Negative Imaginary State Feedback Equivalence for a Class of Nonlinear Systems

TL;DR

This paper establishes conditions under which certain nonlinear systems are equivalent to nonlinear negative imaginary systems via state feedback, enabling stabilization and control design for systems with specific nonlinearities.

Contribution

It provides necessary and sufficient conditions for state feedback equivalence to nonlinear NI systems with positive definite storage functions, including stabilization methods.

Findings

Systems are equivalent to nonlinear NI systems if and only if they are weakly minimum phase.

The approach allows asymptotic stabilization against nonlinear OSNI uncertainties.

Numerical example demonstrates the control and stabilization process.

Abstract

In this paper, we investigate the necessary and sufficient conditions under which a class of nonlinear systems are state feedback equivalent to nonlinear negative imaginary (NI) systems with positive definite storage functions. The nonlinear systems of interest have a normal form of relative degree less than or equal to two. The nonlinearity of the system is restricted with respect to a subset of the state variables, which are the state variables that have external dynamics. Under mild assumptions, such systems are state feedback equivalent to nonlinear NI systems and nonlinear output strictly negative imaginary (OSNI) systems if and only if they are weakly minimum phase. Such a state feedback control approach can also asymptotically stabilize the systems in question against nonlinear OSNI system uncertainties. A numerical example is provided to show the process of the state feedback equivalence control and stabilization of uncertain systems.