Ferroelectric and ferromagnetic materials are employed as both actuators and sensors in a wide variety of applications including fluid pumps, nanopositioning stages, sonar transducers, vibration control, ultrasonic sources, and high-speed milling. They are attractive because the resulting transducers are solid-state and often very compact. However, the coupling of field to mechanical deformation, which makes these materials effective transducers, also introduces hysteresis and time-dependent behavior that must be accommodated in device designs and models before the full potential of compounds can be realized. In this article, we present highly efficient modeling techniques to characterize hysteresis and constitutive nonlinearities in ferroelectric and ferromagnetic compounds and model inversion techniques which permit subsequent linear control designs.