A Parameterization of Stabilizing Controllers over Commutative Rings

In this paper we are concerned with 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, 1-D as well as n-D systems, etc.[1]. The factorization approach was patterned after Desoer et al.[2] and Vidyasagar et al.[1]. In this approach, when problems such as feedback stabilization are studied, one can focus on the key aspects of the problem under study rather than be distracted by the special features of a particular class of linear systems. A transfer matrix of this approach is considered as the ratio of two stable causal transfer matrices. For a long time, the theory of the factorization approach had been founded on the coprime factorizability of transfer matrices, which is satisfied by transfer matrices over the principal ideal domains or the Bezout domains. Sule in [3, 4] has presented a theory of the feedback stabilization of strictly causal plants for multi-input multi-output transfer matrices over commutative rings with some restrictions. This approach to the stabilization theory is called “coordinate-free approach” in the sense that the coprime factorizability of transfer matrices is not required. Recently, Mori and Abe in [5, 6] have generalized his theory over commutative rings under the assumption that the plant is causal. They have introduced the notion of the generalized elementary factor, which is a generalization of the elementary factor introduced by Sule[3], and have given the necessary and sufficient condition of the feedback stabilizability. Since the stabilizing controllers are not unique in general, the choice of the stabilizing controllers is important for the resulting closed loop. In the classical case, that is, in the case where there exist the right-/left-coprime factorizations of the given plant, the stabilizing controllers can be parameterized by the method called “YoulaKučera parameterization”[1, 2, 7, 8]. However, it is known that there exist models in which some stabilizable transfer matrices do not have their right-/left-coprime factorizations[9]. In such models, we cannot employ the Youla-Kučera parameterization directly. In this paper we will give a parameterization of the stabilizing controllers over commutative rings by using the results given by Sule[3], and Mori and Abe[5, 6]. Here we briefly outline how the parameterization, which is different from the YoulaKučera parameterization, will be obtained. Let H(P,C) denote the transfer matrix of