A Negative Imaginary Approach to Modeling and Control of a Collocated Structure

A transfer-function is said to be negative imaginary if the corresponding frequency response function has a negative definite imaginary part (on the positively increasing imaginary axis). Negative imaginary transfer-functions can be stabilized using negative imaginary feedback controllers. Flexible structures with compatible collocated sensor/actuator pairs have transfer-functions that are negative imaginary. In this paper a model structure that typically represents a collocated structure is considered. An identification algorithm which enforces the negative imaginary constraint is proposed for estimating the model parameters. A feedback control technique, known as integral resonant control (IRC), is proposed for damping vibrations in collocated flexible structures. Conditions for the stability of the proposed controller are derived, and shown that the set of stabilizing IRCs is convex. Finally, a flexible beam with two pairs of collocated piezoelectric actuators/sensors is considered. The proposed identification scheme is used determining the transfer-function and an IRC is designed for damping the vibrations. The experimental results obtained are reported.