Discretization of hybrid CPPS data into timed automaton using restricted Boltzmann machines

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Abstract

Cyber–Physical Production Systems (CPPSs) are hybrid systems composed of a discrete and continuous part. However, most of the applied machine learning algorithms handle the dynamics of the two parts separately and in different fashions: for the discrete part, the notion of discrete events (and their timings) is essential (e.g. when learning automata or rules), while the dynamics of the continuous part is often defined by differential equations or time-series models. Reconciling the different nature of the two is a major challenge for machine learning. One solution is to express continuous behavior in discrete terms, i.e. the explicit events are extracted. Then, at the cost of information loss caused by discretization, the overall behavior can be jointly analyzed.

This paper proposes a novel machine learning discretization approach called DENTA (Deep Network Timed Automaton) which solves the aforementioned challenges through the construction of an (overall) deterministic timed automaton from the original hybrid data. First, it hierarchically extracts new features from the continuous data using a deep network of stacked restricted Boltzmann machines (RBMs). We show that high-level RBM abstractions can further be used to automatically detect meaningful discrete events in continuous system behavior. Finally, a discrete representation of overall system behavior in the form of a timed automaton is created, which allows a joint timing analysis of the whole system. The model is verified by the anomaly detection on a synthetic and a real-world dataset and the results show clear advantages of the approach for a specific class of systems.

Introduction

A Cyber–Physical Production System (CPPS) (Monostori, 2014, Lee, 2008, Rajkumar et al., 2010, Evans and Annunziata, 2012, Muhuri et al., 2019) is defined by collaborating computational elements and physical processes. These computational components provide a set of value-discrete and value-continuous signals carrying information about hybrid system behavior.

Analyzing this data by means of machine learning leads to new services such as condition monitoring (Morgan and O’Donnell, 2018, Andonovski et al., 2018), fault detection (Bayar et al., 2015), diagnosis and root cause identification (Pulido et al., 2019, Bunte et al., 2019, Diedrich et al., 2019) and alarm management (Wang et al., 2016, Vasquez Capacho et al., 2017). In many cases the knowledge about system causalities and timings is essential: the changes of discrete signals may lead to a different behavior of continuous signals; event sequences can be classified as incorrect; timing changes hint at upcoming problems; system dependencies can be used to identify the root causes based on alarms; and known system timings outline potential system optimizations. However, the concepts of causalities and timings are closely associated with discrete events, sequences of events and timings between these events—all undefined for the continuous part of the system.

The existing approaches often solve these challenges by learning hybrid automata which use discrete variables to identify system modes and, in the second step, they learn continuous models for each mode (Niggemann et al., 2012, Maier, 2014). This approach neglects the dependencies between discrete and continuous signals within each of the modes. Other approaches treat both types of signals in one learning step, for example by using suitable neural networks (Goh et al., 2017, Eiteneuer and Niggemann, 2018). However, they never discretize the result and, therefore, fail to identify the causalities and timings.

Discretizing the resulting model, for instance by turning it into a timed automaton in this work, leads to other advantages as well. First of all, the result can be understood by a human expert more easily. Furthermore, it helps to solve the classical stability–flexibility problem of machine learning: the automata are kept stable until the states are explicitly modified, for example when error thresholds are reached or by means of user interaction.

The following research questions are addressed in this work:

(RQ i) What is a reasonable definition of a discretized system state? Here, a latent representation of a probabilistic model and the capability of predicting signals is used to derive the states.

(RQ ii) How can the information from both the discrete and continuous part be handled equally and within one algorithmic framework? The proposed model is based on a deep network of stacked restricted Boltzmann machines. The input network layer represents raw continuous observations (time series), while the top layer is a distributed binary vector representation of a high level of abstraction which is then treated jointly and equally as originally discrete observations.

(RQ iii) How can we turn the result into a timed hybrid automaton which explicitly represents states, timing and signal dependencies? The most abstract network layer defines states and produces events, and is, therefore, interpreted as an automaton. It also handles the events from discrete data, which enables what we call the holistic timing and state analysis.

(RQ iv) The resulting automaton must be verified. This work introduces an anomaly-detection-based verification approach: any rejected automaton path implies an anomaly. There is no ultimate verification procedure of automaton correctness, but anomaly detection (AD) allows the analysis of a link between model complexity and AD performance. Here, two cases are studied: a simulation and a real-world conveyor system.

The structure of the paper is the following. In Section 2, a discussion of state-of-the-art approaches is given. The DENTA is defined and theoretically analyzed in Section 3. Section 4 provides an experimental evaluation of the approach, while the conclusion and future work are given in Section 5.

Section snippets

State-of-the-art approaches

In general, there are various methods for the discretization of continuous observations (Dougherty et al., 1995, Dimitrova et al., 2010). The simplest are the equal width interval and equal frequency interval described in Appendix C. For complex multidimensional problems, such approaches might not be useful as they do not consider the dependencies between the variables and thus cannot represent the structure of the data well.

However, machine learning approaches often yield better results.

Demonstration example

Consider a vertical conveyor transporting items from position A to B and vice versa. A central controller communicates with other components in a broader system and controls the movement between these two positions. The conveyor can be immobile at the top or bottom position, or it can move upwards or downwards. Furthermore, it can carry an item or not. Let us assume that the behavior of the conveyor at this abstract level can be described using a timed automaton with eight states (Fig. 1). The

Evaluation

To verify the learned automaton, we apply it to anomaly detection, i.e. we check whether the learned automaton can be used to detect non-normal timings, non-normal signal sequences and non-normal signal values. The anomaly detection procedure is described in Section 3. To evaluate the DENTA we compare it to the several other methods in terms of anomaly detection performance. These methods are shortly described in Appendix C. Two systems are used in the analysis: a real-world conveyor system is

Conclusion

The paper introduced a novel method called DENTA for the discretization of multivariate mixed-type time series into a timed automaton representation. It enables a holistic timing analysis of the discrete and continuous part, which can be essential for anomaly detection, diagnosis or the optimization of complex hybrid systems such as Cyber–Physical Production Systems (CPPSs). The construction of the automaton is based on a deep neural network comprised of stacked restricted Boltzmann machines

CRediT authorship contribution statement

Nemanja Hranisavljevic: Conceptualization, Methodology, Software, Data curation, Validation, Investigation, Resources, Writing - original draft. Alexander Maier: Methodology, Supervision, Writing - original draft, Writing - review & editing, Project administration, Funding acquisition. Oliver Niggemann: Conceptualization, Methodology, Investigation, Supervision, Validation, Writing - original draft, Writing - review & editing, Project administration, Funding acquisition.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was developed within the Fraunhofer Cluster of Excellence “Cognitive Internet Technologies”.

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