A step-wise procedure for reduced order approximation in the ν-gap metric

Recent results on reduced-order approximation in the ν-gap metric, characterised in terms of a non-convex feasibility problem, are investigated further. A detailed analysis of the properties of the constituent rank constrained linear matrix inequalities, when the nominal system has an LQG-balanced state-space realisation, reveals that it is possible to construct a feasible point directly for a particular choice of reduced order and ν-gap error. This gives rise to a step-wise procedure, based on constructing an optimal approximant at each step. While as of yet, the freedom in the parameterisation of optimal approximants has not been exploited, the new step-wise technique developed in this paper appears to perform well for numerical examples, yielding approximants between the upper and lower bounds for approximation in the ν-gap metric.