Multivariable Control of Smart Timoshenko Beam Structures Using POF Technique
Active Vibration Control (AVC) is an important
problem in structures. One of the ways to tackle this problem is to
make the structure smart, adaptive and self-controlling. The objective
of active vibration control is to reduce the vibration of a system by
automatic modification of the system-s structural response. This
paper features the modeling and design of a Periodic Output
Feedback (POF) control technique for the active vibration control of
a flexible Timoshenko cantilever beam for a multivariable case with
2 inputs and 2 outputs by retaining the first 2 dominant vibratory
modes using the smart structure concept. The entire structure is
modeled in state space form using the concept of piezoelectric
theory, Timoshenko beam theory, Finite Element Method (FEM) and
the state space techniques. Simulations are performed in MATLAB.
The effect of placing the sensor / actuator at 2 finite element
locations along the length of the beam is observed. The open loop
responses, closed loop responses and the tip displacements with and
without the controller are obtained and the performance of the smart
system is evaluated for active vibration control.
[1] O. J. Aldraihem, R. C. Wetherhold, and T. Singh, "Distributed control of
laminated beams : Timoshenko Vs. Euler-Bernoulli Theory," J. of
Intelligent Materials Systems and Structures, vol. 8, pp. 149-157, 1997.
[2] H. Abramovich, "Deflection control of laminated composite beam with
piezoceramic layers-Closed form solutions," Composite Structures, vol.
43, no. 3, pp. 217-131, 1998.
[3] O. J. Aldraihem, and K. A. Ahmed, "Smart beams with extension and
thickness-shear piezoelectric actuators," Smart Materials and Structures,
vol. 9, no. 1, pp. 1-9, 2000.
[4] A. K. Ahmed, and J. A. Osama, "Deflection analysis of beams with
extension and shear piezoelectric patches using discontinuity functions"
Smart Materials and Structures, vol. 10, no. 1, pp. 212-220, 2001.
[5] L. E. Azulay, and H. Abramovich, "Piezoelectric actuation and sensing
mechanisms-Closed form solutions," Composite Structures J., vol. 64,
pp. 443-453, 2004.
[6] T. Baily, and J. E. Hubbard Jr., "Distributed piezoelectric polymer active
vibration control of a cantilever beam," J. of Guidance, Control and
Dynamics, vol. 8, no.5, pp. 605-611, 1985.
[7] A. Benjeddou, M. A. Trindade, and R. Ohayon, "New shear actuated
smart structure beam finite element," AIAA J., vol. 37, pp. 378-383,
1999.
[8] E. F. Crawley, and J. De Luis, "Use of piezoelectric actuators as
elements of intelligent structures," AIAA J., vol. 25, pp. 1373-1385,
1987.
[9] K. Chandrashekhara, and S. Varadarajan, "Adaptive shape control of
composite beams with piezoelectric actuators," J. of Intelligent
Materials Systems and Structures, vol. 8, pp. 112-124, 1997.
[10] B. Culshaw, "Smart Structures : A concept or a reality," J. of Systems
and Control Engg., vol. 26, no. 206, pp. 1-8, 1992.
[11] C. R. Cooper, "Shear coefficient in Timoshenko beam theory," ASME J.
of Applied Mechanics, vol. 33, pp. 335-340, 1966.
[12] S. B. Choi, C. Cheong, and S. Kini, "Control of flexible structures by
distributed piezo-film actuators and sensors," J. of Intelligent Materials
and Structures, vol. 16, pp. 430-435, 1995.
[13] A. B. Chammas, and C. T. Leondes, "Pole placement by piecewise
constant output feedback," Int. J. Contr., vol. 29, pp. 31-38, 1979.
[14] A. B. Chammas, and C. T. Leondes, "On the design of LTI systems by
periodic output feedback, Part-I, Discrete Time pole assignment," Int. J.
Ctrl., vol. 27, pp. 885-894, 1978.
[15] Chammas, A. B. and C. T. Leondes, "On the design of LTI systems by
periodic output feedback, Part-II, Output feedback controllability," Int.
J. Ctrl., vol. 27, pp. 895-903, 1978.
[16] C. Doschner, and M. Enzmann, "On model based controller design for
smart structure," Smart Mechanical Systems Adaptronics SAE
International USA, pp. 157-166, 1998.
[17] P. Donthireddy, and K. Chandrashekhara, "Modeling and shape control
of composite beam with embedded piezoelectric actuators," Comp.
Structures, vol. 35, no. 2, pp. 237-244, 1996.
[18] J. L. Fanson, and T. K. Caughey, "Positive position feedback control for
structures," AIAA J., vol. 18, no. 4, pp. 717-723, 1990.
[19] J. C. Geromel, C. C. De Souza, and R. E. Skeleton, "LMI Numerical
solution for output feedback stabilization," Proc. American Contr. Conf.,
pp. 40-44, 1994.
[20] P. Gahnet, A. Nemirovski, A. J. Laub, and M. Chilali, "LMI Tool box
for Matlab", The Math works Inc., Natick MA, 1995.
[21] S. Hanagud, M. W. Obal, and A. J. Callise, "Optimal vibration control
by the use of piezoceramic sensors and actuators," J. of Guidance,
Control and Dyn., vol. 15, no. 5, pp. 1199-1206, 1992.
[22] W. Hwang, and H. C. Park, "Finite element modeling of piezoelectric
sensors and actuators", AIAA J., vol. 31, no. 5, pp. 930-937, 1993.
[23] J. B. Kosmataka, and Z. Friedman, "An improved two-node Timoshenko
beam finite element", Computers and Struct., vol. 47, no. 3, pp. 473-
481, 1993.
[24] W. S. Levine, and M. Athans, "On the determination of the optimal
constant output feedback gains for linear multivariable systems," IEEE
Trans. Auto. Contr., vol. AC-15, pp. 44-48, 1970.
[25] T. C. Manjunath, and B. Bandyopadhyay, "Modeling and fast output
sampling feedback control of a smart Timoshenko cantilever beam,"
International Journal of Smart Structures and Systems, vol. 1, no. 3,
ISSN 1738-1584, pp. 283-308, Sep. 2005.
[26] T. C. Manjunath, and B. Bandyopadhyay, "Vibration control of a smart
flexible cantilever beam using periodic output feedback," Asian Journal
of Control, vol. 6, no. 1, pp. 74 - 87, Mar. 2004.
[27] T. C. Manjunath, and B. Bandyopadhyay, "Fault tolerant control of
flexible smart structures using robust decentralized periodic output
sampling feedback technique," International Journal of Smart Materi.
and Struct., vol. 14, no. 4, pp. 624-636, Aug. 2005.
[28] T. C. Manjunath, and B. Bandyopadhyay, R. Gupta, and M. Umapathy,
"Multivariable control of a smart structure using periodic output
feedback control technique," Proc. of the Seventh International
Conference on Control, Automation, Robotics and Computer Vision,
ICARCV 2002, Singapore, Paper No. 2002P1283, pp. 1481-1486, Dec.
2-5, 2002.
[29] Manjunath, T.C., Bandyopadhyay, B. and Janardhanan, S.,
"Multivariable control of a smart structure using periodic output
feedback control technique," Proc. 3rd International Conference on
System Identification and Control Problems, SICPRO 2004, Institute of
Control Sciences, Moscow, Russia, Paper No. 23016, pp. 1300-1312,
Jan. 28 -30, 2004.
[30] S. Raja, G. Prathap, and P. K. Sinha, P.K., "Active vibration control of
composite sandwich beams with piezoelectric extension-bending and
shear actuators," Smart Materials and Structures, vol. 11, no. 1, pp. 63-
71, 2002.
[31] S. Rao, and M. Sunar, "Piezoelectricity and its uses in disturbance
sensing and control of flexible structures : A survey," Applied
Mechanics Rev., vol. 47, no. 2, pp. 113-119, 1994.
[32] C. T. Sun, and X. D. Zhang, "Use of thickness-shear mode in adaptive
sandwich structures," Smart Materials and Structures J., vol. 3, no. 4,
pp. 202-206, 1995.
[33] V. L. Syrmos, P. Abdallah, P. Dorato, and K. Grigoriadis, "Static output
feedback : A survey," Automatica, vol. 33, no. 2, pp. 125-137, 1997.
[34] P. Seshu, "Textbook of Finite Element Analysis," 1st Ed. Prentice Hall
of India, New Delhi, 2004.
[35] J. Thomas, and B. A. H. Abbas, "Finite Element Methods for dynamic
analysis of Timoshenko beam," J. of Sound and Vibration, vol. 41, pp.
291-299, 1975.
[36] M. Umapathy, and B. Bandyopadhyay, "Control of flexible beam
through smart structure concept using periodic output feedback," System
Science Journal, vol. 26, no. 1, pp. 23-46, 2000.
[37] H. Werner, and K. Furuta, "Simultaneous stabilization based on output
measurements," Kybernetika, vol. 31, no. 4, pp. 395-411, 1995.
[38] H. Werner, H., "Robust multivariable control of a turbo-generator by
periodic output feedback," Proc. American Contr. Conf., New Mexico,
pp. 1979-1983, 1997.
[39] Y. C. Yan, J. Lam, and Y. X. Sun, "Static output feedback stabilization:
An LMI approach," Automatica, vol. 34, no. 12, pp. 1641-1645, 1998.
[40] X. D. Zhang, and C. T. Sun, "Formulation of an adaptive sandwich
beam," Smart Mater. and Struct., vol. 5, no.6, pp. 814-823, 1996.
[41] T. C. Manjunath, and B. Bandyopadhyay, "Modeling and fast output
sampling feedback control of a smart Timoshenko cantilever beam",
Smart Structures and Systems, vol. 1, no. 3, pp. 283-308, Sept. 2005.
[1] O. J. Aldraihem, R. C. Wetherhold, and T. Singh, "Distributed control of
laminated beams : Timoshenko Vs. Euler-Bernoulli Theory," J. of
Intelligent Materials Systems and Structures, vol. 8, pp. 149-157, 1997.
[2] H. Abramovich, "Deflection control of laminated composite beam with
piezoceramic layers-Closed form solutions," Composite Structures, vol.
43, no. 3, pp. 217-131, 1998.
[3] O. J. Aldraihem, and K. A. Ahmed, "Smart beams with extension and
thickness-shear piezoelectric actuators," Smart Materials and Structures,
vol. 9, no. 1, pp. 1-9, 2000.
[4] A. K. Ahmed, and J. A. Osama, "Deflection analysis of beams with
extension and shear piezoelectric patches using discontinuity functions"
Smart Materials and Structures, vol. 10, no. 1, pp. 212-220, 2001.
[5] L. E. Azulay, and H. Abramovich, "Piezoelectric actuation and sensing
mechanisms-Closed form solutions," Composite Structures J., vol. 64,
pp. 443-453, 2004.
[6] T. Baily, and J. E. Hubbard Jr., "Distributed piezoelectric polymer active
vibration control of a cantilever beam," J. of Guidance, Control and
Dynamics, vol. 8, no.5, pp. 605-611, 1985.
[7] A. Benjeddou, M. A. Trindade, and R. Ohayon, "New shear actuated
smart structure beam finite element," AIAA J., vol. 37, pp. 378-383,
1999.
[8] E. F. Crawley, and J. De Luis, "Use of piezoelectric actuators as
elements of intelligent structures," AIAA J., vol. 25, pp. 1373-1385,
1987.
[9] K. Chandrashekhara, and S. Varadarajan, "Adaptive shape control of
composite beams with piezoelectric actuators," J. of Intelligent
Materials Systems and Structures, vol. 8, pp. 112-124, 1997.
[10] B. Culshaw, "Smart Structures : A concept or a reality," J. of Systems
and Control Engg., vol. 26, no. 206, pp. 1-8, 1992.
[11] C. R. Cooper, "Shear coefficient in Timoshenko beam theory," ASME J.
of Applied Mechanics, vol. 33, pp. 335-340, 1966.
[12] S. B. Choi, C. Cheong, and S. Kini, "Control of flexible structures by
distributed piezo-film actuators and sensors," J. of Intelligent Materials
and Structures, vol. 16, pp. 430-435, 1995.
[13] A. B. Chammas, and C. T. Leondes, "Pole placement by piecewise
constant output feedback," Int. J. Contr., vol. 29, pp. 31-38, 1979.
[14] A. B. Chammas, and C. T. Leondes, "On the design of LTI systems by
periodic output feedback, Part-I, Discrete Time pole assignment," Int. J.
Ctrl., vol. 27, pp. 885-894, 1978.
[15] Chammas, A. B. and C. T. Leondes, "On the design of LTI systems by
periodic output feedback, Part-II, Output feedback controllability," Int.
J. Ctrl., vol. 27, pp. 895-903, 1978.
[16] C. Doschner, and M. Enzmann, "On model based controller design for
smart structure," Smart Mechanical Systems Adaptronics SAE
International USA, pp. 157-166, 1998.
[17] P. Donthireddy, and K. Chandrashekhara, "Modeling and shape control
of composite beam with embedded piezoelectric actuators," Comp.
Structures, vol. 35, no. 2, pp. 237-244, 1996.
[18] J. L. Fanson, and T. K. Caughey, "Positive position feedback control for
structures," AIAA J., vol. 18, no. 4, pp. 717-723, 1990.
[19] J. C. Geromel, C. C. De Souza, and R. E. Skeleton, "LMI Numerical
solution for output feedback stabilization," Proc. American Contr. Conf.,
pp. 40-44, 1994.
[20] P. Gahnet, A. Nemirovski, A. J. Laub, and M. Chilali, "LMI Tool box
for Matlab", The Math works Inc., Natick MA, 1995.
[21] S. Hanagud, M. W. Obal, and A. J. Callise, "Optimal vibration control
by the use of piezoceramic sensors and actuators," J. of Guidance,
Control and Dyn., vol. 15, no. 5, pp. 1199-1206, 1992.
[22] W. Hwang, and H. C. Park, "Finite element modeling of piezoelectric
sensors and actuators", AIAA J., vol. 31, no. 5, pp. 930-937, 1993.
[23] J. B. Kosmataka, and Z. Friedman, "An improved two-node Timoshenko
beam finite element", Computers and Struct., vol. 47, no. 3, pp. 473-
481, 1993.
[24] W. S. Levine, and M. Athans, "On the determination of the optimal
constant output feedback gains for linear multivariable systems," IEEE
Trans. Auto. Contr., vol. AC-15, pp. 44-48, 1970.
[25] T. C. Manjunath, and B. Bandyopadhyay, "Modeling and fast output
sampling feedback control of a smart Timoshenko cantilever beam,"
International Journal of Smart Structures and Systems, vol. 1, no. 3,
ISSN 1738-1584, pp. 283-308, Sep. 2005.
[26] T. C. Manjunath, and B. Bandyopadhyay, "Vibration control of a smart
flexible cantilever beam using periodic output feedback," Asian Journal
of Control, vol. 6, no. 1, pp. 74 - 87, Mar. 2004.
[27] T. C. Manjunath, and B. Bandyopadhyay, "Fault tolerant control of
flexible smart structures using robust decentralized periodic output
sampling feedback technique," International Journal of Smart Materi.
and Struct., vol. 14, no. 4, pp. 624-636, Aug. 2005.
[28] T. C. Manjunath, and B. Bandyopadhyay, R. Gupta, and M. Umapathy,
"Multivariable control of a smart structure using periodic output
feedback control technique," Proc. of the Seventh International
Conference on Control, Automation, Robotics and Computer Vision,
ICARCV 2002, Singapore, Paper No. 2002P1283, pp. 1481-1486, Dec.
2-5, 2002.
[29] Manjunath, T.C., Bandyopadhyay, B. and Janardhanan, S.,
"Multivariable control of a smart structure using periodic output
feedback control technique," Proc. 3rd International Conference on
System Identification and Control Problems, SICPRO 2004, Institute of
Control Sciences, Moscow, Russia, Paper No. 23016, pp. 1300-1312,
Jan. 28 -30, 2004.
[30] S. Raja, G. Prathap, and P. K. Sinha, P.K., "Active vibration control of
composite sandwich beams with piezoelectric extension-bending and
shear actuators," Smart Materials and Structures, vol. 11, no. 1, pp. 63-
71, 2002.
[31] S. Rao, and M. Sunar, "Piezoelectricity and its uses in disturbance
sensing and control of flexible structures : A survey," Applied
Mechanics Rev., vol. 47, no. 2, pp. 113-119, 1994.
[32] C. T. Sun, and X. D. Zhang, "Use of thickness-shear mode in adaptive
sandwich structures," Smart Materials and Structures J., vol. 3, no. 4,
pp. 202-206, 1995.
[33] V. L. Syrmos, P. Abdallah, P. Dorato, and K. Grigoriadis, "Static output
feedback : A survey," Automatica, vol. 33, no. 2, pp. 125-137, 1997.
[34] P. Seshu, "Textbook of Finite Element Analysis," 1st Ed. Prentice Hall
of India, New Delhi, 2004.
[35] J. Thomas, and B. A. H. Abbas, "Finite Element Methods for dynamic
analysis of Timoshenko beam," J. of Sound and Vibration, vol. 41, pp.
291-299, 1975.
[36] M. Umapathy, and B. Bandyopadhyay, "Control of flexible beam
through smart structure concept using periodic output feedback," System
Science Journal, vol. 26, no. 1, pp. 23-46, 2000.
[37] H. Werner, and K. Furuta, "Simultaneous stabilization based on output
measurements," Kybernetika, vol. 31, no. 4, pp. 395-411, 1995.
[38] H. Werner, H., "Robust multivariable control of a turbo-generator by
periodic output feedback," Proc. American Contr. Conf., New Mexico,
pp. 1979-1983, 1997.
[39] Y. C. Yan, J. Lam, and Y. X. Sun, "Static output feedback stabilization:
An LMI approach," Automatica, vol. 34, no. 12, pp. 1641-1645, 1998.
[40] X. D. Zhang, and C. T. Sun, "Formulation of an adaptive sandwich
beam," Smart Mater. and Struct., vol. 5, no.6, pp. 814-823, 1996.
[41] T. C. Manjunath, and B. Bandyopadhyay, "Modeling and fast output
sampling feedback control of a smart Timoshenko cantilever beam",
Smart Structures and Systems, vol. 1, no. 3, pp. 283-308, Sept. 2005.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:49821", author = "T.C. Manjunath and B. Bandyopadhyay", title = "Multivariable Control of Smart Timoshenko Beam Structures Using POF Technique", abstract = "Active Vibration Control (AVC) is an important
problem in structures. One of the ways to tackle this problem is to
make the structure smart, adaptive and self-controlling. The objective
of active vibration control is to reduce the vibration of a system by
automatic modification of the system-s structural response. This
paper features the modeling and design of a Periodic Output
Feedback (POF) control technique for the active vibration control of
a flexible Timoshenko cantilever beam for a multivariable case with
2 inputs and 2 outputs by retaining the first 2 dominant vibratory
modes using the smart structure concept. The entire structure is
modeled in state space form using the concept of piezoelectric
theory, Timoshenko beam theory, Finite Element Method (FEM) and
the state space techniques. Simulations are performed in MATLAB.
The effect of placing the sensor / actuator at 2 finite element
locations along the length of the beam is observed. The open loop
responses, closed loop responses and the tip displacements with and
without the controller are obtained and the performance of the smart
system is evaluated for active vibration control.", keywords = "Smart structure, Timoshenko theory, Euler-Bernoulli
theory, Periodic output feedback control, Finite Element Method,
State space model, Vibration control, Multivariable system, Linear
Matrix Inequality", volume = "1", number = "6", pages = "271-17", }
