|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 one-dimensional 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 IO-behavior 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 IO-system such that
the output feedback system is well-posed and stable.
We characterize stability and stabilizability and construct all stabilizing
compensators of a stabilizable IO-system (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
one-dimensional case. In contrast to the existing literature the theorems
and proofs of this paper do not need or employ the so-called 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 one-dimensional systems (to the author).
An important mathematical tool, new in systems theory, is P. Gabrielís
localization theory which only in the case of ideal-convex (S. Shankar,
V.R. Sule) unstable regions coincides with the usual one. Algorithmic
tests for stability, stabilizability and ideal-convexity 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
|Keywords||stability, stabilization, multidimensional system, behavior, stable,transfer matrix|
University of Innsbruck|
Institut für Mathematik (Universität Innsbruck)|