A Real Representation Of Complex Polynomial Matrices In The Framework Of Conjugate Product

Polynomial matrix equations play an important role in analysis and design ofcontrol systems. In addition, in some fields such as system identification andgeneralized predictive control, polynomial matrix equations are frequentlyencountered. Therefore, polynomial matrix equations have been widely researched bymany scholars.The concept of conjugate product for complex polynomial matrices wasproposed to solve a class of polynomial matrix equations. On the basis of conjugateproduct, we propose a newly defined operator called a real representation of complexpolynomial matrices. The real representation maps complex polynomial matrices toreal polynomial matrices, which simplifies conjugate product operations. Rank anddeterminant of matrices in the framework of conjugate product are obtained with thereal representation. Although conjugate product and left conjugate product areformally related, they are independent in some important properties. Thus the realrepresentation in the framework of left conjugate product is introduced, and therelationship between these two conjugate product is proved by means of the realrepresentation.Rational fraction matrices are frequently encountered when complex coefficientlinear systems are involved, thus the concept of rational fractions is established byextending polynomial matrices in the framework of conjugate product. Sum andconjugate product of rational fractions are proposed and further investigated. It isshown that the set of established rational fractions with the defined operations of sumand conjugate is a division ring. Rational fractions in the framework of conjugateproduct give a new way for the analysis and synthesis of complex control systems. Atlast, the real representation of rational fractions in the framework of conjugate productis presented. Some important properties of rational fractions such as similarity andconsimilarity are explored. The real representation simplifies operations of rationalfractions in the framework of conjugate product.