This work revisits recent results on maximal multiplicity induced-dominancy for spectral values in reduced-order time-delay systems and extends it to the general class of second-order retarded differential equations. A parametric multiplicity-induced-dominancy property is characterized, allowing to a delayed stabilizing design with reduced complexity. As a matter of fact, the approach is merely a delayed-output-feedback where the candidates’ delays and gains result from the manifold defining the maximal multiplicity of a real spectral value, then, the dominancy is shown using the argument principle. Sensitivity of the control design with respect to the parameters uncertainties/variation is discussed. Various reduced order examples illustrate the applicative perspectives of the approach.
Keywords: Time-delay systems, stability and stabilization, exponential decay, pole-placement, control design
@article{COCV_2020__26_1_A57_0,
author = {Boussaada, Islam and Niculescu, Silviu-Iulian and El-Ati, Ali and P\'erez-Ramos, Redamy and Trabelsi, Karim},
title = {Multiplicity-induced-dominancy in parametric second-order delay differential equations: {Analysis} and application in control design},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2020},
publisher = {EDP Sciences},
volume = {26},
doi = {10.1051/cocv/2019073},
mrnumber = {4147584},
zbl = {1453.34098},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2019073/}
}
TY - JOUR AU - Boussaada, Islam AU - Niculescu, Silviu-Iulian AU - El-Ati, Ali AU - Pérez-Ramos, Redamy AU - Trabelsi, Karim TI - Multiplicity-induced-dominancy in parametric second-order delay differential equations: Analysis and application in control design JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2019073/ DO - 10.1051/cocv/2019073 LA - en ID - COCV_2020__26_1_A57_0 ER -
%0 Journal Article %A Boussaada, Islam %A Niculescu, Silviu-Iulian %A El-Ati, Ali %A Pérez-Ramos, Redamy %A Trabelsi, Karim %T Multiplicity-induced-dominancy in parametric second-order delay differential equations: Analysis and application in control design %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2019073/ %R 10.1051/cocv/2019073 %G en %F COCV_2020__26_1_A57_0
Boussaada, Islam; Niculescu, Silviu-Iulian; El-Ati, Ali; Pérez-Ramos, Redamy; Trabelsi, Karim. Multiplicity-induced-dominancy in parametric second-order delay differential equations: Analysis and application in control design. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 57. doi: 10.1051/cocv/2019073
[1] , Complex Analysis, McGraw-Hill, Inc., New York (1979). | Zbl | MR
[2] , , and , On qualitative properties of low-degree quasipolynomials: Further remarks on the spectral abscissa and rightmost-roots assignment. Bull. Math. Soc. Sci. Math. Roumanie 61 (2018) 361–381. | MR | Zbl
[3] and , Differential-difference Equations. Academic Press, New York (1963). | MR | Zbl
[4] , and , Robust Control: The Parametric Approach, Prentice-Hall information and system sciences series. Prentice Hall PTR (1995). | Zbl
[5] , Stability with respect to the delay: On a paper of k. l. cooke and p. van den driessche. J. Math. Anal. Appl. 228 (1998) 293–321. | Zbl | MR | DOI
[6] , and , Inverted pendulum stabilization: Characterization of codimension-three triple zero bifurcation via multiple delayed proportional gains. Syst. Contr. Lett. 82 (2015) 1–9. | MR | Zbl | DOI
[7] and , Computing the codimension of the singularity at the origin for delay systems: The missing link with Birkhoff incidence matrices. 21st International Symposium on Mathematical Theory of Networks and Systems, 1–8 (2014).
[8] and Characterizing the codimension of zero singularities for time-delay systems. Acta Applicandae Mathematicae 145 (2016) 47–88. | Zbl | MR | DOI
[9] and , Tracking the algebraic multiplicity of crossing imaginary roots for generic quasipolynomials: A Vandermonde-based approach. IEEE Trans. Autom. Contr. 61 (2016) 1601–1606. | MR | Zbl | DOI
[10] and , On the dominancy of multiple spectral values for time-delay systems with applications. IFAC-PapersOnLine 51 (2018) 5–60. | DOI
[11] , , and , On the coalescence of spectral values and its effect on the stability of time-delay systems: Application to active vibration control. Procedia IUTAM 22 (2017) 75–82. | DOI
[12] , and , Toward a decay rate assignment based design for time-delay systems with multiple spectral values, in Proceeding of the 23rd International Symposium on Mathematical Theory of Networks and Systems (2018) 864–871.
[13] , , , and , 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 Appl. 542 (2018) 589–604. | MR | Zbl | DOI
[14] , , , and , Proceedings of the 20th ILAS Conference, Leuven, Belgium (2016).
[15] , and , Multiplicity and stable varieties of time-delay systems: A missing link, in Proceeding of the 22nd International Symposium on Mathematical Theory of Networks and Systems (2016).
[16] , On characteristic roots and stability charts of delay differential equations. Int. J. Robust Nonlinear Contr. 22 (2012) 892–917. | MR | Zbl | DOI
[17] , Stability analysis for a vector disease model. Rocky Mountain J. Math. 9 (1979) 31–42. | MR | Zbl | DOI
[18] and , On zeroes of some transcendental equations. Funkcial. Ekvac. 29 (1986) 77–90. | MR | Zbl
[19] and , On stability of lms methods and characteristic roots of delay differential equations. SIAM J. Numer. Anal. 40 (2003) 629–650. | MR | Zbl | DOI
[20] , and , Stability of Time-Delay Systems, Birkhauser Boston, Inc., Cambridge, MA, (2003). | MR | Zbl | DOI
[21] and , Introduction to functional differential equations, Applied Mathematics Sciences, Vol. 99, Springer Verlag, New York, 1993. | MR | Zbl | DOI
[22] , Counting roots of the characteristic equation for linear delay-differential systems. J. Differ. Equ. 136 (1997) 222–235. | MR | Zbl | DOI
[23] , Roots of the transcendental equation associated with a certain difference-differential equation. J. London Math. Soc. 25 (1950) 226–232. | MR | Zbl | DOI
[24] and , Semi-Discretization for Time-Delay Systems: Stability and Engineering Applications, Applied Mathematics Sciences. Springer, Berlin (2011). | MR | Zbl | DOI
[25] , , and , Static output feedback stabilization: necessary conditions for multiple delay controllers. IEEE Trans. Automat. Contr. 50 (2005) 82–86. | MR | Zbl | DOI
[26] and , Robust stability at the swallowtail singularity. Front. Phys. 1 (2013) 4. | DOI
[27] , , and , Dynamics of an inverted pendulum with delayed feedback control. SIAM J. Appl. Dyn. Syst. 4 (2005) 333–351. | MR | Zbl | DOI
[28] , Invariant factors assignment for a class of time-delay systems. Kybernetika 37 (2001) 265–275. | MR | Zbl
[29] and , Delay margin of low-order systems achievable by pid controllers. IEEE Trans. Automat. Contr. 64 (2019) 1958–1973. | MR | Zbl | DOI
[30] , Feedback controllers for a wind tunnel model involving a delay: Analytical design and numerical simulation. IEEE Trans. Automat. Contr. 29 (1984) 1058–1068. | MR | DOI
[31] and , Finite spectrum assignment problem for systems with delays. IEEE Trans. Automat. Contr. 24 (1979) 541–552. | MR | Zbl | DOI
[32] , Geometry of Polynomials, Number 3 in Geometry of Polynomials. American Mathematical Society, Providence, Rhode Island, USA, 1949. | MR | Zbl
[33] , and , An explicit formula for the splitting of multiple eigenvalues for nonlinear eigenvalue problems and connections with the linearization for the delay eigenvalue problem. SIAM J. Matrix Anal. Appl. 38 (2017) 599–620. | MR | Zbl | DOI
[34] , , and , Continuous pole placement for delay equations. Automatica 38 (2002) 747–761. | MR | Zbl | DOI
[35] and , Stability and stabilization of time-delay systems, Advances in Design and Control. SIAM, Philadelphia, USA (2007). | MR | Zbl
[36] , The methods of harmonie analysis in the theory of control. Avtomat. i Telemekh 3 (1938) 27–81.
[37] and , Finite spectrum assignment for input delay systems. IFAC Proc. 34 (2001) 201–206.
[38] , and , On an estimate of the decay rate for stable linear delay systems. Int. J. Contr. 36 (1982) 95–97. | MR | Zbl | DOI
[39] and , Stability of . IEEE Trans. Automat. Contr. 34 (1989) 460–462. | MR | Zbl
[40] and , Stabilizing a chain of integrators using multiple delays. IEEE Trans. Automat. Contr. 49 (2004) 802–807. | MR | Zbl | DOI
[41] , , and , Delay Effects on Output Feedback Control of Dynamical Systems. Springer Berlin Heidelberg, Berlin, Heidelberg (2010) 63–84. | Zbl
[42] , Regeneration theory. Bell Syst. Techn. J. 11 (1932) 126–147. | Zbl | DOI
[43] and , An exact method for the stability analysis of time delayed linear time-invariant (lti) systems. IEEE Trans. Automat. Contr. 47 (2002) 793–797. | MR | Zbl | DOI
[44] and , An exact method for the stability analysis of time-delayed lti systems. IEEE Trans. Automat. Contr. 47 (2002) 793–797. | MR | Zbl | DOI
[45] , Nyquist criterion for systems with distributed delays. Ann. DAAAM Proc. (2011) 485–487. | DOI
[46] , and , A stability test for control systems with delays based on the nyquist criterion. Int. J. Math. Mod. Meth. Appl. Sci. 5 (2011) 1213–1224.
[47] and . Problems and Theorems in Analysis, Vol. 1, Integral Calculus, Theory of Functions, Springer-Verlag, New York, Heidelberg, Berlin (1972). | MR | Zbl
[48] , , and , Design of proportional-integral-retarded (pir) controllers for second-order lti systems. IEEE Trans. Automat. Contr. 99 (2015) 1–6. | MR | Zbl
[49] , A stability test for systems with delays. Joint Automat. Contr. Conf. 17 (1980) 39.
[50] , Delay differential equations in single species dynamics, in Delay Differential Equations and Applications, Springer, Berlin (2006) 477–517. | MR | Zbl | DOI
[51] , Real and complex analysis, In Mathematics series, McGraw-Hill. New York (1987). | Zbl | MR
[52] and , Bifurcation analysis of an inverted pendulum with delayed feedback control near a triple-zero eigenvalue singularity. Nonlinearity 17 (2004) 85–103. | MR | Zbl | DOI
[53] and , Extending the permissible control loop latency for the controlled inverted pendulum. Dyn. Syst. 20 (2005) 189–199. | MR | Zbl | DOI
[54] , , , and , Stability and stabilization of systems with time delay. IEEE Contr. Syst. 31 (2011) 38–65. | MR | Zbl | DOI
[55] , Retarded Dynamical Systems: Stability and Characteristic Functions, Pitman research notes in mathematics series. Longman Scientific and Technical, London (1989). | MR | Zbl
[56] and , Proportional minus delay controller. IEEE Trans. Automat. Contr. 24 (1979) 370–372. | MR | Zbl | DOI
[57] , , and , A nonsmooth optimisation approach for the stabilisation of time-delay systems. ESAIM: COCV 14 (2008) 478–493. | MR | Zbl | Numdam
[58] and , Mapping based algorithm for large-scale computation of quasi-polynomial zeros. IEEE Trans. Automat. Contr. 54 (2009) 171–177. | MR | Zbl | DOI
[59] and , Direct method for tds stability analysis. IEE Proc. D Contr. Theor. Appl. 134 (1987) 101–107. | Zbl | DOI
[60] , Stability criteria and the real roots of a transcendental equation. J. Soc. Ind. Appl. Math. 9 (1961) 136–148. | MR | Zbl | DOI
[61] , and , Delay-dependent stability analysis by using delay-independent integral evaluation. Automatica 70 (2016) 153–157. | MR | Zbl | DOI
[62] , and , Time-Delay Systems. World Scientific, Singapore (2010). | MR | Zbl | DOI
[63] , and , Dimensional analysis approach to dominant three-pole placement in delayed pid control loops. J. Process Contr. 23 (2013) 1063–1074. | DOI
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The content of this paper was partially presented in The 23rd International Symposium on Mathematical Theory of Networks and Systems July 16-20, 2018. Hong Kong.
