Implementation of a nonlinear controller in Phase-Locked Loop experiments for nonlinear structure identification

Experimental continuation methods are used to retrieve and identify nonlinear characteristics of vibrating structures. Among the available methods, Phase-Locked Loop (PLL) allows for an easy-to-implement yet efficient method to continue nonlinear solutions such as backbone curves or frequency response functions. The PLL automatically locks onto the prescribed phase and thanks to a linear (proportional-integral) controller, can stabilize unstable periodic orbits. However, the tuning of the different parameters to be used in such a loop are seldomly documented in the literature, which in turn might lead to long duration tests. To ease the tuning effort and reduce the experimenting time, a nonlinear controller is here proposed as a way to improve the efficacy of Phase-Locked Loop testing. Thanks to the proposed design, named NCPLL (Nonlinear Controller PLL), most of the parameters are tuned easily, while a rapid locking to the prescribed state is at hand. The nonlinear gain can be easily adapted to reach a locked state rapidly. The efficacy of the NCPLL is first demonstrated on simple numerical examples including nonlinear oscillators with smooth restoring forces and Coulomb friction, and a finite element beam model with localized nonlinearities. Then the method is deployed on two different experimental test rigs. First, the case of smooth nonlinearity is tackled thanks to a cantilever beam vibrating in the magnetic field created by two magnets. Finally, the case of friction is addressed by considering an assembled beam with friction joints. In all the tested cases, the NCPLL shows excellent performance, requiring minimal tuning efforts whilst leading to fast measurements.