A natural observer-based adaptive controller for structurally perturbed second order distributed parameter systems

In this note we consider a class of second order distributed parameter systems with structured perturbations ζ̈(t)+D ζ̇(t)+K ζ(t) = Bu+B f1(yp)+B f2(yv) and with partial position and velocity measurements: yp(t) = Cpζ(t), yv(t) = Cvζ̇(t). It is desired to design a controller so that the plant follows the reference model ζ̈m +Dmζ̇m(t)+Kmζm(t) = Br(t). The proposed model reference adaptive controller u(t) = −Gpζ̂(t)−Gv ̇̂ ζ(t)+ r(t)− f̂1(yp)− f̂2(yv) utilizes both state position and velocity estimates ζ̂(t), ̇̂ ζ(t), and the on-line approximators f̂1(yp), f̂2(yv) of the unknown perturbation functions f1(yp), f2(yv). As argued in [5], a natural second order observer is required in order to provide estimates of the state position and state velocity with the added property that the estimate of the velocity equals the derivative of the state position. We thus propose a natural second order adaptive observer in order to implement the model reference controller ̈̂ ζ(t)+D ̇̂ ζ(t)+K ζ̂(t) = Bu+B f̂1(yp)+B f̂2(yv)−Lp(ŷp − yp)−Lv(ŷv − yv) ŷp(t) = Cpζ̂(t), ŷv(t) = Cv ̇̂ ζ(t). The well posedness of the above observer and adaptive controller are examined along with the adaptive policy for updating the on-line approximators and the subsequent stability of the closed loop system.