion A for ll N (39) A for ll N (4) gurn ponnil sbiliy sup( ) Dno In [5] h following horm N hs bn provd Thorm For ch boundd squnc A N of mrics hr iss consn C> such h for ny w hv

4 4 KNWS A ( C A (4) In h proof of h min rsul of his scion w will us h following discr vrsion of Gronwll's inquliy [] Thorm 3 Suppos h for wo squncs u( ) N nd f ( ) N of rl numbrs h following inquliy holds for crin i u ( ) p q u( i) f ( i) (4) p, q R nd ll N, hn i u ( ) p qf ( i) (43) for ll N For fid () cn b rwrin in h form ) ) ) A A ) (44) nd for is soluion is givn by l ( ) A A l A l ( ) A (45) Hnc by (4) ) C l C A l A l (4) Muliplying his inquliy by C yilds C ) C l C A l A (47) Applying Gronwll's inquliy nd ing ino ccoun h ) ) M w obin l ( ) C M ( A) C ( A) M (48) W hv provd h following horm Thorm 4 If M, hn sysm () is ponnilly sbl 4 NUMERICAL EXAMPLE Considr sysm () wih 8 A (49) I cn b show h mri hs h form A 8 4, so w cn wri A A A, whr A 8 4 nd A I is h r of chng of ) Thn A, hus spcrl 8 rdius is A nd chosn mri norm 8 A s uppr bounds 4 For mri ) h sbiliy condiion using Thorm 4 is givn by: sup A N 8 M 58 (5) 4 8 Th discr im sysm whr w swichd bwn dfind mrics of h form ) is ponnilly sbl bcus h sbiliy condiion holds 5 CONCLUSION For sionry discr nd coninuous sysms in h lirur hr r dscribd h sbiliy condiions For compl sysm whr s dpnds on swiching signl, h sbiliy condiions r no so obvious W considr sbiliy condiions for slowly vrying sysms whr h prmr vriion of, ()) is "slow nough" In his ppr no only quliiv bu lso quniiv sbiliy condiions for such sysms r drmind Vry imporn is h drivd condiions us only informion bou mrics For chc, if discr sysm is ponnilly sbl, w nd only informion bou ignvlu spcrl rdius nd mri norm Imporn is h, h drivd sbiliy condiion for discr im sysm dosn dpnd on ordr of mrics REFERENCS [] Agrwl RP, Diffrnc quions nd inquliis Thory, mhods nd pplicions Nw Yor: Mrcl Dr, [] Coppl WA, Dichoomis in sbiliy hory, Lc Nos Mh, 9, Springr-Vrlg, Brlin l, 978 [3] Dlcii JuL, Krin MG, Sbiliy of soluions of diffrnil quions in Bnch spcs, AMS, Providnc, Rhod Islnd, 974 [4] Dsor CA, Slowly vrying sysm _ = (, IEEE Trnscions on Auomic Conrol, vol4, 99, p [5] Fuchs JJ, On h good us of h spcrl rdius of mri, IEEE Trnscions on Auomic Conrol, vol7, 98, p34-35 [] Guo D, Rugh W, A sbiliy rsul for linr prmr-vrying sysms, Sysms nd Conrol Lr vol4, 995, p -5

5 KNWS 43 [7] Hoppnsd FC, Singulr prurbions on h infini inrvl, Trns Am Mh Soc, vol3, 9, p5-535 [8] Ilchmnn A, Owns DH, Przl-Wolrs D, Suficin condiions for sbiliy of linr imvrying sysms, Sysms nd Conrol Lr vol9, 987, p57-3 [9] Krisslmir G, An pproch o sbl indirc dpiv conrol, Auomic, vol, 985, p45 43 [] Przyłusi K, Rmrs on h sbiliy of linr infini dimnsionl discr im sysms, Journl of Diffrnil Equion vol7, 988, p89- [] Rosnbroc HH, Th sbiliy of linr im dpndn conrol sysms, In Journl Elcr Conrol, vol5, 93, p73 8 [] Wu MY, A no on sbiliy of linr im-vrying sysms, IEEE Trns Auom Conrol, vol9,, 974 [3] Zhng J, Gnrl lmms for sbiliy nlysis of linr coninuous im sysms wih slowly im vrying prmrs, Inrnionl Journl of Conrol, vol58, 993, p mgr inż An Szyd Polichni Śląs Wydził Auomyi, Elronii i Informyi Insyu Auomyi ul Admic 44- Gliwic l: (3) mil: nbl@polslpl