Negative Imaginary Systems Theory for Nonlinear Systems: A Dissipativity Approach

Negative imaginary (NI) systems theory constitutes a well-established theoretical framework for analyzing and designing linear-time-invariant (LTI) control systems. This paper extends NI systems theory to the nonlinear domain using time-domain interpretations of the linear NI property. We formally define the nonlinear negative imaginary (NNI) property for general nonlinear systems using dissipativity notions with respect to a relevant work rate. A nonlinear negative imaginary system is shown to be dissipative/passive from the input to the time derivative of the output. Using Lyapunov stability theory, we derive a nonlinear generalization of the most general NI robust stability result. Specifically, we establish a robust stability result for the positive feedback interconnections of two NNI systems, serving as a counterpart to the passivity theorem for nonlinear systems. To demonstrate the applicability of this NNI stability result, we provide an example of a control design for a nonlinear mass-spring system, where the input is a force and the output is the mass displacement. Additionally, we extend our framework to include NNI systems with free motion and present sufficient conditions under which a cascade connection of an affine-input and an integrator nonlinear system is guaranteed to be NNI.