The Fractional Representation Approach to Synthesis Problems: An Algebraic Analysis Viewpoint Part I: (Weakly) Doubly Coprime Factorizations

In this paper, we show how to reformulate the fractional representation approach to analysis and synthesis problems within an algebraic analysis framework. In terms of modules, we give necessary and sufficient conditions so that a system admits (weakly) left/right/doubly coprime factorizations. Moreover, we explicitly characterize the integral domains A such that every plant— defined by means of a transfer matrix whose entries belong to the quotient field of A—admits (weakly) doubly coprime factorizations. Finally, we show that this algebraic analysis approach allows us to recover, on the one hand, the approach developed in [M. C. Smith, IEEE Trans. Automat. Control , 34 (1989), pp. 1005–1007] and, on the other hand, the ones developed in [K. Mori and K. Abe, SIAM J. Control Optim., 39 (2001), pp. 1952–1973; V. R. Sule, SIAM J. Control Optim., 32 (1994), pp. 1675–1695 and 36 (1998), pp. 2194–2195; M. Vidyasagar, H. Schneider, and B. A. Francis, IEEE Trans. Automat. Control , 27 (1982), pp. 880–894; M. Vidyasagar, Control System Synthesis: A Factorization Approach, MIT Press, Cambridge, MA, 1985].