A nonsmooth optimisation approach for the stabilisation of time-delay systems

Vandewalle, Stefan; Michiels, Wim; Verheyden, Koen; Vanbiervliet, Joris

ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 3, pp. 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.

Classification : 65Q05, 65K10, 90C26
Mots clés : stabilisation, delay differential equations, nonsmooth optimisation, bundle gradient methods

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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

[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

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

[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 | Zbl

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

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