Impulse Response Model Identification in the Case of a Multi-Period ZOH Input

Abstract

This paper considers the problem of impulse response identification for a linear sampled-data system where the input signal is held constantly within a multiple of the sampling period of the output signal. To improve identification accuracy, this paper proposes a new identification approach by employing the Haar scaling and wavelet functions. At first, we point out that the discrete-time impulse response of a linear system with zero-order hold (ZOH) input is actually the piecewise-constant approximation of the continuous-time impulse response. Based on the close relation between piecewise-constant approximation with Haar scaling and wavelet functions, a hierarchical identification procedure is proposed which identifies the system impulse response from a coarse resolution level to a fine resolution level successively. At each resolution level, the BIC is utilized to determine the length of the decomposed impulse response in the corresponding subspace, so that some redundant parameters in the high frequency-domain which are sensitive to the noise effects are discarded. Since the identified impulse response model is not smooth when it is represented by some Haar scaling functions of different widths, we can replace each Haar scaling function in the preidentified impulse response model by a Gaussian basis function with corresponding position and width. Then an improved identification method is also proposed to achieve smooth continuous-time impulse response model from sampled data. It is shown through simulation study that the proposed methods yield accurate estimate of the impulse response even in the ill-conditioned cases.