The Vinnicombe Metric For Nonlinear Operators

Abstract

This paper describes an extension of the Vinnicombe metric on linear operators to a pseudometric on nonlinear operators. A metric for finite-dimensional time-varying operators is shown to be capable of guaranteeing stability and performance robustness and reduces to the standard Vinnicombe metric for the time-invariant operator case, which is known to be less conservative than the gap metric. The analysis exploits the time-varying operator equivalents of unstable poles and normalized coprime fractional descriptions. In addition, a time-varying operator equivalent of the winding number is defined.