Optimal Actuator Placement for Active Vibration Control With Known Disturbances

Abstract

Two cost functions, based on the disturbance sensitivity grammian, are presented that can be used to find the optimal position of an active vibration control actuator. The disturbance sensitivity grammian is a novel parameter in the field of active vibration control, and by minimizing the "size" of this matrix, one can maximize the effectiveness of the actuator to control the structure in a closed-loop configuration. Two examples—a pinned-pinned beam and an L-shaped membrane—are used to demonstrate the applicability of the criterion. It turns out that the optimal position is very close to where the disturbance acts on the system. The criterion is also not sensitive to the assumed damping included in the model.

Keywords

Vibration control actuator placement optimal control disturbance sensitivity grammian

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References

Arbel, A. , 1981, "Controllability measures and actuator placement in oscillatory systems," International Journal of Control 33(3), 565-574.

Chen, C. , 1984, Linear System Theory and Design, Holt, Rinehart, and Winston, New York.

Franklin, G.F. , Powell, J.D. , and Workman, M.L. , 1990, Digital Control of Dynamic Systems, Addison-Wesley, Reading, MA.

Friedland, B. , 1986, Control System Design: An Introduction to State-Space Methods, McGraw-Hill, New York. Furuta, K. , Sano, A. , and Atherton, D. , 1988, State Variable Methods in Automatic Control, Wiley, Chichester, UK.

Hac, A. and Liu, L. , 1993, "Sensors and actuator location in motion control of flexible structures," Journal of Sound and Vibration 167(2), 239-261.

Hamdan, A.M.A. and Nayfeh, A.H. , 1989, "Measures of modal controllability and observability for first and second-order linear systems," Journal of Guidance 12(3), 421-428.

Hughes, P.C. and Skelton, R.E. , 1980, "Controllability and observability of linear matrix-second-order systems," ASME Journal of Applied Mechanics 47, 415-420.

Kalman, R.E. , Ho, Y.C. , and Narenda, K.S. , 1962, "Controllability of linear dynamical systems," Contribution to Differential Equations 1(2), 189-213.

Lim, K.B. , 1992, "Method for optimal actuator and sensor placement for large flexible structures," Journal of Guidance 15(1), 49-57.

10.

Maghami, P.G. and Joshi, S.M. , 1993a, "Sensor/actuator placement for flexible space structures ," IEEE Transactions on Aerospace and Electronic Systems 29(2), 345-351.

11.

Maghami, P.G. and Joshi, S.M. , 1993b, "Sensor-actuator placement for flexible structures with actuator dynamics," Journal of Guidance, Control and Dynamics 16(2), 301-307.

12.

Meirovitch, L. , 1967, Analytical Methods in Vibrations, Prentice Hall, Englewood Cliffs, NJ.

13.

Schulz, G. and Heimbold, G. , 1983, "Dislocated actuator/sensors positioning and feedback design for flexible structures," Journal of Guidance 6(5), 361-367.

14.

van Niekerk, J.L. , 1995, "An approximate method to determine the optimal position of sensors and actuators for structures with repeated eigenfrequencies ," Journal of the Southern African Acoustics Institute 6, 1-12.