A novel class of controllers, called resetting virtual absorbers, is proposed as a means for achieving energy dissipation. A general framework for analyzing resetting virtual absorbers is given, and stability of the closed-loop system is analyzed. Special cases of resetting virtual absorbers, called the virtual trap-door absorber and the virtual one-way absorber, are described, and some illustrative examples are given.
Bainov, D.D. and Simeonov, P.S., 1989, Systems with Impulse Effect, Ellis Horwood Series in Mathematics and its Applications. Ellis Horwood Limited, England.
2.
Bupp, R.T., Bemstein, D.S., Chellaboina, V., and Haddad, W.M., 1996, "Finite settling time control of the double integrator using a virtual trap-door absorber," in Proceedings of the IEEE Conference on Control Applications, Dearborn, MI, September, pp. 179-184.
3.
Bupp, R.T., Bemstein, D.S., and Coppola, V.T., 1996, "Experimental implementation of integrator backstepping and passive nonlinear controllers on the RTAC testbed," in Proceedings of the IEEE Conference on Control Applications, Dearborn, MI, September, pp. 279-284.
4.
Bupp, R.T., Corrado, J.R., Coppola, V.T., and Bernstein, D.S., 1995, "Nonlinear modification of positive-real LQG compensators for enhanced disturbance rejection and energy dissipation," in Proceedings of the American Control Conference, Seattle, WA, June, pp. 3224-3228.
5.
Den Hartog, J.P., 1956, Mechanical Vibrations, 4th edition, McGraw-Hill, New York.
6.
Duquette, A.P., Tuer, K.L., and Golnaraghi, M.F. , 1993, "Vibration control of a flexible beam using a rotational internal resonance controller, Part I: Theoretical development and analysis," Journal of Sound and Vibration167, 41-62.
7.
Frahm, H., 1909, "Device for damping vibration of bodies," U.S. Patent No. 989958.
8.
Golnaraghi, M.F., Tuer, K.L., and Wang, D., 1994, "Development of a generalized active vibration suppression strategy for a cantilever beam using internal resonance," Nonlinear Dynamics5, 131-151.
9.
Golnaraghi, M.F., Tuer, K.L., and Wang, D., 1995, "Towards a generalized regulation scheme for oscillatory systems via coupling effects," IEEE Transaction on Automatic Control40, 522-530.
10.
Haddad, W.M., Bernstein, D.S., and Wang, YW, 1994, "Dissipative H2/H∞ controller synthesis ," IEEE Transactions on Automatic Control39, 827-831.
11.
Haddad, W.M. and Chellaboina, V., 1997, "Nonlinear fixed-order dynamic compensation for passive systems," in Proceedings of the American Contrnl Conference, Albuquerque, NM, June, pp. 2160-2164.
12.
Hu, S., Lakshmikantham, V., and Leela, S., 1989, "Impulsive differential systems and the pulse phenomena ," Journal of Mathematical Analysis and Applications137, 605-612.
13.
Juang, J. and Phan, M., 1992, "Robust controller designs for second-order dynamic systems: A virtual passive approach," Journal of Guidance, Control, and Dynamics15(5), 1192-1198.
14.
Kishimoto, Y., Bernstein, D.S., and Hall, S.R., 1995, "Energy flow control of interconnected structures: I. Modal subsystems, II. Structural subsystems," Control Theory and Advanced Technology10(4), 1563-1618.
15.
Korenov, B.G. and Reznikov, L.M., 1993, Dynamic Vibration Absorbers: Theory and Technical Applications , Wiley, UK.
16.
Kulev, G.K. and Bainov, D.D., 1989, "Stability of sets for systems with impulses," Bulletin of the Institute of Mathematics Academia Sinica17(4), 313-326.
17.
Lai, J.S. and Wang, K.W., 1996, "Parametric control of structural vibrations via adaptable stiffness dynamic absorbers," Journal of Vibration and Acoustics118, 41-47.
18.
Lakshmikantham, V., Bainov, D.D., and Simeonov, P.S., 1989, Theory of Impulsive Differential Equations, volume 6 of Series in Modern Applied Mathematics, World Scientific , Singapore.
19.
Lakshmikantham, V., Leela, S., and Kaul, S., 1994, "Comparison principle for impulsive differential equations with variable times and stability theory," Nonlinear Analysis, Theory, Methods and Applications22(4), 499-503.
20.
Lakshmikantham, V., Leela, S., and Martynyuk, A.A., 1989, Stability Analysis of Nonlinear Systems, Marcel Dekker, New York.
21.
Lakshmikantham, V. and Liu, X., 1989, "On quasi stability for impulsive differential systems ," Nonlinear Analysis, Theory; Methods and Applications13(7), 819-828.
22.
Liu, X., 1988, "Quasi stability via Lyapunov functions for impulsive differential systems," Applicable Analysis31, 201-213.
23.
Lozano-Leal, R. and Joshi, S.M., 1988, "On the design of dissipative LQG-type controllers," in Proceedings of the IEEE Conference on Decision and Control , Austin, TX, December.
24.
Ormondroyd, J. and Den Hartog, J.P. , 1928, "The theory of the dynamic vibration absorber," Transactions of the ASME50.
25.
Ouceini, S.S. and Golnaraghi, M.F. , 1996, "Experimental implementation of the internal resonance control strategy," Journal of Sound and Vibration191, 377-396.
26.
Ouceini, S.S., Tuer, K.L., and Golnaraghi, M.F. , 1995, "Regulation of a two-degree-of-freedom structure using resonance," Journal of Dynamic Systems, Measurement and Control117, 247-251.
27.
Phan, M., Juang, J., Wu, S., and Longman, R.W., 1993, "Passive dynamic controllers for nonlinear mechanical systems," Journal of Guidance, Control, and Dynamics16(5), 845-851.
28.
Quan, R. and Stech, D., 1996, "Time varying passive vibration absorption for flexible structures," Journal of Vibration and Acoustics118, 36-40.
29.
Siddiqui, S.A.Q. and Golnaraghi, M.F. , 1996, "New vibration regulation strategy and stability analysis for a flexible gyroscopic system," Journal of Sound and Vibration193, 465-481.
30.
Simeonov, P.S. and Bainov, D.D., 1985, "The second method of Lyapunov for systems with an impulse effect," TamkangJournal of Mathematics16(4), 19-40.
31.
Simeonov, P.S. and Bainov, D.D., 1987, "Stability with respect to part of the variables in systems with impulse effect, " Journal of Mathematical Analysis and Applications124, 547-560.
32.
Snowdon, J.C., 1968, Vibration and Shock in Damped Mechanical Systems, John Wiley, New York.
33.
Sun, L.Q., Jolly, M.R., and Norris, M.A., 1995, "Passive, adaptive and active tuned vibration absorbers - A survey," Journal of Vibration and Acoustics117, 234-242.
34.
Tuer, K.L., Golnaraghi, M.F., and Wang, D., 1994, "Regulation of a lumped parameter cantilever beam via internal resonance using nonlinear coupling enhancement," Dynamics and Control4, 73-96.
35.
Wan, C., Bernstein, D.S., and Coppola, V.T., 1996, "Global stabilization of the oscillating eccentric rotor ," Nonlinear Dynamics10, 49-62.
36.
Watts, P., 1883, "On a method of reducing the rolling in ships at sea," Transactions of the Institute of Naval Architecture24, 165-190.
