Negative imaginary lemmas for improper descriptor systems

Abstract

This paper studies the negative imaginary property for improper descriptor systems. Firstly, it is demonstrated that a descriptor system with a negative imaginary transfer function cannot be impulse-free, that is, the negative imaginary transfer function is improper. Then some necessary conditions and sufficient conditions are established to characterize the negative imaginary property of improper descriptor systems based on the quasi-Weierstrass form. Furthermore, the lossless negative imaginary property of improper descriptor systems is derived by the relationship between the lossless positive realness and the lossless negative imaginariness. Several examples are given to illustrate the obtained results. In addition, this paper presents a generalized Kalman–Yakubovich–Popov lemma for the negative imaginary property of descriptor systems, which is achieved independently of the relationship between positive realness and negative imaginariness.