Modelling and stabilizability of piezoelectric materials

This research explores the modelling and stabilizability of piezoelectric materials, such as beams, actuators, and composites, focusing on their applications in high-precision positioning and shape control. A piezoelectric actuator consists of a piezoelectric layer between two electrodes, which responds to electric input signals by changing shape. Interconnecting a piezoelectric actuator with a purely mechanical layer results in a piezoelectric composite that can be shaped. The study contrasts different electromagnetic assumptions—dynamic, quasi-static, and static—and examines how each affects the stability and performance of piezoelectric composites.

Novel voltage-controlled and current-controlled composite models are developed, with the current-controlled models leveraging a new "combined Lagrangian" method that integrates mechanical and electromagnetic dynamics through so-called traditors, which couple force and flow balance equations in a non-energetic manner. We show that voltage-controlled piezoelectric composites are stabilisable under certain system parameters, and for current-controlled piezoelectric composites, this holds only under the fully dynamic electromagnetic assumption. Finally, a novel Passivity-Based Control (PBC) design is introduced, from which we derive two control methodologies –output shaping and input shaping- using a Krasovskii Lyapunov function as the storage function for a piezoelectric beam and applicable to a large class of (electro-)mechanical systems. These advancements provide a foundation for more efficient, stable control of piezoelectric-based systems in engineering.