Adaptive Identification Of Two Unstable Pdes With Boundarysensing And Actuation

摘要

In this paper we consider a problem of on-line parameter identification of parabolic partial differential equations (PDEs). In the previous study, on the actuation side, both distributed (SIAM J. Optim. Control 1997; 35:678-713; IEEE Trans. Autom. Control 2000; 45:203-216) and boundary (IEEE Trans. Autom. Control 2000; 45:203-216) actuations were considered in the open loop, whereas for the closed loop (unstable plants) only distributed one was addressed. On the sensing side, only distributed sensing was considered. The present study goes beyond the identification framework of (SIAM J. Optim. Control 1997; 35:678-713; IEEE Trans. Autom. Control 2000; 45:203-216) by considering boundary actuation for the unstable plants, resulting in the closed-loop identification, and also introducing boundary sensing. This makes the proposed technique applicable to a much broader range of practical problems. As a first step towards the identification of general reaction-advection-diffusion systems, we consider two benchmark plants: one with an uncertain parameter in the domain and the other with an uncertain parameter on the boundary. We design the adaptive identifier that consists of standard gradient/least-squares estimators and backstepping adaptive controllers. The parameter estimates are shown to converge to the true parameters when the closed-loop system is excited by an additional constant input at the boundary. The results are illustrated with simulations.