Backstepping observer design for parabolic PDEs with measurement of weighted spatial averages

摘要

This paper is concerned with the observer design for one-dimensional linear parabolic partial differential equations whose outputis a weighted spatial average of the state over the entire spatial domain. We focus on the backstepping approach, which provides asystematic procedure to design an observer gain for systems with boundary measurement. If the output is not a boundary value of thestate, the backstepping approach is not directly applicable to obtaining an observer gain that stabilizes the error dynamics. Therefore,we attempt to convert the error system into another system to which backstepping is applicable. The conversion is successfullyachieved for a class of weighting functions, and the resultant observer realizes exponential convergence of the estimation error withan arbitrary decay rate in terms of the L2 norm. In addition, an explicit expression of the observer gain is available in a special case.The effectiveness of the proposed observer is also confirmed by numerical simulations.