Abstract
The stability and boundedness of mechanical system have been one of important research topics. In this paper ultimate boundedness of a dry friction oscillator, belonging to nonsmooth mechanical system, is investigated by proposing a controller design method. Firstly a sufficient condition of the stability for the nominal system with delayed state feedback is derived by constructing a Lyapunov–Krasovskii function. The delayed feedback gain matrix is calculated by applying linear matrix inequality method. Secondly on the basis of the delayed state feedback, a continuous function is designed by Lyapunov redesign to ensure that the solutions of the friction oscillator system are ultimately bounded under the overall control. Moreover, the ultimate bound can be adjusted in practice by choosing appropriate parameter. Accordingly friction-induced vibration or instability can be controlled effectively. Numerical results show that the proposed method is valid.
Similar content being viewed by others
References
Leine R.I., Van Campen D.H., Van De Vrande B.L.: Bifurcations in nonlinear discontinuous systems. Nonlinear Dyn. 23, 105–164 (2000)
Kowalczyk P., Bernardo M.di.: Two-parameter degenerate sliding bifurcations in Filippov systems. Phys. D 204, 204–229 (2005)
Awrejcewicz J., Lamarque C.-H.: Bifurcation and Chaos in Nonsmooth Mechanical Systems. World Scientific, Singapore (2003)
Richard J.P.: Time-delay systems: An overview of some recent advances and open problems. Automatica 39, 1667–1694 (2003)
Tadmor G.: The standard H ∞ problem in systems with a single input delay. IEEE T. Automat. Contr. 45, 382–397 (2000)
Mirkin L., Tadmor G.: H ∞ control of system with I/O delay: A review of some problem-oriented methods. IMA J. Math. Control Inf. 19, 185–199 (2002)
Kolmanovskii V.B., Niculescu S.I., Richard J.P.: On the Liapunov–Krasovskii functionals for stability analysis of linear delay systems. Int. J. Control 72(4), 374–384 (1999)
Fridman E., Shaked U.: Delay-dependent stability and H ∞ control: constant and time-varying delay. Int. J. Control 76, 48–60 (2003)
Lien C.H., Sun Y.J., Hsieh J.G.: Global stabilizability for a class of uncertain systems with multiple time-varying delays via linear control. Int. J. Control 72, 904–910 (1999)
Yue D., Han Q.L.: Delayed feedback control of uncertain systems with time-varying input delay. Automatica 41, 233–240 (2005)
Jiang X.F., Han Q.L.: Delay-dependent robust stability for uncertain linear systems with interval time-varying delay. Automatica 42, 1059–1065 (2006)
Sun J., Liu G.P.: State feedback and output feedback control of a class of nonlinear systems with delayed measurements. Nonlinear Anal. 67, 1623–1636 (2007)
Li H.N., Li J., Song G.B.: Improved suboptimal Bang–Bang control of aseismic buildings with variable friction dampers. Acta Mech. Sin. 23(1), 101–109 (2007)
Popp K., Rudolph M.: Vibration control to avoid stick-slip motion. J. Vib. Control 10, 1585–1600 (2004)
Elmer F.J.: Controlling friction. Phys. Rev. E 57, 4903–4906 (1998)
Das J., Mallik A.K.: Control of friction driven oscillation by time-delayed state feedback. J. Sound Vib. 297, 578–594 (2006)
Hassan K.K.: Nonlinear Systems, 3rd edn. Prentice Hall, New Jersey (2002)
Zhang X.M., Wu M., She J.H., He Y.: Delay-dependent stabilization of linear systems with time-varying state and input delays. Automatica 41, 1405–1412 (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
The project supported by the National Natural Science Foundation of China (10672007).
Rights and permissions
About this article
Cite this article
Li, Z., Wang, Q. & Gao, H. Control of friction oscillator by Lyapunov redesign based on delayed state feedback. Acta Mech Sin 25, 257–264 (2009). https://doi.org/10.1007/s10409-008-0181-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10409-008-0181-y