Mathematical Sciences:Topis in Interpolation and System Theory

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This research will explore some basic new directions in systems theory. It has been known for some time that the standard problem of H-infinity control can be converted to a Nevanlinna- Pick interpolation problem. Recent work has illuminated a far-reaching analogue of point evaluation and a whole theory for associated analogues of Nevanlinna-Pick interpolation for time-varying systems. This investigation will pursue further elaboration of these ideas, in particular, the parametrization of a system in terms of zero and pole data for the time-variant case, and the connections of Nevanlinna-Pick interpolation for time-variant systems with Nevanlinna-Pick interpolation for time-invariant systems having infinite -dimensional input-output and state spaces. In addition, the PI will continue work on nonlinear systems and nonlinear H-infinity control. In particular, the construction of an inner-outer factorization for a nonlinear plant is a basic requirement in a number of important applications but is well-understood only in special situations. Finally, work on various interpolation problems involving meromorphic matrix functions on closed Riemann surfaces will be carried out. Zero-pole interpolation problems are closely related to classical Riemann-Hilbert problems and Wiener-Hopf factorization as well as to the classification of holomorphic vector bundles over the surface. In addition, such zero-pole interpolation ideas provide a new tool for completely understanding of work done in the 1970's on Nevanlinna-Pick interpolation for functions analytic on a finitely connected planar domain. A basic tool is the realization theory for meromorphic bundle maps on an algebraic curve with determinantal representation as the transfer function of a 2-dimensional linear system. A basic paradigm in control theory is what is now called H-infinity control. This approach provides a way to formulate control objectives mathematically in a way which guarantees reliable results when the mathematical model of the physical system under consideration is not a completely accurate representation of the true system. The discrepancies may be caused by uncertainty in certain physical parameters, or the effect of outside disturbances not taken into account in the mathematical model. One of the earliest solution procedures for the simplest H- infinity control problems was through the classical mathematical theory of Nevanlinna-Pick interpolation. Spurred by the needs of the emerging H-infinity theory, various researchers improved, extended and made this classical theory computationally practical. The PI has been instrumental in extending this approach to time-variant systems and nonlinear systems (where the basic physical law s no longer satisfy a superposition principle). Although it has now known how to unify time-variant theory conceptually with the time-invariant case, further work is needed to obtain a better understanding of decomposability and robust stability properties for eventually time-invariant linear systems and for nonlinear systems. Applications to which this research is expected to contribute include airplane flight control, improvements in automobile suspension systems and compact disk players, and chemical process control, to mention just a few. Research on interpolation problems for functions defined on topologically more complicated surfaces called Riemann surfaces will also be carried out. This work is expected be useful in applications to control systems involving transfer functions having algebraic singularities.