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Control based bifurcation analysis for experiments
Sieber, J.; Krauskopf, B.
published in
Nonlinear Dynamics
2008
DOI (link to publisher)
10.1007/s11071-007-9217-2
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Sieber, J., & Krauskopf, B. (2008). Control based bifurcation analysis for experiments. Nonlinear Dynamics,
51(3), 365-377. https://doi.org/10.1007/s11071-007-9217-2
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Nonlinear Dyn (2008) 51:365–377
DOI 10.1007/s11071-007-9217-2
ORIGINAL ARTICLE
Control based bifurcation analysis for experiments
Jan Sieber · Bernd Krauskopf
Received: 5 July 2006 / Accepted: 9 January 2007 / Published online: 6 February 2007
C
Springer Science +Business Media B.V. 2007
Abstract We introduce a method for tracking nonlin-
ear oscillations and their bifurcations in nonlinear dy-
namical systems. Our method does not require a math-
ematical model of the dynamical system nor the ability
to set its initial conditions. Instead it relies on feedback
stabilizability, which makes the approach applicable
in an experiment. This is demonstrated with a proof-
of-concept computer experiment of the classical au-
tonomous dry-friction oscillator, where we use a fixed
time step simulation and include noise to mimic exper-
imental limitations. For this system we track in one pa-
rameter a family of unstable nonlinear oscillations that
forms the boundary between the basins of attraction of
a stable equilibrium and a stable stick-slip oscillation.
Furthermore, we track in two parameters the curves of
Hopf bifurcation and grazing-sliding bifurcation that
form the boundary of the bistability region.
The research of J.S. was supported by EPSRC grant
GR/R72020/01, and that of B.K. by an EPSRC Advanced
Research Fellowship.
J. Sieber (
)
Department of Engineering, King’s College, University of
Aberdeen, Aberdeen AB24 3UE, UK
e-mail: j.sieber@abdn.ac.uk
B. Krauskopf
Bristol Centre for Applied Nonlinear Mathematics,
Department of Engineering Mathematics, Queen’s
Building, University of Bristol, Bristol BS8 1TR, UK
Keywords Bifurcation analysis
.
Numerical
continuation
.
Hybrid experiments
.
Hardware-in-the-loop
PACS 05.45-a
.
02.30.Oz
.
05.45.Gg
Mathematics Subject Classification (2000) 37M20
.
37G15
.
37M05
1 Introduction
The simplest and most frequently encountered type of
self-excited nonstationary behavior in nonlinear dy-
namical systems are periodic oscillations. A common
and typical scenario is a loss of stability of a stable
equilibrium in a Hopf bifurcation when a complex con-
jugate pair of eigenvalues of the linearization crosses
the imaginary axis under variation of a single param-
eter. At the critical parameter value a family of oscil-
lations is born. If the system is nonlinear then the os-
cillations are nonlinear as well, and, close to the Hopf
bifurcation, they are either stable (supercritical Hopf
bifurcation) or unstable (subcritical Hopf bifurcation).
This scenario is well understood theoretically and can
be found in standard textbooks such as [15]. If one
tracks the emerging family of periodic orbits (which
correspond to the nonlinear oscillations), other bifur-
cations may be encountered. If the dynamical system
is described mathematically in the form of a system
of differential equations then there are numerical con-
tinuation tools available [7, 15] that can track families
Springer
366 Nonlinear Dyn (2008) 51:365–377
of periodic orbits regardless of their dynamical stabil-
ity. What is more, these tools can also track curves
of bifurcations in two-dimensional parameter planes.
These curves form the boundaries of parameter re-
gions with qualitatively different long-time behavior of
the dynamical system (for example, regions of stable
steady states, stable nonlinear oscillations, or regions
with more complicated behavior such as quasiperiodic
or chaotic motions). Thus, numerical continuation is a
tremendously useful method if oneknows the equations
of motion and wants to understand the qualitative be-
havior of a nonlinear system and how it depends on the
system’s parameters. For example, references [7, 15]
discuss and illustrate this approach with a long list of
classical examples and provide an entry point into the
extensive literature.
By contrast, if the dynamical system is given in the
form of an experiment then the task of tracking periodic
oscillations and their stability boundaries is quite chal-
lenging. One approach is to run the experiment close
to its stability boundary for a sufficiently long time to
determine if the transients decay or grow. See [17] for a
successful and systematic application of this approach
in an experiment with electronic circuits. However, the
decay or growth of transients is typically very slow
close to the stability boundary. Moreover, the polyno-
mial effects caused by non-normality [24] interfere for
long periods of time if the exponential decay/growth
is weak. Finally, small perturbations are amplified due
to the lack of damping in the critical directions. Thus,
the approach of observing transients is time-consuming
and may produce results of low accuracy.
In this paper we present an alternative method for
the tracking of unstable periodic orbits and their bifur-
cations in experiments. The fundamental assumptions
for our method are, first, that the system is feedback sta-
bilizable (as explained in Section 4) and, second, that
the feedback control input and the system parameters
of interest can be varied automatically with a preci-
sion that corresponds to the accuracy of the desired
results. The core algorithm is a continuation routine
which provides an iterative computational method that
prescribes a sequence of control inputs and parameter
values. This sequence eventually converges toward a
noninvasive control input. The computations have to
be performed parallel to running the experiment, but
not in real time. Importantly, it is not necessary to set
initial values of the internal state variables of the ex-
periment explicitly (which would involve stopping and
reinitializing the experiment). Moreover, the dynami-
cal system always remains in a stable regime with a
closed stabilizing feedback loop, so that our method
does not require to run the experiment freely close to
its stability boundary.
The work presented here followson from [20] where
we introduced a control-based continuation scheme as
an embedding of extended time-delayed feedback con-
trol (see also Section 2) and demonstrated its use for
the continuation of periodic orbits of a dynamical sys-
tem where only the output of a simulation is available.
Here we extend this work to the continuation of pe-
riodic orbits and their bifurcations. Furthermore, we
use a projection onto Fourier modes to obtain a suf-
ficiently robust continuation method that can be used
even in the presence of limited precision and noise in
the experimental measurement.
The performance of our method is demonstrated in
a proof of concept with a classical mechanical system,
namely the dry-friction oscillator, which we run as a
computer experiment. To mimic the restrictions faced
by experimenters we disturb the numerical simulation
and its output by a small amount of noise and restrict
the output to a discrete time series of fixed sampling
step size. This allows an initial estimate of how well
the algorithm is able to cope with the limited preci-
sion that is characteristic for experiments. Specifically,
we continue a family of unstable nonlinear oscillations
in one parameter and curves of Hopf bifurcation and
grazing-sliding bifurcation in two parameters.
Our methodisideallysuitedforcomputer-controlled
experiments, especially for hybrid tests such as real-
time dynamic substructured testing of mechanical and
civil engineering systems [4, 16] (also called hardware-
in-the-loop testing). These tests couple an experimental
test specimen of a poorly understood or critical com-
ponent in real time (and bidirectionally) to a computer
simulation of the remainder of the structure. One of
the central goals of hybrid experiments is the track-
ing of stability boundaries. The automatic and precise
variation of parameters and the feedback stabilizabil-
ity, which our method requires, are particularly easy to
achieve in hybrid tests (since they are in part run as a
computer simulation).
The outline of the paper is as follows. We briefly
discuss existing approaches for finding unknown pe-
riodic orbits and bifurcations in dynamical systems
without model in Section 2. Section 3 introduces the
dry-friction oscillator, the example used throughout the
Springer
Nonlinear Dyn (2008) 51:365–377 367
paper. In Section 4 we give the details of the general
control-based continuation algorithm for periodic or-
bits, and show in Section 5 how this procedure works
for the family of unstable periodic orbits in the dry-
friction oscillator. In Section 6 we extend the core
continuation to enable it to track Hopf and grazing-
sliding bifurcations directly, and we demonstrate the
tracking of bifurcations in the dry-friction oscillator in
Section 7. We conclude and outline further work in
Section 8.
2 Background on related methods
One approach for finding periodic orbits and equilibria
in dynamical systems without knowledge of equations
of motion is stabilization with extended time-delayed
feedback control (ETDFC). This method aims to stabi-
lize unstable periodic orbits without prior knowledge of
their time profile [10, 18, 19, 25]. Its essential idea is to
feed back a difference between the current state and the
state from one or more periods ago, which necessitates
prior knowledge of the period of the orbit. The method
can also be applied to stabilizing unknown equilibria
[13], in which case adaptive (or washout) filters [1, 11]
are a simpler alternative. The main advantage of this
method is that its implementation is possible for anyex-
periment with feedback stabilizable periodic orbits and
equilibria. Furthermore, the method avoids running the
free dynamical system close to its stability boundary.
However, the question is still open under which con-
ditions the extended time-delayed or washout filtered
feedback converges [2, 26]. Feedback stabilizability of
the periodic orbit is necessary but not sufficient. The
approach in [20] is to embed ETDFC into a contin-
uation scheme, which allows for the continuation of
periodic orbits in more general, but still limited situa-
tions, for example, through fold (saddle-node) bifurca-
tions. As will become clear in Section 4, the method
described in this paper has been inspired directly by
the (continuation embedding of) ETDFC, but modifies
it substantially in order to make it more robust. This ad-
ditional robustness comes at the price of an increased
computational effort.
A second method, known as equation-free coarse-
grained (or time-stepper based) bifurcation analysis,
is well established in the context of microscopic sim-
ulations such as kinetic Monte Carlo simulations; see
[14] for a survey and references. This approach as-
sumes that a small number of macroscopic quantities
(such as the first few moments) already satisfy a closed
system of ordinary differential equations (ODEs). The
right-hand side of this ODE and its partial derivatives
are then evaluated by running appropriately initialized
short bursts of a microscopic simulator. This proce-
dure relies fundamentally on the ability to initialize
the microscopic simulator “at will,” and on the imple-
mentation of a “lifting operator” that maps the values
of the macroscopic quantities to initial values of the
ensembles of microscopic particles. If these two ingre-
dients are given then a number of high-level tasks can
be performed on the macroscopic level, including bi-
furcation analysis, optimization, controller design, and
control [21]. Coarse-grained bifurcation analysis has
been demonstrated successfully in the analysis of equi-
librium dynamics of multiparticle systems. Since the
microscopic simulator is treated as a “computer exper-
iment” the approach could, in principle, be applied to
real experiments. A potentially limiting factor in the
adoption of this time-stepper based approach to real
experiments is the impossibility (or impracticality) to
initialize the real experiment at will. In fact, initializing
the system is particularly difficult if one aims to find
nonequilibrium dynamics such as periodic orbits. In
the continuation approach of our method the previous
solution is used as seed in the quasi-Newton step, so
that the need for initializing the system does not arise.
Finally, both ETDFC and the method presented here
rely on the existence of a successful feedback control.
There exist standard techniques in control engineering
that aim to identify the right-hand side of the system
locally by probing systematically various control in-
puts. The resulting linear identified system can be used
subsequently to obtain bifurcation diagrams. This ap-
proach has been demonstrated in [5] for an implemen-
tation of the Colpitts oscillator as an electronic circuit.
There it is assumed that the internal state of the dy-
namical system can be measured. In general, system
identification is an inverse problem and, as such, inher-
ently ill-posed. The method presented here, as well as
ETDFC and coarse-grained bifurcation analysis, avoid
the need for solving an inverse problem.
3 The dry-friction oscillator with constant forcing
As our illustrating example we consider a dry-friction
oscillator with constant forcing as sketched in Fig. 1.
Springer
368 Nonlinear Dyn (2008) 51:365–377
M
K
C
v
x
Fig. 1 Setup of an idealized dry-friction oscillator
The friction between the running belt and the mass in-
duces a force on the mass, pushing it against the damped
spring that is fixed to the wall. The overall force on the
mass at position x is
−Kx − C
˙
x − F
f
(
˙
x −v) (1)
where x is measured with respect to the reference posi-
tion of the relaxed spring, K is the spring constant, C is
the damping, F
f
is the force exerted by the friction, and
v is the velocity of the running belt. Thus, the dynamics
of this nonlinear single-mass-spring-damper system is
governed by an equation of motion of the form
M
¨
x +C
˙
x + F
f
(
˙
x −v) + Kx = 0. (2)
It has been observed experimentally by [12] that the
dynamics of the dry-friction oscillator changes quali-
tatively when one varies the system parameter v. The
setup of [12] uses a mass fixed to a bending beam on
a large rotating disc to prevent disturbances by lateral
degrees of freedom; see also [22]. Figure 2 shows the
sequence of experimental measurements from [12] for
different values of the velocity v. The figure shows
that for large v the rest state (equilibrium) x
0
,given
by x
0
=−F
f
(−v)/K and
˙
x = 0, is stable. At a criti-
cal velocity v
h
≈ 0.188 the rest state x
0
loses its sta-
bility. If one decreases v gradually from above v
h
to
below v
h
one observes a sudden transition to large-
amplitude stick-slip oscillations. Similarly, when in-
creasing v in the stick-slip oscillation regime the sys-
tem jumps to a stable equilibrium at a critical velocity
v
d
>v
h
. Figure 2 clearly shows this bistability in the
parameter region [v
h
,v
d
] ≈ [0.188, 0.52].
If one makes assumptions about all parameters in (2)
the model can be analyzed numerically using standard
numerical methods such as AUTO [7] and S
LIDECONT
[6]. For example, a friction model that supports the
presence of the smooth linear instability at v
h
is
F
f
(w)=[F
c
+ (F
s
− F
c
) exp(−|w|) + F
v
|w|]sgn(w),
(3)
which gives the velocity-force curve shown in Fig. 3.
The argument w is positive in (3) for non-sticking
trajectories. The analysis of the numerical model (2)
completes the diagram in Fig. 2 and predicts the fol-
lowing [9]:
(H) the event of loss of stability of the equilibrium is
a subcritical Hopf bifurcation,
120
140
100
80
40
20
0
60
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
conjectured unstable
oscillations (UP)
measured stable stick-slip
(GS)
Amplitude [arbitrary units]
Pulling velocity [m/s]
(critical velocity = 0.188 m/s)
measured stable equilibria
(H)
Fig. 2 Diagram of the
experimental results from
[12] showing the velocity v
versus the amplitude of
oscillations. Dots show the
amplitude of measured
stick-slip oscillations (fitted
with a solid curve) and
squares refer to measured
equilibria. The dashed curve
connecting the transitions is
the conjectured family (UP)
of dynamically unstable
periodic orbits. (Reprinted
with kind permission from
G. St´ep´an; translations from
the Hungarian original
courtesy of G. Orosz)
Springer
![Fig. 2 Diagram of the experimental results from [12] showing the velocity v versus the amplitude of oscillations. Dots show the amplitude of measured stick-slip oscillations (fitted with a solid curve) and squares refer to measured equilibria. The dashed curve connecting the transitions is the conjectured family (UP) of dynamically unstable periodic orbits. (Reprinted with kind permission from G. Stépán; translations from the Hungarian original courtesy of G. Orosz)](/figures/fig-2-diagram-of-the-experimental-results-from-12-showing-3s0hlvg9.webp)