This project is joint with C. I. Byrnes, Washington University, St. Louis, and T. T. Georgiou, University of Minnesota. Several important applied problems in circuit theory, robust stabilisation 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 a complete parameterisation of all such solutions in terms of the zero structure of a certain function appearing naturally in several applications, and this parameterisation can be used as a design instrument. We have developed algorithms to determine any such solution by solving a convex optimisation problem, which is the dual of the problem to maximise a certain generalized entropy criterion. Software based on state space concepts and homotopy continuation have been developed, and the computational methods are applied to several problems in systems and control, including sensitivity shaping of the frequency response of a closed-loop system and robust regulation with robust stability.
We have introduced a Kullback-Leibler type distance between spectral density functions of stationary stochastic processes and solve the problem of optimal approximation of a given spectral density by one that is consistent with prescribed second-order statistics obtained from data produced by a bank of filters.
Studies have begun to apply our methods to the classical Youla's problem of optimal power transfer.
In collaboration with A. Megretski, MIT, we have applied the basic ideas of this project to generalized interpolation in H-infinity.
Interpolation, Optimization, Power systems optimisation, Robust control and stabilization, Stochastic systems, Partial stochastic realization, Spectral methods, Operator theory