A nonsmooth optimisation approach for the stabilisation of time-delay systems
ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 3, p. 478-493

This paper is concerned with the stabilisation of linear time-delay systems by tuning a finite number of parameters. Such problems typically arise in the design of fixed-order controllers. As time-delay systems exhibit an infinite amount of characteristic roots, a full assignment of the spectrum is impossible. However, if the system is stabilisable for the given parameter set, stability can in principle always be achieved through minimising the real part of the rightmost characteristic root, or spectral abscissa, in function of the parameters to be tuned. In general, the spectral abscissa is a nonsmooth and nonconvex function, precluding the use of standard optimisation methods. Instead, we use a recently developed bundle gradient optimisation algorithm which has already been successfully applied to fixed-order controller design problems for systems of ordinary differential equations. In dealing with systems of time-delay type, we extend the use of this algorithm to infinite-dimensional systems. This is realised by combining the optimisation method with advanced numerical algorithms to efficiently and accurately compute the rightmost characteristic roots of such time-delay systems. Furthermore, the optimisation procedure is adapted, enabling it to perform a local stabilisation of a nonlinear time-delay system along a branch of steady state solutions. We illustrate the use of the algorithm by presenting results for some numerical examples.

DOI : https://doi.org/10.1051/cocv:2007060
Classification:  65Q05,  65K10,  90C26
Keywords: stabilisation, delay differential equations, nonsmooth optimisation, bundle gradient methods
@article{COCV_2008__14_3_478_0,
     author = {Vandewalle, Stefan and Michiels, Wim and Verheyden, Koen and Vanbiervliet, Joris},
     title = {A nonsmooth optimisation approach for the stabilisation of time-delay systems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {14},
     number = {3},
     year = {2008},
     pages = {478-493},
     doi = {10.1051/cocv:2007060},
     zbl = {1146.65056},
     mrnumber = {2434062},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2008__14_3_478_0}
}
Vandewalle, Stefan; Michiels, Wim; Verheyden, Koen; Vanbiervliet, Joris. A nonsmooth optimisation approach for the stabilisation of time-delay systems. ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 3, pp. 478-493. doi : 10.1051/cocv:2007060. http://www.numdam.org/item/COCV_2008__14_3_478_0/

[1] D. Breda, Solution operator approximation for delay differential equation characteristic roots computation via Runge-Kutta methods. Appl. Numer. Math. 56 (2005) 318-331. | MR 2207591 | Zbl 1095.65072

[2] D. Breda, S. Maset and R. Vermiglio, Computing the characteristic roots for delay differential equations. IMA J. Numer. Anal. 24 (2004) 1-19. | MR 2027286 | Zbl 1054.65079

[3] D. Breda, S. Maset and R. Vermiglio, Pseudospectral differencing methods for characteristic roots of delay differential equations. SIAM J. Sci. Comput. 27 (2005) 482-495. | MR 2202230 | Zbl 1092.65054

[4] J. Burke, A. Lewis and M. Overton, Approximating subdifferentials by random sampling of gradients. Math. Oper. Res. 22 (2002) 567-584. | MR 1926659 | Zbl 1082.49019

[5] J. Burke, A. Lewis and M. Overton, A nonsmooth, nonconvex optimization approach to robust stabilization by static output feedback and low-order controllers, in Proceedings of ROCOND 2003, Milan, Italy (2003).

[6] J. Burke, A. Lewis and M. Overton, A robust gradient sampling algorithm for nonsmooth, nonconvex optimization. SIAM J. Opt. 24 (2005) 567-584. | Zbl 1078.65048

[7] J. Burke, D. Henrion, A. Lewis and M. Overton, HIFOO - A matlab Package for Fixed-Order Controller Design and H-infinity optimization, in Proceedings of ROCOND 2006, Toulouse, France (2006).

[8] J. Burke, D. Henrion, A. Lewis and M. Overton, Stabilization via nonsmooth, nonconvex optimization. IEEE Trans. Automat. Control 51 (2006) 1760-1769. | MR 2265983

[9] O. Diekmann, S. Van Gils, S.V. Lunel and H.-O. Walther, Delay Equations. Appl. Math. Sci. 110, Springer-Verlag (1995). | MR 1345150 | Zbl 0826.34002

[10] K. Engelborghs and D. Roose, On stability of LMS methods and characteristic roots of delay differential equations. SIAM J. Numer. Anal. 40 (2002) 629-650. | MR 1921672 | Zbl 1021.65040

[11] K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL. ACM Trans. Math. Softw. 28 (2002) 1-21. | MR 1918642 | Zbl 1070.65556

[12] K. Gu, V. Kharitonov and J. Chen, Stability of time-delay systems. Birkhauser (2003). | Zbl 1039.34067

[13] J. Hale and S.V. Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences 99. Springer-Verlag, (1993). | MR 1243878 | Zbl 0787.34002

[14] V. Kolmanovskii and A. Myshkis, Introduction to the theory and application of functional differential equations, Math. Appl. 463. Kluwer Academic Publishers (1999). | MR 1680144 | Zbl 0917.34001

[15] T. Luzyanina and D. Roose, Equations with distributed delays: bifurcation analysis using computational tools for discrete delay equations. Funct. Differ. Equ. 11 (2004) 87-92. | MR 2056700 | Zbl 1064.34057

[16] W. Michiels and D. Roose, An eigenvalue based approach for the robust stabilization of linear time-delay systems. Int. J. Control 76 (2003) 678-686. | MR 1979889 | Zbl 1039.93059

[17] W. Michiels, K. Engelborghs, P. Vansevenant and D. Roose, Continuous pole placement for delay equations. Automatica 38 (2002) 747-761. | MR 2133350 | Zbl 1034.93026

[18] S.-I. Niculescu, Delay effects on stability: A robust control approach, LNCIS 269. Springer-Heidelberg (2001). | MR 1880658 | Zbl 0997.93001

[19] J.-P. Richard, Time-delay systems: an overview of some recent and open problems. Automatica 39 (2003) 1667-1694. | MR 2141765 | Zbl 1145.93302

[20] R. Seydel, Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos, Interdisciplinary Applied Mathematics 5. Springer-Verlag, 2nd edn. (1994). | MR 1314200 | Zbl 0806.34028

[21] K. Verheyden and D. Roose, Efficient numerical stability analysis of delay equations: a spectral method, in Proceedings of the IFAC Workshop on Time-Delay Systems 2004 (2004) 209-214.

[22] K. Verheyden, K. Green and D. Roose, Numerical stability analysis of a large-scale delay system modelling a lateral semiconductor laser subject to optical feedback. Phys. Rev. E 69 (2004) 036702.

[23] K. Verheyden, T. Luzyanina and D. Roose, Efficient computation of characteristic roots of delay differential equations using LMS methods. J. Comput. Appl. Math. (in press). Available online 5 March 2007. | MR 2391684 | Zbl 1135.65349

[24] T. Vyhlídal, Analysis and synthesis of time delay system spectrum. Ph.D. thesis, Department of Mechanical Engineering, Czech Technical University, Czech Republic (2003).