Model Reduction for Complex Adaptive Structures

Abstract

A complex adaptive structure for the purposes of this paper is treated as a structure with embedded piezoelectric sensors and actuators that are connected through a control loop so that the structure can adaptively or optimally respond to an external disturbance [1–4]. Such structures, also called smart structures, present special challenges in analytical and numerical modeling. The piezoelectric devices involve coupled electric and elastodynamic fields, the devices are multiple in number, the size of the structure is typically much larger in size relative to the sensors and actuators, and in real applications the geometry of the structure is complex. The response of such a structure to an external excitation and the response of the embedded sensors can only be simulated numerically. The finite element method has been successfully used to solve such problems, but the size of the resulting matrices becomes very large even for simple geometries [5–9]. To give an example, to obtain the transient response of a simple clamped plate with five pairs of sensors and actuators it is necessary to retain the first 50 structural modes and this results in an 800 × 800 matrix. In order to then interface such a numerical model with a control algorithm such as an H_∞ robust controller including modeling and device uncertainties, may challenge even the fastest computers to provide the required actuator excitations for real time controlled response of the adaptive structure. The objective of this paper is to address this issue and present a technique to condense or reduce the system model while still retaining the essential dynamical features of the smart structure [7].