In this paper, we present necessary and sufficient conditions under which a linear time-invariant (LTI) system is state feedback equivalent to a negative imaginary (NI) system. More precisely, we show that a minimal LTI strictly proper system can be rendered NI using full state feedback if and only if it can be output transformed into a system, which has relative degree less than or equal to two and is weakly minimum phase. We also considered the problems of state feedback equivalence to output strictly negative imaginary systems and strongly strict negative imaginary systems. Then we apply the NI state feedback equivalence result to robustly stabilize an uncertain system with strictly negative imaginary uncertainty. An example is provided to illustrate the proposed results, for the purpose of stabilizing an uncertain system.

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This paper presents a framework to address the robust output feedback consensus problem for networked heterogeneous nonlinear Negative-Imaginary (NI) systems with free body dynamics. The aim of this paper is to complete and extend the results in previous papers on robust output feedback consensus for multiple heterogeneous nonlinear NI systems so that the systems in the network are allowed to have free body motion. A subclass of NI systems called Output Strictly Negative-Imaginary (OSNI) systems are applied as controllers to ensure that the outputs of the nonlinear NI plants converge to the same limit trajectory. The definitions of nonlinear NI systems and nonlinear OSNI systems are extended and a new stability result is developed for the interconnection of a single nonlinear NI system and a single nonlinear OSNI system. Robust output feedback consensus is addressed by establishing a similar stability result for the interconnection of networked NI systems and networked OSNI systems.

This paper provides a state feedback stabilization approach for nonlinear systems of relative degree less than or equal to two by rendering them nonlinear negative imaginary (NI) systems. Conditions are provided under which a nonlinear system can be made a nonlinear NI system or a nonlinear output strictly NI (OSNI) system. Roughly speaking, an affine nonlinear system which has a normal form with relative degree less than or equal to two after possible output transformation can be rendered nonlinear NI and nonlinear OSNI. In addition, if the internal dynamics of the normal form is input-to-state stable, then there exists a state feedback input that stabilizes the system. This stabilization result is then extended to achieve stability for system with a nonlinear NI uncertainty.

A robust output feedback consensus problem for networked identical nonlinear negative-imaginary (NI) systems is investigated in this paper. Output consensus is achieved by applying identical linear output strictly negative-imaginary (OSNI) controllers to all the nonlinear NI plants in positive feedback through the network topology. First, we extend the definition of nonlinear NI systems from single-input single-output (SISO) systems to multipleinput multiple-output (MIMO) systems and also extend the definition of OSNI systems to nonlinear scenarios. Asymptotic stability is proved for the closed-loop interconnection of a nonlinear NI system and a nonlinear OSNI system under reasonable assumptions. Then, an NI property and an OSNI-like property are proved for networked identical nonlinear NI systems and networked identical linear OSNI systems, respectively. Output feedback consensus is proved for a network of identical nonlinear NI plants by investigating the stability of its closed-loop interconnection with a network of linear OSNI controllers. This closed-loop interconnection is proposed as a protocol to deal with the output consensus problem for networked identical nonlinear NI systems and is robust against uncertainty in the individual system's model.