Imitating a Tuned Vibration Absorber With an Euler-Lagrange Controller: Comparing Different Stability Proofs

Vibrations in mechanical systems are often undesirable. They could lead to failure and/or disruption of proper operation of the system. Hence, mitigation of vibrations is indispensable. Two major classes can observed: passive and active vibration mitigation. The former does not rely on sensors and actuators and is for that reason assumed to be less complex, more reliable with regard to failure, and more intuitive to tune. The latter allows more design flexibility, can be adaptive to changes over time, and is more compact.
In the class of active vibration absorption, a feedback loop containing a control strategy is used to connect the sensors with the actuators. Many complex, mathematical controllers already exist that have proven to be very effective to decrease vibrations in a system. However, interpretability of the controller parameters can be lacking and, thus, impede an intuitive tuning strategy, because of its abstract nature and the rapid increasing number of controller parameters. The flexibility of active control allows to imitate a passive tuned vibration absorber with extra design freedom: all types of nonlinearities and interconnections can be created. This could lead to a vibration control strategy that combines the advantages of passive and active vibration mitigation.
The first step in designing an active nonlinear controller is to guarantee stability. Therefore, it is necessary to proof the ranges of the controller parameters to achieve an asymptotically stable system. Different methods to proof stability can be used and will lead to different limitations for the controller.
In this work, the direct method of Lyapunov is used to prove stability based on an extended Lure type Lyapunov function and a straightforward energy based Lyapunov function. Both stability proofs will be compared with respect to the interconnection system/EL-controller they yield and the corresponding controller design freedom they offer.