Robust boundary control approaches to the stabilization of the Euler–Bernoulli beam under external disturbances

Abstract

In this paper, the boundary feedback control problem for the Euler–Bernoulli beam with unknown time-varying distributed load and boundary disturbance is investigated. Based on the Lagrangian–Hamiltonian mechanics, the model of the beam is derived as a partial differential equation. To suppress the external disturbance, two disturbance rejection control approaches are adopted in the control design. Firstly, a disturbance observer is designed to estimate the boundary disturbance online. Thus, the effect of the boundary disturbance can be canceled directly in the feedback loop. A time-varying function is applied in the disturbance observer to prevent an excessively large control input. Secondly, a new observer is developed to estimate the upper bound of the disturbance and a sign function is introduced to suppress the influence of the disturbance without demanding the boundedness of the derivative of the boundary disturbance. The well-posedness and the uniform boundedness of the closed-loop system are proved using the operator semigroup theory and the Lyapunov method. Numerical comparisons with existing results are made for demonstrating the advantages of the proposed approaches.

Robust adaptive vibration control of a gantry crane system with flexible cable

“…It follows from () and the similar arguments as in References 49; Chaps. 5,6 and 50 that () has a unique weak solution $$$ U={\left[w(x),\kern0.3em z(x),\kern0.3em \phi, \kern0.3em {u}_a,\kern0.3em \tilde{\Theta},\kern0.3em \psi \right]}^T\in D(A) $$$. Step 3: To prove that the domain $$$ D(A) $$$ is dense in $$$ \mathscr{H} $$$, it is a direct conclusion for maximal monotone operator in corollary 2.4.3, Reference 51 applying to the operator $$$ -A:D(A)\to \mathscr{H} $$$.…”