TY - JOUR

T1 - Global finite-dimensional observer-based stabilization of a semilinear heat equation with large input delay

AU - Katz, Rami

AU - Fridman, Emilia

N1 - Publisher Copyright:
© 2022 Elsevier B.V.

PY - 2022/7

Y1 - 2022/7

N2 - We study global finite-dimensional observer-based stabilization of a semilinear 1D heat equation with globally Lipschitz semilinearity in the state variable. We consider Neumann actuation and point measurement. Using dynamic extension and modal decomposition we derive nonlinear ODEs for the modes of the state. We propose a controller that is based on a nonlinear finite-dimensional Luenberger observer. Our Lyapunov H1-stability analysis leads to LMIs, which are shown to be feasible for a large enough observer dimension and small enough Lipschitz constant. Next, we consider the case of a constant input delay r>0. To compensate the delay, we introduce a chain of M sub-predictors that leads to a nonlinear closed-loop ODE system, coupled with nonlinear infinite-dimensional tail ODEs. We provide LMIs for H1-stability and prove that for any r>0, the LMIs are feasible provided M and the observer dimension N are large enough and the Lipschitz constant is small enough. Numerical examples demonstrate the efficiency of the proposed approach.

AB - We study global finite-dimensional observer-based stabilization of a semilinear 1D heat equation with globally Lipschitz semilinearity in the state variable. We consider Neumann actuation and point measurement. Using dynamic extension and modal decomposition we derive nonlinear ODEs for the modes of the state. We propose a controller that is based on a nonlinear finite-dimensional Luenberger observer. Our Lyapunov H1-stability analysis leads to LMIs, which are shown to be feasible for a large enough observer dimension and small enough Lipschitz constant. Next, we consider the case of a constant input delay r>0. To compensate the delay, we introduce a chain of M sub-predictors that leads to a nonlinear closed-loop ODE system, coupled with nonlinear infinite-dimensional tail ODEs. We provide LMIs for H1-stability and prove that for any r>0, the LMIs are feasible provided M and the observer dimension N are large enough and the Lipschitz constant is small enough. Numerical examples demonstrate the efficiency of the proposed approach.

KW - Distributed parameter systems

KW - Nonlinear systems

KW - Observer-based control

KW - Time-delay systems

UR - http://www.scopus.com/inward/record.url?scp=85131132309&partnerID=8YFLogxK

U2 - 10.1016/j.sysconle.2022.105275

DO - 10.1016/j.sysconle.2022.105275

M3 - מאמר

AN - SCOPUS:85131132309

VL - 165

JO - Systems and Control Letters

JF - Systems and Control Letters

SN - 0167-6911

M1 - 105275

ER -