DOI: 10.17586/2226-1494-2017-17-6-1025-1032


D. V. Efimov, A. S. Kremlev

Read the full article 
Article in Russian

For citation: Efimov D.V., Kremlev A.S. Robust properties of systems with negative degree of homogeneity with respect to delay. Scientific and Technical Journal of Information Technologies, Mechanics and Optics , 2017, vol. 17, no. 6, pp. 1025–1032 (in Russian). doi: 10.17586/2226-1494-2017-17-6-1025-1032

The paper considers robustness problems of a class of weighted homogeneous systems with negative homogeneity degree in relation to the delay. It is shown that in the case of global asymptotic stability of a nonlinear weighted homogeneous system with negative homogeneity degree, in the presence of a delay in the system, all the trajectories converge asymptotically to some compact set containing the origin. In the absence of delay, such systems reach their equilibrium position in a finite time. The robustness analysis also covers cases of variable and multiple delays. The presented analysis is based on the application of the Lyapunov methods for delayed systems (the Lyapunov-Razumikhin function method) and the theory of weighted homogeneous systems. Computer simulation was performed to verify the analysis of system robustness with a negative degree of homogeneity in relation to the delay. A stabilizing system that represents a double integrator is used as an example. This system is weighted homogeneous with a negative degree when using a nonlinear state feedback control law that ensures that the system achieves its equilibrium position for the desired finite time. During computer simulation, the state vector was available for measurement with some delays. The computer simulation has confirmed the effectiveness of the presented theoretical results.   

Keywords: weighted homogeneous systems, time-delay, robustness, nonlinear systems

Acknowledgements. This work is supported by the Russian Science Foundation under grant No.17-19-01422 and carried out in ITMO University.

 1.      ErneuxT. Applied Delay Differential Equations. NY, Springer, 2009, 204 p. doi: 10.1007/978-0-387-74372-1
2.      Gupta R.A., Chow M.Y. Networked control system: overview and research trends. IEEE Transactions on Industrial Electronics, 2010, vol. 57, no. 7, pp. 2527–2535. doi: 10.1109/tie.2009.2035462
3.      Chiasson J., Loiseau J.J. Applications of Time Delay Systems. Berlin, Springer, 2007, 355 p. doi: 10.1007/978-3-
4.      Fridman E. Introduction to Time-Delay Systems: Analysis and Control. Springer, 2014, 378 p.
5.      Richard J.-P. Time-delay systems: an overview of some recent advances and open problems. Automatica, 2003, vol. 39, no. 10, pp. 1667–1694. doi: 10.1016/s0005-1098(03)00167-5
6.      Efimov D., Polyakov A., Perruquetti W., Richard J.-P. Weighted homogeneity for time-delay systems: finite-time and independent of delay stability. IEEE Transactions on Automatic Control, 2016, vol. 61, no. 1, pp. 210–215. doi: 10.1109/tac.2015.2427671
7.      Kolmanovskii V.B., Nosov V.R. Stability of Functional Differential Equations. Academic Press, 1986, 217 p.
8.      Moulay E., Dambrine M., Yeganefar N., Perruquetti W. Finite-time stability and stabilization of time-delay systems. Systems & Control Letters, 2008, vol. 57, no. 7, pp. 561–566. doi: 10.1016/j.sysconle.2007.12.002
9.      Gu K., Kharitonov V.L., Chen J. Stability of Time-Delay Systems. Boston, Birkhauser, 2003, 353 p. doi: 10.1007/978-1-4612-0039-0
10.   Bacciotti A., Rosier L. Lyapunov Functions and Stability in Control Theory. London, Springer, 2005, 237 p.
11.   Efimov D., Perruquetti W. Homogeneity for time-delay systems. IFAC Proceedings Volumes, 2011, pp. 3861–3866. doi: 10.3182/20110828-6-it-1002.03195
12.   Hale J.K. Theory of Functional Differential Equations. Springer, 1977,365 p.doi: 10.1007/978-1-4612-9892-2
13.   Zubov V.I. On systems of ordinary differential equations with generalized-homogeneous right-hand sides. Izvestiya Vuzov. Matematika, 1958, no. 1, pp. 80–88. (In Russian)
14.   Rosier L. Homogeneous Lyapunov function for homogeneous continuous vector field. Systems & Control Letters, 1992, vol. 19,no. 6, pp. 467–473.doi: 10.1016/0167-6911(92)90078-7
15.   Zimenko K.A., Polyakov A.E., Efimov D.V., Kremlev A.S. Finite-time stability of system of series-connected integrators. Journal of Instrument Engineering, 2015, vol. 58, no. 9, pp. 681–686. (In Russian) doi: 10.17586/0021-3454-2015-58-9-681-686

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License
Copyright 2001-2019 ©
Scientific and Technical Journal
of Information Technologies, Mechanics and Optics.
All rights reserved.