Research Article s Volume 6 • Issue 2 • 1000234 J Electr Electron Syst, an open access journal ISSN: 2332-0796 Open Access Research Article Journal of Electrical & Electronic Systems J o u r n a l o f E l e c tr i c a l & E l e c t r o n i c S y s t e m s ISSN: 2332-0796 Mohsen et al., J Electr Electron Syst 2017, 6:2 DOI: 10.4172/2332-0796.1000234 *Corresponding author: Edris Mohsen, Department of Electrical and Computer Engineering Western University, London, Ontario, Canada, Tel: +1 519-661-2111; E-mail: emohsen2@uwo.ca Received July 14, 2017; Accepted July 25, 2017; Published July 27, 2017 Citation: Mohsen E, Brown LJ, Chen J (2017) A Real time Alternative to the Hilbert Huang Transform Based on Internal Model Principle. J Electr Electron Syst 6: 233. doi: 10.4172/2332-0796.1000234 Copyright: © 2017 Mohsen E, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. A Real time Alternative to the Hilbert Huang Transform Based on Internal Model Principle Edris Mohsen*, Lyndon J Brown and Jie Chen Department of Electrical and Computer Engineering Western University, London, Ontario, Canada Keywords: Internal model principle; Frequency identifcation; Adaptive multiple notch flters; Periodic disturbance; State variables; Bandpass flter; Instantaneous Fourier decomposition Introduction In this article, we are interested in the problem of identifying signals of the following form 1 1 () ( )sin () () i m n ij ij i j dt A t t nt φ = = = + ∑∑ (1) 0 () () (0) t ij i ij t jw t dt φ φ = + ∫ (2) and n(t) is measurement noise. Tese are signals that are the sums of n periodic components with each component composed of m i harmonics. Te periods, the harmonic amplitudes and relative phases can vary slowly in time. By identifcation, we mean determining the values ω i , Ā ij and φ ij - φ 11 . Several techniques have been developed in the literature to solve this problem. Te most traditional technique is the fast Fourier transform. Newer techniques include wavelet analysis. Tese approaches sufer from not allowing continuous estimations of the frequencies and have difcult trade-ofs between time and frequency resolutions. Other approaches are based on the use of adaptive notch flters [1] and output regulation [2]. A new approach that has been widely applied is the Hilbert Huang Transform (HHT) [3]. Control engineers treat similar problems where exact tracking of reference signals or rejection of disturbances is required. Approaches that accomplish this include repetitive controllers [4] and adaptive feed-forward cancellation (AFC) [5]. Te repetitive controller is based on a fundamental control theory principle called the internal model principle (IMP). Tis principle was presented by Francis and Wonham and states that the output error can be driven asymptotically to zero by placing a model of exogenous signals in a stable feedback loop [6]. Unfortunately small errors in this model can lead to signifcant degradation in the performance of internal model principle controllers. Tis problem of uncertainty in the signal model can be overcome with adaptive controllers [7]. In achieving asymptotically perfect rejection of disturbances it is inherent that the disturbance is completely identifed. Tus, these types of controllers can be turned into signal processing algorithms by replacing the process to be controlled with tuning functions [8]. Unfortunately, to successfully implement this algorithm requires being able to tune a stable feedback control loop for the entire range of possible frequencies in the model given by equation (1). Fortunately, it has been shown that in the signal processing framework, the simplest tuning solution, i.e. selecting all of the gains to be one, is guaranteed to be stable. Tis algorithm has been successfully applied to the problem of the repeatable disturbances seen in disk drive head control [9]. Unfortunately, by resorting to this simple tuning approach, there is no control over the dynamics and noise rejection characteristics of the algorithm. When the frequencies are known a priori, the report [10] shows how the dynamics of the algorithm can be completely specifed. Unfortunately this article requires solving a set more than 2 2 t i n m = ∑ coupled linear equations which are a function of the signal’s frequencies. Unless the sample rate is less than 1Hz this will not be feasible to do each sample. Tis article shows how these parameters can be explicitly solved by simply evaluating some frequency response functions at certain frequencies. In Section II, an instantaneous Fourier decomposition (IFD) algorithm [11] that is similar in approach to the HHT is presented. In Section III an updated formula for calculating the instantaneous frequencies are given. In Section IV, the new realtime tuned algorithm is presented. In Section V, the ability of the proposed algorithm to identify the periodic signal with uncertain frequencies is demonstrated. Conclusions are drawn in Section VI. A preliminary version of this article was presented at the 30th annual IEEE Canadian Conference on Electrical and Computer Engineering (IEEE 2017 CCECE) in Windsor [12]. Abstract This article presents a new tuning approach for an adaptive internal-model-principle based signal identifcation algorithm whose computational costs are low enough to allow a realtime implementation. The algorithm allows an instantaneous Fourier decomposition of non-stationary signals that have a strongly predictable component. The algorithm is implemented as a feedback loop resulting in a closed loop system with a frequency response of a bandpass flter with notches at the frequencies of the Fourier decomposition. This is achieved through real time selection of the coeffcients of the transfer functions in the feedback loop. Previously these coeffcients were selected by solving a large set of coupled linear equations. Rules for explicitly solving for these parameters are given that only involve evaluating frequency responses at the frequencies of the instantaneous Fourier decomposition. This allows realtime implementation on a low cost lap top with sampling rates up to 10 kHz.