State-Space Formulas For The Computation Of The Gap

We consider the computation of the gap between the graphs of rational transfer functions. The gap is related to the norm of a parallel projection, which can be computed as a regular indefinite linear quadratic optimal control problem with partially specified initial conditions. The use of coprime factorizations is avoided in this way, and Riccati equations are derived directly in terms of the state-space realizations. The computation needed to check whether the gap is smaller than some fixed number gamma is shown to be an eigenvector/eigenvalue problem for a hamiltonian matrix in which the state-space parameters of minimal realizations of the transfer functions are stacked.