STABILITY CRITERION FOR HYBRID SYSTEMS WITH DELAY. Laviniu Bejenaru

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1 STABILITY CRITERION FOR HYBRID SYSTEMS WITH DELAY Lavu Bejearu PhD sude, Deparme of Auomac Corol, Uversy of Craova, Romaa Emal: yahoo.com Tel: Absrac: Ths paper preses hybrd sysems wh delay ad oe crero for sably of hese. Usg me-doma echque, I use he Lyapuov aspec volved sably. The paper preses a Lyapuov-Razumkh sably resul emphaszes characersc of Lyapuov-Razumkh fucos. Keywords: Hybrd sysems, sably, delay, Lyapuov-Razumkh fucos. INTRODUCTION Dffere aspecs from real lfe ca be modeled by hybrd sysems. Ths kd of sysem has bee kow a grea aeo he las decade because he applcao may felds, a specal aeo gvg o corol. Geerally speakg, a hybrd sysem s a mure of couous ad dscree saes. Oe characersc of hese s ha may cases he couous saes chacg by a dscree law. The rader ca read dffere aspecs of he hybrd sysems (va der Schaf ad Schumacher, 2; Lberzo, 23) or, lke pecewse smooh dyamcal sysems ame (d Berardo e al., 23). The eamples ca be draw from a wde rage of applcao areas, cludg process corol (Learso e al., 996), cosraed mechacal sysems (Broglao, 999), robocs (Spog ad Vdyasagar, 989; Proe, 22), power sysems (Hskes, 24), ad power elecrocs (Yua e al., 998). I fac, ay physcal devce ha ehbs hyseress, or corol loop wh a-wd-up lms (Goodw e al., 2), s effecvely a hybrd sysem. More deals abou hybrd sysems ad may refereces ca be foud (DeCarlo e al., 2). The presece of he algorhms hybrd sysems gves effcecy hose ad here are may algorhms, see (Aa e al., 25), ad he refereces from here. There are may resuls o sably ad sablzao hybrd sysems ad some of hese ca be read (DeCarlo e al., 2) bu, however, here are few papers o swched (lear) sysems wh delay. I (Guagmg e al., 22) s preseed a corollably aalyss for swched lear sysems wh delay he corol sgal ad (Zha e al., 23) s suded sably ad L2 ga aalyss for swchg symmerc sysems wh me delay. Sably ad sablzao of swched lear sysems wh sae delay s suded (Guagmg ad Wag, 24). A varao sably sudy s made usg Lyapuov-Krasovsk fucoals (Balakrsha e al., 24). I (Praja ad Jadbabae, 25) he Lyapuov-Razumkh fucos are used for safey verfcao of me-delay sysems. Ths paper s orgazed as follows. I Seco 2 I prese some oos, Seco 3 s preseed he resul of hs paper ad Seco 4 s he cocluso. 2. PRELIMINARIES Dffere aspecs of he hybrd sysems are sudyg. We eed oe mahemacal model for sudy ad dffere oos ad hose are preseed here. 2. Sources of delays Modelg he hybrd sysems usg problems from real lfe, we epec o oba some delays because oly oe deal model we do have ay fluece of he physcal characerscs of he compoes of

2 he pla. We have: -delays rasmsso: he physcal characerscs fluece he rasmsso me. I area of compuer ad commucaos eworks we mus ake o cosderao whe desgg corol laws he delays mposed by he commucao ework ha we use o rasm/receve formao for he dusral process uder corol. For eample, a hybrd sysem operaes over a dusral real-me commucao ework, usually amed eworked hybrd sysem, see (Davrazos ad Koussoulas, 22). Ths s a resul of usg a dscree eve supervsory coroller commucag wh he local corollers hrough a commucao ework. As a resul of hs, he formao, ha he process eceeds some predefed lms sae varables, such as emperaure, s perceved by he supervsor wh a delay, whch depeds o he commucao ework. Somemes here are sysems wh a lower boud o he delay me. Those ca model commucao or compuaoal delays ha preve o make dsc, dscree acos quck successo. The placeme of such delay cosras s ofe used o preve he syhess of Zeo corollers whch sasfy he safey propery oly by vrue of eforcg fely may eves fe me. -sldg mode: he rajecory ca come from a Ω I subse of he sae space ad o swch se ca have a sldg mode for a me τ before leaves ha swch se ad goes o Ω j subse as ca see Fg.. I Fg. 2 we have a suao where he swch map s o he boh sdes of he swch se ad we have a delay because he sldg mode o he swch se. Tryg o model hs behavor we ca cosder he e maps: : ~ a : ~ a ~ + Σ ; f : a ; Σ ; f : ~ a ( Σ o f ) o ( Σ o f ad he applcao s. The map Σ could has he e form derved from sldg A S flow: Σ = e ~ τ ~ ( ), where A = T A, wh T a erm whch depeds o he form of he source sysem. τ Fg. The sldg mode for a rajecory has as effec a delay sae of he hybrd sysem wh a value equals wh me sldg mode + ) S ~ ~ Fg. 2 Oe delay geeraes by he sldg mode. The swch maps are o he boh sdes of he swch se. 2.2 Mahemacal model Oe of he characerscs of he hybrd sysems s ha here s o a geeral model, bu we have may models for dffere problems ad applcaos. You ca see more deals abou models (DeCarlo e al., 2). I he geeral class of hybrd sysems we cosder delays o saes obag he e mahemacal model whch has he τ value of he delay: & ( = f ( ) + D ) + α(, = ϕ(, [,], ) = ϕ() = = s(, )) () ude α( = sau α( = BB u(. We ca have f () = A for swched lear sysems: ~ & = Aq ( ) + D ) ) = ϕ(, [,] = s(, )) whch s a smplfed form of (). 2.3 Lyapuov-Razumkh fucos (2) The codos for sably s ofe gve by a fuco whch has he propery ha s posve defed ad has he dervaves alog s rajecores egave defed ad s ame s Lyapuov fuco. A smlar aspec of he sably s by usg Lyapuov-Razumkh fucos. The Razumkh fucos meas ha oe eeds he egavy of he Lyapuov-Razumkh caddae o for all rajecores, bu oly for some crcal oes defed by he evoluo of he sysem over oe delay erval [-τ, ]. I he lear case, f oe assumes a smple quadrac Lyapuov caddae, hs sably resul s srogly relaed wh he S - procedure of

3 Yakubovch, see (Yakubovch, 977). Le C([-τ, ], R ) deoes he Baach space of couous vecor fucos mappg he erval [-τ, ] o R wh he opology of uform covergece. We have from (Nculescu, 2): Theorem (Razumkh Sably Theorem) Suppose ha he fuco f:r C([-τ, ], R ) R akes bouded ses of C([-τ, ], R ) bouded ses of R ad suppose u; v; w: R + R + are couous, odecreasg fucos such ha u(s), v(s), w(s) posve for s >, u() = v() =. Assume ha here ess a couous fuco V:R R R such ha: u( ) V (; ) v( ) for R, R. The followg saemes hold: a) V & (, ) ) f V(+θ, +θ)) < p(v(, )), θ [-τ, ], he he rval soluo of & ( = f (, ),, wh he al codo ( θ ) = φ( θ ), θ [,], s uformly sable. b) If here ess a couous odecreasg fuco p: R + R + ; p(s) > s, such ha V & (, ) ) f V(+θ, +θ)) < p(v(, )), θ [-τ, ], he he rval soluo of & ( = f (, ),, wh he al codo ( θ ) = φ( θ ), θ [,], s uformly asympocally sable. If u(s) + as s +, he he rval soluo s globally asympocally sable. 2.4 Swch ses Le Q be a dscree lo, Q = {, 2, 3,, Q } The fuco s:h(=r Q) Q by s(, ) = j, where R ad,j Q, meas ha we have a swch of dscree sae from sae o sae j. To pu ogeher all couous saes whch are volved hs chage, we defe swch se. Oe mage of he swch ses s (Rubesso ad Learso, 2). I (Bracky, 995) he oo of swch se s replaced by swch surface. The swch se S, j s S { R s(, j}, j = ) = ( ) (3) For each Q, he vecor feld f, : R R s assumed o be locally Lpschz couous. Ths codo s ecessary o assure some properes of he vecor feld. for he de fed. The swch se ca be gve by swch fucos. So, f a swch fuco s a map s, j : R R, he he swch se ca be defed as S, j = { s, j () = }. Geerally, he swch fucos represe hyperplaes he eeded sae space,.e. s, j ( ) = C, j + D, j. Le assume ha we have for our sysem m swch ses. 3. LYAPUNOV-RAZUMIKHIN STABILITY CRITERION IN HYBRYD SYSTEMS The problem of sably for hybrd sysems wh delays s mpora he coe whch he delays for sysems appear from dffere measures of echcal characerscs of he plas. The sably resuls are eesos of Lyapuov heory where he esece of a absrac eergy fuco sasfyg cera properes verfes sably. 3. Sably crero Here I gve oe sably crero for hybrd sysems wh delays. I use he characersc of Lyapuov- Razumkh fucos ad he ma dea s ha I mpose sably codos o rajecory whch swch from oe paro compoe of sae space o aoher by swch ses o delay erval. Le cosderae a paro of sae space = R, UΩ where for every Ω we have a V Lyapuov- Razumkh fuco assocaed. We suppose u, v, w: R + R + are couous, odecreasg fucos such ha u(s), v(s), w(s) posve for s >, u() = v() =. Theorem 2. For a hybrd sysem wh delay () we cosderae ha we have above paro ad he V :R + R R fucos wh he e characerscs: a. u( ) V (, ) v( ), R +, Ω R b. here are he couous odecreasg fucos p : R + R +, p (s) > s such ha V & (, ) ) f V (+θ, +θ)) < p (V (, )), θ [-τ, ]. c. V j () V () f swch from sae oward j sae coform wh S j. The he equlbrum po s sable. Proof: Tryg o esablsh sably equlbrum po he sese of Lyapuov, mus be show ha for ay R > (ay R > such ha he ball BBR() s cluded Ω ) here ess r(r) > such ha < r mples ha ( < R for all. Le he al sae (, ) H be chose such ha < r, where H H s he subse of he al saes, H = R Q as subseco 2.4. Due o he frs codo, ad

4 he couy of he u, v fucos, for ay R > here ess r(r) > such ha v(r) < u(r) as ca be see Fg. 3. Le k deoe he cosecuve mes whe he rajecory passes from oe rego o aoher. Because τ s he legh of he delay, we ca spl he posve as he ervals [ h -τ, h ], where h = + h τ. For h = ad =, we have he erval [-τ,]. Le he rajecory sars from Ω for (=) ad whch have he propery V (+θ, +θ)) < p (V (, )), θ [-τ, ]. If he rajecory moves from Ω o Ω j o [ h -τ, h ] he he c. codo from heorem assure he decreasg of he Lyapuov fuco values or, wh oher words, of he eergy. If Ω for (, ) he V() = V (), (, ). By egrao we oba: V() V()) - w(, (, ). If s fe, he he rajecory ever leaves Ω ad soluo s sable, usg a. ad b. properes coform Theorem, Razumkh sably heorem from (Hale ad Verduy Luel, 993) or (Nculescu, 2), so he cocluso. Oherwse, le he rajecory says Ω j for ( k, k+ ). Coform wh he verfcao sep from he mahemacal duco mehod, we assume ha we have (4) for ( k, k+ ). We assume ha he me for swch from Ω j o aoher rego Ω h s k+2, so for ( k+, k+2 ) he rajecory s sde of Ω j. If we egrae o ( k+, k+2 ), we have V() V( k+ )) - w(, k + From he c. codo we have V j ( k+ +ε)) V ( k+ - ε)) wh ε >, ε. So, usg (5) ad he (4) wh = k+, resuls: V() V( k+ )) - w( k k + V())- + w( - w( = V()) - k + (4) (5) w( for ( k+, k+2 ). Ad from duco resuls ha we have (4) for ay (=). We have w s posve ad hece V() V( ) for all. Fally, we have u( ( ) V() V( ) v( ) v(r) u(r) ad resul ha ( < R for all. I hs way we have he codo for sably. v(r) u(r) r v( ) R V() u( ) Fg. 3 The vsualzao of v(r) < u(r). For ay R> here s r(r)> wh above relao The sably characersc usg Lyapuov- Razumkh ad hs uly s for a hybrd sysems wh delay gve by ha he rajecory fellow he codo oly o delay erval [-τ, ]. O he oher had, we ca replace V & (, ) ) wh V& (, ). 4. CONCLUSIONS The paper preses hybrd sysem wh delay ad oe crero for sably of hs. Usg me-doma echque, I use he Lyapuov aspec volved sably. The paper preses a Lyapuov-Razumkh sably resul emphaszes characersc of Lyapuov-Razumkh fucos. The eeso sably o hybrd sysems s aurally ad ca be usefull. REFERENCES Aa, S.A., M. Alamr ad C. Caudas de W (25) Subopmal Corol of swched olear sysems uder locao ad swchg cosras I: Proceedgs of he IFAC World Cogress. Praha, Czech Republc. Balakrsha, H., Dukkpa N., McKeow N. ad Toml, C.J. (24). Sably Aalyss of Swched Hybrd Tme-Delay Sysems Aalyss of he Rae Corol Proocol. Workg paper. hp // su-valley. Saford. edu / hybrd/. Bracky, M. S. (995). Sudes hybrd sysems: Modelg, aalyss ad corol. PhD dsserao. Dep. of elecrcal ad Compuer Egeerg, MIT. Broglao, B. (999) Nosmooh Mechacs: Models, Dyamcs ad Corol, 2d edo. Sprger- Verlag, New York.

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