A Finite-Dimensional Controller for Robust Output Tracking of an Euler–Bernoulli Beam

In this paper, we consider robust output tracking problem of an undamped Euler-Bernoulli beam with boundary control and boundary observation. In particular, we study a cantilever beam which has control and observation at the free end. As our main result, we construct a finite-dimensional, internal model based controller for the output tracking of the beam system. In addition, we consider a case where the controller achieves the robust output tracking for the cantilever beam with distributed control and observation. Numerical simulations demonstrating the effectiveness of the controller are presented.

I. Introduction

In this paper, we consider output tracking of an Euler Bernoulli beam with conservative clamped boundary conditions at one end and control at the other end. The beam system we study is given by

$$\begin{equation*}\begin{array}{l} {\rho (\xi ){w_{tt}}(\xi,t) + {{\left( {EI(\xi ){w_{\xi \xi }}} \right)}_{\xi \xi }}(\xi,t) = 0,0 < \xi < 1,t > 0,} \\ {w(0,t) = 0,{w_{\xi} }(0,t) = 0,} \\ {\left( {EI(\xi ){w_{\xi \xi }}} \right)(1,t) = 0,} \\ { - {{\left( {EI(\xi ){w_{\xi \xi }}} \right)}_{\xi} }(1,t) = u(t),} \\ {y(t) = {w_t}(1,t),} \\ {w(\xi,0) = {w_0}(\xi ),{w_t}(\xi,0) = {w_1}(\xi ),0 < \xi < 1,} \end{array}\tag{I.1} \end{equation*}$$ where w(ξ,t) is the transverse displacement of the beam at position ξ and time t, w_t(ξ,t) and w_ξ(ξ,t) denote time and spatial derivatives of w(ξ,t), respectively, ρ(ξ) and EI(ξ) are linear density and flexural rigidity of the beam, respectively, u(t) is an external boundary input and y(t) is a boundary observation. The parameters ρ(ξ) and EI(ξ) satisfy the conditions $$\begin{equation*}\rho (\cdot),EI(\cdot) \in {C^4}([0],[1]),\quad \rho (\xi ),EI(\xi ) > 0\quad \forall \xi \in [0],[1].\tag{I.2} \end{equation*}$$