Skip to main content

Tipp

Weitere Kapitel dieses Buchs durch Wischen aufrufen

2023 | OriginalPaper | Buchkapitel

Counting Characteristic Roots of Linear Delay Differential Equations. Part II: From Argument Principle to Rightmost Root Assignment Methods

verfasst von : Silviu-Iulian Niculescu, Islam Boussaada

Erschienen in: Controlling Delayed Dynamics

Verlag: Springer International Publishing

Aktivieren Sie unsere intelligente Suche um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

This chapter briefly presents some “user-friendly” methods and techniques (frequency-domain approaches) for the analysis and control of linear dynamical systems in presence of delays. The presentation is as simple as possible, focusing more on the main intuitive (algebraic, geometric) ideas to develop theoretical results, and their potential use in practical applications. To fix better the ideas, scalar and second-order examples are largely discussed. Next, a particular attention will be paid to the existing links between the maximal allowable multiplicity of the characteristic roots and the spectral abscissa of the dynamical system. The underlying property—multiplicity induced dominancy— will be particularly useful in constructing low-complexity controllers by partial pole placement. Such an idea is particularly exploited in vibration control.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Fußnoten
1
More precisely, the estimation of the upper bound of the number of the characteristic roots of exponential polynomials located on a horizontal strip.
 
2
Based on the same Cauchy’s argument principle.
 
3
Whose origins date back to the work of Evans in the late 1950s, Evans (1950), MacFarlane (1979).
 
4
Such links have been previously highlighted in Mazanti et al. (2021a) for a particular class of systems and then generalized in Boussaada et al. (2022).
 
5
It should be mentioned that, for reading facility, some specific notations are introduced later on.
 
6
Belonging to the Banach space of continuous functions \(\mathcal {C}([-\tau ,0],\mathbb {R}^n)\).
 
7
In terms of existence of solutions for the corresponding Cauchy problem.
 
8
That is the set of eigenvalues of the corresponding characteristic matrix function.
 
9
Depending on the parameters \(\overrightarrow{p}\) and \(\tau \).
 
10
It should be noted that, even in the simple case of second-order linear DDEs of retarded type, the corresponding 5 parameters induce a large computational effort to perform such an analysis.
 
11
The reader may find the proof of this theorem in any standard textbook on the theory of analytic functions.
 
12
In our case, we deal with quasipolynomial functions.
 
13
Quasipolynomials are analytic functions.
 
14
Called also the argument variation.
 
15
Which is strongly inspired from Stépán’s result.
 
16
More precisely, Pólya and G. Szegö (1972, Problem 206.2, p. 144 and p. 347).
 
17
Namely, the well-known Birkhoff interpolation problem.
 
18
Bogdanov-Takens singularity corresponds to equilibrium point admitting a double spectral value at zero for which corresponds a single eigenvector. Near a Bogdanov-Takens singularity a codimension 2 bifurcation may occur in two dimensional parameters space.
 
19
Indeed, in this case, we have \(N=2\), \(\tau _1=0,\,\tau _2=\tau \) and \(d_1=1,\,d_2=0\).
 
20
In other words, the sum of the degrees of the polynomials involved in the quasi-polynomial plus the number of polynomials involved minus one is called the degree of a given quasi-polynomial. Further discussions on such a notion can be found in Wielonsky (2001).
 
21
Such a property does not hold for scalar systems with one delay, for which increasing the delay leads to instability, see for instance Walton and Marshall (1987), Niculescu et al. (2010).
 
22
The software is freely available for download on https://​cutt.​ly/​p3delta, where installation instructions, video demonstrations, and the user guide are also available.
 
Literatur
Zurück zum Zitat Ahlfors, L. V. (1979). Complex analysis. McGraw-Hill, Inc. Ahlfors, L. V. (1979). Complex analysis. McGraw-Hill, Inc.
Zurück zum Zitat Balogh, T., Boussaada, I., Insperger, T., & Niculescu, S. -I. (2022). Conditions for stabilizability of time-delay systems with real-rooted plant. The International Journal of Robust and Nonlinear Control, (to appear). Balogh, T., Boussaada, I., Insperger, T., & Niculescu, S. -I. (2022). Conditions for stabilizability of time-delay systems with real-rooted plant. The International Journal of Robust and Nonlinear Control, (to appear).
Zurück zum Zitat Banks, H. T., del Rosario, R. C. H., & Tran, H. T. (2002). Proper orthogonal decomposition-based control of transverse beam vibrations: Experimental implementation. IEEE Transactions on Control Systems Technology,  10(5), 717–726. CrossRef Banks, H. T., del Rosario, R. C. H., & Tran, H. T. (2002). Proper orthogonal decomposition-based control of transverse beam vibrations: Experimental implementation. IEEE Transactions on Control Systems Technology,  10(5), 717–726. CrossRef
Zurück zum Zitat Barker, L. K. (1979). Mikhailov stability criterion for time-delayed systems. Technical Memorandum TM-78803, NASA, Washington D.C., USA. Barker, L. K. (1979). Mikhailov stability criterion for time-delayed systems. Technical Memorandum TM-78803, NASA, Washington D.C., USA.
Zurück zum Zitat Bellman, R., & Cooke, K. (1963). Differential-difference equations. New York: Academic Press. Bellman, R., & Cooke, K. (1963). Differential-difference equations. New York: Academic Press.
Zurück zum Zitat Benarab, A., Boussaada, I., Trabelsi, K., Mazanti, G., & Bonnet, C. (2020). The MID property for a second-order neutral time-delay differential equation. In Proceedings of the 24th International Conference on System Theory, Control and Computing, pp. 202–207. Benarab, A., Boussaada, I., Trabelsi, K., Mazanti, G., & Bonnet, C. (2020). The MID property for a second-order neutral time-delay differential equation. In Proceedings of the 24th International Conference on System Theory, Control and Computing, pp. 202–207.
Zurück zum Zitat Bini, D. A., & Boito, P. (2010). A fast algorithm for approximate polynomial GCD based on structured matrix computations. In D. A. Bini et al., (Ed.), Numerical methods for structured matrices and applications (Vol. 199, pp. 155–173) OT. Birkhäuser Basel. Bini, D. A., & Boito, P. (2010). A fast algorithm for approximate polynomial GCD based on structured matrix computations. In D. A. Bini et al., (Ed.), Numerical methods for structured matrices and applications (Vol. 199, pp. 155–173) OT. Birkhäuser Basel.
Zurück zum Zitat Boese, F. G. (1998). Stability with respect to the delay: On a paper of K. L. Cooke and P. van den Driessche. Journal of Mathematical Analysis and Applications, 228(2), 293–321. Boese, F. G. (1998). Stability with respect to the delay: On a paper of K. L. Cooke and P. van den Driessche. Journal of Mathematical Analysis and Applications, 228(2), 293–321.
Zurück zum Zitat Boussaada, I., & Niculescu, S. -I. (2014). Computing the codimension of the singularity at the origin for delay systems: The missing link with Birkhoff incidence matrices. Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems, pp. 1–8. Boussaada, I., & Niculescu, S. -I. (2014). Computing the codimension of the singularity at the origin for delay systems: The missing link with Birkhoff incidence matrices. Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems, pp. 1–8.
Zurück zum Zitat Boussaada, I., & Niculescu, S.-I. (2016a). Characterizing the codimension of zero singularities for time-delay systems. Acta Applicandae Mathematicae,  145(1), 47–88. Boussaada, I., & Niculescu, S.-I. (2016a). Characterizing the codimension of zero singularities for time-delay systems. Acta Applicandae Mathematicae,  145(1), 47–88.
Zurück zum Zitat Boussaada, I., & Niculescu, S. I. (2016b). Tracking the algebraic multiplicity of crossing imaginary roots for generic quasipolynomials: A Vandermonde-based approach. IEEE Transactions on Automatic Control,  61, 1601–1606. Boussaada, I., & Niculescu, S. I. (2016b). Tracking the algebraic multiplicity of crossing imaginary roots for generic quasipolynomials: A Vandermonde-based approach. IEEE Transactions on Automatic Control,  61, 1601–1606.
Zurück zum Zitat Boussaada, I., & Niculescu, S.-I. (2018). On the dominancy of multiple spectral values for time-delay systems with applications. IFAC-PapersOnLine,  51(14), 55–60. CrossRef Boussaada, I., & Niculescu, S.-I. (2018). On the dominancy of multiple spectral values for time-delay systems with applications. IFAC-PapersOnLine,  51(14), 55–60. CrossRef
Zurück zum Zitat Boussaada, I., Mazanti, G., Niculescu, S. I., Huynh, J., Sim, F., & Thomas, M. (2022). Partial pole placement via delay action: A Python software for delayed feedback stabilizing design. Boussaada, I., Mazanti, G., Niculescu, S. I., Huynh, J., Sim, F., & Thomas, M. (2022). Partial pole placement via delay action: A Python software for delayed feedback stabilizing design.
Zurück zum Zitat Boussaada, I., Irofti, D. A., & Niculescu, S. I. (2014). Computing the codimension of the singularity at the origin for time-delay systems in the regular case: A Vandermonde-based approach. In Proceedings of the 13th European Control Conference, pp. 1–6. Boussaada, I., Irofti, D. A., & Niculescu, S. I. (2014). Computing the codimension of the singularity at the origin for time-delay systems in the regular case: A Vandermonde-based approach. In Proceedings of the 13th European Control Conference, pp. 1–6.
Zurück zum Zitat Boussaada, I., Unal, H. U., & Niculescu, S. I. (2016). Multiplicity and stable varieties of time-delay systems: A missing link. In Proceedings of the 22nd International Symposium on Mathematical Theory of Networks and Systems, pp. 1–6. Boussaada, I., Unal, H. U., & Niculescu, S. I. (2016). Multiplicity and stable varieties of time-delay systems: A missing link. In Proceedings of the 22nd International Symposium on Mathematical Theory of Networks and Systems, pp. 1–6.
Zurück zum Zitat Boussaada, I., Niculescu, S.-I., Tliba, S., & Vyhlídal, T. (2017). On the coalescence of spectral values and its effect on the stability of time-delay systems: Application to active vibration control. Procedia IUTAM,  22, 75–82. CrossRef Boussaada, I., Niculescu, S.-I., Tliba, S., & Vyhlídal, T. (2017). On the coalescence of spectral values and its effect on the stability of time-delay systems: Application to active vibration control. Procedia IUTAM,  22, 75–82. CrossRef
Zurück zum Zitat Boussaada, I., Tliba, S., Niculescu, S. I., Ünal, H. U., & Vyhlídal, T. (2018). Further remarks on the effect of multiple spectral values on the dynamics of time-delay systems: Application to the control of a mechanical system. Linear Algebra Application,  542, 589–604. Boussaada, I., Tliba, S., Niculescu, S. I., Ünal, H. U., & Vyhlídal, T. (2018). Further remarks on the effect of multiple spectral values on the dynamics of time-delay systems: Application to the control of a mechanical system. Linear Algebra Application,  542, 589–604.
Zurück zum Zitat Boussaada, I., Niculescu, S. I., El-Ati, A., Pérez-Ramos, R., & Trabelsi, K. (2020). Multiplicity-induced-dominancy in parametric second-order delay differential equations: Analysis and application in control design. ESAIM: COCV, 26, 57. Boussaada, I., Niculescu, S. I., El-Ati, A., Pérez-Ramos, R., & Trabelsi, K. (2020). Multiplicity-induced-dominancy in parametric second-order delay differential equations: Analysis and application in control design. ESAIM: COCV, 26, 57.
Zurück zum Zitat Boussaada, I., Mazanti, G., & Niculescu, S. I. (2022). The generic multiplicity-induced-dominancy property from retarded to neutral delay-differential equations: When delay-systems characteristics meet the zeros of Kummer functions. Comptes Rendus de Mathématique. https://​hal.​archives-ouvertes.​fr/​hal-03298718. Boussaada, I., Mazanti, G., & Niculescu, S. I. (2022). The generic multiplicity-induced-dominancy property from retarded to neutral delay-differential equations: When delay-systems characteristics meet the zeros of Kummer functions. Comptes Rendus de Mathématique. https://​hal.​archives-ouvertes.​fr/​hal-03298718.
Zurück zum Zitat Buchholz, H. (1969). The confluent hypergeometric function with special emphasis on its applications. Translated from the German by H. Lichtblau and K. Wetzel. Springer Tracts in Natural Philosophy (Vol. 15). Springer New York Inc., New York. Buchholz, H. (1969). The confluent hypergeometric function with special emphasis on its applications. Translated from the German by H. Lichtblau and K. Wetzel. Springer Tracts in Natural Philosophy (Vol. 15). Springer New York Inc., New York.
Zurück zum Zitat Chen, C. F., & Tsay, Y. T. (1976). A general frequency stability criterion for multi-input-output, lumped and distributed-parameter systems. International Journal of Control,  23(3), 341–359. MathSciNetCrossRef Chen, C. F., & Tsay, Y. T. (1976). A general frequency stability criterion for multi-input-output, lumped and distributed-parameter systems. International Journal of Control,  23(3), 341–359. MathSciNetCrossRef
Zurück zum Zitat Chen, W., Buehler, M., Parker, G., & Bettig, B. (2004). Optimal sensor design and control of piezoelectric laminate beams. IEEE Transactions on Control Systems Technology,  12(1), 148–155. CrossRef Chen, W., Buehler, M., Parker, G., & Bettig, B. (2004). Optimal sensor design and control of piezoelectric laminate beams. IEEE Transactions on Control Systems Technology,  12(1), 148–155. CrossRef
Zurück zum Zitat Cooke, K. L. (1979). Stability analysis for a vector disease model. Rocky Mountain Journal of Mathematics,  9, 31–42. MathSciNetCrossRef Cooke, K. L. (1979). Stability analysis for a vector disease model. Rocky Mountain Journal of Mathematics,  9, 31–42. MathSciNetCrossRef
Zurück zum Zitat El’sgol’ts, L. E., & Norkin, S. B. (1973). Introduction to the theory and application of differential equations with deviating arguments. Academic Press. El’sgol’ts, L. E., & Norkin, S. B. (1973). Introduction to the theory and application of differential equations with deviating arguments. Academic Press.
Zurück zum Zitat Erdélyi, A., Magnus, W., Oberhettinger, F., & Tricomi, F. (1981). Higher transcendental functions (Vol. I). Melbourne, USA: Robert E. Krieger Publishing Co. Erdélyi, A., Magnus, W., Oberhettinger, F., & Tricomi, F. (1981). Higher transcendental functions (Vol. I). Melbourne, USA: Robert E. Krieger Publishing Co.
Zurück zum Zitat Evans, W. E. (1950). Control system synthesis by root locus method. AIEE Transactions,  69, 66–69. Evans, W. E. (1950). Control system synthesis by root locus method. AIEE Transactions,  69, 66–69.
Zurück zum Zitat Gérardin, M., & Rixen, D. (1997). Mechanical vibrations: Theory and application to structural dynamics (2nd ed.). Wiley. Gérardin, M., & Rixen, D. (1997). Mechanical vibrations: Theory and application to structural dynamics (2nd ed.). Wiley.
Zurück zum Zitat Ha, T. T., & Gibson, J. A. (1980). A note on the determinant of a functional confluent Vandermonde matrix and controllability. Linear Algebra and its Applications,  30, 69–75. MathSciNetCrossRef Ha, T. T., & Gibson, J. A. (1980). A note on the determinant of a functional confluent Vandermonde matrix and controllability. Linear Algebra and its Applications,  30, 69–75. MathSciNetCrossRef
Zurück zum Zitat Hassard, B. D. (1997). Counting roots of the characteristic equation for linear delay-differential systems. Journal of Differential Equations,  136(2), 222–235. MathSciNetCrossRef Hassard, B. D. (1997). Counting roots of the characteristic equation for linear delay-differential systems. Journal of Differential Equations,  136(2), 222–235. MathSciNetCrossRef
Zurück zum Zitat Hayes, N. D. (1950). Roots of the transcendental equation associated with a certain difference-differential equation. The Journal of the London Mathematical Society, s1–25(3), 226–232. Hayes, N. D. (1950). Roots of the transcendental equation associated with a certain difference-differential equation. The Journal of the London Mathematical Society, s1–25(3), 226–232.
Zurück zum Zitat Kharitonov, V. L., Niculescu, S.-I., Moreno, J., & Michiels, W. (2005). Static output feedback stabilization: Necessary conditions for multiple delay controllers. IEEE Transactions on Automatic Control,  50(1), 82–86. MathSciNetCrossRef Kharitonov, V. L., Niculescu, S.-I., Moreno, J., & Michiels, W. (2005). Static output feedback stabilization: Necessary conditions for multiple delay controllers. IEEE Transactions on Automatic Control,  50(1), 82–86. MathSciNetCrossRef
Zurück zum Zitat Kirillov, O., & Overton, M. (2013). Robust stability at the swallowtail singularity. Frontiers in Physics,  1, 24. CrossRef Kirillov, O., & Overton, M. (2013). Robust stability at the swallowtail singularity. Frontiers in Physics,  1, 24. CrossRef
Zurück zum Zitat Komzsik, L. (2005). What every engineer should know about computational techniques of finite element analysis. Taylor & Francis Group, NW, FL: CRC Press. Komzsik, L. (2005). What every engineer should know about computational techniques of finite element analysis. Taylor & Francis Group, NW, FL: CRC Press.
Zurück zum Zitat Krall, A. M. (1968). Stability techniques for continuous linear systems. New York: Gordon and Breach. Krall, A. M. (1968). Stability techniques for continuous linear systems. New York: Gordon and Breach.
Zurück zum Zitat Kögl, M., & Bucalem, M. L. (2005). Analysis of smart laminates using piezoelectric MITC plate and shell elements. Computers & Structures,  83, 1153–1163. CrossRef Kögl, M., & Bucalem, M. L. (2005). Analysis of smart laminates using piezoelectric MITC plate and shell elements. Computers & Structures,  83, 1153–1163. CrossRef
Zurück zum Zitat Ma, D., Boussaada, I., Bonnet, C., Niculescu, S. I., & Chen, J. (2020). Multiplicity-Induced-Dominancy extended to neutral delay equations: Towards a systematic PID tuning based on Rightmost root assignment. In Proceedings of the 2020 American Control Conference, Denver, USA. Ma, D., Boussaada, I., Bonnet, C., Niculescu, S. I., & Chen, J. (2020). Multiplicity-Induced-Dominancy extended to neutral delay equations: Towards a systematic PID tuning based on Rightmost root assignment. In Proceedings of the 2020 American Control Conference, Denver, USA.
Zurück zum Zitat MacFarlane, A. F. G. (1979). Frequency-response methods in control systems. New York: IEEE Press. MacFarlane, A. F. G. (1979). Frequency-response methods in control systems. New York: IEEE Press.
Zurück zum Zitat Marden, M. (1949). Geometry of polynomials. AMS: Providence. Marden, M. (1949). Geometry of polynomials. AMS: Providence.
Zurück zum Zitat Mazanti, G., Boussaada, I., Niculescu, S. I., & Vyhlídal, T. (2020). Spectral dominance of complex roots for single-delay linear equations. In Proceedings of the 21st IFAC World Congress, IFAC-PapersOnLine, Berlin, Germany. Mazanti, G., Boussaada, I., Niculescu, S. I., & Vyhlídal, T. (2020). Spectral dominance of complex roots for single-delay linear equations. In Proceedings of the 21st IFAC World Congress, IFAC-PapersOnLine, Berlin, Germany.
Zurück zum Zitat Mazanti, G., Boussaada, I., & Niculescu, S. -I. (2021a). Multiplicity-induced-dominancy for delay-differential equations of retarded type. The Journal of Differential Equations, 286, 84–118. Mazanti, G., Boussaada, I., & Niculescu, S. -I. (2021a). Multiplicity-induced-dominancy for delay-differential equations of retarded type. The Journal of Differential Equations, 286, 84–118.
Zurück zum Zitat Mazanti, G., Boussaada, I., Niculescu, S. I., & Chitour, Y. (2021b) Effects of roots of maximal multiplicity on the stability of some classes of delay differential-algebraic systems: The lossless propagation case. In Proceedings of the 24th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2021) (Vol. 54. pp. 764–769). IFAC-PapersOnLine. Cambridge, UK. Mazanti, G., Boussaada, I., Niculescu, S. I., & Chitour, Y. (2021b) Effects of roots of maximal multiplicity on the stability of some classes of delay differential-algebraic systems: The lossless propagation case. In Proceedings of the 24th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2021) (Vol. 54. pp. 764–769). IFAC-PapersOnLine. Cambridge, UK.
Zurück zum Zitat Michiels, W., & Niculescu, S. -I. (2007). Stability and stabilization of time-delay systems (Vol. 12) Advances in design and control. SIAM. Michiels, W., & Niculescu, S. -I. (2007). Stability and stabilization of time-delay systems (Vol. 12) Advances in design and control. SIAM.
Zurück zum Zitat Mori, T., & Kokame, H. (1989). Stability of \(\dot{x}(t) = {A}x(t) + {B}x(t-\tau )\). IEEE Transactions on Automatic Control,  34, 460–462. MathSciNetCrossRef Mori, T., & Kokame, H. (1989). Stability of \(\dot{x}(t) = {A}x(t) + {B}x(t-\tau )\). IEEE Transactions on Automatic Control,  34, 460–462. MathSciNetCrossRef
Zurück zum Zitat Mori, T., Fukuma, N., & Kuwahara, M. (1982). On an estimate of the decay rate for stable linear delay systems. International Journal of Control,  36(1), 95–97. MathSciNetCrossRef Mori, T., Fukuma, N., & Kuwahara, M. (1982). On an estimate of the decay rate for stable linear delay systems. International Journal of Control,  36(1), 95–97. MathSciNetCrossRef
Zurück zum Zitat Niculescu, S.-I., & Michiels, W. (2004). Stabilizing a chain of integrators using multiple delays. IEEE Transactions on Automatic Control,  49(5), 802–807. MathSciNetCrossRef Niculescu, S.-I., & Michiels, W. (2004). Stabilizing a chain of integrators using multiple delays. IEEE Transactions on Automatic Control,  49(5), 802–807. MathSciNetCrossRef
Zurück zum Zitat S-Niculescu, S. I., Michiels, W., Gu, K., & Abdallah, C. T. (2010). Delay effects on output feedback control of dynamical systems (pp. 63–84). Springer Berlin. S-Niculescu, S. I., Michiels, W., Gu, K., & Abdallah, C. T. (2010). Delay effects on output feedback control of dynamical systems (pp. 63–84). Springer Berlin.
Zurück zum Zitat Niculescu, S. I., Boussaada, I., Li, X. G., Mazanti, G., & Méndez-Barrios, C. -F. (2021). Stability, delays and multiple characteristic roots in dynamical systems: A guided tour. In Proceedings of the 16th IFAC Workshop Time Delay Systems (Vol. 54, pp. 222–239), IFAC-PapersOnLine. Guangzhou, China. Niculescu, S. I., Boussaada, I., Li, X. G., Mazanti, G., & Méndez-Barrios, C. -F. (2021). Stability, delays and multiple characteristic roots in dynamical systems: A guided tour. In Proceedings of the 16th IFAC Workshop Time Delay Systems (Vol. 54, pp. 222–239), IFAC-PapersOnLine. Guangzhou, China.
Zurück zum Zitat Niculescu, S. -I., Li, X. -G., & Çela, A. (2023). Counting characteristic roots of linear delay-differential equations. Part I: frequency-sweeping stability tests and applications. In D. Breda (Ed.), Controlling Delayed Dynamics: Advances in Theory, Methods and Applications, CISM Lecture Notes (pp. 117–156). Wien-New York: Springer. Niculescu, S. -I., Li, X. -G., & Çela, A. (2023). Counting characteristic roots of linear delay-differential equations. Part I: frequency-sweeping stability tests and applications. In D. Breda (Ed.), Controlling Delayed Dynamics: Advances in Theory, Methods and Applications, CISM Lecture Notes (pp. 117–156). Wien-New York: Springer.
Zurück zum Zitat Nyquist, H. (1932). Regeneration theory. Bell System Technical Journal,  11(1), 126–147. CrossRef Nyquist, H. (1932). Regeneration theory. Bell System Technical Journal,  11(1), 126–147. CrossRef
Zurück zum Zitat Obreschkoff, N. (1928). Nullstellen linearer kombinationen von exponentialfunktionen. Jahresbericht der Deutschen Mathematiker-Vereinigung,  37, 81–84. Obreschkoff, N. (1928). Nullstellen linearer kombinationen von exponentialfunktionen. Jahresbericht der Deutschen Mathematiker-Vereinigung,  37, 81–84.
Zurück zum Zitat Olver, F. W. J., Lozier, D. W., Boisvert, R. F., & Clark, C. W. (Eds.). NIST handbook of mathematical functions. Cambridge: Cambridge University Press. Olver, F. W. J., Lozier, D. W., Boisvert, R. F., & Clark, C. W. (Eds.). NIST handbook of mathematical functions. Cambridge: Cambridge University Press.
Zurück zum Zitat Pinney, E. (1958). Ordinary difference-differential equations. Berkeley: University California Press. Pinney, E. (1958). Ordinary difference-differential equations. Berkeley: University California Press.
Zurück zum Zitat G. Pólya, G., & Szegő, G. (1972). Problems and theorems in analysis (Vol. I) Integral calculus, theory of functions. Springer, New York. G. Pólya, G., & Szegő, G. (1972). Problems and theorems in analysis (Vol. I) Integral calculus, theory of functions. Springer, New York.
Zurück zum Zitat Ruan, S. (2006). Delay differential equations in single species dynamics. In Delay Differential Equations and Applications (pp. 477–517). Springer. Ruan, S. (2006). Delay differential equations in single species dynamics. In Delay Differential Equations and Applications (pp. 477–517). Springer.
Zurück zum Zitat Satche, M. (1949). Discussions on “stability of linear oscillating systems with constant time lag." Journal of Applied Mechanics, Transactions ASME, 16(4), 419–420. Satche, M. (1949). Discussions on “stability of linear oscillating systems with constant time lag." Journal of Applied Mechanics, Transactions ASME, 16(4), 419–420.
Zurück zum Zitat Stépán, G. (1979). On the stability of linear differential equations with delay (pp. 971–984). Colloquia Mathematica Societas Janos Bolyai: In Qualitative Theory of Differential Equations. Stépán, G. (1979). On the stability of linear differential equations with delay (pp. 971–984). Colloquia Mathematica Societas Janos Bolyai: In Qualitative Theory of Differential Equations.
Zurück zum Zitat Stépán, G. (1989). Retarded dynamical systems: Stability and characteristic functions. London: Longman Scientific. Stépán, G. (1989). Retarded dynamical systems: Stability and characteristic functions. London: Longman Scientific.
Zurück zum Zitat Tliba, S. (2004). Contrôle actif des vibrations dans des structures mécaniques minces instrumentés de transducteurs piézoélectriques (in French). Ph.D. Thesis, ENS Cachan, France. Tliba, S. (2004). Contrôle actif des vibrations dans des structures mécaniques minces instrumentés de transducteurs piézoélectriques (in French). Ph.D. Thesis, ENS Cachan, France.
Zurück zum Zitat Tliba, S. (2011). Control of a vibrating axisymmetric membrane using piezoelectric transducers. In Proceedings of the 18th IFAC World Congress (pp. 7713 (6 p.)), Milano, Italy. Tliba, S. (2011). Control of a vibrating axisymmetric membrane using piezoelectric transducers. In Proceedings of the 18th IFAC World Congress (pp. 7713 (6 p.)), Milano, Italy.
Zurück zum Zitat Tliba, S., Abou-Kandil, H., & Prieur, C. (2005). Active vibration damping of a smart flexible structure using piezoelectric transducers: \(\cal{H}_\infty \) design and experimental results. In Proceedings of the 16th IFAC World Congress, Prague. Tliba, S., Abou-Kandil, H., & Prieur, C. (2005). Active vibration damping of a smart flexible structure using piezoelectric transducers: \(\cal{H}_\infty \) design and experimental results. In Proceedings of the 16th IFAC World Congress, Prague.
Zurück zum Zitat Tsypkin, Y. Z. (1946). Stability of systems with delayed feedback. Avtomathika i Telemech,  7, 107–129. Tsypkin, Y. Z. (1946). Stability of systems with delayed feedback. Avtomathika i Telemech,  7, 107–129.
Zurück zum Zitat Vyhlídal, T., & Zitek, P. (2009). Mapping based algorithm for large-scale computation of quasi-polynomial zeros. IEEE Transactions on Automatic Control,  54(1), 171–177. MathSciNetCrossRef Vyhlídal, T., & Zitek, P. (2009). Mapping based algorithm for large-scale computation of quasi-polynomial zeros. IEEE Transactions on Automatic Control,  54(1), 171–177. MathSciNetCrossRef
Zurück zum Zitat Walton, K., & Marshall, J. E. (1987). Direct method for TDS stability analysis. EE Proceedings D—Control Theory and Applications, 134(2), 101–107. Walton, K., & Marshall, J. E. (1987). Direct method for TDS stability analysis. EE Proceedings D—Control Theory and Applications, 134(2), 101–107.
Zurück zum Zitat Wielonsky, F. (2001). A Rolle’s theorem for real exponential polynomials in the complex domain. Journal de Mathématiques Pures et Appliquées,  4, 389–408. MathSciNetCrossRef Wielonsky, F. (2001). A Rolle’s theorem for real exponential polynomials in the complex domain. Journal de Mathématiques Pures et Appliquées,  4, 389–408. MathSciNetCrossRef
Zurück zum Zitat Wynn, P. (1973). On the zeros of certain confluent hypergeometric functions. Proceedings of the American Mathematical Society,  40, 173–182. MathSciNetCrossRef Wynn, P. (1973). On the zeros of certain confluent hypergeometric functions. Proceedings of the American Mathematical Society,  40, 173–182. MathSciNetCrossRef
Metadaten
Titel
Counting Characteristic Roots of Linear Delay Differential Equations. Part II: From Argument Principle to Rightmost Root Assignment Methods
verfasst von
Silviu-Iulian Niculescu
Islam Boussaada
Copyright-Jahr
2023
DOI
https://doi.org/10.1007/978-3-031-01129-0_6

    Marktübersichten

    Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.