Control of a thermoelastic beam using heat actuation

This paper explores the idea of using heat as an actuator to simultaneously control vibration and temperature of a thermoelastic beam. We first model the beam as a slender, uniform cantilever beam of rectangular cross-section subject to heat through heat patches on the lower and upper surfaces at some discrete spanwise locations. The governing equations of the model are two coupled partial differential equations: one governing the elastic bending displacement and one governing the two-dimensional heat conduction of the beam. Through a discretization, the partial differential equations are replaced by a set of ordinary differential equations in a compact state-space form. We show that the coupling is actually between elastic displacement and those components of temperature contributing to the thickness-wise gradient at the midplane. The linear quadratic regulator in conjunction with the Kalman–Bucy filter is used for the control design to simultaneously damp out the displacement and the gradient. In a numerical example, we show the presence of thermoelastic damping due to the coupling. We also show that the displacement and gradient can simultaneously be controlled by using displacement measurements only, and that for less control effort it is also necessary to include some temperature measurements in the feedback.