Controller Design for Euler-Bernoulli Smart Structures Using Robust Decentralized POF via Reduced Order Modeling
This paper features the proposed modeling and design
of a Robust Decentralized Periodic Output Feedback (RDPOF)
control technique for the active vibration control of smart flexible
multimodel Euler-Bernoulli cantilever beams for a multivariable
(MIMO) case by retaining the first 6 vibratory modes. The beam
structure is modeled in state space form using the concept of
piezoelectric theory, the Euler-Bernoulli beam theory and the Finite
Element Method (FEM) technique by dividing the beam into 4 finite
elements and placing the piezoelectric sensor / actuator at two finite
element locations (positions 2 and 4) as collocated pairs, i.e., as
surface mounted sensor / actuator, thus giving rise to a multivariable
model of the smart structure plant with two inputs and two outputs.
Five such multivariable models are obtained by varying the
dimensions (aspect ratios) of the aluminum beam, thus giving rise to
a multimodel of the smart structure system. Using model order
reduction technique, the reduced order model of the higher order
system is obtained based on dominant eigen value retention and the
method of Davison. RDPOF controllers are designed for the above 5
multivariable-multimodel plant. The closed loop responses with the
RDPOF feedback gain and the magnitudes of the control input are
observed and the performance of the proposed multimodel smart
structure system with the controller is evaluated for vibration control.
[1] M. Aoki, "Control of large scale dynamic systems by aggregation,"
IEEE Trans. Auto. Contr., vol. AC-13, pp. 246 - 253, 1968.
[2] 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, pp, 605 - 611, 1985.
[3] E. F. Crawley, and J. De Luis, "Use of piezoelectric actuators as
elements of intelligent structures," AIAA J, vol. 25, pp. 1373 - 1385,
1987.
[4] A. B. Chammas, and C. T. Leondes, "Pole placement by piecewise
constant output feedback," Int. J. Contr., vol. 29, pp. 31 - 38, 1979.
[5] 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.
[6] A. B. Chammas, 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.
[7] B. Culshaw, "Smart Structures : A concept or a reality," J. of Systems
and Control Engg., vol. 26, no. 206, pp. 1 - 8, 1992.
[8] 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.
[9] E. J. Davison, "A method for simplifying linear dynamical systems,"
IEEE Trans. Auto. Contr., vol. AC-11, pp. 93 - 101, 1966.
[10] J. L. Fanson, and T. K. Caughey, "Positive position feedback control for
structures," AIAA J., vol. 18, no. 4, pp. 717 - 723, 1990.
[11] P. Gahnet, A. Nemirovski, A. J. Laub, and M. Chilali, "LMI Tool box
for Matlab", The Math works Inc., Natick MA, 1995.
[12] W. Hwang, and H. C. Park, "Finite element modeling of piezoelectric
sensors and actuators", AIAA J., vol. 31, no. 5, pp. 930 - 937, 1993.
[13] 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.
[14] S. S. Lamba, and S. Vittal Rao, "On the suboptimal control via the
simplified model of Davison," IEEE Trans. Auto. Contr., vol. AC-19,
pp. 448 - 450, 1974.
[15] 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.
[16] J. Mark Balas, "Feedback control of flexible structures," IEEE Trans.
Automat. Contr., vol. AC-23, pp. 673 - 679, 1978.
[17] T. C. Manjunath, and B. Bandyopadhyay, "Vibration control of a smart
flexible cantilever beam using periodic output feedback control
technique," Proc. Fourth Asian Control Conference ASCC-2002, paper
no. 1679, pp. 1302 - 1307, Sept. 25-27, 2002.
[18] 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.
[19] 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 Mater.
and Struct., vol. 14, no. 4, pp. 624 - 636, Aug. 2005.
[20] 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.
[21] M. S. Mahmoud, and G. M. Singh, "Large scale systems modeling,"
Pergamon Press, Oxford, 1981.
[22] 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.
[23] V. L. Syrmos, P. Abdallah, P. Dorato, and K. Grigoriadis, "Static output
feedback : A survey," Automatica, vol. 33, no. 2, pp. 125 - 137, 1997.
[24] P. Seshu, "Textbook of Finite Element Analysis," 1st Ed. Prentice Hall
of India, New Delhi, 2004.
[25] 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.
[26] H. Werner, and K. Furuta, "Simultaneous stabilization based on output
measurements," Kybernetika, vol. 31, no. 4, pp. 395 - 411, 1995.
[27] H. Werner, "Robust multivariable control of a turbo-generator by
periodic output feedback," vol. 31, no. 4, pp. 395 - 411, 1997.
[28] 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.
[1] M. Aoki, "Control of large scale dynamic systems by aggregation,"
IEEE Trans. Auto. Contr., vol. AC-13, pp. 246 - 253, 1968.
[2] 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, pp, 605 - 611, 1985.
[3] E. F. Crawley, and J. De Luis, "Use of piezoelectric actuators as
elements of intelligent structures," AIAA J, vol. 25, pp. 1373 - 1385,
1987.
[4] A. B. Chammas, and C. T. Leondes, "Pole placement by piecewise
constant output feedback," Int. J. Contr., vol. 29, pp. 31 - 38, 1979.
[5] 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.
[6] A. B. Chammas, 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.
[7] B. Culshaw, "Smart Structures : A concept or a reality," J. of Systems
and Control Engg., vol. 26, no. 206, pp. 1 - 8, 1992.
[8] 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.
[9] E. J. Davison, "A method for simplifying linear dynamical systems,"
IEEE Trans. Auto. Contr., vol. AC-11, pp. 93 - 101, 1966.
[10] J. L. Fanson, and T. K. Caughey, "Positive position feedback control for
structures," AIAA J., vol. 18, no. 4, pp. 717 - 723, 1990.
[11] P. Gahnet, A. Nemirovski, A. J. Laub, and M. Chilali, "LMI Tool box
for Matlab", The Math works Inc., Natick MA, 1995.
[12] W. Hwang, and H. C. Park, "Finite element modeling of piezoelectric
sensors and actuators", AIAA J., vol. 31, no. 5, pp. 930 - 937, 1993.
[13] 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.
[14] S. S. Lamba, and S. Vittal Rao, "On the suboptimal control via the
simplified model of Davison," IEEE Trans. Auto. Contr., vol. AC-19,
pp. 448 - 450, 1974.
[15] 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.
[16] J. Mark Balas, "Feedback control of flexible structures," IEEE Trans.
Automat. Contr., vol. AC-23, pp. 673 - 679, 1978.
[17] T. C. Manjunath, and B. Bandyopadhyay, "Vibration control of a smart
flexible cantilever beam using periodic output feedback control
technique," Proc. Fourth Asian Control Conference ASCC-2002, paper
no. 1679, pp. 1302 - 1307, Sept. 25-27, 2002.
[18] 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.
[19] 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 Mater.
and Struct., vol. 14, no. 4, pp. 624 - 636, Aug. 2005.
[20] 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.
[21] M. S. Mahmoud, and G. M. Singh, "Large scale systems modeling,"
Pergamon Press, Oxford, 1981.
[22] 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.
[23] V. L. Syrmos, P. Abdallah, P. Dorato, and K. Grigoriadis, "Static output
feedback : A survey," Automatica, vol. 33, no. 2, pp. 125 - 137, 1997.
[24] P. Seshu, "Textbook of Finite Element Analysis," 1st Ed. Prentice Hall
of India, New Delhi, 2004.
[25] 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.
[26] H. Werner, and K. Furuta, "Simultaneous stabilization based on output
measurements," Kybernetika, vol. 31, no. 4, pp. 395 - 411, 1995.
[27] H. Werner, "Robust multivariable control of a turbo-generator by
periodic output feedback," vol. 31, no. 4, pp. 395 - 411, 1997.
[28] 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.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:57965", author = "T.C. Manjunath and B. Bandyopadhyay", title = "Controller Design for Euler-Bernoulli Smart Structures Using Robust Decentralized POF via Reduced Order Modeling", abstract = "This paper features the proposed modeling and design
of a Robust Decentralized Periodic Output Feedback (RDPOF)
control technique for the active vibration control of smart flexible
multimodel Euler-Bernoulli cantilever beams for a multivariable
(MIMO) case by retaining the first 6 vibratory modes. The beam
structure is modeled in state space form using the concept of
piezoelectric theory, the Euler-Bernoulli beam theory and the Finite
Element Method (FEM) technique by dividing the beam into 4 finite
elements and placing the piezoelectric sensor / actuator at two finite
element locations (positions 2 and 4) as collocated pairs, i.e., as
surface mounted sensor / actuator, thus giving rise to a multivariable
model of the smart structure plant with two inputs and two outputs.
Five such multivariable models are obtained by varying the
dimensions (aspect ratios) of the aluminum beam, thus giving rise to
a multimodel of the smart structure system. Using model order
reduction technique, the reduced order model of the higher order
system is obtained based on dominant eigen value retention and the
method of Davison. RDPOF controllers are designed for the above 5
multivariable-multimodel plant. The closed loop responses with the
RDPOF feedback gain and the magnitudes of the control input are
observed and the performance of the proposed multimodel smart
structure system with the controller is evaluated for vibration control.", keywords = "Smart structure, Euler-Bernoulli beam theory,
Periodic output feedback control, Finite Element Method, State space
model, SISO, Embedded sensors and actuators, Vibration control,
Reduced order model", volume = "1", number = "6", pages = "293-17", }
