On Rational State Space Realizations

Forsman, Krister

doi:10.1016/b978-0-08-041901-5.50061-0

“…We achieve only local observability but guarantee the existence of a realization defined on an affine space. Note that our result is sharp in the sense that there exist realizable DAEs without observable realizations by a system of the state-space form, see [4,Section 4].…”

Section: Introductionmentioning

confidence: 93%

“…Two popular classes considered in this paper are rational functions and input-affine rational functions as in [11,12,17], but one could also consider algebraic, analytic, or smooth functions [14,19,20,22]. From the constructive standpoint, the case of single-output-no-input systems (for which rational and inputaffine rational functions coincide) has been considered by Forsman [4]. He has shown that a DAE in can be realized by a rational system in the state-space form if and only if the corresponding hypersurface is unrational.…”

Section: Introductionmentioning

confidence: 99%

On realizing differential-algebraic equations by rational dynamical systems

Pavlov,

Pogudin

2022

Preprint

0

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Real-world phenomena can often be conveniently described by dynamical systems (that is, ODE systems in the state-space form). However, if one observes the state of the system only partially, the observed quantities (outputs) and the inputs of the system can typically be related by more complicated differential-algebraic equations (DAEs). Therefore, a natural question (referred to as the realizability problem) is: given a differential-algebraic equation (say, fitted from data), does it come from a partially observed dynamical system? A special case in which the functions involved in the dynamical system are rational is of particular interest. For a single differential-algebraic equation in a single output variable, Forsman has shown that it is realizable by a rational dynamical system if and only if the corresponding hypersurface is unirational, and he turned this into an algorithm in the first-order case.In this paper, we study a more general case of single-input-singleoutput equations. We show that if a realization by a rational dynamical system exists, the system can be taken to have the dimension equal to the order of the DAE. We provide a complete algorithm for first-order DAEs. We also show that the same approach can be used for higher-order DAEs using several examples from the literature. CCS CONCEPTS• Computing methodologies → Symbolic calculus algorithms.

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“…Two popular classes considered in this paper are rational functions and input-affine rational functions as in [12,13,21], but one could also consider algebraic, analytic, or smooth functions [18,23,24,26]. From the constructive standpoint, the case of single-output-no-input systems (for which rational and input-affine rational functions coincide) has been considered by Forsman [4]. He has shown that a DAE in 𝑦 can be realized by a rational system in the state-space form if and only if the corresponding hypersurface is unrational.…”

Section: Introductionmentioning

confidence: 99%

On Realizing Differential-Algebraic Equations by Rational Dynamical Systems

Павлов

¹

,

Pogudin

²

2022

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation

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Real-world phenomena can often be conveniently described by dynamical systems (that is, ODE systems in the state-space form). However, if one observes the state of the system only partially, the observed quantities (outputs) and the inputs of the system can typically be related by more complicated differential-algebraic equations (DAEs). Therefore, a natural question (referred to as the realizability problem) is: given a differential-algebraic equation (say, fitted from data), does it come from a partially observed dynamical system? A special case in which the functions involved in the dynamical system are rational is of particular interest. For a single differential-algebraic equation in a single output variable, Forsman has shown that it is realizable by a rational dynamical system if and only if the corresponding hypersurface is unirational, and he turned this into an algorithm in the first-order case.In this paper, we study a more general case of single-input-singleoutput equations. We show that if a realization by a rational dynamical system exists, the system can be taken to have the dimension equal to the order of the DAE. We provide a complete algorithm for first-order DAEs. We also show that the same approach can be used for higher-order DAEs using several examples from the literature. CCS CONCEPTS• Computing methodologies → Symbolic calculus algorithms.

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“…In addition, they have recently acquired additional practical importance in the area of Computer-Aided Geometric Design for airplanes, automobiles, and household appliances, which would be unthinkable without computational help and the development of effective methods for the manipulation of algebraic curves and also surfaces [18][19][20][21]. The parameterizations of these entities also play an important role in other aspects, such as line integral, representation, distribution of polynomial interpolation nodes [22], control theory [23], etc.…”

Section: Introductionmentioning

confidence: 99%

Some New Symbolic Algorithms for the Computation of Generalized Asymptotes

Campo-Montalvo

¹

,

Sevilla

²

,

Magdalena‐Benedito

³

et al. 2022

Symmetry

3

0

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We present symbolic algorithms for computing the g-asymptotes, or generalized asymptotes, of a plane algebraic curve, C, implicitly or parametrically defined. The g-asymptotes generalize the classical concept of asymptotes of a plane algebraic curve. Both notions have been previously studied for analyzing the geometry and topology of a curve at infinity points, as well as to detect the symmetries that can occur in coordinates far from the origin. Thus, based on this research, and in order to solve practical problems in the fields of science and engineering, we present the pseudocodes and implementations of algorithms based on the Puiseux series expansion to construct the g-asymptotes of a plane algebraic curve, implicitly or parametrically defined. Additionally, we propose some new symbolic methods and their corresponding implementations which improve the efficiency of the preceding. These new methods are based on the computation of limits and derivatives; they show higher computational performance, demanding fewer hardware resources and system requirements, as well as reducing computer overload. Finally, as a novelty in this research area, a comparative analysis for all the algorithms is carried out, considering the properties of the input curves and their outcomes, to analyze their efficiency and to establish comparative criteria between them.

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On Rational State Space Realizations

Cited by 3 publications

References 17 publications

On realizing differential-algebraic equations by rational dynamical systems

On realizing differential-algebraic equations by rational dynamical systems

On Realizing Differential-Algebraic Equations by Rational Dynamical Systems

Some New Symbolic Algorithms for the Computation of Generalized Asymptotes

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