A HMM-Based Fault Detection Method for Piecewise Stationary ...

A HMM-Based Fault Detection Method for Piecewise Stationary Industrial Processes Stefan Windmann, Florian Jungbluth, Oliver Niggemann Fraunhofer IOSB-INA Email: {stefan.windmann, florian.jungbluth, oliver.niggemann}@iosb-ina.fraunhofer.de

Abstract—In this paper, fault detection in piecewise stationary industrial processes is investigated. Such processes can be modeled as sequences of distinct system modes in which the respective expectation values and variances of process variables do not change. In particular, piecewise stationary processes with autonomous transitions between system modes are considered in this work, i.e. processes without observable trigger events such as on/off signals. A Hidden Markov Model (HMM) is employed as underlying system model for such processes. System modes are modeled as hidden state variables with given transition probabilities. Continuous process variables are assumed to be Gaussian distributed with constant second order statistics in each system mode. A novel HMM-based fault detection method is proposed which incorporates the Viterbi algorithm into a fault detection method for hybrid industrial processes. Experimental results for the proposed fault detection method are presented for a module of the Lemgo Smart Factory.

symptoms. Model-based approaches do not have this problem since system models take all inputs and compute the outputs, i.e. they work in direction of the physical causality. Since general, phenomenological approaches are chosen for local compact devices while model-based approaches are chosen for complex, distributed plants.

I. I NTRODUCTION Reliable fault detection allows costs and risks to be reduced through the early detection of faults and problems in the process and by preventing component failures and, in extreme cases, a production stop of the entire plant. Typical goals of such approaches for complex and distributed industrial automation systems are the detection of faults, suboptimal energy consumption, or wear (see e.g. [1], [2], [3], [4]). Generally speaking two classes of algorithmic approaches exist for the detection of anomalous situations [5]: • Phenomenological Approach: Here, the systems output including its sensors, data and its energy consumption is directly classified as correct or anomalous (see e.g. [6], [7]). • Model-based Approach: Model-based approaches employ a model to simulate the normal behavior of a plant. For this, the simulation model needs all input of the plant, e.g. product information, plant configuration, plant status, etc. If the actual measurements vary significantly from the simulation results, the behavior is classified as anomalous (see e.g. [2], [3], [4]). While phenomenological approaches are often more straightforward and do not require a system model, they have one major inherent drawback: They must deduce against the direction of causality since they deduce from measurements (i.e. symptoms) to faults. For complex distributed systems with many interdependencies between components and complex causalities, this is a hard task because a high number of classification rules is needed to discriminate between all possible combinations of

Fig. 1.

Model-based fault detection.

The principle of model-based fault detection consists of comparing the actual process behavior with the behavior as predicted from the process models (see Fig. 1). The purpose is to detect deviations in system behavior from the normal state (for example too high or too low energy consumption of a conveying system). In doing so, significant outliers are detected and can be displayed in a human machine interface. The process models can be learned from process data collected about the plant and its components in normal, fault-free operation, as described in section II-B. This allows the detection of, on one hand, discrete events that are incompatible with the learned process models [3], and, on the other, deviations of the continuous behavior from the predicted process behavior. A lot of research has been conducted with respect to fault detection in industrial processes. Clustering-based fault detection methods create groups of strongly related objects and objects which do not strongly belong to any cluster [8]. Neural networks and regression-based methods have been used to approximate the functional dependency between continuous process variables and time [9]. Sensor signals are predicted according to this functional dependency and significant deviations of predicted signal values from the observations are reported as faults. Stochastic approaches to fault detection are predominantly based on building a probability distribution model and considering how likely objects are under that model [10]. Statistical

tests are used to assess the likelihood of faults (see e.g. [4], [11]). In most of the stochastic approaches, state variables are employed for modeling the temporal transitions of hidden process variables which are related to the measurements with a measurement model. Common approaches are Kalman filters (e.g. [12], [13], [14]) and particle filters [15]. Reference [4] is an example in which sequences of stochastic state space models are employed to model the energy consumption in the modes of hybrid automata. The outlined methods approximate the involved probability distributions by second order statistics or particles. Kalman filter and particle filter approaches are based on state space models for continuos process variables. The main contribution of this paper is a novel fault detection method for piecewise stationary industrial processes which is based on a Hidden Markov Model. In this approach, discrete hidden state variables are employed to model system modes (states s0 and s1 in Fig 1). Continuos process behaviour is assumed to be a function of the hidden state variables. Thereby, a Markov Model is established for the temporal dependencis between system modes instead of the state space model for continuous process variables in Kalman and particle filters. The state space models in Kalman or particle filters are based on the immediate relations between subsequent samples and are for this reason appropriate to model short-term dependencies in the continuous process behavior. In contrast, HMMs are suitable to consider long-term dependencies in the continuous process behavior because the system modes are persistent for some time instances. HMMs were first applied to speech recognition technology [16] and are still the core technology of many commercial speech recognition systems (see e.g. [17]). Further application fields are cryptanalysis [18] and bioinformatics [19]. HMMs are strongly related to the hybrid automata investigated in [5] and [20] which have been used for fault detection in hybrid industrial systems. However, the main difference is that hybrid automata are based on observable discrete events which trigger system mode transitions (e.g. between system mode s0 and s1 in Fig 1). In contrast, the proposed HMM is more suitable for systems where discrete trigger events are not observable so that only the impact on continuous process variables is observable. In general, different model learning and fault detection methods are required for hybrid automata and HMMs. However, a fault detection method for hybrid automata forms the basis of the proposed HMM-based fault detection method [21].

II. M ODEL - BASED FAULT D ETECTION The probabilistic framework for model-based fault detection [4] is shown in Fig. 2.

Fig. 2.

Probabilistic framework for model-based fault detection.

Measurements x(k) at discrete time instances k are predicted according to a stochastic process model with input u(k). A common approach to fault detection is the evaluation of the residuals r(k) = x(k) − x ¯(k) between measurements x(k) and predictions x ¯(k). Methods such as log-likelihood ratios, generalized log-likelihood ratios, sequential probability ratio tests, or cumulative sum test statistics are used in literature to detect changes in the residuums (see [22] for an overview). A related approach is the evaluation of x(k) with respect to the conditional probability density p(x(k)|x(0) . . . x(k − 1)) (see e.g. [4], [11]). Observations are assumed to be improbable once measurement x(k) at discrete time k drops out of a given confidence interval. In this paper, this approach is slightly extended by evaluation of p(x(k−l+1) . . . x(k)|x(0) . . . x(k−l)) instead of p(x(k)|x(0) . . . x(k − 1)). In doing so, unlikely observation sequences of length l are detected instead of single unlikely observations. Evaluation of p(x(k − l + 1) . . . x(k)|x(0) . . . x(k − l)) requires an adequate model of the process behavior which essentially determines in which way the probability density p(x(k − l + 1) . . . x(k)|x(0) . . . x(k − l)) is obtained. In this paper, a Hidden Markov Model is employed for this purpose (see section II-A). Parameter estimation and fault detection with the HMM are considered in section II-B and II-C, respectively. A. Hidden Markov Model In this paper, physical systems with piecewise stationary process behavior are considered. A convenient representation for such systems is the Hidden Markov Model (HMM) [23].

The remaining part of the paper is structured as follows: Section II-C outlines the probabilistic framework for fault detection in piecewise stationary industrial processes. The underlying HMM and parameter estimation for the HMM are considered in section II-A and section II-B, respectively. The novel fault detection scheme is introduced in section II-C. The proposed method is evaluated in a given application scenario (see section III). Section IV gives a conclusion. Fig. 3.

Hidden Markov Model.

Pn−1

k=0 γi (k)x(k) P n−1 γi (k) s Pk=0 n−1 ¯ i )2 k=0 γi (k)(x(k) − µ σ ¯i = . Pn−1 k=0 γi (k)

The HMM depicted in Fig. 3 is a 4-tupel λ = (S, π, T, Θ), with [23]: • • • •

A finite set of states S = {s0 , s1 . . . sm−1 }. Prior probabilities π = {π0 , π1 . . . πm−1 }. Transition matrix T = {α00 , α01 . . . αm−1,m−1 } Observation models Θ = {Θ0 , Θ1 . . . Θm−1 }.

System modes are modeled as finite set of states S = {s0 , s1 . . . sm−1 }. At each time instance k, the system is assumed to be in each system Pm−1mode si , i = 0 . . . m − 1, with probability P (si (k)), i=0 P (si (k)) = 1. Prior probability distribution at time instance k = 0 is denoted as π. Transition matrix T = {αij |i = 0 . . . m − 1, j = 0 . . . m − 1}

(1)

comprises the transition probabilities between system modes si , i = 0 . . . m − 1, and sj , j = 0 . . . m − 1. Furthermore, in contrast to [24] where an HMM is applied as system model for a discrete event system, continuous observation models Θi = (µi , σi ) are assumed in each system mode si , i = 0 . . . m − 1. In this paper, we assume a stationary Gaussian distribution pΘi (x(k)) = bi (x(k)) = N (µi , σi )

(2)

in each system mode si , i = 0 . . . m − 1 with expectation value µi and standard deviation σi , i.e. observation models Θi = (µi , σi ), respectively. B. Parameter Estimation

µ ¯i =

(11)

¯ has converged to a The recursion stops, as soon as P (X|λ) ¯ maximum, i.e. for λ ≈ λ. Inputs: Initial Hidden Markov Model λ = (S, π, T, Θ) ¯ = (S, ¯ π ¯ Outputs: Improved Hidden Markov Model λ ¯ , T¯, Θ) (1) do (2) Estimation step (3) αi (0) = πi bi (x(0)), i = 0 . . . m − 1 (4) for k = 0 . . . n − 2 //Forward procedure Pm−1 (5) αj (k + 1) = bj (x(k + 1)) i=0 αi (k)αij , j = 0 . . . m − 1 (6) βi (n − 1) = 1, i = 0 . . . m − 1 (7) for k = n − P2m. . . 0 //Backward procedure (8) βi (k) = j=0 βj (k + 1)αij bj (x(k + 1)), i = 0 . . . m − 1 αi (k)βi (k) (9) γi (k) = Pm−1 αj (k)βj (k)

j=0

αi (k)αij βj (k+1)

(10)

ξij (k) = Pm−1

(11) (12)

Maximization step π ¯i = γiP (0)

(13)

α ¯ ij = Pk=0 n−1

i=0

n−1

αi (k)βi (k)

ξij (k)

(14)

γi (k) k=0 Pn−1 γi (k)x(k) k=0 µ ¯i = P n−1 s Pk=0 γi (k)

(15)

σ ¯i =

n−1

γ (k)(x(k)−µ ¯ i )2

i Pn−1

k=0

k=0

Parameter estimation for the HMM is accomplished by application of the Baum-Welch algorithm (algorithm 1 in Fig. 4) [23]. The Baum-Welch algorithm finds the optimal parameter vector ¯ = maxλ P (X|λ) λ (3)

(10)

γi (k)

¯ (16) while P (X|λ) < P (X|λ)) Fig. 4.

Algorithm 1 Baum-Welch algorithm [23]

C. Fault detection for the given observations X = (x(0), . . . x(k − 1)). This is accomplished by recursive estimation of P (X|λ) and maximization of λ (see Fig. 4). The E-step involves the computation of probabilities γi (k) = P (s(k) = i|X, λ)

(4)

ξij (k) = P (s(k) = i, s(k + 1) = j|X, λ).

(5)

Both probabilites are obtained as function of forward variables αi (k) = p(x(0), . . . , x(k), s(k) = i|λ)

(6)

and backward variables βi (k) = p(x(k + 1), . . . , x(n − 1)|s(k) = i, λ)

ϑi (k) = maxs(0)...s(k), {p(x(0) . . . x(k); s(0) . . . s(k)|γ)} , s(k)=i

(7)

¯ is obtained which are recursively computed in lines (2) - (7). λ from γi (k) and ξij (k) as follows [23]: π ¯i = λi (0) Pn−2 ξij (k) α ¯ ij = Pk=0 n−2 k=0 γi (k)

The proposed fault detection method is based on ANODA [21], a fault detection method for deterministic hybrid automata. In contrast to ANODA, in the application case of this paper no discrete events are observable which determine the sequence of system modes. The proposed fault detection method shown in Fig. 5 obtains observations of continuous process variables x(k) for time instances k, k = 0, . . . n−1. System modes are implicitly determined by application of the Viterbi algorithm (see [23]). The Viterbi algorithm computes probabilities

(8) (9)

(12) i.e. the probabilities of the most probable state sequences responsible for the first k observations x(k) with i as final state. Computation of ϑi (k) is accomplished by the following recursion [23]: ϑi (0) = πi bi (x(0)) ϑi (k) = maxj=0...m−1 {αji ϑj (k − 1)} .

(13)

Furthermore, Viterbi approximation p(x(0) . . . x(k)) ≈ maxs(0)...s(k) p(x(0) . . . x(k); s(0) . . . s(k)|γ).

(14)

is applied, i.e. the assumption that the total likelihood p(x(0) . . . x(k)) can be approximated as the total likelihood of the single most likely state sequence. In the proposed method, fault detection for a frame of length l is accomplished by evaluation of the probability density p(x(k − l + 1) . . . x(k)|x(0) . . . x(k − l)) p(x(0) . . . x(k)) = p(x(0) . . . x(k − l)) maxs(0)...s(k) {p(x(0) . . . x(k); s(0) . . . s(k)|γ)} ≈ maxs(0)...s(k−l) {p(x(0) . . . x(k − l); s(0) . . . s(k − l)|γ)} maxi=0...m−1 {ϑi (k)} = . (15) maxj=0...m−1 {ϑj (k − l)} In (15), Bayes formula, Viterbi approximation (14) and definition (12) of ϑi (k) have been applied, respectively. For numerical reasons, ϑi (k) is replaced in this work by Λi (k) = log(ϑi (k)).

(16)

In doing so, (13) becomes the following recursion (lines (1)(3) of Algorithm 2 in Fig. 5): Λi (0) = Πi + Bi (x(k)) Λi (k) = Bi (x(k)) + maxj=0...m−1 (Aji + Λj (k − 1)) (17) with Πi = log(πi )

(18)

Bi (x(k)) = log(bi (x(k))) 1 1 = − log(2πσi ) − 2 2 Aij = log(αij ).

Inputs: (1) Hidden Markov Model λ = (S, π, T, Θ) (obtained with Algorithm 1 from training data) (2) Observations o(k) = (k, y(k)) (periodically obtained) Output: Detected anomalies f1 , f2 (if existing, else “OK”) (1) Λi (0) = Πi + Bi (x(0)) (2) for k = 0, . . . n − 1 do (3) Λi (k) = Bi (x(k)) + maxj=0...m−1 (Aji + Λj (k − 1)) (4) Λ(1) (k) = maxi Λi (k) − maxj Λj (k − 1). (5) if Λ(1) (k) < T (1) : (6) Report anomaly f1 . (7) end if (8) Λ(5) (k) = maxi Λi (k) − maxj Λj (k − 5) (9) if Λ(5) (k) < T (5) : (10) Report anomaly f2 . (11) end if (12) end for Fig. 5.

Algorithm 2 HMM-based fault detection

product packing system, a bearing robot, and a lid robot. These modules comprise a number of components including: distributed PLCs, communication buses, a popping machine, several conveyor belts etc. In our evaluations, we focus our attention on the popping machine depicted in Fig. 6. The popping machine of the LSF receives the raw material (corn), and activates the air heater. A fan blows the hot air toward the raw material container. Popcorn is obtained as a result of this processing. The described process represents one production cycle.

(19) 

x − µi σi

2 (20) (21)

Furthermore, the following expression for Λ(l) (k) = log(p(x(k−l+1), . . . x(k)|x(0), . . . x(k−l))) (22) is obtained from (15): Λ(l) (k) = maxi Λi (k) − maxj Λj (k − l).

(23)

For fault detection, it is evaluated whether, the likelihood Λ(l) (k) is beyond a given threshold T (l) . In the proposed fault detection method, Λ(l) (k) is evaluated for l = 1 and l = 5 and for given thresholds T (1) , T (5) (lines (4)-(11)). III. E VALUATION AND D ISCUSSION Evaluations have been conducted on the Lemgo Smart Factory (LSF). This evaluation platform is described in detail in [3] and [21]. The LSF represents a plant for storing, transporting, processing and packing raw material (corn). It has a modular design, i.e. it consists of eight modules, namely: a storage system, a few transportation systems, a weighting station, a bottle filling mechanism, a production facility, a

Fig. 6.

The popping machine of the Lemgo Model Factory.

For learning the behavior model of the popping machine, we used data recorded in a production cycle with training data. Data sampling rate was 1Hz. In the experiments, the active power of the popping machine was considered as output of the behavior model. The power range of the popping machine is 0 − 1150W . The model predicts this output, while the fault detection algorithm detects its discrepancies from the observed signal values (see Fig. 1). Fig. 7 shows the time diagram of the output signal, taken in one of the recorded production examples (with the mean value and the standard deviation

276.97 ± 16.51W ). The output signal can be considered as piecewise stationary with expectation values µ0 ≈ 292W for time instances k = [0, 100] and µ1 ≈ 267W for time instances k = [100, 175], and small variances σ0 , σ1 in these time segments, respectively. The peaks in time segment k = [175, 220] can be considered as sequences of system modes with larger variances. The algorithm described in section II-B was applied for parameter estimation of the HMM. The learnt HMM with states si and state transition probabilities for the given module is depicted in Fig. 8. It is worth noting that only 14 of the 72 transitions have weights αij > ,  = 0.005. This leads to almost deterministic transitions in some system modes (e.g. α02 ≈ 1).

three faults. In table I, results obtained in [4] with switching Kalman filters (SKF) for signal drops and jumps by 1%, 1.5% and 2% are used as reference. With the novel anomaly detection scheme significant improvements compared to [4] were achieved. On average, accuracy was improved from 79.29% to 85.24%. TABLE I ACCURACY FOR THE THREE FAULTS Fault f1 (2.5%) f2 (1.5%) f2 (1%) f2 (2.5%) f3 (1.5%) f3 (1%) f3

SKF 100% 100.00% 87.50% 52.50% 100.00% 75% 40%

HMM 100% 100.00% 93.33% 66.67% 100.00% 66.67% 70.00%

IV. C ONCLUSION

Fig. 7.

Energy consumption of the popping machine.

In the present work, fault detection for piecewise stationary industrial processes has been investigated. The proposed fault detection method is based on a Hidden Markov Model. Experimental results showed significant improvements for the novel fault detection method compared to existing methods where the continuous process behavior is modeled with switching linear Kalman filters, neural networks or multilinear regression. On average, accuracy was increased from 79.29% to 85.24% compared to Switching Kalman Filters which in turn outperformed neural networks and multilinear regression methods. A PPENDIX : S WITCHING K ALMAN F ILTERS

Fig. 8.

Hidden Markov Model of the popping machine.

Once the model was learned, the anomaly detection algorithm described in section II-C was executed on additional test cycles. The switching Kalman filters (SKF) introduced in [4] were used as baseline method (see appendix). In [4], SKFs were reported to make better approximations than neural networks (NNs) which in turn outperform multiple linear regression models [21] due to the high nonlinearity of the approximated signal (see Fig. 7). The fault detection was performed targeting the following faults: Zero value of the signal (f1), signal drops (f2) and signal jumps (f3). The faults f1, f2 and f3 were detected when the process (continuous) output variable was out of range. Table I shows the average accuracy of fault detection calculated over 100 runs that were performed for each of the

The application of Switching Kalman Filters for fault detection in industrial processes has been investigated in [4]. The underlying system behavior is modelled in this approach with a deterministic automaton. The automaton consists of discrete system states which determine the system behavior. Transitions between system states are triggered by discrete events, e.g. control signals such as ”‘switch on/off”’ or ”‘open/close valve”’. Model learning algorithms such as OTALA [20] can be applied to learn the automaton. The principle of the OTALA algorithm consists in creating a new state for each combination of discrete signals which occurs in the data. Transitions are inserted between states which correspond to subsequent combinations of discrete signals in the input data. The energy consumption in each particular mode of a hybrid automaton can be modelled as a sequence of linear state space models. Switches between the linear state space models account for non-linearities or switches in the energy consumption. The learning of state space models ME1 , ME2 , . . . consists of two steps (see Fig. 9): In step 1, splitting points are detected (vertical dotted lines in Fig. 9) where the state space models are assumed to switch. In this work, a time series analysis based on a moving average (MA) method [25] is employed to obtain predictions yb(k) for the measured energy consumption y(k). Splitting

Fig. 9. Learning the energy consumption in each mode of the hybrid automaton.

point candidates are detected for |b y (k) − y(k) > yS | with the given threshold yS . In order to reduce the number of splitting points, adjacent splitting point candidates are removed. More precisely, splitting point candidates with a distance less than a given parameter δS are merged. In doing so, the first and the last splitting point in a time series of adjacent candidates is preserved in each case. In step 2, linear state space models are learned for each time segment between two splitting points. The energy consumption in each time segment s is described with a linear state space model MEs (a standard random walk model): x(k) = x(k − 1) + us (k) y(k) = x(k) + vs (k)

(24)

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