An upper bound to the gap between two unstable systems

In [1-3] the gap metric has been introduced for approximating unstable systems. It has been shown that "small" uncertainties in gap preserve closed loop stability and a prespecified bound on the norm of the sensitivity operator. Here, we deal with single-input single-output frequency responses, bounded on the jw-axis, and provide an upper bound to the gap between two such systems in terms of the gap between their domains (which can be determined from the unstable poles) and the distance between their jw-axis frequency responses. An illustrative example is given.