Robust performance analysis for uncertain negative-imaginary systems

Negative-imaginary systems are important in engineering practice as this class of systems appears quite often in practical problems, for example, lightly damped flexible structures with collocated position sensors and force actuators. In this paper, an analytical framework for robust performance of uncertain negative-imaginary systems is proposed. The results are obtained by transforming negative-imaginary systems into a bounded-real framework via the positive-real property. This paper deals with all the significant technical difficulties that appear due to the transformation and the punctured j ω-axis frequency condition of negative-imaginary systems. The problem is equivalently cast into a structured singular value condition that gives a quantitative performance test for this class of systems. This result also gives an analytical framework for robust stability when the perturbations are mixture of bounded-real and negative-imaginary uncertainties. A numerical example is presented to show the usefulness of the proposed methods.