Tracking Control for Hybrid Systems via Embedding of Known Reference Trajectories

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1 Tacking Conol fo Hybid Sysems via Embedding of Known Refeence Tajecoies Ricado G. Sanfelice, J. J. Benjamin Biemond, Nahan van de Wouw, and W. P. Mauice H. Heemels Absac We sudy he poblem of designing conolles o ack ime-vaying sae ajecoies fo plans modeled as hybid dynamical sysems, which ae sysems wih boh coninuous and discee dynamics. The efeence ajecoies ae given by funcions ha may exhibi jumps. The class of conolles consideed ae also modeled as hybid sysems. These ae designed o guaanee sabiliy of acking and ha he diffeence beween he plan s sae and he efeence ajecoy conveges o zeo. Using ecenly developed ools fo he sudy of asympoic sabiliy in hybid sysems, we ecas he acking poblem as he poblem of sabilizing a closed se and deive condiions fo he design of acking conolles fo hybid efeence ajecoies wih he popey ha he jump imes of he plan coincide wih hose of he given efeence ajecoies. The appoach is illusaed in examples. I. INTRODUCTION The lieaue on sabiliy analysis and sabilizaion of equilibia fo sysems wih sae jumps is elaively well developed. On he ohe hand, in many conol poblems, such as acking, oupu egulaion, synchonizaion, and obseve design, he goal consiss of sabilizing ime-vaying ajecoies (e.g., ime-vaying efeence ajecoies fo acking). To effecively ackle such poblems fo hybid sysems, esuls on he sabiliy and sabilizaion of ime-vaying ajecoies of such sysems ae impeaive. Unfounaely, geneal esuls fo sabilizing impulsive/disconinuous, o, moe geneally, hybid ajecoies ae no cuenly available. Noable specific soluions o sabilizaion of such ajecoies ae given by he wok in [], [], [4], in which he sae esimaion and acking poblems fo paicula classes of mechanical sysems wih impacs ae addessed, he wok in [7], in which he obseve design poblem fo a class of complemenaiy sysems is sudied, he wok in [9], in which he acking conol fo a class of measue diffeenial inclusions is solved, he wok in [], [2], [3], in which he acking poblem fo classes of mechanical sysems wih unilaeal consains is addessed, and he wok in [6] consideing he juggling poblem as a acking poblem. In his pape, we pesen sufficien condiions chaaceizing conolles solving a sae acking conol poblem. We R. G. Sanfelice is wih he Depamen of Aeospace and Mechanical Engineeing, Univesiy of Aizona, 3 N. Mounain Ave, AZ 8572, USA. sicado@u.aizona.edu J. J. B. Biemond, Nahan van de Wouw, and W. P. M. H. Heemels ae wih he Depamen of Mechanical Engineeing, Eindhoven Univesiy of Technology, P.O. Box 53, 56 MB Eindhoven, The Nehelands. j.j.b.biemond,n.v.d.wouw,m.heemels@ue.nl. J. J. B. Biemond and Nahan van de Wouw ae suppoed by he Nehelands oganisaion fo scienific eseach (NWO). conside plans given in ems of a consained flow equaion ξ = f p (ξ,u) (ξ,u) C p () and a consained jump inclusion ξ + G p (ξ,u) (ξ,u) D p, (2) wih oupu y = ξ. Fo his class of hybid sysems, a conolle assigning he inpu u and measuing ξ is o be designed so ha he diffeence beween ξ and he efeence ajecoy, which may boh flow and jump, is well behaved. Wihou being pecise abou a noion of acking, i should be expeced ha he acking conolle guaanees boh sabiliy and aaciviy popeies elaive o he efeence ajecoy. The fome consiss of he popey ha soluions o he plan saing close o he efeence say close o i while he lae consiss of he popey ha he disance beween he plan s soluion componen and he efeence deceases asympoically. As we indicae in Secion II, his is a difficul ask when he jumps of he efeence and of he plan do no coincide. The poposed appoach in his noe consiss of ecasing a sae acking poblem fo hybid sysems, which is defined in Secion IV, as he sabilizaion of a closed se ha embeds he efeence ajecoy. Exploiing sufficien condiions fo asympoic sabiliy of closed ses fo hybid sysems, in Secion V we pesen sufficien condiions fo a class of hybid acking conolles enfocing ha he jump imes of he plan o coincide wih hose of he given efeence ajecoy. The appoach is illusaed in examples in Secion VI. II. MOTIVATION Conside a scala, single-valued hybid plan as in ()- (2) wih y = ξ and he efeence ajecoy o be acked given by he sawooh signal shown in Figue, which has disconinuiies when eaching. Tajecoies ξ o he plan can be defined as funcions defined on hybid ime domainsdomξ, which ae subses of R N := [,+ ) {,,2,...} and paameeize he ajecoies by flow ime and jump ime j [6]; see Secion III fo moe deails. A ypical appoach used in acking conol of coninuous-ime and discee-ime plans consis of defining he acking eo and hen analyzing he esuling ime-vaying eo dynamics. Following his appoach, he efeence ajecoy on he hybid ime domain dom is given by (,j) = j [ j, j+], (3) whee j = j, j N. Noe ha (,j) [,] fo all (,j) dom, whee dom is he union of [ j, j+ ] {j} fo evey

2 j N. Le T := j N > ( j,j ) (4) denoe he fixed values of (,j) a which jumps. Then, he dynamics of he acking eo e := y (,j) = ξ (,j) ae given by he flow equaion when ė = f p (e+(,j),u) (5) (e+(,j),u) C p and [ j, j+ ], (6) and by he jump equaion when e + = G e (e+(,j),u,,j) (7) (e+(,j),u) D p o (,j) T, (8) whee G e is defined a evey poin saisfying (8) as G p (e+,u) (e+,u) D p,(,j) T G e = e+ (e+,u) D p,(,j) T G p (e+,u) (e+,u) D p,(,j) T. (Fo noaional convenience, we emoved he agumens of some of he funcions.) 2 2 (a) Hybid ac. j 2 3 (b) Pojecion ono R. Fig.. Refeence ajecoy fo he acking conol poblem in Secion II. (ξ,,g e ) = (.99,,.99) G e * (ξ,,g e ) = (,,) Fig. 2. A esuling jump map G e fo he eo sysem in he acking conol poblem of Secion II. The map G e is defined fo each (ξ,) (R {}) ({} [,]). Fis, noe ha, in paicula, he consains (6) and (8) canno be wien in ems of he acking eo solely. Now, suppose ha a feedback law u = κ c (y,) is designed o map ξ he eo o zeo when boh he plan s sae ξ and jump simulaneously, ha is, he hid case in he definiion of G e is zeo. I is possible ha, fom poinsξ in C p ha ae neaby D p and imes(,j) T,G e updaese o e + = e+, which is fa fom zeo. In fac, Figue depics a paicula map G e as a funcion of ξ and (,j) when he jumps of he plan occu when ξ =, ha is, D p := {(ξ,u) : ξ = }, and wih G p (ξ,u) = ξ +u, κ c (y,) = ξ. Since, fo he given efeence, condiion (,j) T is equivalen o (,j) =, he jump map G e is wien as a funcion of (ξ,) only and is defined a evey poin saisfying (8), which is he se of poins (ξ,) in (R {}) ({} [,]). Noe ha when (,j) T (equivalenly, (,j) = ) if ξ = hen e + = bu if ξ is slighly below, hen e + will be close o afe he jump. This peaking phenomenon, which is due o he jump insans of plan and efeence no coinciding, has also been ecognized in [8], [], [3] and imposes a difficuly in guaaneeing ha he nom of e conveges o zeo. We conside acking conolles ha avoid he issue of an inceasing eo signal by ensuing ha jumps of he plan occu a he same insan as he jumps of he efeence ajecoies. Fo he illusaive example above, a conolle designed wih he said appoach will assign u so ha he jumps of he plan and he efeence ajecoy occu joinly. Fo his pupose, we ecas he acking conol poblem as he sabilizaion of a closed se which embeds he imevaying efeence ajecoy. Fo he design of he acking conolles we exploi sufficien condiions fo asympoic sabiliy of hybid sysems in [5] (see also [4] and [5]). An alenaive appoach based on geneaing he efeence ajecoies fom an exosysem was poposed in [3]. III. PRELIMINARIES Below, given a se S, S denoes is closue; given a veco x R n, x denoes he Euclidean veco nom; given a se S R n and a poin x R n, and x S := inf y S x y. A funcion α : R R is said o belong o class-k (α K) if is coninuous, zeo a zeo, and sicly inceasing and o belong o class-k (α K ) if i belongs o class-k and is unbounded.pd denoes he se of eal-valued posiive definie funcions. A hybid sysem H wih sae x, inpu u, and oupu y is modeled as ẋ = f(x,u) (x,u) C H x + G(x,u) (x,u) D (9) y = h(x), whee R n is he space fo he sae x, U R m is he space fo inpus u, he se C R n U is he flow se, he funcion f : C R n is he flow map, he se D R n U is he jump se, G : D R n is he jump map, and h : R n R p is he oupu map. The daa of he hybid sysem H is given by (C,f,D,G,h). Soluions o hybid sysems H ae defined by hybid acs on hybid ime domains, which ae funcions defined on subses of R N given by he union of inevals of he fom [ j, j+ ] {j}, j j+ ; see [6] fo moe deails.

3 We define sabiliy and Lyapunov funcions fo closed hybid sysems (no inpus and oupus) given by { ẋ = f(x) x C H x + () G(x) x D. The following definiion inoduces sabiliy fo geneal ses of he sae space. Given φ(,) R n, SH(φ(,)) denoes he se of maximal soluions φ o H wih φ(,). Definiion 3. (sabiliy): A se A R n is said o be unifomly globally sable (UGS) if hee exiss α K such ha each soluion φ SH(φ(,)) saisfies φ(,j) A α( φ(,) A ) fo all (,j) domφ; unifomly globally aacive (UGA) if fo each ε > and λ > hee exiss T > such ha, fo any soluion φ SH(φ(,)) wih φ(,) A λ, (,j) domφ and +j T imply φ(,j) A ε; unifomly globally asympoically sable (UGAS) if i is boh unifomly globally sable and unifomly globally aacive. Definiion 3.2 (Lyapunov funcion candidae): A funcion V : domv R is said o be a Lyapunov funcion candidae fo he hybid sysem H = (C,f,D,G) wih espec o he closed se A if he following condiions hold: ) C D G(D) domv, 2) V is coninuously diffeeniable on an open se conaining C, whee C denoes he closue of C. The following esul fo asympoic sabiliy of closed ses will be employed in he design of hybid conolles fo acking. I is a Lyapunov sabiliy heoem fo hybid sysems. Theoem 3.3: (Lyapunov heoem [5]) Le H = (C,f,D,G) be a hybid sysem and le A R n be closed. If V is a Lyapunov funcion candidae fo H wih espec o A and hee exis α,α 2 K, and a posiive definie and coninuous funcion ρ such ha α ( x A ) V(x) α 2 ( x A ) x C D G(D), (a) V(x),f(x) ρ( x A ) x C, (b) V(g) V(x) ρ( x A ) x D, g G(x), (c) hen A is unifomly globally asympoically sable fo H. The following esul inoduces elaxed Lyapunov condiions. Coollay 3.4: (elaxed Lyapunov condiions [5]) Le H = (C,f,D,G) be a hybid sysem and le A R n be closed. Suppose ha V is a Lyapunov funcion candidae fo H wih espec o A and hee exis α,α 2 K, and a coninuous ρ PD such ha (a) and eihe A) o B) below holds: A) Condiion (c) holds, V(x),f(x) x C, (2) and, fo each λ >, hee exis γ λ K, N λ such ha fo evey soluion φ o H wih φ(,) A (,λ] we have ha (,j) domφ, + j T imply j γ λ (T) N λ ; B) Condiion (b) holds, V(g) V(x) x D, g G(x), (3) and, fo each λ >, hee exis γ λ K, N λ such ha fo evey soluion φ o H wih φ(,) A (,λ] we have ha (,j) domφ, + j T imply γ λ (T) N λ ; hen A is unifomly globally asympoically sable. This coollay saes ha unifom aaciviy can be asseed as long as he Lyapunov funcion deceases, along soluions, ove sufficienly long hybid ime inevals. Moe pecisely, A) is abou he Lyapunov funcion being noninceasing duing flows bu sicly deceasing duing jumps and he jumps occu fequenly enough while B) is abou he Lyapunov funcion being noninceasing duing jumps bu sicly deceasing duing flows and he flows occu fo long enough. IV. PROBLEM STATEMENT We conside plans modeled as hybid sysems H p wih sae ξ R np, inpu u R mp, and oupu y = ξ given by H p { ξ = fp (ξ,u) (ξ,u) C p ξ + G p (ξ,u) (ξ,u) D p (4) wih daa (C p,f p,d p,g p ). We conside hybid acs : dom R np defining efeence ajecoies o be acked. The following class of acking hybid conolles wih sae η R nc and daa (C c,f c,d c,g c,κ c ) is consideed η = f c (η,y,) (η,y,) C c H c η + G c (η,y,) (η,y,) D c (5) u = κ c (η,y,). The inpu of H c has been assigned o (y,) while is oupu u o he inpu of he plan H p. The closed-loop sysem (4)- (5) esuling fom he ineconnecion of H p and H c is denoed H cl, has sae x := (ξ,η) R np R nc and is given by } ξ = f p (ξ,κ c (η,ξ,)) (ξ,κc (η,ξ,)) C p η = f c (η,ξ,) } and (η,ξ,) C c ξ + G p (ξ,κ c (η,ξ,)) η + (ξ,κ = η c (η,ξ,)) D p } ξ + = ξ η + (η,ξ,) D G c (η,ξ,) c, (6) whee, fo noaional simpliciy, we have omied he agumen (,j) of he ime-vaying efeence. Using he above definiions, we sae a acking conol poblem fo hybid sysems. Tacking Conol Poblem ( ): Given a plan H p and a complee efeence ajecoy design he daa

4 (C c,f c,d c,g c,κ c ) of he conolle H c so ha he se of poins ξ saisfying ξ = (,j) (7) is unifomly globally asympoically sable. Poblem ( ) asks fo a conolle such ha he (imevaying) se of poins (7) has he UGS and UGA popeies (see Definiion 3.) fo he closed-loop sysem. The aaciviy popey implies ha complee soluions o H cl saisfy lim ξ(,j) (,j) =. +j Moeove, he sabiliy popey implies ha soluions o he plan wih iniial condiions ξ(,) = (,), if hey exis, saisfy ξ(,j) = (,j) fo all (,j) domξ. Noe ha unless fuhe condiions ae imposed on, he se in (7) is ime vaying and no compac. Fuhemoe, boundedness of he sae of he conolle is no guaaneed by UGAS of (7) and has o be esablished sepaaely. V. A CLASS OF HYBRID CONTROLLERS FOR STATE TRACKING WITH KNOWN REFERENCE TRAJECTORIES A. Main Appoach In smooh sysems, a well-known appoach is o inoduce he coodinae ansfomaion e = ξ and hen analyze he esuling sysem. This appoach is used fo sysems wih ime-iggeed sae jumps in [2]. Howeve, in geneal, he flow and jump ses as well as he flow and jump maps of he eo dynamics become ime dependen. To avoid his issue, we ecas Poblem ( ), which peains o he sabilizaion of a ime-vaying se, as he sabilizaion of a closed, no necessaily bounded, ime-invaian se. To his end, given a efeence : dom R np, following (4), we define he se T collecing all of he poins (,j) in he domain of a which jumps, ha is, evey poin ( j,j) dom fo which ( j,j+) dom. Auxiliay vaiables τ R and k N ae incopoaed as saes o paameize a given efeence ajecoy. Tha is, τ evolves coninuously accoding o he flow ime paamee, while k evolves disceely accoding o he jump ime paamee j a jumps of. In his seing, he se o be sabilized is given by A = {(x,τ,k) R np R N : ξ = (τ,k) }. (8) Fo insance, fo he example of Secion II, he se o be sabilized wih he poposed appoach is given by { (x,τ,k) : ξ + k = τ [ k, k+ ],( k,k) (,) T }, whee T is given in (4). This se is closed and unbounded in he τ and k componens. The nex ingedien of he appoach is o guaanee, by design of he conolle, ha he jumps of he plan and of he efeence ajecoy occu simulaneously. This will be a consain in he design of he conolle, which, while i The definiion of UGAS fo a ime-vaying sysem follows Definiion 3.; see [4]. esics he ype of sysems fo which he acking poblem can be solved, i allows fo a soluion o ceain acking poblems as Secion VI illusaes. Wih a conolle saisfying such a popey, ou appoach is o ecas he poblem unde sudy as he sabilizaion of he se A fo he esuling closed-loop sysem ξ = f p (ξ,κ c (η,ξ,(τ,k))) (ξ,κ η = f c (η,ξ,(τ,k)) c (η,ξ,(τ,k))) C p and (η,ξ,(τ,k)) C τ = c and τ [ k k =, k+ ],k N ξ + G p (ξ,κ c (η,ξ,(τ,k))) η + = η, τ + (ξ,κ = τ c (η,ξ,(τ,k))) D p k + and (τ,k) T = k + ξ + = ξ η + G c (η,ξ,(τ,k)) (η,ξ,(τ,k)) D c. τ + = τ, k + = k (9) This closed-loop sysem is denoed Hcl. Is daa is given by (C,f,D,G) whee C := {(x,τ,k) : (ξ,κ c (η,ξ,(τ,k))) C p, τ [ k, k+ ],k N,(η,ξ,(τ,k)) C c}, f p (ξ,κ c (η,ξ,(τ,k))) f(x,τ,k) := f c (η,ξ,(τ,k)), D := D D 2 D := {(x,τ,k) : (ξ,κ c (η,ξ,(τ,k))) D p, (τ,k) T } D 2 := {(x,τ,k) : (η,ξ,(τ,k)) D c }, G p (ξ,κ c (η,ξ,(τ,k))) G (x,τ,k) := η τ k + (x,τ,k) D \D 2, ξ G(x,τ,k) := G 2 (x,τ,k) := G c (η,ξ,(τ,k)) τ k (x,τ,k) D 2 \D, {G (x,τ,k),g 2 (x,τ,k)} (x,τ,k) D D 2. Then, asympoic sabiliy of A can be asseed using he sufficien condiions povided by Theoem 3.3 and Coollay 3.4. B. Chaaceizaion of Hybid Conolles The daa (C c,f c,d c,g c,κ c ) is designed so ha: Thee exis a Lyapunov funcion candidae V : R np R nc R N R fohcl wih espec oa, funcions α,α 2 K, and a coninuous ρ PD such ha α ( (x,τ,k) A ) V(x,τ,k) α 2 ( (x,τ,k) A ) (x,τ,k) C D G(D), V(x,τ,k),f(x,τ,k) ρ( (x,τ,k) A ) (x,τ,k) C, (2) (2)

5 V(g) V(x,τ,k) ρ( (x,τ,k) A ) (x,τ,k) D \D 2, g G (x,τ,k), V(g) V(x,τ,k) ρ( (x,τ,k) A ) (x,τ,k) D 2 \D, g G 2 (x,τ,k), (22) (23) V(g) V(x,τ,k) ρ( (x,τ,k) A ) (x,τ,k) D D 2, g {G (x,τ,k),g 2 (x,τ,k)}. (24) Remak 5.: The condiions above imply ha complee soluions o he closed-loop sysem ae such ha (x,τ,k)(,j) A as +j, ha is, ξ(,j) (τ(,j),k(,j)) as +j. This includes all possible soluions wih unconsained iniial condiions of τ and k, in paicula, τ(,) = k(,) =, fo which (τ(,j),k(,j)) = (,j) and, consequenly, ξ(,j) (,j) as +j. Noe ha complee soluions o H cl have he popey ha τ(,j)+k(,j) is unbounded as +j. Fuhemoe, i implies ha ξ(,j) = (,j) on he domain of definiion of soluions saing fom ξ(,) = (,),τ(,) = k(,) =, when soluions fom such poins exis. While he condiions above could have been expessed in ems of he acking eo e, i is aely he case ha is dynamics can be wien as a funcion of e and η only. The daa of he hybid conolle has o be chosen so ha (2)-(24) hold. In paicula, condiion (2) depends on f c,c c and κ c ; (22) depends on κ c ; and (23) depends on G c and D c, which ae all o be chosen in he design. The following esul summaizes he discussion above. Theoem 5.2: Suppose ha fo a given complee efeence ajecoy : dom R np he se A is closed. If hee exiss a hybid conolle H c guaaneeing ha he jumps of and H p occu simulaneously and hee exis a Lyapunov funcion candidae V : R np R nc R N R fo H cl wih espec o A as in (8), funcions α,α 2 K, and a posiive definie and coninuous funcion ρ such ha (2)-(24) hold, hen H c povides a soluion o Poblem ( ). Remak 5.3: The condiions in 2) of Theoem 5.2 can be elaxed accoding o iems A) and B) of Coollay 3.4. Theoem 5.2 chaaceizes conolles solving he acking poblem. We foesee ha fo specific classes of hybid sysems (such as hose wih linea flow and jump maps), consucive conolle design echniques can be developed. The simple examples in he nex secion illusae he feasibiliy of he design of conolles saisfying he condiions of he heoem. VI. EXAMPLES Example 6. (Tacking a squae wave signal): Conside he hybid plan H p ξ = aξ +u ξu, ξ >, (25) ξ + = b+u 2 ξu, ξ >, (26) whee a,b >, and conside he poblem of acking he squae wave signal (,j) = ( ) j+ [ j, j+ ], j N, j = j. Then, following he appoach poposed in Secion V, he goal is o solve Poblem ( ) wih A given by { (ξ,τ,k) : ξ = ( ) k+,(τ,k) j N ([j,j +] {j}) }. Fo his pupose, we conside he saic conolle [ ] [ ] u a(τ,k) = κ u c (ξ,(τ,k)) =, 2 b (τ,k)+λ(ξ (τ,k)) wih λ [,). I follows ha, fo evey ξ(,) <, evey jump of igges a jump of he plan. In fac, if ξ(,) <, since u = a(τ,k), we have ha aξ(,)(,) > and soluions iniially flow. Flows of ξ will no igge a jump since he sign of ξ emains consan. Jumps of he closed-loop sysem occu only when changes sign, which is a (,j) s in T, T = {(,),(2,),(3,2),...}. Then, he closed-loop sysem H = (C,f,D,G) given by } ξ = a(ξ +(τ,k)) aξ(τ,k), ξ >, τ =, k = τ [ k, k+ } ],k N ξ + = (τ,k)+λ(ξ (τ,k)) aξ(τ,k), τ + = τ, k + = k + ξ >,(τ,k) T capues all of he soluions o he oiginal sysem wih iniial condiions ξ(,) <, τ(,) = k(,) =. To esablish asympoic sabiliy of A, conside he Lyapunov funcion candidae V(ξ,τ,k) = 2 (ξ (τ,k))2, fo which condiion (2) holds ivially. Fo each (ξ,τ,k) saisfying aξ(τ,k), ξ >,τ [ k, k+ ], k N we have V(ξ,τ,k),f(ξ,τ,k) = 2aV(ξ,τ,k); and fo each (ξ,τ,k) saisfying aξ(τ,k), ξ >,(τ,k) T we have V(G(ξ,τ,k)) V(ξ,τ,k) = ( λ 2 )V(ξ,τ,k). Then, Theoem 5.2 implies unifom global asympoic sabiliy of A fo he closed-loop sysem. Figue 3(a) depics a closed-loop sysem ajecoy conveging o he efeence asympoically, boh along flows and jumps. Figue 3(b) depics he Lyapunov funcion along he ajecoy. ξ, ξ (a) (b) Fig. 3. Refeence and closed-loop sysem ajecoy fo Example 6.. The Lyapunov funcion along he ajecoies is also shown. Paamees: a = b =, λ =.9. V

6 Example 6.2 (Tacking fo a moion conol sysem): Conside a paicle wih mass M acuaed by a foce inpu u. The posiion of he paicle is denoed by ξ and is velociy by ξ 2. The conolle foce u conains a Lebesgue inegable pa u and an impulsive pa u 2 wih impulses a insans i. Hence, he plan is impulsive and modeled as ] ] ξ = [ ξ2 u M when i, ξ + = ξ +[ u 2 M when = i, whee M >, he sae ξ is compleely measued. The inpu u will[ be] designed, such ha he sae ξ follows a efeence =, given in Figue 4. The componen 2 jumps 2 a imes (,j) T = j N (j +,j). Such a efeence ajecoy can be desiable fo he posiion of he end effeco of a obo sysem. A conolle ha sabilizes he se A is given by u = k (ξ ) k 2 (ξ 2 2 ), (,j) T, M, (,j) +3,4k+2) (4k +4,4k+3) u 2 = k N(4k M, (,j) k N(4k +,4k) (4k +2,4k+), whee k,k 2 >. Using coodinaes z = ξ, he closedloop sysem H = (C,f,D,G) can be wien as [ ] } τ [ ż = z, τ =, k = k, k+ ] k N k M k2 M z + = z, τ + = τ, k + = k + } (τ,k) T. The feedfowad signal u 2 assues ha z is no affeced by he jumps of he efeence. Fuhemoe, if he iniial condiions ae ξ(,) = (,), τ(,) = k(,) =, hen he soluion saisfies (,j) = ξ(,j) fo all (,j) dom. Take he quadaic Lyapunov funcion V(z,τ,k) = z Pz wih P = P > such ha V(z,τ,k),f(z,τ,k) V(z,τ,k). Such a maix P is guaaneed o exis due o he coninuous dynamics of z. Since z does no change a jumps, we ge V(G(z,τ,k)) V(z,τ,k) = (27) fo evey (z,τ,k). By he popeies of V, hee exis funcions α,α 2 and ρ such ha boh (2) and (2) ae saisfied. Moeove, he hybid ime domain of each soluion o he closed-loop sysem is unbounded in he diecion. Hence, ξ, (a) ξ ξ2, Fig. 4. Refeence and closed-loop ajecoy fo Example 6.2. Paamees: M =, k =, and k 2 =.5. (b) 2 ξ2 following Remak 5.3, Theoem 5.2 guaanees global unifom asympoic sabiliy of he se A fo he closed-loop sysem. In Figue 4, a closed-loop ajecoy is shown fo paamees M =, k = and k 2 =.5. VII. CONCLUSION Fo a class of hybid sysems given in ems of hybid inclusions, we sae a acking conol poblem fo acking of efeence signals wih jumps. The poposed echnique consiss of embedding he efeence ajecoy ino a se and hen apply Lyapunov sabiliy ools o he closed-loop sysem. The class of conolles consideed have o guaanee he sic popey of jump imes of he plan coinciding wih hose of he given efeence ajecoies. Relaxaion of his singen condiion is pa of cuen eseach. REFERENCES [] J. M. Bougeo and B. Bogliao. Tacking conol of complemenaiy Lagangian sysems. In. J. Bif. Chaos, 5(6): , 25. [2] B. Bogliao, S. Niculescu, and P. Ohan. On he conol of finiedimensional mechanical sysems wih unilaeal consains. IEEE Tansacions on Auomaic Conol, 42(2):2 25, 997. [3] B. Bogliao, S.I. Niculescu, and M. Moneio-Maques. On acking conol of a class of complemenay-slackness hybid mechanical sysems. Sysems and Conol Lees, 39(4): , 2. [4] S. Galeani, L. Menini, and A. Tonambe. 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