On the Parametrization of All Regularly Implementing and Stabilizing Controllers

Abstract

In this paper we deal with problems of controller parametrization in the context of behavioral systems. Given a full plant behavior, a subbehavior of the manifest plant behavior is called regularly implementable if it can be achieved as the controlled behavior resulting from the interconnection of the full plant behavior with a suitable controller behavior, in such a way that the controller does not impose restrictions that are already present in the plant. We establish a parametrization of all controllers that regularly implement a given behavior. We also obtain a parametrization of all stabilizing controllers.

The decentralized implementability problem

“…Theorem 5.2 shows that K can be any behavior of L w that is wedged in between the given behaviors Nw(P full ) and (P full )w. The implementability problem was also studied in [6], [15] and [9]. In particular, the question when a particular controlled behavior can be implemented by a feedback processor remains a very important one, and was discussed e.g.…”

“…Then the following are equivalent: 1) K is regularly implementable by full interconnection with respect to P, 2) there exists a polynomial matrix F with F (λ) full row rank for all λ ∈ C such that R = F K. Having obtained necessary and sufficient conditions for implementability and regular implementability of a given desired behavior K, we now aim at establishing characterizations of all controllers C that (regularly) implement it. Parameterizations of these controllers have been established before in [9]. In the following two lemmas we formulate alternative characterizations: Lemma 3.7: Let P ∈ L w and let K ∈ L w .…”

“…The following result characterizes regular implementability in terms of kernel representations (see [9]): Proposition 3.6: Let P ∈ L w and K ∈ L w . Let P = ker(R) and K = ker(K) be minimal kernel representations of plant and desired behavior.…”

Behavioral Systems Theory: A Survey

“…The wide areas of stability and stabilization (Oberst, 2006) and of controller synthesis (Praagman et al, 2007) in the behavioral context were not even touched. The same holds for the equally significant topics of observability and observer design (Valcher and Willems, 1999).…”

On the regular implementability of nD systems

“…An alternative characterization of regular implementability by partial interconnection was given for the 1D case in [8], and for the general nD case in [9]. This characterization is in terms of the canonical controller: A second aim of this paper is to re-investigate for nD behaviors the role of the canonical controller in the problem of (regular) implementability.…”