Reduced order approximation in the ν-gap metric

Most model order reduction techniques involve measures of approximation error that reflect differences in open-loop behaviour. Within the context of feedback compensator design, however, it is arguably more important to measure approximation error in terms of the difference in behaviour when in closed-loop. The gap metric and its variants are known to capture the difference between open-loop systems in terms of closed-loop behaviour. In this paper, we consider an order reduction problem in which the approximation error is quantified using the nu-gap metric. In particular, a characterisation of when a fixed-order model lies within a specified nu-gap distance of a nominal full-order model is obtained in terms of the feasibility of two LMIs and a rank constraint. A numerical example is presented to illustrate an application of the main ideas.