On Necessary and Sufficient Conditions for Incremental Stability of Hybrid Systems Using the Graphical Distance Between Solutions

Transcription

1 On Necessary and Sufficien Condiions for Incremenal Sabiliy of Hybrid Sysems Using he Graphical Disance Beween Soluions Yuchun Li and Ricardo G. Sanfelice Absrac This paper inroduces new incremenal sabiliy noions for a class of hybrid dynamical sysems given in erms of differenial equaions and difference equaions wih sae consrains. Incremenal sabiliy is defined as he propery ha he disance beween every pair of soluions o he sysem has sable behavior (incremenal sabiliy) and approaches zero asympoically (incremenal araciviy) in erms of graphical convergence. Basic properies of he class of graphically incremenally sable sysems are considered as well as hose implied by he new noions are revealed. Moreover, several sufficien and necessary condiions for a hybrid sysem wih such a propery are esablished. Examples are presened hroughou he paper o illusrae he noions and resuls. I. INTRODUCTION In recen years, incremenal sabiliy-like properies have been used in he sudy of synchronizaion [], [2], [3], observer design [4], [5], conrol design [6], as well as he sudy of convergen sysems [7]. Unforunaely, he incremenal sabiliy noions based on Euclidean and Riemannian disance canno be applied direcly o sysems wih variables ha can change coninuously and, a imes, jump discreely. These sysems, known as hybrid sysems, are capable of modeling a wide range of complex dynamical sysems, including roboic, auomoive, and power sysems as well as naural processes. Alhough se sabiliy heory in erms of Lyapunov funcions is available (see [8], [9]), incremenal sabiliy noions for such sysems would enable he sudy of properies similar o he curren noion for coninuousime sysems. While he iniial effor in [] defines an incremenal sabiliy noion ha prioriizes ordinary ime, he noion herein migh be difficul o guaranee for hybrid sysems ha conain he so called peaking phenomenon as recognized in, e.g., [], [2]. In his paper, we inroduce conceps for he sudy of incremenal graphical sabiliy of hybrid sysems ha may have he peaking phenomenon. In paricular, for a class of hybrid sysems in he framework of [8], we inroduce new noions of incremenal sabiliy and incremenal araciviy based on he graphical disance beween soluions. Moreover, sufficien condiions based on hose found in conracion heory and finie-ime sabiliy heory wihin a neighborhood of he jump se are esablished. Furhermore, i is shown ha finie-ime sabiliy wihin a neighborhood of he jump se is also a necessary condiion for a hybrid sysem o be graphically incremenally sable or aracive. To he bes Y. Li and R. G. Sanfelice are wih he Deparmen of Compuer Engineering, Universiy of California, Sana Cruz, CA yuchunli,ricardo@ucsc.edu This research has been parially suppored by he Naional Science Foundaion under CAREER Gran no. ECS- 536 and by he AFOSR under Gran no. FA of our knowledge, he noion of incremenal sabiliy and is properies for hybrid sysems have no been horoughly sudied, only discussed briefly in [3] for a class of ransiion sysems in he conex of bisimulaions, and in [] for a paricular class of hybrid sysems prioriizing ordinary ime. The remainder of his paper is organized as follows. Secion II briefly reviews he hybrid sysem framework used and is basic properies. Secion III inroduces he noion of graphical incremenal sabiliy and araciviy, and illusraes hem in wo examples. Secion IV presens he main resuls, which consis of several sufficien and necessary condiions. Examples are discussed hroughou he paper o illusrae he resuls. Due o he space limiaions, complee proofs will be published elsewhere. A. Noaion II. PRELIMINARIES ON HYBRID SYSTEMS AND GRAPH NOTIONS Given a se S R n, he closure of S is he inersecion of all closed ses conaining S, denoed by S; S is said o be discree if here exiss δ > such ha for each x S, (x+δb) S={x}; cons is he closure of he convex hull of he se S. R :=[, ) and N := {,,...}. Given vecors ν R n, w R m, ν defines he Euclidean vecor norm ν = ν ν and [ν w ] is equivalen o (ν,w). Given a funcion f : R m R n, is domain of definiion is denoed by dom f, i.e., dom f := {x R m : f(x) is defined}. The range of f is denoed by rge f, i.e., rge f := {f(x) : x domf}. The righ limi of he funcion f is defined as f + (x) := lim ν + f(x+ν) if i exiss. Given a poin y R n and a closed se A R n, y A := inf x A x y. A funcion α : R R is a class-k funcion, also wrien α K, if α is zero a zero, coninuous, sricly increasing, and unbounded; α is posiive definie, also wrien α PD, if α(s)> for all s> and α()=. Given a real number x R, floor(x) is he closes ineger o x from below. A funcion V : R n R is called a Lyapunov funcion wih respec o a se A if V is coninuously differeniable and such ha c ( x A ) V(x) c 2 ( x A ) for all x R n and some funcions c,c 2 K. B. Preliminaries In his paper, a hybrid sysem H has daa (C,f,D,g) and is defined by ż = f(z) z C, z + () = g(z) z D,

2 where z R n is he sae, he map f defines he flow map capuring he coninuous dynamics and C defines he flow se on whichf is effecive. The map g defines he jump map and models he discree behavior, while D defines he jump se, which is he se of poins from where jumps are allowed. A soluion φ o H is paramerized by (,j) R N, where denoes ordinary ime and j denoes jump ime. The domain dom φ R N is a hybrid ime domain when i saisfies [8, Definiion 2.3]. A soluion o H is called maximal if i canno be exended, i.e., i is no a runcaed version of anoher soluion. I is called complee if is domain is unbounded. A soluion is Zeno if i is complee and is domain is bounded in he direcion. A soluion is precompac if i is complee and bounded. The se S H conains all maximal soluions o H, and he se S H (ξ) conains all maximal soluions o H from ξ. A hybrid sysem H is said o saisfy he hybrid basic condiions if i saisfies [8, Definiion 6.5]. The propery of pre-forward compleeness of a hybrid sysem H is characerized in [8, Definiion 6.2]. The graph of a hybrid arc is defined in [8, Definiion 5.2], and he disance of graphs of wo hybrid arcs is measured by ε-closeness noion in [8, Definiion 4.]. In order o characerize he propery of hybrid arcs graphically converging o each oher, we inroduce he following noion. Definiion 2.: Given ε >, wo hybrid arcs φ and φ 2 are evenually ε-close if here exiss T > such ha (a) for each (,j) domφ such ha + j > T, here exiss (s,j) domφ 2 saisfying s < ε and φ (,j) φ 2 (s,j) < ε, (2) (b) for each (,j) domφ 2 such ha + j > T, here exiss (s,j) domφ saisfying s < ε and φ 2 (,j) φ (s,j) < ε. (3) We refer he reader o [8] and [9] for more deails on hese noions and he hybrid sysems framework. III. INCREMENTAL GRAPHICAL STABILITY In his paper, for hybrid sysems H as in (), we are ineresed in characerizing he incremenal sabiliy propery, namely, he noion ha he graphical disance beween every pair of maximal soluions o he sysem has sable behavior and approaches zero asympoically. To highligh he inricacies of his propery in he hybrid seing, a canonical example of hybrid sysems, he so-called bouncing ball sysem, is considered. Every maximal soluion o he bouncing ball sysem is Zeno and converges o he origin; see [8, Example. and 2.2] for more deails. Consider wo soluions o his sysem, given by φ and φ 2, from iniial condiions φ (,) = (5,) (ball iniialized a a posiive heigh wih zero velociy) and φ 2 (,)=(,3) (ball iniialized a he ground wih a posiive velociy). Figure (a) shows he posiion (firs) componen (φ i for i {,2}) of hese wo soluions, and Figure (b) shows he velociy (second) componen (φ 2 i for i {,2}) of hem. The Zeno behavior of he soluions makes i A soluion is Zeno if i is complee and is domain is bounded in he direcion j φ φ 2 (a) The firs componen (heigh φ i ) of soluions from φ (,) = (5,) and φ 2 (,) = (,3). 6 8 d( φ 2, φ 2 2 ) (b) The projecion of he second componen (velociy φ 2 i ) of soluions from φ (,) = (3,3) and φ 2 (,) = (3,3.) on he direcion. Fig.. Soluions o he bouncing ball sysem. The Euclidean disance, which is d( φ 2 (,j ), φ 2 2 (,j 2)) := φ 2 (,j ) φ 2 2 (,j 2) for all (,j ) dom φ and (,j 2 ) dom φ 2, has repeiive large peaks. exremely difficul o analyze incremenal propery of hese wo soluions. In fac, since graphical incremenal sabiliy requires comparing he graphs beween wo soluions, if one soluion (φ ) reaches he Zeno ime sooner han he second soluion (φ 2 ), hen he graphical disance beween hese wo soluions canno be evaluaed afer one of hem has approached he Zeno ime. This siuaion is shown in Figure (a), where φ 2 approaches Zeno a abou = 6sec while φ is sill describing he moion of he ball bouncing. Such exreme difference in he domains of he soluions makes i difficul (if no impossible) o compare φ and φ 2. Furhermore, a similar siuaion is encounered if, insead, he poinwise disance is used o measure he disance beween soluions. As shown in Figure (b), he poinwise disance of velociy (second) componens of wo soluions ( φ 2 i for i {,2}) has repeiive large peaks, even hough hey are iniialized very close o each oher. To avoid such possible dramaic differences on he hybrid ime domains and disances beween soluions o a hybrid sysem, hroughou he paper, we consider a class of hybrid sysems ha saisfies he following assumpion. Assumpion 3.: The hybrid sysem H = (C,f,D,g) is such ha ) each maximal soluion o H has a hybrid ime domain ha is unbounded in he direcion; 2) he flow map f is coninuously differeniable; 3) here exiss γ > such ha for each maximal soluion o H, he flow ime beween wo consecuive jumps is lower bounded by γ. Lemma 2.7 in [4] provides a sufficien condiion for condiion 3). As we show laer on, hese assumpions are no resricive, in fac, some of hem are necessary for esablishing sufficien condiions for a hybrid sysem o be graphically incremenally sable. The noion of incremenal sabiliy used in his work measures he graph disance beween soluions o hybrid sysems. I is defined as follows. Definiion 3.2: Consider a hybrid sysem H wih sae z R n. The hybrid sysem H is said o be ) incremenally graphically sable (δs) if for every ε > here exiss δ > such ha, for any wo maximal φ 2 φ 2 2

3 soluions φ,φ 2 o H, φ (,) φ 2 (,) δ implies φ and φ 2 are ε-close; 2) incremenally graphically locally aracive (δla) if here exissµ > such ha for everyε >, for any wo maximal soluionsφ,φ 2 oh, φ (,) φ 2 (,) µ implies ha φ and φ 2 are evenually ε-close; 3) incremenally graphically globally aracive (δga) if for every ε >, and for any wo maximal soluions φ,φ 2 o H, φ and φ 2 are evenually ε-close; 4) incremenally graphically locally asympoically sable (δlas) if i is boh δs and δla. Remark 3.3: Noe ha he δla noion is differen from he δs noion. The former requires ha every pair of maximal soluions o H iniialized close converge o each oher graphically when complee, while he laer requires ha every pair of wo maximal soluions o H iniialized close say close graphically. The following examples illusrae some of he incremenal sabiliy noions (δs and δlas) in Definiion 3.2. Example 3.4: (A imer) Consider he hybrid sysem H ż = z [,], (4) z + = z =. (5) Noe ha every maximal soluion o H is complee. Consider wo maximal soluions φ, φ 2 o he sysem. To show δs, for a given ε >, le < δ < ε and assume φ (,) φ 2 (,) < δ. Wihou loss of generaliy, we furher suppose φ (,) > φ 2 (,). Then, soluion φ jumps before φ 2. For each j N \{}, le j = max (,j ) domφ domφ 2 and j = min (,j) domφ domφ 2. Then, we have ha for each [, ], here exiss (s,) domφ 2 such ha s = and φ (,) φ 2 (,) = φ (,)+ φ 2 (,) δ<ε. (6) For each [, ], φ (,) φ 2 (,) = φ 2 (,) (7) φ (,) φ 2 (,) δ < ε, where we used he fac ha φ (,)+ =. Moreover, for each [, ], φ (,) φ 2 (,) = δ < ε. (8) Noe ha δ < ε holds for all [, ]. In fac, for each [ i, i ], where i N\{,}, φ (,i ) φ 2 (,i ) = (φ ( i,i ) φ 2( i,i )) δ < ε. Moreover, for each [ i, i ], where i N\{,}, φ ( i,i ) φ 2 (,i ) = φ ( i,i ) φ 2 ( i,i ) ( i ) < ε, and φ (,i) φ 2 ( i,i) = i δ < ε. Therefore, he sysem is δs. Moreover, since he disance beween φ and φ 2 does no converge o zero, φ,φ 2 are no evenually ε-close and hus he sysem is neiher δla nor δga. As shown in Figure 2(a), he domains of wo soluions o he imer sysem are differen from each oher. In such case, he Euclidean disance may no be a good candidae of a disance funcion for he sudy of incremenal properies, φ φ (a) The projecions of wo soluions from φ (,) = and φ 2 (,) =.2 on he direcion. d(φ, φ2) dg(φ, φ2) (b) Comparison beween Euclidean disance (op) and graphical disance (boom) beween φ and φ 2. Fig. 2. Two soluions φ and φ 2 o he imer sysem in Example 3.4. Unlike he Euclidean disance, which is d(φ (,j ),φ 2 (,j 2 )) := φ (,j ) φ 2 (,j 2 ) for all (,j ) domφ and (,j 2 ) domφ 2, which assumes he value.8 for.2 seconds afer every.8 seconds, he graphical disance saisfies d g(φ (,j),φ 2 ) := min{ φ (,j) φ 2 (s,j) : s.2,(,j) domφ,(s,j) domφ 2 }, which is upper bounded by.2. as shown in he op subfigure of Figure 2(b). Noe ha no maer how close he wo maximal soluions are iniialized, he peak always exiss for he Euclidean disance beween hem. However, he graphical disance beween he soluions, as shown in he boom subfigure of Figure 2(b), is bounded by.2 for all hybrid ime in he respecive domain of he soluions. We will presen a sysemaic way o check he propery laer on in his paper. 2 Example 3.5: Consider a hybrid sysem H wih daa f(z) : = z z C := [i,i+] i {2k:k N} (9) g(z) : = z z D := {2k : k N\{}}. Noe ha every maximal soluion o H is complee. Given ε >, consider wo maximal soluions φ and φ 2 such ha φ (,) φ 2 (,) < δ, where δ = min{,ε}. Then, i is guaraneed ha J := sup (,j) domφ j = sup (,j) domφ2 j <. For eachj N\{}, le j = max (,j ) domφ domφ 2 and j = min (,j) domφ domφ 2. Wihou loss of generaliy, assume φ 2 (,) > φ (,) 2, hen φ jumps firs. Then, we have ha for each [, ], here exiss (s,) domφ 2 such ha s = and φ (,) φ 2 (,) = φ (,)e φ 2 (,)e δ<ε. () For each [, ], φ (,) φ 2 (,) = e φ (,) e φ 2 (,) e φ (,) e φ 2 (,) δ < ε, () where we used he propery e φ (,)=floor(φ (,))= floor(φ 2 (,)) = e φ2 (,). Noe ha = ln(φ (,)) ln(floor(φ (,))) and = ln(φ 2 (,)) ln(floor(φ 2 (,))), herefore, = ln(φ 2 (,)) ln(φ (,)). Furhermore, by mean value heorem, here exiss φ [φ (,),φ 2 (,)] such ha = φ φ (,) φ 2 (,) φ (,) φ 2 (,) δ < ε. Similarly, for each [, ], φ (,) φ 2 (,) φ 2(,) φ (,) δ. (2) If φ (,),φ 2 (,), we have ha φ (,) 2 I may be possible o consruc an alernaive disance funcion ha is decreasing along soluions using he ideas in [5].

4 φ φ (a) The projecions of wo soluions from φ (,) = 4.5 and φ 2 (,) = 4.2 on he direcion. d(φ, φ2) dg(φ, φ2) (b) Comparison beween graphical disance and Euclidean disance beween φ and φ 2. Fig. 3. Two soluions φ and φ 2 o he sysem H. Unlike he Euclidean disance, which is d(φ (,j ),φ 2 (,j 2 )) := φ (,j ) φ 2 (,j 2 ) for all (,j ) domφ and (,j 2 ) domφ 2, which does no decrease along soluions, he graphical disance saisfies d g(φ (,j),φ 2 ) := min{ φ (,j) φ 2 (s,j) : s.2,(,j) domφ,(s,j) domφ 2 }, which decreases along soluions. φ 2 (,) e φ (,) φ 2 (,) δ for all (,) domφ = domφ 2. In fac, lim,(,) domφ φ (,) φ 2 (,) =. Therefore, he sysem H is δlas. As shown in Figure 3(a), he domains of wo soluions o he sysem H are differen from each oher. The Euclidean disance beween φ and φ 2 is shown in he op subfigure of Figure 3(b), while he graphical disance beween hem, as shown in he boom subfigure of Figure 3(b), converges o zero asympoically. IV. SUFFICIENT AND NECESSARY CONDITIONS FOR GRAPHICAL INCREMENTAL STABILITY NOTIONS In his secion, we presen several sufficien and necessary condiions of incremenal graphical sabiliy properies for hybrid sysems ha saisfy Assumpion 3.. The following resul highlighs a necessary propery of he hybrid ime domains of wo hybrid arcs ha are graphically close. In paricular, i holds for soluions o a hybrid sysem H ha is δs or δga. Lemma 4.: Given ε > and wo hybrid arcs φ and φ 2, he following hold: ) If φ and φ 2 are ε-close, hen sup j = sup j. (3) (,j) domφ (,j) domφ 2 2) If φ and φ 2 are complee and evenually ε-close, hen (3) holds. Example 4.2: Consider he Example 3.5, and wo soluions φ and φ 2 o H from φ (,)=4.5 and φ 2 (,)=, respecively. Soluionφ jumps wice while soluionφ 2 never jumps. Therefore, φ and φ 2 is no evenually ε-close which prevens he sysem from being δga even hough i is shown o be δlas in Example 3.5. The following resul esablishes ha uniqueness is a necessary condiion for δs. In urn, in paricular, i jusifies he choice of single-valued flow and jump maps in he definiion of H in (). Proposiion 4.3: (uniqueness of soluions under δs) Consider a hybrid sysem H wih sae z R n. Suppose H saisfies Assumpion 3. and H is δs. Then, every maximal soluion o H is unique. Based on Proposiion 4.3, assuming uniqueness of soluions o H is no a all resricive when sudying incremenal graphical sabiliy. Hence, in he following resuls we impose he following uniqueness of soluions assumpion. Assumpion 4.4: The hybrid sysem H = (C,f,D,g) is such ha each maximal soluion φ o H is unique. A sufficien condiion for guaraneeing uniqueness of maximal soluions requires f o be locally Lipschiz and no flow from C D, see [8, Proposiion 2.]. When he jump se D is a discree se, he following sufficien condiions for a hybrid sysem H o be δlas are esablished. In paricular, he graphical disance beween any wo maximal soluions o a hybrid sysem H sricly decreases during flows. Theorem 4.5: (δlas hrough flow wih D being a discree se) Consider a hybrid sysem H = (C,f,D,g) wih sae z R n. Suppose H saisfies Assumpion 3., Assumpion 4.4, and he hybrid basic condiions. Moreover, suppose D is a discree se. If here exis β > and δ > such ha H saisfies ) f(z)+ f (z) 2βI for all z conc; 2) for each δ [,δ ], each maximal soluion φ o H from φ(, ) saisfying φ(,) C, φ(,) D = δ (4) is such ha here exiss s [,δ] saisfying φ(s,) D =, φ(,) D δ [,s], (5) and he maximal soluion φ o H from g(φ(s,)) saisfies hen, H is δlas. φ(,) g(φ(s,)) δ [,s]; (6) Skech of he Proof: Given ε >, and using δ as in he iem 2) of assumpion and γ as in Assumpion 3., consider wo maximal soluions φ, φ 2 o H such ha φ (,) φ 2 (,) < δ, where δ is chosen such ha < δ min{ε,δ,γ} and for each z D, (z+δb) D = {z}. Ifφ (,),φ 2 (,) C and no jump occurs o eiherφ or φ 2, hen, by he generalized mean value heorem for vecorvalued funcions and iem ), for all (, ), we have d d φ (,) φ 2 (,) 2 2β φ (,) φ 2 (,) 2. (7) Then, by he comparison lemma, for all [, ), φ (,) φ 2 (,) exp( β) φ (,) φ 2 (,) δ. (8) Now consider he case when eiher φ or φ 2 jump. Assume φ jumps firs and J =. Furhermore, for each j N \ {}, le j = max (,j ) domφ domφ 2 and j = min (,j) domφ domφ 2, and =. Case I: When φ (,),φ 2 (,) C, similarly as in (7), for all [, ], we have ha φ (,) φ 2 (,) δ. (9) When =, since φ jumps firs, φ (,) D and φ (,) = g(φ (,)). Noe ha under iem 3) of Assumpion 3., g(d) D =. Therefore, by using (7) and iem 2), we obain

5 a) for each j N\{} and each [ j, j+ ]: φ (,j) φ 2 (,j) exp( β( j+ j )) φ (,) φ 2 (,) ε, where j := j k= ( k k ), b) for each j N\{} and each [ j, j ]: φ 2 (,j ) φ ( j,j ) exp( β j ) φ (,) φ 2 (,) ε, c) for each j N\{} and each [ j, j ]: (2) (2) φ (,j) φ 2 ( j,j) (22) exp( β j ) φ (,) φ 2 (,) ε. Therefore, φ and φ 2 are ε-close. Case II If φ (,),φ 2 (,) D, since D is a discree se and φ (,) φ 2 (,) δ, φ (,) = φ 2 (,). By he uniqueness of soluions, we have ha φ (,j) = φ 2 (,j) for all (,j) domφ. Thus, φ and φ 2 are ε-close. Case III If φ (,) C,φ 2 (,) D, he argumens follows similarly as in Case I. Therefore, by combining argumens in hree cases, i is proved ha φ and φ 2 are ε-close which implies H is δs. Noe ha he case when J < follows similarly wih [ J, ) {J} domφ domφ 2 and above. On he oher hand, he δla propery follows from (2), (2) and (22). Remark 4.6: The firs condiion in Theorem 4.5 guaranees sric decrease of he graphical disance beween wo maximal soluions on he inersecions of heir hybrid ime domains. The second condiion in Theorem 4.5 implies ha on he mismached pars of heir hybrid ime domains, he graphical disance beween hem does no grow. The condiion proposed in iem 2) of Theorem 4.5 can be guaraneed by he following sufficien condiion. Proposiion 4.7: Consider a hybrid sysem H = (C,f,D,g) wih sae z R n. Suppose H saisfies Assumpion 3., Assumpion 4.4, and he hybrid basic condiions. Moreover, suppose D is a discree se. Then, iem 2) of Theorem 4.5 holds if here exiss δ > such ha, for any z D, he following hold: here exis c,c 2 >, c 2 (,c ], and α (,) such ha ) for V (z) = z z 2, we have V (z),f(z) + c V α(z) < and z z 2α c ( α) for all z C ((z +δ B)\D), 2) for V 2 (z) = z g(z ) 2, we have V 2 (z),f(z) c 2 V2 α(z) < for all z C (g(z )+δ B). The following example illusraes he sufficien condiion in Theorem 4.5. Example 4.8: Consider he sysem in Example 3.5 and he se M = [,M], where M >. For he hybrid sysem H M = (C M,f,D M,g), he condiions in Theorem 4.5 can be verified as follows. Each maximal soluion o H M is complee and is domain is unbounded in he direcion. Moreover, he flow map is coninuously differeniable on con(c M). Furhermore, for any maximal soluion φ o he sysem H M from φ(,) (C D) M, denoe d := max{x : x C,x φ(,)}. If φ(,), hen φ never jumps and he jump ime beween wo consecuive jumps is bounded below by. If d 2, he flow ime beween wo consecuive jumps of φ is bounded below by ln d d ln floor(m+2) floor(m+). For all z con(c M), f(z)+ f(z) = 2, so iem ) in Theorem 4.5 is saisfied wih β =. Moreover, given z D M, he Lyapunov funcion V (z) = z z 2 saisfies V (z),f(z) = 2(z z )( z) 2z (z z ) = 2z V 2 (z) for z (M C) ((z + ln d d B) \ (M D)), where we used he propery ha z z for all z (M C) ((z + ln d d B)\(M D)). Furhermore, he Lyapunov funcion V 2 (z) = z g(z ) 2 saisfies V 2 (z),f(z) = 2(z g(z ))( z) 2z (g(z ) z) = 2g(z )V 2 2 (z) for z (g(z )+ln d d B) (C M), where we used he propery ha z g(z ) for all z (g(z ) + ln d d B) (C M) and g(z ) = z < z. Then, by Proposiion 4.7 and Theorem 4.5, we have ha H M is δlas. I follows ha he finie-ime convergence propery in iem 2) of Theorem 4.5 is a necessary condiion for a hybrid sysem H o be δs or δla. Indeed, wihou he finie-ime convergence propery nearby D and g(d), he graphs of he soluions would no be close. Theorem 4.9: (necessary condiion for δs and δla) Consider a hybrid sysem H wih sae z R n. Suppose H saisfies Assumpion 3., Assumpion 4.4, and he hybrid basic condiions. Furhermore, suppose H is δs or δla. Then, here exiss δ > such ha each maximal soluion φ o H from φ(,) saisfying φ(,) D δ andφ(,) C converges o D wihin finie ime, i.e., here exiss s > such ha φ(s,) D =. Under furher assumpions, he above resuls for a discree jump se can be exended o he case of a generic se D. For his purpose, we inroduce he following forward invarian noion. Definiion 4.: (forward invariance from away of D) A se A R n is said o be forward invarian for H from away of D if for each soluion φ o H from φ(,) A \ D, φ(,) A for all (,) domφ. Now, we are ready o presen sufficien condiions for a generic jump se. Theorem 4.: (δlas hrough flow for generic D) Consider a hybrid sysem H = (C,f,D,g) wih sae z R n. SupposeH saisfies Assumpion 3., Assumpion 4.4, and he hybrid basic condiions. If here exis β > and δ > such ha H saisfies ) f(z)+ f (z) 2βI for all z conc; 2) for each δ [,δ ], each maximal soluion φ o H saisfying φ(,) C, φ(,) D = δ (23) is such ha here exiss s [,δ] saisfying φ(s,) D = (24) and he se φ(s,)+δb is forward invarian from away

6 of D, and each maximal soluion φ o H from φ(,) g(φ(s,))+δb (25) saisfies φ(,) g(φ(s,))+δb (26) for all [,s]; 3) he jump map g is locally Lipschiz on D wih Lipschiz consan L [,], 3 i.e., g(z ) g(z 2 ) L z z 2 for all z,z 2 D such ha z z 2 δ ; hen, H is δlas. The following resul esablishes a sufficien condiion for a hybrid sysem H o be δlas hrough jumps. In paricular, he graphical disance beween any wo maximal soluions o a hybrid sysem H sricly decreases during jumps. Theorem 4.2: (δlas hrough jumps for generic D) Consider a hybrid sysem H = (C,f,D,g) wih sae z R n. SupposeH saisfies Assumpion 3., Assumpion 4.4, and he hybrid basic condiions. If here exis δ,l,l 2 > such ha ) f(z)+ f(z) for all z conc; 2) for each δ [,δ ], each maximal soluion φ o H from φ(, ) saisfying φ(,) C, φ(,) D = δ saisfies φ(s,) D = for some s [,δ]; 3) for each z D and each δ [,δ ], he se z +δb is forward invarian for H from away of D; 4) he jump map g is locally Lipschiz on D wih Lipschiz consan L, i.e., g(z ) g(z 2 ) L z z 2 for all z,z 2 D such ha z z 2 δ ; 5) f is bounded on conc wih bound L 2, i.e., f(z) L 2 for all z conc; 6) L +L 2 ; hen, H is δs. Furhermore, if he domain of each maximal soluion o H is unbounded in he j direcion, and L and L 2 can be chosen such ha L +L 2 <, hen, H is δlas. The following example illusraes he condiions in Theorem 4.2. Example 4.3: Consider he imer sysem in Example 3.4. Each maximal soluionφ o i has a domain ha is unbounded in he and j direcion. Moreover, he flow ime beween wo consecuive jumps of φ is lower bounded by. The condiion in iem ) of Theorem 4.2 can be verified as f(z)+ f(z) = for all z conc. The condiion in iem 2) can be verified according o Proposiion 4.7. Consider δ (,) and he Lyapunov funcion V(z) = z 2 D. For each z (D+δ B) C\D, we have V(z)=(z ) 2 and V(z),f(z) = 2( z)= 2V 2(z), where we used he propery ha z for all z (D+δ B)\D. Iem 3) of Theorem 4.2 follows from he fac D={} is a singleon and V(z),f(z) = 2( z)< for allz (D+δ B) C\D. Iem 4) of Theorem 4.2 is saisfied wih c =, and iem 5) of Theorem 4.2 is saisfied wih c 2 =. Therefore, he imer sysem in Example 3.4 is δs. 3 Such g is also known as a weak conracion map. V. CONCLUSION In his paper, we show ha graphical incremenal sabiliy is a key noion for he sudy of incremenal sabiliy for hybrid sysems. Oher noions based on poinwise Euclidean disance fall shor when applied o sysems ha exhibi he peaking phenomenon, which is a ypical behavior in racking and observer design for hybrid sysems. Several sufficien and necessary condiions for a hybrid sysem o be graphically incremenally sable and graphically incremenally aracive were provided and illusraed in examples. REFERENCES [] C. Cai and G. Chen. Synchronizaion of complex dynamical neworks by he incremenal ISS approach. Physica A: Saisical Mechanics and is Applicaions, 37(2): , 26. [2] Q. Pham, N. Tabareau, and J. J. Sloine. A conracion heory approach o sochasic incremenal sabiliy. IEEE Transacions on Auomaic Conrol, 54(4):86 82, April 29. [3] A Hamadeh, G. B San, and J. Goncalves. Consrucive synchronizaion of neworked feedback sysems. In Proc. 49h IEEE Conference on Decision and Conrol, pages , Dec 2. [4] R. G. Sanfelice and L. Praly. Convergence of nonlinear observers on R n wih a Riemannian meric (Par I). IEEE Transacions on Auomaic Conrol, 57(7):79 722, July 22. [5] A. P. Dani, S. J. Chung, and S. Huchinson. Observer design for sochasic nonlinear sysems using conracion analysis. In Proc. 5s Annual Conference on Decision and Conrol, pages , Dec 22. [6] M. Zamani and P. Tabuada. Backsepping design for incremenal sabiliy. IEEE Transacions on Auomaic Conrol, 56(9): , Sep 2. [7] A. Pavlov, N. van de Wouw, and H. Nijmeijer. Frequency response funcions for nonlinear convergen sysems. IEEE Transacions on Auomaic Conrol, 52(6):59 65, June 27. [8] R. Goebel, R. G. Sanfelice, and A. R. Teel. Hybrid Dynamical Sysems: Modeling, Sabiliy, and Robusness. Princeon Universiy Press, New Jersey, 22. [9] R. Goebel, R. G. Sanfelice, and A. R. Teel. Hybrid dynamical sysems. IEEE Conrol Sysems Magazine, 29(2):28 93, April 29. [] Y. Li, S. Phillips, and R. G. Sanfelice. Resuls on incremenal sabiliy for a class of hybrid sysems. In Proc. of IEEE 53rd Annual Conference on Decision and Conrol, pages , Dec 24. [] R. G. Sanfelice, J. J. B. Biemond, N. van de Wouw, and W. P. Maurice. H. Heemels. An embedding approach for he design of saefeedback racking conrollers for references wih jumps. In. J. Robus Nonlinear Conrol, 24:585 68, 24. [2] S. Galeani, L. Menini, and A. Poini. Robus rajecory racking for a class of hybrid sysems: An inernal model principle approach. IEEE Transacions on Auomaic Conrol, 57(2): , Feb 22. [3] P. Prabhakar, J. Liu, and R. M. Murray. Pre-orders for reasoning abou sabiliy properies wih respec o inpu of hybrid sysems. In Proc. of he Inernaional Conference on Embedded Sofware (EMSOFT), pages, Sep 23. [4] R. G. Sanfelice, R. Goebel, and A. R. Teel. Invariance principles for hybrid sysems wih connecions o deecabiliy and asympoic sabiliy. IEEE Transacions on Auomaic Conrol, 52(2): , 27. [5] J. J. B. Biemond, N. van de Wouw, W. P. M. H. Heemels, and H. Nijmeijer. Tracking conrol for hybrid sysems wih sae-riggered jumps. IEEE Transacions on Auomaic Conrol, 58(4):876 89, April 23.