Part of the motivation of this paper comes from the recent paper [W. Guo and H.-Z. Zhou, SIAM J. Control Optim., 57 (2019), pp. 1890–1928], where an adaptive control approach was used to estimate in real time all amplitudes and phases of the external harmonic sinusoidal disturbance with known frequencies to achieve output regulation for a 1-d Euler–Bernoulli beam equation. In our paper, we change the landscape in a completely different way for the output regulation of the Euler–Bernoulli beam equation for which the disturbances are generated from a completely unknown finite-dimensional exosystem; i.e., the frequencies of the sinusoidal signals that appear in the reference signal and disturbances are completely unknown. In addition, the number of frequencies is also assumed to be unknown yet has a known upper bound. We apply the technique of the Sylvester equation to decouple the ODE of the exosystem and the PDE of the plant first, and then adopt the adaptive observer to estimate all possible unknown frequencies that have entered into the tracking error. By designing two different adaptive observers to estimate the unknown frequencies as well as the state of the plant and exosystem, we propose two tracking error based feedback controls for achieving output regulation for this PDE.  

　　Publication:  

　　SIAM Journal on Control and Optimization, Volue 61, Issue 4 (2023)  

　　http://dx.doi.org/10.1137/22M1501805  

　　Author:  

　　Bao-Zhu Guo  

　　School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China, and Key Laboratory of System and Control, Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, China, and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China.  

　　Email: bzguo@iss.ac.cn