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One of the main pillars of science and technology is determinism i.e., the possibility of prediction. This means that we try to lay the foundations of complex systems and unravel the laws of mechanics, electromagnetism and other branches of science, by offering a tool that enables predictions.

Given a mechanical system, be it a machine tool or a vehicle, one can write down the equations governing their way of functioning and potential degradation. If one knows the present condition and operation of the system, one should simply solve a bunch of equations in order to determine the future condition and operational status. Of course, solving equations is not always a simple matter and sometimes we need numerical analysis and other simplification tools, but this implies at least the principle of determinism. The present situation determines the future and therefore predictive maintenance, and prognosis of systems and components is built on the basis of this determinism.

Sensitivity to Initial Conditions

In determinism there is one missing ingredient that cannot be neglected: The initial conditions. In predictive maintenance we claim that same causes will always produce the same effects trying to identify root causes to mitigate the risk of potential undesired effects. However, we must define what we mean by the same causes and the same effects, since it is manifest that no event ever happens more than once, so that the causes and effects cannot be the same in all respects. Indeed, when we say ‘That cause produces those effects’, this is only true when small variations in the initial circumstances produce only small variations in the final state of the system. In a great many physical phenomena this condition is satisfied; but there are other cases in which a small initial variation may produce a great change in the final state of the system, as when the displacement of shafts or gears in rotating equipment after a minor adjustment produce an extremely different vibration pattern and makes the maintenance analyst totally crazy.

Prognosis and chaotic butterflies

Can we conclude that if a tiny change of direction of a parameter is sufficient to cause a variation of the final outcome then prediction is not possible? So far, the idea that some physical systems could be complicated and sensitive to small variations of the initial conditions—making predictions impossible in practice deserves the attention of the maintenance and reliability community.

Thanks to Lorenz, Poincare and others, we are aware of systems and characteristics that increase the vulnerability of our predictions and compromise our way of performing prognosis. If a single flap of a butterfly’s wing can be instrumental in generating a tornado, so all the previous and subsequent flaps of its wings introduce continuous distortions in the predictions and they are not discrete ones.

Therefore, predictability is sensitive to the uncertainty in the initial condition and without knowledge of such conditions prediction might be rubbish and well-known deterministic systems can be very unpredictable: small errors in the initial condition can grow exponentially in time. This phenomenon, now known colloquially as the butterfly effect, is clearly influencing predictive maintenance performance since our forecasts are impacted by chaos theory.

This influence creates the so-called term prognosis obstruction i.e., the prognosis horizon where our predictions are acceptable in terms on uncertainty. This concept implies that not all timelines are valid for predictability. Predictability is often quantified in terms of the growth rate of errors in the initial condition. We can compute the time needed for small errors in the initial condition to double in size. In this way, a different growth rate of initial conditions will modify prediction. Therefore, prediction of failures in times close to the time of origin will be difficult due to our ignorance about initial conditions; in short, do not try to predict the next second of a vibration if you do not really know all boundaries and initial conditions.

In the long-term the situation is similar; growth rates in the initial conditions show that for longer lead times the error growth follows a power law which systematically depends on the initial size of the error. For even longer lead times the error will be large. This unfortunately means that long term prediction make no sense and our prognosis horizon has some time range where predictive maintenance can be performed, and prognostics delivered. Again, do no try to predict a vibration in five years because an error in the prediction will kill any outcome of your algorithm.

The digital butterfly and digital transformation

The butterfly effect is a concept that small, seemingly trivial events may ultimately result in something with much larger consequences, in simple words, it may have a non-linear impact on a very complex system. Butterfly effect and chaos theory are in existence much earlier than digital transformation and the impacts of the above mentioned in digitized maintenance is even more dramatic. Digital transformation in maintenance has been a very complex process, and a fundamental aspect of Industry 4.0 requires digitization and digitalization extended throughout the industries, making all sectors instrumented, interconnected and intelligent to some extent where every player both affects and is affected by the other players in the digital ecosystem.

New maintenance models are emerging all the time in different sectors, with digital transformation as a key underlying factor. Indeed, Digital transformation means going beyond digitization so it is not simply about applying new technologies to old maintenance processes. That is why digital transformation tries to confront the annoying butterfly effect and its limitation which is "the Sword of Damocles" when forecasting failures. Is there a chance to get rid of this effect thanks to the 4th industrial revolution and smart application of Industrial AI?

In this connected and instrumented industrial world, maintenance has found new opportunities arising from unexpected connections and dependencies between assets within an industry or from different industrial sectors. Indeed, failure forecasting capabilities have increased dramatically thanks to the availability of huge amounts of disparate data being able to be fused and merged so predictions do not depend on one-dimension datasets anymore.

Confronting the digital butterfly with AI

The industries have been exposed to digital disruption during the whole industry 4.0 decade, even though they have different levels of innovation. Maintenance has not changed and the function of maintenance has been under the spotlight to be reinvented and increase performance by using all the enabling technologies such as AI, IoT,big data, machine learning etc., whether the customers are the same or different. This will make our industrial assets more robust and resilient. This maintenance transformation has combined the traditional problems and tools with other newcomers to the sector with a hybrid approach as the outcome where tradition combines high tech. This is rather unique since maintenance has not been demolished by digital tools to create a new maintenance or define a new paradigm, but has been refurbished keeping key elements and ways of working powered by new technologies and creating a new way to do business in a natural way. Indeed, unsolved issues such as prognosis obstruction are benefited from upcoming technologies.

We have seen that for decades the chaos theory states that “the flutter of a butterfly’s wing in one specific location can cause an earthquake far away”. Talking in terms of industrial systems we can conclude that as per the chaos theory, a very tiny change in one industry can cause big changes in a system level which are most likely negative, especially in terms of a lack of predictability and prognosis obstruction popping up as a limiting barrier for predictive maintenance. This means that digital ecosystems are fragile in terms of performance due to a potential digital butterfly flapping its wings, producing small change in a player and leading to disruption across other players and other industries. A single digital wing beat could make maintenance prediction meaningless and could compromise the whole maintenance strategy making a traditional company’s long-standing approach towards maintenance of no value at a time threatening their existence – a classic case is rail track in UK.

Maintenance in a highly instrumented and connected industrial sector can be a victim of chaos theory where the “butterfly effect” makes long-term prediction impossible or at least less reliable. Even the smallest perturbation to a complex system can touch off a series of events that leads to a dramatically divergent future. The inability to pin down the state of these systems precisely enough to predict how they will play out, means we live under a veil of uncertainty.

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Artificial intelligence is supporting maintenance to overcome this issue and make predictions feasible and certain, but we need to deeply review our prediction methods. The usual approach to predicting a chaotic or unknown situation is to measure its conditions at one moment as accurately as possible, use this data to calibrate a physical model, and then evolve the model forward. That is why machine learning is “a very useful and powerful approach,”. Deep learning, while being more complicated and computationally intensive than traditional machine learning algorithms, will also work well for tackling chaos. However, to confront the butterfly effect you really need to be sure that your model is big enough comprising the boundary and initial conditions of your system. If by mistake we miss in our little model a relevant part of the influencing reality on the system, we will be totally blind to perceive any variation of such reality in my model and therefore prediction will not only be inaccurate but erroneous.

Prognosis in the long term and short term are in actual fact compromised with chaos theory. The shorter it is, the touchier or more prone to the butterfly effect a system is, with similar states departing more rapidly for disparate futures. The longer it is, a growing error makes the prediction totally uncertain and erroneous. Chaotic systems are everywhere in industry, and can go haywire relatively quickly. Yet strangely, chaos itself is hard to pin down. It is controversial when people start saying something is chaotic and more so when we talk about chaos in our failure forecasting, but it grabs people’s attention while having no agreed-upon mathematical definition or necessary and sufficient conditions. It is not easy; in some cases, tuning a single parameter of a system can make it go from chaotic to stable or vice versa. The only solution to keep up with such complexity therefore, is to be sure that the model for prediction you are using is big enough and you do not neglect relevant facts or contextual properties which may modify substantively the behaviour of your system without noticing.

The digital butterfly effects will be more visible as more and more industrial systems and associated maintenance processes are digitalized and go through the digital transformation. The task of prediction of failures and other unwanted events will be faced with new challenges both from an engineering and business point of view.

Diego Galar, PhD, Professor of Condition Monitoring, Luleå University of Technology
Uday Kumar, PhD, Chair Professor and Head of the Division of Operation and Maintenance Engineering, Luleå University of Technology
Ramin Karim, PhD, Professor of Operation and Maintenance, Luleå University of Technology

References
• Lorenz, E. (1961). Chaos theory.
• Galar, D., Goebel, K., Sandborn, P., & Kumar, U. (2021). Prognostics and Remaining Useful Life (RUL) Estimation: Predicting with Confidence. CRC Press.