An adaptive control scheme for a class of second order distributed parameter systems with structured perturbations

In this work, we consider a class of second order distributed parameter systems that can nominally be stabilized via static output feedback. It is assumed that the adaptive estimates of the unknown feedback gains can be utilized for the convergence of state position and velocity. The requisite stability analysis utilizes a parameter-dependent Lyapunov function specifically constructed for the system in its natural second order formulation. By further assuming that structured perturbations of the damping and stiffness operators, often describing modelling uncertainties, are present in the system, a modification to the proposed adaptive control law is included in order to address these unmodelled uncertainties. On-line approximators of the structured perturbations are included in the adaptive control law and it is shown that they result in convergence of the state to zero while guaranteeing that all other signals are bounded in appropriate norms.