Model-free Continuation of Periodic Orbits in Certain Nonlinear Systems Using Continuous-Time Adaptive Control

TL;DR

This paper extends noninvasive adaptive control methods to continue periodic orbits in certain nonlinear systems, providing theoretical guarantees and practical validation for control-based continuation of both stable and unstable orbits.

Contribution

It introduces a generalized adaptive control approach for nonlinear systems with known structure, enabling continuation of periodic orbits with guaranteed performance bounds.

Findings

Guaranteed asymptotic convergence to periodic responses under persistent excitation

Successful continuation of stable and unstable periodic orbits using COCO software

Validation through numerical simulations confirming theoretical predictions

Abstract

This paper generalizes recent results by the authors on noninvasive model-reference adaptive control designs for control-based continuation of periodic orbits in periodically excited linear systems with matched uncertainties to a larger class of periodically excited nonlinear systems with matched uncertainties and known structure. A candidate adaptive feedback design is also proposed in the case of scalar problems with unmodeled nonlinearities. In the former case, rigorous analysis shows guaranteed performance bounds for the associated prediction and estimation errors. Together with an assumption of persistent excitation, there follows asymptotic convergence to periodic responses determined uniquely by an a priori unknown periodic reference input and independent of initial conditions, as required by the control-based continuation paradigm. In particular, when the reference input equals the sought periodic response, the steady-state control input vanishes. Identical conclusions follow for the case of scalar dynamics with unmodeled nonlinearities, albeit with slow rates of convergence. Numerical simulations validate the theoretical predictions for individual parameter values. Integration with the software package COCO demonstrate successful continuation along families of stable and unstable periodic orbits with a minimum of parameter tuning. The results expand the envelope of known noninvasive feedback strategies for use in experimental model validation and engineering design.