11.1.1 Coherent Detection

In signal processing, one of the well-known and most widely used detection methods is classified in two parts: coherent detection and noncoherent detection [5]. Coherent detection basically uses measured frequency and phase distortion of a signal, which is compensated in the subsequent stages of the fault detection. On the other hand, noncoherent detection is applied without knowing the phase information. Since precise measurement of inspected low amplitude fault signatures is a challenging task, noncoherent detection is a more practical tool for fault diagnosis applications. Indeed, once the necessary information is accurately provided, the coherent detection usually performs better than noncoherent detection as it utilizes more signal information, which increases the complexity [5]. The noncoherent detection yields more reliable detection under severely noisy conditions where inaccurate information is available as its performance is not dependent on the distortion factor.

A simplified coherent detection is presented by Akin et al. [2]. As shown in Figure 11.1, the fault amplitude and phase can be monitored using a phase detection procedure. Compared to the techniques detailed by Bellini et al. [4] and Benbouzid et al. [7], phase-sensitive detection has reduced computational complexity and made it possible to perform a large amount of data processing using a low-cost DSP. However, the performance of coherent detection depends on the phase accuracy of the fault signature as depicted in Figure 11.2. Therefore, these kinds of techniques are applicable in conditions where phase ambiguities are negligible.

11.1.2 Noncoherent Detection (Phase Ambiguity Compensation)

The noncoherent detection is basically the amplitude detection procedure based on phase elimination and hence is inherently immune to the phase ambiguities of fault signatures. The elimination of the phase estimation stage reduces the computational burden of the noncoherent detection technique. Even though it shows lower performance than coherent detection at a steady state when the phase information is provided, it is well known that the detection method allows the user to obtain more reliable detection results under noisy or dynamic system conditions [5]. The block diagram of noncoherent detection is briefly given in Figure 11.3. So depending on the detection environment of fault SNR, the fault diagnosis method is to be determined between coherent and noncoherent detection.

11.1.3 Fault Frequency Offset Compensation

Rotor speed is one of the most critical variables that needs to be monitored continuously both for motor control and fault detection. The speed feedback is measured either by an encoder or estimated without speed sensors in the DSP code. Unlike the motor control, very precise speed

information is needed to identify the severity of the faults. However, in practical applications, a small mismatch between the speed feedback and the actual speed is commonly observed due to encoder resolution, inaccurate speed estimation algorithms, noise interferences, or slow response of sensors to unexpected transient condition, and so on. In addition to phase ambiguities, even a small amount of fault frequency offset yields erroneous fault detection results. Assuming the fault frequency offset w_offset = w_fault – w_ref ≠ 0 and phase delay φ_ref – φ_fault = φ_offset, the cross-correlation output signal will be

where ϖ is the motor current noise.

In Figure 11.2, the normalized decibel loss of coherent detection versus fault frequency offset is simulated using MATLAB where the phase offset percentage is defined between zero and 2π. It is clearly shown that fault frequency offset, phase offset, or a random combination of these two can truly suppress the fault signature, which typically has –40 to –80 dB amplitude.

However, great complexity will also be required if all of the expected offsets are monitored. Here, the current signal, I_cross(n), is expected to provide high enough resolution for fault detection even if it is averaged in time or down sampled with noise elimination through averaging. The information of fault signature is expected to remain, as the low fault frequency offset will not be interfered with in the low pass filtering such as the averaging operation if it is appropriately designed based on the Nyqist theorem.

Applying an offset detection technique to an averaged signal with small samples will reduce the complexity. Maximum likelihood (ML) detection is used to estimate the sinusoid at offset frequency, which is the maximum of the periodogram [1] and is given by

where

is the averaged signal, N₁ is the number of samples averaged, N₂ is assumed the physical DSP buffer size used for this purpose where the relation between parameters is as follows:

The tracking bound without aliasing is given by

The maximum bound comes from the Nyquist sampling theorem. If the offset (w_offset = w_fault – w_ref ≠ 0) is assumed, the aliasing will not be observed, practically.

The computational complexity of ML detection in Equation (11.2) depends on N and the frequency range f_range = f_max – f_min. Since the ML algorithm application in this study has high complexity, it needs modification for real-time DSP applications. These parameters will be limited through the averaged (effectively down sampled) signal with reduced N and limited frequency range where the maximum fault frequency offset between the reference signal and the fault signal frequencies is assumed to be less than 1 Hz (= f_range < 1Hz) for simplicity. Since the frequency error is fundamentally caused by motor speed feedback error, it can be adaptively adjusted depending on the performance of a speed estimator in industry application. In this way, the ML estimator can effectively be utilized in a DSP for on-line fault diagnosis.

The frequency resolution of ML-based offset detection in Equation (11.2) is determined as follows:

where N_tri is the number of applications of ML trials within f_range.

11.2.1 Adaptive Threshold Design (Noise Ambiguity Compensation)

Reliability is one of the major challenges facing fault diagnostic systems because the decision should be made for a small fault signature in a highly noisy industrial environment. In fact, the detection algorithm applied at fault frequencies detects noise signatures even with healthy motors, the amplitudes of which are usually hard to be discriminated from small fault signatures. One of the practical design considerations of the threshold encountered is how the detected signature can be reliably decided as the existing fault signature. The diagnostic decision making based on the threshold trained to the motor line current noise variation can evaluate the reliability of detected signature in DSP applications.

Here, the threshold is derived using the statistical decision theory [1] with the hypothesis of H₀ and H₁ for decision tests, which are as follows:

where H₀ is the hypothesis of having only noise without any faults; H₁ is the hypothesis of existing fault signature with amplitude I_fault in white Gaussian noise, ϖ, channel; and N(0,σ²) means zero mean noise with variance σ². I_fault is assumed reliably detected under the phase and frequency errors of a signal, which are the major errors in diagnostic signal processing. Therefore, the hypothesis in Equation (11.7) becomes possible by advantaging the techniques in previous sections independently derived from any control scheme of a motor assuming major error conditions.

A decision rule is made based on the optimal statistical test with a likelihood-ratio test (LRT) of the two distributions of this hypothesis, which is as follows:

where γ is the temporary threshold. With Gaussian distribution of noise, Equation (11.8) is derived as follows:

where γ′ is the threshold and N is the number of samples of current signal used for detection.

Let T=1N∑nN=1Istator in Equation (11.10). Then, the statistics of averaged stator current signal, T, is calculated as follows:

The performance with threshold γ′ applied to the averaged signal T with statistics shown in Equation (11.11) can be derived using the detection probability in Equation (11.12) and false alarm event probability in Equation (11.13), which are as follows:

where Q is the Q-function, which is detailed in a later section.

With range of allowable error (false alarm), P_FA, a threshold is calculated from Equation (11.12) and Equation (11.13):

The threshold provides a reliable decision-making tool for small signature detection in a noisy channel. Signature-based fault diagnosis performed with reliably detected signatures through the threshold will lead to more accurate condition monitoring while discriminating its results from random noise interference signatures. From Equation (11.14), the threshold is dependent on the number of samples and the noise variance estimated. These are independently determined from the motor operating point parameters (i.e., the fundamental stator current level, torque, rotor speed, motor specifications). This is a desirable feature of the fault diagnosis algorithm applicable for general purposes. It becomes possible since the complicated motor environments are generally reflected in line current noise, which is measured for threshold design. It implies that the diagnostic process is simplified without considering various reference estimations of different motor conditions, which will result in increased system complexity and prior knowledge of these variations.

The only unknown parameter in the threshold Equation (11.14) is the noise variance. The instantaneous line current noise is effectively measured for the threshold parameter using the method described later in this section.

11.2.2 Q-Function

Figure 11.4 (top) shows the probability distribution curve of noise and signature amplitude assuming an additive zero mean Gaussian noise channel. The area under each probability curve is one. By assuming an arbitrary decision threshold, γ_a, the probability distribution of decision-making errors can be identified in the shaded area as type I error. The reliability of small signal detection mainly depends on how the type II error (false detection) is suppressed. The Q-function is used to measure the error probability of false detection, which is the right side of the shaded area in Figure 11.4 (top).

The Q-function is defined as follows:

The term Q^–1(P_FA) in Equation (11.14) is effectively a weighting factor. With a greater weighting factor, the threshold in Equation (11.14) is to be increased and the false detection rate is decreased; this relationship is computed through Equation (11.15) and plotted in Figure 11.4 (bottom). Once the allowable false error rate P_FA is determined and, hence, the weighting factor, diagnostic decision making can be performed with a constant false alarm probability independent from the random noise conditions of the line current signal. This is because the threshold in Equation (11.14) is adaptively determined based on instantaneous noise condition and decouples the effect of noise on decision-making performance.

The optimization of threshold level and parameter depends on the diagnostic requirement of a specific system. An ideal threshold simultaneously minimizes false detection and missing detection probability. In small signal detections, minimizing the false alarm is commonly of more concern. Based on assumed noise conditions and allowed error probability, threshold parameters can be adaptively designed and optimized for a target system.

11.2.3 Noise Estimation

Noise variance can be estimated via the mean squared error (MSE) criterion. The MSE estimation is performed assuming infinite estimation time and zero mean noises from uniformly distributed signal distortion. Since harmonic signals are approximately periodic in the stationary operation of a motor and averaged to zero, noise content remains as follows:

Let μ^=1N∑k=1Nϖk be an unbiased estimator of ϖμ=E(μ^) whose mean is E(μ^)=0. From MSE criterion, noise statistics are derived as follows:

where μ−E(μ^)=0. Noise variance is derived from Equation (11.18).

11.3.1 Modeled MATLAB Simulation result

A typical stator current is modeled with fault conditions of a broken rotor bar. The distorted current signal is established assuming –15 dB noise, and 11% total harmonic distortion (THD) with 5th and 7th harmonics. The broken rotor bar signature of –40dB amplitude is inserted based on the fault equation assuming slip s = 0.016 pu where the excitation frequency is 60 Hz. The simulation is performed with the modeled signal in Figure 11.5. (All the experiments/simulations are performed assuming steady-state operation of a motor.) In the simulation, P_FA, frequency tracking range and the available buffers of a DSP are assumed the same as shown in Table 11.1. The signal with 50K samples is utilized for each simulation result. The fundamental signal is assumed filtered in the simulation.

The frequency tracked amplitudes and the threshold measured are shown simultaneously in Figure 11.5 with offsets varying from 0 to 1 Hz. In the figure, negative (–) frequency values are simply replicas of positive (+) offset results for convenience since noncoherent detection cannot discriminate polarity of frequency. Zero frequency is the point where the tracking scheme is not applied. In the figure, it is shown that the fault frequency offset inserted is accurately tracked at the frequency of the maximum normalized amplitude in all trials. One can also determine that the maximum points are above threshold while signals are below each threshold at the point without frequency tracking. The assumed tracking resolution 0.04 Hz shows sufficient performance to discriminate maximum points.

Table 11.1 Experiment Environment

11.3.2 Off-Line Experiments

The experiments are run utilizing line current data obtained by a 1.25 MS/s, 12-bit resolution data acquisition system, which is set to produce a 25 KHz sampling frequency. The 3-hp induction motors are loaded by the direct current DC generator, which is assumed open-loop controlled in all experiments. The acquired off-line data are processed through MATLAB.

In Figure 11.6, the stator current from the data acquisition card is shown with eccentricity (Figure 11.6a) and mixed fault signatures and unknown signatures simultaneously through fast Fourier transform (FFT) analysis (Figure 11.6b). Instantaneous fault frequencies are measured based on the fault equation. The motor is designed with mixed fault conditions for performing the experiments in a practical environment.

In Table 11.2 and Table 11.3, DF is the detection flag, DA is the detected amplitude, and TH is the threshold. The definition with subscript T is the result obtained through the frequency tracking operation. All amplitudes are shown in decibels.

From an FFT spectrum analyzer, the eccentricity signature monitored is –41.2 dB at 20% torque. It tends to decrease in the high torque range and around –55.45 dB at 40%~100% torques. For the broken rotor bar signature, it is –45.7 dB at 50% torque. Unlike the eccentricity, the broken rotor bar signatures increase with load and –41.8 dB at 100% torque. These results are taken to evaluate the accuracy of detection in the off-line fault diagnosis.

11.3.3 On-Line Experimental Results

The induction motor is fed by the inverter. The voltage-to-frequency (v/f) motor control and on-line fault diagnosis service routine are simultaneously implemented on a 32-bit fixed-point, 12-bit ADC, 150-MHz DSP of TMS320F2812.

In Figure 11.9 and Figure 11.10, the zero frequency is the fault signature frequency measured by the DSP from the fault equation. In Figure 11.9, the DSP measures the fault signature frequency correctly showing a maximum at zero frequency, that is, –40.2 dB. In Figure 11.10, 0.24 Hz fault frequency offset between the expected and the existing fault signature frequency is monitored for the broken rotor bar signature.

The changes in detected amplitude and thresholds in time are shown in Figure 11.11 and Figure 11.12. In both figures, the detected signature hardly varies after 2 seconds. The threshold measured is unstable initially and becomes stabilized after about 8 seconds. After becoming stabilized, it tends to decrease since one of the threshold parameters, effective noise variance, σ²/N, decreases as the number of samples used increases, which confirms careful derivation in Equation (11.14).

The latency time of about 10 seconds in fault diagnosis is assumed to be acceptable because condition monitoring is performed in a relatively long period of time, especially with a mechanical type of fault such as broken rotor bar or eccentricity.

In on-line experiments, the threshold applied is designed to keep false detection errors strictly within 0.097% as shown in Table 11.1. That is why the signatures are usually detected close to threshold within 5~10 dB. The thresholds can be further decreased to detect small signatures by reducing the weighting factor in Equation (11.14). This can be done based on the relation shown in Figure 11.4 (bottom) from the trade-off of detection performance. The resolution of signature amplitude tracking can also be further improved by intentionally adding known frequency bias, which lets the detection achieved be more precise as the relatively high frequency signal can be identified in a relatively shorter time period.

The fault detection and decision-making capability of the robust fault diagnosis algorithm are demonstrated in this chapter by mathematical verifications and off-line/on-line experiments. It is observed that ambiguities such as the fault signature frequency mismatch, the phase of the fault vector, and changes in the noise level of fault signatures can be efficiently handled using a simple algorithm capable of frequency tracking, phase eliminating detection, and adaptive threshold.