Boundary Control of Undamped Beam using Square-Root Velocity

The coprime factorization approach is used to design a linear time-invariant feedback controller to stabilize an undamped beam in the bounded-input-bounded-output (BIBO) sense. Transfer functions are factored into coprime ratios of elements from a ring of stable causal transfer functions that includes transcendental functions of ¿S. We consider a pinned-free beam with a control torque at the pinned end. It is shown that feedback involving "square-root velocity" (= displacement × ¿S) leads to BIBO-stability. However, this stable system is "ill-posed" in the sense that it can be destabilized by the introduction of an arbitrarily small delay in the feedback loop.