Feedback Stability of Negative Imaginary Systems

This paper extends the robust feedback stability theorem of negative imaginary systems by removing restrictive assumptions on the instantaneous gains of the systems that were imposed in the earlier literature, and it further generalizes this robust analysis result into the case that allows negative imaginary systems to have poles at the origin. In doing so, we extend the class of negative imaginary systems for which this robust stability theorem is applicable. We also show that this new generalized necessary and sufficient result specializes to the earlier theorems under the same assumptions. We additionally prove that the previously known dc gain condition is not only necessary and sufficient for robust feedback stability under the earlier specified instantaneous gain assumptions, but is also necessary and sufficient for robust feedback stability under new, different and equally simple assumptions. The general robust feedback stability theorem for negative imaginary systems with free body dynamics (i.e., poles at the origin) derived in this paper also specializes to the case that is only applicable for the negative imaginary system without poles at the origin. Since the results for negative imaginary systems with free body dynamics developed in this paper depend on the existence of a matrix Ψ with certain properties, we also propose a systematic construction of this matrix Ψ and show that construction of one such Ψ is sufficient for determining the feedback stability of the closed-loop system. Finally, examples are used to demonstrate the applicability of the results.