Catching Critical Transition in Engineered Systems

A variety of engineered systems can encounter critical transitions where the system suddenly shifts from one stable state to another at a critical threshold. 'e critical transition has aroused vital concerns for its potentially disastrous impacts. We validate an often taken-for-granted hypothesis that the failure of engineered systems can be attributed to the respective critical transitions and show how early warning signals are closely associated with critical transitions. We demonstrate that it is feasible to use early warning signals to predict system failures. Our findings open a new path to forecast failures of engineered systems with a generic method and provide supporting evidence for the universal existence of critical transition in dynamical systems at multiple scales.


Introduction
Critical transition, i.e., the dramatic shift of states in complex dynamical systems, is increasingly recognized as a vital concern for its potentially disastrous impacts in many socialecological systems [1,2]. Despite the difficulty in predicting such transitions, recent years witnessed significant progresses in identifying early warning signals [3][4][5][6][7][8][9] by analyzing the observed time series [10][11][12]. For example, critical slowing down (CSD) [10] as well as other indicators like the changing skewness [13] have been found to be present in a wide spectrum of complex systems ranging from ecosystems [14][15][16], neurons in mammalian cortices [17], and climates [18] to financial markets [19] when approaching the tipping point. A detailed review of past research dedicated to CSD indicators and associated tipping points and promising research directions for future research has been conducted by Nazarimehr et.al. [20]. Besides, some researchers have explored the approaches of predicting the tipping points [21] and discussed the new scientific understanding of social problems through a complex systems perspective, such as crowd disasters, crime, terrorism, wars, and epidemics [22]. It is thus intriguing to explore the viability of exploiting the early warning signals to detect the tendency toward failure in engineered systems, which also exhibits intricate stochastic behavior under complex working conditions. However, it is still an open problem to establish the universality of the above early warning indicators in both natural and engineered systems. Here we demonstrate the feasibility of using early warning signals to predict system failures in a variety of engineered systems from various domains. We show how early warning signals, i.e., CSD and deviating skewness, which can be extracted by performing a Fluctuation Mining (FM) procedure to capture the inherent stochastic behaviors, are closely associated with critical transitions at the points of systems approaching failures. Our results suggest that early warning signals are prevalent in dynamic systems. We anticipate that finding early warning signals through critical transition analysis will open a new path to predict the failures of engineered systems by offering a generic method.
Complex dynamical systems can undergo critical transitions when arriving at a specific threshold [23,24]. e existence of such tipping points is the result of the inherent nonlinearity and stochasticity in the dynamical systems [25]. Due to the global concern on the sustainability of the complex social-ecological systems, intensive research efforts have been dedicated to identifying early warning signals for critical transitions [26]. It is increasingly recognized that a wide range of systems exhibit CSD and deviating skewness, which are caused by the fact that it is harder for a system approaching the critical transition to return to the equilibrium upon perturbations. Various studies suggest that these early warning signals can be recognized in the intrinsic stochastic fluctuations of many social-ecological systems and these signals provide an essential solution to predict potential natural disasters [2].
Despite their seemingly vast difference from socialecological systems, engineered systems such as jet engines, locomotives, and bearings are also dynamical systems demonstrating nonlinear behaviors and stochastic fluctuations under the complex interplay of internal and environmental factors [27,28]. e failure of engineered systems may incur disastrous impacts since the operation of modern society hinges on their smooth functioning. One example is the tragic high-speed train accident [29] happening in Eschede (Germany) in 1998, which was caused by the fatigue-induced crack of a wheel. Intuitively, the process from the suddenly accelerated worsening of cracking to the final derailment bears significant similarity to the critical transition in natural systems. Here, we explore the universality of early warning signals in engineered systems.
Discovering potential failures in engineered systems is extremely challenging. Although these systems are designed to offer well-defined behaviors, the complex interactions within their components and with the working environment tend to be too complex for exhaustive analytical modeling [30]. e problem can be more involved if a positive loop of feedback exists in the system. Even if the system dynamics approaching failure can be derived through physics-offailure or reliability analysis [31], the working conditions and the environment cannot be fully reproduced for a complete treatment of various idiosyncrasies in the system. e globalization of the manufacturing process also hinders a comprehensive understanding of the domain knowledge. Precise prediction of failures in engineered systems in a timely fashion is still a worldwide challenge. It is thus appealing to devise effective generic early warning indicators that require less or even no domain knowledge.

Methods and Results
Hypothesizing that the critical transition is strongly correlated with the failure of engineered systems, we examine four datasets from different engineered systems, that is, an airbrake system of diesel locomotive (a pneumatic system), a turbofan engine (a mechatronic system), an IGBT (a power electronic system), and a rolling bearing (a mechanic system), to investigate whether early warning signals can be identified. e airbrake dataset is gathered from a commercially operating locomotive, while the remaining three datasets are public prognostics benchmarks available by NASA Prognostics Center of Excellence (PCoE) [32]. In the NASA PCoE Datasets, the turbofan engine data are generated in an industry-strength simulation environment [33], whereas the IGBT and bearing datasets are produced through aging experiments [34,35]. All the datasets are univariate or multivariate time series collected by sensors during the process from normal operating to running into failure. With respect to data availability, the airbrake dataset is gathered from a commercially operating locomotive, while the remaining three are public prognostics benchmarks made available by NASA Prognostics Center of Excellence (PCoE) at the website: https://ti.arc.nasa.gov/tech/dash/ groups/pcoe/prognostic-data-repository/.
With proper preprocessing, we derive the variance, autocorrelation, and skewness characteristics of the time series. Although they are from considerably different domains, all the four systems unanimously exhibit the typical symptoms of CSD and deviating skewness ( Figure 1 and Supplementary Figure S1). Such an observation seems to suggest that early warning indicators can serve as the precursor of system failure.
e dynamic behaviors of an engineered system are generally reflected by one or more observable signals, which can be discretely sampled into time series. e values in a time series are determined by both an operational trend (i.e., main trend) and stochastic fluctuations. In the case of the complex dynamic social-ecological system, a detrending [11] procedure is often necessary to extract the stochastic components for analyzing early warning signals [1]. As rather obvious variations can be observed in the curves of variance, autocorrelation, and skewness ( Figure 1), we devise a Fluctuation Mining (FM) framework to ascertain stochastic fluctuations that is not directly associated with the designed operation of engineered systems (Figures 2(a)-2(c)). e framework consists of a data cleansing step that removes the failure-insensitive components in the observed signals and a detrending step that eliminates the dominant trend, i.e., the overall tendency of change, in the time series. For instance, the application of the FM on the IGBT dataset successfully reveals a much more obvious propensity of increasing variance and autocorrelation as well as deviating skewness (Figures 2(f )-2(h) and Supplementary Figure S2). In addition, although early warning signals can be consistently identified in all the four engineering systems from different domains, we still need to verify that the observations with a 50% window size are not a coincidence. us, the sensitive test is implemented that analyzed the indicators with different window sizes ( Figure S3). e findings so far have validated the existence of early warning signals in engineered systems with forthcoming failures. e correspondence between the critical transition and the system failure, however, is still an open problem. We endeavored Student's t-test [36], the Gaussian kernel-based probability density test [36], the Akaike Information Criterion-(AIC-) based ARIMA model analysis [37], and the phase space analysis [38] on the four engineered systems before and after they reach their failures (Figure 3 and Supplementary Figure S4). e analysis results consistently suggest that the systems undergo an abrupt change of state. For example, the bearing system remains stable under normal operation and diverges to an unstable state upon failure (Figures 3(b)-3(e)), the turbofan engine witnesses a complete change of the underlying dynamical function (Figure 3(g)), and in the case of the IGBT system, the normal functioning stage and failure stage reveal significant varying limit cycles in the phase portrait (Figures 3(i) and 3(k)). e above analysis shows strong evidence on the equivalence  . e times series of the collector current mainly be two components: the switch-on current signal (which is failure-insensitive) at around 8 A and the switch-off current signal (which is failure-sensitive) exponentially converging to zero (d(1)). We show how to derive a more evident tendency in the early warning signals by focusing on failure-sensitive components in Figure 2.   [36]. It shows that the left section has only one stable equilibrium point, while the right section has two, of which the one at the negative axis potentially drives the system to unstable (i.e., the corresponding system failure). h(1-3) and j(1-3), two snippets of the three selected signals of the IGBTsystem at the normal functioning stage and the close-to-failure stage. (i, k) e phase portraits of the signals in h(1-3) and j (1)(2)(3).
e phase portrait of the normal functioning stage has a clear limit cycle serving as the basin of attractor, while the portrait of the close-tofailure stage reveals divergent behaviors indicating the occurrence of critical transition. More analysis of critical transition can be found in the supplemental material. e above analysis of the engineered systems suggests that the systems undergo certain critical transitions when running into system failures.
between the critical transition and the system failure in the engineered systems. Such correspondence further demonstrates the applicability of the early warning signals in predicting incoming system failures.
Our findings are the first to recognize the critical transition as a common property in engineered systems, and early warning signals can be effectively used in predicting failures in such systems. e discoveries have implications in Considering the nonlinear phenomenon like hysteresis, the above hypothesis is likely to hold. (d) e proposed three phases, a nominally designed working phase that is equivalent to the basin of attractions, a predefined safety-oriented phase when the system initializes the self-protection after certain failures happened, and an unstable failure phase exhibiting out-of-control performance, in engineered systems with dynamical behaviors. e catastrophic bifurcation, i.e., critical transition, occurs when the system departs from the basin of attraction and loses its resilience as the system fails. 8 Mathematical Problems in Engineering several aspects. First, we offer evidence that the phenomenon of early warning signals is prevalent in both natural and engineered systems. It is probably not surprising that there are no absolute boundaries between the two as the complex dynamics are fundamentally universal. Second, we validate an often taken-for-granted hypothesis that the failure of engineered systems can be attributed to the respective critical transitions. ird, it is feasible to catch disastrous bifurcations with early warning indicators, which can be identified by investigating the fluctuations in an engineered system. Such a discovery paves the way toward a generic methodology to predict and prevent potential disasters caused by the failure of engineered systems.

Discussion
Critical transition, as shown to have a strong correlation with system failure, is likely to be the aftermath of a catastrophic bifurcation [39]. Taking the bearing system and the IGBT system as examples, the sudden shift in the fluctuation signals exhibits the features of bifurcations (Figures 4(a) and 4(b)). In fact, nonlinear phenomena like hysteresis, which are closely related to bifurcations, can be found in many engineered systems, e.g., hysteretic actuators in control systems, positive feedback-based circuits in electronics, airstream in aerodynamics, elastic deformation of materials, and magnetic and electrical hysteresis in ferromagnetic materials [40]. Hence, we anticipate that the catastrophic bifurcation, which can be triggered by a small perturbation of a latent parameter, to be a common phenomenon in engineered systems approaching failures (Figure 4(c)). We propose that the engineered systems with dynamical behaviors can be abstracted to have three phases: a nominal design phase that is equivalent to the basin of attraction, a predefined safety-oriented phase when the system initializes the self-protection after certain faults happened, and an unstable failure phase exhibiting out-of-control performance ( Figure 4(d)). e critical transition occurs when the system departs from the basin of attraction and loses its resilience [41] as the system fails.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.