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2017 | 27 | 1 | 19-32
Tytuł artykułu

Construction of algebraic and difference equations with a prescribed solution space

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
This paper studies the solution space of systems of algebraic and difference equations, given as auto-regressive (AR) representations A(σ)β(k) = 0, where σ denotes the shift forward operator and A(σ) is a regular polynomial matrix. The solution space of such systems consists of forward and backward propagating solutions, over a finite time horizon. This solution space can be constructed from knowledge of the finite and infinite elementary divisor structure of A(σ). This work deals with the inverse problem of constructing a family of polynomial matrices A(σ) such that the system A(σ)β(k) = 0 satisfies some given forward and backward behavior. Initially, the connection between the backward behavior of an AR representation and the forward behavior of its dual system is showcased. This result is used to construct a system satisfying a certain backward behavior. By combining this result with the method provided by Gohberg et al. (2009) for constructing a system with a forward behavior, an algorithm is proposed for computing a system satisfying the prescribed forward and backward behavior.
Rocznik
Tom
27
Numer
1
Strony
19-32
Opis fizyczny
Daty
wydano
2017
otrzymano
2016-05-25
poprawiono
2016-09-14
zaakceptowano
2016-10-12
Twórcy
  • Department of Mathematics, Faculty of Sciences, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece
  • Department of Mathematics, Faculty of Sciences, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece
Bibliografia
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  • Gantmacher, F.R. (1959). The Theory of Matrices, Vols. 1, 2, Chelsea Publishing Co., New York, NY.
  • Gohberg, I., Lancaster, P. and Rodman, L. (2009). Matrix Polynomials, Reprint, SIAM, Philadelphia, PA.
  • Hayton, G., Pugh, A. and Fretwell, P. (1988). Infinite elementary divisors of a matrix polynomial and implications, International Journal of Control 47(1): 53-64.
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  • Karampetakis, N.P. (2015). Construction of algebraic-differential equations with given smooth and impulsive behaviour, IMA Journal of Mathematical Control and Information 32(1): 195-224.
  • Karampetakis, N.P. and Vologiannidis, S. (2003). Infinite elementary divisor structure-preserving transformations for polynomial matrices, International Journal of Applied Mathematics and Computer Science 13(4): 493-503.
  • Karampetakis, N., Vologiannidis, S. and Vardulakis, A. (2004). A new notion of equivalence for discrete time AR representations, International Journal of Control 77(6): 584-597.
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  • Vardulakis, A. (1991). Linear Multivariable Control. Algebraic Analysis and Synthesis Methods, John Wiley & Sons, Chichester.
  • Vardulakis, A., Limebeer, D. and Karcanias, N. (1982). Structure and Smith-MacMillan form of a rational matrix at infinity, International Journal of Control 35(4): 701-725.
  • Willems, J.C. (1986). From time series to linear system, II: Exact modelling, Automatica 22(6): 675-694.
  • Willems, J.C. (1991). Paradigms and puzzles in the theory of dynamical systems, IEEE Transansactions on Automatic Control 36(3): 259-294.
  • Willems, J.C. (2007). Recursive computation of the MPUM, in A. Chiuso et al. (Eds.), Modeling, Estimation and Control, Springer, Berlin, pp. 329-344.
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  • Zerz, E., Levandovskyy, V. and Schindelar, K. (2011). Exact linear modeling with polynomial coefficients, Multidimensional System Signal Processing 22(1-3): 55-65.
Typ dokumentu
Bibliografia
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