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

Abstract

Abstract. 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) Introduction. Using the algebraic analysis viewpoint of the fractional representation approach to analysis and synthesis problems [5,28,29], developed in the first part of the paper [17], we give necessary and sufficient conditions for internal stabilizability. Moreover, using these results, we prove that every multi-input multi-output (MIMO) plant-defined by means of a transfer matrixT are matrices whose entries belong to an integral domain A of single input single output (SISO) stable plants-is internally stabilizable iff A is a Prüfer domain [6,23]. From the fact that the intersection between coherent Sylvester domains (see [17] for more details) and Prüfer domains are just Bézout domains, we also recover the result of Vidyasagar [29]: every MIMO plant admits doubly coprime factorizations iff A is a Bézout domain. Hence, if the algebra A is a Prüfer domain but not a Bézout domain, there exist plants which are internally stabilizable but fail to admit doubly coprime factorizations. Therefore, it is not possible to parametrize all their stabilizing controllers by means of the Youla-Kučera parametrization [4,28]. These results allow us to explain the counterexamples exhibited in [1,12]. We prove that, over a projective-free domain A (e.g., H ∞ (C + ), RH ∞ ), every stabilizable system admits doubly coprime factorizations. Finally, we show that the previous results allow us to recover, on the one hand, the results of [25] and, on

Polynomial Solutions and Annihilators of Ordinary Integro-Differential Operators

“…Linear systems are usually described by means of finite matrices with entries in a certain ring D. As explained in [18], if D is a coherent ring, an algebraic systems theory can be developed as if D was a noetherian ring. Hence, Theorem 3 shows that an algebraic systems theory can be developed over I.…”

Noncoherent uniform algebras in $\mathbb {C}^n$

“…To conclude, let me point out that the coherence of rings of stable transfer functions of multidimensional systems, such as A(B n ) or A(D n ), plays a role in the stabilization problem in Control Theory via the factorization approach; see [17].…”

On a General Structure of the Stabilizing Controllers Based on Stable Range

“…In that framework, we can study internal stabilization (existence of an internally stabilizing controller), parametrization of all stabilizing controllers, strong stabilization (possibility of stabilizing a plant by means of a stable controller), simultaneous stabilization (possibility of stabilizing a set of plants by means of a single controller), metrics of robustness (gap or graph topologies), H ∞ or H 2 -optimal controllers, etc. See [2, 6, 42] for more details.Recently, the reformulation of the fractional representation approach to analysis and synthesis problems within an algebraic analysis approach has allowed us to obtain new necessary and sufficient conditions for internal stabilizability and for the existence of (weakly) left/right/doubly coprime factorizations in the general setting [25,26,24]. Moreover, all the rings of SISO stable plants (used in this framework) over which one of the previous properties is satisfied were completely characterized [25,26,24].…”

“…Recently, the reformulation of the fractional representation approach to analysis and synthesis problems within an algebraic analysis approach has allowed us to obtain new necessary and sufficient conditions for internal stabilizability and for the existence of (weakly) left/right/doubly coprime factorizations in the general setting [25,26,24]. Moreover, all the rings of SISO stable plants (used in this framework) over which one of the previous properties is satisfied were completely characterized [25,26,24].…”

“…Moreover, all the rings of SISO stable plants (used in this framework) over which one of the previous properties is satisfied were completely characterized [25,26,24]. In [27,28], a new parametrization of all stabilizing controllers of a stabilizable plant was developed.…”