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Control of chaos in nonlinear systems with time-periodic coefficients
S.C Sinha
()
Alexandra Dvid
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Control of chaos in nonlinear systems with
time-periodic coefficients
BY S. C. SINHA* AND ALEXANDRA DA VID
In this study, some techniques for the control of chaotic nonlinear systems with periodic
coefficients are presented. First, chaos is eliminated from a given range of the system
parameters by driving the system to a desired periodic orbit or to a fixed point using a
full-state feedback. One has to deal with the same mathematical problem in the event
when an autonomous system exhibiting chaos is desired to be driven to a periodic orbit.
This is achieved by employing either a linear or a nonlinear control technique. In the
linear method, a linear full-state feedback controller is designed by symbolic
computation. The nonlinear technique is based on the idea of feedback linearization.
A set of coordinate transformation is introduced, which leads to an equivalent linear
system that can be controlled by known methods. Our second idea is to delay the onset of
chaos beyond a given parameter range by a purely nonlinear control strategy that
employs local bifurcation analysis of time-periodic systems. In this method, nonlinear
properties of post-bifurcation dynamics, such as stability or rate of growth of a limit set,
are modified by a nonlinear state feedback control. The control strategies are illustrated
through examples. All methods are general in the sense that they can be applied to
systems with no restrictions on the size of the periodic terms.
1. Introduction
The phenomenon of chaos in nonlinear systems has been investigated extensively
in the last two decades. Recently, there has been a significant interest in
controlling chaotic systems. Among the control strategies used to suppress chaos,
perhaps the most well known is the OttGrebogiYorke method (Ott et al. 1990).
This method relies on the facts that chaotic systems are very sensitive to initial
conditions and that there is typically an infinite number of unstable periodic
orbits embedded in the chaotic attractor. One of these unstable orbits is
stabilized by controlling perturbations. The method uses the Poincare map of the
system. Some of its limitations are that the parameter changes can only be
discrete and only certain embedded orbits can be stabilized. Another control
One contribution of 15 to a Theme Issue Exploiting chaotic properties of dynamical systems for
their control.
method has been suggested by Pyragas (1992). In his approach, the stabilization
is achieved either by periodic external perturbation or by feedback control with
time delay. A detailed review of chaos control can be found in the work by
Fradkov & Evans (2002).
Most of the work, however, has dealt with autonomous systems. In the physics
literature, a non-autonomous system is often converted into an autonomous,
discrete system using a Poincare map in which the fixed points of the Poincare
map correspond to periodic orbits of the system. In this case, since it is not easy
to obtain an analytical expression for the Poincare map, one often needs to
obtain an approximate expression based on numerical methods or experimental
data. Local stabilization of periodic orbits in chaotic systems by feedback control
has been dealt with in previous works by several authors (Cheng & Dong
1993a,b; Chen 1996). However, in all these papers, the non-autonomous nature of
the problems was ignored and the RouthHurwitz criterion was applied as if the
systems were autonomous. Since the problem of stabilization of a periodic orbit
leads to a non-autonomous system with time-periodic coefficients, Floquet theory
must be used to guarantee stability of the system. Sinha et al. (2000) have
proposed a general approach in the design of active controllers for nonlinear
systems exhibiting chaos. In this method, it is shown that a system exhibiting
chaos can be driven to a desired periodic motion by combining a feed-forward
and a feedback controller. The feedback controller is designed using a method
proposed by Sinha & Joseph (1994), where a time-invariant auxiliary system is
constructed and stabilized with pole placement method. However, this method
uses a least-square approximation approach and in certain parameter ranges, the
stability of the periodic orbits may not be guaranteed.
In this paper, three approaches are suggested for local chaos control in
nonlinear systems with time-periodic coefficients. The first one is based on the
symbolic computation of the Floquet transition matrix (FTM) associated with
the linear part of the system. By using the technique introduced by Sinha &
Butcher (1997), the state transition matrix (STM) of the resulting system is
calculated symbolically. The symbolic expression of the STM contains the
unknown control gains and these can be assigned by placing the Floquet
multiplier (...truncated)