Greatest Common Divisors via Generalized syl;ester and Bezout Matrices

We present new methods for comoutiw the ereatest common . " " right divisor of polynomial matrices. These methods imolve the recently shldied generalized Sylvester and eeneralized Bezoutian resultant matrices,~hich require no polynomial o&ations. They can provide a row proper -test common right divisor, test for coprimeness and calculate dual d&cal indices. The generalized resultant matrices are developments of the d a r Sylvester and Bemutian resultants and many of the familiar properties of these latter matrices are demonsbated to have analogs with the properties of the generaJized resultant makices for mabix polynomials. Greatest common divisors (gcd's) of polynomial matrices play an important part in the theory and application of general differential systems as studied extensively by Rosenbrock [I], 121, Wolovich [3], and others. For example, they are useful in 1) obtaining irreducible matrixfraction descri~tions (and hence minimal stale-mace realizations) of transfer-function matrices, 2) studying decoupling zeros and uncontrollable and unobservable modes of given systems, and 3) obtaining the . pole-zero structure of given multivariable systems. Most of the system-theory literature in this area has focused on the somewhat mare restricted problem of devising tests for the coprimeness of matrix polynomials-see, e.g., [4]-[Ill, or in obtaining irreducible MFD's by more direct methods-see, e.g., [I2]-[14]. These methods can in principle often also lead to a gcd, as we explain now. First note that 1151 a areoresf common r iahr dioisor' (gcrd) of two ~olynomial matrices . . . . . C(s) and D(s), having the same number of columns, is any polynomial matrix R(s) such that I) R(s) is a right divisor of (C(s),D(s)), i.e., Manvreripl received January 13, 1978: rsviard August 28. 1978. Paper rccommcndcd by M. Sain. Chairman of the Linear Systems Commitfcc. This work war supported by the U.S. Army Rerearch Office under Grant DAAG29-77.C-W42, by Lhc Australian Rssearch Oran. Cammittcs, Natlanal Science Foundation under the U.S.-Australian Cooperative Scicncc Program, by the Air Force Officc of Scientific Reresrch, Air Force Systems Command, under Contract AF44-62G74-C-M8, and by the National Science Foundation under Grant ENG75-10533. R. R. Bilmead and 8. D. 0. Anderson are with the Department of Electrical Engineer. ing, University of Ncwcantle. Ncw Sauth Wales, Australia. S:Y. Kung is with the Department of Electrical Enginscring, Uniucrsily of Sauthcrn California, Lor Angeles, CA 90WI. 7. Kailalh is with the lnformstion System Laboracoiy. S~anford University, S~anlord, C& 9d'40< . lSimilar delinilions and ruulls apply to grcatcrt ~ o m m o n left divisors (geld's), no that we shall canfine our direussions lo gcrd's. 0018-9286/78/1200-1043$00.75 01978 IEEE 1044 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-23, NO. 6, DECBMBBR 1978 for some polynomial matrices {C($),D(s)); and 2) it is divisible by any other right divisor, say R,(s), of {C(s),D(s)], i.e., R(s)=M(s)R,(s), for some polynomial matrix M(s). gcrd's are clearly not unique, since they can differ by unimodular factors, i.e., polynomial matrices with constant (nonzero) determinants. For nondegeneracy, we shall also require that the matrix F(s)= [D'(s)C'(s)l' has full rank (for almost all s) because otherwise we wuld have gcrd's of arbitrarily high degree. In system theory, this condition is often assured by having one of the matrices, say C(s), be square and nonsingular. For example, the pair [C(s),D(s)) often arises as a socalled right MFD (matrix-fraction description) of a matnix transfer function H(s),