March 18, 2002
Joe Ball
Title: Multivariable Nevanlinna-Pick interpolation and H-infinity control for multidimensional systems
We review recent results of Zhiping Lin (Multidimensional Systems and Signal Processing 9 (1998), 149-172) which, under certain circumstances, convert the internal stabilization problem for a discrete-time, linear, multidimensional system to model matching form. Here, ``multidimensional'' means that the ``time'' coordinate along which the recursive system equations are expressed is taken to be () rather than . As a consequence, the -control problem for such a system becomes a Nevanlinna-Pick interpolation problem for Schur-class functions on the polydisk ; interpolation nodes may be isolated points, or whole subvarieties of dimension less than . The original result of Agler for interpolation on the polydisk leads to a sufficient (also necessary if ) condition for a solution in terms of existence of a solution to a certain Linear Matrix Inequality (LMI) rather than positivity of a single Pick matrix as in the classical case. For the case of interpolation along a variety of dimension , we arrive at an infinite LMI; how best to solve such an infinite LMI is a topic which remains to be explored. Besides direct applications to multidimensional systems, there are connections with adaptive, robust control for single-variable systems as has been emphasized by Bill Helton (see IEEE Trans. Auto. Control AC-46 (2001), 2038-2093).