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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
##### Keywords:
time-delay; disturbance; ADRC; dynamic feedback control
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