Structural Stability of Hybrid Systems

STRUCTURAL STABILITY OF HYBRID SYSTEMS

¢ £ ¡ Slobodan N. Simic´ ¡ , Karl H. Johansson , John Lygeros , and Shankar Sastry

¤ Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, ¦

CA 94720-1770, U.S.A., ¥ simic|sastry @eecs.berkeley.edu § Department of Signals, Sensors and Systems, Royal Institute of Technology,

100 44 Stockholm, Sweden, [email protected] ¨ Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, U.K., [email protected]

Keywords: Stability; Hybrid automata; Hybrifold; Zeno states; from the “true” system.1 The importance of structural stabil- Dynamical systems ity for smooth (continuous and discrete time) dynamical systems was reemphasized in the influential work of Smale [19], Abstract Peixoto [15], and others during the 1960s. It was hoped that in some sense “most” systems were structurally stable, as in We study hybrid systems from a global geometric perspective dimension two. However, this hope was soon shattered by the as piecewise smooth dynamical systems. Based on an earlier discovery of very large sets of non-structurally stable systems work, we define the notion of the hybrifold as a single piece- in dimensions greater than two. Despite this, robustness of its wise smooth state space reflecting the dynamics of the origi- qualitative behavior is still a very important piece of informa- nal system. Structural stability for hybrid systems is introduced tion about a system. and analyzed in this framework. In particular, it is shown that The classical theory of structural stability of smooth dynami- a Zeno state is locally structurally stable and that a standard cal systems states, for instance, that hyperbolic equilibria are equilibrium on the boundary of a domain implies structural in- locally structurally stable. This result extends directly to hy- stability. brid systems in the sense that if a hyperbolic equilibrium of a hybrid system is in the interior of a domain, then the equilib- 1 Introduction rium is locally structurally stable. A standard equilibrium on the boundary of a domain, however, is not locally structurally The study of various dynamical properties of hybrid systems stable. The main contribution of the paper is to show that Zeno has been quite intensive the last decade, e.g., [20, 6, 13, 21, 8, states, which necessarily lie on the boundary of a domain [18], 10, 4, 12, 9, 22]. Recently we suggested a geometric approach are locally structurally stable. This means that it may be diffi- for the analysis of a class of hybrid systems [17, 18]. We in- cult to remove Zenoness by perturbing the system. troduced the notions of the hybrifold and hybrid flow in order to study the hybrid system as a single piecewise smooth dy- The outline of the paper is as follows. Hybrid systems and namical system. This study is continued in the current paper; their executions are defined in Section 2, together with hybri- in particular, structural stability for hybrid systems is discussed folds and hybrid flows. In Section 3 we introduce the notions in a geometric framework. of topologically equivalence and structural stability for hybrid systems, and show that the Zeno state is locally structurally sta- Roughly speaking, a smooth dynamical system is structurally ble. Some discussion on related work is given in Section 4. stable if it is topologically equivalent to every system which is sufficiently close to it. We will generalize this idea to hybrid systems. To make the notion precise, we will specify what is 2 Preliminaries meant by “close” and “topologically equivalent.” Structural sta- We start by giving the definition of a hybrid system and its exe- bility for continuous time dynamical systems (i.e., flows) was cution. Then we describe how so called regular hybrid systems introduced by Andronov and Pontryagin [2], who proved that without branching can be studied on a quotient space called the “most” smooth systems on a two-dimensional disk are struc- hybrifold. See [17] and [18] for more details. turally stable. The importance of this property for models of real-life systems is clear: if a system is structurally stable, its qualitative properties are insensitive to small perturbations of 2.1 Hybrid system and execution the system. If it is not, even a small error in measurement can Definition 1 (Hybrid system) An © -dimensional hybrid sys- produce a system which qualitatively looks entirely different tem is a 6-tuple ! where

1This work was supported by the NASA grant NAG-2-1039, EPRI grant EPRI-35352-6089, ONR under N00014-97-1-0946, DARPA under F33615- 1This is not to say that all important systems are structurally stable. Take, 98-C-3614, and ARO under DAAH04-96-0341. for example, the Hamiltonian ones.

¥ £¢¤¢£¢ ¦¥ ¦ \ G #8 8r \

is the set of discrete states of , where An execution is called inﬁnite, if or ;

G #8 K 8r \ ¥¨§ is an integer; Zeno, if and ; and maximal if it is not

a strict preﬁx of any other execution. The last statement means

is the set of edges;

#8A@ 7]@ I^@ 8

that there exists no other execution \@ such that

8Z@ ^ b^@ 8 ^ T^@

- -

is a strict preﬁx of and on (in the sense that

¥ ¦

is the set of domains of , where

9 $X#8ZY

on - for all ).

¥¦

8 7] 7^ K

We say that an execution \ starts at a point

¥ ¦

^ 8 8 E is the set of vector ﬁelds such that

- < < <

- if and . It passes through if

T^ d $lX#8ZY d9 d&8 is Lipschitz on for all ;

- <

- for some , , .

¥ !" ¦

is the guards, where for each A hybrid system is called deterministic if for every

#%$ ©& , ; there exists at most one maximal execution starting from . It

is called non-blocking if for every " there is at least one in-

¥('*)"+ ¦ is the set of resets, where for each ﬁnite execution starting from . Necessary and sufﬁcient condi-

)

%$ , '

, is a relation between elements of tions for a hybrid system to be deterministic and non-blocking

)

- ' ©. / 0 1- and elements of , i.e., . can be found in [10]. Roughly speaking, resets have to be func-

tions, guards have to be mutually disjoint and whenever a con-

Given , the basic idea is that starting from a point in some do- tinuous trajectory of one of the vector ﬁelds in is about to

main we ﬂow according to until (and if) we reach some exit the domain in which it lies, it has to hit a guard.

#2$ '43 #5

guard , then switch via the reset -76 , continue flowing 4- in 1- according to and so on. The hybrid time trajectory, 2.2 Hybrifold and hybrid flow defined next, is interpreted as the time instants when discrete transitions from one domain to another take place. Unless specified otherwise, we will from now on assume that

is regular. In short, this means that is deterministic and non-

Deﬁnition 2 (Hybrid time trajectory) A (forward) hybrid blocking and that all the building blocks of are piecewise

smooth. Furthermore, each continuous time orbit always exits

8 ¥9 ¦(: - time trajectory is a sequence (ﬁnite or inﬁnite) -7;=<

a domain through a guard which lies on the boundary of the

9 ?> 8 78A@ $D§FE

- - of intervals such that -CB , for all if the

domain, and enters it through the image of a reset map which

G 9 H> 8 I8A@

- - - sequence is inﬁnite; if is ﬁnite, then B for all

¡ is a homeomorphism and takes values on the boundary of the

EKJL$ JMGON 9 > 8 78A@

and is either of the form B or

: :

: “next” domain. For a more detailed and precise formulation of

> 8 78A@ 8P- 8A@ 8Q-RJS8A@ T8Q-7UWV -

. The sequences and - satisfy: , for : : the notion of regularity, see [17] and [18].

all $ .

0 _h © <

Given , deﬁne a map < , (where R will

¥E £¢£¢¤¢ 7G 8 ¦ G #8

We use X 8ZY to denote the set if is ﬁnite, be speciﬁed later) as follows. Let be arbitrary. Because

E Q[ £¢¤¢£¢¦ G 8

and ¥ if is inﬁnite. of the assumption that is deterministic and non-blocking,

¡ #8 ¦] I^

there exists a unique inﬁnite execution \ starting

E¢J d£8r #\ ¤ $S

Deﬁnition 3 (Execution) A (forward) execution of a hybrid at . For any there exist a unique

d> 8Q- 78A@

\ #8 ¦] I^ 8

system is a triple , where is a hybrid time such that . Then deﬁne

^ ¥¤^Z-`$abX 8ZY¦

trajectory, ]¨X#8ZY_ is a map, and is

d! T^ d P¢

d9P- ^e-Rf9Q-g_h iP3

maps such that for all , -76 satisﬁes

^ #d TkiQ3 #^ #d C¢

- -

-76

#d! d #d!/ ¥¨¦ IN§d!

To deﬁne for negative , set ,

¨ where is the reverse hybrid system [18, 17]. Let < be the

Furthermore, for all $klX 8ZY , we have 0

largest subset of R S on which is deﬁned. It can be

¥E ¦§

< ¡

¡ shown [17, 18] that contains a neighborhood of int

] m$ 7] m$n ^A- #8 0 /] o$ !¦] o$pn

" 0 E in R . Moreover, for all , , and

and

#d!P 2© / ª #d!nS© / !

^e- 8 !7^e-7UWV #8P-7UWV "'R3qiQ3 iP3 ¢

-76 -7UWVI6#6 -

whenever both sides are deﬁned.

#8 ¦] I^ 8(r #\ For an execution \ , denote by its execution time: The basic idea in construction of the hybrifold from a hybrid

system is simple: “glue” each guard to the image of the cor-

3os 6

responding reset via the reset map. More precisely, let « be

@ @

¬«®') ¡ !

8¤r #\ #8 N+8Q- uovmw 8 Na8C

- the equivalence relation on generated by for all

3os

-¦x 6

-7;=<

l and . Here is the closure of the guard

/ '*) 'p)

2 and is the extended reset which coincides with on

|}m~QZ ~W a}m/q¤=

If a reset relation zW{ is actually a map , with ,

}o/!*zW{ / / we write zW{¤}of instead of . but is deﬁned on a neighborhood of where it is a

piecewise smooth homeomorphism onto its image. The exis- to prove the following results [17, 18]: if each vector ﬁeld

)

tence of is ensured by regularity of [17, 18]. Collapse in is smooth (in addition to being globally Lipschitz), then

d_ #^

each equivalence class to a point to obtain the quotient space for each ^ the map is continuous and

smooth except at countably many points in ; each map

ª ¢¡!«¢

* p * ^

is injective; whenever both sides are deﬁned

* #^

U ; and there is an open and dense subset of on which

£ £¡«

Deﬁnition 4 (Hybrifold) We call the hybrifold *

is smooth. Observe that the local ﬂow of a smooth vector

of . ﬁeld satisﬁes these properties.

A point is called an -limit point of if

Denote by ¤ the natural projection , which as-

umvow §xRr* #^ d ¥_ 8¤r #^

, for some sequence . The

¥¡«

signs to each its equivalence class . Put the quotient

^ ^ set of all -limit points of is called the -limit set of and is

topology on . Recall that this is the smallest topology that

denoted by .

¦D©

makes ¤ continuous, i.e., a set is open if and only

©¦

if ¤¨§ is open in . Some basic properties of the hybri-

fold include [17, 18]: the hybrifold is a topological © - Deﬁnition 6 (Zeno state) A point is called a Zeno

^ #¢ #^ 8 ^ 0T

manifold with boundary; both and its boundary are piece- state for if and r .

_ ¤

wise smooth; and the restriction ¤ int int is

int

a diffeomorphism. It is not difﬁcult to see that can be nat- V

¤§ ©#

We will also refer to points in as Zeno states in . If the

urally equipped with a distance function which makes ¤ into a

^ execution starting from ^" is Zeno, then consists of piecewise isometry. The hybrifold enables us to study the dy-

exactly one Zeno state for ^ and

namics of a hybrid system on a single phase space.

¤ #^ 0© $

eT_

Definition 5 (Hybrid flow) The hybrid flow of ,

3,+

)&%('*) 6

, is given by

#^ ©

see [17, 18]. Here r denotes the set of discrete

#d!¤ ¡ ¤W #d!/ P¢ transitions which occurs inﬁnitely many times in the execution

starting from ^ .

¥ #d!¤ ¤ a d! ¬£< ¦

Here . In other words, orbits

* § ¤

of are obtained by projecting orbits of by . By the The following example illustrates Zeno.

§ d!

-orbit of we mean the collection of points for all

d #d!/ "0<

possible d (i.e., all such that ). Example 1 (Water tank system [1, 8, 18, 17])

Consider the hybrid system -/. , where

#d!¤ ¤

¡ ¡

In general, may not be a single point. Therefore, ¡

¥ Q[ ¦ ¥ ¦[ ! 2[ ¦ ¥¤¦ > E Q > E P

, , ,

assume that is without branching which, roughly speaking, ¡

¦[ , means the following: if is the equivalence class of some point

lying on the boundary of a domain, then there exists at most one

`V "0bN1V ¤N2143 65 73 IN21V 0TN8143 65

" which can be reached from its corresponding domain by

a trajectory of the corresponding vector ﬁeld, and at most one

¡ ¡:9

Q[ ¥ ^ V I^;3 0 kV ^<3 =>3 ¦ ] whose trajectory enters the corresponding domain. If

the hybrid system satisﬁes this property, then it can be shown

¡ 9

2[ ¥ /[ ^ V I^;3 0 ?3 ^ V =V ¦

* #d!¤ ¤ d! 0 that is indeed a single point, for all . For details, please see [17].

and

(9 9

Next we establish some basic properties of the hybrid ﬂow for ¡

'13 ^=V =>3 /[ ^ V 6=>3

V 376

a regular hybrid system without branching.

9 ¡(9

'13 2[ =V I^<3 =V I^;3 C¢

3 V 6 ^¬

For each d R and , let

The water tank interpretation is as follows. For p , de-

d ¥k #d! ¦

is deﬁned

notes the volume of water in tank and the constant ﬂow

= of water out of tank . The desired minimum volume of wa-

and

ter in tank is to be achieved by dedicating the constant in-

#^ ¥© /© I^ ¦y¢

R is deﬁned

ﬂow 0 exclusively to one tank at a time. The control strategy

^ ¤ ¤ #^ &> E Q L> E I8 ^

Observe that if , then r ,

@=V

is to switch the inﬂow to the ﬁrst tank whenever ^!V and

8 ^ \ ¤

where r is the execution time of , the unique execu-

^¥3 =>3

to the second tank whenever (assuming that the ini-

d.E d

tion of starting at . Also, for , contains all points

(V =>3

tial volumes are greater than = and , respectively). Assume

^ ¤ ¡ 8 #^ 0.d

such that r .

7A4B C1V 143 gD0 E1V nF143 =(V G=>3 E

w and, for simplicity, .

-/.

#d d *_

If is not empty, denote by the time Then, every inﬁnite execution of is Zeno. Moreover, the

^ * #d!I^

d map of deﬁned by . It is possible “origin” of is a Zeno state. 5

3 Structural Stability of Hybrid Systems Deﬁnition 8 (Structural stability) A hybrid system

£ c

is said to be -structurally stable if it has a neighborhood

Intuitively speaking, a smooth dynamical system is structurally £ in such that every element of is topologically equivalent stable if it is topologically equivalent to every system which is to . sufﬁciently close to it. We will generalize this idea to hybrid systems. We claim that the water tank system is structurally stable.

Deﬁnition 7 (Topological equivalence) Two regular hybrid Example 1 (Cont’d)

¥4§

systems without branching and are called topologically To see that -/. is -structurally stable, for any , denote 3

equivalent if their hybrid ﬂows are topologically equivalent. by the angular variable in the polar coordinate system in R

. _

¦[

That is, there exists a homeomorphism car- (see Fig. 1) and observe that for ,

¢¡

rying orbits of the hybrid ﬂow of to those of , preserving

E JSJ T

the orientation, but not necessarily preserving the time.

¥ /E ¦E ¦

on N . Here denotes the derivative of along

¬@

In other words, and are topologically equivalent if their

! " # , i.e., . Let be the set of all

dynamics are qualitatively the same.

c¤£

@ @ @

# We now deﬁne -topology on the space of regular hybrid sys-

tems without branching. Relative to this topology we say that

a@ /E ¦E %$ '&

such that for all , ,

"@ is close to if they have exactly the same states, discrete

transitions, domains and guards; further, vector ﬁelds and reset U

¡ ¡

E4 &

"@ maps of should be close to the corresponding vector ﬁelds

and reset maps of , and the corresponding reset maps should

=N ¥ /E 7E ¦ '@ '*) R on , im ) im , for all , and

have exactly the same images. Let us make this more precise.

( (

c¤£

' N+') )P)

)

M§ G

For ¥ and two -manifolds and , denote by

c¦£ c¤£

7G G U

the space of all maps from to equipped U

c¦£ c¦£

¡ ¡

§ §

* )P)

where , and will be speciﬁed later.

§ ©¨& ¦G . §

with the -topology. So are close if +

. ¨ ^ E.J¥bJ ¥ .¦ §

+ +

+ ) _ is close to , for all and , where For , consider the map deﬁned as the

¥ § ^

denotes the derivative (or tangent map) of at . For more ﬁrst-return map for of the hybrid ﬂow of . It is of class

c¦£

/E ¦E

details, see [7]. and its Lipschitz constant at is less than or equal to

Let denote the set of all regular hybrid systems without ,

.- -

V0/WV 31/ 3

branching with all components of

c¤£

class . That is, all vector ﬁelds in , guards in , and re-

- 2- ' 3 ¦'R3

V 3

5 5

V 376 3 VI6

£ £ £ c

c where are Lipschitz constants of , respec-

set maps in are of class . We deﬁne a topology on

E 7E

V / 3 tively, at , and / are the norms of the derivatives

by specifying a collection of sets which we declare to be its ¡

E ¦E 'R3 _ ¦[

V 3

5 VI6 at of the maps im 3 and

basis. Here is how we do that. ¡

'R3 _ /[

3¦6

im V , deﬁned by ﬂowing from the image

k 1

Let . For each , choose of a reset to the guard along the corresponding vector ﬁeld

c¦£

/ .g &

a neighborhood of in . For each , (cf., [11]).

) ) )

/ ! ' '

choose a neighborhood of in im . Let be

+p@ R_ /

Let ) be the corresponding ﬁrst-return map

¬@ p@ R@ @ @ @ `@ 4

the set of all hybrid systems

£ "@ for . That it is well deﬁned is guaranteed by the above con-

such that

a@ a@ 3

dition on V and (this is not difﬁcult to check). The map

c¦£

+p@ E ¦E

) is and, as above, its Lipschitz constant at is less

@ @ @ @

than or equal to

@ @ @ @ @

.- -

/ /

V V 3 3

and for every and ,

-A@ 0-e@ @ @ ,@

/ /

3 V 3

where V are the corresponding numbers for

@ @

)

' ¢

)

e@ 2-e@ 'p@ 7'@

and -

V 3

3 3

5 5

3¦6 3 V 6

(i.e., are Lipschitz constants of V , respec-

E ¦E tively, at , etc.)

Using all possible such sets as a basis, we generate a topol-

£ £ ,

We know that . It is not difﬁcult to see that we can choose

ogy on .

, , ,

¡ ¡

3 2)Q) @ @ to make sufﬁciently close to so that .

The following result is an immediate consequence of the deﬁ-

+ ª / +R@ 3 ¤ _ E 7E Then for every and , ) , as

nition. ,

@ *

¥l_ , exponentially fast, in fact, as . Thus every exe-

¬@§4 E 7E

cution of every converges to . Therefore, every

Proposition 1 If is close to relative to , then

5 -/. -/.

is topologically equivalent to , so is struc-

. turally stable.

PSfrag replacements

# ^¬

constant Theorem 1 Suppose that and let . Suppose

constant

^ ¤ V (a) for some , int is a hyperbolic equilibrium

for ,

3 or

- ^

(b) ^ is a Zeno sink, i.e., there is a neighborhood of

& &

in such that ^ is the Zeno state for every execution

Figure 1: The water tank example. starting in - .

Then ^ is locally structurally stable.

/E 7E

Note that in concluding this it does not matter whether is @

a Zeno state for . However, we now show that every execu- Proof. (a) Follows from the classical theory of dynamical sys- "@

tion of is indeed Zeno. tems; see, for example, Theorem 4.11 in [14].

#2$ R 8() Let and consider a real-valued function de-

(b) The proof is completely analogous to that for -/. . The only

) )

' 8 ¡ ﬁned as follows: for im , let be the amount of

difﬁculty is to ﬁnd an analog of the function which was read-

time it takes the - -trajectory of to ﬁrst reach the boundary

ily available in -/. because of the special structure of domains

1- m$ I a@ E 7E &$

of (that is, to reach ). Since - , the Im- and guards. However, in general, we can still do it locally: for )

plicit Function Theorem guarantees that 8 is a smooth func-

every domain with vector ﬁeld we can always ﬁnd a

) )

¤ '

tion. Let . be the ﬁrst-return time of im . That is,

function § deﬁned in some neighborhood of in , where

)

. ¤ b83 ] ¡ n,83 ]

5 5

- 6

-76 , where is the ﬁrst intersection of

¡ ^ + ¢ §

¤ and , such that relative to on , has

)

,@ o$ I .

the - -orbit with . Then is a smooth function and let ¡ properties analogous to those of relative to the vector ﬁelds

be its Lipschitz constant on some neighborhood of & . Clearly, ¤

in -/. . One way to do this is the following. Let (represent-

!) & bE . . Therefore, ing an “entry set” into a domain, i.e., the image of some reset)

and ¥ (representing an “exit set” from a domain, i.e., a guard)

@ @

) ) )

. + ¡ . + ¤ N. & >

) )

¡ are two smooth hypersurfaces in R which meet transversely

J + ¤ N &

)

¦¤D§¥

and let T (representing a Zeno state). Assume is

J ¢

¤ ¥ a smooth vector ﬁeld on R transverse to both and which in particular means that it has no equilibria. That this is a fair Therefore,

assumption was shown in [17]: vector ﬁelds in a hybrid sys-

r r

¢ t

t tem never vanish at its Zeno state. Let be a sufﬁciently small

@ @

8 ¤ .!) + ¤ J T

)

neighborhood of in R . Then there exists a diffeomorphism

;=< ;=<

R such that , , and

. (This is an exercise in elementary

" ') §

for all im . We will soon see that this turns out not to be

#^ A A ^ ¡(^ V

differential geometry.) Let and set

a coincidence.

* § § ¢ ¤§¥

§ . Then is a smooth function on ,

3.1 Structural stability of hybrid equilibria Furthermore, by the Flow Box theorem [14], we can choose

¢ ¢ so small that the ﬂow of on looks approximately like

Recall that an equilibrium of a hybrid system is a point

a collection of straight lines from ¤ to . Then we also get

^ which does not move under the hybrid ﬂow, i.e.,

J&§"JU.b ¢

E on , as desired.

* d!I^ ^ dp ^

, for all . Hence, a point in the interior § of the domain for which the vector ﬁeld vanishes is an equilibrium of the hybrid system. Equilibria also include Zeno states Note that in Theorem 1 the only equilibria considered on the which make no time progress. boundary of domains are Zeno states. However, we will soon see without difﬁculty that a standard equilibrium on the bound-

Deﬁnition 9 (Local structural stability) For a hybrid system ary of a domain implies structural instability. Recall [17] that a

standard equilibrium of is a point at which all rele-

^¬

, an equilibrium is called locally structurally

¤§ #^ ¥7 £¢¤¢£¢ / ¦

vant vector ﬁelds vanish, i.e., if V where

stable if there exists a neighborhood of in such that

4- ¤e- '&

A- 1- , then .

¬@1' ^

every is locally topologically equivalent to at .

@ @

That is, there exists an equilibrium of and neighbor- £

Proposition 2 Let and suppose that has a standard

£¢p@ ^ I^@

hoods ¢ of in , respectively, such that

¢¡

¤! ( 1-%$ #"

equilibrium in - . Then is structurally unstable.

,@ ¢@

the restriction of the hybrid flow of to is topologically ¢ equivalent to the restriction of the hybrid flow of to . Proof. A small perturbation of the vector fields will cause the We now state the main result of the paper. removal of the equilibrium from the boundary. In other words,

¤ § #^ ¥7=V £¢¤¢£¢ / ¦ if ^ is a standard equilibrium and [8] K. H. Johansson, M. Egerstedt, J. Lygeros, and S. Sastry.

where A-,ª 1- , then it is not difﬁcult to perturb each vector On the regularization of Zeno hybrid automata. Systems

- ,@ ,@ @

- -

ﬁeld to - so that has an equilibrium in the inte- & Control Letters, 38:141–150, 1999.

4- e- rior of 1- and no equilibria on the boundary of near . The new hybrid system is clearly not topologically equivalent [9] M. Lemmon. On the existence of solutions to controlled to the original one. hybrid automata. In Hybrid Systems: Computation and Control, Pittsburgh, PA, 2000.

4 Conclusions [10] J. Lygeros, K. H. Johansson, S. Sastry, and M. Egerstedt. On the existence of executions of hybrid automata. In Structural stability for hybrid systems was introduced in the IEEE Conference on Decision and Control, Phoenix, AZ, paper. It is an important property, because a structural stable 1999. system is robust to modeling errors. A Zeno state is an equi- [11] J. Lygeros, K. H. Johansson, S. N. Simic,´ J. Zhang, and librium of a hybrid system, which arises from the interaction S. Sastry. Dynamical properties of hybrid automata. IEEE of the continuous and the discrete dynamics. It was shown that Transactions for Automatic Control, 2001. Submitted. Zeno states are locally structurally stable. This means that Zeno may be difficult to remove by perturbing the system. Similarly, [12] A. S. Matveev and A. V. Savkin. Qualitative Theory of in analogy to the classical case, hyperbolic hybrid closed or- Hybrid Dynamical Systems. Birkhauser¨ , Boston, MA, bits [16] can be shown to be locally structurally stable as well, 2000. as will be presented elsewhere. [13] A. S. Morse. Control using logic-based switching. In Al- Many issues on structural stability of hybrid systems remain to berto Isidori, editor, Trends in Control. A European Per- be studied. For example, an important question is: in dimen- spective, pages 69–113. Springer, 1995. sion two, do structurally stable systems form a “large” set (in analogy with the case of smooth systems)? An affirmative an- [14] J. Palis Jr. and W. de Melo. Geometric Theory of Dynam- swer, in a slightly different context, was given by [3]. Namely, ical Systems. Springer Verlag, New York, 1982. even if “sliding” in the sense of Filippov [5] is allowed, then the [15] M. M. Peixoto. Structural stability of two-dimensional generic piecewise smooth vector field on a smooth, orientable, manifolds. Topology, 1:101–120, 1962. boundaryless, compact surface is structurally stable. Moreover, structural stability is completely characterized by a set of four [16] S. N. Simic,´ K. H. Johansson, J. Lygeros, and S. Sastry. conditions, which we do not discuss here. Hybrid limit cycles and hybrid poincare-bendixson.´ Sub- mitted to CDC’01. References [17] S. N. Simic,´ K. H. Johansson, J. Lygeros, and S. Sastry. [1] R Alur and T A Henzinger. Modularity for timed and Towards a geometric theory of hybrid systems. Mathe- hybrid systems. In A. Mazurkiewicz and J. Winkowski, matics of Control, Signals, and Systems, 2001. Submit- editors, CONCUR 97: Concurrency Theory, Lecture ted. Notes in Computer Science 1243, pages 74–88. Springer- Verlag, 1997. [18] S. N. Simic,´ K. H. Johansson, S. Sastry, and J. Lygeros. Towards a geometric theory of hybrid systems. In Hy- [2] A. A. Andronov and L. Pontryagin. Systemes` grosseris. brid Systems: Computation and Control, volume 1790 of Dokl. Akad. Nauk. SSSR, 14:247–251, 1937. Lecture Notes in Computer Science, pages 421–436, Pitts- burgh, PA, 2000. Springer-Verlag. [3] M. Broucke, C. Pugh, and S. N. Simic.´ Structural stability of piecewise smooth systems. Computational and [19] S. Smale. Differentiable dynamical systems. Bull. Amer. Applied Math., 2001. To appear. Math. Soc, 73:747–817, 1967. [4] R. A. Decarlo, M. S. Branicky, S. Pettersson, and [20] L. Tavernini. Differential automata and their discrete sim- B. Lennartson. Perspectives and results on the stability ulators. Nonlinear Analysis, Theory, Methods & Applica- and stabilizability of hybrid systems. Proc. of the IEEE, tions, 11(6):665–683, 1987. 88(7):1069–1082, 2000. [21] A. J. van der Schaft and J. M. Schumacher. Complemen- [5] A. F. Filippov. Differential Equations with Discontinuous tarity modeling of hybrid systems. IEEE Transactions on Righthand Sides. Kluwer Academic Publishers, 1988. Automatic Control, 43(4):483–490, April 1998. [6] J. Guckenheimer and S. Johnson. Planar hybrid systems. [22] J. Zhang, K. H. Johansson, J. Lygeros, and S. Sastry. Dy- In Hybrid Systems and Autonomous Control Workshop, namical systems revisited: Hybrid systems with Zeno ex- Cornell University, Ithaca, NY, 1994. ecutions. In Hybrid Systems: Computation and Control, [7] M. W. Hirsch. Differential Topology. Springer-Verlag: Pittsburgh, PA, 2000. New York, 1976.