Finding Real Rational Weighting Functions for Robust Control Analysis and Design; Connections to Boundary Interpolation and Circuit Synthesis∗

We consider the problem of generating a matrix transfer function with prescribed strict bistability, biproperness, and positivity properties from a sequence of positive definite sample values at points on the frequency axis. We show how this arises in problems from multi-input/multioutput control analysis and design. This is a matrix interpolation problem with additional conditions, both on the nature of the data and the function class of the interpolant solution. The main result is to establish that it is always possible to find a solution to this problem, which has import for the validity of using sampled experimental data in determining non-conservative multi-input/multioutput stability margins via weighting functions. The full class of solutions in presented. The methods of the paper make strong contact with matrix Nevanlinna-Pick interpolation theory and with circuit theory, notably the methods of multi-port circuit synthesis.