March 18, 2002

Joe Ball

Title: Multivariable Nevanlinna-Pick interpolation and H-infinity control for multidimensional systems

Abstract:

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 ${\mathbb Z}^{d}$ ($d>1$) rather than ${\mathbb Z}$. As a consequence, the $H^{\infty}$-control problem for such a system becomes a Nevanlinna-Pick interpolation problem for Schur-class functions on the polydisk ${\mathbb D}^{d}$; interpolation nodes may be isolated points, or whole subvarieties of dimension $r$ less than $d$. The original result of Agler for interpolation on the polydisk leads to a sufficient (also necessary if $d=2$) 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 $d=1$ case. For the case of interpolation along a variety of dimension $r \ge 1$, 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).