A parameterization of stabilizing controllers over commutative rings

Concerns the factorization approach to control systems, which has the advantage that it embraces, within a single framework, numerous linear systems, such as continuous-time as well as discrete-time systems, lumped as well as distributed systems, 1D as well as n-D systems, etc. We begin with the mathematical preliminaries, set up the feedback stabilization problem and present previous results. To obtain the set ℋ of transfer matrices of the standard feedback system with all stabilizing controllers, we use both right- and left-coprime factorizations over the ring of fractions of the set of stable causal transfer functions. In order to establish the existence of such right/left-coprime factorizations, we present the one-to-one correspondence between the sets of radicals of the generalized elementary factors of the plant and its transposed plant. A parameterization of the stabilizing controllers is then presented. We consider a multidimensional system with structural stability as an example, and we present the parameterization of its stabilizing controllers. Our method gives a solution to an open problem about the parameterization of the stabilizing controllers for a multidimensional system with structural stability