Guanrong Chen
University of Houston, USA, gchen@uh.edu
Jorge L. Moiola
Universidad del Sur, Argentina, comoiola@criba.edu.ar
Hua O. Wang
Duke University, USA, hua@ee.duke.edu
Abstract
Various bifurcations exist in nonlinear
dynamical systems such as complex circuits,
networks, and devices. Bifurcations
can be important and beneficial if they are
under appropriate control. Bifurcation
control and anti-control deal with modification
of system bifurcative characteristics
by a designed control input. Typical bifurcation
control and anti-control objectives
include delaying the onset of an inherent bifurcation,
stabilizing a bifurcated solution
or branch, changing the parameter value of an existing
bifurcation point, modifying the
shape or type of a bifurcation chain, introducing a
new bifurcation at a preferable
parameter value, monitoring the multiplicity, amplitude,
and/or frequency of some limit
cycles emerging from a bifurcation mechanism,
optimizing the system performance
near a bifurcation point, or a combination of some
of these. This article offers a
brief overview of this emerging and promising field of
research, putting the main subject
of bifurcations control and anti-control into perspective.
1. Introduction
Bifurcation control refers to the
task of designing a controller to suppress or reduce
some existing bifurcation dynamics
of a given nonlinear system, thereby achieving some
desirable dynamical behaviors.
Anti-control of bifurcations, as opposed to the direct
control, is to create some intended
bifurcations at some preferable time or parameter
values by means of various control
methods [4]. Typical bifurcation control and
anti-control objectives include
delaying the occurrence of an inherent bifurcation,
introducing a new bifurcation phenomenon
at a preferable time or parameter value,
changing the parameter set or values
of an existing bifurcation point, modifying the
shape or type of a bifurcation
chain, stabilizing a bifurcated solution or branch,
monitoring the multiplicity, amplitude
and/or frequency of some limit cycles emerging
from a bifurcation mechanism, optimizing
the system performance near a bifurcation
point, or a combination of some
of these objectives [2].
It is now known that bifurcation
properties of a system can be modified via various
feedback control methods. Representative
approaches employ linear or nonlinear
state-feedback controls, apply
a washout filter-aided dynamic feedback controller, use
harmonic balance approximations
in (time-delayed) feedback, and utilize quadratic
invariants in normal forms, etc.
Bifurcation control and anti-control
with various objectives have been implemented in
some experimental systems or tested
by using numerical simulations in a great number
of engineering, biological, and
physiochemical systems. Examples can be found in
electrical, mechanical, chemical,
and aeronautical engineering, as well as in biology,
physics, chemistry, and meteorology,
to name just a few. Bifurcation control not only
is important in its own right,
but also suggests a viable and effective way for chaos
control [1], because bifurcation
and chaos are usually ``twins'' and, in particular,
period-doubling bifurcation is
a typical route to chaos in many nonlinear dynamical
systems [4].
2. Bifurcations in Nonlinear Circuits and Systems
Even very simple nonlinear circuits
are rich sources of bifurcation phenomena.
Chua's circuit is a typical example.
In fact, there are many nonlinear circuits, need
not be very complex in structure,
that can display various bifurcation properties.
Coupled circuits, circuit arrays,
and circuit networks are more interesting but more
difficult to analyzed and applied.
In the study of bifurcations, circuits provide a
unique paradigm that is self-unified
and self-contained.
Bifurcations certainly exist almost
everywhere within the realm of nonlinear dynamical
systems, much beyond the territory
of circuitry. For instance, power systems generally
have rich bifurcation phenomena.
In particular, when the consumer demand for power
reaches its peaks, the dynamics
of an electric power network may move to its stability
margin, leading to oscillations
and bifurcations. This may quickly result in voltage
collapse. Such chaotic networks
include some cellular neural networks, laser networks,
and communication networks.
Mechanical systems provide another
playground for bifurcations. A road vehicle under
steering control can have Hopf
bifurcation when it loses stability, which may also develop
chaos and even hyperchaos. A hopping
robot, even a simple two-degree-of-freedom
flexible robot arm, can produce
unusual vibrations and undergo period-doubling
bifurcations which eventually lead
to chaos. An aircraft stalls during flight, either below
a critical speed or over a critical
angle-of-attack, can respond various bifurcations.
Dynamics of aeroengine compressors,
vehicals, ships, etc. can exhibit bifurcations
according to vibration or wave
frequencies that are close to the natural frequency of the
machine, creating oscillations,
bifurcations, and chaotic motions that may cause
catastrophe. Many chemical reaction
and fluid dynamic processes also have similar
behaviors, not to mention the weather
dynamics and biological population dynamics.
Bifurcations are ubiquitous in physical
systems, even subject to controls. It is now
known that various bifurcations
can occur in many nonlinear systems including, perhaps
unexpectedly, some closed-loop
systems under feedback or adaptive controls. This seems
to be counterintuitive; however,
local instability and complex dynamical behavior can
indeed result from such controlled
systems. Chances are, in these systems, one or more
poles of the closed-loop transfer
function of the linearized system may move to cross over
the stability boundary when feedback
or adaptation mechanisms are not robust enough,
potentially leading to signal divergence
as the control process continues. This sometimes
may not lead to a global unboundedness
in a complex nonlinear system, but rather, to
some self-excited oscillations,
bifurcations, and even chaos. Examples include the
popular automatic gain control
loops and various controlled or uncontrolled pendula.
Adaptive control systems, on the
other hand, are more likely to produce bifurcations due
to stability changes of some system
components such as the estimator and the adaptor.
3. Challenges from Bifurcation Control --- Two Examples
Controlling and anti-controlling
bifurcations have foreseen a tremendous impact on
real-world applications, and its
significance in both dynamics analysis and systems
control will not only be enormous,
but actually be both profound and far-reaching.
Before getting into more technical
details, it is illuminating to discuss some control
problems of two representative
examples --- the discrete-time logistic map and a
continuous-time model of an electric
power system --- to appreciate the challenge of
bifurcation control and anti-control.
3.1 The logistic map
The well-known logistic map is described by
x(k+1)=f(x_k,p):=p x(k) (1-x(k)) (1)
where p>0 is a real variable parameter.
This map has two equilibria, x^*=0 and
x^*=(p-1)/p. With 0<p<1,
the point x^*=0 is stable. However, it is interesting to
observe that, for 1<p<3,
all initial points of the map converge to x^*=(p-1)/p in the
limit. The dynamical evolution
of the system behavior, as p is gradually increased
from 3.0 to 4.0 by small steps,
is shown in Fig. 1. This figure (the bifurcation diagram)
shows that at p=3, a stable period-two
orbit is born out of $x^*$, which becomes
unstable at the moment, so that
in addition to $0$ there emerge two more stable
equilibria. When p increases to
the value of 3.44948... each of these two points
bifurcates into two new points,
as can be seen from the figure. These four points
together constitute a period-four
solution of the map (at p=3.44948...). As p moves
through a sequence of values: 3.54409...,
3.5644..., ..., an infinite series of bifurcations
is created by such "period-doubling",
which eventually leads to chaos:
period 1 -> period 2 -> period 4 -> ... -> period 2^k ... -> ... chaos
At this point, several control oriented
problems may be asked: Is it possible (and, if so,
how) to find a simple (say, linear)
control sequence, u_k, to be added to the right-hand
side of the logistic map, such
that
(i) the limiting chaotic behavior
of the period-doubling bifurcation process is suppressed?
(ii) the first bifurcation is delayed,
or this and the subsequent bifurcations are changed
either
in form or in stability?
(iii) the asymptotic behavior of
the system becomes chaotic (if chaos is beneficial), for a
parameter
value of p that is not in the chaotic region without control?
3.2 An electric power model
A simple yet representative electric
power system is shown in Fig. 2, where theta is the
rotational angle of the power generator.
In this power system, the load is represented by
an induction motor, M_I, in parallel
with a constant PQ (active-reactive) load. The variable
reactive power demand, p, at the
load bus is used as the primary system parameter. Also in
the power system, the load voltage
is V_L \angle(theta_L), with magnitude V_L and \angle
(theta_L), the slack bus has terminal
voltage E \angle(0) (a phasor), and the generator has
terminal voltage denoted E_m \angle(theta).
When the system parameter p is gradually
increased or decreased, with appropriate values
of the other system parameters,
very complex dynamical phenomena can be observed [5].
These are shown in Fig. 3, where
On the left-hand side:
p=10.818, a turning point of periodic orbit occurs;
p=10.873, first period-doubling bifurcation occurs;
p=10.882, second period-doubling bifurcation occurs;
p=10.946, a subcritical Hopf bifurcation occurs;
On the right-hand side:
p=11.410, a saddle-node bifurcation occurs;
p=11.407, a supercritical Hopf bifurcation occurs;
p=11.389, first period-doubling bifurcation occurs;
p=10.384, second period-doubling bifurcation occurs.
In this figure, (1) denotes stable
equilibria, (2) stable limit cycles, (3) and (4) different types
of unstable equilibria, and (5)
and (6) different types of unstable limit cycles. The dynamics
of this system, with varying a
second parameter (machine damping), have shown the
connection of the two Hopf bifurcation
points with a degenerate Hopf bifurcation and the
disappearance of the chaotic behavior.
Similar to the logistic map discussed above, a few interesting control problems are:
(i) can the limiting chaotic behavior of the period-doubling bifurcation process be suppressed?
(ii) can the first bifurcation be
delayed in occurrence, or this and the subsequent bifurcations
be changed
either in form or in stability?
(iii) can the voltage collapse be avoided or delayed through bifurcation or chaos control?
Nonconventional control problems
like these have posed a real challenge to dynamics analysts,
control engineers, and circuit
specialists.
4. Various Bifurcation Control Methods
As mentioned above, bifurcations
can be modified (controlled or anti-controlled) via various
feedback control methods. Representative
approaches employ linear or nonlinear state-feedback
controls, apply a washout filter-aided
dynamic feedback controller, use harmonic balance
approximations in (time-delayed)
feedback, and utilize quadratic invariants in normal forms, etc.
4.1 State feedback controls
State feedback can be used for determining
the one-dimensional transcritical, pitchfork, and
saddle-node types of bifurcations,
as well as the stabilities of the equilibria. The period-doubling
bifurcation can also be controlled
in a similar way.
Hopf bifurcation exists in higher-dimensional
(>1) systems, but can also be controlled by state
feedback. A Hopf bifurcation corresponds
to the situation where, as the parameter p is varied
to pass a critical value p_0, the
system Jacobian has one pair of complex conjugate eigenvalues
moving from the left-half plane
to the right, crossing the imaginary axis, while all the other
eigenvalues remain stable. At that
moment of crossing, the real parts of the two eigenvalues
become zero, and the stability
of the existing equilibrium changes from being stable to unsable.
Also, at the moment of crossing,
a limit cycle is born. These phenomena are completely
characterized by the classical
result of Hopf bifurcation theorem.
To design a controller for bifurcation
modification purpose, Taylor expansion, and sometimes
linearization, of the given nonlinear
dynamical system is a common approach. Since
bifurcations are closely related
by the eigenvalues of the linearized model, controlling the
behaviors of these eigenvalues
in an appropriate way is key to many bifurcation control
objectives.
4.2 Controllers designed based on normal forms
The general theory of bifurcations
in nonlinear dynamical systems is built on the basis of normal
forms. Systems with the same normal
form have equivalent bifurcations. Therefore, bifurcations
can be classified according to
equivalent systems in normal forms. Thus, development of a
systematic design technique for
bifurcation control requires a unified basis -- a set of normal
forms for control systems.
A set of normal forms is a family
of simple nonlinear control systems, such that many system in
a general form can be transformed
into a unique system in that family. For dynamical systems
without control, Poincare developed
a framework of normal forms for autonomous systems.
The normal form theory for control
systems differs from the theory of Poincare in the following
two aspects:
(i) In a dynamical system without
control, a single vector field is involved. However, there are
two vector fields (the nonlinear
system f and the nonlinear control gain g) in a controlled system
to be simplified simultaneously.
(ii) In the Poincare theory of normal
forms, the transformations used are changes of coordinates.
The transformation group for control
systems consists of both changes of coordinates and state
feedbacks.
Because of these two differences,
the study of bifurcations for control systems requires a set of
normal forms for both functions
f and g, under the transformation group consisting of changes
of coordinates as well as state
feedbacks. This poses some real challenges for further studies.
4.3 Controls via harmonic balance approximations
As is well known, limit cycles are
associated with bifurcations. In fact, one type of degenerate
(or singular) Hopf bifurcations
determines the appearance of multiple limit cycles under system
parameter variation. Therefore,
the birth and the amplitudes of multiple limit cycles can be
controlled by monitoring the corresponding
degenerate Hopf bifurcations. This task can be
accomplished in the frequency domain
setting [6].
For continuous-time systems, limit
cycles generally do not have analytic forms, and so have to
be approximated in applications.
For this purpose, the harmonic balance approximation technique
is very efficient. This technique
is useful in controlling bifurcations, such as for delaying and
stabilizing the onset of period-doubling
bifurcations.
As an example, consider again the
electric power model shown in Fig. 2. The amplitudes of the
system limit cycles can be controlled
(e.g., to zero) by using a state-feedback controller designed
based on this frequency domain
approach employing the harmonic balance approximation
technique.
5. Potential Applications of Bifurcation Control
Bifurcation control is useful in
many engineering applications. Due to the vast and still rapidly
growing array of information on
potential applications of bifurcation control in engineering systems,
it is literally impossible to give
an all-rounded and comprehensive coverage of these materials in
one single section of this article.
Therefore, only a few selected topics are presented here.
5.1 Application in power network control and stabilization
Nonlinearity is an inherent and
essential characteristic of electric power systems, especially in
heavily loaded operation. Historically,
power systems were designed and operated conservatively
and, as a result, systems were
normally operated within a region where system dynamical behaviors
were fairly linear. Only occasionally
would systems be forced to the limits where nonlinearities
could begin to have significant
impacts on the system behaviors.
Notably, the recent trend has different
promises. Economic and environmental factors, along with
the current trend towards an open
access market, have strongly demanded that power systems be
operated much closer to their limits
as they become more heavily loaded. Ultimately, there will be
greater dependence on control methods
that can enable the system capability rather than on
expensive physical system expansion.
It is therefore vital to gain greater understanding of the
nonlinear phenomena of an operational
power system.
In studying the electric power system
shown in Fig. 2, voltage collapse refers to an event in which
the voltage magnitudes in AC power
systems decline to some unacceptably low levels that can lead
to system blackout. The power system
model exhibits rich nonlinear phenomena, including
bifurcations and chaos.
One bifurcation control approach
to the problem of controlling voltage collapse in this power
system model is to add a control,
u, to the system, where the control occurs in the excitation
system and involves a purely
electrical controller [7,8]. Feedback signals, which are some
dynamic functions of the angular
velosity, are widely used in power system stabilizers (PSS).
A nonlinear bifurcation control
law of the form u = k_n (\dot theta)^3 transforms the subcritical
Hopf bifurcation to a supercritical
bifurcation. It also ensures a sufficient degree of stability of the
bifurcated periodic solutions,
so that chaos and crises are eliminated. This control law allows stable
operation very close to the point
of impending collapse (saddle node bifurcation). Figure 4 shows
a bifurcation diagram for the closed-loop
system with control gain k_n=0.5.
Another linear bifurcation control
law, u = k_l (\dot theta), involves changing the critical parameter
value, at which the Hopf bifurcations
occur, by a linear feedback control. This linear feedback law
eliminates the Hopf bifurcations
and the resulting chaos and crises [7,8]. Therefore, the linearly
controlled system can operate at
a stable equilibrium up to the saddle node bifurcation.
In summary, although the relative
importance of the effects of the nonlinear phenomena in general
power systems under stressed conditions
is still a topic for further research, the bifurcation control
approach appears to be a viable
technique for controlling these systems.
5.2 Applications in axial flow compressor and jet engine control
Another application of bifurcation
control is in the hearts of aeroengines: the axial flow compressors.
Recent years have witnessed a flurry
of research activities in axial flow compressor dynamics, both
in terms of analysis of stall phenomena
and their control. This interest is due to the increased
performance that is potentially
achievable in modern gas turbine jet engines by operating near the
maximum pressure rise. The increased
performance comes at the price of a significantly reduced
stability margin. Specifically,
axial flow compressors are subject to two distinct aerodynamic
instabilities, rotating stall and
surge, which are associated with bifurcations. Both of these instabilities
are disruption of the normal operating
condition that is designed for steady and axisymmetric flow,
and both can bring catastrophic
consequences to jet airplanes. Because these instabilities occur
at the critical operating point
of the highest pressure rise, the compressors are forced to operate at
a much lower pressure rise in order
to provide adequate stability margin which limit greatly
the performance of axial flow compressors.
Due to the design constraint, there
has been much work on enhancing compression system stability
using active control. Many of the
early control strategies were designed to extend the stable
axisymmetric operating range by
delaying the onset of stall. The application of bifurcation control to
compression system has initiated
a promising paradigm aiming at solving this challenging problem.
These bifurcation control approaches
look for controllers to enhance the operability of the
compression system by modifying
the nonlinear stability characteristics of the compression system.
Using the popular third-order Moore-Greitzer
model, it was found that the first stalled flow solution
is born through a subcritical bifurcation.
The practical importance of the subcritical stall
bifurcation is that when the axisymmetric
flow operating point becomes subject to perturbations,
the system will jump to a large-amplitude,
fully developed stall cell. Subcritical bifurcations
also imply hysteresis, and so returning
the throttle to its original position may not bring the system
out of stall.
One control strategy seeks to transform
the hard subcritical bifurcation at the onset of stall into a
soft supercritical bifurcation,
thereby eliminating the hysteresis associated with rotating stall. The
compressor stall application is
an excellent example for illustration (both theory and experimental
validation) of a guiding philosophy
in bifurcation control. It relates to stabilization, or ``softening,''
of bifurcations, with implications
to improving system performance and robustness. Other
approaches employed more conventional
control approaches such as backstepping technique to
arrive at control laws for surge
and rotating stall.
Some other approaches on bifurcation
control of compression systems involves output feedback,
under the assumption that the unstable
modes corresponding to the critical eigenvalue of the
linearized system are not linearly
controllable. Some stabilizability conditions can be derived for
the situation where the critical
mode is linearly observable through output measurement that
includes state-feedback as a special
case. It is shown that linear controllers are adequate for
stabilization of transcritical
bifurcation, and quadratic controllers are adequate for stabilization of
pitchfork and Hopf bifurcations,
respectively.
5.3 Application in cardiac alternans and rhythms control
One interesting application of bifurcation
control is the control of pathological heart rhythms. The
rhythm of the heart is determined
by a wave of electrical impulses (in the form of action potential),
which travels in the heart condition
pathway. Arrhythmias in the heart such as fibrillation and
ectopic foci are life threatening.
Understanding the mechanism leading to arrhythmias is an
important medical problem with
enormous impact. Within this context, an even more challenging
problem is the control and curing
of such abnormal biological disorders. For a control engineer,
a natural question is concerning
with the role of feedback in such situations. From a bifurcation
control point of view, what is
interesting about arrhythmias is that they have been closely linked
to a variety of bifurcations, both
static and dynamic, and chaos. This connection enables bifurcation
control methods to be used for
controlling heart rhythms.
As an application, dynamic bifurcation
control has been applied to suppression of pathological
rhythm (cardiac alternans) in an
atrioventricular modal conduction model. It has been shown that
this theoretical model, which incorporates
physiological concepts of recovery, facilitation and
fatigue, can accurately predicts
a variety of experimental observed complex rhythms of nodal
conduction.
5.4 Other examples of bifurcation control applications
A list of potential applications
of bifurcation control can be continued. In some physical systems,
such as the stressed system, delay
of bifurcations offers an opportunity to obtain stable operating
conditions for the machine beyond
the margin of operability at the normal situation. Sometimes,
it is desirable that the stability
of bifurcated limit cycles can be modified, with application to some
conventional control problems such
as thermal convection experiments. Other examples include
stabilization via bifurcation control
in tethered satellites and magnetic bearing systems; delay of
bifurcation in rotating chains
via external periodic forcing, and in various mechanical systems
such as robotics and electronic
systems such as laser machines and nonlinear circuits.
6. To Probe Further
When leaving the idealized mathematical
domain and looking around the natural world, one
certainly finds a very interesting
and realistic phenomenon --- there is almost nothing that is linear
but is not man-made out there,
is it? The nonlinear nature of the real world, and of the real life,
have brought up a great deal of
technological challenges to scientists and engineers --- the most
difficult yet also most exciting
complexities in dynamics, for which bifurcations, chaos, and
fractals alike all get to interplay
within a common ground of the mathematical as well as physical
wonderland.
The field of bifurcation control
is still very much in a rapidly evolving phase. This is the case not
only in deeper and wider theoretical
studies but also in many newly found real-world applications.
It calls for further efforts and
endeavors from the communities of engineering, physics, applied
mathematics, and biological as
well as medical sciences. New results and new publications on the
subject of bifurcation control
continue to appear, leaving a door widely open to every individual
who has the desire and courage
to pursue further in this stimulating and promising direction of
new research.
References
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[4] Chen, G. and Dong, X., From
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[5] Chiang, H. -D., Dobson, I.,
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[6] Moiola, J. L. and Chen, G.,
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\end
This Article appears in the IEEE
Circuits and Systems Society Newsletter, pp.1-10, June/July 1999
(the complete version with figures
is copyrighted to the IEEE)