Gain margin constrained H₂ and H_∞ optimal Positive Position Feedback control for piezoelectric vibration suppression

Positive Position Feedback (PPF) is a simple resonant control technique widely used to suppress vibrations. Several different tuning methods exist for this type of controller. However, the previously developed tuning methods often rely on simplifications and assume that the system can be reduced to a single mass spring system that exhibits roll-off after the resonance that is to be suppressed. This assumption is not generally valid as the response may also be influenced by a higher frequency resonance, or may not exhibit roll-off at all. This happens for example in systems with collocated piezoelectric sensing and actuation. Furthermore, robustness against gain variations and limited available actuation power are difficult to take into account with the available tuning methods. This paper presents the analytical solution to the H₂ optimal tuning problem and a reliable numerical solution to the H_∞ optimal tuning problem of PPF controllers for a general class of systems. The system model consists of a second order transfer with direct feed-through terms that are able to capture the response close to an undamped resonance accurately, even if it is influenced by higher frequency modes or if the system does not exhibit roll-off. The method leaves the open loop gain as a free tuning parameter, which can be set to adhere to a gain margin constraint or limit the utilised actuation power. Furthermore, experimental results are included to demonstrate the effectiveness, robustness and flexibility of the proposed optimal tuning approach.