Details:
Title  Stability and Stabilization of Multidimensional Input/Output Systems  Author(s)  Ulrich Oberst  Text  AMS subject classification: 93D15, 93D25, 93C20, 93C35  Type  Technical Report, Misc  Abstract  In this paper we discuss stability and stabilization of continuous and
discrete multidimensional input/output (IO) behaviors (of dimension
r) which are described by linear systems of complex partial differential
resp. difference equations with constant coefficients, the signals being
taken from various function spaces, in particular from those of polynomialexponential
functions. Stability is defined with respect to a disjoint decomposition
of the r−dimensional complex space into a stable and an
unstable region, the standard stable region in the onedimensional continuous
case being the set of complex numbers with negative real part.
A rational function is called stable if it has no poles in the unstable region.
An IObehavior is called stable if the characteristic variety of its
autonomous part has no points in the unstable region. This is equivalent
with the stability of its transfer matrix and an additional condition. The
system is called stabilizable if there is a compensator IOsystem such that
the output feedback system is wellposed and stable.
We characterize stability and stabilizability and construct all stabilizing
compensators of a stabilizable IOsystem (parametrization).
The theorems and proofs are new, but essentially inspired and influenced
by and related to the stabilization theorems concerning multidimensional
input/output maps as developed, for instance, by N.K. Bose, J.P. Guiver,
S. Shankar, V.R. Sule, Z. Lin, E. Zerz and A. Quadrat and, of course,
also to the seminal papers of M. Vidyasagar, D.C. Youla et al. in the
onedimensional case. In contrast to the existing literature the theorems
and proofs of this paper do not need or employ the socalled fractional
representation approach, ie. various matrix fraction descriptions of the
transfer matrix, thus avoid the often lengthy matrix computations and
seem to be of interest even for onedimensional systems (to the author).
An important mathematical tool, new in systems theory, is P. Gabriel’s
localization theory which only in the case of idealconvex (S. Shankar,
V.R. Sule) unstable regions coincides with the usual one. Algorithmic
tests for stability, stabilizability and idealconvexity and the algorithmic
construction of stabilizing compensators are addressed, but still encounter
many difficulties, in particular the open problems listed by L. Xu, Z. Lin
et al.  Keywords  stability, stabilization, multidimensional system, behavior, stable,transfer matrix  Length  41 
Language  English  Year  2005  Month  August  Translation 
No  Refereed 
No  Organization 
University of Innsbruck  Institution 
Institut für Mathematik (Universität Innsbruck) 
