Control-Based Exploration of Bifurcation Diagrams of Dynamical Systems by Partial Reference Fourierization*

Control-based continuation (CBC) is a systematic method for exploring the dynamics of nonlinear systems and tracing bifurcation diagrams directly during experimental tests. To find a possible natural response of an underlying system under test, CBC iterates on a control reference signal (usually approximated using Fourier series) to make the controller noninvasive. The many Fourier modes required to accurately represent possible responses of the system lead to high testing times. The main contribution of this paper is to propose a control strategy that can asymptotically track the possible natural responses of a system in a noninvasive manner whilst using approximate reference signals described by short Fourier series. Our approach relies on a time-varying control strategy where noninvasiveness is guaranteed by design, and the mechanism of the controller compensates for the errors in the reference signals due to partial Fourierization. Accordingly, asymptotic tracking of the full Fourier series (i.e., the possible natural responses of the system) based on short Fourier series as reference signals is guaranteed. The proposed control strategy is validated by rigorous analysis and a simulation example.