Adaptive boundary control of the size-dependent behavior of Euler-Bernoulli micro-beams with unknown parameters and varying disturbance

In this paper, modeling and vibration control of a strain-gradient clamped-free Euler-Bernoulli micro-beam exposed to varying disturbance is studied. A strain-gradient model of the Euler-Bernoulli micro-beam is represented in this paper and consisted of one partial differential equation and six ordinary equations as governing motion equation and boundary conditions, respectively. A boundary controller is proposed to suppress the system's vibration. The controller is derived based on the direct Lyapunov method. An adaptation law is devised to assure system's stability under parametric uncertainties. With the proposed adaptive robust boundary control, uniform boundedness under environmental and input disturbances could be achieved. The state of the system is proven to converge to a small neighborhood of zero by appropriately choosing design parameters. Simulations are provided to illustrate the applicability and effectiveness of the proposed controller.