Mastering Uncertainty in Mechanical Engineering
Peter F. Pelz, Peter Groche, Marc E. Pfetsch and Maximilian Schaeffner
Peter F. Pelz
2 Types of Uncertainty
Peter F. Pelz, Marc E. Pfetsch, Sebastian Kersting, Michael Kohler,
Alexander Matei, Tobias Melz, Roland Platz, Maximilian Schaeffner
and Stefan Ulbrich
3 Our Speciϐic Approach on Mastering Uncertainty
Peter F. Pelz, Robert Feldmann, Christopher M. Gehb, Peter Groche,
Florian Hoppe, Maximilian Knoll, Jonathan Lenz, Tobias Melz, Marc
E. Pfetsch, Manuel Rexer and Maximilian Schaeffner
4 Analysis, Quantiϐication and Evaluation of Uncertainty
Maximilian Schaeffner, Eberhard Abele, Reiner Anderl,
Christian Bölling, Johannes Brötz, Ingo Dietrich, Robert Feldmann,
Christopher M. Gehb, Felix Geßner, Jakob Hartig, Philipp Hedrich,
Florian Hoppe, Sebastian Kersting, Michael Kohler, Jonathan Lenz,
Daniel Martin, Alexander Matei, Tobias Melz, Tuğrul Öztürk, Peter
F. Pelz, Marc E. Pfetsch, Roland Platz, Manuel Rexer, Georg Staudter,
Stefan Ulbrich, Moritz Weber and Matthias Weigold
5 Methods and Technologies for Mastering Uncertainty
Peter Groche, Eberhard Abele, Nasssr Al-Baradoni, Sabine Bartsch,
Christian Bölling, Nicolas Brötz, Christopher M. Gehb, Felix Geßner,
Benedict Götz, Jakob Hartig, Philipp Hedrich, Daniel Hesse,
Martina Heßler, Florian Hoppe, Laura Joggerst, Sebastian Kersting,
Hermann Kloberdanz, Maximilian Knoll, Michael Kohler,
Martin Krech, Jonathan Lenz, Michaela Leštáková, Kevin T. Logan,
Daniel Martin, Tobias Melz, Tim M. Müller, Tuğrul Öztürk,
Peter F. Pelz, Roland Platz, Andrea Rapp, Manuel Rexer,
Maximilian Schaeffner, Fiona Schulte, Julian Sinz, Jörn Stegmeier,
Matthias Weigold and Janine Wendt
6 Strategies for Mastering Uncertainty
Marc E. Pfetsch, Eberhard Abele, Lena C. Altherr, Christian Bölling,
Nicolas Brötz, Ingo Dietrich, Tristan Gally, Felix Geßner,
Peter Groche, Florian Hoppe, Eckhard Kirchner,Hermann Kloberdanz, Maximilian Knoll, Philip Kolvenbach,
Anja Kuttich-Meinlschmidt, Philipp Leise, Ulf Lorenz,
Alexander Matei, Dirk A. Molitor, Pia Niessen, Peter F. Pelz,
Manuel Rexer, Andreas Schmitt, Johann M. Schmitt, Fiona Schulte,
Stefan Ulbrich and Matthias Weigold
Peter F. Pelz, Peter Groche, Marc E. Pfetsch and
One of three dimensions of a system’s quality besides
effort and availability. Formal acceptability is reached by conformity
with explicit legal constraints or any implicit conventions; informal
acceptability is fostered by functional quality and minimal social costs.
An energy converter generating potential inϐluence on a
Finite sequence of (computer-)instructions to solve a
Predictive process (and system) change with the aim of
reducing uncertainty. Anticipating is one of four abilities/functions of a
One of three dimensions of a system’s quality besides
effort and acceptability. Availability measures the relative usability of a
technical system in time.
Metric for evaluating the resilience of a technical
system. The buffering capacity of a technical system measures the
amount of structural change for which the fulϐilment of a
predetermined required minimum of functional performance can still
be guaranteed. Depending on the context, the buffering capacity can
attain continuous or integer values. (Example: In case of integer values,
it describes the maximum number of components that can fail while
still maintaining the required minimum of functional performance.) A
system has a buffering capacity of k if it guarantees the required
minimum of functional performance within a predetermined range of
inϐluencing factors for all possible failure scenarios of up to k
Synonym for an assembly or single item.
A data-induced conϐlict exists when the
interpretation and usage of uncertain data from more than one source
leads to contradictory statements about the appropriate design of
processes or products.
Requirement for the design and usage/operation of a sociotechnical system.
Generic term for a quantiϐiable system value.
Data shall be ϐindable, accessible, interoperable,
reusable (FAIR). As such, FAIR data management fosters transparency
and hence acceptability. It is the prerequisite for quality key
performance indicators (quality KPI).
The nature of data uncertainty depends on the form
in which data are available. If data are stochastically distributed, there
is stochastic uncertainty. If they are known to be within limits, but not
stochastically distributed, there is incertitude. Unnoticed or ignored
uncertainty occurs when there is neither stochastic uncertainty nor
Methodical procedure from the ϐirst idea through planning,
conception and development to the virtual elaboration of a (loadbearing) product.
A technical system is designed for a speciϐic design point.
If the system designer strives for a robust design, considerations not
only comprise one design point, but also an uncertainty area around it.
The function of a system can only be realised within a
certain design space. The design space is limited by physical laws as
well as by the available resources or resource materials, components
and technologies. The design space can be expanded by innovations or
restricted by banning technologies, e.g. by the requirement for carbonfree energy supply. Systems that enable the same function usually differ
in quality. The task of Sustainable Systems Design is to select from these
competing systems one with an optimal quality within the design
A diagnosis is used to ϐind the cause of disturbances. If a
disturbance is not directly observable but only its effect on the system,
only the symptoms of the disturbance can be observed. The diagnosis
allows conclusions from these symptoms on the cause. In particular, it
serves to ϐind causes in data-induced conϔlicts.
A disturbance leads to unexpected, unauthorised
deviations of at least one system value. This can lead to a malfunction
or failure of the system.
One of three dimensions of a system’s quality besides
acceptability and availability. Effort measures the investment costs and
social costs given by energy and material consumption to achieve a
desired system function.
A ϐlexible system is characterised by the fact that it fulϐils
multiple predeϐined functions with accepted functional quality.
Flexibility is used as a strategy to master uncertainty during the
product life cycle of technical systems.
Desired relationship between a system’s input and output
with the aim of fulϐilling a task.
Between stakeholders agreed and predeϐined
function of a system.
Metric for evaluating the resilience of a technical system.
Its behaviour may be described as “graceful degradation” at the
boundary of its performance range towards the loss of the required
minimum degree of the functional performance. Mathematically, it is
deϐined by the directional derivative of the functional performance
curve in the direction of a given inϐluencing factor (or a vector of
multiple inϐluencing factors). In the case of non-differentiability, it is
given by the limit from the direction of the design point.
Hardware-in-the-Loop (HiL) tests investigate
the behaviour of real components connected to real-time simulated
systems and allow the stepwise integration of a technical module or
component into a real system by combining cyber world and real
Disregarded but relevant reality. The effect of uncertainty
is unknown or only suspected. Ignorance is associated with model
uncertainty and with ignored possible manifestations of a product,
system or process. No statement can be made about the probability
distributions of an unfolding uncertain property.
Limit values of an emerging uncertain product
characteristic can be assumed. Furthermore, no probability
distributions have to be presumed. There are known or estimated
membership functions in fuzzy analysis or intervals in interval
analysis. The variability is uncertain.
Information is derived from the interpretation of data
and serves as the basis for decisions. Interpretation may be performed
Reduction of model uncertainty and data uncertainty
through permanent model identiϐication and model adaptation during
the product life cycle. Learning is one of four abilities/functions of a
Metric for evaluating the resilience of a technical system. The
margin of a technical system is the distance of the actual functional
performance to the system’s required minimum of functional
Abstract image of an object in form of a mathematical model or
other, such as imaginary on the basis of intuition. A mathematical
model is substantiated in axiomatic (white-box model) or empirical
(black-box model) terms or both (grey-box model). Mathematical
models represent a functional relationship between input and output
data, model parameters and internal variables, like states.
Model derived from measurements of a process or
the experience of a user. In the ϐirst case, these models are called datadriven models today. In the second case, the deposited model is part of
an expert system.
Model that combines axiomatic and empirically
derived relationships as well as (expert) user experience.Model horizon
Model derived by deduction from axioms. Model uncertainty in whitebox models arises from an impermissible model structure or
impermissible simpliϐications, e.g. an assumption of quasi-stationary
system behaviour, inadmissible constitutive equations as well as
inadmissible initial and boundary conditions.
Boundary of the relevant reality represented by the
Model uncertainty arises from an incomplete
mapping of the object. Parts of the relevant reality are ignored. In the
case of model uncertainty, the functional relationship is suspected,
unknown, incomplete or ignored—ignorance prevails in all cases.
Function-oriented group of components of a technical entity
or algorithm; each with clear interfaces.
Sensing a process by means of data acquisition and data
analysis via models to obtain information. Monitoring is the one of four
abilities/functions of a resilient system.
Generic term for product, system or process.
Target for the design and usage/operation of a sociotechnical system.
An operator provides an effective quantity to be able to
carry out a process. The effective quantity is the purpose of the
operator and thus causes the desired change of state. In production, for
example, the operator comprises machines and the necessary auxiliary
material for the production process of the load-bearing structure. In
the process of usage, however, the operator refers, for example, to the
used load-bearing system.
Model parameters are brought into a functional
relationship in a model. They are a data component. Model parameters
are derived from empirical data, literature or model analysis.
Basis for assessing the resilience of a technical
system. The performance range describes the range of inϐluencing
factors in which a technical system is able to achieve a predeϐined
required minimum functional performance. The performance rangeProcess
Process chain, resilient
Product life cycle
can be mathematically expressed by the so-called “superlevel set” of
the functional performance curve at the level of the required minimum
A process transforms a primary state into a ϐinal state. The
process is associated with an individual process or a process chain.
A process started at a time always shows the
same behaviour. It can start at any time without a change of result. The
parameters of its mathematical description and transfer functions of a
controller are for example invariable in time (invariant).
A process started at a time shows different
behaviour over time, see time-invariant process.
A process chain is the combination of individual
processes. They transform a primary state into a ϐinal state, with the
operand going through various intermediate states. Process chains can
be modelled across life cycle phases. A process chain can also be used
to represent a component structure.
In a resilient process chain, monitoring,
responding, learning and anticipating internal and external
disturbances (for example machine failure, manufacturing uncertainty,
slump in demand, and uncertainty in product usage) can be used to
Common and applicable mode of communication for
the various areas of expertise to deϐine a process chain incorporating
systematically and transparently the individual processes and the
A product is an object that did not originate naturally, but is
produced by man himself for other people, and that is used or
consumed in the context of purpose-oriented action in usage processes.
The product life cycle describes the process chain:
sourcing, production, usage and reuse/recycling.
Properties of products or systems are divided into
function and quality.
The process of making products, components or systems.Radius of performance
Resilient process chain
Measures—in the tradition of Taguchi—the effort with which a
function is achieved. The effort is measured in economic and social
costs. In addition, there is the availability and the acceptability.
Metric for evaluating the resilience of a
technical system. It is connected to the technical system’s performance
range. The radius of performance measures the minimum distance of
the design point to the speciϐic value of an inϐluencing variable for
which the required minimum level of functional performance is no
The part of reality that is not necessary to answer a
The part of reality necessary to answer a question.
The feature of a product to not fail with a certain
probability under stated functional and environmental conditions
during a speciϐied period of time.
process chain, resilient
Process intervention based on information with the aim of
reducing uncertainty. Responding is one of four abilities/functions of a
Speciϐic operational measures to deal with uncertainty
at the real process and product level. Risk analysis is limited to the
identiϐication and description of risks. The deϐinition and application of
speciϐic measures are covered by risk management.
Robust Design is an engineering design methodology
also known as Taguchi methods. In Robust Design (i) uncertainty is
replaced by stochastic uncertainty using the concept of quality loss
functions; (ii) sourcing, design and production phases are hollistically
treated by the concept of off-line quality control. The basis of off-line
quality control is ϐirstly the Design of Experiments (DoE) and secondly
the robust optimisation in the so-called parameter design. Taguchi
developed the methodology based on previous works of Ronald Fisher
on DoE. In modern Robust Design the concept of perceived quality, i.e.
customer experience, and social costs as quality measures are already
Sustainable Systems Design
A product is designed and optimised in such a
way that, even with unavoidable inϐluence of disturbances and
variations of input variables within the model horizon, the user
expectations are completely fulϐilled.
A robust system proves to be insensitive or only
insigniϐicantly sensitive to deviations in system properties or varying
usage. Robustness is used as a strategy to master uncertainty from the
different perspectives of mathematical optimisation, product or system
design and production.
In contrast to technical systems or technoeconomic systems that only include technical components and their
interaction regarding the ϐlux of energy, material or information
including money, socio-technical systems take into account the human
being and the technical components. The effects of the interaction of
humans and technology play a decisive role in the analysis of sociotechnical systems.
Model-based acquisition of information at the component
and/or system level. The target value is not measured directly, but
determined based on a model. As such, soft sensors are familiar with
state observers and Kalman ϐilters based on Bayesian methods.
The state variables, according to the state space
representation of system theory, describe the current state of a system,
regardless of its origin, e.g. force, speed, etc.
Only a part of all possible structures of the
design space is evaluated, i.e. the remaining part of the design space is
Combination of functions (functional structure),
components (component structure) or process chains to fulϐil a
Engineering design methodology
representing the design process as a constraint optimisation process:
functional requirements and design space form the constraints, qualitySystem
dimensions give the objectives. The three quality dimensions are
minimal effort, maximal availability and maximal acceptability.
The system describes the totality of all elements considered.
Setting a system boundary deϐines the object or product, respectively,
the objects or products.
An active system is characterised by the supply of
external energy to inϐluence a process. The external energy always
inϐluences the process via the operator. The term external energy does
not include energy that is available to fundamentally necessary
operations within the process, in particular no supply energy.
A technical system that can be adjusted to the
particularities of various situations, due to its technical characteristics.
Adaptivity is the prerequisite for a resilient system.
A passive system is characterised by the fact that
external energy is only provided for the processes that are
fundamentally necessary in the process, i.e. in particular as supply
A resilient technical system guarantees a
predetermined minimum of functional performance even in the event
of disturbances or failure of system components and a subsequent
possibility of recovering at least the setpoint function. Resilience can
be increased by adjusting the system state via monitoring, responding,
learning and/or anticipating, as well as by systematically designing the
A semi-active system is characterised by the
supply of external energy to inϐluence the operator. In this case, any
properties of the operator can be inϐluenced by the external energy.
The process itself is only affected indirectly by the external energy. The
term external energy does not include any energy that is available to
fundamentally necessary operations within the process, in particular
no supply energy.
In contrast to technical systems that only
include technical components and their interaction, techno-economic
systems take into account the ϐlux of money and economic measures
such as proϐit.Time-invariant process
Uncertainty occurs when the usage properties and
process characteristics of a system cannot, or can only be partially
Natural, random and irreducible uncertainty.
Uncertainty due to incomplete scientiϐic
knowledge. Epistemic uncertainty can be reduced by new insights.
Partial to complete details on probability
distributions of an emerging uncertain product characteristic are
available. There are known or estimated probability density functions;
the variability is always determined.
Usage or operation of a component, product, system or process.
Analysis to what extent a model after calibration is suitable
for the description of a relevant functional relationship by comparison
of reality and model. Furthermore, evaluation to what extent a product
meets the predeϐined quality and functional constraints and to what
extent a product is accepted by the customer and the society.
Review whether the model is consistent and has been
correctly solved. Furthermore, evaluation to what extent the design
and production methods and technologies are selected correctly.
A system’s vulnerability or violability.
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