A decomposition technique for nonlinear dynamical system analysis

IEEE Transactions on Automatic Control

Volumn 57, Issue 6, 2012, Pages 1516-1521

  Anderson, James ^a   Papachristodoulou, Antonis ^a  

^a UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Large scale systems; nonlinear systems; sum of squares (SOS)

Indexed keywords

COMPOSITE LYAPUNOV FUNCTIONS; COMPUTATIONAL ANALYSIS; COMPUTATIONAL SAVINGS; DECOMPOSITION APPROACH; DECOMPOSITION TECHNIQUE; SUM OF SQUARES; SUM-OF-SQUARES PROGRAMMING;

NONLINEAR DYNAMICAL SYSTEMS; NONLINEAR SYSTEMS; SYSTEMS ANALYSIS;

LYAPUNOV FUNCTIONS;

EID: 84861744673     PISSN: 00189286     EISSN: None     Source Type: Journal    
DOI: 10.1109/TAC.2011.2175058     Document Type: Article

References (27)

1. F. N. Bailey, "The application of lyapunov's second method to interconnected systems," J. SIAM Control, vol. 3, no. 3, pp. 443-462, 1966.
2. M. Araki, "Stability of large-scale nonlinear systems-quadratic- order theory of composite-system method using M-matrices," IEEE Trans. Autom. Control, vol. AC-23, no. 2, pp. 129-142, Apr. 1978. (Pubitemid 8595796)
3. P. J. Moylan and D. J. Hill, "Stability criteria of large-scale systems," IEEE Trans. Autom. Control, vol. AC-23, no. 2, pp. 143-149, Apr. 1978.
4. J. C. Willems, "Dissipative dynamical systems. II. Linear systems with quadratic supply rates," Arch. Rational Mech. Anal., vol. 45, pp. 352-393, 1972.
5. J. Xiang and G. Chen, "On the -stability of complex dynamical networks," Automatica, vol. 43, pp. 1049-1057, 2007.
6. H. K. Khalil, Nonlinear Systems, 3rd ed. Englewood Cliffs, NJ: Prentice Hall, Inc., 2001.
7. A. I. Zečević and D. D. Šilak, Control of Complex Systems: Structural Constraints and Uncertainty. New York: Springer, 2010.
8. P. A. Parrilo, "Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization," Ph.D. dissertation, Caltech, Pasadena, CA, 2000.
9. G. Chesi, "Estimating the domain of attraction for uncertain polynomial systems," Automatica, vol. 40, no. 11, pp. 1981-1986, 2004.
10. A. Papachristodoulou and S. Prajna, "A tutorial on sum of squares techniques for systems analysis," in Proc. Amer. Contol Conf., 2005, pp. 2686-2700. (Pubitemid 41189020)
11. G. Chesi, "LMI techniques for optimization over polynomials in control: A survey," IEEE Trans. Autom. Control, vol. 55, no. 11, pp. 2500-2510, Nov. 2010.
12. P. Parrilo, "Exploiting algebraic structure in sum of squares programming," in Positive Polynomials in Control. New York: Springer-Verlag, 2005.
13. J. Löfberg, "Pre-and post-processing sum-of-squares programs in practice," IEEE Trans. Autom. Control, vol. 54, no. 5, pp. 1007-1011, May 2009.
14. S. Prajna, A. Papachristodoulou, P. Seiler, and P. A. Parrilo, "SOSTOOLS: Sum of squares optimization toolbox for MATLAB," 2012 [Online]. Available: http://www.eng.ox.ac.uk/control/sostools
15. J. F. Sturm, "Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones," Optim. Methods Softw., vol. 11-12, pp. 625-653, 1999.
16. U. Topcu and A. Packard, "Local stability analysis for uncertain nonlinear systems," IEEE Trans. Autom. Control, vol. 54, no. 5, pp. 1042-1047, May 2009.
17. H. Wolkowicz and Q. Zhao, "Semidefinite programming relaxations for the graph partitioning problem," Discrete Appl. Math., vol. 96-97, pp. 461-479, 1999. (Pubitemid 129563014)
18. G. Karypis and V. Kumar, "A fast and high quality multilevel scheme for partitioning irregular graphs," SIAM Sci.Comput., vol. 20, no. 1, pp. 359-392, 1998. (Pubitemid 128689516)
19. F. Alizadeh, "Interior point methods in semidefinite programming with applications to combinatorial optimization," SIAM J. Optim., vol. 5, pp. 13-51, 1995.
20. S. E. Karisch, F. Rendl, and J. Clausen, "Solving graph bisection problems with semidefinite programming," INFORMS J. Comp., vol. 12, no. 3, pp. 177-191, 2000.
21. C. Godsil and G. Royle, Algebraic Graph Theory Inc.. New York: Springer-Verlag, 2000.
22. J. Anderson and A. Papachristodoulou, "A network decomposition approach for efficient sum-of-squares programming based analysis," in Proc. Amer. Control Conf., Baltimore, MD, 2010, pp. 4492-4497.
23. D. G. Feingold and R. S. Varga, "Block diagonally dominant matrices and generalizations of the Gerschgorin Circle Theorem," Pacific J. Math, vol. 12, pp. 1241-1250, 1962.
24. S. Lall, J. E. Marsden, and S. Glava?ski, "A subspace approach to balanced truncation for model reduction of nonlinear control systems," Int. J. Robust Nonlin. Control, vol. 12, no. 6, pp. 519-535, 2002. (Pubitemid 34520056)
25. M. Ikeda and D. D. Šiljak, "Generalized decompositions of dynamic systems and vector Lyapunov functions," IEEE Trans. Autom. Control, vol. AC-26, no. 5, pp. 1118-1125, Oct. 1981. (Pubitemid 12439162)
26. J. Anderson and A. Papachristodoulou, "Dynamical system decomposition for efficient sparse analysis," in Proc. 49th IEEE Conf. Decision Control, 2010, pp. 6565-6570.
27. T. Zhang, P. Brazhnik, and J. Tyson, "Computational analysis of dynamical responses to the intrinsic pathway of programmed cell death," Biophys. J., vol. 97, pp. 415-434, 2009.