Compensation for Distributed Hysteresis Operators in Active Structural Systems

A class of integral, hysteretic control in uence operators is derived for the representation of structural systems exhibiting hysteresis due to active materials. The hysteretic control in uence operator is deŽ ned in terms of a probability distribution, or more generally a measure, that describes the concentration of a particular hysteresis kernel. SpeciŽ cally, two types of hysteresis kernels are studied in detail: the ideal relay kernel leading to a Preisach integral hysteresis operator and a generalized play kernel leading to a Krasnosel’skii–Pokrovskii operator. When combined with the thermal and structural dynamics equations, the governing equations are history-dependent integropartial differential equations, coupled in a cascade structure. Existence, uniqueness, and convergence of Galerkin approximation methods have been derived for the forward simulation and model identiŽ cation. The cascade structure of the coupled, nonlinear partial differential equations is advantageous in that model reference control and model reference adaptive control strategies can be derived for the systems under consideration. Numerical and experimental results that validate the theoretical developments are presented.