Adaptive control designs for control-based continuation of periodic orbits in a class of uncertain linear systems

This paper proposes two novel adaptive control designs for the feedback signals used in the control-based continuation paradigm to track families of periodic orbits of periodically excited dynamical systems, including black box simulation models and physical experiments. The proposed control designs rely on modifications to the classical model reference adaptive control framework and the more recent $${\mathscr {L}}_1$$ adaptive control architecture, in which an additional low-pass filter is used to ensure guaranteed transient performance and robustness to time delays in the control input even in the limit of arbitrarily large adaptive gains. In contrast to the proportional control formulations that have been used in the literature on control-based continuation, the proposed control designs achieve stable performance with a minimum of parameter tuning. In the context of a class of linear systems with matched uncertainties, the paper demonstrates the successful integration of adaptive control feedback in control-based continuation. Specifically, the control designs are shown to ensure that the control input stabilizes the sought periodic orbits of the uncontrolled system and vanishes along these orbits, provided that an a priori unknown reference input is chosen appropriately. Numerical results obtained using the coco software package demonstrate how the combination of a nonlinear solver (Newton’s method) with the pseudo-arclength parameter continuation scheme can be used to trace the correct choice for the reference input under variations in an excitation parameter.