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A nonsmooth optimisation approach for the stabilisationof time-delay systems

Published online by Cambridge University Press:  21 November 2007

Joris Vanbiervliet ,
Koen Verheyden ,
Wim Michiels  and
Stefan Vandewalle
Joris Vanbiervliet
Affiliation:
Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium; [email protected]
Koen Verheyden
Affiliation:
Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium; [email protected]
Wim Michiels
Affiliation:
Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium; [email protected]
Stefan Vandewalle
Affiliation:
Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium; [email protected]
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Abstract

This paper is concerned with the stabilisation of linear time-delay systems by tuning a finite number of parameters. Such problems typically arise in thedesign 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 rightmostcharacteristic root, or spectral abscissa, in function of the parameters to be tuned. In general, the spectral abscissa is a nonsmooth and nonconvex function, precludingthe 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 toinfinite-dimensional systems. This is realised by combining the optimisation method with advanced numerical algorithms toefficiently 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.

Keywords

Stabilisationdelay differential equationsnonsmooth optimisationbundle gradient methods
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 3 , July 2008 , pp. 478 - 493
Copyright
© EDP Sciences, SMAI, 2007

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References

Breda, D., Solution operator approximation for delay differential equation characteristic roots computation via Runge-Kutta methods. Appl. Numer. Math. 56 (2005) 318–331. CrossRef
Breda, D., Maset, S. and Vermiglio, R., Computing the characteristic roots for delay differential equations. IMA J. Numer. Anal. 24 (2004) 1–19. CrossRef
Breda, D., Maset, S. and Vermiglio, R., Pseudospectral differencing methods for characteristic roots of delay differential equations. SIAM J. Sci. Comput. 27 (2005) 482–495. CrossRef
Burke, J., Lewis, A. and Overton, M., Approximating subdifferentials by random sampling of gradients. Math. Oper. Res. 22 (2002) 567–584. CrossRef
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).
Burke, J., Lewis, A. and Overton, M., A robust gradient sampling algorithm for nonsmooth, nonconvex optimization. SIAM J. Opt. 24 (2005) 567–584.
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).
Burke, J., Henrion, D., Lewis, A. and Overton, M., Stabilization via nonsmooth, nonconvex optimization. IEEE Trans. Automat. Control 51 (2006) 1760–1769. CrossRef
O. Diekmann, S. van Gils, S.V. Lunel and H.-O. Walther, Delay Equations. Appl. Math. Sci. 110, Springer-Verlag (1995).
Engelborghs, K. and Roose, D., On stability of LMS methods and characteristic roots of delay differential equations. SIAM J. Numer. Anal. 40 (2002) 629–650. CrossRef
Engelborghs, K., Luzyanina, T. and Roose, D., Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL. ACM Trans. Math. Softw. 28 (2002) 1–21. CrossRef
K. Gu, V. Kharitonov and J. Chen, Stability of time-delay systems. Birkhauser (2003).
J. Hale and S.V. Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences 99. Springer-Verlag, (1993).
V. Kolmanovskii and A. Myshkis, Introduction to the theory and application of functional differential equations, Math. Appl. 463. Kluwer Academic Publishers (1999).
Luzyanina, T. and Roose, D., Equations with distributed delays: bifurcation analysis using computational tools for discrete delay equations. Funct. Differ. Equ. 11 (2004) 87–92.
Michiels, W. and Roose, D., An eigenvalue based approach for the robust stabilization of linear time-delay systems. Int. J. Control 76 (2003) 678–686. CrossRef
Michiels, W., Engelborghs, K., Vansevenant, P. and Roose, D., Continuous pole placement for delay equations. Automatica 38 (2002) 747–761. CrossRef
S.-I. Niculescu, Delay effects on stability: A robust control approach, LNCIS 269. Springer-Heidelberg (2001).
Richard, J.-P., Time-delay systems: an overview of some recent and open problems. Automatica 39 (2003) 1667–1694. CrossRef
R. Seydel, Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos, Interdisciplinary Applied Mathematics 5. Springer-Verlag, 2nd edn. (1994).
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.
Verheyden, K., Green, K. and Roose, D., Numerical stability analysis of a large-scale delay system modelling a lateral semiconductor laser subject to optical feedback. Phys. Rev. E 69 (2004) 036702. CrossRef
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.
T. Vyhlídal, Analysis and synthesis of time delay system spectrum. Ph.D. thesis, Department of Mechanical Engineering, Czech Technical University, Czech Republic (2003).