Shear force feedback control of a single-link flexible robot with a revolute joint

Non-dissipative boundary feedback for Rayleigh and Timoshenko beams

“…Besides the above asymptotics, [4] also showed that -as in the dissipative case-the eigenvalues of the closed loop system are all in the open left-half plane. However, for partial differential equations certain pathologies may occur that prevent the stability of a system to be determined from the location of its eigenvalues.…”

“…The feedback control strategies used are often of the static output feedback kind and the input and output are usually chosen to make the closed loop system dissipative. An intriguing non-dissipative control strategy was however chosen in [4]. We refer to that article for the physical interpretation of their choice of feedback.…”

“…[3]). The eigenvalues of the nondissipative closed-loop system were shown in [4] to be asymptotic to the parts of the parabolas Imλ = ±c (Reλ) 2 in the left-half plane (see figure 1(a)). This indicates that high frequencies are much better damped by the non-dissipative feedback than by dissipative feedbacks, a very attractive property.…”

“…However, for partial differential equations certain pathologies may occur that prevent the stability of a system to be determined from the location of its eigenvalues. Due to this, [4] only managed to show the exponential stability of the closed-loop system for smooth initial conditions in spite of the fact that all its eigenvalues are in the open left half-plane and are bounded away from the imaginary axis. Using estimates of the Green function [2] showed that the closed-loop system is a Riesz spectral system and since for Riesz spectral systems the location of the eigenvalues does determine the stability, exponential stability followed (also for non-smooth initial data).…”

“…As mentioned, [4] chose an Euler-Bernoulli beam model (and the subsequent articles mentioned followed suit). This neglects the fact that the beam has a moment of inertia (and probably less importantly it neglects shear effects and non-linear effects).…”