The Computation of Common Infinity-norm Lyapunov Functions for Linear Switched Systems

Transcription

1 ISS Egd UK Jour of Iformto d Comutg Scece Vo. 6 o The Comutto of Commo Ifty-orm yuov Fuctos for er Swtched Systems Zheg Che Y Go Busess Schoo Uversty of Shgh for Scece d Techoogy Shgh 93 Ch Fcuty of Scece gbo Uversty of Techoogy gbo 356 Ch Receved r 9 cceted y 7 ) bstrct. Ths er studes the robem of the comutto of commo fty-orm yuov fuctos. For set of cotuous-tme TI systems or dscrete-tme TI systems whose system mtrces re uer trgur form or ower trgur form t s roved tht there exst commo fty-orm yuov fuctos for them. The four gorthms of comutg commo fty-orm yuov fuctos re reseted. Fy sever exmes re sted. Keywords: Commo yuov fuctos fty-orm swtched systems. Itroducto swtched systems s oe tht combes cotuous or dscrete ) dymcs wth ogc-bsed swtchg mechsm tht determes brut mode swtches the system oerto t vrous ots tme []. ost reserch hs bee devoted to the stbty of swtched systems [-4]. s we ow yuov fuctos y mortt roe the stbty theory of cotro systems for some tme. I vew of ths cosderbe mout of recet wor hs focused o yg smr des to swtched systems. ost recety my uthors hve derved codtos for the stbty of er swtched systems bsed o the exstece of commo yuov fuctos for ther costtuet systems [5-7]. For umerc d rctc resos commo qudrtc yuov fuctos re usuy chose [8-]. Howeverqudrtc yuov fuctos c be too coservto d efforts hve bee devoted to the deveomet of other tyes of commo yuov fuctos. Commo fty-orm yuov fuctos re mortt oe whch hve bee studed to cosderbe extet [-]. How to comute commo yuov fuctos s of mortce becuse ths w rovde some megfu resuts of cotro systems. I ths er we gve gorthms of comutg commo fty-orm yuov fuctos.. Premres Throughout ths ote the foowg otto s used: et R deote re dmeso sce. m R deotes the set of m re mtrces. s the coverse of R. The orms x re defed by d x x = = ) Ths wor ws suorted by to Scece Foudto of Ch uder grt: ) Shgh edg dsce rojectuder grt:s35) Scetfc reserch rojects of Zhejg Educto Dertmet uder grt:y675) Pubshed by Word cdemc Press Word cdemc Uo

2 6 Zheg Che et : The Comutto of Commo Ifty-orm yuov Fuctos for er Swtched Systems The fty orm of mtrx Cosder fmy of er systems Defto fucto x mx x. = s defed by R mx = j= j x& = x x R R =. ) V x) = Wx W R s sd to be commo yuov fucto of the er systems ) f there exst mtrces such tht d m = Q R W = QW ) m q jj q j 3) for j m. q j etres of the mtrx Q. Gve set of stbe dscrete-tme TI systems descrbed by the foowg equtos = j x t ) = x t) x R R =. 4) Defto The fucto of the vector orm form V x) = Wx W R m R W ) = s sd to be commo fty-orm yuov fucto for the set of systems4) f there exst mtrces m Q R = such tht w hve the mtrx retos d for. W = QW 5) Q 6) 3. Comutto of commo fty-orm yuov fuctos or et us cosder = ) or 4) hve the form s foows The foowg we gve the theorem beow.. JIC em for cotrbuto: edtor@jc.org.u

3 Jour of Iformto d Comutg Scece Vo. 6 ) o JIC em for subscrto: ubshg@wu.org.u 63 Theorem et R = be Hurwtz such tht re uer trgur mtrces the systems ) shre commo fty-orm yuov fucto = Wx x V ) where = W d > >. Proof. From ) we hve = W W Q = = =. 7) otcg 3) we hve ) 3 3 =. 8) Sce re Hurwtz 8) re wys true. By ) ) ) we get ) ) ). So c be equ to y umber )) m ) ) ) d we c fx. Smry geer we ) d ) m ).

4 64 Zheg Che et : The Comutto of Commo Ifty-orm yuov Fuctos for er Swtched Systems Fy we obt. Ths cometes the roof. Bsed o the roof bove we c get the gorthm of comutg commo fty-orm yuov fuctos. gorthm Deote = m ) =. ) ) ) ) Ste Comute m ). For y m )) outut ) ) Ste Set = Ste 3 Comute for y ) outut Ste 4 = Ste 5 If goto ste otherwse sto Ste 6 utut V x) = Wx. I wht foows we gve exme. Suose = = ). By comutg we c get 8 W =. Whe re ower trgur mtrces we hve the foowg theorem. Theorem et R = be Hurwtz such tht re ower trgur mtrces the systems ) shre commo fty-orm yuov fucto V x) = Wx where d W = > >. The roof of ths theorem s smr to Theorem so the roof s omtted here. I wht foows we gve the gorthm. gorthm JIC em for cotrbuto: edtor@jc.org.u

5 Jour of Iformto d Comutg Scece Vo. 6 ) o Deote B = m ) ) ) ) ) =. Ste Comute m ) for y m )) Ste Set = Ste 3 Comute B. For y B ) outut Ste 4 = Ste 5 If goto ste otherwse sto Ste 6 utut V x) = Wx. I wht foows we gve exme. Suose outut = = ). By comutg we c get W =. 8 ext we cosder dscrete-tme TI systems. Theorem 3 et R = be Schur such tht re uer trgur mtrces the systems 4) shre commo fty-orm yuov fucto V x) = Wx Where d Proof: Smr to Theorem Q = otcg 3) we hve W =. > > =. JIC em for subscrto: ubshg@wu.org.u

6 66 Zheg Che et : The Comutto of Commo Ifty-orm yuov Fuctos for er Swtched Systems Sce 3 3 9) ) re Schur 9) re wys true. By we get ) ) ) ) ) ) ) ). So c be equ to y umber m )) d we c fx. Smry geer we Fy we obt ) cometes the roof. Bsed o the roof bove we c get the gorthm 3. gorthm 3 Deote C = m ) d ) m ). otcg re Schur so > =.Ths =. ) ) Ste Comute m ) for y m )) outut ) ) Ste Set = Ste 3 Comute C. For y C ) outut Ste 4 = Ste 5 If goto ste otherwse sto Ste 6 utut V x) = Wx I wht foows we gve exme. ). JIC em for cotrbuto: edtor@jc.org.u

7 Jour of Iformto d Comutg Scece Vo. 6 ) o Suose = = 4). By comutg we c get 3 W =. 4 Smry we c obt Theorem 4. Theorem 4 et R = be Schur such tht re ower trgur mtrces the systems 4) shre commo fty-orm yuov fucto V x) = Wx where d W =. > > Bsed o the roof bove we c get the gorthm 4. gorthm 4 Deote D = m =. ) Ste Comute m ). For y m )) outut Ste Set = Ste 3 Comute D for y D ) outut Ste 4 = Ste 5 If goto ste otherwse sto Ste 6 utut V x) = Wx. I wht foows we gve exme. JIC em for subscrto: ubshg@wu.org.u

8 68 Zheg Che et : The Comutto of Commo Ifty-orm yuov Fuctos for er Swtched Systems Suose 4. Cocusos = 3 4 = 4). By comutg we c get 4 4 W =. 8 3 I ths er we de wth the robem of the comutto of commo fty-orm yuov fuctos for cotuous-tme TI systems or dscrete-tme TI systems. For systems mtrces re uer trgur form or ower trgur form we reset ytc methods d fesbe gorthms re ddressed. 5. Refereces [] D. berzo. Swtchg systems d cotro []. Brhuser Bosto Ju 3. [] D. berzo.s.orse. Bsc robems o stbty d desg of swtched systems [J]. IEEE Cotro System gze ):59-7. [3] T Huqg Zog. Stbzto of swtched systems v comoste qudrtc fuctos[j]. IEEE Trsctos o utomtc Cotro. 8 53): [4]. Gurvts R. Shorte d. so. the stbty of swtched ostve er systems [J]. IEEE Trsctos o utomtc Cotro. 7 56): [5] D.Cheg. Guo d J.Hug. qudrtc yuov fuctos[j]. IEEE Trsctos o utomtc Cotro ): [6] F. Kor. so d R. Shorte. coostve er yuov fuctos for sets of er ostve systems. utomtc ): [7] T HuZog. Proertes of the comoste qudrtc yuov fuctos[j]. IEEE Trsctos o utomtc Cotro ): [8]. so. Swtched Systems covex coes d commo yuov fuctos. Ph.D Thess. UI yooth 4. [9] R.. Shorte K.S. redr d. so. resut o commo qudrtc yuov fuctos [J]. IEEE Trsctos o utomtc Cotro. 3 48): -3. []. so d R.. Shorte commo qudrtc yuov fuctos for stbe dscrete-tme TI systems[j]. I Jour of ed themtcs. 4 59: []. Pos. fty orm s yuov fuctos for er systems [J]. IEEE Trsctos o utomtc Cotro ): []. Pos. yuov fucto costructo by er rogrmmg [J]. IEEE Trsctos o utomtc Cotro ):3-6. JIC em for cotrbuto: edtor@jc.org.u