Multiscale Wavelet-LQR Controller for Linear Time Varying Systems

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.