The Fractional Representation Approach to Synthesis Problems: An Algebraic Analysis Viewpoint Part II: Internal Stabilization

In this second part of the paper [A. Quadrat, SIAM J. Control Optim., 40 (2003), pp. 266–299], we show how to reformulate the fractional representation approach to synthesis problems within an algebraic analysis framework. In terms of modules, we give necessary and sufficient conditions for internal stabilizability. Moreover, we characterize all the integral domains A of SISO stable plants such that every MIMO plant—defined by means of a transfer matrix whose entries belong to the quotient field K = Q(A) of A—is internally stabilizable. 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; S. Shankar and V. R. Sule, SIAM J. Control Optim., 30 (1992), pp. 11–30; V. R. Sule, SIAM J. Control Optim., 32 (1994), pp. 1675–1695; 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].