Negative imaginariness indices in feedback stability analysis

We introduce the notions of negative imaginariness input and output indices (NI I/O-indices) for describing systems that do not manifest the negative imaginary property on certain frequency bands, thereby characterising classes of linear time-invariant systems that are larger than those of negative imaginary systems without poles at the origin. We show that if the total frequency-dependent negative imaginariness in a feedback interconnection is positive as measured by
the NI indices, whereby any deficiency in negative imaginariness in one open-loop component can be compensated for by a surplus of negative imaginariness in another, then the feedback interconnection is stable if and only if the static (a.k.a. DC) loop gain is less than unity. The result covers the feedback interconnection of a negative imaginary system and a strictly negative imaginary system, which naturally gives rise to positive total negative imaginariness. Importantly, we derive the NI indices based condition from a more general robust stability result established herein involving quadratic frequency dependent inequalities that may be used to characterise system properties beyond negative imaginariness. The proof relies on the multivariable Nyquist stability criterion and does not make use of state-space realisations of the underlying systems.