Vibration control is an effective alternative to conventional feedback and feedforward control. Motivated by its important application in physical systems and few results on general oscillatory tracking control, we consider tracking control of a class of nonlinear systems using oscillation in the paper. We propose a new oscillatory control design using general averaging analysis for the tracking problem. Based on the oscillation functions associated with accessible vibrating components of the system, oscillatory control is designed to track a desired trajectory. Comparing to existing oscillatory tracking control, our approach is robust to initial conditions. We show the effectiveness of the proposed method by two simulation examples, which include a second-order nonholonomic integrator and the inverted pendulum system. For the inverted pendulum system, we show that our designed oscillatory control does not need state feedback to track a desired trajectory, which is desirable for systems where state measurement is not feasible.

References

References
1.
Meerkov
,
S. M.
, 1980, “
Principle of Vibrational Control: Theory and Applications
,”
IEEE Trans. Autom. Control
,
25
(
4
), pp.
755
762
.
2.
Bellman
,
R. E.
,
Bentsman
,
J.
, and
Meerkov
,
S. M.
, 1986, “
Vibrational Control of Nonlinear Systems: Vibrational Stability
,”
IEEE Trans. Autom. Control
,
31
(
8
), pp.
710
716
.
3.
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
.
4.
Bentsman
,
J.
, 1987, “
Vibrational Control of a Class of Nonlinear Systems by Nonlinear Multiplicative Vibrations
,”
IEEE Trans. Autom. Control
,
32
(
8
), pp.
711
716
.
5.
Jeon
,
S.
,
Thundat
,
T.
, and
Braiman
,
Y.
, 2007, “
Effect of Normal Vibration on Friction in the Atomic Force Microscopy Experiment
,”
Appl. Phys. Lett.
,
88
(
1
), p.
214102
.
6.
Nijimeijer
,
H.
, and
van der Schaft
,
A. J.
, 1990,
Nonlinear Dynamical Control Systems
,
Springer-Verlag
,
New York
.
7.
Isdori
,
A.
, 1995,
Nonlinear Control Systems
,
3rd ed.
,
Springer-Verlag
,
New York
.
8.
Khalil
,
H. K.
, 2002,
Nonlinear Systems
,
3rd ed.
,
Prentice Hall
,
NJ
.
9.
Baillieul
,
J.
, 1993, “
Stable Average Motions of Mechanical Systems Subject to Periodic Forcing
,”
Dynamics and Control of Mechanical Systems: The Falling Cat and Related Problems
,
M. J.
Enos
ed., vol.
1
,
Springer Verlag
,
New York, NY
, pp.
1
23
.
10.
Baillieul
,
J.
, 1998, “
The Geometry of Controlled Mechanical Systems
,”
Mathematical Control Theory
,
J.
Baillieul
and
J. C.
Willems
, eds.,
Springer Verlag
,
New York, NY
, pp.
322
354
.
11.
Baillieul
,
J.
, and
Weibel
,
S.
, 1998, “
Scale Dependence in the Oscillatory Control of Micromechanisms
,”
Proceedings of IEEE Conference on Decision and Control
, Tampa, FL, pp.
3058
3063
.
12.
Shapiro
,
B.
, and
Zinn
,
B. T.
, 1997, “
High-Frequency Nonlinear Vibrational Control
,”
IEEE Trans. Automa. Control
,
42
(
1
), pp.
83
90
.
13.
Sussmann
,
H. J.
, and
Liu
,
W. S.
, 1991, “
Limits of Highly Oscillatory Controls and the Approximation of General Paths by Admissible Trajectories
,”
Proceedings of IEEE Conference on Decision and Control
, pp.
437
442
.
14.
Sussmann
,
H. J.
, and
Liu
,
W. S.
, 1992, “
Lie Bracket Extensions and Averaging: The Single Bracket Case
,”
Nonholonomic Motion Planning
,
Z. X.
Li
and
J. F.
Canny
, eds.,
Kluwer
,
Massachusetts
, pp.
109
148
.
15.
Martinez
,
S.
,
Cortes
,
J.
, and
Bullo
,
F.
, 2003, “
Analysis and Design of Oscillatory Control Systems
,”
IEEE Trans. Automa. Control
,
48
(
7
), pp.
1164
1177
.
16.
Morgansen
,
K. A.
, and
Brockett
,
R. W.
, 1999, “
Nonholonomic Control Based on Approximate Inversion
,”
Proceedings of IEEE American Control Conference
, San Diego, CA, pp.
3515
3519
.
17.
Brockett
,
R. W.
, 1982, “
Control Theory and Singular Riemannian Geometry
,”
New Directions in Applied Mathematics
,
P.
Hilton
and
G.
Young
eds.,
Springer-Verlag
,
New York, NY
, pp.
11
27
.
18.
Blekhman
,
I. I.
, 2000,
Vibrational Mechanics: Nonlinear Dynamic Effects, General Approach, Applications
,
World Scientific
,
Singapore
.
19.
Bogolyubov
,
N. N.
, 1945, “
On Certain Statistical Methods in Mathematical Physics
,” AN USSR, Kiev (in Russian).
20.
Weibel
,
S.
,
Kaper
,
T. J.
, and
Baillieul
,
J.
, 1997, “
Global Dynamics of a Rapidly Forced Cart and Pendulum
,”
Nonlinear Dyn.
,
13
, p.
131170
.
You do not currently have access to this content.