The equivalent estimation procedure as in the PM case can be used for parameter estimation of the AM signal. The model for the discrete analytical AM signal corresponding to (4.4) is:
iam,a[n, κ] = κ1[1 + κ2cos (2πκ3n − κ4)] exp j (2πκ5n − κ6) (4.88) with κ = [κ1, . . . , κ6] the parameter vector. The parameter of interest for fault detection is the AM modulation index κ2.
However, the simplifications for reduction of the number of parameters with the PM signal cannot be applied in this case. Hence, a time-consuming optimization
100 200 300 400 500 9.98 9.99 10 10.01 10.02x 10 −3
Data record length N
E
[ˆθ3
]
(a) Mean estimated PM index
100 150 200 250 300 350 400 450 500 0 1 2 3 4x 10 −7
Data record length N
M S E [ˆθ3 ] MSE[ ˆθ3] CRLB
(b) Mean square estimation error
Figure 4.35: Simulation results: mean estimated PM index vs. N and mean square estimation error together with CRLB vs. N , optimization with fixed grid search.
must be carried out with respect to 6 parameters or 5 with previous demodula- tion. An alternative is the use of the instantaneous amplitude aiam,a[n, κ
0] i.e. the absolute value of the analytical signal which is given by the following expression in our case:
aiam,a[n, κ
0] = |i
am,a[n, κ]| = κ1[1 + κ2cos (2πκ3n − κ4)] (4.89) with κ0 = [κ1κ2κ3κ4].
Using the instantaneous amplitude, the problem of estimating κ2 is now equiv- alent to estimating the amplitude of a sinusoid with an additional DC component. κ2 is the relative amplitude of the sinusoidal component with respect to the DC level κ1. It is well known that the MLE for this problem is the maximum of the periodogram (see [Kay88]). The estimation procedure is therefore the following:
• Estimation of the DC level ˆκ1 = N1 PN −1n=0 aiam,a[n, κ
0] • ˆκ2 is the maximum of the periodogram of (aiam,a[n, κ
0] − ˆκ
1), normalized with respect to ˆκ1. Zero-padding can be used for greater accuracy and the search of the maximum can be limited to an interval [fc,min, fc,max].
4.4.2.1 Simulation Results
In order to test the AM modulation index estimator, numerical simulations are carried out. The generated real AM signals are of the following form (see (4.88)):
iam[n, κ] = κ1[1 + κ2cos (2πκ3n − κ4)] cos (2πκ5n − κ6) + w[n] (4.90) where w[n] is zero-mean white Gaussian noise of variance σ2. The chosen SNR is 50 dB and the parameters are κ = √2 0.01 0.125 π/4 0.25 0.
The estimator is tested for different values of the data record length N . For each value of N , 1000 independent realizations of iam[n] are generated and ˆκ2 is calculated. The results are displayed in Fig. 4.36. Fig. 4.36(a) shows the mean
100 200 300 400 500 0.0099
0.01 0.0101
Data record length N
E
[ˆκ
2
]
(a) Mean estimated AM index
100 200 300 400 500 0 0.5 1 1.5 2 2.5 3 3.5x 10 −7
Data record length N
M
S
E
[ˆκ2
]
(b) Mean square estimation error
Figure 4.36: Simulation results: mean estimated AM index vs. N and mean square estimation error vs. N .
Signal Processing Steady State Transient State
Method PM AM Discr. PM AM Discr.
Spectral Estimation X X - - -
Instantaneous Frequency X - - X - -
Spectrogram X X X X X -
Pseudo WD X X X X X X
Parameter Estimation X X X - - -
Table 4.1: Summarized performances of the discussed signal processing meth- ods in steady and transient state: Columns PM / AM indicate the capability of detecting the modulation and Discr. the capability of discriminating PM and AM.
estimated modulation index E[ˆκ2] with respect to the data record length where N varies from 64 to 512. It can be seen that the relative error is always inferior to 0.5% which signifies a low estimation bias. Fig. 4.36(b) shows the obtained MSE for the AM modulation index. It decreases approximately proportional to 1/N .
4.5
Summary
In this chapter, the previously presented signal processing methods were applied to the faulty stator current models with PM and AM. The signatures of the faulty cur- rent signals with the non-parametric analysis methods have been derived through theoretical calculus and simulations. Hence, the different methods and their per- formances can be compared. Evaluation criteria for the different methods are the capability to detect and distinguish AM and PM in steady or transient state. The performances of the parametric and non-parametric methods are summarized in Table 4.1.
The first analysis method, spectral estimation, was found to be useful only for detection purposes in steady state. The two types of modulation cannot be distinguished under the particular conditions of this application. Since spectral
estimation supposes stationary signals, it cannot be used during transients. Instantaneous frequency estimation can effectively be used in steady and tran- sient state. Since AM has no effect on the IF, only PM can be detected. The extraction of fault indicators is possible employing e.g. the spectrogram.
The direct use of the spectrogram itself yields acceptable results in steady state. AM and PM can be detected and discriminated if a priori knowledge about the modulation frequency is available. Since the performance of the spectrogram depends heavily on the window length, the latter must be chosen with respect to the modulation period. During transients, PM and AM can be detected with an adequate window length but their discrimination is nearly impossible with the considered signals. In general, the spectrogram suffers from a lack of time and frequency resolution.
The Wigner Distribution and PWD overcome this problem since they offer good time-frequency resolution. Certain undesirable interferences can be controlled by smoothing. It has been shown through theoretical demonstrations and simulations that PM and AM lead to characteristic outer interferences that can be used for detection and discrimination. Since the WD perfectly localizes linear chirp signals it performs well in steady state and during linear transients.
Finally, a parameter estimation approach based on the AM and PM signal models was presented for use with stationary signals. Maximum likelihood es- timation leads to a practically realizable estimator. Two methods for numerical optimization were effectively used. The fixed grid search and the evolution strategy offer tradeoffs between computation time and accuracy. The estimated PM and AM modulation indices can be directly used as fault indicators. The estimator performs well even for short data records. Note that in this work, the parame- ter estimation approach has only been used for stationary signals. However, the models can theoretically be extended to take into account transient signals.