Negative Imaginary State Feedback Equivalence for Systems of Relative Degree One and Relative Degree Two

Abstract

This paper presents necessary and sufficient conditions under which a linear system of relative degree either one or two is state feedback equivalent to a negative imaginary (NI) system. More precisely, we show for a class of linear time-invariant strictly proper systems, that such a system can be rendered minimal and negative imaginary using full state feedback if and only if it is controllable and weakly minimum phase. A strongly strict negative imaginary (SSNI) state feedback equivalence result is also provided. The NI state feedback equivalence result is then applied in a robust stabilization problem for an uncertain system with a strictly negative imaginary (SNI) uncertainty.

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

“…We need to find the state feedback matrices K 1 ∈ R p×m and K 2 ∈ R p×p such that the system (44) is SSNI. Lemma 17: [28] Suppose the system (44) has (A 00 , A 01 ) controllable. Then the following statements are equivalent:…”

“…In this paper, we consider the general case which allows the system to have mixed relative degrees. Also, the relative degree condition is an assumption in [28], while it is a part of the necessary and sufficient conditions in this paper. This makes the present paper a complete result for the NI state feedback equivalence problem.…”

“…The contribution of this paper is to provide conditions under which a system is state feedback equivalent to an NI system, an OSNI system or an SSNI system. This work, together with the preliminary conference paper [28], is the first in the literature where NI state feedback equivalence is investigated. In [28],…”

“…This work, together with the preliminary conference paper [28], is the first in the literature where NI state feedback equivalence is investigated. In [28],…”