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

ESAIM - Control, Optimisation and Calculus of Variations

Volumn 14, Issue 3, 2008, Pages 478-493

  Vanbiervliet, Joris ^a   Verheyden, Koen ^a   Michiels, Wim ^a   Vandewalle, Stefan ^a  

^a UNIVERSITY OF LEUVEN   (Belgium)

Author keywords

Bundle gradient methods; Delay differential equations; Nonsmooth optimisation; Stabilisation

Indexed keywords

BESSEL FUNCTIONS; BOOLEAN FUNCTIONS; DELAY CONTROL SYSTEMS; DIFFERENCE EQUATIONS; DIFFERENTIAL EQUATIONS; DIFFERENTIATION (CALCULUS); FUNCTION EVALUATION; NUMERICAL METHODS; ORDINARY DIFFERENTIAL EQUATIONS; PARAMETER ESTIMATION; STANDARDS; SYSTEM STABILITY;

APPLIED (CO); CONTROLLER DESIGNS; FINITE NUMBERS; FIXED ORDER; GENERAL (CO); INFINITE DIMENSIONAL SYSTEMS; LINEAR TIME-DELAY SYSTEMS; NON CONVEX FUNCTIONS; NONSMOOTH; NUMERICAL ALGORITHMS; NUMERICAL EXAMPLES; OPTIMISATION; OPTIMISATION ALGORITHMS; PARAMETER SETS; REAL PART; STABILISATION; STEADY STATE SOLUTIONS; TIME DELAY SYSTEMS; TIME DELAYING;

TIME DELAY;

EID: 46749158807     PISSN: 12928119     EISSN: 12623377     Source Type: Journal    
DOI: 10.1051/cocv:2007060     Document Type: Article

References (24)

1. D. Breda, Solution operator approximation for delay differential equation characteristic roots computation via Runge-Kutta methods. Appl. Numer. Math. 56 (2005) 318-331.
2. D. Breda, S. Maset and R. Vermiglio, Computing the characteristic roots for delay differential equations. IMA J. Numer. Anal. 24 (2004) 1-19.
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.
4. J. Burke, A. Lewis and M. Overton, Approximating subdifferentials by random sampling of gradients. Math. Oper. Res. 22 (2002) 567-584.
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.
7. J. Burke, D. Henrion, A. Lewis andM. 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.
9. O. Diekmann, S. van Gils, S.V. Lunel and H.-O. Walther, Delay Equations. Appl. Math. Sci. 110, Springer-Verlag (1995).
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.
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.
12. K. Gu, V. Kharitonov and J. Chen, Stability of time-delay systems. Birkhauser (2003).
13. J. Hale and S.V. Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences 99. Springer-Verlag, (1993).
14. V. Kolmanovskii and A. Myshkis, Introduction to the theory and application of functional differential equations, Math. Appl. 463. Kluwer Academic Publishers (1999).
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.
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.
17. W. Michiels, K. Engelborghs, P. Vansevenant and D. Roose, Continuous pole placement for delay equations. Automatica 38 (2002) 747-761.
18. S.-I. Niculescu, Delay effects on stability: A robust control approach, LNCIS 269. Springer-Heidelberg (2001).
19. J.-P. Richard, Time-delay systems: an overview of some recent and open problems. Automatica 39 (2003) 1667-1694.
20. R. Seydel, Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos, Interdisciplinary Applied Mathematics 5. Springer-Verlag, 2nd edn. (1994).
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.
24. T. Vyhlídal, Analysis and synthesis of time delay system spectrum. Ph.D. thesis, Department of Mechanical Engineering, Czech Technical University, Czech Republic (2003).