Realization Theory for Rational Systems: Minimal Rational Realizations

Němcová, Jana; Schuppen, Jan H. van

doi:10.1007/s10440-009-9464-y

Cited by 26 publications

(23 citation statements)

References 31 publications

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“…Moreover, it shows that if we consider as a field F from Proposition 5.14 the observation field of a response map to be realized, then the algorithm for constructing a rational realization from Proposition 5.14 gives as a result a rationally observable rational realization. Additional results which we got by studying minimal rational realization can be found in [15]. For some remarks on algebraic reachability of rational systems, see [14].…”

Section: Discussionmentioning

confidence: 99%

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Realization Theory for Rational Systems: The Existence of Rational Realizations

Němcová¹,

Schuppen²

2009

SIAM J. Control Optim.

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Abstract. In this paper we solve the problem of the existence of rational realizations of response maps. Sufficient and necessary conditions for a response map to be realizable by a rational system are presented. We provide also the characterization of the existence of rationally observable and canonical rational realizations for a given response map.Key words. rational systems, realization theory, algebraic reachability, rational observability AMS subject classifications. 93B15, 93B27, 93C10 DOI. 10.1137/080714506 1. Introduction. The motivation to investigate realization theory of rational systems is the use of rational systems as models of phenomena in life sciences, in particular, in systems biology. For example, rational systems occur as models of metabolic, genetic, and signaling networks. They can be found also in engineering, physics, and economics. Moreover, as Bartosiewicz stated in [3], the theory of rational systems could be simpler and more powerful, once it is developed, than the theory of smooth systems.The realization problem for rational systems considers a map from input functions to output functions and asks whether there exists a finite-dimensional rational system with an initial condition such that its input/output map is identical to the considered map. Such a system is then called a realization of the considered input/output map. A generalization, which is not addressed in this paper, is to regard any relation between observed variables and ask for a realization as a rational system. Another goal of realization theory is to characterize certain properties of realizations. One wants to find the conditions under which the systems realizing the considered map are observable, controllable, or minimal. The relations between realizations having these properties are also of interest, since they can be applied in control and observer synthesis and in system identification.Polynomial and rational systems are a special class of nonlinear systems admitting a more refined algebraic structure. Realization theory for discrete-time polynomial systems was formulated by Sontag in [16]. Later,in [18], Wang and Sontag published their results on realization theory for polynomial and rational continuous-time systems based on the approach of formal power series in noncommuting variables and on the relation of two characterizations of observation spaces.Another approach to realization theory for polynomial continuous-time systems, motivated by the results of Jakubczyk in [11] for nonlinear realizations, is introduced by Bartosiewicz in [1,4]. This approach is based on [16]. Furthermore, in [3], Bartosiewicz introduces the concept of rational systems and deals with the problem of immersion of smooth systems into rational systems. Since this problem is similar to

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Section: Discussionmentioning

confidence: 99%

“…. , α k ), α j ∈ U, k ∈ N, we get that ϕ = in [15]. They can be implemented by using already existing computer algebra packages.…”

Section: Canonical Rational Realizationsmentioning

confidence: 99%

Realization Theory for Rational Systems: The Existence of Rational Realizations

Němcová¹,

Schuppen²

2009

SIAM J. Control Optim.

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“…These assumptions make it possible to study structural and global identifiability of parametrizations of parametrized polynomial and parametrized rational systems by means of realization theory developed in [5] for polynomial and in [26,25] for rational systems.…”

Section: Structural and Global Identifiabilitymentioning

confidence: 99%

“…We refer to the results of realization theory for rational systems in [26,25]. The main result which is applied to obtain this characterization is the following: This theorem corresponds to Theorem 4.2 in polynomial case.…”

Section: Structural Identifiability Of Rational Systemsmentioning

confidence: 99%

Structural identifiability of polynomial and rational systems

Němcová¹

2010

Mathematical Biosciences

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“…Further, Bartosiewicz (1987) established that a smooth system may be immersed into a rational system if the observation field is a finitely generated extension of R, and stated that rational systems could be simpler and more powerful than smooth systems. In addition, the existence of rational realizations of response maps was investigated in Němcová & van Schuppen (2009, 2010. It was shown that if a response map is realized by a rational system, then there also exists a minimal rational realization of the map (Němcová & van Schuppen, 2010).…”

Section: Introductionmentioning

confidence: 99%