On α- and -negative imaginary systems

This paper is concerned with α- and -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 -negative imaginary transfer functions. Finally, the developed results are illustrated by numerical examples.

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