  Negative imaginary (NI) systems theory is a well-established system theoretic
framework for analysis and design of linear-time-invariant (LTI) control
systems. In this paper, we aim to generalize negative imaginary systems theory
to a class of nonlinear systems. Based on the time domain interpretation of the
NI property for LTI systems, a formal definition in terms of a dissipativity
with an appropriate work rate will be used to define the nonlinear negative
imaginary (NNI) property for a general nonlinear system. Mechanical systems
with force actuators and position sensors are nonlinear negative imaginary
according to this new definition. Using Lyapunov stability theory, we seek to
establish a nonlinear generalization of the NI robust stability result for
positive feedback interconnections of NNI systems. An example of a nonlinear
mass-spring-damper system with force as input and displacement of the mass as
output will be presented to illustrate the applicability of the NNI stability
result. Furthermore, the case of NI systems with free motion will be
investigated in the nonlinear domain based on the dissipativity framework of
NNI systems.
