Linear control theory is used to develop an improved localized control scheme for spatially extended chaotic systems, which is applied to a coupled map lattice as an example. The optimal arrangement of the control sites is shown to depend on the symmetry properties of the system, while their minimal density depends on the strength of noise in the system. The method is shown to work in any region of parameter space and requires a significantly smaller number of controllers compared to the method proposed earlier by Hu and Qu [Phys. Rev. Lett. 72, 68 (1994)]. A nonlinear generalization of the method for a 1D lattice is also presented

Cross, M. C.

Grigoriev, R. O.

Schuster, H. G.

English

Caltech Authors - Main

VOLUME 79, NUMBER 15 P HY S I CA L REV I EW LE T T ER S 13 OCTOBER 1997Pinning Control of Spatiotemporal Chaos
R.O. Grigoriev, M. C. Cross, and H. G. Schuster
Condensed Matter Physics 114-36 and Neural Systems Program 139-74, California Institute of Technology,
Pasadena, California 91125
(Received 29 April 1997)
Linear control theory is used to develop an improved localized control scheme for spatially extended
chaotic systems, which is applied to a coupled map lattice as an example. The optimal arrangement
of the control sites is shown to depend on the symmetry properties of the system, while their minimal
density depends on the strength of noise in the system. The method is shown to work in any region
of parameter space and requires a significantly smaller number of controllers compared to the method
proposed earlier by Hu and Qu [Phys. Rev. Lett. 72, 68 (1994)]. A nonlinear generalization of the
method for a 1D lattice is also presented. [S0031-9007(97)04237-3]
PACS numbers: 05.45.+b, 02.30.WdControlling chaos in high-dimensional systems [1]
and spatiotemporal chaos especially is a very important
problem with numerous applications to turbulence [2],
instabilities in plasma [3], multimode lasers [4], and
reaction-diffusion systems [5].
The present Letter represents an effort to develop a gen-
eral control algorithm for spatiotemporally chaotic systems
using the methodology of linear control theory, which al-
ready proved to be fruitful [6]. Clarifying a number of
issues will have direct bearing on this. For instance, it is
not clear how many parameters are required for successful
control. If the control is applied locally, what is the mini-
mal density of controllers and how should they be arranged
to obtain optimal performance? What are the limitations of
the linear control scheme and how can they be overcome?
Consider the coupled map lattice (CML), originally
introduced by Kaneko [7], in an alternative form:
zt11i ­ Fsz
t
i21, z
t
i , z
t
i11d
­ fss1 2 2edzti 1 esz
t
i21 1 z
t
i11dd , (1)
with i ­ 1, 2, · · · , L and periodic boundary conditions
(BC), zti1L ­ zti imposed. We also assume that the local
map fsz, ad is a nonlinear function with parameter a, such
that fszp, ad ­ zp.
To be specific, we choose
fszd ­ azs1 2 zd , (2)
but emphasize that all the major results hold independent
of this choice. This CML has a homogeneous steady state
zp ­ 1 2 1ya, which is unstable for a . 3.0. Our goal
is to stabilize it using a minimal number of controllers.
The first attempt in this direction was undertaken by
Hu and Qu [8]. The authors tried to stabilize the homo-0031-9007y97y79(15)y2795(4)$10.00geneous state by controlling an array of M periodically
placed pinning sites hi1, · · · , iMj with appropriately cho-
sen control utm
zt11i ­ Fsz
t
i21, z
t
i , z
t
i11d 1
MX
m­1
dsi 2 imdutm . (3)
This however required a very dense array with distance
between controllers Lp ­ LyM # 3 in the physically
interesting interval of parameters 3.57 , a , 4.0.
The reason for this is the spatial periodicity of the pin-
nings. Since the system is spatially uniform, its eigen-
modes are just Fourier modes and the control does not
affect the modes whose nodes happen to lie at the pinnings,
i.e., modes with periods equal to 2Lp , 2Lpy2, 2Lpy3, etc.,
provided those are integer. The control scheme worked
only when all such modes were stable.
It is however not necessary to destroy the periodicity
completely to achieve control: that would complicate the
analysis unnecessarily. Instead we add one more pinning
site between each of the existing ones. Not all positions
are good, but some do solve the problem—previously
uncontrollable modes become controllable.
In order to understand how the pinnings should be
placed and see whether we achieve improved performance
by introducing additional controllers, we have to use a few
results of the linear control theory [9]. We will start with
linearizing Eq. (3) about the homogeneous steady state
zt ­ szp, · · · , zpd in both the state vector and control to
obtain the following standard equation
xt11 ­ Axt 1 But , (4)
where we denoted the displacement x ­ z 2 zp. If we
define a ­ ›fsxp, ady›x, then the L 3 L Jacobian A is
given byA ­ a
0BBBBBB@
1 2 2e e 0 · · · e
e 1 2 2e e · · · 0
0 e 1 2 2e · · · 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
e 0 0 · · · 1 2 2e
1CCCCCCA (5)© 1997 The American Physical Society 2795
VOLUME 79, NUMBER 15 P HY S I CA L REV I EW LE T T ER S 13 OCTOBER 1997and the L 3 M control matrix Bij ­
P
m ds j 2
mddsi 2 imd depends on how we place the pinning sites.
If we use synchronous linear feedback ut ­ 2Kxt ,
Eq. (4) becomes
xt11 ­ sA 2 BKdxt , (6)
and the solution x ­ 0 can be made stable by a suitable
choice of the feedback gain matrix K , if the controllability
condition rank sCd ­ L is satisfied. The controllability
matrix C is defined via
C ­ sB AB · · · AL21Bd . (7)
One can easily verify that the matrix B calculated for
a periodic array of pinning sites does not satisfy the
controllability condition and therefore the homogeneous
steady state is not controllable. It can be stabilized if
the weaker stabilizability condition is satisfied, i.e., all
uncontrollable modes are stable. However this imposes
excessive restrictions on the pinning density.
The condition for stabilizability can be obtained from
the spectrum of eigenvalues of the matrix (5)
gi ­ af1 2 2es1 2 cosskidg , (8)
where k1 ­ 0, ki ­ ki11 ­ piyL for i ­ 2, 4, 6, . . . and,
for L even, KL ­ p and a ­ 2 2 a. Specifically, we
need jsa 2 2d f1 2 2es1 2 cosspjyLpddgj , 1 for all
j ­ 1, . . . , L 2 2, such that Lpyj is integer. Using this
criterion one can obtain the relation between the minimum
coupling, the distance between controllers and parameter
a of the local chaotic map for a stabilizable system. For
instance, j ­ 1 yields
e ­
a 2 3
2sa 2 2d f1 2 cosspyLpdg
. (9)
The numerically obtained results of Hu and Qu (Fig. 1)
are seen to lie exactly on the curves, given by Eq. (9).
It is possible however to extend the limits of the control
scheme quite substantially by making the system control-
lable as opposed to stabilizable. This is easily achieved
by choosing a different matrix B, i.e., placing the pinning
FIG. 1. Periodic array of single pinning sites: Critical cou-
pling ecr as a function of parameter a. The data points repre-
sent the numerical results from Fig. 2 of [8], with e rescaled by
a factor of 2 to make it compatible with our definition.2796sites differently. Doing so will enable us to control the
system anywhere in the parameter space at the same time
using a smaller density of controllers.
First one has to determine the dimensionality of the
matrix B, in other words determine the minimal number
of parameters required to control the CML (1) of an
arbitrary length. It can be shown [10] that the minimal
number of parameters required to control a system with
degenerate Jacobian is equal to the greatest multiplicity of
its eigenvalues.
Since the system considered has parity symmetry, the
eigenvalues (8) of its Jacobian are in fact doubly degener-
ate, so the minimal number of control parameters yielding
a controllable system in our case is two, meaning at least
two pinning sites are required. One can easily verify that
the controllability condition for an L 3 2 matrix
Bij ­ ds j 2 1ddsi 2 i1d 1 ds j 2 2ddsi 2 i2d (10)
is indeed satisfied for a number of arrangements hi1, i2j.
The restrictions on the mutual arrangement of the con-
trollers are again given by the condition of controllability:
L should not be a multiple of ji2 2 i1j; otherwise the mode
with the period 2ji2 2 i1j becomes uncontrollable.
The next step in the algorithm is to determine the
feedback gain K. Pole placement techniques based on
Ackermann’s method [11] are inapplicable to the problem
of controlling spatially extended systems because they are
numerically unstable [12] and break down rapidly for
problems of order greater than 10.
Instead we use the method of the linear-quadratic (LQ)
control theory [9], applicable to the unstable periodic
trajectories as well as fixed points. This method is not
only numerically stable, but also allows one to optimize
the control algorithm to increase convergence speed, and
at the same time minimize the strength of control. As
we will see below, decreasing control enlarges the basin
of attraction, which has very important consequences for
the time to achieve control (capture the chaotic trajectory).
The optimal solution is obtained by minimizing the cost
functional
V sx0d ­
‘X
t­0
sxtyQxt 1 utyRutd , (11)
where Q and R are the weight matrices that can be chosen
as any positive-definite square matrices.
The minimum of (11) is reached when
K ­ sR 1 ByPBd21ByPA , (12)
where P is the solution to the discrete-time algebraic
Ricatti equation
P ­ sQ 1 AyPAd 2 AyPyBsR 1 ByPBd21ByPA .
(13)
Numerical simulations show that the CML (1), (2) can
indeed be stabilized by this linear control scheme in a
wide range of parameters a and e. The solution for K is
presented in Fig. 2 for a ­ 4.0, e ­ 0.33, and L ­ 8 with
VOLUME 79, NUMBER 15 P HY S I CA L REV I EW LE T T ER S 13 OCTOBER 1997FIG. 2. Feedback matrix: Feedback gain vectors K1j and K2j
for left and right controllers placed at the sides of the lattice
(i1 ­ 1, i2 ­ 8) as functions of the lattice site for a ­ 4.0 and
e ­ 0.33.
Q ­ I838 and R ­ I232. The steady homogeneous state
zp ­ 0.75 has three unstable and five stable directions and
we use two pinning sites to control it.
The contribution 2Kmixti from the site i far away from
the pinning site im is larger, as expected: since the feedback
is applied indirectly through coupling to the neighbors,
the perturbation introduced by the controllers decays with
increasing distance to the pinning sites.
Noise limits our ability to locally control arbitrarily large
systems with local interactions. We estimate the largest
length of the system using the controllability condition,
which in our case determines whether there exists control
ut , . . . , ut1L21, bringing an arbitrary initial state xt to an
arbitrary final state xt1L. Without noise
xt1L ­ Akxt 1
L21X
t­0
2X
m­1
AL212tbm ut1tm , (14)
where bm is the mth column of B.
However, if there is noise of relative magnitude s at
time t, it will be amplified by a factor of g per iteration,
where jgj ­ expslmaxd is the largest eigenvalue (8) of
the Jacobian. At time t 1 L its magnitude will be sjgjL.
For the control to work, all terms AL212tbiut1ti should
be of the same order as or exceed the amplified noise.
Indeed, the perturbation dxtim introduced by the con-
troller im affects the dynamics of the remote site j only
after propagating a distance D ­ jim 2 jj in time T ­ D,
decaying by a factor of ae per iteration. Consequently the
dynamics of site j at time t 1 L will be affected by control
applied only at times t, . . . , t 1 L 2 T 2 1. This dictates
Osdxt1L2T21im saed
T d * sjgjL.
Because of the periodic BC 0 # D # Ly2. On the
other hand, jdxti j , dx , 1, where dx determines the
range, where the linear approximation (4) is valid. As a
result we obtain the estimates on the size of the control-
lable system, similar to those obtained by Aranson et al.
for the lattice with asymmetric coupling [cf. Eq. (15) ofRef. [13] ]. The tightest bounds are given by D ­ 0:
Ls1dmax ­ 2
lnssd
lmax
, 0.5 , e , 1 , (15)
and D ­ Ly2:
Ls2dmax ­
2 lnssd
lnsed 2 lmax
, 0 , e , 0.5 . (16)
The maximal length of the system that can actually be
stabilized by the LQ method with two pinning sites, placed
next to each other, is obtained numerically by choosing
the fixed point as the initial condition and monitoring the
evolution of the system in the presence of noise under
control (12), calculated for Q ­ IL3L and R ­ I232. This
length is quite large for the moderate level of noise (Fig. 3)
and is rather close to the values (15) and (16) where the
controllability breaks down.
The problem of controlling a large one-dimensional
system with the length L . Lmaxssd can be easily reduced
to the problem of independently controlling a number
of smaller systems with the length Lp , Lmaxssd. We
partition the entire lattice hzt1, . . . , ztLj into M ­ LyLp
subdomains hztsm21dLp11, . . . , z
t
mLp j, and control it with an
array of pinning sites im1 ­ sm 2 1dLp 1 1, im2 ­ mLp ,
m ­ 1, . . . , M, positioned periodically at the boundaries of
these subdomains.
The stabilization can be achieved by choosing
utim1 ­ Fsz
t
im2 , z
t
im1 , z
t
im111d 2 Fsz
t
im121, z
t
im1 , z
t
im111d
1
LpY
i­1
usdxi 2 jxtsm21dLp1ijd
LpX
i­1
K1ix
t
sm21dLp1i
utim2 ­ Fsz
t
im221, z
t
im2 , z
t
im1d 2 Fsz
t
im221, z
t
im2 , z
t
im211d
(17)
1
LpY
i­1
usdxi 2 jxtsm21dLp1ijd
LpX
i­1
K2ix
t
sm21dLp1i ,
where usxd is a step function.
FIG. 3. The largest length of the lattice which can be
controlled with two pinning sites: Theoretical estimates (solid
lines) and numerical results (dots) obtained with the uniformly
distributed noise of amplitude s ­ 10214 as functions of
coupling e for a ­ 4.0.2797
VOLUME 79, NUMBER 15 P HY S I CA L REV I EW LE T T ER S 13 OCTOBER 1997This arrangement effectively carries two functions. We
use control (17) to (nonlinearly) decouple the subdomains,
simultaneously imposing periodic boundary condition for
each subdomain (the first two terms) to make the system
controllable. Then we stabilize each subdomain asynchro-
nously by applying a linear (in deviation xti ­ zti 2 zp)
feedback (the last term), inside the neighborhood of the
fixed point determined by dxi. Since the linear approxi-
mation (4) is valid only if
dxi ¿ jKmij21, m ­ 1, . . . , M , (18)
strong feedback significantly decreases the size of the cap-
ture region, which makes the capture time large. Reduc-
tion of the capture time can be achieved by minimizing
the feedback strength using the LQ method (12), (13).
We demonstrate this approach by stabilizing the homo-
geneous stationary state of the CML defined by Eqs. (1)
and (2) with a ­ 4.0, e ­ 0.33. L ­ 128 sites were di-
vided into M ­ 16 subdomains of length Lp ­ 8, each
controlled by two pinning sites. The results presented in
Fig. 4 show the evolution of the system from the initial
condition chosen to be a collection of random numbers in
the interval f0, 1g.
Equations (15) and (16) now give the minimal density
of pinning sites that yields the controllable fixed point
solution. It is indeed seen to be much lower than that
given by (9), e.g., 2yLp ­ 1y20 (2y37 from the numerics;
see Fig. 3) as opposed to 1yLp ­ 1y2 for the choice a ­
4.0, e ­ 0.4 and the precision of calculations given by
s ­ 10214.
Although the resulting control scheme becomes non-
linear (and therefore requires full knowledge of the evo-
lution equations), it has the additional benefit, that the
capture time is determined by the length Lp ¿ L and is
typically many orders of magnitude smaller than that ob-
FIG. 4. Stabilizing uniform steady state: A large lattice
(L ­ 128) is controlled by an array of double pinning sites,
placed at the boundaries of subdomains with length Lp ­ 8.
The state of the system was plotted at each 10 000th step.2798tained for the linear control scheme [obtained by lineariz-
ing (17)], which requires only the Jacobian to be known.
In fact our computational resources were insufficient to
observe even a single capture for L . 40 with the lin-
earized control. Generalizing this nonlinear approach to
higher-dimensional systems remains a challenge.
To summarize, we have shown that the restrictions on
the minimal density of periodically placed single pinning
sites obtained by Qu and Hu [8] as a result of numerical
simulations can in fact be obtained analytically from the
stabilizability condition.
The efficiency of the control scheme can be improved
significantly if one uses double pinnings instead of single
ones. The homogeneous steady state becomes controllable
for any values of the control parameters and the minimal
density of pinning sites is reduced substantially. It is
shown that the maximal distance between the pinnings
depends on the strength of noise in the system and can
be estimated analytically.
The appropriately chosen (using the LQ technique) feed-
back can decrease the capture time for the chaotic trajec-
tory by enlarging the capture region. The introduction of
nonlinearity into the control scheme can decrease this time
even more significantly by effectively decoupling the large
lattice into a number of smaller subdomains.
The authors thank J. C. Doyle for many fruitful discus-
sions and J. Socolar and D. Egolf for valuable comments
that led to improvements to the derivation of Eqs. (15)
and (16). This work was partially supported by the
NSF through Grant No. DMR-9013984. H. G. S. thanks
C. Koch for the kind hospitality extended to him at Cal-
tech and the Volkswagen Foundation for financial support.
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Pinning control of spatiotemporal chaos

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Pinning control of spatiotemporal chaos

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