  Control-based continuation is technique for tracking the solutions and
bifurcations of nonlinear experiments. The basic idea is to apply the method of
numerical continuation to a feedback-controlled physical experiment. Since in
an experiment it is not (generally) possible to set the state of the system
directly, the control target is used as a proxy for the state. The challenge
then becomes to determine a control target such that the control is
non-invasive, that is, it stabilises a steady-state (or periodic orbit) of the
original open-loop experiment without altering it otherwise.
  Once implemented, control-based continuation enables the systematic
investigation of the bifurcation structure of a physical system, much like if
it was numerical model. However, stability information (and hence bifurcation
detection and classification) is not readily available due to the presence of
stabilising feedback control. This paper uses methods from the system
identification community to extract stability information in the form of
Floquet multipliers from the closed-loop experiment, thus enabling the direct
detection of bifurcations. In particular, it is shown that a periodic
auto-regressive model with exogenous inputs (ARX) can be constructed that
approximates the time-varying linearisation of the experiment around a
particular periodic orbit. This method is demonstrated using a physical
nonlinear tuned mass damper.
