These troublesome features are due to the presence of bifurcations. For the
example of Fig. 3.18, which is quite typical for applications, there are two
saddle-node bifurcations, one at each of the two curve points with vertical tangent.
Performing a frequency sweep corresponds to using sequential continuation
(Sect. 5.11.1). And as described in Sect. 5.11.1, and illustrated in Fig. 5.11(a), this
breaks down at turning points, and thus at saddle-node bifurcations. In purely
numerical analysis this can be overcome by using pseudo-arclength continuation
(Sect. 5.11.2 and Fig. 5.11(b)). But how should something similar be accomplished
in a lab experiment? And in particular – how should it be possible to continuate the
response curve at the unstable branches? Even if the system somehow could be
pushed onto a point on the unstable branch, by the definition of stability it would
only stay there very shortly, so there would be no chance of continuating to a
neighbor point.
Only rather recently has it become possible to tackle the abovementioned serious
obstacles for proper lab measurement of nonlinear frequency responses, so that these
can be observed in full, i.e. with the peak top included, without hysteresis, and
including even the unstable branches. The techniques go by names such as
control-based continuation (in experiments), experimental bifurcation analysis, and
experimental tracking of bifurcations. The actual implementation of these involves
an advanced mix of experimental, numerical, mathematical, statistical, and signal
processing techniques. Here we just summarize the main ideas, and otherwise refer
to the specialized literature, e.g., Sieber and Krauskopf (2008), Sieber et al. (2011),
Barton et al. (2012), Bureau et al. (2013, 2014), Barton (2017), Renson et al. (2019).
Basically control-based continuation, in lab experiments, works by proceeding
from one equilibrium state to a neighboring one by using the same kind of
tangent-predictor and corrector technique as with pseudo-arclength numerical
continuation (Sect. 5.11.2). That is, except for some important differences: There is
no mathematical model of the experimental system involved, so the gradient
information needed for the Jacobian has to be estimated from measured data. Also,
though at stable branches the correction step is not difficult (the system automatically ends up at the equilibrium state, if within the attraction zone), at unstable
branches automatic feedback control is needed. The condition for this control to be
effective is that the control force vanishes at the equilibrium (stable or unstable), i.e.
it should be non-invasive, when at the equilibrium.
With control-based continuation, since there is no mathematical model, there is
no model zero-problem such as (5.34). Instead an equivalent zero-problem is formulated based on the control force, specifying an equilibrium as a state where the
control force is zero, i.e. is non-invasive. A feedback control is then devised which
will change the system state so as to reduce the applied control force to zero; thus
the control force acts as the “reference signal”, as a proxy for the (unknown) system
equilibrium state. The control target for the system is then determined in terms of
the Fourier coefficients of the response that will reduce the control force to zero.
The control turns both stable and unstable equilibrium states into asymptotically
stable ones. The controller does not change the equilibrium solution itself, but only
its linearization, in a way so as to stabilize otherwise unstable orbits.
306
5 Bifurcation Analysis
example of Fig. 3.18, which is quite typical for applications, there are two
saddle-node bifurcations, one at each of the two curve points with vertical tangent.
Performing a frequency sweep corresponds to using sequential continuation
(Sect. 5.11.1). And as described in Sect. 5.11.1, and illustrated in Fig. 5.11(a), this
breaks down at turning points, and thus at saddle-node bifurcations. In purely
numerical analysis this can be overcome by using pseudo-arclength continuation
(Sect. 5.11.2 and Fig. 5.11(b)). But how should something similar be accomplished
in a lab experiment? And in particular – how should it be possible to continuate the
response curve at the unstable branches? Even if the system somehow could be
pushed onto a point on the unstable branch, by the definition of stability it would
only stay there very shortly, so there would be no chance of continuating to a
neighbor point.
Only rather recently has it become possible to tackle the abovementioned serious
obstacles for proper lab measurement of nonlinear frequency responses, so that these
can be observed in full, i.e. with the peak top included, without hysteresis, and
including even the unstable branches. The techniques go by names such as
control-based continuation (in experiments), experimental bifurcation analysis, and
experimental tracking of bifurcations. The actual implementation of these involves
an advanced mix of experimental, numerical, mathematical, statistical, and signal
processing techniques. Here we just summarize the main ideas, and otherwise refer
to the specialized literature, e.g., Sieber and Krauskopf (2008), Sieber et al. (2011),
Barton et al. (2012), Bureau et al. (2013, 2014), Barton (2017), Renson et al. (2019).
Basically control-based continuation, in lab experiments, works by proceeding
from one equilibrium state to a neighboring one by using the same kind of
tangent-predictor and corrector technique as with pseudo-arclength numerical
continuation (Sect. 5.11.2). That is, except for some important differences: There is
no mathematical model of the experimental system involved, so the gradient
information needed for the Jacobian has to be estimated from measured data. Also,
though at stable branches the correction step is not difficult (the system automatically ends up at the equilibrium state, if within the attraction zone), at unstable
branches automatic feedback control is needed. The condition for this control to be
effective is that the control force vanishes at the equilibrium (stable or unstable), i.e.
it should be non-invasive, when at the equilibrium.
With control-based continuation, since there is no mathematical model, there is
no model zero-problem such as (5.34). Instead an equivalent zero-problem is formulated based on the control force, specifying an equilibrium as a state where the
control force is zero, i.e. is non-invasive. A feedback control is then devised which
will change the system state so as to reduce the applied control force to zero; thus
the control force acts as the “reference signal”, as a proxy for the (unknown) system
equilibrium state. The control target for the system is then determined in terms of
the Fourier coefficients of the response that will reduce the control force to zero.
The control turns both stable and unstable equilibrium states into asymptotically
stable ones. The controller does not change the equilibrium solution itself, but only
its linearization, in a way so as to stabilize otherwise unstable orbits.
306
5 Bifurcation Analysis