A Negative Imaginary Lemma and the stability of interconnections of linear negative imaginary systems

The note is concerned with linear negative imaginary systems. First, a previously established Negative Imaginary Lemma is shown to remain true even if the system transfer function matrix has poles on the imaginary axis. This result is achieved by suitably extending the definition of negative imaginary transfer function matrices. Secondly, a necessary and sufficient condition is established for the internal stability of the positive feedback interconnections of negative imaginary systems. Meanwhile, some properties of linear negative imaginary systems are developed. Finally, an undamped flexible structure example is presented to illustrate the theory.