This paper discusses the averaging, control authority, and vibrational control of mechanical control-affine systems with high-frequency, high-amplitude inputs. The inputs have different frequencies of the same order. This work is an extension of the existing averaging method for high-frequency mechanical systems with single-frequency inputs. Vibrational control authority of mechanical control-affine systems is introduced, and the effects of inputs' waveform and frequency on vibrational control authority are investigated. The results show that, in general, using multifrequency inputs may result in lower control authority of mechanical systems compared to single-frequency inputs, especially when using harmonic inputs. The results on vibrational control authority of the systems with multifrequency inputs are demonstrated using vibrational control of a horizontal pendulum with two inputs. This paper also discusses the averaging of multiple-time-scale control systems.
Issue Section:
Research Papers
References
1.
Bogoliubov
, N. N.
, and Mitropolsky
, Y. A.
, 1961
, Asymptotic Methods in the Theory of Non-Linear Oscillations
, Hindustan Publishing Corporation
, Delhi, India
.2.
Mitropolsky
, Y. A.
, 1967
, “Averaging Method in Non-Linear Mechanics
,” Int. J. Nonlinear Mech.
, 2
(1
), pp. 69
–96
.3.
Guckenheimer
, J.
, and Holmes
, P.
, 1983
, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences
, Springer-Verlag
, New York
.4.
Sanders
, J. A.
, and Verhulst
, F.
, 1985
, Averaging Methods in Nonlinear Dynamical Systems. Applied Mathematical Sciences
, Springer-Verlag
, New York
.5.
Meerkov
, S. M.
, 1980
, “Principle of Vibrational Control: Theory and Applications
,” IEEE Trans. Autom. Control, AC
, 25
(4
), pp. 755
–762
.6.
Bellman
, R. E.
, Bentsman
, J.
, and Meerkov
, S. M.
, 1986
, “Vibrational Control of Nonlinear Systems: Vibrational Stabilizability
,” IEEE Trans. Autom. Control
, 31
(8
), pp. 710
–716
.7.
Bellman
, R. E.
, Bentsman
, J.
, and Meerkov
, S. M.
, 1986
, “Vibrational Control of Nonlinear Systems: Vibrational Controllability and Transient Behavior
,” IEEE Trans. Autom. Control
, 31
(8
), pp. 717
–724
.8.
Kapitza
, P. L.
, 1965
, “Dynamical Stability of a Pendulum When Its Point of Suspension Vibrates
,” Collected Papers of P. L. Kapitza
, Vol. 2, P. L.
Kapitza
, and D.
ter Haar
, ed., Pergamon
, Oxford, UK, pp. 714
–725
.9.
Hong
, K. S.
, 2002
, “An Open-Loop Control for Underactuated Manipulators Using Oscillatory Inputs: Steering Capability of an Unactuated Joint
,” IEEE Trans. Control Syst. Technol.
, 10
(3
), pp. 469
–480
.10.
Bentsman
, J.
, and Hong
, K. S.
, 1991
, “Vibrational Stabilization of Nonlinear Parabolic Systems With Neumann Boundary Conditions
,” IEEE Trans. Autom. Control
, 36
(4
), pp. 501
–507
.11.
Bentsman
, J.
, Hong
, K. S.
, and Fakhfakh
, J.
, 1991
, “Vibrational Control of Nonlinear Time Lag Systems: Vibrational Stabilization and Transient Behavior
,” Automatica
, 27
(3
), pp. 491
–500
.12.
Bentsman
, J.
, and Hong
, K. S.
, 1993
, “Transient Behavior Analysis of Vibrationally Controlled Nonlinear Parabolic Systems With Neumann Boundary Conditions
,” IEEE Trans. Autom. Control
, 38
(10
), pp. 1603
–1607
.13.
Tahmasian
, S.
, and Woolsey
, C. A.
, 2015
, “A Control Design Method for Underactuated Mechanical Systems Using High Frequency Inputs
,” ASME J. Dyn. Syst., Meas., Control
, 137
(7
), p. 071004.14.
Tahmasian
, S.
, and Woolsey
, C. A.
, 2016
, “Flight Control of Biomimetic Air Vehicles Using Vibrational Control and Averaging
,” J. Nonlinear Sci.
, 27
(4), pp. 1193–1214.15.
Bullo
, F.
, 2002
, “Averaging and Vibrational Control of Mechanical Systems
,” SIAM J. Control Optim.
, 41
(2
), pp. 542
–562
.16.
Blekhman
, I. I.
, 2000
, Vibrational Mechanics
, World Scientific Publishing Co
, Singapore
.17.
Crouch
, P. E.
, 1981
, “Geometric Structures in Systems Theory
,” IEE Proc. D (Control Theory Appl.)
, 128
(5
), pp. 242
–252
.18.
Bullo
, F.
, and Lewis
, A. D.
, 2005
, Geometric Control of Mechanical Systems. Texts in Applied Mathematics
, Springer
, New York
.19.
Weibel
, S.
, Baillieul
, J.
, and Kaper
, T. J.
, 1995
, “Small-Amplitude Periodic Motions of Rapidly Forced Mechanical Systems
,” Conference on Decision and Control
(CDC
), New Orleans, LA, Dec. 13–15, pp. 533
–539
.20.
Fidlin
, A.
, and Thomsen
, J. J.
, 2008
, “Nontrivial Effects of High-Frequency Excitation for Strongly Damped Mechanical Systems
,” Int. J. Non-Linear Mech.
, 43
(7
), pp. 569
–578
.21.
Khalil
, H. K.
, 1996
, Nonlinear Systems
, Prentice Hall
, Upper Saddle River, NJ
.22.
Agrachev
, A. A.
, and Gamkrelidze
, R. V.
, 1979
, “The Exponential Representation of Flows and the Chronological Calculus
,” Math. USSR: Sb.
, 35
(6
), pp. 727
–785
.23.
Tahmasian
, S.
, Allen
, D. W.
, and Woolsey
, C. A.
, 2016
, “On Averaging and Input Optimization of High-Frequency Mechanical Control Systems
,” J. Vib. Control
, 24
(5), pp. 937–955.24.
Tahmasian
, S.
, Jafari
, F.
, and Woolsey
, C. A.
, 2016
, “On Vibrational Stabilization of a Horizontal Pendulum
,” ASME
Paper No. DSCC2016-9675.25.
Tsakalis
, K. S.
, and Ioannou
, P. A.
, 1993
, Linear Time-Varying Systems
, Prentice Hall
, Upper Saddle River, NJ
.Copyright © 2018 by ASME
You do not currently have access to this content.
