Polynomial Approximation of Quasipolynomials Based on Digital Filter Design Principles
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.
KeywordsApproximation Bilinear transformation Digital filter MATLAB Polynomials Pre-warping Quasipolynomials
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.
- 2.Sipahi, R., Vyhlídal, T., Niculescu, S.-I., Pepe, P.: Time Delay Systems: Methods, Applications and New Trends. LNCIS, vol. 423. Springer, New York (2012)Google Scholar
- 4.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)Google Scholar
- 6.Zítek, P., Víteček, A.: Control Design of Time-Delay and Nonlinear Subsystems. CTU Publishing (1999) (in Czech)Google Scholar
- 9.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)Google Scholar
- 13.Pekař, L.: On a controller parameterization for infinite-dimensional feedback systems based on the desired overshoot. WSEAS Trans. Syst. 12, 325–335 (2013)Google Scholar
- 14.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)Google Scholar
- 17.Oppenheim, A.: Discrete Time Signal Processing. Pearson Higher Education, Upper Saddle River, NJ (2010)Google Scholar
- 19.Balátě, J.: Automatic Control. BEN Publishing, Prague (2004). (in Czech)Google Scholar
- 21.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)Google Scholar
Open Access This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 2.5 International License (http://creativecommons.org/licenses/by-nc/2.5/), which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.
The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.