Multiscale Wavelet-LQR Controller for Linear Time Varying Systems
Publication: Journal of Engineering Mechanics
Volume 136, Issue 9
Abstract
This paper proposes a multiresolution based wavelet controller for the control of linear time varying systems consisting of a time invariant component and a component with zero mean slowly time varying parameters. The real time discrete wavelet transform controller is based on a time interval from the initial until the current time and is updated at regular time steps. By casting a modified optimal control problem in a linear quadratic regulator (LQR) form constrained to a band of frequency in the wavelet domain, frequency band dependent control gain matrices are obtained. The weighting matrices are varied for different bands of frequencies depending on the emphasis to be placed on the response energy or the control effort in minimizing the cost functional, for the particular band of frequency leading to frequency dependent gains. The frequency dependent control gain matrices of the developed controller are applied to multiresolution analysis (MRA) based filtered time signals obtained until the current time. The use of MRA ensures perfect decomposition to obtain filtered time signals over the finite interval considered, with a fast numerical implementation for control application. The proposed controller developed using the Daubechies wavelet is shown to work effectively for the control of free and forced vibration (both under harmonic and random excitations) responses of linear time varying single-degree-of-freedom and multidegree-of-freedom systems. Even for the cases where the conventional LQR or addition of viscous damping fails to control the vibration response, the proposed controller effectively suppresses the instabilities in the linear time varying systems.
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Acknowledgments
The financial support by Rice University and Trinity College Dublin for this research work is gratefully acknowledged. This work was carried out during the visit of the first writer to Rice University. The paper was submitted earlier to the date indicated, as the manuscript had to be switched from the old system to the new on-line Editorial Manager system.
References
Daubechies, I. (1992). Ten lectures on wavelets, Society of Industrial and Applied Mathematics, Philadelphia.
Den Hartog, J. P. (1956). Mechanical vibrations, McGraw-Hill, New York.
Dimentberg, M. F. (1988). Statistical dynamics of non-linear and linear time varying systems, Wiley, New York.
Ibrahim, R. A. (1985). Parametric random vibration, Wiley, New York.
Ioannou, P., and Sun, J. (1996). Robust adaptive control, Prentice-Hall, New York, ⟨http://www-rcf.usc.edu/~ioannou/Robust_Adaptive_Control.htm⟩ (June 7, 2010).
Kamen, E. W. (1988). “The poles and zeros of a linear time-varying system.” Linear Algebr. Appl., 98, 263–289.
Shinozuka, M., and Sato, Y. (1967). “Simulation of nonstationary random processes.” J. Eng. Mech., 93(1), 11–40.
Sun, J., and Ioannou, P. (1992). “Robust adaptive LQ control schemes.” IEEE Trans. Autom. Control, 37(1), 100–106.
Tsakalis, K. S., and Ioannou, P. (1993). Linear time varying plants: Control and adaptation, Prentice-Hall, New York.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 136 • Issue 9 • September 2010
Pages: 1143 - 1151
Copyright
© 2010 ASCE.
History
Received: Sep 4, 2009
Accepted: Mar 11, 2010
Published online: Mar 16, 2010
Published in print: Sep 2010
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