Gap metric performance bounds for linear feedback systems

The difference between linear, possibly infinite-dimensional and time-varying, stable feedback systems, which involve identical controllers, is considered in terms of a performance/robustness related closed-loop mapping. Upper and lower bounds on the induced norm of the difference, in this sense, are derived in terms of the gap metric. To this end, the underlying geometry of the feedback stabilisation problem is exploited. The advantage of such an approach is that it does not rely on any specific system representation. With regard to extending the results to nonlinear systems, a simple example is presented to show that the upper bound does not hold in general when the controller (in common) is nonlinear.