656 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 19, NO. 3, MAY 2011 Identification of Time-VaryingSystems Using Multi-Wavelet Basis Functions Yang Li, Hua-liang Wei, and S. A. Billings Abstract—This brief introduces a new parametric modelling and identification method for linear time-varying systems using a block least mean square (LMS) approach where the time-varying param- eters are approximated using multi-wavelet basis functions. This approach can be applied to track rapidly or even sharply varying processes and is developed by combining wavelet approximation theory with a block LMS algorithm. Numerical examples are pro- vided to show the effectiveness of the proposed method for dealing with severely nonstationary processes. Application of the proposed approach to a real mechanical system indicates better tracking ca- pability of the multi-wavelet basis function algorithm compared with the normalized least squares or recursive least squares rou- tines. Index Terms—B-splines basis functions, block least mean squares (LMS), normalized least mean squares (LMS), parameter estimation, recursive least squares (RLS), system identification, time variation. I. INTRODUCTION M ANY processes are inherently time-varying and can not effectively be characterized using time invariant models. Modelling and analysis of time-varying systems is often a challenging problem. One feature of time-varying sys- tems is that such signals contain nonstationary transient events. One approach to characterize such non-stationary processes is to employ time-varying parametric models for example the time-varying autoregressive with an exogenous (TVARX) model [1], or simply the time-varying autoregressive (TVAR) model [2]. Two main classes of methods can be used to resolve the TVARX and TVAR model estimation problem. The first uses recursive estimation of the time-varying coefficients and the second constrains the evolution of the coefficients to be linear or nonlinear combinations of some basis functions with appropriate properties. These approaches have been called sto- chastic and deterministic regression approaches, respectively [3]. Some of the most popular recursive algorithms are the least mean square (LMS) algorithm, the recursive least square (RLS) algorithm [4], the Kalman filter and the random walk Kalman filter (RWKF) algorithms [5]–[7]. The basis function expansion and regression method is a deterministic parametric Manuscript received March 30, 2010; accepted May 31, 2010. Manuscript re- ceived in final form June 01, 2010. Date of publication July 01, 2010; date of cur- rent version April 15, 2011. Recommended by Associate Editor A. Alessandri. The work of Y. Li was supported by the University of Sheffield under the schol- arship scheme. This work was supported by the Engineering and Physical Sci- ences Research Council (EPSRC), U.K. and the European Research Council (ERC). The authors are with the Department of Automatic Control and Systems Engineering, the University of Sheffield, Sheffield, S1 3JD, U.K. (e-mail: s.billings@sheffield.ac.uk; coq08yl@sheffield.ac.uk; w.hualiang@sheffield.ac. uk). Color versions of one or more of the figures in this brief are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2010.2052257 modelling approach, where the associated time-varying coeffi- cients are expanded as a finite sequence of predetermined basis functions [8]–[10]. Generally, these coefficients are expressed using a linear or nonlinear combination of a finite number of basis functions. The problem is then reduced to time invariant coefficient estimation, where the unknown adjustable model parameters are those involved in the basis expansion. Hence, the initial time-varying modelling problem is simplified to deterministic regression selection and parameter estimation. An attractive approach is to expand the time-varying coef- ficients using wavelets as the basis functions. Wavelets have been proved to be a valuable tool for signal processing and have been shown to possess excellent linear or nonlinear approxi- mation properties which outperform many other approximation schemes and are well suited for approximating general non-sta- tionary signals, even those with very sharp or abrupt discon- tinuities. Wavelets have also successfully been used in system identification and modelling [11]–[14]. In this brief, a new wavelet multi-resolution parametric mod- elling and identification technique for the identification of sys- tems with time-varying parameters is proposed, where the as- sociated time dependent parameters are approximated using a set of multi-wavelet basis functions, which transforms the time- varying identification problem into a time-invariant parametric expansion. The identification of the model parameters can then be achieved by adopting a block LMS algorithm. One advantage of the proposed approach, which combines wavelet approxima- tion theory with a block LMS algorithm, is that the new wavelet based algorithm can be used to track very rapidly or even sharply varying processes. The novel approach proposed can thus track rapid time variation and is more suitable for the estimation of process parameters of inherently non-stationary processes. A multi-wavelet basis function approach is used because of the ability to capture the signals characteristics at different scales. Two examples, one for a synthetic data set and a second for a real mechanical system are given to illustrate the capability and efficacy of the proposed method. It is shown that the pro- posed method can produce much better tracking performance compared with traditional LMS and RLS approaches. II. METHODOLOGY Consider an input-output relationship of a TVARX process which is described by the following equation: (1) where and are the sampled measurable input, output, and prediction error signals, and are the time-varying pa- rameters to be determined, and are the maximum model orders, and represents discrete time. The proposed method ex- pands the time varying parameters and onto multi- 1063-6536/$26.00 © 2010 IEEE