Negative Imaginary Theory for a Class of Linear Time-Varying Systems

This letter introduces the notion of linear time-varying (LTV) negative imaginary systems. LTV negative imaginary systems are defined using a time-domain dissipative supply rate $w$ ( $u,\dot {y}$ ) that depends on input to the system ( $u$ ), time-derivative of the system's output ( $\dot {y}$ ) and an index $\delta \geq 0$. For $\delta > 0$ , it gives rise to a strict subclass within the LTV negative imaginary systems, termed as LTV output strictly negative imaginary systems. For characterizing the proposed class of systems, a set of linear differential matrix inequality conditions is derived based on the given state-space realization. Subsequently, LTV negative imaginary theory is specialized to linear parameter-varying (LPV) cases for which, the differential matrix inequality conditions can easily be avoided by considering the rate of variation of the uncertain parameters as independent LMI variables. Finally, a set of sufficient conditions is derived which ensures that the origin is a globally asymptotically stable equilibrium point of an unforced positive feedback interconnection of two uniformly asymptotically stable LTV negative imaginary systems.