  Control-based continuation (CBC) is a general and systematic method to
explore the dynamic response of a physical system and perform bifurcation
analysis directly during experimental tests. Although CBC has been successfully
demonstrated on a wide range of systems, rigorous and general approaches to
designing a noninvasive controller underpinning the methodology are still
lacking. In this paper, a noninvasive adaptive control strategy for a wide
class of nonlinear systems with unknown parameters is proposed. We prove that
the proposed adaptive control methodology is such that the states of the
dynamical system track a reference signal in a noninvasive manner if and only
if the reference is a response of the uncontrolled system to an excitation
force. Compared to the existing literature, the proposed method does not
require any a priori knowledge of some system parameters, does not require a
persistent excitation, and is not restricted to linearly-stable systems,
facilitating the application of CBC to a much larger class of systems than
before. Rigorous mathematical analyses are provided, and the proposed control
method is numerically demonstrated on a range of single- and
multi-degree-of-freedom nonlinear systems, including a Duffing oscillator with
multiple static equilibria. It is especifically shown that the unstable
periodic orbits of the uncontrolled systems can be stabilized and reached,
noninvasively, in controlled conditions.
