Premium
Adaptive identification of two unstable PDEs with boundary sensing and actuation
Author(s) -
Smyshlyaev Andrey,
Orlov Yury,
Krstic Miroslav
Publication year - 2009
Publication title -
international journal of adaptive control and signal processing
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.73
H-Index - 66
eISSN - 1099-1115
pISSN - 0890-6327
DOI - 10.1002/acs.1056
Subject(s) - backstepping , distributed parameter system , boundary (topology) , control theory (sociology) , benchmark (surveying) , identification (biology) , adaptive control , estimator , parameter identification problem , boundary value problem , partial differential equation , mathematics , computer science , mathematical analysis , control (management) , model parameter , statistics , botany , geodesy , artificial intelligence , biology , geography
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. Copyright © 2008 John Wiley & Sons, Ltd.