Improving Identification Accuracy of Impulse Response Models by Using the Haar Wavelet

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. It is well known that in this case, the identification problem is usually ill-conditioned, if the system under study is discretized with a much shorter sampling period of the output signal. To improve the 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 its continuous-time impulse response. Then 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 (low frequency subspace) to a fine resolution level (high frequency subspace) 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, based on the designed gaussian basis functions. It is shown through simulation study that the proposed method yields accurate estimate of the impulse response even in the ill-conditioned cases.