Adapting Predictive Feedback Chaos Control for Optimal Convergence Speed

1.2 MB·English

by  Christian Bick

Adapting Predictive Feedback Chaos Control for Optimal Convergence
Speed
Christian Bick1,2, Marc Timme1,3, and Christoph Kolodziejski1,4
1NetworkDynamicsGroup, Max PlanckInstituteforDynamicsand Self-Organization (MPIDS),
37077 Go¨ttingen, Germany and Bernstein Center for Computational Neuroscience, Go¨ttingen
2 Institute for Mathematics, Georg–August–Universita¨t Go¨ttingen, 37073 Go¨ttingen, Germany
2 3 Institute for Nonlinear Dynamics, Georg–August–
1
0
Universita¨t Go¨ttingen, 37077 Go¨ttingen, Germany
2

n 4 Third Institute of Physics — Biophysics,
a
J
Georg–August–Universita¨t Go¨ttingen, 37077 Go¨ttingen, Germany
∗
6

Abstract
]
O
A Stabilizingunstableperiodicorbitsinachaoticinvariantsetnotonlyrevealsinformationaboutitsstruc-
.
n ture but also leads to various interesting applications. For the successful application of a chaos control
i
l
n scheme, convergence speed is of crucial importance. Here we present a predictive feedback chaos control
[

method that adapts a control parameter online to yield optimal asymptotic convergence speed. We study
1
v
6 theadaptivecontrolmapbothanalyticallyandnumericallyandprovethatitconvergesatleastlinearlytoa
5
4 valuedeterminedbythespectralradiusofthecontrolmapattheperiodicorbittobestabilized. Themethod
1
. iseasytoimplementalgorithmicallyandmayfindapplicationsforadaptiveonlinecontrolofbiologicaland
1
0
2 engineeringsystems.
1
:
v
PACSnumbers: 05.45.Gg,02.30.Yy
i
X
r
a
∗Electronicaddress: [email protected]
1
I. INTRODUCTION
Somechaoticattractorscontaininfinitelymanyunstableperiodicorbits. Thesecanbeseenasa
“skeleton” for the chaotic attractor, therefore revealing important information about the dynamics
of the system itself. By suitable perturbations the stability of these unstable periodic points can
be changed; a control perturbation renders them stable. Such “chaos control” has applications in
manyfields[1],includingbiological[2]andartificialneuralnetworks[3,4].
In the last twenty years, different methods for stabilizing unstable periodic orbits have been
suggested. The seminal work by Ott, Grebogi, and Yorke (OGY) [5] and its implementations
employ arbitrary small perturbations of a parameter of the system to stabilize a known unstable
periodic orbit of a discrete time dynamical system. A successful application of the OGY method,
however, requires prior knowledge about or online analysis of the dynamics to determine fixed
pointsandtheirstabilityproperties.
Adifferentapproachisgivenbypredictivefeedbackcontrol(PFC)[6,7]whichovercomesthis
disadvantage. Inthisapproachthefuturestateofthedynamicscalculatedfromthecurrentstateis
fedbackintothesystemtostabilizeaperiodicorbit. Thisfeedbackcontrolisnoninvasive,i.e.,the
control strength vanishes upon convergence, and is extremely easy to implement. It is a special
case of a recent effort to stabilize all periodic points of a discrete time dynamical system [8, 9]
whichisalsocloselyrelatedtononlinearsuccessiveover-relaxationmethods[10,11]. Ithasbeen
extensivelystudied[12–16]andextended[17,18]withrespecttoitsoriginalpurposeasatoolfor
examiningthestructureofchaoticattractors.
In any real world application, speed of convergence is of crucial importance. For example, if
a robot is controlled by stabilizing periodic orbits in a chaotic attractor [3], the time it needs to
reacttoachangingenvironmentisboundedbythetimethesystemneedstoconvergetoaperiodic
point of a given period. Hence, in praxis, one desires to tune the control parameter such that the
spectralradiusoftheunknownperiodicpointwhichthesystemconvergestoisminimized. Tothe
best of our knowledge, previous works on chaos control have not considered convergence speed
while maintaining its simplicity in terms of implementation. Adaptation of the control parameter
has an impact on convergence speed. However, existing adaptation mechanisms [3, 19] have two
major shortcomings; they do not optimize for speed and, for adaptation of heuristic nature, may
adapttheparametertoregimeswherestabilizationfails.
Here we introduce an adaptation method that overcomes these shortcomings. It adaptively
2
tunesthecontrolparameteronlinetoachieveoptimalasymptoticconvergencespeed. Thisworkis
organizedasfollows. InthesecondsectionwereviewthePFCmethodandintroducethenotation
that will be used throughout the paper. In the third section, we present the adaptation method and
prove its convergence properties. As an example, the well-known logistic map is studied both
analyticallyandnumericallyinSections4and5beforegivingsomeconcludingremarks.
II. PRELIMINARIES
A differentiable map f : Rn Rn gives rise to a dynamical system through the evolution
→
equation
x = f(x ) (1)
k+1 k
with x Rn for all k Z. The sequence (x ),k N is called an orbit of the dynamical system
k k
∈ ∈ ∈
with initial condition x and if f p(x ) = x for all k 0 we say that the orbit is periodic with
0 ◦ k k+p
≥
periodp. Here,
f p = f f f
◦
(cid:124) ◦ ◦(cid:123)(cid:122)···◦ (cid:125)
ptimes
denotes the p-fold composition of f. Let Fix(f) := x Rn f(x) = x be the set of fixed
{ ∈ | }
points, i.e., periodic points of period one. Note that any periodic orbit is a fixed point of the p-th
iterateofthemapf sowewillusetheexpressionsfixedpointandperiodicorbitinterchangeably.
Let A Rn be a forward invariant subset of Rn with respect to f, i.e., f(A) A. If periodic
⊂ ⊂
points are dense in A and f maps transitively, then we call A a chaotic set. Julia sets [20], as
described below, are examples of such chaotic sets. Let df denote the derivative of f at x Rn
x
| ∈
andid : Rn Rn theidentitymap.
→
Theresultsof[9]arenowsummarizedasfollows:
PropositionII.1. Supposex Fix(f)andthederivativedf anddf idarereal,nonsingu-
∗ x x
∈ | ∗ | ∗−
lar and diagonalizable. Then there exist parameters µ > 0 and an orthogonal matrix C O(n)
k
∈
suchthatx isanattractivefixedpointofthemapg obtainedfromf throughthetransformation
∗ µ
S(µ,C ) : f id+µC (f id) =: g .
k k µ
(cid:55)→ −
Inparticular,S(µ,C )preservesthesetoffixedpoints,thatisFix(f) = Fix(g ).
k µ
In fact, it can be shown that the number of matrices C needed to stabilize all fixed points of
k
a given map f is quite limited. They depend on the local stability properties of fixed points and
3
1
)
µ
(
∗
x
ρ
λ
∞
0 µ 1
∞
Controlparameterµ
Figure1: Sketchofthedependenceofthespectralradiusρ (µ)onµforsomefixedpointx accordingto
x ∗
∗
PropositionII.1.
there are types of fixed points that can be stabilized for C id . For a given x Fix(f)
k ∗
∈ {± } ∈
withχ C,j 1,...,n ,beingtheeigenvaluesofdg wewanttodenoteby
j µ x
∈ ∈ { } | ∗
ρ (µ) := max χ
x j
∗ j 1,...,n {| |}
∈{ }
thespectralradius,i.e.,themaximumoftheabsolutevaluesoftheeigenvaluesofthederivativeof
g atx . Wehave
µ ∗
dg = id+µC (df id). (2)
µ x k x
| | −
for all x Rn. In other words, the proposition above ensures the existence of µ and C (x )
k ∗
∈
for a given x Fix(f) such that the transformation S(µ,C ) gives ρ (µ) < 1, cf. Figure 1.
∗ k x
∈ ∗
Therefore,withtheseparameters,thefixedpointx off isanattractingfixedpointforg .
∗ µ
Theresultsabovearedirectlyrelatedtopredictivefeedbackchaoscontrolmethods. Atransfor-
mationT : f g iscalledachaoscontroltransformationifg canbewrittenasg = f +ηcwith
η
(cid:55)→
control perturbation c : Rn Rn and η R. Note that in case C id the transformations
k
→ ∈ ∈ {± }
S(C ,µ)arechaoscontroltransformationssince
k
g = f +(1 µ)(id f)
µ
∓ −
withη = 1 µ. Therefore,wewillrefertothesetransformationsS(µ,C )asPFCtransformations.
k
−
WithoutlossofgeneralitywerestrictourselvestothecaseC = id.
k
TheresultsofPropositionII.1,however,givelittleinformationaboutthespeedofconvergence,
except for the fact that when decreasing µ towards zero convergence takes longer and longer as
the spectral radius approaches one. In the vicinity of a stabilized fixed point, convergence is at
4
least linear and the rate of convergence is bounded from above by the quantity ρ (µ). In order to
x
∗
obtainanadaptationmethodthatincreasesthespeedofconvergencewethereforehavetominimize
ρ (µ)usingthecontrolparameterµ. Forarandominitialcondition,wedonotknowtowhatfixed
x
∗
pointx (ifany)thetrajectorywillconvergeto. Weonlyhaveaconvergingsequencex x . In
∗ k ∗
→
otherwords,wearelookingforawaytoobtainasequenceµ µ where
k
→ ∞
 (cid:12) 
(cid:12)
 andassumptionsof 
(cid:12)
µ = sup µ > 0 (cid:12) µ˜ > 0 : ρ (µ) ρ (µ˜) (3)
x x
∞  (cid:12)∀ ∗ ≤ ∗ Prop.II.1aresatisfied 
(cid:12)
theoptimalµtominimizeρ (µ). Defineλ := ρ (µ ).
x x
∗ ∞ ∗ ∞
In applications, the control parameter µ plays a double role; on the one hand, it can be used to
turn chaos control on and off, µ = 1, on the other hand it is the crucial parameter to stabilize the
periodicorbitsandtodeterminethespeedofconvergence.
Definetheclassoffunctions
(µ ,p) := f card( x Fix(f p) ρ (µ ) < 1forC = id ) > 0 ,
0 ◦ x 0 k
F { | { ∈ | } }
with parameters p N, and µ > 0 and where card denotes the cardinality of a set. The sets
0
∈
(µ ,p) are the functions f with a chaotic set that have at least one periodic orbit of period p
0
F
whichcanbestabilizedforthegivenparameters.
III. ANADAPTATIONMETHODTOACCELERATECHAOSCONTROL
In this section, suppose f (µ ,p) for some µ > 0 and without loss of generality, p = 1,
0 0
∈ F
since we can replace f with the p-th iterate. Suppose g is the transformed map after applying
µ
S(µ,id). Furthermore we assume that for all times k < 0 the system evolves according to Equa-
tion (1), i.e., with η = 1 µ = 0, along a trajectory of points in the chaotic set A. At time k = 0
−
the control parameter µ is set to µ . Therefore, because of the assumptions on f, there is at least
0
one periodic orbit of period p on the chaotic attractor which is now an attracting periodic orbit.
LetsFix(f)denotethesetofthesestabilizedfixedpoints.
5
A. Closetoafixedpoint
Recall two facts: any differentiable map h 1(U) on an open set U Rn is Lipschitz-
∈ C ⊂
continuousonanycompactK U,thatis
⊂
h(x) h(y) dh x y ,
K
(cid:107) − (cid:107) ≤ (cid:107) (cid:107) ·(cid:107) − (cid:107)
forallx,y K where isthesupremumoftheoperatornormsinducedbyanorm on
K
∈ (cid:107) · (cid:107) (cid:107) · (cid:107)
K anddhisthetotalderivative. Furthermore,foranycontractionhonaBanachspace(X, ),
(cid:107) · (cid:107)
i.e.,amapthatsatisfies
h(x) h(y) L x y
(cid:107) − (cid:107) ≤ (cid:107) − (cid:107)
with a Lipschitz constant L < 1, the Banach Fixed Point Theorem gives the existence of a unique
fixed point x together with the error estimates x x Lk x x and x x
∗ (cid:107) ∗ − k(cid:107) ≤ 1 L (cid:107) 0 − 1(cid:107) (cid:107) k+1 − k(cid:107) ≤
−
L x x . Here,x = h k(x )foraninitialconditionx X.
k k 1 k ◦ 0 0
(cid:107) − − (cid:107) ∈
Let x sFix(f) be fixed. According to Proposition II.1 there exists a λ < 1 for µ = µ
∗ 0 0
∈
sufficiently small such that ρ (µ ) < λ . Therefore, there exists a vector norm such that
x 0 0
∗ (cid:107) · (cid:107)
we have dg λ for the induced operator norm cf. Figure 2(a). We will omit the index
µ x op 0
(cid:107) | ∗(cid:107) ≤
indicating the operator norm when it is clear from the context. Henceforth all norms denote this
vectornormandtheinducedoperatornormrespectively.
Let B(ε,x) denote a ball of radius ε centered at x and by B(ε,x) its closure. Assume that for
ε˜smallenoughthereisaconstantK 0suchthat
≥
dg dg K x x (4)
µ x µ x ∗
|(cid:107) | ∗(cid:107)−(cid:107) | (cid:107)| ≤ (cid:107) − (cid:107)
for all x with x x < ε˜independent of µ. This condition is depicted in Figure 2(b). Now we
∗
(cid:107) − (cid:107)
canchooseε ε˜suchthat dg < 1. Putdifferently,forsufficientlysmallδ > 0there
≤ (cid:107) µ0|x(cid:107)B(ε,x∗) 0
existsanε (0,ε˜)suchthat dg λ +δ =: L < 1. Thechoiceofε(corresponding
∈ (cid:107) µ0|x(cid:107)B(ε,x∗) ≤ 0 0 0
tothesizeoftheballaroundx )dependsonλ ,µ ,andδ .
∗ 0 0 0
Remark III.1. Condition (4) is satisfied in case f has a bounded second derivative on sFix(f)
duetothefunctionaldependenceofthederivativeofg onµasgivenby(2).
µ
Algorithm III.2. For given f (µ ,p),µ ,λ , and K let x B(ε,x ). The convergence
0 0 0 0 ∗
∈ F ∈
accelerationalgorithmconsistsofthefollowingsteps:
Step1 (Iterate): Calculatex = g (x ).
1 µ0 0
6
1 1
L
0
λ λ
0 0
ρ (µ) dg
x µ x
∗ k | ∗k
dg dg
k µ|x∗k k µ|xkk
0 µ 1 0 µ 1
0 0
Controlparameterµ Controlparameterµ
(a) (b)
Figure 2: (a) An appropriately chosen norm approximates the spectral radius from above with ρ (µ )
x 0
∗ ≤
dg λ . (b) According to Condition (4), dg lies in a K x x -tube around dg .
(cid:107) µ0|x∗(cid:107) ≤ 0 (cid:107) µ|xk(cid:107) (cid:107) ∗− k(cid:107) (cid:107) µ|x∗(cid:107)
While iterating, this tube becomes smaller and smaller. As another consequence of (4), we have
dg in a Kε-tube around dg . Therefore, for ε small enough, there is L < 1 such that
(cid:107) µ|x(cid:107)B(ε,x∗) (cid:107) µ|x∗(cid:107) 0
dg L .
(cid:107) µ0|x(cid:107)B(ε,x∗) ≤ 0
Step2 (Optimize µ): Minimize the “cost function” dg with respect to µ (0,1) under the
(cid:107) µ|x1(cid:107) ∈
conditions
(cid:16) (cid:17)
l(µ) := dg + 2KL0 x x < L (5a)
(cid:107) µ|x1(cid:107) 1 L0 (cid:107) 0 − 1(cid:107) 0
−
µmaximal (5b)
whereL = λ +δ asabove,cf. Figure3.
0 0 0
Step3 (Set quantities): If the minimization under constraints of Step 2 returned a result µ
opt
(cid:16) (cid:17) (cid:16) (cid:17)
then set µ := µ ,λ := dg + KL0 x x , δ := KL0 x x and
1 opt 1 (cid:107) µ|x1(cid:107) 1 L0 (cid:107) 0 − 1(cid:107) 1 1 L0 (cid:107) 0 − 1(cid:107)
− −
L := λ +δ . Otherwisesetµ := µ ,λ := λ ,δ := δ andL := L .
1 1 1 1 0 1 0 1 0 1 0
Repeatthestepswithallindicesincreasedbyone.
Forthismethodweobtainthefollowingresults.
LemmaIII.3. WiththeassumptionsofAlgorithmIII.2abovewehavethatforanyinitialcondition
x B(ε,x )AlgorithmIII.2yieldsatrajectoryx x .
0 ∗ k ∗
∈ →
7
1 1
L
0
l(µ) l(µ)
dg dg
µ x µ x
k | ∗k k | ∗k
dg dg
k µ|xkk k µ|xk+Nk
0 µ 1 0 µ 1
0 0
Controlparameterµ Controlparameterµ
(a) (b)
Figure 3: (a) Inequality (5a) does not always have to be satisfied. (b) Since x x we have that after a
k ∗
→
finitetimeN thefunctionl(µ)isbelowL forsomeµ.
0
Proof. If the optimization process does not give a result, convergence is ensured by Proposi-
tion II.1 and the Banach Fixed Point Theorem. Without loss of generality, suppose that optimiza-
tionyieldsaresultfork = 1. Thenbecauseof(4)and(5a)wehave
dg dg +K x x
(cid:107) µ1|x∗(cid:107)B((cid:107)x1−x∗(cid:107),x∗) ≤ (cid:107) µ1|x∗(cid:107) (cid:107) 1 − ∗(cid:107)
dg +2K x x
≤ (cid:107) µ1|x1(cid:107) (cid:107) 1 − ∗(cid:107)
(cid:18) (cid:19)
2KL
dg + 0 x x
≤ (cid:107) µ1|x1(cid:107) 1 L (cid:107) 1 − 0(cid:107)
0
−
= L L < 1.
1 0
≤
Therefore,x iscontainedinaballaroundthefixedpointx onwhichthemapg isacontraction
1 ∗ µ1
with contraction coefficient L . The same calculation is valid for subsequent optimization steps
1
fork > 1.
The lemma above ensures that the adaptation does not compromise convergence against the
stabilizedfixedpoint. Butwilloptimizationactuallytakeplace? ForamapwithK = 0,adaptation
isnotnecessarysince dg = dg andthereforewecansetµstraighttotheoptimalvalue.
(cid:107) µ|x∗(cid:107) (cid:107) µ|xk(cid:107)
Lemma III.4. With the assumptions of Algorithm III.2 Inequality (5a) is satisfied every finitely
manysteps.
Proof. By definition we have dg L = λ + δ with δ > 0. Hence we have
(cid:107) µ0|x∗(cid:107) ≤ 0 0 0 0
dg < L . Let ζ > 0 be such that ζ < L dg . Since x x is a Cauchy
(cid:107) µ0|x∗(cid:107) 0 0 − (cid:107) µ0|x∗(cid:107) (cid:107) k − k+1(cid:107)
8
(cid:16) (cid:17)
sequenceand dg dg thereisanN Nsuchthat 2KL0 x x < ζ and
(cid:107) µ0|xk(cid:107) → (cid:107) µ0|x∗(cid:107) ∈ 1−L0 (cid:107) N − N−1(cid:107) 2
dg dg < ζ. Thus,wehave
|(cid:107) µ0|x∗(cid:107)−(cid:107) µ0|xN(cid:107)| 2
(cid:18) (cid:19)
2KL ζ ζ
dg + 0 x x < dg + + < dg +ζ < L .
(cid:107) µ0|xN(cid:107) 1 L0 (cid:107) N − N−1(cid:107) (cid:107) µ0|x∗(cid:107) 2 2 (cid:107) µ0|x∗(cid:107) 0
−
Therefore,Inequality(5a)willbesatisfiedaftermaximallyN = N steps.
0
Inductively, by increasing all indices above by N, the same argument gives a sequence N ,
l
l N, of indices for which Inequality (5a) is satisfied. This completes the proof of the assertion.
∈
Remark III.5. Although the adaptation method gives a sequence µ that minimizes the norm
k
while ensuring convergence, it is not clear how often optimization yields a result. Additional
conditions on the map f, such as requiring monotonicity of dg in x , influence how often
(cid:107) µ|xk(cid:107) k
the parameter µ will be adapted. On the other hand, additional constraints make the theory less
broadlyapplicable.
If Inequality (5a) is satisfied for some k > 0, then, because of continuity, it holds for a whole
closed neighborhood of µ . This gives µ with dg < dg unless dg is
k k+1 (cid:107) µk+1|x∗(cid:107) (cid:107) µk+1|x∗(cid:107) (cid:107) µ|x∗(cid:107)
constantonthatinterval.
DefinitionIII.6([21]). Amatrixnorm onRn n iscalledminimalforM Rn n ifρ(M) =
× ×
(cid:107) · (cid:107) ∈
M .
(cid:107) (cid:107)
Themainresultsofthissectioncannowbesummarizedinthefollowingtheorem.
Theorem III.7. Suppose f (µ ,p) for µ > 0 such that f satisfies (4) with K 0 in a
0 0
∈ F ≥
neighborhoodofx sFix(f). Furthermore,letε,λ ,andδ bechosenasdescribedabove. Then
∗ 0 0
∈
for any initial condition x B(ε,x ) Algorithm III.2 minimizes an upper bound for the spectral
0 ∗
∈
radiusρ (µ).
x
∗
Inparticular,iftheinducedoperatornorm isminimalfordg ,itconvergesatleast
op µ x
(cid:107) · (cid:107) ∞| ∗
linearlywithasymptoticconvergencespeedλ .
∞
RemarkIII.8. Fordimensionn = 1,theEuclideannormisminimal.
Proof. (of Theorem III.7) Lemmas III.3 and III.4 ensure convergence against the fixed point x
∗
andadaptationofthecontrolparameterµafteramaximumofsomefinitenumberofsteps.
9
By construction, µ tends to a value which minimizes the norm of the derivative of g at x .
k µ ∗
For arbitrary dimension n we have ρ (µ ) dg . If in addition the norm is minimal,
x∗ k ≤ (cid:107) µk|x∗(cid:107)
in the limit the spectral radius is minimized yielding optimal asymptotic convergence speed, i.e.,
µ µ .
k
→ ∞
RemarkIII.9. Onecouldalsouseconvergenceaccelerationtransformations[22]inordertoget
a better approximation to ρ (µ). However, to exploit the acceleration within the framework of
x
∗
thistheory,onewouldhavetohavesuitableerrorestimatesforthetransformedsequence.
ThechoiceofthesizeneighborhoodB(ε,x )dependsonthedesiredestimateofthecontraction
∗
constant. It is clearly bounded from above since we have to make sure that there is a contraction.
On the other hand, it is desirable to take a neighborhood as large as possible to make the method
applicabletoasmanyinitialconditionsaspossible.
B. Fromlocaltoglobal
Wewantto considerthesituationwherethe controlisturnedonat arandompointintime. We
choose indices such that this time is k = 0. In general, the initial condition x for the adaptation
0
method is unknown and it is likely to be outside of a neighborhood B(ε,x ) as defined in the
∗
section above. One possible scenario is the existence of a chaotic attractor that makes up part
of the old chaotic attractor and ending up in its basin of attraction when the control parameter is
turned up. Another likely scenario is to have x close to the boundary of the basin of attraction of
0
one of the stabilized fixed points. An initial condition close to the basin boundary implies a long
transientiterationbeforetheadaptationmethodbecomesapplicable.
We want to quantify the latter scenario. Since we assumed the system to follow the evolution
equation x = f(x ) for all k 0, x is distributed on the attractor according to some f-
k k 1 0
− ≤
invariant measure m on A. Suppose m is an ergodic probability measure on A. So m(U) is the
probabilitythatx U attimek = 0foranymeasurablesubsetU A.
0
∈ ⊂
LetB(ε(x ),x )denotetheneighborhoodsofx sFix(f)forwhichtheaccelerationmethod
∗ ∗ ∗
∈
described in the previous section is applicable for all initial conditions within that neighborhood.
Becauseoff (µ ,p)atleastoneoftheseballsisnotempty. Define
0
∈ F
 
(cid:91)
V = V =  B(ε(x),x) A
0
∩
x sFix(f)
∈
10