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Nonlinear simulation of a string system under boundary robust control. (English) Zbl 1084.93013
In this very important and useful paper, vibration control of a nonlinear flexible system is considered. The dynamic equations governing the motion of the system have the following form: \begin{aligned} M_0[\ddot p_0(t)+ \ddot y_b] &= f_0(t)T(0,t) y_x(0,t)- b_0(p_0, t)\dot p_0(t),\\ M_1[\ddot p_1(t)+\ddot y_b] &= f_1(t)+ T(1,t) u_x(1, t)- b_1(p_1, t)\dot p_1(t)\quad\text{and}\\ m(x){\partial^2 Y(x,t)\over\partial t^2}& = {\partial\over\partial x}\Biggl\{T(x,t){\partial Y(x,t)\over\partial x}\Biggr\},\end{aligned} where $$Y(x, t)= y(x, t)+ y_b(t)$$ represents displacement of the string with respect to the ground. It is assumed that the string tension is nonlinear and of the form $$T(x, t)= T_0(x)+ w(x)y^2_x(x, t)$$, $$T_0(x)> 0$$ for all $$x\in[0,1]$$. In the model, the forces $$f_0(t)$$ and $$f_1(t)$$ are boundary control variables to be designed. Given $$f_0(t)$$ and $$f_1(t)$$, a trajectory of the system can be solved under the initial conditions (initial positions of control mechanisms, and initial displacement and velocity of the string). Here, the problem of using boundary controls to suppress vibration is considered.
Main result: In order to damp out oscillation, a new robust control is designed to compensate for nonlinear uncertainties in string dynamics and in the control mechanism. It is shown (the Lyapunov direct method is used) that the proposed control makes the closed-loop system exponentially stable. Finally, the effectiveness of the control is demonstrated through nonlinear simulation when the finite-difference approach is used.
##### MSC:
 93C20 Control/observation systems governed by partial differential equations 37N35 Dynamical systems in control 93C10 Nonlinear systems in control theory 93D21 Adaptive or robust stabilization 74H45 Vibrations in dynamical problems in solid mechanics 74K05 Strings
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