Onset and stabilization of delay-induced instabilities in piezoelectric digital vibration absorbers

Abstract

The stability of a piezoelectric structure controlled by a digital vibration absorber emulating a shunt circuit is investigated in this work. The formalism of feedback control theory is used to demonstrate that systems with a low electromechanical coupling are prone to delay-induced instabilities entailed by the sampling procedure of the digital unit. An explicit relation is derived between the effective electromechanical coupling factor and the maximum sampling period guaranteeing a stable controlled system. Since this sampling period may be impractically small, a simple modification procedure of the emulated admittance of the shunt circuit is proposed in order to counteract the effect of delays by anticipation. The theoretical developments are experimentally validated on a clamped-free piezoelectric beam.

Figures (15)

• Figure 1: General working principle of the DVA.
• Figure 2: Block diagram representation of the input/output relation in a digital system.
• Figure 3: Signals in the MCU when it operates as a simple unity gain: input signal (), output signal () and continuous average of the output signal ().
• Figure 4: Block diagram representation of the nominal \ref{['fig:feedbackShunt']} and delayed \ref{['fig:feedbackDelayedShunt']} controlled systems.
• Figure 5: Bode plot of the open-loop transfer function \ref{['sfig:openLoopTF']}: $K_c=0.01$ (), $K_c=0.05$ (), $K_c=0.1$ () and $K_c=0.2$ () ; phase margin as a function of $K_c$\ref{['sfig:phaseMargin']}.