On the duality between filtering and Nevanlinna-Pick interpolation

Positive real rational functions play a central role in both deterministic and stochastic linear systems theory, as well as in circuit synthesis, spec tral analysis, and speech processing. For this reason, results about positi ve real transfer functions and their realizations typically have many appli cations and manifestations. In this paper, we study certain manifolds and submanifolds of positive real transfer functions, describing a fundamental geometric duality between fil tering and Nevanlinna Pick interpolation. Not surprisingly, then, this dual ity, while interesting in its own right, has several corollaries which prov ide solutions and insight into some very interesting and intensely research ed problems. One of these is the problem of parameterizing all rational sol utions of bounded degree of the Nevanlinna-Pick interpolation problem, whic h plays a central role in robust control, and for which the duality theorem yields a complete solution. In this paper, we shall describe the duality t heorem, which we motivate in terms of both the interpolation problem and a fast algorithm for Kalman filtering, viewed as a nonlinear dynamical system on the space of positive real transfer functions. We also outline a new proof of the recent solution to the rational Nevanlin na Pick interpolation problem, using an algebraic topological generalizatio n of Hadamard's global inverse function theorem.