Vibration Suppression Control for a Flexible Beam with Sliding Mode Observer
In this study, the vibration suppression of smart structure is performed by using piezoelectric patch structure. The smart structure consists of a beam, as a host structure, and piezoelectric patches, attached to the surface of the beam, as actuation and sensing elements. Two sources of instabilities, namely, the observer spillover and the control spillover, are considered in the current design of the controller based on a reduced order model of the large scale system.
To design a controller, that will attenuate the vibration, the balance realization is used to select the reduced order model that is most controllable and observable. Eight state is selected for the reduced model in the present work.
The sliding mode observer, which based on the equivalent control, is designed to estimate eight states of the reduced model where the state estimation error is proved bounded. By using the estimated state via sliding mode observer an optimal LQR controller is designed that attenuate the vibration of a smart cantilever beam using piezoelectric element. To overcome the control spillover problem, an avoidance condition was derive, that will ensure the asymptotic stability for the proposed vibration control design.
The numerical simulations are preformed to test the vibration attenuation ability of the proposed optimal control. For 10 mm initial tip displacement, the piezoelectric actuator found able to reduce the tip displacement to about 1.3 mm after 15s, while it equal to 7 mm with the open loop case.
The simulations show, also, that the optimal control action is performed with minimum effort where only 30 voltage is required while piezoelectric actuator is saturated at 200 voltage.
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