Issue 
Mechanics & Industry
Volume 18, Number 8, 2017
Experimental Vibration Analysis



Article Number  808  
Number of page(s)  6  
DOI  https://doi.org/10.1051/meca/2017049  
Published online  21 August 2018 
Regular Article
Order tracking using H_{∞} estimator and polynomial approximation
^{1}
PRISME Laboratory, University of Orleans,
21 rue Loignylabataille,
28000
Chartres, France
^{2}
PRISME Laboratory, University of Orleans,
12 rue de Blois,
45067
Orleans, France
^{*} email: amadou.assoumane@etu.univorleans.fr
Received:
12
December
2016
Accepted:
5
December
2017
This paper presents the H_{∞} estimator for discretetime varying linear system combined with the polynomial approximation for order tracking of nonstationary signals. The proposed approach is applied to the gearbox diagnosis under variable speed condition. In this instance, it is well known that the occurrence of a fault on a gear tooth leads to an amplitude and a phase modulation in the vibration signal. The purpose is to estimate this unknown amplitude and phase modulation by tracking orders. To estimate these modulations, the vibration signal is described in state space model. Then, the H_{∞} criterion is used to minimize the worst possible amplification of the estimation error related to both the process and the measurement noises. Such an approach doesn't require any assumption on the statistic properties of the noises unlike the Kalman estimator. A numerical example is given in order to evaluate the performance of the H_{∞} estimator regarding the conventional Kalman estimator.
Key words: Order tracking / estimator / polynomial approximation / nonstationary conditions
© AFM, EDP Sciences 2018
1 Introduction
Order tracking using the state space approach is one of the tools widespread for the processing of nonstationary signals. The socalled the state space model is composed of two equation: the state equation and the measurement equation. The technique the most presented in this area is the Kalman estimator and more precisely the Voldkalman estimator in the area of the mechanical systems diagnosis [1]. Vold et al. present the theoretical basis about this estimator in [2]. This kind of estimator suppose that the measurement noise and the process noise are centered, Gaussian and white with known statistics.
In the literature many works on the Voldkalman estimator for order tracking has been done. Pan and Lin have realized an interesting explorative study on the Voldkalman estimator [3]. Behrouz and al. also applied this estimator to diagnose a bearing default and they have translated the state equation in term of second order autoregressive model [4]. These study have provided conclusive results. However, the unrealistic assumptions on the noises naturally limit the application of this estimator in real cases.
Therefore, the H_{∞} estimator is proposed in this paper to evaluate the amplitude modulation and the phase modulation. To estimate these modulations the vibration signal is described using the state variables. Then these latter are modelled by a Taylor series. This method generalize that of Voldkalman. With the H_{∞} estimator we make no assumption on the noise statistics. They must only be of finite energy. More details on the discrete H_{∞} estimator can be found in the work of Shen and Deng [5].
This paper is structured as follows: Section 2 presents the theoretical foundation about the H_{∞} estimator and Section 3 provides an example of simulation which validated our proposal.
2 Theoretical background
2.1 Problem formulation
In this paper, the gearbox vibration signal is modelled as (1) where A_{i} and φ_{i} are respectively the amplitude and the phase of the ith order, v is the measurement noise which contains the unwanted part of the signal, f_{i} = o_{i}f_{r} is the instantaneous frequency of the order of interest with f_{r} the reference frequency and o_{i} the value of the orderi.
In the discrete form, (Eq. (1)) becomes: (2) where is the angular displacement and f_{s} is the sampling frequency.
The purpose is to estimate the amplitude and the phase of some specific orders of interest using the H_{∞} estimation approach. For this, the problem is formulated in term of estimation of the state variables. Note that the amplitude and the phase modulation are the key features to diagnose the gear state [6].
2.2 State space modelling
Let us consider the formula established in (Eq. (2)). The purpose here is to build the measurement and the state equation.
Linearizing (Eq. (2)) leads to: (3)
where a_{i,c} = A_{i}cosφ_{i} and a_{i,s} = A_{i}sinφ_{i}. Let put and
The amplitudes a_{i,c} and a_{i,s} are unknown. For estimating them, these amplitudes are modeled by a polynomial approximation as follows: (4) (5) and the coefficients of the polynomial by a random walk process such as: (6) (7)where w_{i,.} is a random signal. With those new variables (3) can be rewritten as: (8)with and with Note that A^{T} is the transpose of the matrix A.
Assuming that the measurement matrix is the following measurement equation is obtained: (9) where and v is the measurement noise with a covariance matrix V.
Then the state equation is: (10) where and w(k) = [w_{1,c} w_{1,s} w_{2,c} w_{2,s} ⋯ ⋯ w_{M,c} w_{M,s}]^{T} is the process noise with a covariance matrix W.
2.3 Discrete H_{∞} estimator design
From equations (9) and (10) the following state space model is considered: (11) Let us note the estimation error where is the estimate of x_{k} and E{ . } will stand for the expectation value.
Several facts may be used against the Kalman estimator although it is an attractive and powerful tool to estimate x_{k};

the Kalman estimator minimizes while the user may be interested in minimizing the worstcase error;

the Kalman estimator assumes that the noises are zeromean with Gaussian distribution;

the Kalman estimator assumes also that and are known.
These limitations have led to the statement of the H_{∞} estimation problem. Several formulations exist in the literature. The H_{∞} estimator solution that we present here is originally developed by Ravi Banar [7] and further explored by Shen and Deng [5]. These pioneers define the following cost function: (12) where is an estimate of x_{0}, Q > 0, P_{0} > 0, W > 0 and V > 0 are the weighting matrices and are left to the choice of the designers and depend on the performance requirements. The notation defines the weighted Q − L_{2} norm, i.e,
Problem statement [8]: Given the scalar γ > 0, find estimation strategy that achieve (13) where “ sup” is the supremum value and γ is the desired level of noise attenuation.“
The H_{∞} estimation problem consists of the minimization of the worst possible amplification of the estimation error. This can be interpreted as a “minmax” problem in which the estimation error is to be minimized and the exogenous disturbances (w_{k} and v_{k}) and the error of initialization () is to be maximized.
Remember that unlike the Wiener/Kalman estimator, the H_{∞} estimator deals with deterministic noises and no a priori information on their statistic properties are required. The solution of the H_{∞} estimation problem is given in the theorem below from [5].
Theorem: Let γ > 0 be a prescribed level of noise attenuation. Then, there exists a H_{∞} estimator for x_{k} if and only if there exists a stabilizing symmetric solution P_{k} > 0 to the following discretetime Riccati equation: (14) Then the H_{∞} estimator gives the estimate of x_{k} such as: (15)K_{k} is the gain of the H_{∞} estimator and is given by: (16)
Another way to solve the Riccati equation (14) is presented by Yaesh and Shaked [9]. The method is given as follows:

1.
Form the Hamiltonian
where n is x dimension.

2.
Find the eigenvectors of Z corresponding to the eigenvalues ℰ_{i} (i = 1, … , n) outside the unit circle

3.
Form the matrix of the corresponding eigenvectors denoted by:

4.
Compute
Note that the smaller γ, the more easy the problem is to solve. When γ tends to (the optimal value of γ) the eigenvalues of P tend to infinity and therefore is close to a singular matrix. Shaked and Theodor [10] investigated the behavior of the optimal H_{∞} estimator when γ tends to . They showed that when γ reaches , there exists at least one or more unbounded eigenvalues.
In the special case, where γ → 0, the H_{∞} estimator reduces to a Kalman estimator.
3 Numerical implementation
In this section, a synthetic signal is used to illustrate the performances of H_{∞} estimation approach. The generated signal (see Fig. 1) is described by the following equation. (19) where f_{r} is the instantaneous frequency linearly increasing from 0 to 50 Hz in 5 s, o_{i} contains the order's number and v is the measurement noise. The signal is composed of three orders presented in the Table 1. Figure 2 displays the rpmfrequency spectrum using the conventional windowing Fourier transform that characterizes three orders.
The results presented below have been got using a MonteCarlo simulation based on 400 iterations.
The parameters of the estimator have been taken as follows:

the covariance of the process noise W = 10^{−9};

the covariance of the measurement noise V = 10^{−3};

the initial covariance error P_{0} = 10^{−3};

the level of the noise attenuation γ = γ_{opt} = 10^{0.178}.
γ_{opt} is equal to the greatest value that guarantees the stability of the matrix P. This stability is reached, according to Yaesh and Shaked [11], when Ps eigenvalues are bounded in the unit circle. As plotted in the Figure 3, this stability is reached for γ = 10^{0.178}. Beyond this value there exists at least one or more eigenvalues that are outside the unit circle.
The measurement noise is modelled by a Poisson noise as mentioned in [11]. The Kalman estimator algorithm presented by Dan Simon [12] and the H_{∞} estimator have been applied to the generated vibration signal. The performance of both estimators is measured in term of signal to noise ratio. Table 2 gives the performance got for the two estimators. In both cases the H_{∞} estimator provides a better result than the Kalman estimator. The SNR_{out} value is the signal to noise ratio calculated by: (20) where N is the number of samples, y_{k} is the noiseless signal at times k and is the estimated or filtered signal. The criterion of comparison is improved by about 0.7 dB using the H_{∞} estimator. Therefore the H_{∞} estimator is a good alternative to deal with real situation where the noises are not really Gaussian.
Figures 4–6 show the effectiveness of the H_{∞} estimator for order tracking in nonstationary signal processing. We see in this last figure that the estimated we got by the H_{∞} estimation is closer to the original amplitude than the Kalman estimation.
Fig. 1 Synthetic signal. 
The amplitudes of the synthetic signal.
Fig. 2 Illustration of rpmfrequency spectrum. 
Fig. 3 Maximum of the eigenvalues of the covariance matrix error. 
Performance comparison between Kalman and H_{∞} filtering.
Fig. 4 Amplitude of the 1st order estimated using the H_{∞} and the Kalman estimator. 
Fig. 5 Amplitude of 3rd order estimated using the H_{∞} and the Kalman estimator. 
Fig. 6 Amplitude of the 9th order estimated using the H_{∞} and the Kalman estimator. 
4 Conclusion
Through this paper a method has been developed to estimate order's amplitude based on the H_{∞} estimation in nonstationary operations. This method uses the information of the instantaneous frequency of the signal and makes no assumption on the noises statistics. It takes advantage on the classical Kalman estimation and it can be consider as an extension of this last one. Since the estimator is designed to minimize the worst casedisturbances, the H_{∞} estimation approach is more robust to process any kind of noisy signal. The application of this method in reallife data will concern our future research.
References
 M.C. Pan, C.X. Wu, Adaptive VoldKalman estimation order tracking, Mech. Syst. Signal Process. 21 (2007) 2957–2969 [CrossRef] [Google Scholar]
 M.M. Vold, J. Blough, Theoretical foundations for high performance order tracking with the Voldkalman tracking filter, SAE Pap. (1997) 972007 [CrossRef] [Google Scholar]
 M.C. Pan, Y.F. Lin, Further exploration of VoldKalmanfiltering order tracking with shaftspeed informationII: Engineering applications, Mech. Syst. Signal Process. 20 (2006) 1134–1154 [CrossRef] [Google Scholar]
 B. Safarinejadian, J. Zarei, A. Ramezani, Fault diagnosis of induction motors using a recursive Kalman estimation algorithm, Int. Rev. Electr. Eng. 8 (2013) 96–103 [Google Scholar]
 X.M. Shen, L. Deng, Game theory approach to discrete H_{∞} estimator design, IEEE Trans. Signal Process. 45 (1997) 1092–1095 [CrossRef] [Google Scholar]
 P.D. McFadden, Detecting fatigue cracks in gears by amplitude and phase demodulation of the meshing vibration, J. Vib. Acoust. Stress Reliab. Des. 108 (1986) 165–170 [CrossRef] [Google Scholar]
 R.N. Banavar, J.L. Speyer, A linearquadratic game approach to estimation and smoothing, in: American Control Conference, IEEE, 1991, pp. 2818–2822 [Google Scholar]
 B. Hassibi, A.H. Sayed, T. Kailath, Linear estimation in Krein spaces. II. Applications, IEEE Trans. Autom. Control. 41 (1996) 34–49 [CrossRef] [Google Scholar]
 I. Yaesh, U. Shaked, A transfer function approach to the problems of discretetime systems: H_{∞}optimal linear control and estimation, IEEE Trans. Autom. Control. 36 (1991) 1264–1271 [CrossRef] [Google Scholar]
 U. Shaked, Y. Theodor, A frequency domain approach to the problems of H_{∞}minimum error state estimation and deconvolution, IEEE Trans. Signal Process. 40 (1992) 3001–3011 [CrossRef] [Google Scholar]
 G. Yang, W. Xu, W. Jia, M. He, Random vibrations of Rayleigh vibroimpact oscillator under Parametric Poisson white noise, Commun. Nonlinear Sci. Numer. Simul. 33 (2016) 19–29 [CrossRef] [Google Scholar]
 D. Simon, The Discretetime Kalman filter, in: Optimal State Estimation: Kalman, H infinity, and Nonlinear Approaches, John Wiley & Sons, Hoboken, New Jersey, 2006, pp. 124–129 [Google Scholar]
Cite this article as: A. Assoumane, E. Sekko, C. Capdessus, P. Ravier, Order tracking using H_{∞} estimator and polynomial approximation, Mechanics & Industry 18, 808 (2018)
All Tables
All Figures
Fig. 1 Synthetic signal. 

In the text 
Fig. 2 Illustration of rpmfrequency spectrum. 

In the text 
Fig. 3 Maximum of the eigenvalues of the covariance matrix error. 

In the text 
Fig. 4 Amplitude of the 1st order estimated using the H_{∞} and the Kalman estimator. 

In the text 
Fig. 5 Amplitude of 3rd order estimated using the H_{∞} and the Kalman estimator. 

In the text 
Fig. 6 Amplitude of the 9th order estimated using the H_{∞} and the Kalman estimator. 

In the text 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.