The revision and extension of the RMS ring for time delay systems

• Autorzy: Pekař L. , Prokop R.

Identyfikatory

DOI

10.1515/bpasts-2017-0038

• Języki publikacji: EN

Abstrakty

EN

This paper is aimed at reviewing the ring of retarded quasipolynomial meromorphic functions (RMS) that was recently introduced as a convenient control design tool for linear, time-invariant time delay systems (TDS). It has been found by the authors that the original definition does not constitute a ring and has some essential deficiencies, and hence it could not be used for an algebraic control design without a thorough reformulation which i.a. extends the usability to neutral TDS and to those with distributed delays. This contribution summarizes the original definition of RMS simply highlights its deficiencies via examples, and suggests a possible new extended definition. Hence, the new ring of quasipolynomial meromorphic functions RQM is established to avoid confusion. The paper also investigates and introduces selected algebraic properties supported by some illustrative examples and concisely outlines its use in controller design.

Słowa kluczowe

EN

time delay system; ring; Bézout domain; coprime factorization

PL

system opóźniania; pierścień; domena Bézout

• Wydawca: Polska Akademia Nauk, Wydział IV Nauk Technicznych
• Czasopismo: Bulletin of the Polish Academy of Sciences. Technical Sciences
• Rocznik: 2017
• Tom: Vol. 65, nr 3
• Strony: 341--349
• Opis fizyczny: Bibliogr. 33 poz., wykr.

Twórcy

autor

Pekař L.

• Faculty of Applied Informatics, Tomas Bata University in Zlín, St. Nad Stráněmi 4511, 76005 Zlín, Czech Republic

autor

Prokop R.

• Faculty of Applied Informatics, Tomas Bata University in Zlín, St. Nad Stráněmi 4511, 76005 Zlín, Czech Republic

Bibliografia

• [1] H. Górecki, S. Fuksa, P. Grabowski, and A. Korytowski, Analysis and Synthesis of Time Delay Systems, Wiley, New York, 1989.
• [2] J.K. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993.
• [3] T. Vyhlídal, J.-F. Lafay, and R. Sipahi (eds.), Delay Systems: From Theory to Numerics and Applications, Springer, New York, 2014.
• [4] J. Konopacki and K. Mościńska, “Estimation of filter order for prescribed, reduced group delay FIR filter design”, Bull. Pol. Ac.: Tech. 63 (1), 209–216 (2015).
• [5] Y. Rouchaleau, Linear, Discrete Time, Finite Dimensional Dynamical Systems over Some Classes of Commutative Rings, PhD thesis, Stanford University, 1972.
• [6] E.W. Kamen, “On the algebraic theory of systems defined by convolution operations”, Math. Sys. Theory 9 (1), 57–74 (1975).
• [7] A.S. Morse, “Ring models for delay-differential systems”, Automatica 12 (5), 529–531 (1976).
• [8] E.D. Sontag, “Linear systems over commutative rings: A survey”, Ricer. Autom. 7 (1), 1–34 (1976).
• [9] E.W. Kamen, P.P. Khargonegar, and A. Tannenbaum, “Stabilization of time delay systems using finite dimensional compensators”, IEEE Trans. Autom. Control 30 (1), 75–78 (1985).
• [10] E.W. Kamen, P.P. Khargonegar, and A. Tannenbaum, “Proper stable Bézout stabilization and feedback control of linear timedelay systems”, Int. J. Control 43 (3), 837–857 (1986).
• [11] D. Brethé and J.J. Loiseau, “An effective algorithm for finite spectrum assignment of single-input systems with delays”, Math. Comput. Simul. 45 (4–5), 339–348 (1998).
• [12] J.A. Hermida-Alonso, “On linear algebra over commutative rings”, in Handbook of Algebra, vol. III, pp. 3–61, ed. M. Hazewinkel, 2003.
• [13] R.F. Curtain and K. Morris, “Transfer functions of distributed parameters systems”, Automatica 45 (5), 1101–1116 (2009).
• [14] J. Klamka, A. Babiarz, and M. Niezabitowski, “Banach fixedpoint theorem in semilinear controllability problems – a survey”, Bull. Pol. Ac.: Tech. 64 (1), 21–35 (2016).
• [15] C.A. Desoer, R.W. Liu, J. Murray, and R. Seaks, “Feedback system design: The fractional representation approach to analysis and synthesis”, IEEE Trans. Autom. Control 25 (3), 399–412 (1980).
• [16] M. Vidyasagar, Control System Synthesis: A Factorization Approach, MIT Press, Cambridge, MA, 1985.
• [17] V. Kučera, “Diophantine equations in control – a survey”, Automatica 29 (6), 1361–1375 (1993).
• [18] C. Foias, H. Özbay, and A. Tannenbaum, Robust Control of Infinite Dimensional Systems: Frequency Domain Methods, Springer Verlag, Berlin, 1996.
• [19] M.C. Smith, “On stabilization and the existence of coprime factorization”, IEEE Trans. Autom. Control 34 (9), 1005–1007 (1989).
• [20] R. Prokop and J.P. Corriou, “Design and analysis of simple robust controllers”, Int. J. Control 66 (6), 905–921 (1997).
• [21] P. Zítek and V. Kučera, “Algebraic design of anisochronic controllers for time delay systems”, Int. J. Control 76 (16), 1654–1665 (2003).
• [22] P. Zítek, V. Kučera, and T. Vyhlídal, “Meromorphic stabilization and control of time delay systems”, Proc. 16th IFAC World Congress, IFAC, Prague, 638–638 (2005).
• [23] F.M. Callier and C.A. Desoer, “Simplifications and clarifications on the paper ‘An algebra of transfer functions for distributed linear time-invariant systems’ “, IEEE Trans. Circ. Syst. 27 (4), 320–323 (1980).
• [24] M.V. Carriegos, A. Defrancisco-Iribarren, and R. Samtamaria-Sanchez, “The effective calculation of all assignable polynomials to a single-input system over a Bézout domain”, IEEE Trans. Autom. Control 55 (1), 222–225 (2010).
• [25] J.-L. Hong, “H∞ control for multiple state-delayed systems: A coprime factorization approach”, J. Chinese Inst. Eng. 29 (2), 201–210 (2006).
• [26] J.R. Partington and C. Bonnet, “H∞ and BIBO stabilization of delay systems of neutral type”, Syst. Control Lett. 52 (3–4), 283–288 (2004).
• [27] C. Bonnet, A.R. Fioravanti, and J.R. Partington, “Stability of neutral systems with commensurate delays and poles asymptotic to the imaginary axis”, SIAM J. Control Optim. 49 (2), 498–516 (2011).
• [28] R. Prokop, J. Korbel, and L. Pekař, “Delay systems with meromorphic functions design”, 12th IEEE Int. Conf. on Control & Automation, IEEE Computer Society, Kathmandu, Nepal, 443–448 (2016).
• [29] L. Pekař and R. Prokop, “Algebraic robust control of a closed circuit heating-cooling system with a heat exchanger and internal loop delays”, Appl. Therm. Eng. 113, 1464–1474 (2017).
• [30] L. Pekař, “A ring for description and control of time-delay systems”, WSEAS Trans. Syst. 11 (10), 571–585 (2012).
• [31] J.J. Loiseau, M. Cardelli, and X. Dusser, “Neutral-type timedelay systems that are not formally stable are not BIBO stabilizable”, IMA J. Math. Control Inf. 19, 217–227 (2002).
• [32] T. Vyhlídal and P. Zítek, “QPmR – Quasi-Polynomial Root-Finder: Algorithm update and examples”, in Delay Systems: From Theory to Numerics and Applications, pp. 299–312, eds. T. Vyhlídal, J.-F. Lafay, and R. Sipahi, Springer, New York, 2014.
• [33] J. Carrel, A Group Theoretic Approach to Abstract Linear Algebra, Springer, New York, 2016.

Uwagi

PL

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