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Fundamental important aspects of dissipative chaotic systems are the following:
- the exponential sensitivity of orbits to small perturbations
- the complex structure of the chaotic attractor containing an
infinite number of unstable periodic orbits (UPOs).
The utilization of these properties of chaotic systems can achieve special
advantages in controlling chaotic systems.
For instance, small perturbations can lead to large effects, and
flexible switching is possible between many different periodic orbits without
changing the global configuration of the system. Many feedback control
strategies [1] based on this general idea use small
perturbations in a control parameter to manipulate the behavior of chaotic
systems. These benefits cannot be achieved in non-chaotic systems in which
large effects in behavior typically require large changes in the control
parameter. In addition the control strategy that is discussed and applied
on an impact oscillator system will assume no knowledge of the underlying
model or equations that govern the dynamics and thus the control strategy
in conjunction with the learning algorithm is very robust and can be applied
to achieve control on an experimental data.
A brief outline of the
recursive proportional feedback (RPF) control algorithm developed by
Rollins et al. [2] and the learning algorithm developed by Rhode
et al. [3] which are used to control and track the nonlinear pendulum
impacting a sinusoidally moving wall. A detailed
discussion on the high dimensional RPF [4] control algorithm which is
used to control the linear oscillator impacting two symmetrically placed
stationary walls can be seen here.
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LOW DIMENSIONAL RPF CONTROL |
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The dynamical state of a system is often characterized by measuring the
value of a single variable as a function of time. It is often possible to
simplify the analysis of these continuous time systems by sampling a
measurable variable at discrete times chosen by a recurring event connected
with the dynamics of the system. For example, the values of the variable
being measured are obtained at the instant the trajectory intersects a
surface - called the Poincaré surface - in state space, or at the
period of an external drive, or by taking the successive maxima (or
minima) of the variable. Such a procedure allows the system to be completely
described by a discrete mapping - called the Poincaré mapping -
which preserves the characteristic features of the original continuous system.
A map based control strategy is a parametric control scheme that stabilizes
a chosen UPO, based on a linear approximation of the map in the neighborhood
of a desired periodic state. The control is achieved by adding feedback to
a nominal value of an appropriate system parameter at every cycle. In its
original form, first proposed by OGY [5], the feedback signal is a
linear function of the difference between the current state and the desired
state. Various modifications (for example, targeting the fixed point
directly rather than the stable manifold, and adding a recursive term made
necessary when reconstructing the state space using the delay coordinates)
to the original OGY scheme have been successfully applied to control various
systems, such as electronic circuits, lasers and
chemical reactions.
The RPF control strategy applies changes to the control parameter
determined by the control law given by
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dpi = K (xi - xfp) + R dpi-1, |
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where K and R are control constants. The control is applied
whenever the
system comes within a predetermined window containing the desired fixed point.
To implement the RPF control we must determine the control constants K and
R as well as the fixed point xfp, from the data.
To be able to track and control the system we use the learning algorithm developed
by Rhode et al. [3]. We assume the
position of the fixed point and
the system dynamics in its neighborhood are unknown. The adaptive control
mechanism waits for the system to make a few close returns, that is, measured
values of xi-1, xi, and xi+1 that are within linear
approximation of the fixed point. The data obtained from these close returns are
all that is needed to determine the control constants and the unstable fixed point
which is to be stabilized.
In the actual implementation of the learning algorithm the linearized dynamics
is expressed as
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dpi = wa xi+1 + wb xi + wc + wd dpi-1. |
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where the offset wc determines the unknown fixed point. The fixed point
is given by xfp = - wc/(wa + wb). To determine
the control constants the above equation is iterated once and then set
xi+2 = xfp
and dpi+1 = 0. By comparing the resultant equation with
Eq. 1, the control constants are given by,
The process of determination of the weights (wa, wb, wc, wd)
is performed in the following manner. After N = 10 close returns, Eq. 2
is set up as
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and using least square fit the constants
(wa, wb, wc, wd) are determined.
Using Eq. 3 the control constants are determined and using
Eq. 2 the control signal to be applied in
the next step. The learning process is continuous and weights the most recent
responses of the system most heavily so that the control scheme will be able
to adapt to drifts in the system dynamics that are slow compared to the
period of the controlled orbit.
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SIMULATION: CONTROL OF IMPACT OSCILLATORS |
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In practical applications of real impact system, the systems are modeled
and investigated to identify, and thus avoid, unacceptable responses. For
instance, high velocity impacts will cause the greatest wear or damage to
the system or where chaotic solutions exist, it is desirable to avoid the
irregular nature of the resulting motion. Using techniques of controlling
chaotic system it might be possible to select trajectories with a desirable
sequence of impacts. These unstable trajectories that are embedded within the
chaotic attractor could be stabilized using small perturbations to some
accessible system parameter. This method of stabilizing unstable periodic
orbits using small perturbations to system parameter would be very
advantageous in many technological applications of impact oscillators.
Both the RPF 2 and HRPF (High Dimensional Recursive
Proportional Feedback) control algorithm were applied
to control chaotic impacts of two
different impact oscillator systems described earlier.
We first show the successful
implementation of control algorithm for the model system described.
To simulate an experimental
situation, we consider only a single scalar signal y(t) available for
measurement and use the drive amplitude (A) as one of the accessible
system parameters for control. The time-delay coordinate vector x is
constructed from y(t) according to delay time embedding technique
where tn indicates the
time when the drive phase is some predetermined fixed value. We found that
an embedding [6] dimension of two was enough for the orbit shown in
figure below. We chose an orbit that completes one oscillation
per drive cycle and we get the value of the position (y) at two different
values of the phase within the same cycle and thus the embedding vector in
this particular case reduces to
where (fn1) and (fn2) indicates the nth time
y(t) goes through two different phase values. The delay time t
used in this case is given by (fn2-fn1)/w.
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STATIONARY WALL IMPACT SYSTEM |
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To be able to control a particular orbit of the system, the orbit must be
identified and the control constants need to be determined. It should be
noted that most of the periodic orbits that are stable for a given value
of the system parameters are destroyed at grazing impact as a result
of stable-unstable orbit collision. Consequently, these orbits cannot be
stabilized because they are not embedded within the attractor.
The periodic orbit that we chose to control is a P121 orbit and becomes
unstable as a result of period doubling bifurcation.
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A phase plane projection of the chaotic attractor (red) for
w = 1.9, b = 0.05,
K = 0.7 and A = 4.32 and the embedded unstable P121 orbit (green) that
is stabilized. The inset shows the return map at the Poincaré section using
the embedding variables. The return map shows the
unstable orbit (green), that is stabilized.
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The system is started at parameter values (w = 1.9,
b = 0.05, K = 0.7, A = 3.9) where the chosen orbit
is stable. Since the system will be always in the
neighborhood of this orbit, finding the control weights by estimating the
linearized dynamics can be done in a few iterations. The goal is to control the
chosen orbit when the parameter setting is such that the system exhibits chaotic
motion. This is done by slowly changing A to reach chaotic region while
adapting the control parameters to this change without losing control. The
learning algorithm simultaneously applies control signal and uses the system
response to re-estimate the system dynamics and determine the control weights.
The figure below shows the successful application of controlling the
P121 from A = 3.95 to A = 11.7 and P142 orbit from A = 13.8 to A = 16.0
by applying small perturbations to A. Continuous learning through updating of
the learning matrix ensures the adaptability of the control to the drift of the
system parameter A.
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Controlling and tracking the period 1 orbits of sinusoidally driven linear
oscillator with symmetrically placed walls from A = 3.9 to A = 16.0, using
both low and high dimensional RPF control law. A is used as the control
parameter. Position of the oscillator when drive phase goes through
p versus Drive Amplitude is plotted. Results of two runs, one
with control and learning on, and one with control off, are superimposed. The
orbit shown in 'red' is controlled from A = 3.9 to A = 11.7 and
a different period 1 orbit (shown in 'green') is controlled from
A = 13.8 to A = 16.0.
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To confirm the estimation of the system dynamics by the control algorithm,
we independently perform a stability analysis of the two controlled orbits.
Using the constant phase space Poincaré section, we numerically calculate
the Jacobian around the P121 fixed point by taking differences for
neighboring points. The fixed point is found using the multivariable
Newton-Raphson method. By drifting the system through the parameter space
by small amounts the unstable orbit can be tracked and the eigenvalues
calculated. The result of this method is shown in the figure below,
where the modulus of the eigenvalues at the Poincaré section when the
drive phase is p modulo 2p
for both the controlled orbits are plotted
as drive amplitude is varied from A = 3.9 to A = 16.0.
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Modulus of the eigenvalues at the Poincare section when the drive phase is
p modulo 2p for both the
controlled orbits are plotted as drive amplitude is varied from A = 3.9 to A = 16.0.
The continuous line plot is for data obtained from the follow orbit algorithm and
the data obtained from the control algorithm is plotted using
° symbol.
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It is clear from the figure above that the P121 orbit is becoming
increasingly unstable as A is increased. At A = 11.7, when the control
is lost, the modulus of the eigenvalue is around 45 in the
unstable direction. This sharp increase in the modulus of the eigenvalue
is because the unstable P121 orbit is about to go through grazing impact
with the wall. This is shown in the phase plane projection of the
orbit at A = 12.5 in the figure below.
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The plot shows the phase portrait of the asymmetric unstable orbit at A = 12.5
when the orbit is about to graze with the right wall.
The modulus of the eigenvalue in the unstable direction is around 60.
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MOVING WALL IMPACT SYSTEM |
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Using the low-dimensional RPF control algorithm we stabilize and track a
period-1 orbit by applying small changes in the drive frequency w.
The control/learning algorithm was initiated at a parameter value
w = 7.8 where the period-1 orbit is stable. While
drifting the drive frequency into regions where the period-1 orbit becomes
unstable and the uncontrolled system exhibits chaotic behavior, the control
succeeds in stabilizing the period-one state and tracking it. At
w = 9.0,
the stabilized period-one orbit suddenly disappears as an attractor. At this
point, the attractor collapses to a high period orbit, which is stable.
By further increasing the drive frequency the system again evolves into
chaos. We were able to restart the control algorithm, where the second
period-1 orbit appears, and stabilize and track this orbit for increasing
drive frequency (see figure below).
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Controlling and tracking the period 1 orbits of nonlinear pendulum
impacting a sinusoidally moving wall from w = 7.9 to
w = 12.7 rad/s. Time between impacts versus frequency is
plotted. Results of two runs, one with control and learning on, and one with control
off, are superimposed. The orbit shown in 'red' is controlled from
w = 7.9 to w = 9.0 rad/s and
a different period 1 orbit (shown in 'green') is controlled from
w = 9.9 to w = 12.7 rad/s.
The adaptive RPF control method is used to control and track the period 1 orbits
using the drive frequency as the control parameter.
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We have shown that the chaotic dynamics in impact oscillator
can be stabilized by using only small perturbations , to motion on a desired
periodic orbit with a given sequence of impacts. More specifically, it was
possible to control the model impact systems by applying small perturbations
to the parameters related to the external drive. These results could be of
interest in the technological application of impact systems where the
chaotic response of the system is undesirable.
- [1]
-
T. Shinbrot, Advances in Physics 44, 73 (1995);
T. Shinbrot, C. Grebogi, E. Ott, J. A. Yorke, Nature 363,
411 (1993); G. Chen and X. Dong, Int. J. Bifurcations Chaos
3, 1363 (1993).
- [2]
- R. W. Rollins, P. Parmananda,
and P. Sherard, Phys. Rev. E 47, R780 (1993).
- [3]
- M. A. Rhode, R. W. Rollins, and C. A. Vassiliadis,
Proceedings of the 26th Southeastern Symposium on System
Theory, (The Institute of Electrical Engineers, Los Alamitos,
CA 90720-1264, 1994), pg. 638.
- [4]
-
M. A. Rhode, J. Thomas and R. W. Rollins,
Phys. Rev. E 54, 4880 (1996).
- [5]
- E. Ott, C. Grebogi, and J. A. Yorke,
Phys. Rev. Lett. 64, 1196 (1990).
- [6]
- F. Takens,
Detecting strange attractors in turbulence.
Dynamical Systems and Turbulence, ed. Rand,
D. A. & Young, L.-S., New York: Springer-Verlag.
Lecture Notes in Math. 898. pp. 366-81 (1981).
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John von Neumann
