Nonlinear Dynamics of Circular Dielectric Elastomer Membranes with Proportional-Integral-Derivative Feedback Control

Abstract Achieving large deformations in dielectric elastomers without dielectric breakdown remains a challenge that limits their technological implementation. This work analyzes the performance of proportional-integral-derivative (PID) feedback control for driving voltage-induced deformations in circular membrane actuators. The dynamic model includes hyperelastic material behavior, strain stiffening at large stretches, electro-elastic coupling, inertial nonlinearities, and a PID control law. When driven by open-loop voltages without feedback, the membrane has one equilibrium at low and high voltages. Three equilibria (corresponding to small, intermediate, and large deformations) are possible at moderate voltages. The use of PID feedback control effectively produces small-stretch equilibria at low and moderate voltages. PID control can generate large stretches at moderate applied voltages, although these large stretches are more difficult to control. Interestingly, the use of proportional control only (without integral and derivative gains) generally results in the membrane reaching intermediate stretches when large-stretch commands are given. These intermediate stretches, which are statically unstable, are stabilized by the controller. Precise tuning of the PID controller gains can produce large-stretch equilibria. Divergence and flutter instability occur for larger controller gains. For small-stretch commands, the through-thickness electric fields remain far below breakdown fields, even though time-dependent voltages cause dynamic overshoot in the membrane. Avoiding dielectric breakdown for large-stretch commands requires more careful tuning of the controller gains. PID feedback control may permit dielectric elastomers to achieve large deformations in soft transducer applications.