TY - GEN

T1 - Finite-dimensional control of the Kuramoto-Sivashinsky equation under point measurement and actuation

AU - Katz, Rami

AU - Fridman, Emilia

N1 - Publisher Copyright:
© 2020 IEEE.

PY - 2020/12/14

Y1 - 2020/12/14

N2 - Finite-dimensional observer-based controller design for PDEs is a challenging problem. Recently, such controllers were introduced for the 1D heat equation, under assumption that at least one of the observation or control operators is bounded. This paper suggests a constructive method for such controllers in the case where both the observation and control operators are unbounded. We consider boundary control of the 1D linear Kuramoto-Sivashinsky equation with in-domain point measurement. We employ a modal decomposition approach via dynamic extension, where we use the eigenfunctions of a Sturm-Liouville operator. The controller dimension is defined by the number of unstable modes, whereas the observer's dimension N may be larger than this number. We suggest a direct Lyapunov approach to the full-order closedloop system and provide LMIs for finding N and the resulting exponential decay rate. We prove that the LMIs are always feasible, provided N is large enough. A numerical example demonstrates the efficiency of the method and shows that the resulting LMIs are non-conservative.

AB - Finite-dimensional observer-based controller design for PDEs is a challenging problem. Recently, such controllers were introduced for the 1D heat equation, under assumption that at least one of the observation or control operators is bounded. This paper suggests a constructive method for such controllers in the case where both the observation and control operators are unbounded. We consider boundary control of the 1D linear Kuramoto-Sivashinsky equation with in-domain point measurement. We employ a modal decomposition approach via dynamic extension, where we use the eigenfunctions of a Sturm-Liouville operator. The controller dimension is defined by the number of unstable modes, whereas the observer's dimension N may be larger than this number. We suggest a direct Lyapunov approach to the full-order closedloop system and provide LMIs for finding N and the resulting exponential decay rate. We prove that the LMIs are always feasible, provided N is large enough. A numerical example demonstrates the efficiency of the method and shows that the resulting LMIs are non-conservative.

UR - http://www.scopus.com/inward/record.url?scp=85098044302&partnerID=8YFLogxK

U2 - 10.1109/CDC42340.2020.9304032

DO - 10.1109/CDC42340.2020.9304032

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AN - SCOPUS:85098044302

T3 - Proceedings of the IEEE Conference on Decision and Control

SP - 4423

EP - 4428

BT - 2020 59th IEEE Conference on Decision and Control, CDC 2020

PB - Institute of Electrical and Electronics Engineers Inc.

Y2 - 14 December 2020 through 18 December 2020

ER -