This paper presents modelling, system identification, simulation, and experimental results for passivity-based robust control of piezo-actuated flexible beam. The flexible beam configuration considered is a cantilever aluminum beam with a piezoelectric transducer used as the actuator and tip-accelerometer as the sensor. The actuator and sensor are non-collocated. The Lagrangian formulation is used to obtain mathematical model of the flexible link dynamics with piezo actuator. For control design purposes, a finite dimensional approximate model is derived using assumed modes approach. It is shown that the approximate model compares very well with the experimentally identified model. Since the system is inherently not passive, passification techniques are used to render the system robustly passive which enables the use of passivity-based feedback control design. The controller design is validated both in simulation as well as in experiments. The simulation and experimental results demonstrate the effectiveness of controller in suppressing the tip vibrations of the link. The controller design is shown to be robust to both parametric uncertainties and unmodeled dynamics.

Issue Section:
Technical Papers
Keywords:
piezoelectric actuators, beams (structures), flexible structures, vibrations, piezoelectric transducers
Topics:
Control equipment, Design, Dynamics (Mechanics), Modeling, Robust control, Sensors, Transfer functions, Vibration, Actuators, Piezoelectric actuators, Stability, Uncertainty, Frequency response, Feedback, Piezoelectric transducers, Resonance, Robustness
1.
Bailey
,
T.
, and
Hubbard
,
J. E.
,
1985
, “
Distributed Piezoelectric-polymer Active Vibration Control of a Cantilever Beam
,”
J. Guid. Control Dyn.
,
8
(
5
), pp.
605
–
611
.
2.
Baz
,
A.
, and
Poh
,
S.
,
1988
, “
Performance of an Active Control System with Piezoelectric Actuators
,”
J. Sound Vib.
,
126
(
2
), pp.
327
–
343
.
3.
Meirovitch
,
L.
, and
Baruh
,
H.
,
1983
, “
Comparison of Control Techniques for Large Flexible Systems
,”
J. Guid. Control Dyn.
,
6
(
4
), pp.
302
–
310
.
4.
Callahan
,
J.
, and
Baruh
,
H.
,
1996
, “
Active Control of Flexible Structures by Use of Segmented Piezoelectric Elements
,”
J. Guid. Control Dyn.
,
19
(
4
), pp.
808
–
814
.
5.
Gosavi, S. V. 2000, “Modeling and Control of Piezo-Actuated Flexible Link System,” M. S. Thesis, Kansas State University.
6.
Krishnan, H., and Vidyasagar, M., 1987, “Control of a Single-link Flexible Beam Using a Hankel-norm Based Reduced Order Model,” Proceedings of the IEEE International Conference on Robotics and Automation, pp. 914.
7.
Pota, H. R., and Alberts, T. E., 1992, “Multivariable Transfer Functions for a Slewing Piezoelectric Laminate Beam,” IEEE International Conference on Systems Engineering, pp. 257–260, September 17–19, Kobe, Japan.
8.
Alberts
,
T. E.
, and
Pota
,
H. R.
,
1997
, “
Broadband Dynamic Modification Using Feedforward Control
,”
ASME J. Dyn. Syst., Meas., Control
,
119
(
4
), pp.
700
–
706
.
9.
Pota, H. R., Reza Moheimani, S. O., and Smith, M., 1999, “Resonant Controllers for Flexible Structures,” Proceedings of the 38th IEEE Conference on Decision and Control, Vol. 1, pp. 631–636, December 7–10.
10.
Pota, H. R., Alberts, T. E., and Petersen, I. R., 1993, “H∞ Control of Flexible Slewing Link with Active Damping,” Proceedings of the International Symposium on Smart Structures and Materials ’93, pp. 60–71, February.
11.
Moheimani, S. O. R., Pota, H. R., and Petersen, I. R., 1998, “Spatial Control for Active Vibration Control of Piezoelectric Laminates Decision and Control,” Proceedings of the 37th IEEE Conference, Volume 4, pp. 4308–4313, December 16–18.
12.
Kelkar, A. G., and Joshi, S. M., 1997, “Robust Control of Non-passive Systems via Passification,” Proceedings American Control Conference, Volume 5, pp. 2657–2661, Albuquerque, NM, June 4–6.
13.
Fraser, A. R., and Daniel, R. W., 1991, Perturbation Techniques for Flexible Manipulators, Kluwer Academic Publ., Dordrecht.
14.
Meirovitch, L., 1975, Elements of Vibration Analysis, McGraw-Hill.
15.
Papoulis, A., 1962, The Fourier Integral and its Application, McGraw-Hill.
16.
Bode, H. W., 1945, Network Analysis and Feedback Amplifier Design, D. Van Nostrand Company Inc, New York.
17.
Horowitz, I. M., 1963, Synthesis of Feedback Systems, Academic Press, New York.
18.
Ogata, K., 1997, Modern Control Engineering, Prentice Hall, New Jersy.
19.
Kelkar, A. G., and Joshi, S. M., 1996, Control of Nonlinear Multibody Flexible Space Structures, Volume 221 of Lecture Notes in Control and Information Sciences, Springer-Verlag, August.
20.
Joshi, S. M., 1989, Control of Large Flexible Space Structures. (Vol. 131, Lecture Notes in Control and Information Sciences), Springer-Verlag, Berlin.
21.
Willems
,
J. C.
,
1972
, “
Dissipative Dynamical Systems, Parts I and II
,”
Arch. Ration. Mech. Anal.
,
45
, pp.
321
–
393
.
22.
Desoer, C. A., and Vidyasagar, M., 1975, Feedback Systems: Input-Output Properties, Academic Press, New York.
23.
Hill
,
D. J.
, and
Moylan
,
P. J.
,
1976
, “
The Stability of Nonlinear Dissipative Systems
,”
IEEE Trans. Autom. Control
,
21
, pp.
708
–
711
.
24.
Byrnes
,
C. I.
,
Isidori
,
A.
, and
Willems
,
J. C.
,
1991
, “
Passivity, Feedback Equivalence, and Global Stabilization of Minimum Phase Systems
,”
IEEE Trans. Autom. Control
,
36
, pp.
1228
–
1240
.
25.
Ortega
,
R.
,
1989
, “
Passivity Properties for Stabilization of Cascaded Nonlinear Systems
,”
Automatica
,
27
, pp.
423
–
424
.
26.
Lozano
,
R.
,
Brogliato
,
B.
, and
Landau
,
I. D.
,
1992
, “
Passivity and Global Stabilization of Cascaded Nonlinear Systems
,”
IEEE Trans. Autom. Control
,
37
, pp.
1386
–
1388
.
27.
Hill, D., Ortega, R., and van der Schaft, A., 1994, Nonlinear Controller Design Using Passivity and Small-Gain Techniques, Notes from IEEE CDC Tutorial Workshop.
28.
Newcomb, R. W., 1966, Linear Multiport Synthesis, McGraw-Hill, New York.
29.
Joshi
,
S. M.
, and
Gupta
,
S.
,
1996
, “
On a Class of Marginally Stable Positive-Real Systems
,”
IEEE Trans. Autom. Control
,
41
(
1
), pp.
152
–
155
.
30.
Isidori
,
A.
,
Joshi
,
S. M.
, and
Kelkar
,
A. G.
,
1999
, “
Asymptotic Stability of Interconnected Passive Non-linear Systems
,”
Int. J. Robust Nonlinear Control
,
9
, pp.
261
–
273
.
31.
Joshi
,
S. M.
, and
Kelkar
,
A. G.
,
2001
, “
Passivity-based Robust Control of Systems with Redundant Sensors and Actuators
,”
Int. J. Control
,
74
(
5
), pp.
474
–
481
, March.
32.
Kelkar, A. G., and Joshi, S. M., 1998, “Robust Passification and Control of Non-passive Systems,” Proc. American Control Conference, pp. 3133–3137, Adam’s Mark Hotel, Philadelphia, PA, June 24–26.
33.
Pota
,
H. R.
, and
Kelkar
,
A. G.
,
2001
, “
Modelling and Control of Acoustic Ducts
,”
ASME J. Vibr. Acoust.
,
123
(
1
), pp.
2
–
10
, January.
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