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 LQGbalanced 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 stepwise 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. NOTATION The symbols R and C denote the real and complex numbers, respectively. F denotes an m-row by q-column matrix with entries in F, which denotes either R or C. σ̄(X) and σ(X) respectively denote the maximum and minimum singular values of X ∈ F, while λi(X) represents the ith eigenvalue (in decreasing order of size) of X . The spectral radius, i.e. maximum eigenvalue, is denoted by rad(·). A superscript T denotes matrix transpose, whereas ∗ denotes complex conjugate transpose. The superscript symbol † indicates the Moore-Penrose inverse of a matrix, and ⊥ denotes the orthogonal complement of a matrix. Rm×q denotes the transfer functions P : C → C (a.e.), with a (state-space) realisation of the form P (s) = C(sI − A)B + D, for appropriate matrices A ∈ R , B ∈ R, C ∈ R and D ∈ R (i.e. proper real rational functions). The order of such a realisation, denoted (A,B,C,D) is said to be n and this is minimal if [ λI −A B ]