A Theory of Nonlinear Negative Imaginary Systems

Abstract

In this thesis, we aim to generalize the negative imaginary systems theory to a broad class of nonlinear systems. A formal definition will be given for the negative imaginary property in the nonlinear domain by invoking a new dissipativity notion with an appropriate work rate. This formula is considerably more general than the existing classical dissipativity framework. Flexible structures with colocated force actuators and position sensors are dissipative according to this new definition. Having defined the nonlinear negative imaginary property in a time-domain dissipativity framework, we are able to extend some of the main existing results on negative imaginary systems from the linear to nonlinear domain. First, a Lyapunov-based approach will be used to establish the stability robustness of a positive feedback interconnection of negative imaginary systems in the linear case under a set of theoretical assumptions. Then, these assumptions will be adapted in the nonlinear setup to establish the stability robustness analysis of a positive feedback interconnection of nonlinear negative imaginary systems by making use of Lyapunov stability theory and dissipativity techniques. The applicability of this nonlinear stability result will be illustrated through an example of nonlinear mass spring damper system. Furthermore, the nonlinear negative imaginary systems theory will be extended to the case of free motion. It will be shown that, under suitable assumptions, a cascade connection of an affine nonlinear system and single integrator will lead to a nonlinear negative imaginary system (with integrator). Finally, this thesis is concluded by a summary of current progress and a discussion of possible future developments of the nonlinear negative imaginary systems theory.