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

Abstract

Abstract. In this paper, we prove that some stabilizing controllers of a plant, which admits a left/right-coprime factorization, have a special form where their stable and unstable parts are separated. The dimension of the unstable part depends on the algebraic concept of stable range of the ring A of SISO stable plants. Moreover, we prove that, if the stable range of A is equal to 1, then every plant-defined by a transfer matrix with entries in the quotient field of A and admitting a left/right-coprime factorization-can be stabilized by a stable controller (strong stabilization). In particular, using a result of 1. Introduction. The fractional representation approach to analysis and synthesis problems was developed in the eighties in order to express in a unique mathematical framework several questions on stabilization problems. 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]. In [27,28], a new parametrization of all stabilizing controllers of a stabilizable plant was developed. It generalizes the Youla-Kučera parametrization [42] for stabilizable plants which do not necessarily admit doubly coprime factorizations. All these results show that a natural mathematical framework for the study of stabilization problems is the so-called K-theory [22,32]. See [29] for more details.

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

“…Due to a lack of space, it was not possible to develop here the strong and the simultaneous stabilization problems [29]. We refer the reader to [16,18] for a description of a canonical form, based on the concept of stable range, that certain stabilizing controllers possess. This canonical form allows us to show that, over a ring A of SISO stable plants of stable range 1 (e.g., A = H ∞ (C + )), every plant which admits a doubly coprime factorization is strongly stabilizable (i.e., stabilized by means of a stable controller).…”

Stable controllers for robust stabilization of systems with infinitely many unstable poles

“…On the other hand, if we do not require C ∈ RH ∞ but C ∈ H ∞ allowing complex coefficients, every stabilizable P ∈ F ∞ is strongly stabilizable [19], via a complex-valued controller in general.…”

On the stable rank and reducibility in algebras of real symmetric functions

“…(2) Bass and topological stable ranks of A R (D) play an important role in control theory in the problem of stabilization of linear systems. We refer the reader to [12] and [17] for background on the connection between stable rank and control theory. Definition 3.1.…”