Ieee Transactions On Circuits And Systems I: Fundamental Theory And Applications, 2001, v. 48 n. 8, p

Transcription

1 Tile Robus sabilizaio of sigular-imulsive-delayed sysems wih oliear erurbaios Auhor(s) Gua, ZH; Cha, CW; Leug, AYT; Che, G Ciaio Ieee Trasacios O Circuis Ad Sysems I: Fudameal Theory Ad Alicaios, 2, v ,. -9 Issued Dae 2 URL h://hdl.hadle.e/722/44943 Righs Creaive Commos: Aribuio 3. Hog Kog Licese

2 IEEE TRANSACTIONS ON CIRCUITS AND SYSTES I: FUNDAENTAL THEORY AND APPLICATIONS, VOL. 48, NO. 8, AUGUST 2 k ad L74 for he oeraioal amlifier wih dc bias of 65 V. The uer race shows he iu riagular waveform wih a frequecy of 5 mhz. The boom race shows he recagular waveform a he ouu of he oeraioal amlifier. Thus, a good differeiaio acio is obaied by usig he roosed circui. I was o ecessary o add a resisor bewee he egaive iu ermial of he isolaio oeraioal amlifier ad groud sice a dc iu o a differeiaor roduces a zero ouu volage. D. Comariso Amog he Theoreical, Simulaio, ad Exerimeal Resuls The simulaio ad exerimeal resuls verify he rediced frequecy rages of (7) for he iegraor ad (4) for he differeiaor. Noe also ha he clea waveforms of Figs. ad idicae high sigal o oise raios for boh circuis. I should be oied ou ha he case of ifiie k, wih he feedback resisor kr removed, roduced good exerimeal ad simulaio resuls which were omied for breviy. VI. CONCLUSION A acive-ework syhesis of iverse sysem desig is reseed. The syhesis is geeral ad ca be alied wih differe imedaces. Is alicaio o iver a assive differeiaor resuled i a versaile low-frequecy differeial iegraor. Is alicaio o iver a assive RC iegraor yielded a versaile low-frequecy differeial differeiaor. Each emloys a sigle ime cosa, has a resisive iu, ad a reasoably high Q value. Simulaio ad exerimeal resuls verify he heoreical execaios. The acive-ework syhesis ca be alied o obai oher varied realizaios. The differeial iegraors ad differeiaors could easily be modified o obai iverig ad oiverig iegraors ad differeiaors by simly groudig oe of he wo ius i each of he differeial cofiguraios. Addiioally, he limied badwidhs of he circuis miigaes he coribuio of he oise ad yield ouu waveforms wih large sigal o oise raios. ACKNOWLEDGENT The auhor wishes o hak S. K. ira for rovidig he amoshere coducive o research by iviig him o sed he summer of 997 a he Sigal ad Image Processig Laboraory of Uiversiy of Califoria a Saa Barbara, where his research was iiiaed, ad R. Ferzli, F. El-Zoghe, F, Elias, ad B. Alawieh for heir hel i he roducio of he figures, simulaio, ad exerimeal resuls. REFERENCES [] W. J. Tomkis ad J. G. Webser, Eds., Desig of icrocomuer-based edical Isrumeaio. Eglewood Cliffs, NJ: Preice-Hall, 98. [2] S. K. ira, Digial Sigal Processig, 2d ed. New York: cgraw- Hill, 2. [3]. A. Al-Alaoui, A ovel aroach o desigig a oiverig iegraor wih buil-i low-frequecy sabiliy, high-frequecy comesaio ad high Q, IEEE Tras. Isrum. eas., vol. 38,. 6 2, Dec [4], A sable iverig iegraor wih a exeded high-frequecy rage, IEEE Tras. Circuis Sys. II, vol. 45, , ar [5], A differeial iegraor wih a buil-i high frequecy comesaio, IEEE Tras. Circuis Sys. I, vol. 45, , ay 998. [6], A ovel differeial differeiaor, IEEE Tras. Isrum. eas., vol. 4, , Oc. 99. [7] J. G. Graeme, Alicaios of Oeraioal Amlifiers. Tokyo, Jaa: cgraw-hill, 973. Robus Sabilizaio of Sigular-Imulsive-Delayed Sysems Wih Noliear Perurbaios Zhi-Hog Gua, C. W. Cha, Adrew Y. T. Leug, ad Guarog Che Absrac ay dyamic sysems i hysics, chemisry, biology, egieerig, ad iformaio sciece have imulsive dyamical behaviors due o abru jums a cerai isas durig he dyamical rocess, ad hese comlex dyamic behaviors ca be modeled by sigular imulsive differeial sysems. This aer formulaes ad sudies a model for sigular imulsive delayed sysems wih uceraiy from oliear erurbaios. Several fudameal issues such as global exoeial robus sabilizaio of such sysems are esablished. A simle aroach o he desig of a robus imulsive coroller is he reseed. A umerical examle is give for illusraio of he heoreical resuls. eawhile, some ew resuls ad refied roeries associaed wih he -marices ad ime-delay dyamic sysems are derived ad discussed. Idex Terms Imulsive sysems, oliear erurbaio, robus sabilizaio, sigular sysems, ime-delay, uceraiy. I. INTRODUCTION I rece years, cosiderable effors have bee devoed o he aalysis ad syhesis of sigular sysems (kow also as descrior sysems, semisae sysems, differeial algebraic sysems, geeralized sae-sace sysems, ec.). These sysems arise aurally i various fields icludig elecrical eworks [25], roboics [22], [23], social, biological, ad mulisecor ecoomic sysems [2], [29], dyamics of hermal uclear reacors [26], auomaic corol sysems [27], amog may ohers such as sigular erurbaio sysems. Progress i he ivesigaio of sigular sysems ca be foud i books [], [4], [6], [8] ad survey aers [5], [5], [6]. Alhough mos sigular sysems are aalyzed eiher i he coiuous- or discree-ime seig, may sigular sysems exhibi boh coiuous-ime ad discree-ime behaviors. Examles iclude may evoluioary rocesses, esecially hose i biological sysems such as biological eural eworks ad bursig rhyhm models i ahology. Oher examles exis i oimal corol of ecoomic sysems, frequecy-modulaed sigal rocessig sysems, ad some flyig objec moios. These sysems are characerized by abru chages i he saes a cerai isas [3], [9], [], [], [4]. This ye of imulsive heomea ca also be foud i he fields of iformaio sciece, elecroics, auomaic corol sysems, comuer eworks, arificial ielligece, roboics, ad elecommuicaios []. ay sudde ad shar chages occur isaaeously i sigular sysems, i he form of imulses which cao be well described by a ure coiuous-ime or discree-ime model. For isace, if he iiial codiios is icosise, he a sigular sysem will have a fiie auscri received Seember, 2. This work was suored i ar by he Naioal Naural Sciece Foudaio of Chia uder Gra ad Gra 6749, i ar by he Docorae Foudaio of he Educaio iisry of Chia uder Gra , ad i ar by he Foudaio for Uiversiy Key Teacher, Educaio iisry of Chia. This aer was recommeded by Associae Edior. Gilli. Z.-H. Gua is wih he Dearme of Corol Sciece ad Egieerig, Huazhog Uiversiy of Sciece ad Techology, Wuha, Hubei, 4374, Chia. C. W. Cha is wih he Dearme of echaical Egieerig, The Uiversiy of Hog Kog, Pokfulam Road, Hog Kog. A. Y. T. Leug is wih he Dearme of Buildig ad Cosrucio, Ciy Uiversiy of Hog Kog, Kowloo, Hog Kog. G. Che is wih he Dearme of Elecroic Egieerig, Ciy Uiversiy of Hog Kog, Kowloo, Hog Kog. Publisher Iem Ideifier S /$. 2 IEEE

3 2 IEEE TRANSACTIONS ON CIRCUITS AND SYSTES I: FUNDAENTAL THEORY AND APPLICATIONS, VOL. 48, NO. 8, AUGUST 2 isaaeous jum a he iiial ime [2]. For sigular sysems wih ime delays, ifiie imulses as well as fiie jums ca occur i he soluios of he sysems [6], [7]. Therefore, i is very imora, ad ideed ecessary, o sudy sigular imulsive sysems, erhas also wih ime delays. O o of he sigular, imulsive, ad ime-delayed feaures of such dyamic sysems, here migh also be uceraiies such as erurbaios, usually arisig from modelig errors, daa-measureme errors, chages i eviromeal codiios, ad comoe variaios, ec. These alogeher lead he sysem o a uexecedly comlicaed siuaio, hereby leadig o very comlex dyamical behaviors. I he desig of a coroller for such a comlex sysem, i is imora o esure ha he sysem be sable wih resec o hese uceraiies. Robus sabilizaio is a coce addressig his issue of sabiliy for ucerai sysems. I aricular, robus sabilizaio for a sigular ad delayed sysem has recely araced icreasig ieres (see, e.g., [7], [8], [6], [7], [28] ad he refereces herei). These exisig sudies, however, are o for imulsive ye of corol sysems. Give he above backgroud, his aer aems o sudy he robus sabilizaio roblem for a comlex dyamic sysem of he sigular, imulsive, ad ime-delayed ye wih uceraiies from oliear erurbaios. This work is based o our revious ivesigaios o sigular ad imulsive sysems [], [9]. Basically, i his aer, we firs iroduce a model for sigular imulsive delayed sysem wih oliear erurbaios, ad he sudy is exoeial robus sabilizaio roblem. The aer is orgaized as follows. I Secio II, he sigular imulsive delayed sysem is described ad modeled, ad i Secio III, some relimiary resuls are derived ad discussed. The mai resul of global exoeial robus-sabilizaio crieria for he esablished model is give i Secio IV, where a sysemaic desig rocedure for obaiig a robus imulsive coroller is reseed. For illusraio, a umerical examle is described i Secio V. Fially, some coclusios are draw i Secio VI. II. PROBLE FORULATION Le R =[; ), J =[ ; ), ( ) ad R deoe he -dimesioal Euclidea sace. The orm of z is kzk := i= jzij ad jzj := (jz j;...; jz j) T, where z =(z ;...;z ) T 2 R. Similarly, for A =(a ij ) 2 2 R 2 wih he iiial codiios x i () = i (); i =; 2 (2.2) where x i 2 R ;x=(x ;x 2 ) T sae vecor wih 2 = ; u i 2 R m, u =(u ;u 2 ) T corol vecor wih m m 2 = m; A i, B i, ad C i kow real cosa marices of aroriae dimesios; f i (; x(); x( )) oliear ucerai veror fucio wih f i(; ; ) for all 2 J; > is a cosa. Here, Dx i, Dv i ad Dw i deoe he disribuioal derivaives of he fucios x i 2 R, v i, ad w i, resecively. v i;w i: R! R are fucios of bouded variaio ad righ-coiuous o every comac subierval of J. This imlies ha Dv i ad Dw i ca be ideified wih he Lebesgue Sieljes measure, which have he effec of suddely chagig he sae of he sysem a he ois of discoiuiy of v i ad u i. oreover, i:[ ; ]! R are fucios of bouded variaio ad righ-coiuous. Fially, he iiial codiio is deoed by he vecor 8() =( (); 2 ()) T. I geeral, a fucio of bouded variaio ad righ-coiuous cosiss of wo ars: oe is a absoluely coiuous fucio ad he oher is a sigular fucio. Whe discoiuous ois of he fucio are isolaed ad a mos couable, he sigular ar has he form k= a kh k (). Wihou loss of geeraliy, we herefore assume ha v i () = w i () = k= k= ik H k () ik H k (); i =; 2 (2.3) where ik ad ik are cosas, wih discoiuiy ois < 2 < < k < ; lim k! k = where >, ad H k () are he Heaviside fucios defied by kak =max j i= ja ij j mi (A) = =2 mi (AT A) (A) = max j a jj jaj =(ja ij j) 2 i=;i6=j ja ijj where (A) ad mi(a) are he marix measure ad he miimum eigevalue of A, resecively. We use he oaio A(a ij ) B(b ij ) ad (z ;...;z ) T (y ;...;y ) T o mea ha a ij b ij ad z i y i for all i; j =;...;, resecively. The ideiy marix of order is deoed as I, or simly I, if o cofusio arises. Cosider he followig oliear ucerai ime-delay imulsive ad sigular dyamic sysem: I is easy o see ha H k () = ; < k ; k. Dv i = ik ( k ) k= Dw i = k= ik ( k ) where () is he Dirac imulse fucio. Provided ha all he saes are available, he robus-sae feedback coroller u i () are give by Dx =[A x () B x ( )]Dv C u f (; x(); x( ))Dw =[A 2x 2() B 2x 2( )]Dv 2 C 2u 2 f 2(; x(); x( ))Dw 2 (2.) u i() =K ix i()dv i (2.4) where K i is a cosa marix (called he gai marix hereafer) of aroriae dimesio. Obviously, u i() is a imulsive coroller.

4 IEEE TRANSACTIONS ON CIRCUITS AND SYSTES I: FUNDAENTAL THEORY AND APPLICATIONS, VOL. 48, NO. 8, AUGUST 2 3 Subsiuig (2.4) io (2.) yields a oliear ucerai closed-loo ime-delay sigular ad imulsive dyamical sysems, i he form of where Dx = A x () B x ( ) Dv f (; x(); x( ))Dw = A 2 x 2 () B 2 x 2 ( ) Dv 2 f 2 (; x(); x( ))Dw 2 (2.5) A = A C K A 2 = A 2 C 2K 2: (2.6) Now, he roblem is o fid some codiios for robus corol u i(), give by (2.4), such ha he overall oliear ucerai ime-delay sigular imulsive sysem (2.5) is asymoically sable i he resece of oliear uceraiies. III. TIE-DELAY AND IPULSIVE SYSTES I his secio, some ecessary coces ad refied roeries associaed wih ime-delay sysems ad imulsive sysems are derived ad discussed. Cosider he followig ime-delay sysem: x () =Ax() Bx( ) (3.) Remark 3.2: Lemma 3. ad Defiiio 3. imly ha if sysem (3.) is sable wih a decay rae, he iequaliy (3.5) holds. For he esimae of (3.5), we have he refied resul give i Lemma 3.2 below. Lemma 3.2: If = maxfre(s)j de Q(s) = g, he for ay >, he fudameal soluio X() of (3.) saisfies he iequaliy kx()k ex; : (3.6) Proof: I follows from Lemma 3. ha X() has he exressio of (3.4). For simliciy, he followig oaio is used (c) := 2i ci ci = lim T! 2i cit cit where c is some sufficiely large real umber. Firs, we wa o show ha Q (s) ex(s) ds = (c) Q (s) ex(s) ds (3.7) where c >. Cosider he iegraio of he fucio Q (s)e s alog he boudary of he box i he comlex lae, wih boudary L L 2 2, i he couerclockwise direcio, where x () =Ax() Bx( )f (): (3.2) L : fc i: T T g L 2 : f i: T T g I is well kow ha he asymoic sabiliy of (3.) imlies ha he soluio of equaio de Q() =; Q() :=I A B ex( ) (3.3) saisfies Re() <, ad vice versa. Defiiio 3. [24]: Sysem (3.) is said o be sable wih decay rae, if he soluio of (3.) saisfies Re() < for some >. Remark 3.: There are may resuls associaed wih he esimae of he decay rae for he ime-delay sysem (3.). For isace, if (A) >kbk ex(), he sysem (3.) is sable wih he decay rae [24]. Lemma 3. [3]: The followig ime-delay marix sysem wih iiial codiio X () =AX()BX( ) X() = ; < I ; = has a uique soluio o > give by X() = 2i X() 2 R 2 ci Q (s) ex(s) ds; > (3.4) ci : f it : cg 2 : f it : cg: Sice Q (s) has o zeros i he box ad o is boudary, i follows ha Q (s)e s is a aalyic fucio i he box, leadig o L Thus, (3.7) is verified if we ca show ha L Q (s) ex(s) ds! Q (s) ex(s) ds =: Q (s) ex(s) ds! (3.8) as T!. For a 2 marix P, le kp k 2 := =2 max(p P T ), where P T is he cojugae rasose marix of P ad max (P P T ) is he maximum eigevalue of P P T. The kp k kp k 2. I is ow readily show ha kq (s)k 2 2 = max i i Q (s)(q (s)) T = (3.9) mii i Q (s)(q (s)) T where Q(s) is give by (3.3), c>, = kak kbk, ad X() is also a marix-valued fucio of bouded variaio o every comac subierval of J. I addiio, for ay >maxfre()j de Q() = g, here exiss a cosa = () such ha kx()k ex; : (3.5) I Lemma 3., X() is called he fudameal soluio or fudameal soluio marix of (3.). ad T i Q (s) Q (s) = i (si A B s e ) si A Be s T jsj 2 mi(aa T ) mi(bb T )e 2Re(s) 2kAB T ke Re(s) 2jsjkAk2jsjkBke Re(s) : (3.)

5 4 IEEE TRANSACTIONS ON CIRCUITS AND SYSTES I: FUNDAENTAL THEORY AND APPLICATIONS, VOL. 48, NO. 8, AUGUST 2 Choose T such ha 2 T 2 2 T 2 T 2 =2 (kak kbke ) T 2 [ mi(aa T ) mi (BB T )e 2 2kAB T ke ] 4 for all T T.IfT T ad s 2, he (3.9) ad (3.) imlies ha kq (s)k2 2 [( 2 T 2 ) mi(aa T ) mi(bb T )e 2 2kAB T ke 2 2 T 2 (kakkbke )] 4 T 2 amely, kq (s)k 2 2=T. Therefore Q (s)e s ds max T T ;s2 Q (s)e s ds 2 T ec (c )! 2 kq (s)k 2 je s j as T!. Similarly, k Q (s)e s dsk! as T!. Thus, (3.8) holds. Nex, le T be as above. If G(s) =Q (s) (I=(s )), he G(s) =Q (s) I Q(s) s = Q (s) ds I ABes s kg(s)k 2 2 T 2 [j jkak 2 kbk 2 e ] for s = it; jt jt. Accordigly G(s)e s ds G(s)e s ds max s=it; jt jt 2 kg(s)k 2je s j T (j j kak 2 kbk 2 e )ex! (3.) as T!. Furhermore ds I follows from (3.4), (3.7), (3.) ad (3.2) ha kx()k = = Q (s) ex(s) ds (c) Q (s) ex(s) ds G(s) ex(s) ds (s ) I ex(s) ds ex; : This comlees he roof. Lemma 3.3 [2]: The geeral soluio of (3.) wih iiial codiio is give by x() = k () 2 [ k ; k ] (3.3) x() =X( k ) k ( k ) B X(s) k (s)ds; k where k () is a arbirarily give iiial fucio o [ k ; k ] ad X() is he fudameal soluio marix of sysem (3.). By Lemmas 3. ad 3.3, he followig resul is immediae. Lemma 3.4: The geeral soluio of sysem (3.2) wih iiial codiio (3.3) is give by x() =X( k ) k ( k ) B X( s ) k (s) ds X( s)f (s) ds; k where X() is he fudameal soluio marix of (3.). IV. ROBUST STABILIZATION I his secio, we discuss he robus sabilizaio roblem of sysem (2.), or effecively, he robus-sabiliy roblem of he closed-loo sysem (2.5). Assume ha k k, > kf i (; x(); x( ))k c i kx()k c i2 kx( )k; (s ) Ie s ds (s ) Ie s ds 2 ad sysem i =; 2 (4.) x () =A x () B x ( ) (4.2) max s=it; jt jt k(s ) Ik 2je s j ex; : (3.2) ds is sable wih decay rae >, where c ij are cosas ad A ;B are give by (2.5) ad (2.6). I follows from Remark 3.2 ad Lemma 3.2 ha here exiss a cosa such ha kx ( s)k ex[ ( s)]; s (4.3)

6 IEEE TRANSACTIONS ON CIRCUITS AND SYSTES I: FUNDAENTAL THEORY AND APPLICATIONS, VOL. 48, NO. 8, AUGUST 2 5 where X () is he fudameal soluio marix of sysem (4.2) ad ca be ake as = =. For coveiece, defie he followig oaio: k =mi k = k mi(i A k ); 2 mi(a 2 2k ) (j k jc j 2k jc 2 ) (4.4) k = k [maxfkb k k; kb 2 2k kg (j k jc 2 j 2k jc 22 )] (4.5) P () =( ij ()) 222 (4.6) where ik ad ik are defied by (2.3), ad () = (c c2e ) 2 () = () 2 2 () = (c 2 c 22 e ) mi(a 2) 22 () = 2 mi(a 2) [c 2 (c 22 kb 2 k)e ] wih give by (4.3). Theorem 4.: For he closed-loo sysem (2.5), ad for k =; 2;..., assume ha: ) mi (A 2 ) >, k > ; 2) here exiss a cosa saisfyig << such ha I P () is a -marix; 3) maxfe ; ( k k e )g k c where c is a cosa, k ad k are give by (4.4) ad (4.5). The, (l(c)= ) < imlies ha (2.5) is robusly exoeially sable i he large, ad he soluio of (2.5) has he followig esimae: kx()k k8k ex l(c) ( ) ; where = ( (kb ke = ))k(i P ()) k, ad is give by (4.3). Proof: I follows readily from (2.3) ha v i ad w i exis o [ k ; k ). Thus, for 2 [ k ; k ), (2.5) becomes x () = A x () B x ( ) f (; x(); x( )) = A 2 x 2 () B 2 x 2 ( ) f 2 (; x(); x( )); 2 [ k ; k ): (4.7) Le he iiial codiio of sysem (4.7) be x i () = ik () 2 [ k ; k ] ad 8 k () =( k (), 2k ()) T, where ik () is a fucio of bouded variaio ad righ-coiuous o [ k ; k ]. By Lemma 3.3, i follows from (4.7) wih he associaed iiial codiio ha for 2 [ k ; k ) x () =X ( k ) k ( k ) B X ( s ) k (s)ds X ( s)f (s; x(s); x(s )) ds where X () is he fudameal soluio marix of (4.2), ogeher wih (4.3) ad (4.). This leads o kx ()k kb ke ex[ (s)] ex [ ( k )]k k k [c (kx (s)kkx 2 (s)k)c 2 (kx (s)k kx 2(s )k)] ds (4.8) where 2 [ k ; k ), > ad c ij are give by (4.) ad (4.3), ad k ik k =su k ik ()k. ulilyig boh sides of (4.8) by ex[( k )], where is give by assumio 2), yields Le kx ()k ex[( k )] kb ke k k k ex[( )(s)] fc (kx (s)kkx 2(s)k)ex[(s k )]c 2e y i () = (kx (s)kkx 2(s)k) ex[(s k )]g ds: (4.9) su s The, i follows from (4.8) (4.) ha y () kbke fkx i (s)k ex[(s k )]g; i =; 2: k k k (4.) [(c c 2 e )(y ()y 2 ())]; 2 [ k ; k ): (4.) For 2 [ k ; k ), he secod equaio of (4.7) becomes which leads o Sice A 2 x 2 () =B 2 x 2 ( ) f 2 (; x(); x( )) A 2 x 2 () kb 2 kkx 2 k c 2 kx()k c 22 kxk: ka 2x 2()k mi(a 2 ) 2 kx 2()k ad mi(a 2) >, we have, from (4.2), he followig: 2 kx 2 ()k [kb 2kkx 2 ( )k c 2 kx()k mi(a 2) c 22 kx( )k]: oreover, y 2() 2 mi (A 2 ) f(c2 c22e )y () [c 2 (c 22 kb 2 k)e ]y 2 ()g where y i () are defied by (4.). (4.2) (4.3)

7 6 IEEE TRANSACTIONS ON CIRCUITS AND SYSTES I: FUNDAENTAL THEORY AND APPLICATIONS, VOL. 48, NO. 8, AUGUST 2 Combiig (4.) ad (4.3), we obai y() kb ke 9 k P ()y(); 2 [ k; k ); (4.4) where y() = (y (); y 2 ()) T, 9 k = (k kk; ) T, P () is give by (4.6), ad I P () is a -marix. Thus, from Lemma A3 i he Aedix, (4.4) imlies ha y() kbke (I P ()) 9 k; 2 [ k; k ) which reduces o kx()k k8 kk ex[( k)]; 2 [ k; k ) (4.5) where x() =(x (); x 2 ()) T, = ((kb ke = )) k(i P ()) k. O he oher had, sysem (2.5) imlies ha ad x ( k ) x ( k h) = = h h (A x (s) B x (s )) dv (s) f (s; x(s); x(s )) dw (s) (A 2x 2(s) B 2x 2(s )) dv 2(s) f 2(s; x(s); x(s )) dw 2(s) where h> is sufficiely small, as h!, which reduces o or x ( k ) x ( )=A k k x ( k )B k x ( k ) k f ( k ;x( k );x( k )) =A 2 2k x 2 ( k )B 2 2k x 2 ( k ) 2k f 2( k ;x( k );x( k )) (I A k )x ( k )=x ( k )B k x ( k ) k f ( k ;x( k );x( k )) A 2 2k x 2 ( k )=B 2 2k x 2 ( k ) 2k f 2( k ;x( k );x( k )): I follows from (4.6) ad (4.) ha mi I A k kx ( k )k (4.6) kx ( )k kb k kkkx ( k )k j k j(c kx( k )k c 2 kx( k )k) (4.7) mi A 2 2k kx 2 ( k )k 2 kb 2 2k kkx 2 ( k )k j 2k j(c 2kx( k )k c 22kx( k )k): (4.8) Observe ha kxk = kx k kx 2 k. Therefore, (4.7) ad (4.8) reduce o k kx( k )kkx ( k )k[maxfkb kk; kb 2 2k kgj k jc 2 j 2k jc 22 )]kx( k )k where k is give by (4.4). Based o assumio ), we immediaely arrive a kx( k )k k kx( k )k kkx( k )k (4.9) where k ad k are defied by (4.5). From (4.5) ad (4.9), we ca obai he followig resuls: Whe k =, ake 8 () =8() =( (); 2()) T, 2 [ ; ], so ha kx()k k8k ex[( )]; 2 [ ; ) (4.2) which reduces o kx( )k k8k ex[( )] (4.2) kx( )k kx( )k kx( )k: (4.22) Whe k =2, aurally, ake 8 () =x(), 2 [ ; ], so ha i view of (4.2) (4.22) k8 k = su kx()k k8k ex[( )] maxfe ; e g k8k ex[( )] where is give i assumio 3). Thus, for 2 [ ; 2 ) kx()k k8 k ex[( )] k8k 2 ex[( )]: Geerally, as 2 [ k; k ), kx()k k8k k k ex[( )]: (4.23) Sice k c, k k, ( >) ad c k k (c) k ex l(c) ( k ) ex l(c) ( ) (4.24) where 2 [ k; k ). From (4.23) ad (4.24), we have kx()k k8k ex l(c) ( ) ; : (4.25) This comlees he roof. Remark 4.: For assumio 2) i Theorem 4., if I P is a -marix, he by Lemma A2 i he Aedix here exiss a cosa ; << such ha I P () is a -marix. Noe ha he marices I A k ad A 2 2k may be iverible. To sudy his case, we iroduce he followig oaio: ~ k = c k(i A k ) k kc 2k(A 2 2k ) 2k k ~ k = ~ k k(i A k ) k (4.26) ~ k = ~ k [c 2k(I A k ) k k c 22k(A 2 2k ) 2k k maxfk(i A k ) B k k; ka 2 B 2 kg] (4.27) ~P () =(~ ij()) 222 (4.28) where ~ () = (), ~ 2() = 2(), give by (4.6), ad ~ 2 () =ka 2 k(c 2 c 22 e ) ~ 22 () =~ 2 ()ka 2 B2 ke :

8 IEEE TRANSACTIONS ON CIRCUITS AND SYSTES I: FUNDAENTAL THEORY AND APPLICATIONS, VOL. 48, NO. 8, AUGUST 2 7 If A 2 is iverible, he i follows from (4.7) ha which leads o kx 2()k ka 2 B2kkx2( )k ka 2 k(c2 kx()k c 22 kx( )k) y 2() ~ 2()y () ~ 22()y 2(): oreover, ogeher wih (4.), i yields y() kbke 2 [ k ; k ): 9 k ~ P ()y(); If he marices I A k ad A 2 2k are iverible, he i follows from (4.6) ha x ( k )=(I A k ) [x ( )B k k x ( k ) k f ( k ;x( k );x( k ))] x 2 ( k )=(A 2 2k ) [B 2 2k x 2 ( k ) 2k f 2( k ;x( k );x( k ))]: These imly ha kx( k )k ~ k kx( k )k ~ kkx( k )k rovided ha ~ k >. Similar o he iferece of Theorem 4., we obai he followig resul. Corollary 4.: For he closed-loo sysem (2.5), assume ha ) ~ k >, I A k ad A 2 2k are iverible, k =; 2;...; 2) here exiss a cosa saisfyig << such ha I ~P () is a -marix; 3) maxfe ; ( ~ k ~ k e )g k c, where c is a cosa. The, he coclusio of Theorem 4. holds wih = ( (kb ke = ))k(i P ~ ()) k, where ~ k, ~ k, ~ k, ad P ~ () are give by (4.26) (4.28). Theorem 4.2: If assumio 2) of Theorem 4. is relaced by he followig codiio: 2) here exiss a cosa saisfyig << such ha () = maxf () 2 (); 2 () 22 ()g < (4.29) where ij () are give by (4.6), he he coclusio of Theorem 4. holds wih = =( ())( (kb ke = )). Proof: Similar o he argume used i Theorem 4., we obai he iequaliies (4.) ad (4.3). If we defie y() = maxfy i ()j 2 [ k ; ]; i=; 2g i he, by assumio 2), (4.) ad (4.3), we have amely y() kbke y() () 2 [ k ; k ) k k k ()y() kb ke k k k; which reduces o (4.5) wih = [ =( ())]( (kb ke = )). The res of he roof is similar o ha of Theorem 4., herefore, deails are omied. Corollary 4.2: If assumio 2) of Corollary 4. is relaced by he followig codiio: 2) here exiss a cosa saisfyig << such ha ~() = maxf~ ()~ 2 (); ~ 2 ()~ 22 ()g < ; (4.3) where ~ ij() are give by (4.28), he he coclusio of Corollary 4. holds wih = [ =( ~())]( (kb ke = )). V. CONTROLLER DESIGN WITH AN EXAPLE I his secio, we describe a sysemaic-desig rocedure for he robus imulsive corol law develoed above, for he sigular imulsive delayed sysem (2.). The rocedure is esablished o he basis of he aalysis give i Secio IV above. A examle is give o illusrae his desig rocedure, which also serves for ierreaio of he heoreical resuls obaied i he aer. The suggesed desig rocedure is based o Theorems 4. ad 4.2, ad is summarized as follows. Se ) For he sysem (2.), selec he gai marix K such ha he sysem (4.2) is sable wih decay rae >. For coveiece, oe may ick a cosa > such ha (A ) > kb k ex( ). Se 2) Selec he gai marix K 2 ad comue mi(a 2) ad k, which are defied by (4.4). If mi (A 2 ) > ad k >, he go o Se 3; oherwise, go back o Se. Se 3) Pick a cosa, <<, ad he calculae P () or (), which are give by (4.6) ad (4.29), resecively. If I P () is a -marix or if () <, he go o Se 4; oherwise, go back o Se 2. Se 4) Comue maxfe ; ( k k e )g k, where k ad k are give by (4.5). If k c, he go o Se 5; oherwise, go back o Se 3. Se 5) Calculae he cosa wih he exressios = ( =)( (kb ke = ))k(i P ()) k or = ( =( ()))( (kb ke = )). If (l(c)= ) <, he go o Se 6; oherwise, go back o Se. Se 6) Subsiue he deceralized gai marices K ad K 2, deermied i Se 2, io (2.4) o obai he coroller u i () for he sysem (2.). Remark 5.: From Corollaries 4. ad 4.2, we ca derive a similar corol desig rocedure for sysem (2.); he deails are omied for breviy. Examle: Cosider he ucerai, delayed, sigular ad imulsive sysem (2.) ad (2.2) wih oliear erurbaios, where = 2, 2 =2ad =4 A = 2 3 A 2 = 2 B = B 2 = C = C 2 = ad wih oliear erurbaio f i(; x(); x( )) saisfyig (4.), wih c =2, c 2 =, c 2 =, c 22 =2, v i (), ad w i () give by (2.3), where k k ; ==, ad k = 2k = k 9 2k = 2 [k ]: k = k 4 Now, oe ca easily desig a robus imulsive coroller by followig he above rocedure, such ha sysem (2.) secified above is exoeially sable: Se ) Selec K =( ), ad =2. The A = A C K = = (A ) > kb k ex( )=7:39:

9 8 IEEE TRANSACTIONS ON CIRCUITS AND SYSTES I: FUNDAENTAL THEORY AND APPLICATIONS, VOL. 48, NO. 8, AUGUST 2 Se 2) Selec K 2 =( ). The A 2 = A 2 C 2K 2 = From (4.4), we have mi (A 2 )=46> k = 2:27; k =2 2:58; k =2 : 2N: Se 3) Selec a cosa, =, <<, ad calculae I P () = () 2() 2 () 22 () = :86 :4 :9 :85 which is a marix. The k(i P ()) k =:39. Se 4) Comue maxfe ; ( k k e )g maxf; (:38 :39)g :=c. Se 5) Calculae cosa = ( =)( (kb ke = ))k(i P ()) k =:86. Thus, for ay > l(c) <: Se 6) Based o he above resuls, he corresodig coroller u i() is obaied as u () = u 2 () = x ()Dv x 2 ()Dv 2 which exoeially sabilizes he give oliearly erurbed, ime-delayed, sigular, ad imulsive sysem. VI. CONCLUSIONS I his aer, we have formulaed ad sudied he sabilizaio roblem for a geeral sigular-imulsive delayed sysem wih oliear erurbaios. Such a comlex sysem cao be hadled by radiioal echiques ha ca oly deal wih ure coiuous-ime or discree-ime models, erhas wih eiher ime delays or uceraiies. Some secific roeries rescribig his hybrid model have bee aalyzed. Global exoeial robus sabilizaio of he sysem equilibrium via sabilizig coroller desig has bee ivesigaed. A sysemaic rocedure for desigig he robus imulsive coroller has bee suggesed, alog wih a exlici examle for illusraio. Fuure research alog he same lie will be devoed o ossible egieerig alicaios of he roosed model ad is sabilizaio mehodology. APPENDIX Some basic roeries associaed wih he -marix are give here for he reader s coveiece. A marix A saisfyig ay oe of he six codiios lised i Lemma A below is called a -marix. Lemma A [2]: Le A be a real square marix wih oosiive offdiagoal elemes. The he followig six codiios are equivale. ) The ricial miors of A are all osiive. 2) The leadig ricial miors of A are all osiive. 3) There is a vecor x (or y) whose elemes are all osiive such ha he elemes of Ax (or A T y) are all osiive. 4) A is osigular ad he elemes of A are all oegaive. 5) The real ars of he eigevalues of A are all osiive. 6) There is a diagoal marix D =diag(d ;...;d ), wih d i >, such ha DA A T D is a osiive defiie marix. I wha follows, we will give some resuls cocerig -marices. Lemma A2: Le A() = (a ij()) 2 be a coiuous marix-valued fucio, wih oosiive off-diagoal elemes i (; ), >. IfA = (a ij ) 2 is a -marix, he here exiss a cosa, <<, such ha A() is a -marix-valued fucio o he ierval (; ). Proof: Sice A = (a ij ) 2 is a -marix, from Lemma A, here exiss a vecor x =(x ;...;x ) T >, x 2 R, such ha j= a ij x j >, i =;...;. Le f i () = j= a ij ()x j, i = ;...;. The f i = aij xj j= >. Clearly, fi() is a coiuous fucio o (; ), ad here exiss a cosa, i 2 (; ), such ha f i() = j= a ij()x j > ; Le = mi i f i g. The f i() = a ij ()x j > ; j= i =;...; 2 ( i; i); i=;...;: 2 (; ); <<; imlyig ha A() is a -marix-valued fucio o he ierval (; ). This comlees he roof. Lemma A3: For a vecor iequaliy Ax b, where A =(a ij ) 2, b =(b ;...;b ) T, x =(x ;...;x ) T 2 R,ifA is a -marix, he x A b. Proof: I is readily see from Lemma A ha A is osigular ad elemes of A are oegaive. Thus, he vecor iequaliy Ax b imlies ha x A b. This comlees he roof. REFERENCES [] J. D. Alevich, Imlici Liear Sysems, New York: Sriger-Verlag, 99. [2]. Araki, Sabiliy of large scale oliear sysems Quadraic-order heory of comosie sysem mehod usig -marices, IEEE Tras. Auoma. Cor., vol. 23, , Ja [3] D. D. Baiov ad P. S. Simeoov, Sabiliy Theory of Differeial Equaios wih Imulse Effecs: Theory ad Alicaios. Chicheser, U.K.: Ellis Horwood, 989. [4] K. E. Bream, S. L. Cambell, ad L. R. 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10 IEEE TRANSACTIONS ON CIRCUITS AND SYSTES I: FUNDAENTAL THEORY AND APPLICATIONS, VOL. 48, NO. 8, AUGUST 2 9 [8] J. L. Li ad S. J. Che, Robusess aalysis of ucerai liear sigular sysems wih ouu feedback corol, IEEE Tras. Auoma. Cor., vol. 44, , Oc [9] Y. Q. Liu, Z. H. Gua, ad X. C. We, The alicaio of auxiliary simulaeous equaios o roblem i he sabilizaio of sigular ad imulsive large scale sysems, IEEE Tras. Circuis Sysem. I, vol. 42,. 46 5, Ja [2] W. Q. Liu, W. Y. Ya, ad K. L. Teo, O iiial isaaeous jums of sigular sysems, IEEE Tras. Auoma. Cor., vol. 4, , Se [2] D. G. Lueberger, Dyamic equaio i descrior form, IEEE Tras. Auoma. Cor., vol. AT-22, , Feb [22] N. H. cclamroch, Sigular sysems of differeial equaios as dyamical models for cosraied robo sysems, i Proc. IEEE Roboics ad Auomaio Cof., Sa Fracisco, CA, 986, [23] J. K. ills ad A. A. Goldeberg, Force ad osiio corol of maiulaors durig cosraied moio asks, IEEE Tras. Robo. Auoma., vol. 38,. 3 46, Ja [24] T. ori, N. Fukuma, ad. Kuwahara, O a esimae of he decay rae for sable liear delay sysems, I. J. Corol, vol. 36, o., , 982. [25] R. W. Newcomb, The semi-sae descriio of oliear ime variable circuis, IEEE Tras. Circuis Sys., vol. CAS-28,. 62 7, Ja. 98. [26] D. H. Owes ad R. P. Joes, Ieraive soluio of cosraied differeial algebraic sysems, I. J. Cor., vol. 27, o. 6, , 978. [27] G. C. Verghese, B. C. Levy, ad T. Kailah, A geeralized sae-sace for sigular sysems, IEEE Tras. Auoma. Cor., vol. AT-26,. 8 83, Ar. 98. [28] X. Xie ad Y. Liu, Sabiliy for comosie sigular sysems of differeial equaios wih a delay, Circuis Sys. Sigal Process., vol. 5, o. 5, , 996. [29] E. C. Zeema, Duffig s equaio i brai modelig, J. Is. ah. Al., o. 2, , 976. Fig.. Fig. 2. Physical schemaics of a sigle-elecro uelig jucio. Liear-eriodic volage-charge relaio for a SETJ. A Deermiisic Noliear-Caacior odel for Sigle-Elecro Tuelig Jucios ari Häggi ad Leo O. Chua Absrac Sigle-elecro uelig jucios (SETJs) have iriguig roeries which make hem a rimary aoelecroic device for highly comac, fas, ad low-ower circuis. However, sadard models for SETJs are based o a quaum mechaical aroach which is very imracical for he aalysis ad desig of SETJ-based circuiry, where a simle, referably deermiisic model is a rerequisie. We verify by hysics-based oe Carlo simulaios ha he uelig jucio ca i fac be modeled by a iecewise liear volage-charge relaio, which, from he circui-heoreic ersecive, is a oliear caacior. Idex Terms Noliear caacior, sigle-elecro uelig jucio. I. SINGLE-ELECTRON TUNNELING JUNCTIONS Sigle-elecro uelig jucios (SETJs) are erhas he mos comac of all elecroic devices. I is heoreically ossible o creae double-jucio swiches or logic gaes wihi areas smaller ha auscri received Aril 7, 2; revised Jauary, 2. This work is suored by he arie Uderwaer Research Isiue, Office of Naval Research uder Gra N ad Gra N This aer was recommeded by Associae Edior Y. Park.. Häggi is wih he Uiversiy of Nore Dame, Nore Dame, IN USA. L. O. Chua is wih he Uiversiy of Califoria a Berkeley, Berkeley, CA 9472 USA. Publisher Iem Ideifier S Fig. 3. Curre-biased SET jucio. m 2 corresodig o a desiy of 2 devices er cm 2, ad he small caaciace imlies exremely high swichig seeds. This combiaio of desiy ad seed make i difficul o imagie ay oher aleraive echology ha could mach he log-erm ossibiliies of sigle-elecroics. To exlai sigle-elecro effecs, a orhodox heory based o a heomeological Hamiloia aroach wih a uelig erm ad he elecrosaic eergy has roved successful []. To aalyze circuis wih sigle-elecro jucios (SETJs), however, simlified models of he jucio characerisics are required. Oe examle is he oe Carlo model i which classical elecros uel hrough he jucios sochasically wih a robabiliy ha is a fucio of he emeraure ad he chage i elecrosaic eergy. I he limiig case of zero emeraure ad small average curre, i furher reduces o a deermiisic model where elecro uelig occurs as soo as i decreases he overall elecrosaic eergy of he sysem. Based o hese cosideraios, a deermiisic model for he jucio characerisics has bee roosed which avoids ay uecessary comlexiies due o he sochasic aure of quaum mechaics ad hermal flucuaio [2]. I his model, i is assumed ha a elecro uels whe he jucio volage v j reaches he uelig volage V T = e : () 2C j The behavior of he jucio (show i Fig. ) ca herefore be modeled by a sigle-valued iecewise liear volage charge relaio (Fig. 2). This model has bee alied for he ivesigaio of /$. 2 IEEE