1 INTRODUCTION

Vibration analysis of rotating machines is one of the most used techniques for fault diagnosis and condition monitoring due to its good performance and low cost. Nowadays, modern vibration analysis aims to track behavior changes during varying operation and load conditions. The main challenge is to reduce the influence of these variations in mechanical systems under periodic loading caused by rotating operation. Order tracking OT is a signal processing technique aiming to obtain the features of the harmonics spectral component of the shaft reference speed (called basic order), and capture the dynamics of the measured vibration signals. Besides, OT shows an adequate performance in the analysis of non stationary vibration signals, condition monitoring and failure diagnosis (Bai et al. 2005). This technique allows to identify the rotation speed of the machine and the spectral/order components, which are fundamental to determine the machine and its conforming mechanisms state, during changing loads and speed regimes, (e.g. start and stop of the machine), (Guo et al 2006, Pan and Lin 2006). The most used OT methodologies are based on conventional approaches such as Gabor transform (GOT) (Zhao et al. 2008) and instantaneous frequency (IF) estimation based on time-frequency representations (Gao et al. 2006, Guo et al. 2006), where the shaft speed reference signal is not required, allowing to analyze rotating machines with lower quantity of sensors or when it is not possible to doing that measurement. Nevertheless, it is widely known that windowed Fourier transforms have limited resolution in time and frequency axes, and suffer of increased computational cost and offline processing (Pan and Lin 2007). On the other hand, OT methodologies based on parametric modeling display adequate response when computing spectral/order components in both online and offline conditions. Some estimation strategies such as VKF_OT (Pan and Lin 2006, Pan and Wu 2007), recursive least square (RLS) and Kalman filter (Bai et al. 2005), have increased precision in comparison with conventional methods, but require measuring the shaft speed, which makes the order analysis still complex. The measurement of the shaft speed implies installing additional equipment near to the machine,

Instantaneous frequency estimation and order tracking based on Kalman filters

O. Cardona-Morales, L. D. Avendano-Valencia and G. Castellanos-Domnguez Universidad Nacional de Colombia, Manizales, Colombia

ABSTRACT: Order tracking (OT) is a signal processing technique that allows capturing the dynamics of the vibration signals measured from rotating machines. In this paper, its propose an OT scheme based on extended Kalman filter (EKF), which comprises the advantages of existing OT techniques. Proposed scheme allows estimation of different spectral/order components of the signal and, in addition, estimates the shaft speed, improving the reconstruction of the different waveforms of each particular harmonic component. A comparison with the Vold-Kalman filtering (VKF) scheme is developed using two synthetic signals, to determine the noise sensibility and the ability to separate close order components. Results show that EKF scheme successfully characterizes close order components, improving the outcomes of the VKF scheme reducing the waveform reconstruction error of each spectral component. Additionally, the estimation of the reference speed of the shaft has good performance compared with the signal original frequency.

2

which in certain situations is inconvenient. Therefore, both parametric approaches are more difficult to implement. In this paper, an OT scheme that comprises the drawbacks of the techniques mentioned above is proposed. The problem to determining the fundamental frequency of a vibration signal is difficult due to their non-inherent stationarity associated to several parts of the machine. Therefore, it is necessary to introduce a frequency tracker that allows capturing the signal intrinsic dynamics. An approach consisting on a non-linear least minimum squares algorithm, which estimates the amplitude, frequency and phase of a non-stationary sinusoid is presented by Avendano-Valencia et al. (2007b), but the principal shortcomings, come from their lack of robustness in the estimation procedure. Is introduces in La Scala and Bitmead (1994), Bittanti and Saravesi (2000) a frequency tracker based on an oscillator model whose parameters are calculated by extended Kalman filtering, obtaining the amplitude, phase and mainly, the frequency of a harmonic signal for de-noising in non-stationary signals. However, the tuning of model parameters is complex and requires expertise. Nevertheless, if a delta operator is applied, the robustness of the model is improved for the digital signal processing (Avendano-Valencia et al. 2007a). Although the aforementioned EKF scheme is oriented to de-noising, in this paper, it is demonstrated that it is also ideal for OT and allows estimating the shaft speed. A set of two synthetic signals are used to prove the performance of the proposed scheme, either in robustness and sensibility in to noisy environments as well as to close order components decoupling. A comparison with the VKF scheme is carried out to determine the goodness and weakness of the proposed scheme using the test signals. These characteristics are evaluated using the mean square error between the waveform of original and estimate order components.

2 METHODS

The vibration signal generated by rotating machinery can generally be represented as a superposition of sinusoids. Therefore, the signal ( )y n containing K orders generated by a rotating shaft can be written as

1( ) ( ) cos[ ( ) ( )]

K

k kk

y n a n k n n =

= + (1)

where k (n)a and ( )k n denote the time-varying amplitude and phase, respectively, of the k-th order, and 0( ) 2 ( )n f n pi= is the oscillation frequency with 0 ( )f n the rotation frequency of the shaft. By a trigonometric identity, the Eq. (1) can be rewritten as

[ ] [ ]1 1

( ) cos ( ) sin ( )K K

kI kQk k

y n a k n a k n = =

= (2)

where the amplitude of the k-th order can be expanded into a in-phase ( ) ( ) cos ( )kI k ka n a n n= , and a quadrature ( ) ( ) sin ( )kQ k ka n a n n= component (Bai et al

2005). The amplitude of k-th order can be written as 2 2( ) ( ) ( ) .k kI kQa n a n a n= + The variables k (n),a ( )k n and ( )n are time-varying, and they must have smooth

transitions along time. The Eq. (2) can be simplified just to obtain a single component model, which is equivalent to the following state space representation (Bittanti and Saravesi 2000, Avendano-Valencia et al. 2007)

3

cos ( ) sin ( ) 1( 1) ( ) ( )sin ( ) cos ( ) 0

n nn n n

n n

+ = +

x x w (3a)

[ ]( ) 1 0 ( ) ( )n n v n= +y x (3b)

where 2 1( )n x is the state vector, 2 1( )n w and ( )v n represent respectively the process and measurement noise. The matrix multiplying ( )nx in Eq. (3a) is called state transition matrix and is denoted by 2 2( ) . M Eqs. (3a,b) can be used to defined a sinusoidal component with frequency ( ).n For multiple sinusoidal component (orders), each block defined by Eqs. (3a,b) can be augmented into the following form:

1 1 1 1

2 2 2 2

( 1) ( ) ( ) ( )( 1) ( ) ( ) ( )

( 1) ( ) ( ) ( )K K K K

n n n

n n n

n n n

+ + = +

+

x M 0 0 x wx 0 M 0 x w

x 0 0 M x w

(4)

where ( )k nx represents the k-th state vector corresponding to k-th order component. Eq. (4) can be rewritten in short form as:

( 1) ( 1, ) ( ) ( )n n n n n+ = + +X F X w (5)

where 2 1( ) Kn X is the state vector containing in-phase and quadrature components for each order k (k = 1, 2, , K). 2 2( 1, ) K Kn n + F is the state transition matrix, which defines how the state vector changes through time. 2 1( ) ( , ( )) Kn n w 0 QN is the process noise, where 2 2( ) K Kn Q is the covariance matrix. Likewise, the Eq. (3b) can be rewritten as

( ) [ ]( )( )

( )( )

1

2

k

n

ny n v n

n

= +

x

xh h h

x

(6)

or the symbolized form of Eq. (6)

( ) ( ) ( )y n n v n= +HX (7) where [ ]1 0 ,=h comprises the state measurement matrix 1 2 ,KH R and the measurement noise 1 1( )~ ( , (n)) ,v n 0 r N R where ( ) 1 1r n R is the measurement covariance matrix.

2.1 Instantaneous frequency estimation The IF of a real signal is the derivate of the phase of the associated analytic signal to ( ) ,y n corresponding to the in-phase and quadrature components (Gao et al. 2006). Therefore, if a new state is appended to the model of Eqs. (4) and (6), it is possible to estimate the IF (Hajimolahoseini et al. 2008). Assuming that ( ) ( )1k Kn x n += in the Eq. (4a). The Eqs. (4) and (6) can be augmented as follow

4

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

1 1 11

2 2 21

1

1 1 1

1 ( , )1 ( , 2 )

1 ( , )1 1

K

K

k K KK

K K K

n n nn x 0n n nn x 0

n n nn Kx 0n n n0 0 0

+

+

+

+ + +

+ + = +

+ +

x x wM 0 0x x w0 M 0

x x w0 0 Mx x w

(8a)

( ) [ ]

( )( )

( )( )

( )( )

( )( )

1 1

2 2

1 1

0

K K

K K

n v n

n v n

y nn v n

n v n+ +

= +

x

x

h h hx

x

(8b)

where the matrix 1( , )kn kx +M depends of the instant n. Eq. (8a) can be rewritten as

( ) ( )( ) ( )n 1 n, n n+ = +X f X w (9)

where ( )( )n, nf X is nonlinear state transition matrix function. In this case, estimation of the state vector comprises a non-linear set of equations, so approximate solutions of Kalman filters should be used. The simplest approach consists on the use of Kalman filtering through a linearization procedure, known as Extended Kalman Filter (EKF). The basic idea of EKF is to linearize the Eq. (9) at each time instant around the most recent state estimate, which is taken to be ( )n .X Once a lineal model is obtained, the standard Kalman filter equations are applied. Therefore, the non-linear transition matrix is linearized by

( ) ( )( )( ) ( ) ( )n nn, n

n 1, nn

=

+ =

X X

f XF

X (10)

where ( )nX is estimate of state vector defined Kalman filter recursion (Haykin 2001).

3 RESULTS

Two set of synthetic signals are designated to validate the proposed OT scheme, and compare it with the angular-displacement VKF_OT method. The synthetic signals comprise specific order components, where the synthetic signal 1 is employed to determinate the robustness and sensibility of proposed scheme under the influence of additive white Gaussian noise (AWGN) in different intensity levels, and the synthetic signal 2 is to justify the effectiveness to distinguish close orders. The simulated signals where sampled at 1 kHz. The amplitude, phase and frequency estimation, and the separation of order components of the proposed scheme, are investigated here.

3.1 Robustness and sensibility Synthetic Signal 1 The synthetic signal 1 comprises two order components including 1 and 2. The reference speed

of the shaft linearly increases from 0 to 1800 rpm in 5 seconds. The amplitude of each order

5

0 1 2 3 4 50

2

4

6

8

10

12

Time [s]

Am

plitu

de

1st order2nd order

(a)

0 1 2 3 4 50

2

4

6

8

10

12

Time [s]

Am

plitu

de

1st order2nd order

(b)

Figure 1 : Estimated Amplitude of synthetic signal 1 without noise by (a) EKF, and (b) VKF scheme.

0 1 2 3 4 50

2

4

6

8

10

12

Time [s]

Am

plitu

de

1st order2nd order

(a)

0 1 2 3 4 50

2

4

6

8

10

12

Time [s]

Am

plitu

de

1st order2nd order

(b)

Figure 2 : Estimated amplitude of synthetic signal 1 with 6 dB of AWGN by (a) EKF, and (b) VKF.

component linearly increases from 0 to 10 as well. Fig. 1 shows the estimate amplitude of each order component considering that the signal analyzed dont keep noise. The proposed scheme allows an accuracy estimation of the amplitude of each order component due that the answer is most approximate to a linear slope (Fig. 1a), and the MSE allows confirm the affirmation, while that the VKF scheme presents a little deviations until achieved a stable level (Fig. 1b).

The analysis of the presented OT schemes is evaluated by waveform reconstruction ability of each order component. Fig. 2 shows the estimated amplitude for each order component of the signal with 6 dB of AWGN. The VKF scheme allows an adequate estimation of the amplitude of the order components, presenting an answer almost straight line, while that the proposed scheme is observed a slightly choppy, but the MSE indicate that the better amplitude estimation is given by proposed scheme (see Table 1). Also is possible to observe, of graphic way, that the proposed scheme presents a good estimation since it conserves the values and it linearity of the desired amplitude. In order to the robustness and sensibility of the proposed scheme, the test signal is contaminated with AWGN using six different intensity levels (from -10 dB to 10 dB, with increments of 4 dB). The performance of the analyzed schemes is evaluated by means of mean squared error (MSE). The MSE between the original and reconstructed signal is computed for each level. The Table 1 shows the MSE between the original and estimated parameters, amplitude (A), waveform reconstruction (WR) and IF by the presented schemes. In noiseless scenario, the proposed scheme reconstructs the waveform of the order components with the least error. However, as the AWGN intensity increases, the reconstruction error grows, indicating that the proposed scheme is sensible to noise. Other hand, the VKF scheme conserves the error values in a minimal range, showing that is robust to high levels of noise when the reference speed of the shaft is known. For AWGN levels greater to 10 dB, the proposed scheme becomes unstable, i.e. the estimator fails to converge. Fig. 3 shows the IF estimation carried out by proposed scheme. Either in the signal without AWGN as well as in the signal with a AWGN intensity of 6 dB, the approach displays an accurate estimation of the time varying frequency of the signal tracked, improvement even approach as STFT (see Fig. 4), where the frequency estimation depends of several parameters and have a limited resolution. In fact, it is could thinking that the proposed scheme allows a

6

description of the signal in a 3D map (time, frequency and amplitude), because it estimate the changes in the time of the parameters above mentioned.

0 1 2 3 4 50

20

40

60

80

Time [s]

Freq

uen

cy [H

z]

1st order2nd order

(a)

0 1 2 3 4 50

20

40

60

80

Time [s]

Freq

uen

cy [H

z]

1st order2nd order

(b)

Figure 3 : IF estimated of synthetic signal 1 (a) without noise, and (b) 6dB of AWGN, by using EKF.

Table 1: MSE of the Synthetic Signal 1 VKF EKF

Level Noise dB

Order -

A -

WR -

A -

WR -

IF -

NA 1 0.34 0.087 0.21 0.093 0.18 2 0.29 0.14 0.25 0.087 0.35

-10 1 0.28 0.14 0.18 0.21 0.32 2 0.29 0.17 0.26 0.22 0.64

-6 1 0.31 0.14 0.21 0.21 0.34 2 0.29 0.17 0.30 0.23 0.69

-2 1 0.29 0.15 0.12 0.21 0.33 2 0.32 0.17 0.23 0.27 0.67

2 1 0.30 0.15 0.37 0.29 0.42 2 0.28 0.19 0.38 0.37 0.84

6 1 0.44 0.18 0.28 0.32 0.42 2 0.45 0.25 0.48 0.44 0.84

10 1 0.43 0.16 0.61 0.43 5.51 2 0.41 0.27 0.74 0.7 11.04

3.2 Identifying close orders - synthetic signal 2 The synthetic signal 2 comprises three order components including 1, 4, and 4.2. The reference speed of the shaft and the amplitude linearly increase from 0 to 1800 rpm in 5 second and from 0 to 10, respectively. Fig. 4 shows the order components of the synthetic signal 2 in the time-frequency domain obtained by STFT in similar form that to the synthetic signal 1, where is possible to observe that the components 4 and 4.2 are mixed, to making difficult the separability by approaches based on Fourier transform.

Results show that the close order components are separable with a low reconstruction error (see Table 2), which is useful when is desire separate some signal components and to make an independent analysis of each component. The VKF scheme also allows separate the signal components, under the condition of having the fundamental frequency value of the system.

The Table 2 shows the waveform reconstruction error for each order component of the signal. The proposed scheme overpasses to VKF scheme because the reconstruction error is minor in the distinct order components. Nevertheless, the amplitude estimation is most stable using VKF scheme (see Fig. 5). Also is possible to see in Fig. 5, that evaluated schemes in this work have similar performance, due to that the amplitude of integer order components are correctly estimated, as long as the order 4.2 present a stability time slow finishing in the maxima value desired.

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Time [s]Fr

equ

ency

[H

z]

1 2 3 4 50

50

100

150

200

Figure 4 : Illustration synthetic signal 2 by STFT (hamming, length=1023, overlap = 0.5)

0 1 2 3 4 50

2

4

6

8

10

12

Time [s]

Am

plitu

de

1st order4th order4.2th order

(a)

0 1 2 3 4 50

2

4

6

8

10

12

Time [s]

Am

plitu

de

1st order4th order4.2th order

(b)

Figure 5 : Estimated amplitude of closed orders: (a) EKF, and (b) VKF scheme, respectively.

0 1 2 3 4 50

20

40

60

80

100

120

140

Time [s]

Freq

uen

cy [H

z]

1st order4th order4.2th prder

Figure 6 : Estimated IF of closed orders by using EKF.

Table 2 : Estimated MSE of the Synthetic Signal 2 Parameters Order 1 Order 4 Order 4.2

VKF A 0.33 0.52 1.05

WR 0.13 0.35 0.39

EKF A 0.29 0.39 0.91

WR 0.09 0.29 0.32 IF 0.10 0.41 0.43

Additionally, Fig. 6 shows that the estimation of the shaft speed can be obtaining from the signal and the frequency domain information is available by the individual contributions of the order components. It is also noticed that the behavior of the proposed scheme is stable and distinguish close orders.

4 CONCLUSIONS

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The study proposes shows and implements an OT scheme based on EKF. The comparison with a angular-displacement VKF scheme is developed by two synthetic signals. First, the proposed scheme allows estimating amplitude, phase and frequency of the different order components included in the signal, but it is very sensible to the influence of AWGN, presenting unstable responses to high levels of noise. The VKF scheme is most robust that the proposed scheme, but it requires the measurement of the speed shaft. Second, the proposed scheme overcomes the VKF scheme in separability of close orders due to it least waveform reconstruction error of the components analyzed. Finally, the proposed scheme presents good shaft speed estimation, i.e. IF estimation, either to close order components such as signals with additive noise, indicating that it is useful for OT and analysis of rotating machines. The performance evaluation of the proposed scheme to real applications and development of preprocess stages than improve the response with noisy systems are will considerer in further work.

ACKNOWLEDGEMENTS

This work was partially supported by the project Analisis tiempo-frecuencia de seales de vibracin mecnicas para la deteccin de fallos en mquinas rotativas, financied by DIMA and VRI of Universidad Nacional de Colombia. In addition, it was financially supported by the project with code 1119-425-20795 of COLCIENCIAS.

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