4.4 Simulation results
4.4.4 FDI possibilities
500 1000 1500 2000 2500 3000 3500
−6
−4
−2 0 2 4 6 8 10 12
14x 105 Residual6 (all faults simulated)
r6 [kg m2/N m]
time [s]
Figure 4.13: Residual6,r6 =nom ^. Simulation including all faults and no measurement noise.
4.4 Simulation results 93
500 1000 1500 2000 2500 3000
−0.5
−0.4
−0.3
−0.2
−0.1 0 0.1
Residual1
time [s]
500 1000 1500 2000 2500 3000
−15
−10
−5 0 5 10
Residual2
time [s]
500 1000 1500 2000 2500 3000
−0.8
−0.2 0.4
Residual3
time [s]
500 1000 1500 2000 2500 3000
−3
−2
−1 0 1
Residual4
time [s]
500 1000 1500 2000 2500 3000
−10
−5 0 5 10 15
Residual5
time [s]
500 1000 1500 2000 2500 3000
−5 0 5 10
15x 105 Residual6
time [s]
Figure 4.14: Overview over all six residuals (including all faults and no mea-surement noise). The solid lines show the residuals for the faulty case, while the dashed lines show the residuals for the fault-free case.
100 150 200 250 300
−0.1
−0.05 0 0.05 0.1
Residual1
time [s]
100 150 200 250 300
−1
−0.5 0 0.5 1
Residual2
time [s]
100 150 200 250 300
−0.1
−0.05 0 0.05 0.1
Residual3
time [s]
100 150 200 250 300
−1
−0.5 0 0.5 1 1.5 2
Residual4
time [s]
100 150 200 250 300
−1
−0.5 0 0.5 1
Residual5
time [s]
100 150 200 250 300
−1
−0.5 0 0.5
1x 105 Residual6
time [s]
Figure 4.15: All six residuals; zoom-in forhigh
(180s 210s). The solid lines show the residuals for the faulty case, while the dashed lines show the residuals for the fault-free case. The small deviations aroundt=100sare a result of the initialization phase.
4.4 Simulation results 95
600 650 700 750 800
−0.05 0 0.05
Residual1
time [s]
600 650 700 750 800
−5 0 5 10
Residual2
time [s]
600 650 700 750 800
−0.8
−0.6
−0.4
−0.2 0 0.2
Residual3
time [s]
600 650 700 750 800
−0.01
−0.005 0 0.005 0.01
Residual4
time [s]
600 650 700 750 800
−2 0 2 4 6
Residual5
time [s]
600 650 700 750 800
−5
−4
−3
−2
−1 0 1
2x 105 Residual6
time [s]
Figure 4.16: All six residuals; zoom-in fornhigh
(680s 710s). The solid lines show the residuals for the faulty case, while the dashed lines show the residuals for the fault-free case.
800 1000 1200 1400 1600 1800
−5 0
5x 10−3 Residual1
time [s]
800 1000 1200 1400 1600 1800
−0.05 0 0.05
Residual2
time [s]
800 1000 1200 1400 1600 1800
−0.05 0 0.05
Residual3
time [s]
800 1000 1200 1400 1600 1800
−0.2
−0.1 0 0.1 0.2
Residual4
time [s]
800 1000 1200 1400 1600 1800
−0.05 0 0.05
Residual5
time [s]
800 1000 1200 1400 1600 1800
−5000 0 5000
Residual6
time [s]
Figure 4.17: All six residuals; zoom-in for_inc(800s 1700s). The solid lines show the residuals for the faulty case, while the dashed lines show the residuals for the fault-free case. The small deviations aroundt=800sare a result of the shaft speed faultnhigh.
4.4 Simulation results 97
1800 1850 1900 1950 2000
−0.02
−0.01 0 0.01 0.02
Residual1
time [s]
1800 1850 1900 1950 2000
−1
−0.5 0 0.5 1
Residual2
time [s]
1800 1850 1900 1950 2000
−0.05 0 0.05
Residual3
time [s]
1800 1850 1900 1950 2000
−3
−2
−1 0 1
Residual4
time [s]
1800 1850 1900 1950 2000
−0.5 0 0.5
Residual5
time [s]
1800 1850 1900 1950 2000
−5 0
5x 104 Residual6
time [s]
Figure 4.18: All six residuals; zoom-in forl ow
(1890s 1920s). The solid lines show the residuals for the faulty case, while the dashed lines show the residuals for the fault-free case.
2500 2600 2700 2800
−0.02
−0.01 0 0.01 0.02
Residual1
time [s]
2500 2600 2700 2800
−15
−10
−5 0 5 10
Residual2
time [s]
2500 2600 2700 2800
−0.2
−0.1 0 0.1 0.2 0.3 0.4
Residual3
time [s]
2500 2600 2700 2800
−1
−0.5 0 0.5
1x 10−3 Residual4
time [s]
2500 2600 2700 2800
−10
−5 0 5 10 15
Residual5
time [s]
2500 2600 2700 2800
−5 0 5 10
15x 105 Residual6
time [s]
Figure 4.19: All six residuals; zoom-in fornl ow
(2640s 2670s). The solid lines show the residuals for the faulty case, while the dashed lines show the residuals for the fault-free case.
4.4 Simulation results 99
2900 3000 3100 3200 3300 3400
−0.5
−0.4
−0.3
−0.2
−0.1 0 0.1
Residual1
time [s]
2900 3000 3100 3200 3300 3400
−2
−1 0 1 2
Residual2
time [s]
2900 3000 3100 3200 3300 3400
−0.05 0 0.05
Residual3
time [s]
2900 3000 3100 3200 3300 3400
−1
−0.5 0 0.5
1x 10−3 Residual4
time [s]
2900 3000 3100 3200 3300 3400
−1
−0.5 0 0.5 1
Residual5
time [s]
2900 3000 3100 3200 3300 3400
−1 0 1 2 3
4x 105 Residual6
time [s]
Figure 4.20: All six residuals; zoom-in forky
(3000s 3500s). The solid lines show the residuals for the faulty case, while the dashed lines show the residuals for the fault-free case. The small deviations aroundt=2900sare a result of the shaft speed faultnl ow.
500 1000 1500 2000 2500 3000 3500
−0.04
−0.03
−0.02
−0.01 0 0.01 0.02 0.03 0.04
Residual1 (no faults simulated)
r1 [m/s]
time [s]
Figure 4.21: Residual1 (solid line) simulated for the fault-free case without mea-surement noise. Shaft speed referencenref (dashed line) is shown with different scaling and an offset for illustration.
4.4.4.2 Robustness issues
When looking at the residuals in Section 4.4.4.1 it can be seen that the fault-free residuals are not always equal zero. This can clearly be seen when looking at Residual1, Residual3, and Residual6 in Figure 4.18. In order to investigate this effect Figure 4.21 shows Residual1 for the fault-free case. It shows clearly, when comparing Residual1 with the shaft speed reference signalnref (Figure 4.5) that Residual1 deviates from zero in connection with the transitions and the startup phase. Similar effects can be shown for the other five Residuals. There are only slight differences in the dynamical behavior of the deviations. When looking at the residuals in Figures 4.8 - 4.13 it can be seen that these deviations are smaller in magnitude than the fault effects; especially than those fault effects the residu-als have been designed for.
As shown in Section 4.2 the FPRGs cannot be solved when the disturbances
T
extandQf are considered. Hence, they are neglected in all the residuals simu-lated and shown above. IncludingTextin the simulations as defined by the ship propulsion benchmark (see Figure 4.22) will have a clear impact on the residuals.
Figure 4.23 illustrates this for Residual1 in the fault-free case.
4.4 Simulation results 101
500 1000 1500 2000 2500 3000 3500
−0.5 0 0.5 1 1.5 2 2.5 3
3.5x 104 Disturbance T
ext
Text [kN]
time [s]
Figure 4.22: External force Text as it is implemented in the ship propulsion benchmark simulation package.
500 1000 1500 2000 2500 3000 3500
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
Residual1 (fault−free case including Text)
r1 [m/s]
time [s]
Figure 4.23: Residual1 simulated for the fault-free case without measurement noise including the disturbance Text as given in Figure 4.22. Using the initial condition^n(t=0) =0rad=s, andU^(t=0)=0m=s.
500 1000 1500 2000 2500 3000 3500
−3
−2.5
−2
−1.5
−1
−0.5 0 0.5 1 1.5
Residual4 (all faults and measurement noise simulated)
r4
time [s]
Figure 4.24: Residual4 simulated including all faults and measurement noise.
4.4.4.3 Measurement noise
Until now measurement noise has been omitted to enhance the readability of the simulation results. However, for the final residual evaluation in the next subsec-tion to obtain fault detecsubsec-tion and isolasubsec-tion for the ship propulsion system it has to be considered. In this subsection it is illustrated for Residual4 what happens when the measurement noise is considered and how it can be handled.
Figure 4.24 shows the simulation result for Residual4 when the measurement noise is considered. Obviously, the residual evaluation becomes more difficult in the presence of measurement noise than for the noise-free case given in Figure 4.11. For faults like the pitch sensor faultl owoccurring att=1890sa sim-ple threshold testing (threshold=0:5) would be sufficient for detection. This is due to the abrupt occurrence and the high magnitude. However, fault effects with a lower magnitude might hide in the noise. Furthermore, it becomes more difficult to detect incipient faults like_inc. As can be seen from Figure 4.24 it is not possible to use a simple threshold testing due to the alternating behavior of the signal.
There exist several statistical methods to detect a change of mean value in
4.4 Simulation results 103
stochastic signals (Basseville and Nikiforov (1993, 1994)). They can be applied to solve the problem of residual evaluation for a residual containing measure-ment noise. A simple version of the well-known CUSUM-algorithm has been used for the residual evaluation in this thesis. It is described in detail in Bas-seville and Nikiforov (1993)[Chapter 2], hence, it will only be reviewed briefly in the following.
The cumulative sum (CUSUM) algorithm is used to detect a change of mean value in the residuals. For this purpose the residual signalris considered to be Gaussian (as the measurement noise is Gaussian) having a mean valuerand a variance22, which results in the following probability density:
p
r (r)=
1
r p
2 e
(r
r )
2
2 2
r (4.69)
The interesting question for the residual evaluation is now to decide whether the mean value is zeror =0or higher than a given thresholdr threshold. A powerful measure to test between the two hypothesis:
H
0
:
r
=
r
nofault
=0 and H1 :
r
=
r
fault
=threshold
is the log-likelihood ratio defined as:
s(r)=ln p
r
fault (r)
p
r
nofault (r)
: (4.70)
The log-likelihood ratio is a positive measure when the residual r has a mean value that is closer tor
fault
than tor nofault
, otherwise it is negative. This can be seen easily when drawing the two corresponding Gaussian probability density functions into one diagram.
When the variance is considered to be equal for both hypothesis the log-likelihood ratio takes the following form (using (4.69) and (4.70)):
s
i
=s(r
i )=
r
fault
r
nofault
2
r
r
i
r
nofault +
r
fault
2
:
2As the goal is to detect a change in mean value the variance could be calculated online. For the simulations it was estimated based on the fault-free simulations.
whereridenotes theithsample of the residualr. The next step of the CUSUM-algorithm is to sum up the log-likelihood ratios si for the different samples.
Which leads to the following cumulative sum:
S
k
= k
X
i=1 s
i
Obviously, the functionSk is increasing in the faulty case (when the threshold is chosen properly) and decreasing in the fault-free case. The final step of this version of the CUSUM-algorithm is to calculate the following decision function:
g
k
=S
k m
k where mk
= min
1jk S
j
As this function only becomes significantly larger than zero when the function
S
k has been increasing significantly it can be used to decide whether the mean value of the residual has become significantly different from zero or not. For this purpose a second thresholdhis needed:
g
k
>h : the mean value of the residual is closer torfault 0g
k
h : the mean value of the residual is closer tor nofault
For the residual evaluation it is necessary to check for a positive and a negative change in mean value (rfault) and to choose an appropriate value for the de-cision thresholdh.
In the following the results from applying the CUSUM-algorithm to Residual4 in Figure 4.24 are given to illustrate its applicability. The simulations are based on the following values:
2
r
=0:002, based on simulations ofr4for the fault-free case
neg
r
fault
= 0:01, in order to detect a negative change in mean ofr4
pos
r
fault
=0:01, in order to detect a positive change in mean ofr4
Furthermore, the decision functions gk are reset to zero at t = 50, t = 250,
t=350, andt=1800. This is done to enable the detection of the next fault and to compensate for the summing up from the initialization phase. The obtained decision functions are shown in Figure 4.25. Using a second thresholdh=2:5 leads to the decision about fault or no fault as shown in Figure 4.26. The result shows thathighcan be detected att=182s,_incatt=899, andnl owat
t=1891.
4.4 Simulation results 105
0 500 1000 1500 2000 2500 3000 3500
0 20 40 60 80 100 120 140
time [s]
g k
Decision function g
k testing for pos. mean change
0 500 1000 1500 2000 2500 3000 3500
0 100 200 300 400
time [s]
g k
Decision function g
k testing for neg. mean change
Figure 4.25: Decision functions to detect positive and negative changes in the mean value of Residual4 shown in Figure 4.24.
0 500 1000 1500 2000 2500 3000 3500 0
0.5 1 1.5
time [s]
0 : no fault 1 : fault
Alarm for pos. mean change
0 500 1000 1500 2000 2500 3000 3500
0 0.5 1 1.5
time [s]
0 : no fault 1 : fault
Alarm for neg. mean change
Figure 4.26: Evaluation of the decision functions given in Figure 4.25 using a threshold ofh=2:5.
4.4 Simulation results 107
4.4.4.4 Residual evaluation for FDI
To obtain successful fault detection and isolation (FDI) the next task after residual generation is the residresidual evaluation. This is done in two separate steps -fault detection and -fault isolation. First, -fault detection is considered.
As shown in the previous section, it is possible to detect the pitch faultshigh,
_
inc, andl ow using Residual4. The detection times fulfill also the require-ments given by Table 4.3.
From Figure 4.8, 4.9, and 4.10 it can be seen that also the other three faults
n
high, nl ow, and ky can be detected. Applying a CUSUM-algorithm to Residual2 using2 =0:005,r
fault
=0:2, andh =30leads to the detection times given in Table 4.10. There it can be seen that also the detection times for the shaft speed loop faults fulfill the requirements given by Table 4.3.
Fault Detection time Fault Detection time
high
194s n
l ow
2641s
l ow
1895s n
high
682s
_
inc not detected ky 3003s
Table 4.10: Detection times for the different faults when evaluating Residual2.
The evaluation of the residualsr1andr3by using CUSUM-algorithms leads also to the detection of the shaft speed loop faults. However, the detection times are significantly bigger than the given requirements. They are about10 20seconds slower than the values given in Table 4.3.
The next step is fault isolation. As can be seen from the simulations Residual4 only reacts on the pitch faults, which is also clear from its design, see Section 4.3. During the design it was shown that it is not possible to isolate the two pitch faultssensorand_incfrom each other, when they are considered to be arbitrary. However, from the simulations it can be seen that they show different behavior. As a result Residual2 is only affected by sensor, hence, isolation can be obtained, when using both residuals. The shaft speed loop faults can be isolated by using the residualsr1andr3as shown in Figure 4.8 and 4.10.