Robust output regulation of a non-uniform Euler-Bernoulli beam

For an Euler-Bernoulli beam with variable coefficients, all the disturbances and references are supposed to be from an exosystem. Two observer-based controls are proposed to regulate two non-collocated outputs to track prescribed references. First, we construct an uncertain disturbance-free beam system, which is determined by the inputs, measurable outputs, and unknown initial conditions. New exosystems are then defined to address the undetectability of the original exosystem and the uncertainties of input disturbances. We then propose two observer-based controls through an extended observer for the disturbance-free beam and the new exosystems. For the closed-loop system, the tracking errors are proved to be exponentially regulated toward zero. A simulation example is provided to describe the theoretical result.