Time-domain output negative imaginary systems and its connection to dynamic dissipativity

This paper introduces the notion of time-domain output (not necessarily strictly) negative imaginary systems that encompasses all the existing (not only stable) subsets of the negative imaginary systems class. A time-domain dynamic dissipative framework is developed in this paper to describe the full class of output negative imaginary systems (i.e. including systems having poles on the imaginary axis). This dynamic dissipative framework also leads to an LMI-based state-space characterization which can be conveniently used to test the output negative imaginary property of a given LTI system. It is further established that for time-domain output negative imaginary systems, the proposed dynamic dissipative framework is equivalent to the classical dissipative property of this class of systems with respect to the supply rate $w(u,\dot {\bar {y}})$. Interestingly, the class of time-domain output strictly negative imaginary systems now captures systems that are not necessarily stable – it is found that the majority of the SISO, LTI transfer functions with a simple pole at the origin satisfy this property. Numerical examples are provided throughout the paper for better illustration.