Schulz M. Control Theory in Physics and Other Fields of Science: Concepts, Tools, and Applications
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136 5 Chaos Control
Y = Φ(.....Φ(Φ(Φ(Y,λ),λ),λ),....,λ)
.
/0 1
m-times
, (5.23)
which are connected
6
by (5.22).
The evolution of the system under control follows the recursion law (5.17).
With the special choice u = λ, the trajectory of the system, given by the
set
$
X
(1)
,X
(2)
,...,X
(m)
,...
%
, will diverge from the desired unstable orbit
exponentially even if the initial conditions are nearly identical to one of the
m states Y
(µ)
∈ C
m
. We know that because of the ergodicity, the chaotic
trajectory visits or accesses an arbitrarily small neighborhood of the subse-
quent value Y
(µ+1)
of the desired orbit after a sufficiently long period. But
this knowledge is only helpful in obtaining a suitable starting point at which
the control starts. Obviously, we need an active, i.e., time dependent, control
u
(µ)
in order to stabilize the evolution of the system with respect to the con-
trol aim. We assume that the admissible values of u
(µ)
are located in a small
sphere of radius ε and center λ
u
(µ)
∈ U
if
u
(µ)
− λ
≤ ε, (5.24)
which is a subset of the control space, U
⊂ U.
Since the initially chosen value of the control function u is usually not equal
to the special value λ defining the control aim, the trajectory of the system
will diverge from the desired cycle exponentially even the initial conditions are
identical. Furthermore, we assume, that at time t
µ
, the trajectory falls into the
neighborhood of Y
(η)
. Without any restriction, we may define η = µ mod m.
Then the desired next value, Y
(η+1)
, is given by (5.22) while the next value of
the controlled trajectory, X
(µ+1)
, is given by (5.17). The difference between
both the new values is now
X
(µ+1)
− Y
(η+1)
= Φ(X
(µ)
,u
(µ)
) − Φ(Y
(η)
,λ) (5.25)
or with δX
(µ)
= X
(µ)
− Y
(η)
and δu
(µ)
= u
(µ)
− λ
δX
(µ+1)
= Φ(Y
(η)
+ δX
(µ)
,λ+ δu
(µ)
) − Φ(Y
(η)
,λ) (5.26)
=
∂Φ(Y
(η)
,λ)
∂Y
(η)
δX
(µ)
+
∂Φ(Y
(η)
,λ)
∂λ
δu
(µ)
.
This equation may be rewritten in the standard form of a linear problem
δX
(µ+1)
= A
η
δX
(µ)
+ B
η
δu
(µ)
, (5.27)
where the A
η
are matrices
7
of type N × N and the B
η
are matrices of type
N × n
6
It is necessary to say that the solutions must be connected by (5.22), because
(5.23) is often satisfied for more than one finite sets C
m
of m nonidentical values
Y
(µ)
. For example, the logistic map has two fixed points, x
∗
= 0 and x
∗
=1−λ
−1
,
which are not connected by (5.22), i.e. the fixed points form two seperate sets C
1
.
7
Note that the dimension of the phase space is N while the dimension of the control
space is n.
5.2 Time-Discrete Chaos Control 137
A
η
=
∂Φ(Y
(η)
,λ)
∂Y
(η)
and B
η
=
∂Φ(Y
(η)
,λ)
∂λ
(η =1,...,m) . (5.28)
The aim is now to find a linear control law
δu
(µ)
= K
η
δX
(µ)
(5.29)
with the n × N matrix K
η
so that
δX
(µ+1)
<
δX
(µ)
. In this case, the
unstable cycle becomes stable due to the external control. This is by no means
a trivial problem and its solution requires some theoretical preparations. To
solve this problem, we focus on the special case that the system should be
stabilized to an unstable fixed point. Thus we must take into account only
three matrices, A
1
= A, B
1
= B,andK
1
= K. A generalization to the
stabilization of unstable m-cycles is straightforwardly obtainable [60].
5.2.4 Reachability and Stabilizability
In order to solve the above-introduced problem, we have to explain the term
reachability. That is, roughly speaking, the property that each point of the
phase space P can be reached by series of successive control steps. In the case
of a linear system
X
(µ+1)
= AX
(µ)
+ Bu
(µ)
(5.30)
characterized by the matrices A and B, reachability is similar to the require-
ment that there must exist an input
$
u
(1)
,...,u
(N)
%
that transfers the state
X
(µ)
from the origin X
(1)
= 0 to an arbitrary point of the phase space. In
this case we say the pair (A, B) is also reachable. To decide if a pair (A, B)is
reachable, we consider first a one-dimensional control, n = 1. Then we obtain
X
(2)
= Bu
(1)
X
(3)
= ABu
(1)
+ Bu
(2)
.
.
.
X
(N+1)
= A
N−1
Bu
(1)
+ A
N−2
Bu
(2)
+ ... + ABu
(N−1)
+ Bu
(N)
(5.31)
with N free parameters u
(1)
,...,u
(N)
. Now we have two possibilities: either
the N vectors B, AB,...,A
N−1
B are linearely independent or not. In the
first case, these vectors form a complete basis of the phase space, i.e., each
point of this space is reachable at the latest after N steps. For the second
case we assume that C = A
κ
B with (1 <κ<N−1) is the first vector which
is linearly dependent from the previous vectors, i.e., there exists a relation of
the form
C =
κ−1
i=0
α
i
A
i
B. (5.32)
But in this case, the next vector also depends linearly on A
i
B (i =0,...,κ−1).
In fact, we obtain
138 5 Chaos Control
A
κ+1
B = AC =
κ−1
i=0
α
i
A
i+1
B = α
κ−1
A
κ
B +
κ−1
i=1
α
i−1
A
i
B
= α
κ−1
C +
κ−1
i=1
α
i−1
A
i
B
=
κ−1
i=0
[α
κ−1
α
i
+ α
i−1
] A
i
B (5.33)
with α
−1
= 0. The recursive application of this procedure shows that all subse-
quent vectors are linearly dependent from the first κ vectors. That means, the
infinite large set of vectors A
µ
B (µ =0, 1, 2,...) spans only a κ-dimensional
subspace of the phase space and, consequently, not all points of the phase
space are reachable after an arbitrary number of control steps. Thus, we con-
clude that in the case of a one-dimensional control, a pair (A, B) is reachable
if
rank
A
N−1
B,A
N−2
B,...,AB,B
= N. (5.34)
In the case of more than one control parameter, we may use the same argu-
mentation as before. Especially, we find that the pair (A, B) is reachable if an
arbitrary vector X
(N+1)
must be presented by the superposition
X
(N+1)
= A
N−1
Bu
(1)
+ A
N−2
Bu
(2)
+ ···+ ABu
(N−1)
+ Bu
(N)
(5.35)
otherwise there exists a subspace of P which is unreachable after an arbitrarily
large number of steps, i.e., the pair (A, B) is not reachable. The necessary
condition for the reachability is again (5.34), but now for the N × n matrix
B.
Let us assume that the pair (A, B) is reachable. Then we ask for a control
law u = KX so that the mapping
X
(µ)
→ X
(µ+1)
=(A + BK) X
(µ)
(5.36)
is a contraction. This requires that all eigenvalues ξ
i
, i =1,...,N, of the ma-
trix A + BK are smaller than unity, |ξ
i
| < 1. This problem belongs to the sta-
bilizability of the system under control. We focus again to a one-dimensional
input, n = 1. For higher dimensions we refer to the literature [61, 62, 63].
We get the surprising result that one can arbitrarily choose the spectrum of
the matrix (A + BK) if the pair (A, B) is reachable. For such systems there
always exists a similarity transformation T yielding a transformed pair
A
= T
−1
AT and B
= T
−1
B (5.37)
so that we obtain the controller canonical form [62, 64]
5.2 Time-Discrete Chaos Control 139
A
=
−a
N−1
−a
N−2
−a
N−3
···−a
1
−a
0
100··· 00
010··· 00
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 ··· 0100
0 ··· ··· 01 0
and B
=
1
0
0
.
.
.
0
0
. (5.38)
It is simple to check that a
i
(i =1,...,N) are the coefficients of the charac-
teristic polynomial of A
and because of (5.37) also of A,
det(z − A
) = det(z − A)=z
N
+
N−1
i=0
a
i
z
i
. (5.39)
Transforming the feedback K → K
= KT and the state vector X
(µ)
=
T
−1
X
(µ)
also yields the transformed mapping (5.36)
X
(µ+1)
=(A
+ B
K
) X
(µ)
= T
−1
(A + BK) TX
(µ)
. (5.40)
Thus the transformed matrix A
+ B
K
has the same eigenvalues as A + BK,
since both matrices are related by a similarity transformation. Now we choose
the eigenvalues of A
+B
K
to be {ξ
1
,ξ
2
,...,ξ
N
} with |ξ
i
| < 1fori =1,...,N.
Then, the characteristic polynomial for this free choice is given by
N
2
i=1
(z − ξ
i
)=z
N
+
N−1
i=0
a
i
z
i
, (5.41)
where the coefficients a
i
follows immediately from the algebraic expansion of
the left-hand side. It is clear that by choosing
K
=
a
N−1
− a
N−1
,a
N−2
− a
N−2
,...,a
0
− a
0
, (5.42)
the matrix A
+ B
K
has the same form as A
, but with the coefficients a
i
replaced by a
i
. Obviously, this discussion shows that the eigenvalues of every
reachable pair can be arbitrarily assigned. Simultaneously, we have obtained a
simple method for constructing the feedback K = K
T
−1
. The transformation
T is obtainable in two steps. The first step uses the controllability matrix
T
1
=(A
N−1
B,A
N−2
B,...,AB,B) , (5.43)
which transforms the pair (A, B)to
A
= T
−1
1
AT
1
and B
= T
−1
B (5.44)
with
A
=
0 ···0 −a
0
1
.
.
.
.
.
. −a
1
.
.
.
0
.
.
.
1 −a
N−1
and B
=
1
0
0
0
, (5.45)
140 5 Chaos Control
from where we also obtain the coefficients of the characteristic polynomial. A
further transformation,
A
= T
−1
2
A
T
2
and B
= T
−1
B
(5.46)
with
T
2
=
1 a
N−1
··· a
2
a
1
01
.
.
.
.
.
.
.
.
.
0
.
.
.
a
N−1
a
N−2
.
.
.
1 a
N−1
01
(5.47)
yields the requested form (A
,B
). Thus, we have K
= KT
1
T
2
and therefore
the desired matrix K
K = K
T
−1
2
T
−1
1
, (5.48)
where K
is given by (5.42). The components of K
are directly obtainable
from the characteristic polynomial of the matrix A (5.39) and from the desired
eigenvalues of A + BK via (5.41).
Finally, we remark that these considerations also remain valid for time-
continuous linear systems. We refrain from a presentation of this, in compar-
ison to the discrete version, give only slightly modified proof, and refer to the
literature [61, 62, 63].
5.2.5 Observability
Let us now discuss problems related to a control of system if only a reduced
information about the current system state is available. In contrast to what
was followed in Chap. 6, we suppose that the equations of motion of the sys-
tem are still completely known, but the complete state vector X is no longer
measurable by the available experimental instruments. In other words, the
observability of the system may be restricted to some overall functions of
the current state. That means from a mathematical point of view that the
state vector X ∈ P, projected onto an observation vector Y ∈ O,isanele-
ment of the observation space O ⊂ P. Since the dimension of the observation
space is usually lower than the dimension of the phase space, the projection
X → Y implies a loss of information. This statement applies especially to a
control of the system, since the control law consists now in a relation between
the control function u and the observation vector Y . It is important to note
that observability does not mean the reconstruction of the state X from one
measurement Y but the construction of a control u from all previous observa-
tions Y . We consider here again the discrete linearized law (5.30) extended by
the mapping Y
(µ)
= CX
(µ)
. The initial state may be X
(1)
. Then we obtain
from (5.30) for the free system (u =0)
5.3 Time-Continuous Chaos Control 141
Y
(1)
= CX
(1)
Y
(2)
= CAX
(1)
.
.
.
Y
(N)
= CA
N−1
X
(1)
. (5.49)
The space spanned by the vectors CX
(1)
, CAX
(1)
,...,CA
N−1
X
(1)
consid-
ering all X
(1)
∈ P must have the same dimension N as the phase space in
order to map an arbitrary initial state X
(1)
completely to the set of observa-
tions. If this is not the case, we may argue in a manner similar to the problem
of reachability that then also the consideration of higher vectors CA
N
X
(1)
,
CA
N+1
X
(1)
can never lead to a complete observability. Hence, we can say
that the dynamical process is observable if
rank
C
CA
.
.
.
CA
N−1
= N. (5.50)
Again, this statement can be applied in the present form to the characteriza-
tion of linear evolution equations. We find that a linear system
˙
X = AX + Bu and Y = CX (5.51)
is observable if (5.50) holds. Furthermore, in the case of nonlinear evolution
equations we may linearize these equations with respect to a certain refer-
ence point. This allows us to introduce locally defined observability (and also
reachability) conditions.
5.3 Time-Continuous Chaos Control
5.3.1 Delayed Feedback Control
We have already found that time-continuous trajectories of dynamical sys-
tems may form, in particular, unstable periodic orbits. Such an orbit is char-
acterized by cyclic boundary conditions,
X(t)=X(t + T
orbit
), where T
orbit
is the period. Now we will present a control which stabilizes this orbit. Fur-
thermore, we consider the above-introduced projection concept. The whole
N-dimensional state vector X(t) is not the measurable quantity but the lower
dimensional vector,
Y (t)=β(X(t)) (5.52)
where β : P → O with dim O = N
<N, is. From here we construct the control
function [65]
u(t)=U(Y (t) − Y (t − T )) (5.53)
142 5 Chaos Control
with a freely maneuverable n × N
matrix U defining the coupling between
the measurements and the control, where the basic idea of delayed feedback
control methods enters. The time T is called the delay time. The coupling to
the internal dynamics is given by the equations of motion (2.53) which have
now the time-delayed form
˙
X(t)=F (t, X(t),U(β(X(t)) − β(X(t − T )))) . (5.54)
For the sake of simplicity, we assume that the explicit time-dependence of
the function F is also periodic with T , i.e., F (t, X, u)=F (t + T,X,u). This
condition is helpful in defining experimentally the period of the unstable orbits
especially in case that the orbit periods T
orbit
are integer multiples of the
external driving period. Let us now fix T
orbit
= T for what follows
8
.The
existence of the unstable periodic orbit poses the reference equation
˙
X(t)=F
t, X(t), 0
. (5.55)
Now we consider a small deviation from the unstable orbit, Y (t)=X(t)−
X(t),
and a small external control. This allows us to linearize the evolution equation
for the system under control (5.54) with respect to the unstable periodic orbit
constraint
˙
Y (t)=A(t)Y (t)+B(t)[Y (t) − Y (t − T )] (5.56)
with
A(t)=
∂F(t, X, 0)
∂X
X=X(t)
(5.57)
and
B(t)=
∂F(t,
X(t),u)
∂u
u=0
U
∂β(X)
∂X
X=X(t)
. (5.58)
Obviously, the N ×N matrices A(t)andB(t) are periodic functions with the
period T . The periodically linear time dependent control equation (5.56)can
be decomposed into eigen-functions according to the Floquet theory [66]. To
this aim we use the ansatz
Y (t) = exp {−γt}y(t) , (5.59)
where γ is the characteristic exponent or the Floquet exponent and y (t)isa
periodic function, y(t)=y(t + T ). The substitution of (5.59)in(5.56) yields
˙y(t)=
A(t)+B(t)
1 − e
γT
+ γ
y(t) . (5.60)
This is an eigenequation with eigenvalues γ and eigenfunctions y(t). The so-
lution of these equation requires special knowledge about the structure of the
8
The generalization of the problem to different time scales for the external driving
period, the delay time, and the orbit period (as integer multiples of the driving
period) is, of course, always possible.
5.3 Time-Continuous Chaos Control 143
matrices A and B and therefore about the dynamics of the system [67]. We
avoid the problem and focus on the discussion of some general aspects. To
this aim, we consider first the uncontrolled evolution equation which follows
from (5.60) setting B(t) ≡ 0,
˙y(t)=[A(t)+
γ] y(t) , (5.61)
with
γ an eigenvalue of the uncontrolled problem. The geometrical meaning
of the Floquet exponents
γ
i
(i =1,...,N) is quite clear from (5.59). The real
parts
γ
i
=Reγ
i
define the radial attraction to the unstable orbit while the
imaginary parts
γ
i
=Imγ
i
determine the convolution of the trajectory around
the reference orbit. Since the periodic orbit was assumed to be unstable, at
least one eigenvalue has a negative real part.
Now we are ready to consider the complete equation (5.60). For the mo-
ment we introduce the quantity σ =1− e
γT
as a free parameter [71]. Thus,
the eigenvalues of (5.60) for a well-defined dynamics of the system, given by
the matrix A(t), and a control law defined by the matrix U and contained in
B(t), are simple functions of the parameter σ, i.e., we have
γ
i
= ϕ
i
(U, σ) with ϕ
i
(U, 0) = ϕ
i
(0,σ)=γ
i
. (5.62)
Now we are able to estimate the behavior of the eigenvalues qualitatively. We
expand (5.62) in terms of σ and all components of U up to the first non-trivial
order and we recall that σ is also a function of the Floquet exponent
γ
i
≈ ϕ
i
(0, 0) +
n
α=1
N
β=1
R
αβ
i
U
αβ
σ
=
γ
i
+
n
α=1
N
β=1
R
αβ
i
U
αβ
1 − e
γ
i
T
(5.63)
with
R
αβ
i
=
∂
2
ϕ
i
(U, σ)
∂U
αβ
∂σ
U=0,σ=0
. (5.64)
The coefficients R
αβ
i
contain all the details of the system, i.e., the dynamics
of the state X, the coupling between the control and the system, and the pro-
jection of the state to the measurable quantities Y . These coefficients should,
in principle, be obtainable from suitable experiments. Then we have nN
free
coefficients U
αβ
which must be chosen in such a way that Reγ
i
≥ 0 for all
Floquet exponents. Of course, this is a not always solvable; linear optimiza-
tion problem which can be treated by several standard methods are briefly
discussed in Chap. 10.
Time-delayed feedback control methods may allow us to stabilize periodic
orbits in chaotic systems [65]. It is interesting that the shape of the unstable
trajectory remains unchanged also in the presence of a finite control force.
Several applications of this control concepts have been discussed in the context
144 5 Chaos Control
of balancing by humans [68], the control of mechanical oscillating metal beams
[70], or of CO
2
lasers systems [69].
5.3.2 Synchronization
Synchronization is a widespread phenomenon observed between coupled sys-
tems. For example the well-known Belousov–Zhabotinskii chemical reaction
can be chaotic, but it is spatially uniform [57]. Hence, all spatial regions are
obviously synchronized with each other, even if the basic dynamics is a chaotic
motion. However, synchronization is not universal. In other circumstances the
uniformity of the Belousov–Zhabotinskii reaction becomes unstable and a pro-
nounced spatiotemporal dynamics occurs.
The first quantitative observation of a synchronization phenomenon is at-
tributed to Huygens in 1673 during his experiments for developing improved
pendulum clocks [72]. Two clocks were found to oscillate with the same fre-
quency due to the very weak coupling in terms of the nearly imperceptible
oscillations of the trestle on which both clocks were hanging. But we realize
again that synchronization is not a universal phenomena, because it was ob-
served by Huygens only if the individual frequencies of the clocks are almost
coincided.
The problem of a control of chaotic systems can at least be partially solved
by the application of synchronization effects. This is possible if one of the
coupled systems serves as a controller that is connected to the system. The
goal of the control is to make this system follow a prescribed time evolution,
i.e., a tracking protocol. We may interpret this behavior as a synchronization
of the system under control with the dynamics of the controller.
Here, we will discuss a very simple control mechanism using synchroniza-
tion effects. Let us assume we have a system the dynamics of which is described
by the evolution equation (2.53) and a well-defined optimal control problem.
The solution of this problem are the control equations which are defined by
the optimal evolution equation (2.74), the control law (2.75), and the adjoint
evolution equation (2.70). As discussed in Chap. 2, the control law is usu-
ally a set of algebraic relations between the vectorial control functions u
∗
,
the generalized momenta P
∗
, and the state vector X
∗
which may be used for
the elimination of the control function u
∗
= u
∗
(X
∗
,P
∗
,t). Unfortunately, this
representation is not unique. Because of the fact that the differential equa-
tions (2.74), (2.75), and (2.70) solve the optimal control problem, we obtain
the complete solution (X
∗
(t), P
∗
(t), u
∗
(t)). From here, we have infinite num-
ber of ways to eliminate the time in u
∗
(t) partially or completely by suitable
combinations of X
∗
and P
∗
. Hence, there exist infinite number of different re-
lations u
∗
= u
∗
(X
∗
,P
∗
,t). The main problem of using synchronization effects
for driving a system to the optimal dynamics is to find an appropriate equation
connecting u
∗
, X
∗
,andP
∗
. This usually requires heuristic experience.
Let us assume that we have such a relation. Hence, we obtain the optimal
evolution equations
5.3 Time-Continuous Chaos Control 145
˙
X
∗
= F (X
∗
,u
∗
(X
∗
,P
∗
,t),t)=
F (X
∗
,P
∗
,t) (5.65)
and the corresponding adjoint evolution equations which may be written in
the form
˙
P
∗
= V (X
∗
,u
∗
(X
∗
,P
∗
,t),t)=
V (X
∗
,P
∗
,t) . (5.66)
Both equations may be implemented in a computer (the controller) which
is connected with the real system, described also by the evolution equation
(2.53). The control mechanism of the system may now be constructed in such
a way that the control function u is given by
u = u
∗
(X, P
∗
,t) , (5.67)
where X belongs to the current state of the system while the momenta P
∗
are
the “control signals” coming from the computer
9
. Thus, the system dynamics
is given by
˙
X = F (X, u
∗
(X, P
∗
,t),t)=
F (X, P
∗
,t) . (5.68)
Formally, we now have two coupled systems the dynamics of which is defined
by (5.65) and (5.68). The first, the active part, given by the controller, drives
the second, which is called the passive part
10
.
If the initial conditions of the real system are equal to that of the optimal
trajectory, X(0) = X
∗
(0), the future evolution of the controlled system is, of
course, determined by X(t)=X
∗
(t). But in the more realistic case that the
initial conditions are not equal, synchronization can help drive the system to
the nominal trajectory X
∗
(t).
The evolution of the difference Y (t)=X(t) − X
∗
(t) between the system
evolution and the nominal curve is given by
˙
Y (t)=
F (X
∗
(t)+Y (t),P
∗
(t),t) −
F (X
∗
(t),P
∗
(t),t) . (5.69)
Thus, we conclude that synchronization of the nominal system and the con-
trolled system occurs if (5.69) has a stable fixed point for Y = 0. For small
Y (t) this can often be proved by using the linear stability analysis which
requires that the matrix ∂
F (X
∗
,P
∗
,t) /∂X
∗
is negative definite along the
optimal trajectory.
In order to get an impression about the importance of a appropriate choice
of the control law u
∗
= u
∗
(X
∗
,P
∗
,t), we consider a standard linear quadratic
problem with constant coefficients
11
. Here we have
F (X, u, t)=AX + Bu (5.70)
9
We suppose that the computer is fast enough to communicate with the system in
a real time mode.
10
The physical structure of the coupling suggests that the two coupled systems
apparently represent a special version of an open-loop control. But this is not
correct, because the control function u and the state X are originally connected
via the feedback law (5.67).
11
That is the optimal regulator problem; see Sect. 3.3.