On α- and D-negative imaginary systems

Abstract

This paper is concerned with α- and $\mathcal{D}$-negative imaginary systems. The definition of α-negative imaginary transfer functions is firstly introduced. Then, the relationship between negative imaginary and α-negative imaginary transfer functions is studied. By means of the generalised inverse, an α-negative imaginary lemma is proposed to test the α-negative imaginary property of transfer functions. Also, a necessary and sufficient condition is provided for the α-stability of interconnection of negative imaginary systems. A state-feedback controller design condition is established such that the resulting system is α-negative imaginary. Moreover, the concept of α-negative imaginary transfer functions is extended to that of $\mathcal{D}$-negative imaginary transfer functions. Finally, the developed results are illustrated by numerical examples.

Keywords:

• linear systems
• negative imaginary properties
• stabilisation

Additional information

Funding

The work in this paper was financially supported by National Natural Science Foundation of China [grant number 61374026], Programme for New Century Excellent Talents in University 11-0880, Fundamental Research Funds for the Central Universities WK2100100013.