Abstract
This contribution is aimed at a possible procedure approximating quasipolynomials by polynomials. Quasipolynomials appear in linear time-delay systems description as a natural consequence of the use of the Laplace transform. Due to their infinite root spectra, control system analysis and synthesis based on such quasipolynomial models are usually mathematically heavy. In the light of this fact, there is a natural research endeavor to design a sufficiently accurate yet simple engineeringly acceptable method that approximates them by polynomials preserving basic spectral information. In this paper, such a procedure is presented based on some ideas of discrete-time (digital) filters designing without excessive math. Namely, the particular quasipolynomial is subjected to iterative discretization by means of the bilinear transformation first; consequently, linear and quadratic interpolations are applied to obtain integer powers of the approximating polynomial. Since dominant roots play a decisive role in the spectrum, interpolations are made in their very neighborhood. A simulation example proofs the algorithm efficiency.
References
Chiasson, J., Loiseau, J.J.: Applications of Time Delay Systems. Springer, New York (2007)
Sipahi, R., Vyhlídal, T., Niculescu, S.-I., Pepe, P.: Time Delay Systems: Methods, Applications and New Trends. LNCIS, vol. 423. Springer, New York (2012)
Richard, J.P.: Time-Delay systems: an overview of some recent advances and open problems. Automatica 39, 1667–1694 (2003)
Loiseau, J.J., Michiels, W., Niculescu, S.-I., Sipahi, R.: Topics in Time Delay Systems: Analysis, Algorithm and Control. LNCIS, vol. 388. Springer, Berlin (2009)
Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Applied Mathematical Sciences, vol. 99. Springer, New York (1993)
Zítek, P., Víteček, A.: Control Design of Time-Delay and Nonlinear Subsystems. CTU Publishing (1999) (in Czech)
Gu, K., Kharitonov, V.L., Chen, J.: Stability of Time-Delay Systems. Birkhäuser, Boston (2003)
Breda, D., Maset, S., Vermiglio, R.: Pseudospectral differencing methods for characteristic roots of delay differential equations. SIAM J. Sci. Comput. 27, 482–495 (2005)
Vyhlídal, T., Zítek, P.: Quasipolynomial mapping algorithm rootfinder for analysis of time delay systems. In: Proceedings of the 4th IFAC Workshop on Time-Delay Systems (TDS 2003). Rocquencourt, France (2003)
Vyhlídal, T., Zítek, P.: Mapping based algorithm for large-scale computation of quasipolynomial zeros. IEEE Trans. Autom. Control 54, 171–177 (2009)
Vyhlídal, T., Zítek, P.: QPmR—Quasi-Polynomial Root-Finder: Algorithm Update and Examples. In: Vyhlídal, T., Lafay, J.-F., Sipahi, R. (eds.) Delay Systems: From Theory to Numerics and Applications, pp. 299–312. Springer, New York (2014)
Partington, J.R.: Some frequency-domain approaches to the model reduction of delay systems. Ann. Rev. Control 28, 65–73 (2004)
Pekař, L.: On a controller parameterization for infinite-dimensional feedback systems based on the desired overshoot. WSEAS Trans. Syst. 12, 325–335 (2013)
Seuret, A., Özbay, H., Bonnet, C., Mounier, H.: Low Complexity Controllers for Time Delay Systems. Advances in Delays and Dynamics, vol. 2. Springer, New York (2014)
Middleton, R.H., Goodwin, G.C.: Digital Control and Estimation: A Unified Approach. Prentice Hall, Detroit (1990)
Vyhlídal, T., Zítek, P.: Discrete Approximation of a Time Delay System and Delta Model Spectrum. In: Proceedings of the 16th IFAC World Congress, p. 636. IFAC, Prague (2005)
Oppenheim, A.: Discrete Time Signal Processing. Pearson Higher Education, Upper Saddle River, NJ (2010)
Vanbiervliet, T., Verheyden, K., Michiels, W., Vandewalle, S.: A nonsmooth optimization approach for the stabilization of time-delay systems. ESIAM Control Optim. Ca. 14, 478–493 (2008)
Balátě, J.: Automatic Control. BEN Publishing, Prague (2004). (in Czech)
Zítek, P., Kučera, V., Vyhlídal, T.: Meromorphic observer-based pole assignment in time delay systems. Kybernetika 44, 633–648 (2008)
Pekař, L.: A Simple DDS Algorithm for TDS: An Example. In: Proceedings of the 29th European Conference on Modelling and Simulation (ECMS 2015), pp. 246–251. European Council for Modelling and Simulation (ECMS), Varna, Bulgaria (2015)
Acknowledgments
The work was performed with the financial support by the Ministry of Education, Youth and Sports of the Czech Republic within the National Sustainability Programme project No. LO1303 (MSMT-7778/2014) and also by the European Regional Development Fund under the project CEBIA-Tech No. CZ.1.05/2.1.00/03.0089.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Pekař, L., Navrátil, P. (2016). Polynomial Approximation of Quasipolynomials Based on Digital Filter Design Principles. In: Silhavy, R., Senkerik, R., Oplatkova, Z.K., Silhavy, P., Prokopova, Z. (eds) Automation Control Theory Perspectives in Intelligent Systems. CSOC 2016. Advances in Intelligent Systems and Computing, vol 466. Springer, Cham. https://doi.org/10.1007/978-3-319-33389-2_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-33389-2_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-33387-8
Online ISBN: 978-3-319-33389-2
eBook Packages: EngineeringEngineering (R0)