Fault diagnosis is crucial for guaranteeing safe, reliable and efficient operation of modern engineering systems. These systems are typically hybrid. They combine continuous plant dynamics described by continuous-state variables and discrete switching behavior between several operating modes. This paper presents an integrated approach for online tracking and diagnosis of hybrid linear systems. The diagnosis framework combines multiple modules that realize the hybrid observer, fault detection, isolation and identification functionalities. More specifically, a Dynamic Bayesian Network (DBN)-based particle filtering (PF) method is employed in the hybrid observer to track nominal system behavior. The diagnostic module combines a qualitative fault isolation method using hybrid TRANSCEND, and a quantitative estimation method that again employs a DBN-based PF approach to isolate and identify abrupt and incipient parametric faults, discrete faults and sensor faults in a computationally efficient manner. Finally, simulation and experimental studies performed on a hybrid two-tank system demonstrate the effectiveness of this approach.
The increasing complexity of modern industrial systems
motivates the need for online health monitoring and
diagnosis to ensure their safe, reliable, and efficient operation.
These systems are typical hybrid involving the interplay
between discrete switching behavior and continuous plant
dynamics. More specifically, the system configuration
changes consist of known controlled mode transitions
generated from external supervisory controller and
autonomous mode transitions triggered by internal variables
crossing boundary values. The continuous dynamic
behavior is modeled by continuous-state variables that are a
function of the particular discrete mode of operation. As a
result, tasks like online monitoring and diagnosis have to
seamlessly integrate continuous behaviors interspersed with
discrete transitions that often require model switching to
accommodate the discrete transitions [
For complex hybrid systems, faults will typically affect
the continuous behavior and the discrete dynamics of the
system. Some faults may be parametric, and they directly
affect the continuous behavior, others are discrete, thus they
directly affect the mode of system operation. Both types of
faults also have indirect effects on the other type of
behavior. Moreover, faults can have different time-varying
profiles, such as abrupt faults, intermittent faults and incipient
faults [
Previous model-based diagnosis approaches of hybrid
systems were developed separately for parametric faults or
discrete faults. For example, [
Our goal in this paper is to propose an integrated
model-based approach to diagnose single and persistent
incipient or abrupt parametric faults, discrete faults and sensor
faults in hybrid linear systems. This extends our earlier
work [
The rest of this paper is organized as follows. Section 2 briefly presents the different models employed in our diagnosis approach and some basic definition of the different types of faults. A hybrid two-tank system is used as a running example to explain the hybrid bond graph modeling method and the derivation of temporal causal graph and DBN from hybrid bond graph models. Section 3 gives a brief overview of our diagnosis architecture, and then presents our online tracking and fault detection, qualitative fault isolation and quantitative fault isolation and identification schemes in some detail. Section 4 discusses the results of the application of our algorithm to the hybrid two-tank system. Finally, the discussion and conclusions of this paper are presented in the last section. 2
In this section, we formalize the basic definitions, concepts and notation of the modeling approach that goes in conjunction with our diagnosis architecture. 2.1
Bond graphs (BGs) are a domain-independent
topological-modeling language that captures energy-based
interactions among the processes that make up a physical system
[
Valve1
F1
C1
R12
C2 Valve2
Valve3 Tank1
R1
Tank2
R2
Hybrid bond graphs (HBGs) extend BGs by introducing
switched junctions to enable discrete changes in the system
configuration [
The hybrid two-tank system, shown in Figure 1, is the running example we employ in this paper. This system consists of two tanks connected by a pipe, a source of flow into the first tank, and drain pipes at the bottom of each tank. Three valves valve1, valve2 and valve3 can be turned on and off by commands generated from the supervisory controller. When the liquid level in tanks 1 ( h1 ) and/or 2 ( h2 ) reaches the height at which pipe R12 is placed ( h ), a flow is initiated through pipe R12 . The autonomous mode changes associated with this pipe are triggered when the liquid level in tank1 and/or tank 2 goes above or below the height of the pipe R12 . We assume five sensors: M1 and M 2 measure the outflow from tank 1 and tank 2, respectively. M 3 measures the flow through the autonomous pipe R12 , and M 4 and M 5 measure the liquid pressure in tank 1 and tank 2, respectively.
CSPEC1 Sf 1
De : M4 1
Df : M1 Autonomous pipe R12 1
LS f Left 2 6 3 1
CSPEC4 C : C1 4 5 7 0 1 R : R1
Df : M3 9 10 8
R12
11 R : R12 CSPEC2
CSPEC3 f(x) 1 R : R12
CSPEC5 C : C2 0 1 12 13 15
R : R2 RS f 1 16 14
De : M5 Df : M2 1 Right
Figure 2 illustrates the HBG model for the plant in Figure
1 (The HBG model for autonomous pipe R12 is shown
separately at the bottom part of Figure 2). The tanks and
pipes are modeled as fluid capacitances C and resistances R,
respectively. Measurement points occur at junctions. They
are denoted by elements with symbols De for effort variable
measurements and Df for flow variable measurements.
Moreover, the two-tank system has five switched junctions:
the CSPEC1, CSPEC2 and CSPEC3 describe the control
logic for the three valves. CSPEC4 and CSPEC5 together
capture the autonomous mode transitions of the connecting
pipe between the two tanks. Figure 3 (a) shows the CSPEC
for a valve controlled by the switching signal sw. Figure 3
(b) shows CSPEC4 that describes the state of the left tank.
When the liquid height in tank1 is below that of the
autonomous pipe R12 , that state is OFF. If the liquid level
exceeds the height of the pipe, this CSPEC transitions to the
ON state. Similarly, CSPEC5 denotes the state of the right
tank, and the mode of the autonomous pipe depends on the
combination of these two CSPECs. Table 1 shows the
discrete mode for pipe R12 and the corresponding state of
CSPEC4 and CSPEC5 in detail. The corresponding bond
graph configurations are described in [
t t+1 S1 :ON
S2 :OFF
The temporal causal graph (TCG) is a signal flow
diagram that captures the causal and temporal relations
between system variables, and can also be systematically
derived from a BG [
Definition 1 (Temporal Causal Graph): A TCG is a directed graph that can be denoted by a tuple <V, L, D>. V E F S M is a set of vertices involving effort variables E, flow variables F, discrete fault event S and measurement M in hybrid bond graph model. L is a label set {1, 1, , p, p1, N, Z, p dt, p1 dt} . The propagation type of first seven labels is instantaneous, and the last two are temporal. D V L V is a set of edges.
For lack of space, the TCG for hybrid two-tank system is
not shown in this paper, but the algorithms for deriving
TCGs directly from bond graph model can be found in [
Assuming that the system is Markovian and time-invariant, we can model the system as a two-slice temporal Bayes net that illustrates not only the relations between system variables at any time slice t, but also the across-time relations
When all the valves are ON and the liquid level in tank1 and tank2 are above the height of the autonomous pipe R12 , the nominal DBN model for hybrid two-tank system is shown in Figure 4. This DBN model derived from the TCG as the following random variables: the continuous state variables X e4 , e12 presents the pressures at the bottom of each tank, input variables U f1 denotes the input flow into tank 1, and measured variables Y f6 , f9 , f14 indicates the outflow from tank1, the flow through the autonomous pipe R12 and the outflow from tank 2. t t+1
Figure 5 Single DBN model for both abrupt and incipient parametric fault
Since the discrete faults only influence the system mode, but not parameter variables, the DBN fault model corresponding to discrete fault will be constructed from the TCG in the particular discrete mode. For parametric faults, the DBN fault model is generated on the basis of nominal DBN model by augmenting a new random variable for each fault candidate. Figure 5 shows DBN model with parametric faults represented explicitly for the hybrid two-tank system. f6 f9 f14 f9 f14 f1 e4 e12 f1 e4 e12 R1 f1 e4 e12 f1 e4 e12 R1 f6 f9 f14 f9 f14 The abrupt fault Ra and incipient fault Ri are 1 1 represented in the same model. When the fault occurs, fault parameter R1 becomes the additional state variable that need to be tracked. In this paper, we focus on the diagnosis of persistent single faults. We consider incipient or abrupt parametric faults and discrete faults occurring in hybrid linear systems, as well as sensor faults. The precise definition for these faults can be given as follow.
Definition 2 (Incipient parametric fault): An incipient fault profile is defined by a gradual drift in the corresponding component parameter value p(t) from the fault occurrence time t f . The incipient fault parameter pi (t) can be described by:
p(t)
pi (t)
p(t) d (t) p(t) ip (t t f )
t t f
t t f
(1)
where d (t) ip (t t f ) is a linear function with a
constant slope ip that added to the nominal parameter value
from the time point of fault occurrence. Our approach to
isolation and identification of incipient fault parameters is
to calculate this constant slope ip [
Definition 3 (Abrupt parametric fault): An abrupt parametric fault is characterized by step changes in nominal component parameter value p(t) from the fault occurrence time t f . The abrupt fault parameter pa (t) is given by: p(t) t t f pa (t)
p(t) b(t) p(t) pa p(t) t t f
where b(t) a p(t) is a step function that gets added to
p
the parameter value from the time point of fault occurrence.
pa is the percentage change in the parameter expressed as a
fraction, and our goal is to estimate this value [
Definition 4 (Discrete fault): A discrete fault manifests as
a discrepancy between the actual and expected mode of a
switching element in the model [
Discrete faults occur in discrete actuators, like valves and switches that operate in discrete modes (e.g., on and off). Consider the example of a valve, it may be commanded to close, but remain stuck open. Also, it may unexpectedly open or close without a command. This type of fault manifests as an unexpected system mode change, unlike parametric faults, which cause deviations in continuous behavior.
Definition 5 (Sensor fault): A sensor fault is a discrepancy between the measurement and actual value in the model.
In this paper, we only consider sensor bias fault, which can be represented as:
m(t) mb (t) m(t) b m t t f t t f where m(t) is the true value, and bm is the sensor bias term. (3) (2) 3.1
3
Our integrated diagnosis approach for hybrid linear systems
(See Figure 6) combines the Hybrid TRANSCEND
approach [
Initially, a nominal DBN is constructed from the current TCG model. A hybrid observer uses a PF-based nominal DBN model to track the system behavior in individual modes of operation. At the same time, a finite automata method in hybrid bond graph scheme implements the CSPECs, executes controlled and autonomous mode changes, and determines the system model for hybrid observer.
The fault detection continually monitors the statistically significant deviations between the observation y(t) and estimation yˆ (t) generated by hybrid observer. Once a fault is determined, QFI is triggered to generate the initial fault hypothesis, and refine them as additional deviations are observed. When remaining fault hypothesis set satisfies particular condition, the QFII scheme is invoked to run in parallel with QFI. The goal of this scheme is to refine the fault hypothesis further and estimate the value of the fault parameter. The following subsections describe these steps in more detail.
Since the hybrid system is piecewise continuous, discrete mode changes of the hybrid system have to be detected accurately as the continuous behavior of the system evolves. In our work, we have designed hybrid observers that are based on the nominal DBN-based PF scheme to track the continuous behavior in individual modes of operation. PF is a general purpose Markov chain Monte Carlo method that approximates the belief state using a set of samples or particles, and keeps the distribution updated as new observations are made over time. Moreover, the PF approach for DBNs exploits the sparseness and compactness of the DBN representation to provide computationally efficient solutions, because each measured variable in a DBN typically depends on some but not all continuous state variables.
For discrete mode changes, the finite state machine (FSM) for each switched junction determines mode transitions. Since the continuous behavior and discrete mode changes will interact with each other as system evolves, the FSM needs to execute controlled or autonomous mode changes. Explicit controlled changes are relatively simple, but the autonomous mode changes depend on the internal continuous variables. If mode changes occur, the hybrid observer will regenerate the nominal DBN model from TCG in new mode, and use the PF to continuously track system dynamic behavior. The online tracking algorithm for hybrid systems is shown in Algorithm 1. u(t)
System
y(t) Hybrid
Observer Nominal DBN y(t) y(t)
System Monitoring
r(t) + r(t)
Fault Detection
Temporal Causal Graph Hybrid bond
graph Algorithm 1: Online tracking algorithm Input: Number of particles, N; a initial DBN model
D {X , Z ,U ,Y} For each particle i, from 1 to N do
Sample X 0i from the prior probability distribution Assign Y0i as the measurement at time step 0 End For For each time-step t>0 do
If the controlled or autonomous mode change oc
Regenerate a DBN model D' from TCG in new system configuration
End If Prediction: Sample each particle in DBN model Weighting: Compute the weight considering the observation
Resampling: Normalize the weighted samples, and resample N new samples
Calculate the estimated continuous state variables Xt and Yt at time step t End For
The fault detection module compares the measured
variable y(t) from sensors with its estimate, yˆ (t) computed
by the hybrid observer at each time-step t. Ideally, any
inconsistency r(t) y(t) yˆ(t) implies a fault, and
invokes the qualitative fault isolation module. However, to
account for noise in the measurements and modeling errors,
statistical techniques are employed to determine significant
deviations from zero for the residual. In this paper, a Z-test,
which uses a sliding window to compute the residual mean
and variance, is adopted by reliable fault detection with low
false-alarm rates [
The QFI scheme is based on qualitative fault signature
(QFS) method, which was proposed by Mosterman and
Biswas [
Definition 6 (Qualitative Fault Signature): Given a fault f and measurement m, the qualitative fault signature can be denoted by QFS( f , m) {(s1s2 , s3 ), s1, s2 (, , 0,*), s3 (N , Z , X ,*)} ; where and 0 indicate an increase, decrease, and no change for residual magnitude or slope. N, Z and X imply zero to nonzero, nonzero to zero, and no discrete change behavior in the measurement from the estimate. * denotes the ambiguity in the signatures. Table 2 Selected fault signature for hybrid two-tank system for the mode when all the valves are open and liquid level in both tanks are above the height of the autonomous pipe Fault C a
1 C i
1 Ra 1 Ri
1 v1.off v2.off f6 f6
f6 (, X ) (0, X ) (, X ) (0, X ) (0, X ) (, X ) (0, ) (0, )
f9 (, X ) (0, X ) (0, X ) (0, X ) (0, X ) (0, X ) (00, X ) (00, X )
f14 (0, X ) (0, X ) (0, X ) (0, X ) (0, X ) (0, X ) (00, X ) (00, X )
When measurement deviations are detected, the symbol
generator module in QFI scheme is triggered to calculate
the QFS for the current mode of operation. However, since
the fault may have occurred but not detected in an earlier
mode, the fault hypothesis generation module rolls back to
find the previous modes in which fault may have occurred,
and generate fault hypothesis set F {( fi ,i , qi )} , where
i denotes the deviation of fault parameter value, and qi
indicates the possible modes. The progressive monitoring
module applies the forward propagation algorithm to
continually refine the fault candidates in the fault hypotheses
set. For hybrid systems, the progressive monitoring also has
to include forward propagation through mode changes,
which makes the tracking algorithm much more complex.
Narasimhan and Biswas [
The selected qualitative fault signature for hybrid two-tank system in particular mode is shown in Table 2. For incipient parametric faults, the QFS is shown as (0 , s3 ) , where is the first nonzero symbol in the QFS for the abrupt faults with same system parameter. Sensor faults only affect the measurement provided by the sensor, so other measurements that are not affected are denoted by 00. 3.3
Quant-FII scheme will be activated when any of the following conditions are fulfilled: 1) All the measurements have deviated from nominal, so the remaining fault candidates cannot be refined further only by the Qual-FI scheme; 2) The number of fault candidates has been reduced to a predefined value k; 3) A predefined time l has elapsed. We restrict the length of Quant-FII scheme as a pre-specified value, and assume that no autonomous change occurs during this period.
The steps describing this scheme are illustrated as follows: First, a separate DBN faulty model will be constructed for each remaining fault candidate in the hypothesis set. Second, we combine each switched DBN faulty model with PF method to estimate the system behavior. Similar to fault detection scheme, a Z-test method is employed to detect the inconsistency between estimated values from PF and measurements. Ideally, only the correct true fault model will converge to the observed values of the measurements. Once the deviation is determined, the corresponding fault candidate will be dropped. This scheme runs in parallel with the qualitative fault isolation scheme, and if a controlled mode change occurs, these two schemes need to reload the DBN model for new system mode. This is the big difference between continuous systems and hybrid systems.
If the fault hypothesis cannot be refined further or only a single parametric or sensor fault candidate is left, fault identification scheme will be activated to identify the abrupt or incipient parametric fault in the same model and estimate the fault parameter value. We can use the PF result of the fault parameter to calculate the abrupt parameter fault magnitude pa , incipient parameter fault slope ip or sensor fault bias term b .
m 4
To demonstrate the effectiveness of our approach, we apply it to the hybrid two-tank system in Figure 1. In this plant, the incipient parametric faults are modeled as gradual decrease in tank capacity and gradual increases in pipe resistances and denoted as C1i ,C2i , R1i , R2i and R12i respectively. The abrupt parameter faults are modeled as step decrease in tank capacity and step increases in pipe resistances and represented as C1a ,C2a , R1a , R2a and R12a respectively. We consider discrete faults in each controlled valves including the valve gets stuck and valve changes mode without a command. For sensor faults, bias faults causing abrupt changes in the measurement are considered.
We assume that the tanks are initially empty, and start to fill in at a constant rate. The initial configuration of the system is all the valves are set to open. We will denote the system mode as qijkm , where i, j and k are the modes of valve1, valve2 and valve3 respectively, and m is the mode of autonomous pipe R12 . More specifically, the mode of valves includes S1 : on, S2 : off , S3 : Stuck _ on and S4 : Stuck _ off . Therefore, the initial mode of the system is q1113 . At time step t=6.7s, the liquid level in tank 1 reaches the height of autonomous pipe R12 . The system mode transitions from q1113 into q1111 . Now the autonomous pipe R12 acts as an outflow pipe for the tank 1 but as flow source for the tank 2. As system evolves, the liquid level in tank 2 will also reach the autonomous pipe at time step t=53s. After that, system mode changes into q1114 . The experiments have been run for a total of 400s using a sampling period 0.1s. Gaussian white noise with zero mean and variances 0.018 is added to measurements.
In this first experiment, we present our diagnosis approach for a fault scenario. A 10% rate of increase in pipe R1 is injected as the incipient fault at time step t = 60s.
Figure 7 Observed and estimated result for nominal DBN model
We only consider the measurement M 3 and M 2 for the flow f9 through the autonomous pipe R12 and the output flow f14 from tank 2. At time step t=82s, the fault detection scheme detects an increase in the flow f9 , resulting in the initial fault hypothesis F {(C1a , q1114 ), (C1i , q1114 ), (R1a , q1114 ), (R1i , q1114 ), (v2.off , q1414 ), ( f9 , q1114 )} . At 88.4s, the flow f14 shows an increase above nominal (+). A possible autonomous transition is executed for the current inconsistent candidate ( f9 , q1114 ) . After that, the first order change of flow f9 is determined to decrease and increase in mode q1414 and q1114 at time steps t=94.8s and 97.7s, respectively, and finally the possible fault hypotheses are F {(C1i , q1114 ), (R1a , q1114 ), (R1i , q1114 )} . According to the fault signatures in mode q1114 , these three candidates cannot be refined further using observed deviations. Figure 7 represents observed and estimated result generated by the nominal DBN model. The QFII scheme is initiated at time step t=72s, and two separate DBN fault model using C i and Ra/i are con1 1 structed. As more measurements are obtained, the Z-tests indicate a deviation in the measurement estimates obtained by the fault model C1i , and the estimation generated by possible true fault model Ra/i is consistent with mea1 surement. The quantitative fault identification part estimates the value of R1 , and determines that R1 indeed has an incipient fault. While the actual fault slope is 0.1, the estimated slope is 0.1009. The estimation using two faulty models are shown in Figure 8 and Figure 9 respectively, and the plot for estimated value for R1 is presented in Figure 10. In this subsection, we investigate an unexpected switch fault: valve 2 closes without a command at time step t=80s. We only consider the flow f6 and flow f9 in this experiment.
Figure 11 shows the observed and estimated outputs using nominal DBN model. The fault is detected at time step t=80.1s, and the symbol generator reports a decrease in flow f6 . QFI scheme generates the fault hypothesis set F {(R1a , q1114 ), (R1i , q1114 ), (v1.off , q4114 ), (v2.off , q1414 ), ( f6 , q1114 )} . At time step t=80.6s, the symbol generator determines the flow f6 to Z in mode q1114 and q4114 , because of estimated flow fˆ6 0 and the observation f6 0 . This symbol eliminates all the parametric faults and discrete fault v1.off from current trajectory. At 83.6s, the flow f10 shows a positive deviation (+), so the fault candidate (v2.off , q1414 ) is correctly isolated. In this experiment, the real fault candidate is isolated by the QFI scheme, so the QFII scheme is not invoked.
We also perform several additional experiments with different fault types, fault magnitude, noise level and fault occurrence time, and obtain satisfactory results. For lack of space, we do not discuss these results in detail. 5
In this paper, we presented an integrated approach for online monitoring and diagnosis of incipient or abrupt parametric faults, discrete faults and sensor faults in hybrid linear systems. First of all, we adopt the HBGs to model the system, and construct the diagnosis models, i.e., the TCGs and the DBN models from the HBG model in different modes. A PF method based on the switched DBN model is employed for online monitoring of the system dynamic behavior. Once the discrete finite automaton in the HBGs detects the controlled or autonomous mode changes, HBGs will regenerate the TCGs and DBN model in new mode. These modeling approaches guarantee that the hybrid systems can be tracked correctly. Then, we demonstrate that we can accommodate discrete faults and sensor fault models into the TCG and DBN models that represent dynamic system behavior. As a result, our model-based approach can diagnose parametric, discrete and sensor faults within the same modeling and tracking framework. Finally, QFI scheme using Hybrid TRANSCEND approach and QFII scheme by means of switched DBN-based PF approach are combined together into a common framework, which provides more discriminatory power and less computational complexity.
This work builds on approaches presented in
[
This method has been successfully applied to a hybrid
two-tank system, and experimental results demonstrate the
effectiveness of the approach. However, since the
application in this paper is only a relatively simple hybrid linear
system, our future work will scale up this methodology for
more realistic linear and nonlinear hybrid systems.
Moreover, distributed diagnostics techniques can efficiently
decrease the computational complexity for complex real
systems, so this is also a research direction in future [
This research was supported by China Scholarship Council under contract number 201306020068. The work was performed in Prof. Biswas’ lab at the Institute for Software Integrated Systems (ISIS), Vanderbilt University, USA