Rational Nevanlinna-Pick interpolation with degree constraints

Researchers: Anders Lindquist, Anders Blomqvist , Vanna Fanizza, Yohei Kuroiwa and Johan Karlsson in cooperation with C. I. Byrnes (Washington University, St Louis), T. T. Georgiou (University of Minesota), and R. Nagamune (UC Berkeley).

Sponsors: The Swedish Research Council (VR) and the Göran Gustafsson Foundation.

Several important problems in circuit theory, robust stabilization and control, signal processing, and stochastic systems theory lead to a Nevanlinna-Pick interpolation problem, in which the interpolant must be a rational function of at most a prescribed degree. We have obtained a complete parameterization of all such solutions and developed algorithms for determining them. This work was initiated in a joint paper by Byrnes, Georgiou and Lindquist, for which they were awarded the 2003 George S. Axelby Outstanding Paper Award.

Among other things, these methods can be applied to feedback control design under constraints on the McMillan degree of the feedback system, and we have repeatedly been asked to explain the connection between this approach and the one using linear matrix inequalities, as they both pertain to degree constraints. In [A5][C11] we answer this question and explain the similarities and differences between the two approaches.

R. Nagamune and A. Blomqvist have developed an extensive program on numerical methods for solving the underlying nonlinear equations and optimization problems [T1][A1][A16] basically by homotopy continuation methods. Such methods are also used in [C5] to solve the covariance extension equation. New methods for simultaious covariance and cepstral matching are developed in [T1].

Work on automating the tuning and loop shaping in our methodology is on-going. A first attempt at doing this was made in [C22], where however a nonconvex optimization problem had to be solved.