Necessary and Sufficient Conditions for State Feedback Equivalence to Negative Imaginary Systems

Abstract

In this paper, we present necessary and sufficient conditions under which a linear time-invariant (LTI) system is state feedback equivalent to a negative imaginary (NI) system. More precisely, we show that a minimal LTI strictly proper system can be rendered NI using full state feedback if and only if it can be output transformed into a system, which has relative degree less than or equal to two and is weakly minimum phase. We also considered the problems of state feedback equivalence to output strictly negative imaginary systems and strongly strict negative imaginary systems. Then we apply the NI state feedback equivalence result to robustly stabilize an uncertain system with strictly negative imaginary uncertainty. An example is provided to illustrate the proposed results, for the purpose of stabilizing an uncertain system.

Making Nonlinear Systems Negative Imaginary via State Feedback

“…In [31], conditions are given for linear time-invariant (LTI) systems with relative degree one and relative degree two to be rendered NI or strongly strictly NI (SSNI) using state feedback control. This result is then generalized in the paper [32], which gives necessary and sufficient conditions under which an LTI system is state feedback equivalent to an NI, OSNI or SSNI system. In [32], the system is allowed to have mixed relative degree one and two.…”

“…This result is then generalized in the paper [32], which gives necessary and sufficient conditions under which an LTI system is state feedback equivalent to an NI, OSNI or SSNI system. In [32], the system is allowed to have mixed relative degree one and two. Stabilization results are also provided in [31] and [32] for systems with SNI uncertainties.…”

“…In [32], the system is allowed to have mixed relative degree one and two. Stabilization results are also provided in [31] and [32] for systems with SNI uncertainties.…”

Negative Imaginary State Feedback Equivalence for a Class of Nonlinear Systems

“…In Shi et al (2021b), the NI state feedback equivalence problem is investigated for linear systems of relative degree one and relative degree two separately. This result is then generalized in the paper Shi et al (2021a), which considers the case that a system can have both relative degree one and relative degree two output entries. Shi et al (2021a) provides the necessary and sufficient conditions under which a linear system is state feedback equivalent to an NI, OSNI or strongly strictly negative imaginary (SSNI) system.…”

“…This result is then generalized in the paper Shi et al (2021a), which considers the case that a system can have both relative degree one and relative degree two output entries. Shi et al (2021a) provides the necessary and sufficient conditions under which a linear system is state feedback equivalent to an NI, OSNI or strongly strictly negative imaginary (SSNI) system. Such an NI state feedback equivalence approach can be used to asymptotically stabilize systems with SNI uncertainties.…”