 
Summary: SIAM J. CONTROL AND OPTIMIZATION
Vol. 22, No. 5, September 1984
1984 Society for Industrial and Applied Mathematics
008
THE CANONICAL DIOPHANTINE EQUATIONS WITH APPLICATIONS*
W. A. WOLOVICH'f AND P. J. ANTSAKLISt
Abstract. A fundamental relationship between appropriate pairs of polynomial matrices is presented.
This relationship, termed canonical Diophantine equations, can be used to resolve a number of standard
polynomial matrix problems. Here, the general Diophantine equation is constructively resolved in a unique
minimal way; in addition, prime canonical factorizations of a system transfer matrix are derived from
knowledge of any dual factorization.
Key words, multivariable control systems, linear systems, algebraic system theory, polynomial matrix
algebra
1. Introduction. Polynomial matrices play an important role in many different
aspects of linear system theory, especially when one describes the dynamical behavior
of a given system in terms of either a right or left polynomial matrix factorization of
the transfer matrix which defines the system; i.e. T(s)=R(s)pl(s)=pl(s)Q(s).
Questions such as obtaining statespace realizations of T(s), or state observers associ
ated with T(s), or stabilizing compensators, which perform one or several simultaneous
control functions, have been constructively resolved through the manipulation of
