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Dynamic control of an Euler-Bernoulli equation with time-delay and disturbance in the boundary control. (English) Zbl 1415.93117
Summary: The boundary control problem of a cantilever Euler-Bernoulli equation is considered in this paper. If the control at the right end of the beam is of the form \(w_{xxx}(1,t)=u(t-\tau)+r(t)\), where \(\tau>0\) is the input time-delay and \(r(t)\) is an unknown external disturbance, a dynamic feedback control strategy based on the methods of partial state predictor and active disturbance rejection control is used to stabilize the system. Under some assumptions on \(r(t)\), it is proven that the state of the system exponentially converges to and stays in the compact set \(\Omega=\{ (w,w_t)^\top \mid \| (w,w_t)^\top \|_{\mathcal{H}}\leq\varepsilon\}\). The radius \(\varepsilon\) is determined by the time-delay \(\tau\) and the properties of \(r(t)\). The simulations are provided to compare the influence of \(\tau\) and \(r(t)\) on the radius \(\varepsilon\).
MSC:
93B52 Feedback control
93C20 Control/observation systems governed by partial differential equations
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93D15 Stabilization of systems by feedback
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