Making Nonlinear Systems Negative Imaginary via State Feedback

Abstract

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.

Negative Imaginary State Feedback Equivalence for a Class of Nonlinear Systems

“…The system of interest in Theorem 1 is a particular normal form of a general system (1) in the case that the system (1) has relative degree less than or equal to two. Systems with such a relative degree condition have been investigated in Shi et al (2021a) and Shi et al (2022). The state-space model (4) contains a broader class of systems than that investigated in Shi et al (2021a) because Shi et al (2021a) only considers linear systems while in (4a), linearity is only required in the term A 11 z.…”

“…Considering the nonlinear nature of most control systems, Shi et al (2022) addressed the problem of making affine nonlinear systems nonlinear NI using state feedback control. In addition, for a system that can be made nonlinear NI, if its internal dynamics is input-to-state stable (ISS) (see Sontag (2008); Sontag et al (1989) for ISS systems), then there exists a state feedback control law that stabilizes the system.…”

“…In addition, for a system that can be made nonlinear NI, if its internal dynamics is input-to-state stable (ISS) (see Sontag (2008); Sontag et al (1989) for ISS systems), then there exists a state feedback control law that stabilizes the system. The reason that the ISS condition is needed is because the approach in Shi et al (2022) only makes the resulting system nonlinear NI with a positive semidefinite storage function.…”

“…This paper aims to complement the results in Shi et al (2022) by making a system nonlinear NI with a positive definite storage function. The motivation for guaranteeing the positive definiteness of the storage function of the resulting system is to make it easy to construct a positive definite closed-loop Lyapunov function for feedback controller synthesis (see for example Khalil (2002) for Lyapunov's direct method).…”

“…To avoid distracting the readers from the main results on state feedback equivalence, we introduce the system in question in its normal form. The readers are referred to Shi et al (2022) and references therein for the transformation from a general nonlinear system to its normal form.…”