Abstract
A fundamental issue in perturbation analysis is the study of how the asymptotic stability of a linear differential time-delay system varies with respect to small variations of the delay parameter. A classical approach for studying this problem for retarded type linear differential time-delay systems consists in computing the set of critical pairs of their quasipolynomials, namely the values of the delay parameter and the roots of the corresponding quasipolynomials that lie on the imaginary axis, and then in analyzing the variation of these roots with respect to small variations of the delay. Following this approach and using recent methods for solving polynomial systems, we propose a certified and efficient symbolic-numeric algorithm for computing the set of critical pairs of a quasipolynomial. Moreover, using recent algorithmic results developed by the computer algebra community, we present an efficient algorithm for the computation of Newton–Puiseux series at a critical pair. As explained in Li et al. (Analytic Curve Frequency-Sweeping Stability Tests for Systems with Commensurate Delays. Springer, Berlin 2015), these series play an important role in the stability analysis with respect to variations of the delay.
This work was supported by the ANR-13-BS03-0005 (MSDOS).
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References
Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry, vol. 10, 2nd edn. Algorithms and Computation in Mathematics. Springer, Berlin (2006)
Bellman, R.E., Cooke, K.L.: Differential-Difference Equations. The Rand Corporation, Santa Monica (1963)
Brieskorn, E., Knörrer, H.: Plane Algebraic Curves. Birkhaüser, Boston (1986)
Bouzidi, Y., Lazard, S., Pouget, M., Rouillier, F.: Separating linear forms and rational univariate representations of bivariate systems. J. Symb. Comput. 68, 84–119 (2015)
Bouzidi, Y., Lazard, S., Moroz, G., Pouget, M., Rouillier, F., Sagraloff, M.: Solving bivariate systems using rational univariate representations. J. Complex. 37, 34–75 (2016)
Bouzidi, Y., Poteaux, A., Quadrat, A.: Computer algebra methods for the stability analysis of differential systems with commensurate time-delays. In: Proceedings of the 13th IFAC Workshop on Time Delay Systems, Istanbul (Turkey), 22–24 June 2016
Collins, G.E., Akritas, A.G.: Polynomial real root isolation using Descarte’s rule of signs. In: Proceedings of the third ACM symposium on Symbolic and Algebraic Computation, pp. 272–275. ACM (1976)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms, 3rd edn. Undergraduate Texts in Mathematics. Springer, Berlin (2007)
Daouda, D.N., Mourrain, B., Ruatta, O.: On the computation of the topology of a non-reduced implicit space curve. In ISSAC’08: Proceedings of the Twenty-first International Symposium on Symbolic and Algebraic Computation, pp. 47–54. ACM, New York (2008)
Duval, D.: Rational Puiseux expansions. Compos. Math. 70(2), 119–154 (1989)
Gu, K., Kharitonov, V.L., Chen, J.: Stability of Time-Delay Systems. Birkhäuser, Boston (2003)
Hale, J.K.: Theory of Functional Differential Equations. Springer, Berlin (1977)
Kobel, A., Rouillier, F., Sagraloff, M.: Computing Real Roots of Real Polynomials ... and now for real!. International Symposium on Symbolic and Algebraic Computation (ISSAC) (2016)
Kung, H.T., Traub, J.F.: All algebraic functions can be computed fast. J. ACM 25(2), 245–260 (1978)
Li, X.-G., Niculescu, S.-I., Çela, A., Wang, H.-H., Cai, T.-Y.: On computing Puiseux series for multiple imaginary characteristic roots of LTI systems with commensurate delays. IEEE Trans. Autom. Control 58(5), 1338–1343 (2013)
Li, X.-G., Niculescu, S.-I., Çela, A.: Analytic Curve Frequency-Sweeping Stability Tests for Systems with Commensurate Delays. Springer, Berlin (2015)
Li, X.-G., Niculescu, S.-I., Çela, A., Zhang, L.: Characterizing invariance property of uniformly distributed delay systems, http://web1.lss.supelec.fr/delsys2015/Site/Program_files/Wed7_LI.pdf
Marshall, J.E., Gorecki, H., Korytowski, A., Walton, K.: Time-Delay. Systems: Stability and Performance Criteria with Applications. Ellis Horwood, London (1992)
Niculescu, S.-I.: Delay Effects on Stability: A Robust Control Approach, vol. 269. Lecture Notes in Control and Information Sciences. Springer, Berlin (2001)
Olgac, N., Sipahi, R.: An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems. IEEE Trans. Autom. Control 47(5), 793–796 (2002)
Poteaux, A.: Computing monodromy groups defined by plane algebraic curves. In: Proceedings of the 2007 International Workshop on Symbolic-Numeric Computation, pp. 36–45. ACM (2007)
Poteaux, A.: Calcul de développements de Puiseux et application au calcul de groupe de monodromie d’une courbe algébrique plane. Ph.D. Dissertation, Université de Limoges (2008)
Poteaux, A., Rybowicz, M.: Complexity bounds for the rational Newton-Puiseux algorithm over finite fields. Appl. Algebra Eng. Commun. Comput. 22, 187–217 (2011)
Rekasius, Z.V.: A stability test for systems with delays. In: Proceedings of the Joint Automatic Control Conference (1980)
Rouillier, F.: Solving zero-dimensional systems through the rational univariate representation. J. Appl. Algebra Eng. Commun. Comput. 9(5), 433–461 (1999)
Rouillier, F., Zimmermann, P.: Efficient isolation of polynomial real roots. J. Comput. Appl. Math. 162(1), 33–50 (2003)
Rouillier, F.: Algorithmes pour l’étude des solutions rélles des systèmes polynomiaux. Habilitation Thesis, University Paris 6, https://hal.inria.fr/tel-01435142
Rouillier, F.: RS: Real Roots isolation for algebraic systems with rational coefficients with a finite number of complex roots, https://who.rocq.inria.fr/Fabrice.Rouillier/software.php
Walker, R.J.: Algebraic Curves. Springer, Berlin (1950)
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Bouzidi, Y., Poteaux, A., Quadrat, A. (2019). A Symbolic Computation Approach Towards the Asymptotic Stability Analysis of Differential Systems with Commensurate Delays. In: Valmorbida, G., Seuret, A., Boussaada, I., Sipahi, R. (eds) Delays and Interconnections: Methodology, Algorithms and Applications. Advances in Delays and Dynamics, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-030-11554-8_11
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