A fractional representation approach to the robust regulation problem for SISO systems

Abstract

The purpose of this article is to develop a new approach to the robust regulation problem for plants which do not necessarily admit coprime factorizations. The approach is purely algebraic and allows us dealing with a very general class of systems in a unique simple framework. We formulate the famous internal model principle in a form suitable for plants defined by fractional representations which are not necessarily coprime factorizations. By using the internal model principle, we are able to give necessary and sufficient solvability conditions for the robust regulation problem and to parameterize all robustly regulating controllers.

A fractional representation approach to the robust regulation problem for multi‐input multi‐output systems

“…32 The main results extend the existing ones that are specific to some ring R and use the coprime factorizations 7,15,21,23 to the abstract algebraic setting using no coprime factorizations, thus providing a unifying theory for robust regulation in the frequency domain. The theoretical results of this article extend the frequency domain results of SISO systems presented by Laakkonen and Quadrat 24 to multi-input multi-output (MIMO) systems. Unlike with the SISO systems, regulation does not imply robustness with the MIMO systems making the generalization of the results nontrivial.…”

“…Thus, it is sufficient to check that a given factorization of the controller is weakly coprime when applying the result of Theorem 6. Note that the assumption that $$$ \theta $$$ of the above theorem has a coprime factorization does not follow by the existence of a weak coprime factorization 24 and cannot be done without restricting generality.…”

“…A frequency domain formulation of the internal model principle and solvability conditions for the robust regulation problem were given by Vidyasagar 7 for rational transfer functions. These results were later generalized to specific classes of transfer functions suitable for infinite‐dimensional systems 15,21 and by using the fractional representation approach 23,24 . Frequency domain methods for designing regulating controllers have also been considered by several authors 17,18,21,25 …”

“…These results were later generalized to specific classes of transfer functions suitable for infinite-dimensional systems 15,21 and by using the fractional representation approach. 23,24 Frequency domain methods for designing regulating controllers have also been considered by several authors. 17,18,21,25 In this article, the robust regulation problem is studied using the fractional representation approach.…”

“…In particular, the sufficiency part of the internal model principle now address also robustness whereas the original proof only stated that the internal model implies regulation. The preliminary results 24,33 do not take causality into account, so the results addressing it are all new.…”