Abstract
Control-based continuation is technique for tracking the solutions and bifurcations of nonlinear experiments. The idea is to apply the method of numerical continuation to a feedback-controlled physical experiment such that the control becomes non-invasive. Since in an experiment it is not (generally) possible to set the state of the system directly, the control target becomes a proxy for the state. 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 a periodic auto-regressive model with exogenous inputs (ARX) to approximate the time-varying linearisation of the experiment around a particular periodic orbit, thus providing the missing stability information. This method is demonstrated using a physical nonlinear tuned mass damper.
Highlights
Control-based continuation is a systematic method for performing bifurcation studies on physical experiments
Based on modern feedback control schemes it enables dynamical phenomena to be detected and tracked as system parameters are varied in a similar manner to how nonlinear numerical models can be investigated using numerical continuation
Control-based continuation goes beyond these particular methods to allow the use of almost any feedback control scheme and, as such, it is a general purpose tool applicable to a wide range of physical experiments
Summary
Control-based continuation is a systematic method for performing bifurcation studies on physical experiments. Based on modern feedback control schemes it enables dynamical phenomena to be detected and tracked as system parameters are varied in a similar manner to how nonlinear numerical models can be investigated using numerical continuation. Control-based continuation goes beyond these particular methods to allow the use of almost any feedback control scheme and, as such, it is a general purpose tool applicable to a wide range of physical experiments. Periodic orbits have been tracked through instabilities such as saddle-node bifurcations (folds) . Saddle-node bifurcations (folds) can be detected readily and that is because they are geometric features in the solution surface. Bifurcations such as period-doubling bifurcations are not geometric features and can go undetected due to the stabilising affect of the feedback controller.
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