Construction of algebraic and difference equations with a prescribed solution space

• Autorzy: Moysis L. , Karampetakis N. P.

Identyfikatory

DOI

10.1515/amcs-2017-0002

• 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.

Słowa kluczowe

EN

algebraic equation; difference equation; behavior; exact modeling; autoregressive representation; discrete time system; higher order system

PL

równanie algebraiczne; równanie różnicowe; system czasu dyskretnego; system wyższego rzędu

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2017
• Tom: Vol. 27, no. 1
• Strony: 19--32
• Opis fizyczny: Bibliogr. 26 poz., tab.

Twórcy

autor

Moysis L.

• Department of Mathematics, Faculty of Sciences, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece

autor

Karampetakis N. P.

• Department of Mathematics, Faculty of Sciences, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece

Bibliografia

• [1] Antoniou, E., Vardulakis, A. and Karampetakis, N. (1998). A spectral characterization of the behavior of discrete time AR-representations over a finite time interval, Kybernetika 34(5): 555–564.
• [2] Antoulas, A. and Willems, J. (1993). A behavioral approach to linear exact modeling., IEEE Transactions on Automatic Control 38(12): 1776–1802.
• [3] Antsaklis, P.J. and Michel, A.N. (2006). Linear Systems, 2nd Edn., Birkh¨auser, Boston, MA.
• [4] Bernstein, D.S. (2009). Matrix Mathematics. Theory, Facts, and Formulas, 2nd Edn., Princeton University Press, Princeton, NJ.
• [5] Campbell, S. (1980). Singular Systems of Differential Equations, Vol. 1, Research Notes in Mathematics, Pitman, London.
• [6] Gantmacher, F.R. (1959). The Theory of Matrices, Vols. 1, 2, Chelsea Publishing Co., New York, NY.
• [7] Gohberg, I., Lancaster, P. and Rodman, L. (2009). Matrix Polynomials, Reprint, SIAM, Philadelphia, PA.
• [8] 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.
• [9] Kaczorek, T. (2007). Polynomial and Rational Matrices. Applications in Dynamical Systems Theory, Springer, Dordrecht.
• [10] Kaczorek, T. (2014). Minimum energy control of fractional descriptor positive discrete-time linear systems, International Journal of Applied Mathematics and Computer Science 24(4): 735–743, DOI: 10.2478/amcs-2014-0054.
• [11] Kaczorek, T. (2015). Analysis of the descriptor Roesser model with the use of the Drazin inverse, International Journal of Applied Mathematics and Computer Science 25(3): 539–546, DOI: 10.1515/amcs-2015-0040.
• [12] Karampetakis, N.P. (2004). On the solution space of discrete time AR-representations over a finite time horizon, Linear Algebra and Its Applications 382: 83–116.
• [13] 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.
• [14] 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.
• [15] 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.
• [16] Markovsky, I., Willems, J.C., Van Huffel, S. and De Moor, B. (2006). Exact and Approximate Modeling of Linear Systems. A Behavioral Approach, SIAM, Philadelphia, PA.
• [17] Praagman, C. (1991). Invariants of polynomial matrices, Proceedings of the 1st European Control Conference, Grenoble, France, pp. 1274–1277.
• [18] Vardulakis, A. (1991). Linear Multivariable Control. Algebraic Analysis and Synthesis Methods, John Wiley & Sons, Chichester.
• [19] 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.
• [20] Willems, J.C. (1986). From time series to linear system, II: Exact modelling, Automatica 22(6): 675–694.
• [21] Willems, J.C. (1991). Paradigms and puzzles in the theory of dynamical systems, IEEE Transansactions on Automatic Control 36(3): 259–294.
• [22] Willems, J.C. (2007). Recursive computation of the MPUM, in A. Chiuso et al. (Eds.), Modeling, Estimation and Control, Springer, Berlin, pp. 329–344.
• [23] Zaballa, I. and Tisseur, F. (2012). Finite and infinite elementary divisors of matrix polynomials: A global approach, MIMS EPrint 2012.78, Manchester Institute for Mathematical Sciences, University of Manchester, Manchester.
• [24] Zerz, E. (2008a). Behavioral systems theory: A survey, International Journal of Applied Mathematics and Computer Science 18(3): 265–270, DOI: 10.2478/v10006-008-0024-9.
• [25] Zerz, E. (2008b). The discrete multidimensional MPUM, Multidimensional System Signal Processing 19(3–4): 307–321.
• [26] Zerz, E., Levandovskyy, V. and Schindelar, K. (2011). Exact linear modeling with polynomial coefficients, Multidimensional System Signal Processing 22(1–3): 55–65.

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).