Notes on Contraction Theory

0.16 MB·English

by  Nicolas Tabareau

Notes on Contraction Theory
a b
Nicolas Tabareau Jean-Jacques Slotine
aLaboratoire dePhysiologie delaPerceptionetdel’Action, CNRS-CollègedeFrance,
6
11placeMarcelinBerthelot, 75231ParisCedex05,France.
0
0 bNonlinear SystemLaboratory, Massachusetts InstituteofTechnology,
2
Cambridge,Massachusetts, 02139,USA

n
a
J

6
Abstract

]
O
A Thesenotesderiveanumberoftechnicalresultsonnonlinear contraction theory,acom-
. paratively recent tool for system stability analysis. In particular, they provide new results
n
on the preservation of contraction through system combinations, a property of interest in
i
l
n modellingbiological systems.
[

2
Keywords: contraction theory,centralized feedback, hierarchies, attractors
v
1
1
0
1
0
6
0 1 Introduction
/
n
i
l
n
Nonlinear contraction theory [LohmillerandSlotine,1998] is a comparatively re-
:
v
cent tool for system stability analysis. These notes derive a number of technical
i
X
resultsmotivatedbythetheory.Inparticular,theyprovidenewresultsonthepreser-
r
vation of contraction through system combinations, a property of interest in mod-
a
ellingbiologicalsystems.
Section 2 analyzes the preservation of contraction through generalized negative
feedbackbetweencontractingsystems.Section3describesanewsystemcombina-
tion,centralizedcontraction,whichalsopreservescontractionbyaggregation.Sec-
tion4usesstandardresultsfromcomputersciencetosimplifythegeneralstructure
of arbitrary system combinations, and in particular to exploit intrinsic hierarchi-
cal properties. Section 5 discusses some applications to nonlinear attractors, while
section 6 describes the estimation of the successive derivatives of a vector using
compositevariables.
correspondence to: N. Tabareau, CNRS, Laboratoire de Physiologie de la Perception
∗
et de l’Action, Collège de France, 11 place Marcellin Berthelot, 75231 Paris Cedex 05,
France,,Tel.:[phone], Fax:[phone] E-mail:[email protected]
PreprintsubmittedtoElsevierScience
2 Negativefeedback
This section analyzes feedback connections which automatically give rise to con-
tractionwithregardsto apredefined metric.
Consider two contracting systems, of possibly different dimensions and metrics,
andconnect theminfeedback, insuchaway thattheoverallvirtualdynamicsisof
theform
d δz1  F1 G(z,t)Bδz1
= −
dt δz  G(z,t)T AT F δz 
 2  2  2
withA,Btwosquarematrices. Theoverallsystemiscontractingif
(1) A andB aresymmetricpositivedefinite, and
(2) thereexistsβ > 0 suchthat
A˙ +A.F +F TA β A
1 1
B˙ +B.F +F TB ≤ −β B
2 2
≤ −
Indeed, wecan define themetric
A 0
 
M =
 0 B
 
Wehave
d
(δzTMδz) = MF+FTM+M˙
dt
wherethematrixMF isoftheform,
 AF1 AG(z,t)B
MF = −
BG(z,t)TA B.F 
 2 
Thus
d (δzTMδz) = δzT A˙ +AF1 +F1TA 0 δz βδzTMδz
dt  0 B˙ +BF +F TB ≤ −
 2 2 
byhypothesis,whichimpliesthatδzTMδztendsexponentiallytozero.SinceMis
positivedefinite, thisinturn impliesthatδzTδz tendsexponentiallytozero.
3 Centralized contraction
We extend here the class of combinations of contracting systems described in
[LohmillerandSlotine, 1998]. From a practical point of view, the condition given
2
for feedback combinations may be hard to deal with, whereas hierarchical combi-
nationsaremuchsimplerbutnotgeneralenough.Itisthereforeofinteresttofinda
combination having a strong expressiveness together with an automatic guarantee
ofcontraction.
An almost hierarchical feedback combination. The basic idea lies in the fol-
lowing remark. When looking at the combination depicted in figure 3, the loop
between F and F seems to be illusory, as the domain and co-domain in F are
1 2 2
disjoint.Let’stry toformalizethisidea.
We considertwo contracting systemconnecting in feedback in such a way that the
second systemcan besplitintoz1 and z2 soto write
2 2
δz1  F1 G1 0 δz1
d
δz1 =  0 F11 F12δz1
dt  2  2 2  2
    
    
δz2 G F21 F22δz2
 2  2 2 2  2
Then, a following our idea that this almost represents a hierarchical combination,
weapplythemetric
I 0 0 
Θ = 0 ǫ−1I 0 
 
 
 
0 0 ǫI
 
Thisgivesrisetothegeneralized Jacobian
Existing connection
F
1
Forbidden connection
1
F
2
Co−Domain
Domain
F2 F2
2
Fig.1.Acombination thatseemstogiverisetoautomaticcontraction
3
δz1  F1 ǫG1 0 δz1
d
δz1 =  0 F11 ǫ−2 F12δz1
dt  2  2 2  2
    
    
δz2 ǫ G ǫ2 F21 F22 δz2
 2  2 2 2  2
This says that, as long as G and G are bounded, they are negligible. However,
1 2
wehavenomoreguaranteeonthecontractionofF asthematrixoffeedback F12
2 2
and F21 havebeen perturbed byǫ.
2
This tells us that we have to restrict this intuition to a particular kind of feedback
withinF .
2
3.1 Orientablesystems
The first step is to master the metric used in each local feedback. Indeed, as in the
caseoffeedbackcombination,toapplythecombinationrecursively,weneedsome
guarantees of non interference between the metric. Then idea is to require that the
metric use for each combination only acts on the peripheral system and not on the
centralizer. Thisleadsto thenotionoforientablesystem.
Definition 1 Acombinationbetweentwosystemsissaidtobeorientableifametric
thatmakes thegeneralizedJacobiannegativedefinitecan bewritten
M′ 0
 
M =
 0 I
 
Small gain. Consider two contracting systems, of possibly different dimensions
and metrics, and connect them in feedback, in such a way that the overall virtual
dynamicsisoftheform
d δz1  F1 BG(z,t)δz1
=
dt δz  G(z,t)T AT F δz 
 2  2  2
with A, B two square matrices. Note that in this form, A and B must have the
samedimension.
AssumenowthatA and Bsatisfy
Bis invertible
• AB−1 is constantandsymmetricpositivedefinite
•
4
Wecan then definethemetric
AB−1 0
 
M =
 0 I
 
whichcan berewrittenM = ΘTΘwith
√AB−1 0
 
Θ =
 0 I
 
Usingthefact that
√AB−1B = (√AB−1)−1A
wehave
√AB−1F1(√AB−1)−1 √AB−1BG(z,t)
 G(z,t)T(√AB−1B)T F 
 2 
Applying a standard result for small gain feedback (see [Slotine, 2003]), we can
conclude on the contraction of the system if √AB−1F (√AB−1)−1 is negative
1
definiteand thefollowinginequalityholds
σ2(√AB−1BG(z,t)) < λ((√AB−1F (√AB−1)−1) )λ((F ) ) (1)
1 s 2 s
Wecan concludethatthissystemisan orientablescaling-robustsystem.
Note that the aboveassumptionson A and B are verified in the common case that
A isconstantand symmetricpositivedefiniteand B = λI withconstantλ > 0.
Negativefeedback. In thesameway, forasystemoftheform
d δz1  F1 BG(z,t)δz1
= −
dt δz  G(z,t)T AT F δz 
 2  2  2
wecanconstructaorientablemetricsuchthatthesystemiscontractingif√AB−1F (√AB−1)−1
1
isnegativedefinite.
3.2 Orientablescaling-robustsystems
Definition 2 A combination between two systems is said to be orientable scaling-
I 0
 
robust if transformingthesystem by themetric D = (ǫ > 0) leads to an
ǫ
0 ǫ I
 
5
M 0
orientablecombinationwithmetric ǫ . WefurtherrequirethatM ǫ→0 0.
ǫ
 0 I −−→
 
Remark 3 Thedefinitionabovesaysthatthedynamics
d δx1 F11 ǫ−1 F12δx1
=
dt δx2 ǫ F21 F22δx2
    
areorientableforallǫ.
Smallgainandnegativefeedback. Letuscomebacktothetwopreviousexam-
I 0
 
ples.ItisclearthatapplyingthemetricD = forǫ > 0leadstoanother
ǫ
0 ǫI
 
contractingsystemwithmetric
ǫAB−1 0
 
M =
ǫ
 0 I
 
So an orientable small gain (resp. negative feedback) is automatically scaling-
robust.
3.3 Centralizedcontraction
Assumethat wehave, as in figure 3.3, n systemsconnecting to a particularsystem
calledthecenter insuchawaythateveryconnectiontothecenterisorientableand
scaling-robust. Assume also that the connection between the different peripheral
systemsis hierarchical.
That kindofsystemcan berewritten
δz F X X X X X F2 δz
 6   6 6 6 
 δz   0 F X X X X F2 δz 
 5   5 5 5 
    
 δz4   0 0 F4 X X X F24 δz4 
d     
 δz  =  0 0 0 F X X F2 δz 
dt  3   3 3 3 
    
    
 δz   0 0 0 0 F X F2 δz 
 2   2 2 2 
    
 δz1   0 0 0 0 0 F1 F21 δz1 
    
δz  F1 F1 F1 F1 F1 F1 C δz 
 C  6 5 4 3 2 1  C
6
1 6
∼ ∼
2 Center 5
∼ ∼
∼ ∼
3 4
Fig.2.Acentralized combination ofcontracting systems
Forthatparticularvirtualsystem,let ususethemetric
I 0 0 0 0 0 0
 
0 ǫ−1I 0 0 0 0 0 
 
 
0 0 ǫ−2I 0 0 0 0 
 
 
0 0 0 ǫ−3I 0 0 0 
 
 
0 0 0 0 ǫ−4I 0 0 
 
 
0 0 0 0 0 ǫ−5I 0 
 
 
0 0 0 0 0 0 ǫI
 
whichleads tothesystem
δz F ǫX ǫ2X ǫ3X ǫ4X ǫ5X ǫ−1F2 δz
 6   6 6 6 
 δz   0 F ǫX ǫ2X ǫ3X ǫ4X ǫ−2F2 δz 
 5   5 5 5 
    
 δz4   0 0 F4 ǫX ǫ2X ǫ3X ǫ−3F24 δz4 
d     
 δz  =  0 0 0 F ǫX ǫ2X ǫ−4F2 δz 
dt  3   3 3 3 
    
 δz   0 0 0 0 F ǫX ǫ−5F2 δz 
 2   2 2 2 
    
 δz1   0 0 0 0 0 F1 ǫ−6F21 δz1 
    
δz  ǫF1 ǫ2F1 ǫ3F1 ǫ4F1 ǫ5F1 ǫ6F1 C δz 
 C  6 5 4 3 2 1  C
7
Since all the couple (F1,F2) are assumed to be orientable scaling-robust, for any
i i
smallǫ,thereisaconstantmetricMsuchthatthesymmetricpartofthegeneralized
Jacobian can bewritten,whenǫ tendstozero, as
H 0 0 0 0 0 K
 6 6
 0 H 0 0 0 0 K 
 5 5
 
 
 0 0 H4 0 0 0 K4
 
 
 0 0 0 H 0 0 K 
 3 3
 
 
 0 0 0 0 H 0 K 
 2 2
 
 
 0 0 0 0 0 H1 K1
 
 
KT KT KT KT KT KT C 
 6 5 4 3 2 1 
wherethematrices
H K
 i i
KT C 
 i 
are allnegativedefinite.
Applyingabasicresultofmatrixanalysisthusyieldsthecondition
C < KTH−1K
X i i i
i
whichis equivalentto
λ(KTH−1K C−1) < 1
X i i i
i
A sufficientconditionis thus
σ(K )2λ(H )−1 < λ(C)
X i i
i
oreventhelessgeneral buteasier toverifyinequality
σ(K )2 < λ(C)minλ(H )
X i i i
i
Wecall thiscasecentralizedcontraction.
3.4 Goingfurther
Thestructureofcentralizedcontractionisageneralschemethatcanbeextendedto
morecomplexstructures.We presenthere twobasicextensions.
8
Multiplelayers Wecanapplytheresultofcentralizedcontractionevenifthepe-
ripheralsystemiscomposedofdifferentlayers.Forexample,thesystemdescribed
in figure 3 is automatically contracting providing that the red connections are ori-
entablescalingrobust.
Fig.3.Centralized contraction formultiplelayers
Multiple centers In biological systems, it is often of interest to consider a cen-
tralizer which is composed of multiple systems. In that case, the contraction can
be guaranteed if all connections to the center considered as a whole are orientable
scalingrobust.
4 Strongly connected components
In computational neuroscience as in many biological fields, we have to deals with
largesystems.Hereweexploitastandardalgorithmfromcomputerscience[Knuth,1997]
todecomposealargesystemintosub-systems,inasuchwaythatthecontractionof
theoverallsystemcanbededucedfromthecontractionofthesmallersub-systems.
Definition 4 A strongly connected component of a directed graph G = (V,E) is
a maximal set of vertices U V such that for all u,v U, u is reachable from v
⊂ ∈
andv isreachablefromu.
Proposition5 Any directed graph is a union of strongly connected components
plusedges tojointhecomponentstogether.
Thus, we are able to distinguish between micro-systems which are connected in
feedback combinationornot. Indeed,we can statetheproposition:
9
Proposition6 Two sub-systems of large systems are in feedback combination iff
thosetwo systemsbelongsto thesamestronglyconnected component.
Let usnowdescribethealgorithmtocomputesuch adecomposition.
Algorithm. Strongly_connected_components(G)
(1) Use the Depth-First-Search (DFS) algorithm to compute f[U] the finishing
timeofu
(2) ComputeGT = (V,E) where ET = (u,v) (v,u) E
{ | ∈ }
(3) Execute DFS on GT by grabbing vertices in the order of decreasing f[u] as
computedinstep 1.
(4) Outputtheverticesofeachtreeinthedepth-firstforestofstep3.asaseparate
stronglyconnected component
Complexity. ThisalgorithmrunstwicethetimeofDFS(G)whichisΘ( V + E )
| | | |
Fig.4.Thestrongly connected components ofalargesystem
10