×

Finite-time convergent gradient flows with applications to network consensus. (English) Zbl 1261.93058

Summary: This paper introduces the normalized and signed gradient dynamical systems associated with a differentiable function. Extending recent results on nonsmooth stability analysis, we characterize their asymptotic convergence properties and identify conditions that guarantee finite-time convergence. We discuss the application of the results to consensus problems in multi-agent systems and show how the proposed nonsmooth gradient flows achieve consensus in finite time.

MSC:

93A14 Decentralized systems
93C83 Control/observation systems involving computers (process control, etc.)
34A60 Ordinary differential inclusions
90B10 Deterministic network models in operations research
93B40 Computational methods in systems theory (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bacciotti, A.; Ceragioli, F., Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions, ESAIM. control, optimisation & calculus of variations, 4, 361-376, (1999) · Zbl 0927.34034
[2] Bertsekas, D.P.; Tsitsiklis, J.N., Parallel and distributed computation: numerical methods, (1997), Athena Scientific Belmont, MA, ISBN 1886529019
[3] Bhat, S.P.; Bernstein, D.S., Finite-time stability of continuous autonomous systems, SIAM journal on control and optimization, 38, 3, 751-766, (2000) · Zbl 0945.34039
[4] Clarke, F.H., Optimization and nonsmooth analysis, (), ISBN 047187504X · Zbl 0727.90045
[5] Clarke, F. H. (2004). Lyapunov functions and feedback in nonlinear control. In M. S. de Queiroz, M. Malisoff, P. Wolenski (Eds.), Optimal control, stabilization and nonsmooth analysis. Lecture notes in control and information sciences (Vol. 301, pp. 267-282). New York: Springer. · Zbl 1079.93037
[6] Clarke, F.H.; Ledyaev, Y.; Rifford, L.; Stern, R., Feedback stabilization and Lyapunov functions, SIAM journal on control and optimization, 39, 25-48, (2000) · Zbl 0961.93047
[7] Coron, J.M., On the stabilization in finite time of locally controllable systems by means of continuous time-varying feedback laws, SIAM journal on control and optimization, 33, 804-833, (1995) · Zbl 0828.93054
[8] Cortés, J.; Bullo, F., Coordination and geometric optimization via distributed dynamical systems, SIAM journal on control and optimization, 44, 5, 1543-1574, (2005) · Zbl 1108.37058
[9] Cortés, J.; Martínez, S.; Bullo, F., Spatially-distributed coverage optimization and control with limited-range interactions, ESAIM. control, optimisation & calculus of variations, 11, 4, 691-719, (2005) · Zbl 1080.90070
[10] Filippov, A. F. (1988). Differential equations with discontinuous righthand sides, Mathematics and its applications. Vol. 18, Dordrecht: Kluwer Academic Publishers. The Netherlands.
[11] Gazi, V.; Passino, K.M., Stability analysis of swarms, IEEE transactions on automatic control, 48, 4, 692-697, (2003) · Zbl 1365.92143
[12] Helmke, U.; Moore, J.B., Optimization and dynamical systems, (1994), Springer New York, ISBN 0387198571 · Zbl 0943.93001
[13] Hirsch, W.M.; Smale, S., Differential equations, dynamical systems and linear algebra, (1974), Academic Press New York · Zbl 0309.34001
[14] Ögren, P.; Fiorelli, E.; Leonard, N.E., Cooperative control of mobile sensor networks: adaptive gradient climbing in a distributed environment, IEEE transactions on automatic control, 49, 8, 1292-1302, (2004) · Zbl 1365.93243
[15] Olfati-Saber, R., Fax, J. A., & Murray, R. M. (submitted). Consensus and cooperation in multi-agent networked systems. Proceedings of the IEEE, submitted for publication. · Zbl 1376.68138
[16] Olfati-Saber, R.; Murray, R.M., Consensus problems in networks of agents with switching topology and time-delays, IEEE transactions on automatic control, 49, 9, 1520-1533, (2004) · Zbl 1365.93301
[17] Ren, W., Beard, R. W., & Atkins, E. M. (2005). A survey of consensus problems in multi-agent coordination. In American control conference (pp. 1859-1864), Portland, OR, June 2005.
[18] Ryan, E.P., Finite-time stabilization of uncertain nonlinear planar systems, Dynamics and control, 1, 83-94, (1991) · Zbl 0742.93068
[19] Shevitz, D.; Paden, B., Lyapunov stability theory of nonsmooth systems, IEEE transactions on automatic control, 39, 9, 1910-1914, (1994) · Zbl 0814.93049
[20] Tanner, H., Jadbabaie, A., & Pappas, G. J. (2003). Stable flocking of mobile agents, Part I: Fixed topology. In IEEE conference on decision and control, (pp. 2010-2015), Maui, HI, December 2003.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.