Abstract
This paper considers the problem of stabilizing a class of nonlinear systems with unknown bounded delayed feedback wherein the time-varying delay is 1) piecewise constant 2) continuous with a bounded rate. We also consider application of these results to the stabilization of rigid-body attitude dynamics. In the first case, the time-delay in feedback is modeled specifically as a switch among an arbitrarily large set of unknown constant values with a known strict upper bound. The feedback is a linear function of the delayed states. In the case of linear systems with switched delay feedback, a new sufficiency condition for average dwell time result is presented using a complete type Lyapunov-Krasovskii (L-K) functional approach. Further, the corresponding switched system with nonlinear perturbations is proven to be exponentially stable inside a well characterized region of attraction for an appropriately chosen average dwell time. In the second case, the concept of the complete type L-K functional is extended to a class of nonlinear time-delay systems with unknown time-varying time-delay. This extension ensures stability robustness to time-delay in the control design for all values of time-delay less than the known upper bound. Model-transformation is used in order to partition the nonlinear system into a nominal linear part that is exponentially stable with a bounded perturbation. We obtain sufficient conditions which ensure exponential stability inside a region of attraction estimate. A constructive method to evaluate the sufficient conditions is presented together with comparison with the corresponding constant and piecewise constant delay. Numerical simulations are performed to illustrate the theoretical results of this paper.
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Notes
We denote the delay at initialization to be τ 1 without loss of generality.
References
Sun, X., Zhao, J., Hill, D.: Stability and L 2-gain analysis for switched delay systems: A delay-dependent method. Automatica 42(10), 1769–1774 (2006)
Vu, L., Morgansen, K.A.: Stability of time-delay feedback switched linear systems. IEEE Trans. Autom. Control 55(10), 2385–2390 (2010)
Yan, P., Ozbay, H.: Stability Analysis of Switched Time-Delay Systems. In: Proceedings of the 47th IEEE Conference on Decision and Control, pp 2740–2745 (2008)
Kim, S., Campbell, S.: Stability of a class of linear switching systems with time delay. IEEE Trans. Circ. Syst. I: Reg Pap. 53(2), 384–393 (2007)
Song, B., Sun, J.-Q.: Supervisory control of dynamical systems with uncertain time delays. J. Vib. Acoust. 132(6) (2010)
Ailon, A., Segev, R., Arogeti, S.: A simple velocity-free controller for attitude regulation of a spacecraft with delayed feedback. IEEE Trans. Autom. Control 49(1), 125–130 (2004)
Chunodkar, A., Akella, M.: Attitude stabilization with unknown bounded delay in feedback control implementation. J. Guid. Control Dyn. 34(2), 533–542 (2011)
Chunodkar, A., Akella, M.: Rigid Body Attitude Stabilization with Unknown Time-varying Delay in Feedback. In: Proceedings of the 20th AAS/AIAA Space Flight Mechanics Meeting, pp 2740–2745 (2010)
Hespanha, J., Morse, A.: Stability of Switched Systems with Average Dwell-Time. In: Proceedings of the 38th IEEE Conference on Decision and Control, pp 2655–2660 (1999)
Kharitonov, V., Zhabko, A.: Lyapunov-Krasovskii approach to robust stability analysis of time-delay systems. Automatica 39(1), 15–20 (2003)
Gu, K., Niculescu, S.-I.: Additional dynamics in transformed time-delay systems. IEEE Trans. Autom. Control 45(3), 572–575 (2003)
Gu, K., Niculescu, S.-I.: Further remarks on additional dynamics in various model transformations of linear delay systems. IEEE Trans. Autom. Control 46(3), 497–500 (2003)
Melchor-Aguilar, D., Niculescu, S.-I.: Estimates of attraction region for a class of nonlinear time-delay systems. IMA J. Math. Control Appl. 24, 523–550 (2007)
Yoneyama, T.: On the 3/2 stability theorem for one-dimensional delay-differential equations. J. Math. Anal. Appl. 125, 161–173 (1987)
Yoneyama, T.: Uniform asymptotic stability for n-dimensional delay-differential equations. Funcialaj Ekvacioj 34, 495–504 (1991)
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Chunodkar, A.A., Akella, M.R. Stability of Nonlinear Systems with Unknown Time-varying Feedback Delay. J of Astronaut Sci 60, 278–302 (2013). https://doi.org/10.1007/s40295-015-0052-2
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DOI: https://doi.org/10.1007/s40295-015-0052-2