Not every component in a technical system can be equipped with sensors. Hence, when a fault occurs, available sensors often detect it only after its effects have propagated. This makes the isolation of faults difficult. We approach this problem by introducing virtual sensors. For each combination of component and fault type, we let external knowledge describe the fault-sensitive variables. We introduce a novel neural observer that is only trained on normal data. For fault isolation, it uses available sensor measurements where it is possible and reconstructs missing variables as virtual sensors to predict normal behavior. It detects deviations between current values and nominal predictions and computes local symptoms, which are then compared by onset time. The result is a set of root causes describing the faulty behavior. We use Gödel logic to encode physical knowledge and guide training. We evaluate our approach on synthetic networks, the Tennessee Eastman Process, and datasets of two modular production plants.
Figures & tables
Figure 1: Neural observer and diagnosis. The current trajectory uses measurements where available and virtual sensors otherwise. The nominal trajectory rolls the same network forward from t−δt without measurements. Both losses apply only during training on normal data.
Prior knowledge
Example relation
Topology
Connected variables have compatible normal reconstruction errors.
Qualitative relations
Variables change in the same or opposite directions under specified conditions.
Temporal relations
Responses follow within a specified delay.
Physical flow
Signed flows balance and mass flow agrees with volume flow through density.
Modes and state dynamics
Modes are exclusive. Filling implies a nondecreasing tank level.
Sensor physics
Measurement equations relate states and outputs, such as level and pressure.
Table 1: A priori knowledge for training, with process-industry examples. Experiments use the applicable rule groups.
Setup
Rules
Mask
Unique diagnosis
Detection
Healthy alarms
30
None
2
99.17
100.00
13.02
30
Full
2
100.00
100.00
11.46
30
None
4
99.58
100.00
9.38
30
Full
4
100.00
100.00
11.46
120
None
2
96.77
100.00
43.75
120
Full
2
97.71
100.00
38.54
Table 2: Linear networks, eight seeds. Setup: component count or combined stress at 30 components. Mask: hidden outputs per component. Full: value, direction, and delay rules. Rates in percent over 192 healthy and 240 fault recordings per condition (960 faults at 120 components).
Rules
Mask
Unique diagnosis
Detection
Healthy alarms
None
2
72.40
83.20
11.72
Structure
2
70.96
82.16
13.67
Mode
2
52.73
61.33
12.50
Both
2
56.51
64.97
10.94
None
4
64.97
82.29
13.28
Structure
4
62.24
80.99
10.55
Table 3: Hybrid automata. Rates in percent. Mask: hidden measurements per component. Both states are reconstructed. Rules affect training only.
Gödel rules
Mask
Correct
Potential
Hit@3
Detection
Healthy alarms
None
0
64.74
81.23
69.82
93.86
n.e.
Full
0
64.39
79.65
67.72
93.68
n.e.
Random
0
7.86
30.79
10.20
–
–
None
3
66.84
80.88
69.65
93.77
n.e.
Full
3
66.32
79.82
68.86
93.33
n.e.
Random
3
8.65
30.94
10.30
–
–
Table 4: TEP means (%), three model seeds and 380 fault cases. Full combines topology and qualitative rules. Mask counts additionally hidden channels. Correct allows ties in the earliest symptom group and is not unique isolation. Detection counts a persistent symptom anywhere in a fault window. Healthy-recording alarm rates were not evaluated (n.e.). Random matches the no-rule output sizes and has no detector.
Dataset
Training
Correct pump
Detection
Healthy alarms
Reconstruction MAE
Forecast MAE
ds5
Recursive
4/4
4/4
0/1
7.644
9.745
ds5
One-step
4/4
4/4
0/1
9.314
12.728
ds6
Recursive
6/6
6/6
0/1
6.160
9.859
ds6
One-step
6/6
6/6
0/1
9.811
16.307
ds7
Recursive
3/4
3/4
0/1
7.440
10.784
ds7
One-step
2/4
4/4
1/1
8.543
22.046
Table 5: Results for ReDiVO on the HAI-CPPS v2.1 benchmark. All rows use Gödel regularization ( β=0.01 ), state specific calibration, and separate regression heads. Healthy alarms count one healthy episode. Reconstruction and forecast MAE use normal-validation pump speeds in pump rpm, weighting all pumps and operating phases equally.