Table Of ContentHierarchical stability of nonlinear hybrid systems
Mario Sassanoa, Luca Zaccarianb,c
aDipartimento di Ingegneria Civile e Ingegneria Informatica, “TorVergata”, Via del Politecnico 1, 00133, Rome, Italy.
bCNRS, LAAS, 7 avenue du colonel Roche, F-31400 Toulouse, Univ. de Toulouse, LAAS, F-31400 Toulouse, France.
6 cDipartimento di Ingegneria Industriale, Universityof Trento, Italy.
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Abstract
J
6 Inthis shortnoteweproveahierarchicalstabilityresultthatappliestohybriddynamicalsystemssatisfying
the hybrid basic conditions of (Goebel et al., 2012). In particular, we establish sufficient conditions for
]
Y uniform asymptotic stability of a compact set based on some hierarchical stability assumptions involving
twonestedclosedsetscontainingsuchacompactset. Moreover,mimickingthewellknownresultforcascaded
S
systems, we provethatthe basin ofattractionofsuchcompactset coincides with the largestset fromwhich
.
s
all solutions are bounded. The result appears to be useful when applied to several recent works involving
c
[ hierarchical control architectures.
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v
1. Introduction we do not insist on a cascaded nature of the overall
1
system.
7
2 It is well known (see [14, 13] for the global result The result finds applications in several contexts.
1 or [18, Theorem 3.1] for the local result) that for a The authors of this note have been using it as a tool
0 cascadeinterconnectionoftwo nonlinear continuous- toshowstabilitypropertiesinsystemswheredifferent
.
1 time systems, global asymptotic stability (GAS) of substatesconvergetodesirablemanifoldsandhierar-
0 the originofthe upper subsystemand globalasymp- chicallyreachadesirablefinalclosedset[1,10,17,2].
6 toticstabilitywithzeroinput(0-GAS)oftheoriginof Severaladditionalapplicationcanbefoundinthelit-
1
the lower subsystem implies local asymptotic stabil- erature,forexample itmaybe usedasanalternative
:
v ity(LAS)oftheoriginofthecascade,withdomainof way to prove the results in [9].
Xi attractioncoincidingwiththesetofinitialconditions Notation. B is the open unit ball centered at the
from which all trajectories are bounded. In particu- origin. We denoteX+Y ={z :z =x+y,x∈X,y ∈
r
a lar, denoting by x1 and x2, respectively, the state of Y}. We denote the distance |z|M of a point z from
the upper and lower subsystems, GAS of the closed the set M as |z|M :=infw∈M|z−w|.
set M = {(x ,x ) : x = 0} plus GAS of the ori-
1 2 1
gin for initial conditions restrictedto M, plus global
2. Preliminaries and Definitions
boundedness (GB) guarantees GAS of the origin of
the cascade. The extension discussed in this note is Using the formalism in [4], we consider a hybrid
threefold. First, we consider nonlinear hybrid sys- dynamical system described by
tems. Then, we extend this result to the case where
wedonotnecessarilyinsistonthe originand,finally, x∈C, x˙ ∈F(x),
H: (1)
(cid:26) x∈D, x+ ∈G(x).
Email addresses: [email protected] (Mario We suppose here that (C,F,D,G) satisfy the hybrid
Sassano), [email protected] (LucaZaccarian) basic conditions given in [4, Ass. 6.5], which are re-
ported here for completeness. 4. globally attractive if (2) holds for all δ >0;
5. stronglyforwardinvariantifallsolutionsstarting
Assumption 1. (Hybrid basic conditions)
in X remain in X for all times.
(i) C and D are closed subsets of Rn; A compact set X ⊂Rn is
(ii) F : Rn ⇒ Rn is outer semicontinuous and lo- c
1. uniformlyattractivefromacompactsetK ⊂Rn,
cally bounded relative to C, and it is non-empty
K ⊃ X , if for each ǫ > 0, there exists T such
and convex for each x∈C; c
(iii) G : Rn ⇒ Rn is outer semicontinuous and lo- that all solutions ϕ to H satisfy
cally bounded relative toD, and it is non-empty ϕ(0,0)∈K ⇒ |ϕ(t,j)|X ≤ǫ,
c
for each x∈D.
for all (t,j)∈domϕ such that t+j ≥T;
Remark 1. TheconditionsinAssumption1aresuf- 2. uniformly (locally) asymptotically stable (UAS)
ficienttoimplynominalandrobustwell-posednessof if it is stable anduniformly attractive fromeach
system (1), see [4, Def. 6.2]. Note that special cases compact subset of its basin of attraction BX ;
c
correspond to continuous-time differential equations 3. uniformly globally asymptotically stable (UGAS)
x˙ = f(x) where f is continuous (this corresponds to if it is UAS with BX =Rn.
the case C = Rn and D = ∅) and discrete-time dif- c
ference equations x+ = g(x) where g is continuous Given a closed forward invariant set Y for the hy-
(this correspondstothe caseD =Rn andC =∅). In brid system (1), each one of the above properties
holds relative to the set Y if itholds for initialcondi-
these two special cases, all the presented results ap-
tions restricted to Y.
plyandthesolutionconceptintroducedin[4]reduces
to the classicalsolutionconcepts forcontinuous-time Definition 2. Given system H in (1) and a com-
and discrete-time systems, respectively. y pact set K ⊂ Rn, solutions (or trajectories) of H
are uniformly bounded from K if there exists ∆ > 0
We begin the section by introducing some defini-
such that all solutions ϕ are such that ϕ(0,0) ∈ K
tions necessary to make the statements above more
implies |ϕ(t,j)| < ∆ for all (t,j) ∈ domϕ. More-
precise.
over, given a subset X of Rn, solutions (or trajecto-
Definition 1. Given hybrid system H in (1), a ries) of H are uniformly bounded from X if they are
closed set X ⊂Rn is uniformly bounded from each compact subset of X.
If X = Rn then trajectories are uniformly globally
bounded (UGB).
1. stable for (1) if for each pair ǫ>0, ∆>0, there
existsδ >0suchthatallsolutionsϕtoHsatisfy: Remark 2. The UGAS property in Definition 1
does not explicitly require the UGB propertyof Def-
ϕ(0,0)∈((X +δB)∩∆B) ⇒ ϕ(t,j)∈X +ǫB,
inition 2. This is because X compact and f contin-
c
for all (t,j)∈domϕ; uous, together with local stability of Xc and global
convergence to X implies uniform global bounded-
2. attractive if there exists δ >0 suchthatallsolu- c
ness (see [5, Prop. 6.3]). Nevertheless, it has been
tions ϕ to H satisfy
shownin[16]thatfortime-varyingnonlinearsystems
ϕ(0,0)∈(X +δB) ⇒ lim |ϕ(t,j)|X =0; (inotherwords,unboundedattractors)localstability
t+j→+∞
and uniform global convergence does not guarantee
(2)
UGB. Inturns,UGB is necessaryto enforceabound
3. (locally)asymptotically stable (AS) if it is stable on the overshoots of the trajectories, thereby being
and attractive. Moreover, its basin of attraction able to guarantee a KL bound (see [8, Lemma 4.5]
BX is the largest set of initial conditions from or [4, Thm 7.12] for equivalence between UGAS and
which all solutions to (1) converge to X; existence of a global KL bound). y
3. Main stability result From [3, Corollary 19] we conclude that the set M◦
is globally asymptotically stable (GAS) for the re-
The following lemma extends the result of [13, 14]
stricted hybrid dynamics H.
oncascadedsystems. Some relatedresultshavebeen
UAS of M◦ relative to Me. Recalling that (global)
presented for the continuous-time case in [6, 11, 12]
AS of M◦ comprises its stability and noticing that,
and their relation with the lemma are clarified next.
from (3), M◦ is in the interior of C ∪D relative to
Lemma 1. Consider hybrid systemH in (1) satisfy- Me∩(C∪D),thenforanyscalarM¯ >0,stabilityof
ingthebasicconditionsinAssumption1,andassume M◦ for H, together with strong forward invariance
that: of Me, implies that there is a small enough δ > 0
1. the closed set M ⊂ Rn is strongly forward in- such that all solutions to H starting in (M◦+δB)∩
e Me are contained in (M◦ +MB)∩Me, namely all
variant;
solutions to H starting in this set are also solutions
2. the closed set M ⊂ M is stable and globally
i e to the restricteddynamics H. Then AS of M◦ for H
attractive relative to M ;
e relative to Me implies AS of M◦ for H relative to
3. the compact set M◦ ⊂Mi is stable and globally Me. Moreover,since M◦ is compactand H satisfied
attractive relative to M .
i Assumption 1, then [5, Prop. 6.2] (see also [4, Thm
Then the set M◦ is UAS relative to Me for H, with 7.21]) implies that AS of M◦ is uniform in Me.
basin of attraction BM◦ such that BM◦ ∩Me coin- Basin of attraction. First note that for any bounded
cides with the largest subset of M from which all solution in M there exists a large enough selection
e e
solutions are bounded. In particular, if solutions are of M¯ such that C ∪D ∪G(D) in (3) contains that
UGB relative to Me, then the set M◦ is UGAS for solution (and [3, Corollary19]establishes its conver-
H relative to M . Finally, if the above stated attrac- gence). Therefore,anysuchsolutionisguaranteedto
e
tivity properties only hold locally, then the set M◦ is be contained in the set BM◦ ∩Me. Conversely, any
UAS relative to M . solution which is not (uniformly) bounded is not in
e
thedomainofattractionbecause[5,Prop. 6.3]estab-
Proof. Firstnoticethatthepropertyatitem3iswell
lishesthatUASofacompactsetM◦ impliesthatso-
definedbecause,fromitem2,thesetM is(strongly)
fPorrewliamrdinianrvyarSiatenpt.. The lemma is provien using [3, sluetqiuoennscaer,ebuyntihfoermarlbyitbroaurinndeesds ofrfoMm¯,BsMet◦B. MAs◦a∩cMone-
coincides with the largest subset of M from which
e
Corollary19],inasimilarwaytowhatisdonearound
all solutions are bounded. (cid:3)
[3, eqs. (23), (24)]. In particular,we apply [3, Corol-
lary 19] by intersecting M with arbitrarily large Remark 3. Applying [5, Thm 6.5] we have that
i
compact subsets of M . Namely, given an arbitrary both in the local and global cases the result of
e
positivenumberM¯,weapply[3,Corollary19]tohy- Lemma 1 implies the existence of a class KL esti-
brid system H := (C,F,D,G) and compact sets A mateforthe distanceofthesolutionfromtheattrac-
1
and A2 selected as:1 tor M◦. In particular, when Me = Rn and all solu-
tions are bounded, so that the lemma implies UGAS
C :=C∩Me∩(M◦+M¯B) (namely BM◦ = Rn), there exists β ∈ KL such that
F :=F
DG::==GD∩∩(MMe◦∩+(MM¯B◦)+, M¯B) (3) aRfolnlrsatohllleu(ttbi,oojnu)sn∈dsadhtoiosmlfdysϕ|ϕ.w(Witt,hhjt)eh|nMeB◦dMi≤st◦aβin(sc|ϕae(s|0t·,r|Mi0c)t◦|Msrue◦bp,sltae+tcejod)f
A1 =Mi∩(C∪D) by aproperindicatorfunctionofM◦ withrespectto
A2 =M◦. BM◦. Finally, [5, Prop. 6.4] implies that BM◦ is an
open subset of Rn containing M◦. y
1Note that itisnot necessary tointersect G withMe due Remark 4. The result in Lemma 1 is relevant
totheassumedstrongforwardinvarianceproperty. mostly due to its simple formulation and broad ap-
plicability. From the point of view of the proof tech- ordinates exists does not seem to be an easy task to
nique, itis asimple extensionofexisting results. For accomplish, in general. Indeed, even if only want-
the case M = Rn (so that item 1 is trivially guar- ing to define this change of coordinates locally in a
e
anteed) and continuous-time systems x˙ = f(x), an neighborhood Uo of a point xo ∈Rn, its existence is
alternativeproofcanbe obtainedusingtheresultsin related to invariance of the completely integrable dis-
[6] if one adds the extra assumption that f is locally tribution generated by the columns of Ker(∇h(x)T)
Lipschitz and the non-uniform attractivity property with respect to the vector field f, which is a much
in (2) is strengthened to a uniform one. In this stronger hypothesis of invariance of the submanifold
strengthened case, applying the converse Lyapunov M with respect to f, see [7, Lemma 1.6.1] for more
i
theorem in [15] (see also [19]), item 2 implies the ex- detailed discussions. y
istence of a smooth positive semidefinite Lyapunov
function VM establishing global asymptotic stability Acknowledgments. The authors would like to
of M and then we can apply [6, Corollary 1] with thank Andy Teel for useful suggestions.
i
M = M∗ = M = M . Indeed, using the notation
0
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