Dynamic Measuring Systems: Fundamentals and application of time-dependent measurements 9783110713107, 9783110713039

Table of contents :
Foreword
Editorial
Acknowledgment
Contents
List of Contributing Authors
Fundamentals of dynamic measurement analysis for LTI systems
Modeling dynamic measurements in metrology and propagation of uncertainties
Traceable calibration for the dynamic measurement of mechanical quantities
Traceable measurements with dynamically calibrated hydrophones
Dynamic measurement analysis and the internet of things
Index

Citation preview

Dynamic Measuring Systems

De Gruyter Series in Measurement Sciences

�

Edited by Klaus-Dieter Sommer and Thomas Fröhlich

Dynamic Measuring Systems �

Fundamentals and application of time-dependent measurements Edited by Sascha Eichstädt

Editor Dr. Sascha Eichstädt Physikalisch-Technische Bundesanstalt 2-12 Abbestr. 10587 Charlottenburg Berlin Germany [email protected]

ISBN 978-3-11-071303-9 e-ISBN (PDF) 978-3-11-071310-7 e-ISBN (EPUB) 978-3-11-071313-8 ISSN 2510-2974 Library of Congress Control Number: 2023940867 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2024 Walter de Gruyter GmbH, Berlin/Boston Cover image: Wassily and the World of Metrology, Susanna Beyer, llmenau, Deutschland Aus der Sammlung: Measurement lmpressions (Privatbesitz) Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Foreword The De Gruyter Book Series in Measurement Science (DGSMS) includes monographs ranging from the mathematical foundations of metrology, the link between metrology and information theory, and dynamic measurements to recent developments such as quantum sensing and cognitive sensors and measurement systems. The present volume “Dynamic Measuring Systems,” which has been produced under the leadership of the internationally renowned metrologists Sascha Eichstaedt (Physikalisch-Technische Bundes-anstalt, Germany), is one of the first monographs in the DGSMS. The other planned volumes of the book series will be available within a period of 24 months after the publication of this volume. Analysis and correct consideration of the time and frequency dependence of sensors and measurement processes is one of the fundamental tasks and scientific areas of metrology and practical measurement. Classical application areas of the theory of measurement dynamics are energy and communications engineering. In recent years, dynamic measurement systems have gained new importance; on the one hand, due to the modern calculation of measurement uncertainty and, on the other hand—very topically—the strongly networked digital measurement and information systems. Moreover, comparability between discrete and analogues sensors must be analyzed and evaluated. Without consideration and synchronization of the individual dynamic characteristics, the networked digital systems would not work successfully, and metrological traceability could hardly be derived. The authors and the volume editor focus this monograph on the modeling and state space description of dynamic measurement systems and signals and the propagation of measurement uncertainties that can be derived from them. With the help of numerous practical examples from metrology, this succeeds extraordinarily clearly. The link to the classical literature on dynamic measurement systems is successfully established in Chapter “Traceable calibration for the dynamic measurement of mechanical quantities,” which deals with the dynamics of mechanical measurands, and in Chapter “Traceable measurements with dynamically calibrated hydrophones,” which deals with ultrasound measurements. The step into the digital modern age or the IoT related metrology takes place in Chapter “Dynamic measurement analysis and the internet of things,” which deals with the dynamics of cognitive systems and digital metrological twins. The consistently chosen structure of the book starting with the fundamentals of dynamic measurement analysis, the modeling of dynamic measurements and uncertainty propagation, calibration examples up to current developments in the digitalization of metrology make the book an important companion and reference work for all metrologists worldwide. This book should be at least as important for university teaching in metrology and information technology as for their industrial application.

https://doi.org/10.1515/9783110713107-201

VI � Foreword We firmly believe that this book will become a standard work in metrology. We congratulate the editor and the authors on a successful and highly topical monograph and wish it a wide readership. Klaus-Dieter Sommer Technische Universitaet Ilmenau (Germany) Editor of the DG Book Series in Measurement Science Frank Haertig Vice President of the Physikalisch-Technische Bundesanstalt (Germany) President of the International Measurement Federation (IMEKO)

Editorial S. Eichstädt The analysis and study of measuring systems in scenarios with time-dependent measurements dates back at least to the 19th century. Since then several technological and scientific developments and breakthroughs have been accomplished. These developments have changed the way measurements are performed, analyzed and used in science, technology and society. For instance, with the use of computing technologies and low-cost measuring devices, the acquisition of time series measurement data has become common practice. However, in metrology—the science of measurement—the analysis of dynamic measuring systems has not been addressed substantially until the early 21st century. First historical examples of dynamic measurements include the study of the motion of a galvanometer by A. Cornu [1], the modeling and analysis of an oscilloscope by A. Blondel [2] or the investigations of oscillating laboratory balances by D. I. Mendeleev [3]. In these and other early scientific developments, some elementary aspects of dynamic measurements were addressed. For instance, A. N. Krylov [4] considered an expression for estimating the measurement error as a function of time. Later in the 20th century, N. Wiener [5] considered stochastic processes, which can also be applied to model time-dependent measurement errors. In the middle of the 20th century, Kalman [6] established the analysis, modeling and optimal treatment of dynamic measuring systems, taking random variations into account. The Kalman filter is used until today in various applications and modifications. With the increasing use of digital measuring systems, the application of digital signal processing has become common practice in most areas of engineering and a whole new scientific field was established. Metrology for dynamic measuring systems is mostly based on signal processing, system theory and time-series analysis as can be seen in Chapters “Fundamentals of dynamic measurement analysis for LTI systems” and “Modeling dynamic measurements in metrology and propagation of uncertainties.” The range of application areas and examples of dynamic measurements have greatly expanded over the course of the 20th century. Chapter “Traceable calibration for the dynamic measurement of mechanical quantities” gives an introduction to stateof-the-art methods relevant to the traceable dynamic measurement of mechanical quantities. In Chapter “Traceable measurements with dynamically calibrated hydrophones,” the area of ultrasound measurements with hydrophones is considered. Figure 1 shows an example setup for the dynamic calibration of hydrophones. Such measuring instruments are used, for instance, for the characterization and assessment of medical ultrasound devices. The recent developments in industry, science and society, e. g., due to the digital transformation, also affect the view on dynamic measuring systems. For instance, the https://doi.org/10.1515/9783110713107-202

VIII � Editorial

Figure 1: Setup for the calibration of hydrophones for ultrasound measurements [7]. For the measurement, a short ultrasound pulse is used. The ultrasonic signal is reflected at the water surface resulting in a displacement, which is related to the particle velocity of an ultrasonic wave propagating inside the water. The laser vibrometer measures the velocity of the surface, which is shown at the right diagram. From the surface velocity, the ultrasonic pressure inside the water can be calculated and this information is used to calibrate hydrophones in a second step by exposing the hydrophone to the same ultrasonic signal.

development of Industry 4.0 as an interconnected production environment, is based on the increasing use of monitoring measurements in the production process and by the effective use of measurement information. These measurements are dynamic by nature, and thus the importance of dynamic measurements in the development of Industry 4.0 technologies is increasing; see also Chapter “Dynamic measurement analysis and the internet of things.” A further effect of the digital transformation is the rapid increase in the number of sensors in place and the increasing use of sensor networks. This makes it relevant to calibrate sensors on site. Such an in situ calibration is particularly challenging for dynamic calibration as it requires properly designed sensor excitations and a reliable time base. Some early approaches to this end are proposed in the literature, but no widely used standards and guidelines are available yet. The technologies and solutions for these future challenges will largely be based on methods and infrastructure for dynamic measuring systems. For instance, the authors in [8] propose the extension of an existing setup for the dynamic calibration of accelerometers such that it can be used for the calibration of digital sensors. In [9], methods for the analysis of dynamic measurements are applied for industrial sensor networks. Thus, the core concepts of metrology for dynamic measuring systems are the basis for the application of metrology in Industry 4.0, Smart Cities and many more. This book intends to provide an introduction and practical guidance for metrologists toward the use and further development of methodologies in dealing with dy-

Editorial

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namic measuring systems. Chapter “Fundamentals of dynamic measurement analysis for LTI systems” gives an introduction to the basic vocabulary and the method toolbox required. Chapter “Modeling dynamic measurements in metrology and propagation of uncertainties” outlines the state-of-the-art in uncertainty evaluation for dynamic measuring systems. Chapters “Traceable calibration for the dynamic measurement of mechanical quantities” and “Traceable measurements with dynamically calibrated hydrophones” provide practical examples of the dynamic calibration for metrological traceability of dynamic measurements. Finally, Chapter “Dynamic measurement analysis and the internet of things” outlines the use of dynamic measurement methodologies for applications in the (industrial) internet of things (IIoT).

X � Editorial

Bibliography [1] A. Cornu. Sur la condition de stabilite du movement d’un systeme oscilant soumis a une liaison synchrinique pendulaire. C. R. Acad. Sci., 104(22):1463–1470, 1887. [2] A. Blondel. Oscillographes, nouveaux apparelis pour l’etude des oscillations electriques lentes. C. R. Acad. Sci., 116(10):502–506, 1893. [3] D. I. Mendeleev. About methods of precise or metrological weighing. Papers on metrology, VNIIM Standart gid, pages 67–164, 1936. [4] A. N. Krylov. Some notes about crushers and indicators. Izvestia S. Peterburg, Sboi A. N., pages 623–655, 1909. [5] N. Wiener. Extrapolation, Interpolation, and Smoothing of Stationary Time Series. Technology press of MIT and Wiley, New York, 1949. [6] R. E. Kalman. A new approach to linear filtering and prediction problems. J. Basic Eng., 82:35–45, 1960. [7] M. Weber and V. Wilkens. A comparison of different calibration techniques for hydrophones used in medical ultrasonic field measurement. IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 68:1919–1929, 2021. [8] B. Seeger and Th. Bruns. Primary calibration of mechanical sensors with digital output for dynamic applications. ACTA IMEKO, 10(3):177–184, 2021. [9] M. Gruber, T. Dorst, A. Schütze, S. Eichstädt, and C. Elster. Discrete wavelet transform on uncertain data: Efficient online implementation for practical applications. In Advanced Mathematical and Computational Tools in Metrology and Testing XII, volume 90 of Series on Advances in Mathematics for Applied Sciences, pages 249–261, World Scientific, 2021.

Acknowledgment We would like to express our sincere gratitude to Anupam Prasad Vedurmudi for his invaluable contribution to the manuscript. Anupam Prasad Vedurmudi’s exceptional skills in proofreading and correcting the text have significantly improved the quality and clarity of this scientific book. His attention to detail and meticulous approach to editing have made a significant impact on the final product. We appreciate his dedication and commitment to this project, and we are grateful for his time and effort in helping us to refine and improve our work.

https://doi.org/10.1515/9783110713107-203

Contents Foreword � V Editorial � VII Acknowledgment � XI List of Contributing Authors � XV S. Eichstädt and M. Gruber Fundamentals of dynamic measurement analysis for LTI systems � 1 M. Gruber, S. Eichstädt, and C. Elster Modeling dynamic measurements in metrology and propagation of uncertainties � 35 Th. Bruns, L. Klaus, and M. Kobusch Traceable calibration for the dynamic measurement of mechanical quantities � 73 V. Wilkens and M. Weber Traceable measurements with dynamically calibrated hydrophones � 99 S. Eichstädt Dynamic measurement analysis and the internet of things � 117 Index � 127

List of Contributing Authors Th. Bruns Physikalisch-Technische Bundesanstalt (PTB) 100 Bundesallee 38116 Braunschweig Germany E-mail: [email protected]

L. Klaus Physikalisch-Technische Bundesanstalt (PTB) 100 Bundesallee 38116 Braunschweig Germany E-mail: [email protected]

S. Eichstädt Physikalisch-Technische Bundesanstalt Metrology for Digital Transformation 2-12 Abbestraße 10587 Berlin Germany E-mail: [email protected]

M. Kobusch Physikalisch-Technische Bundesanstalt (PTB) 100 Bundesallee 38116 Braunschweig Germany E-mail: [email protected]

C. Elster Physikalisch-Technische Bundesanstalt Data Analysis and Measurement Uncertainty 2-12 Abbestraße 10587 Berlin Germany E-mail: [email protected] M. Gruber Physikalisch-Technische Bundesanstalt Metrology for Digital Transformation 2-12 Abbestraße 10587 Berlin Germany E-mail: [email protected]

M. Weber University of Helsinki Department of Physics 2 Gustaf Hällströmin katu FIN-00014 Helsinki Finland E-mail: [email protected] V. Wilkens Physikalisch-Technische Bundesanstalt Ultrasonics Working Group 1.62 100 Bundesallee 38116 Braunschweig Germany E-mail: [email protected]

S. Eichstädt and M. Gruber

Fundamentals of dynamic measurement analysis for LTI systems This chapter introduces the fundamental principles, vocabulary, and mathematics for the analysis of dynamic measurements. We focus on linear time-invariant (LTI) systems, which are most often used in metrology to model the characteristics of a measuring system used in a dynamic measurement. Furthermore, we focus on principles and notation used in the later chapters. There is a large amount of literature on system and signal theory, system identification, and related topics for further reading. Thus, this chapter only introduces basic elements and refers to other sources for the interested reader.

1 Fundamental principles and vocabulary The analysis of time-dependent measurements has a long history. It combines elements from several disciplines and application areas. The basic mathematical approaches originate from the area of analysis (in particular differential equation theory), and system and signal theory. This section introduces some of the basic terms used in this and the following chapters. A quantity for which the value changes over time in a significant way is called dynamic quantity. For instance, the velocity of an accelerating vehicle measured by an accelerometer has a time-dependent value v(t). The term signal is then typically used for the measured value of a quantity, when that value depends on some independent quantity such as time. Note that signals can also be multivariate and may depend on space, frequency, or other independent quantities instead of time. For simplicity and ease of notation, we here focus on time-dependent signals, also because the term “dynamic” is usually associated with variations in time. Multivariate signals are denoted as x(t), univariate signals as x(t). An example for a multivariate time-dependent signal is an object moving in three-dimensional space, thus x(t) denoting the position in space at time t. The term system denotes the (mathematical) model of a (measurement) process. The mathematical model describes the relation between the signals that enter the system (input signals) and the signals that leave the system (output signals). For instance, an ordinary differential equation (ODE) may describe the signal y(t) that is indicated by an accelerometer depending on the excitation a(t), y(2) (t) + by(1) (t) + cy(t) = ρa(t),

(1)

where the exponent y(k) (t) denotes the kth derivative w. r. t. time. In this example, the process is the measurement of a time-dependent acceleration with an accelerometer. https://doi.org/10.1515/9783110713107-001

2 � S. Eichstädt and M. Gruber The ODE (1) is the mathematical model for this measurement process. The acceleration a(t) is the input signal and the sensor signal y(t) is the output signal; b and c are the system model parameters. A system that contains differential equations to model the measurement process is called a dynamic measuring system and the corresponding process a dynamic measurement. The quantity of interest in this setting is called the dynamic measurand. The ODE description (1) allows to model the dynamic behavior and dynamic properties of the measuring instrument in the measurement process considered as the relation between excitation and response, i. e., between input and output signal. The dynamic behavior of a measuring instrument differs from static behavior in that it depends on the frequency content of the excitation. Examples for characteristic elements of dynamic behavior are ringing (due to a resonance frequency), attenuation of high-frequency, or low-frequency components or a time delay between output and input signal; see Figure 1 for an example. It is worth noting that we often refer to the measuring system as “sensor,” even though we refer to the instrument as a whole. Hence, when we refer to “sensor input signal,” “sensor output signal,” “sensor behavior,” and so on, we always mean the measuring instrument. In cases where we actually refer to the sensing element, we indicate this directly.

Figure 1: Example of input and output signal to an accelerometer, modeled by the ODE equation (1).

Algebraic equations such as y(t) = ρa(t) + γ cannot describe dynamic behavior, because the relation between input and output signal does not allow to model frequencydependent characteristics. Hence, if the measurement process can be described using algebraic equations only, then this system is called a static measuring system and the corresponding measurement process a static measurement. This distinction is not as strict as it may appear at first glance. A static system could be considered a dynamic system in a steady state. Thus, in principle any system can be described using differential equations. In fact, the whole mathematical theory for the analysis of dynamic

Fundamentals of dynamic measurement analysis for LTI systems

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systems applies to the steady state, too. The differential equations then reduce to algebraic equations. For this reason, static systems (i. e., measurement processes that do not require modeling of dynamic behavior) are modeled using algebraic equations from the start. For example, the same accelerometer could be applied in a dynamic measurement or a static measurement. In principle, in both cases an ODE model could be applied as a mathematical model. However, in practice the static measurement is described using algebraic equations as the one above for simplicity. Measurement processes are typically continuous in time, i. e., measurements are carried out continuously instead of at discrete points in time. The signals are then represented by continuous functions of time x(t) and are often referred to as analog signals. On the other hand, when measurements are considered at separate, discrete time instances, the signals are represented by discrete functions of time x[tn ], with tn a specific time instant. These are also referred to as digital signals. For ease of presentation, the function x[tn ] is usually denoted as x[n]. In practice, the analysis of a measurement is carried out using computer systems and software. This software usually requires discrete time signals to work with. Therefore, continuous-time signals are converted from analog (continuous time) to digital (discrete time)—also called AD conversion or ADC. An increasing number of measuring instruments provide discrete-time (digital) signals as output only. Such devices have an internal AD conversion, translating the continuous signal into discrete-time output signals. The measurement of a dynamic quantity with a properly calibrated measuring instrument, followed by a proper derivation of the measurement result (estimate of the measurand’s value with associated uncertainty) is called a traceable dynamic measurement. Traceability here refers to the possibility of tracing the unit of measurement back to the international system of units (SI) via an unbroken chain of calibrations. Hence, when considering metrology for dynamic measurements, sometimes referred to as dynamic metrology, one has to address the association of measurement uncertainty with signals and systems. In Chapter 2, a detailed introduction to this topic is given. Here, we briefly outline the basic terminology and general concepts. We distinguish between the uncertainty associated with a measuring instrument (system) and the uncertainty associated with a dynamic measurand (signal). The uncertainty associated with a measuring instrument is related to a certain representation form (system) and the corresponding system parameters. That is, let the vector θ of parameters encode the dynamic system for a certain dynamic model. Then the uncertainty denoted by Uθ associated with this vector of parameters models the available knowledge about the values that can reasonably be attributed to the parameters. Hence, the dynamic uncertainty of a measuring system is the uncertainty associated with its parameters, determined from a calibration. This is no different to a static measurement. Similarly, the uncertainty Ux associated with a discrete-time dynamic measurand, i. e., a signal, is given as x = (x[t0 ], x[t1 ], . . . , x[tN ]) and encodes the knowledge about the values that can reasonably be attributed to x.

4 � S. Eichstädt and M. Gruber Sometimes the term dynamic uncertainty is used to refer to the component of an uncertainty budget that originates from an uncompensated dynamic behavior of the measuring system. This interpretation is related to the notion of naming uncertainty budget components in accordance with their origin or cause. However, it is not recommended to use dynamic uncertainty for this component, because it can easily be confused with the uncertainty associated with a dynamic measurand. Instead, one may use dynamic uncertainty component or dynamic error. In Section 6, an introduction to the concept of the dynamic error is given.

2 Characteristics of continuous-time measurements Continuous, time-invariant dynamic systems are naturally described by linear differential equations (ODE) with constant coefficients: an y(n) (t) + an−1 y(n−1) (t) + ⋅ ⋅ ⋅ + a1 y(1) (t) + a0 y(t) = bm x

(m)

(t) + b(m−1) x

(m−1)

(1)

(t) + ⋅ ⋅ ⋅ + b1 x (t) + b0 x(t)

(2) (3)

where x(t) is the input signal of the measuring system; y(t) is the corresponding output signal; an , an−1 , . . . , a0 and bm , bm−1 , . . . , b0 are constant coefficients. The exponent y(k) (t) denotes the kth derivative of y(t) w. r. t. time t. The coefficients of the ODE are constant, i. e., time-invariant. Hence, the system described by the ODE does not change over the time period considered. The system is also linear in its inputs. That is, for two different input signals x1 (t) and x2 (t) with corresponding output signals y1 (t) and y2 (t), it holds ℋ{αx1 (t) + βx2 (t)} = αℋ{x1 (t)} + βℋ{x2 (t)} = αy1 (t) + βy2 (t),

(4)

where ℋ denotes the system acting on its inputs. Such systems are called linear, timeinvariant (LTI). The above differential equation is one possible representation of an LTI system. Other possibilities are – step response function – impulse response function – frequency response function – transfer function All the aforementioned representations are equivalent to each other. That is, for the calibration of dynamic measuring system with LTI characteristics, any of these characterizations can be considered. It is worth noting here that any calibration that results in the identification of one of these representations is called a dynamic calibration. In the following, we briefly describe these different representations and their relation to each

Fundamentals of dynamic measurement analysis for LTI systems

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other. We focus on aspects of relevance for the later chapters in this book. For further information on LTI system theory, we refer interested readers, for instance, to [1, 2]. When studying an LTI system by means of calibration experiments, the frequency range covered by the excitation signal (i. e., the input signal) is of great importance. Signals with slow variations only cover a narrow range of frequencies. Thus, these signals can only characterize the measuring system in a narrow range of frequencies. For the theoretical basis of the step response and the impulse response, one therefore considers ideal signals of infinitely fast variations. Hence, theoretically when considered as input signals to the measuring system, these signals fully characterize the LTI system. For the step response of an LTI system, we consider the ideal unit step function as input signal 0

t 0. Then the discrete Fourier transform for x[n] is defined as ∞

X(ω) = ∑ x[n]e−jωn n=−∞

(36)

where again ω = ΩT. The corresponding inverse Fourier transform is given by π

x[n] =

1 ∫ X(ω)ejωn dω. 2π

(37)

−π

A special case is when x[n] = 0 for all n < 0: ∞

X(ω) = ∑ x[n]e−jωn n=0

(38)

This is called the one-sided Fourier transform, which is a special case of the Laplace transform when q = jω. Expressing equation (38) in terms of real and imaginary parts X(ω) = ℛ(ω) + jℐ (ω) results in ∞

ℛ(ω) = ∑ x[n] cos(ωn)

(39)

ℐ (ω) = ∑ x[n] sin(ωn)

(40)

n=0 ∞ n=0

Thus, the real part is an even function and the imaginary part is an odd function. Therefore, the values of X(ω) corresponding to positive and negative values are complex conjugated X(−ω) = X(ω).

(41)

With (39) and (40), it then follows that the discrete one-sided Fourier transform X(ω) is a periodic function of ω with period 2π and is completely determined by its values for ω ∈ [0, π]. Noting that ω = 2πTf , with T the sampling period and f denoting frequency, and with sampling frequency defined as Fs = 1/T, it follows that X(ω) = X(2πfT) is fully determined for f ∈ [0, Fs /2]. The relation between the Fourier transform X(ω) of x[n] and the Fourier transform Xc (Ω) of the corresponding continuous-time x(t) is given by

14 � S. Eichstädt and M. Gruber

X(ω) =

ω 2πr 1 ∞ ). ∑ X( + T r=−∞ c T T

(42)

Hence, the function X(ω) consists of periodic copies of the function Xc (Ω). With (39) and (40), the same holds true for the real and imaginary parts. Let denote ω0 = 2πFs = 2π/T the radial sampling frequency. Then equation (42) becomes X(ω) =

1 ∞ ∑ X (ωFs + ω0 r). T r=−∞ c

(43)

Thus, if Xc (Ω) has components beyond ω0 /2, the Fourier transform X(ω) of the discretetime signal x[n] is distorted by the overlapping content between the periodic copies of Xc (Ω). As a result, the spectral content of Xc (Ω) for higher frequencies is mapped to lower-frequency components of the spectral content of X(ω). Conversely, it can also be said that the sampling frequency Fs = 1/T for sampling the continuous-time signal must be twice as large as the highest frequency (spectral) component of the continuoustime signal in order to obtain a distortion-free discrete-time representation of it. This is also known as the fundamental sampling theorem of signal processing and goes back to Nyquist and others, and the frequency Fs /2 is also called the “Nyquist frequency.” This relation can also be formulated as follows. Consider the continuous function x(t) with Fourier transform X(ω). If |X(ω)| ≈ 0 for all |ω| > ωc , then x(t) can be approximately reconstructed from the discrete x[nT] provided that T ≤ π/ωc . That is, the discrete spectrum then contains all information about the continuous spectrum. The accuracy of the reconstruction depends on how accurately the condition |X(ω)| ≈ 0 for |ω| > ωc is met. In practice, these conditions are enforced by application of a low-pass filter to the continuous-time signal before sampling. The low-pass filter attenuates all frequency components that could result in a distorted discrete-time signal. Therefore, a suitable cut-off frequency ωc has to be defined corresponding to the sampling frequency and the low-pass filter characteristics in order to avoid distortions in the discretization process. Such a low-pass filtering is also applied before downsampling a discrete-time signal for the same reasons and is called antialiasing. The corresponding filter is then called antialias filter [1].

5 Discretization of analog systems There are several methods for the discretizing of an analog system. One is the method of impulse invariance [1]. In principle, this method samples the continuous-time impulse response such that hd [n] = Thc (nT)

(44)

where the sampling period length T is used to scale the discrete-time impulse response hd such that the frequency response Hd magnitude matches that of the continuous-time

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frequency response function Hc . In practice, however, the impulse invariance method is not applied to the impulse response, but rather to an analog transfer function model [1]. By transforming the poles and zeros of the transfer function, a representation of the discrete-time transfer function is obtained. The bilinear transform is a discretization method that is also applied to the analog transfer function. The discrete-time transfer function is obtained with the bilinear transform by substituting s in the transfer function model by [1], s=

2 z−1 T z+1

(45)

The bilinear transform is a highly nonlinear method where the positive imaginary axis in the continuous s-domain is transformed into the upper-half unit circle in the z-domain with z = esT . A result of the nonlinearity is a phenomenon known as “frequency warping” [1]. Some software packages allow the specification of a so-called “matched frequency,” which is used to rescale the frequency values in a process called “prewarping.” With this approach, the discrete-time system’s frequency responses matches that of the analog system at the “matched frequency.” As for all discretization methods, care has to be taken of the choice of the sampling period length to avoid aliasing. Figure 5 shows a simulated example of a second-order analog system, which is discretized using the method of impulse invariance (left-hand side) and the bilinear transform (right-hand side).

Figure 5: Examples for the effect of aliasing when discretizing an analog representation of a dynamic system. Left: using the method of impulse invariance. Right: using the bilinear transform.

For the smallest sampling period Ts = 1e − 5, Figure 5 shows that the discrete-time system cannot be distinguished from the analog one on this scale. For the larger sampling period Ts = 1e − 4, both discretization methods show a clear deviation between the discrete and the analog system’s frequency response. It is worth noting, though, that the result obtained from the impulse invariance method matches the resonance frequency behavior better than the bilinear method. At the same time, the result obtained from the

16 � S. Eichstädt and M. Gruber impulse invariance method shows a strong aliasing effect where the analog system’s frequency response values beyond fs /2, with fs = 1/Ts , are mapped onto lower frequency values in the discrete-time system’s frequency response. This effect becomes even more visible for the smallest sampling period Ts = 5e − 4 with the bilinear transform. Note that some of the plots are not continued beyond a certain frequency, because this is their sampling frequency limit, i. e., they are plotted for [0, fs /2] only.

6 Compensation of dynamic effects In the analysis of dynamic measurements, it is often desired to compensate for certain unwanted behaviors of the measuring system that results from the time-dependent properties of the measurand. For instance, the spectral content of the measurand may have resulted in a ringing or a phase distortion in the system’s output signal. To this end, the known properties of the measuring system can be utilized to compensate for these effects. In general, an input signal x(t) to a measuring system with system model H(jω) results in an output signal y(t) that differs significantly from x(t). In particular, the input signal x(t) can usually not be reconstructed from y(t) using algebraic operations only. Instead, the design of a compensation system is typically carried out. Such a compensation system is itself a dynamic system, because it has time-dependent inputs (the measuring system output) and outputs (the estimate of the value of the measurand). That is, the compensation system is applied to y(t) and calculates an estimate of x(t). This process is also known as dynamic error correction or dynamic input estimation, because it improves the quality of the signal estimate by reducing errors introduced by the dynamic behavior of the measuring system. In general, the compensation (also known as deconvolution [10]) is an ill-posed inverse problem. That is, small changes in the input of the compensation system may result in disproportionally large changes in the compensation result. For instance, small (numerical) noise in y(t) may be amplified by the compensation system such that the estimate of x(t) consists mainly of noise components; see [11] for an example. To this end, a so-called regularization has to be carried out to render the ill-posed problem stable. Therefore, the compensation is often considered in two steps: the design of an inverse dynamic system that compensates the dynamic effects of the measuring system, and subsequently the design of a type of regularization.

6.1 Design of an inverse dynamic system In general, a perfect inverse system with frequency response Hi (jω) has the property that 1 = Hs (jω)Hi (jω),

(46)

Fundamentals of dynamic measurement analysis for LTI systems

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and hence Hi (jω) =

1 . Hs (jω)

(47)

Figure 6 shows the result in the frequency domain. Here, the resonance frequency of the measuring system is perfectly compensated. Note that the magnitude of the spectrum of the measuring system goes to zero asymptotically. Consequently, the amplitude spectrum of the ideal compensation filter goes to infinity for higher frequencies, because of (47). Hence, the inverse system also compensates the high-frequency attenuation of the measuring system. The inverse system (red curve in Figure 6) thus amplifies highfrequency content. When applying this inverse system to the observed output signal of the measuring system, it will thus amplify noise significantly. Even if we were to ignore the noise amplification in y(t) when applying the inverse system, the condition |Hi (jω)| → ∞ for ω → ∞ is not realizable in practice. To this end, one usually selects an upper frequency limit ωc up to which condition (46) must be satisfied by the compensation system Hc (jω): 1 = Hs (jω)Hc (jω)

∀ω ∈ [0, ωc ].

(48)

Figure 6: Frequency characteristic of a measuring system, its ideal inverse and the regularized compensation result.

The interval [0, ωc ] is then called the compensation bandwidth. The value of ωc must match the spectral properties of x(t) in order to avoid systematic errors. At the same time, ωc should be chosen such that noise is not amplified too much. This balancing is the key task of regularization; see Section 6.2. The frequency response of the compensation system can be used for the estimation of the input signal, i. e., the value of the dynamic

18 � S. Eichstädt and M. Gruber measurand in the analysis of a dynamic measurement. A detailed example is given in Chapter 4. As a short summary, this approach works as follows: 1. The system’s output signal y is Fourier transformed to the frequency domain to obtain Y . 2. The compensation system’s frequency response Hc and Y are transformed to the same set of frequencies, e. g., by interpolation or by adjusting the length of y before the Fourier transform. 3. The estimate of the input signal x is calculated in the frequency domain as X̂ = Hc ⋅Y . 4. The estimate x̂ of the input signal x is calculated by applying the inverse Fourier transform to X.̂ The evaluation of uncertainties for this approach is described in Chapter 2. A software implementation for such an uncertainty evaluation is provided in the Python library PyDynamic [12]. The frequency response of the ideal compensation system can also be used to design a digital filter as an inverse filter or deconvolution filter. This digital filter is obtained by fitting a chosen filter model to the frequency response of the ideal compensation system, typically within the desired compensation bandwidth. The resulting filter is then applied to the discrete-time output y[n] of the measuring system. The filter output signal is thus a discrete-time estimate of x(t); see, for instance, [13] for a tutorial. Here, we briefly outline the basic concepts for the design of a digital compensation filter. There are two types of digital filters: finite impulse response (FIR) and infinite impulse response (IIR) type systems. An FIR filter is described by its filter coefficients c = (c0 , c1 , . . . , cN )T in the z-domain as N

HFIR (z) = ∑ ck z−k . k=0

(49)

Note that the filter coefficient vector c is also the discrete-time impulse response of the system modeled by the FIR filter. Hence, an application of an FIR filter to a discrete-time signal x[n] is given as N

y[n] = ∑ ck x[n − k]. k=0

(50)

An IIR filter is described by in the z-domain as ∞

HIIR (z) = ∑ hk z−k . k=−∞

(51)

In practice, IIR type filters are written in a recursive form using two vectors of filter coefficients b = (b0 , b1 , . . . , bNb )T and a = (a0 , a1 , . . . , aNa )T in the z-domain as

Fundamentals of dynamic measurement analysis for LTI systems

HIIR (z) =

N

b bk z−k ∑k=0

N

a al z−l ∑l=0

.

� 19

(52)

Hence, the application of an IIR filter to a discrete-time signal x[n] is given as y[n] =

N

N

a b 1 ( ∑ bk x[n − k] − ∑ al y[n − l]), a0 k=0 l=0

(53)

which is usually simplified by normalizing the filter coefficients such that a0 = 1. After choosing the filter type and order, the design of a digital compensation filter is carried out as a regression problem, e. g., least-squares minimization. That is, the filter coefficients are determined by solving an optimization problem such that the filter’s frequency response corresponds to the reciprocal of the system’s frequency response. For an FIR filter as digital compensation filter, this results in ̃ − Dc) ̃ T (H ̃ − Dc), ̃ min(H

(54)

̃ = (ℛ(H −1 (ω)), ℐ (H −1 (ω)))T H s s

(55)

Ek,⋅ = (1, e−jωk T , . . . , e−jωk NT ).

(56)

where

̃ = (ℛ(E), ℐ (E))T with and D

In practice, a time delay of nd samples is added to the reciprocal of Hs (ω) before fitting the filter coefficients to achieve a causal system and improve the quality of the fit. When an IIR filter is used as digital compensation filter, the time delay also plays an important role for stabilizing the resulting filter [14]. The evaluation of uncertainties for this approach is addressed in Chapter 2. The software PyDynamic can be applied for its easy implementation [12]. Instead of the above described approach in the frequency domain, the inverse system can also be designed in the Laplace domain. Let the system’s transfer function have the form Hs (s) =

Y (s) bm sm + bm−1 sm−1 + ⋅ ⋅ ⋅ + b1 s + b0 = . X(s) an sn + an−1 sn−1 + ⋅ ⋅ ⋅ + a1 s + a0

(57)

Let λi , i = 1, . . . , n be the roots of the denominator polynomial and γj , j = 1, . . . , n the roots of the numerator polynomial, i. e., λi the poles and γj the zeros of the transfer function. The transfer function (57) can then also be written in pole-zero representation as Hs (s) = Ks

(s − γ1 )(s − γ2 ) . . . (s − γm ) (s − λ1 )(s − λ2 ) . . . (s − λn )

(58)

20 � S. Eichstädt and M. Gruber Consequently, the pole-zero representation of the ideal compensation system is given by Hc (s) = Kc

(s − λ1 )(s − λ2 ) . . . (s − λn ) . (s − γ1 )(s − γ2 ) . . . (s − γm )

(59)

Although mathematically feasible, this system is unrealizable in practice. First of all, for real systems it holds that m ≤ n, which results in more zeros than poles in the ideal compensation system. Moreover, the system’s transfer function zeros γi located in the right semi-half of the complex domain make the compensation system unstable as they become its poles. Therefore, the actual compensation transfer function has to have the form Hc (s) = Kc

(s − λ1 )(s − λ2 ) . . . (s − λn ) (s − μ1 )(s − μ2 ) . . . (s − μn+r )

(60)

with r > 0 and all roots μk in the left semiplane. Note that the same holds true for the z-transform as the discrete counterpart of the Laplace transform. That is, the actual compensation transfer function in the z-domain has to have the form Hc (z) = Kc

(z − λ̃1 )(z − λ̃2 ) . . . (z − λ̃p ) (z − μ̃ 1 )(z − μ̃ 2 ) . . . (z − μ̃ q )

(61)

with q ≥ p and all roots μ̃ j located inside the unit circle in the complex domain. The construction of the roots μj and μ̃ j , respectively, is also known as stabilization. An approach of stabilization in the z-domain is described, for instance, in [14]. To summarize the different approaches for the design of an inverse system, we consider the following scenarios: (I) The system’s frequency response is given in terms of a complex-valued vector for the whole frequency range (i. e., from 0 to Nyquist). (II) A parametric model of the system’s transfer function is given, either in the s-domain or z-domain. (III) The system’s impulse response is given in terms of a real-valued vector in the time domain or as a parametric model. (IV) The system’s step response is given in terms of a real-valued vector in the time domain or as a parametric model. Note that there are several other possible scenarios. However, the above are most typical for practical dynamic metrology settings. Figure 7 illustrates how for the above scenarios (I)–(IV) a representation of the inverse system can be obtained. For scenario (I), the inverse system’s frequency response is calculated as the reciprocal of the measuring system’s frequency response. Here, it is important that the frequency response values are given for the whole frequency range. Otherwise the inverse system’s frequency response cannot be applied for input estimation. If this is not the case—for instance due to limitations of the calibration setup—

Fundamentals of dynamic measurement analysis for LTI systems

� 21

Figure 7: Example representation forms of the compensation system and their derivation from different types of knowledge about the measuring system.

extrapolation can be considered; cf. Chapter 4. Another possibility is to fit a parametric model of the frequency response, i. e., the s-domain or z-domain transfer function, to the available system’s frequency response values. This is usually applied for dynamic calibration with sinusoidal excitation; cf. Chapter 3. The result of the fit, e. g., by means of a least-squares regression, leads to scenario (II). For scenario (II), the inverse system’s frequency response can be calculated simply by evaluating the corresponding expression at the respective frequency points. For scenario (III), one calculates the (discrete time) Fourier transform of the impulse response to obtain scenario (I). For scenario (IV), one calculates the numerical integration of the step response to obtain the corresponding impulse response, and thus, scenario (III). Hence, for all scenarios (I)–(IV) an essential step is the calculation of the inverse system’s frequency response. This can be applied directly for input estimation as in Chapter 4. Alternatively, one can fit an FIR or IIR filter model to the inverse system’s frequency response to obtain the so-called “inverse filter” or “deconvolution filter”; cf. Chapter 2. For further insights into the derivation of a digital deconvolution filter, we refer the interested reader to [13]. For the optimal design of the inverse system’s parametric model, a suitable optimization criterion has to be defined. This criterion should take into account the compensation property (48) as well as the stability criterion for the resulting transfer function. To this end, the compensation error can be employed. For instance, an upper bound for the dynamic error can be derived as follows: Let the dynamic error in the discrete time domain be written as ̂ Δ[n] = x[n] − x[n],

(62)

̂ with x[n] the discrete-time output of the compensation system Hc and Ts the sampling ̂ interval length. Due to x[n] = (hc ∗ y)[n] with hc the impulse response of the compensation system, it follows with the definition of the inverse Fourier transform that

22 � S. Eichstädt and M. Gruber π

Δ[n] =

1 ̂ − X(ω))dω ∫ ejωn (X(ω) 2π

(63)

1 ∫ ejωn (Hc (jω)Y (ω) − X(ω))dω, 2π

(64)

−π π

=

−π

where ω = ΩT = 2πfT, and hence Ω = ωFs . From the relation between the discrete-time and the continuous-time Fourier transform, it follows that Y (ω) =

1 ∞ ∑ H (j(Ω + 2πkFs ))Xc (Ω + 2πkFs ) Ts k=−∞ s

(65)

with Fs = 1/T and Xc (Ω) the Fourier transform of x(t). Assuming that no aliasing is present, i. e., Xc (Ω) = 0 for all Ω ≥ πFs , this reduces to Y (ω) =

1 H (jΩ)Xc (Ω). T s

(66)

From this, an upper bound on Δ[n] can be derived as 󵄨π 󵄨󵄨 󵄨󵄨 1 󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 jωn 󵄨󵄨Δ[n]󵄨󵄨 = 󵄨 ∫ e (Hc (jω)Hs (jωFs ) − 1)Xc (ωFs )dω󵄨󵄨󵄨 󵄨󵄨 2πT 󵄨󵄨󵄨󵄨 󵄨 −π

(67)

π

1 󵄨 󵄨󵄨 󵄨 ≤ ∫ 󵄨󵄨H (jω)Hs (jωFs ) − 1󵄨󵄨󵄨󵄨󵄨󵄨Xc (jωFs )󵄨󵄨󵄨dω 2πT 󵄨 c

(68)

1 󵄨 󵄨 ∫ 󵄨󵄨H (jω)Hs (jωFs ) − 1󵄨󵄨󵄨B(ωFs )dω 2πT 󵄨 c

(69)

−π π

≤

−π

where B(Ω) ≥ |X(Ω)| ∀Ω > 0 is an upper bound for the amplitude spectrum of the assumed measuring system input signal, i. e., the measurand. Note that the upper bound (69) can also take into account errors due to discretization by using the full sum from (65) instead of the reduced one. The above derived expression can also be used to evaluate the dynamic error of the measuring system. The dynamic error is defined as ϵ(t) =

1 y(t) − x(t), K0

(70)

with K0 the static gain of the measuring system. That is, the dynamic error is defined as the deviation of the system output from a steady-state response. When the system does not introduce time-dependent errors then ϵ(t) = 0, because only the static gain K0 has to be compensated. Otherwise, the dynamic error ϵ(t) > 0. Consider the case that in expression (69) the compensation system consists of the reciprocal static gain only, i. e., Hc (jω) = 1/K0 . Then the upper bound on Δ[n] is given as

Fundamentals of dynamic measurement analysis for LTI systems π 󵄨󵄨 1 󵄨󵄨󵄨 1 󵄨 󵄨 󵄨󵄨 ∫ 󵄨󵄨󵄨 Hs (jωFs ) − 1󵄨󵄨󵄨B(ωFs )dω 󵄨󵄨Δ[n]󵄨󵄨󵄨 ≤ 󵄨󵄨 2πT 󵄨󵄨 K0 −π

� 23

(71)

It is worth noting that sometimes this or similar derivations of an (worst case) estimate of the dynamic error is considered as the “dynamic uncertainty”; see, for instance, [15]. However, as explained above, this terminology should be avoided to not be confused with the actual uncertainty associated with the value of a dynamic quantity. Instead one should use “dynamic uncertainty component” when this value is used in an uncertainty budget. This is useful, for instance, when a compensation of the system’s dynamic behavior is not possible for whatever reason.

6.2 Regularization of the inverse system The compensation system is designed to reduce, or ideally remove, the unwanted dynamic effects by minimizing the dynamic error of the combined system Hc (s)Hs (s). That is, after the application of the compensation system to the output of the measuring system, the remaining dynamic error should be minimized. In any case, the compensation or deconvolution is an ill-posed inverse problem. In [11] and [16], an insight from a metrological perspective is given. The author in [10] provides an overview on the general problem of deconvolution. 6.2.1 Tikhonov regularization A regularization first proposed by A. N. Tikhonov [17] is based on the solution of the convolution integral t

y(t) = ∫ x(τ)h(t − τ)dτ.

(72)

0

Assuming that x(t) = 0 for all t < 0 and h(t − τ) = 0 for all τ > t, we can expand this to the general convolution integral ∞

y(t) = ∫ x(τ)h(t − τ)dτ.

(73)

−∞

The measured output signal y(t) is modeled as y(t) = y0 (t) + ν(t),

(74)

where ν(t) is a random noise process and y0 (t) denotes the ideal, noise-free output signal. Application of the Fourier transform to the convolution integral (73) yields

24 � S. Eichstädt and M. Gruber X(ω)H(jω) = Y0 (ω) + V (ω).

(75)

This can be transformed to X(ω) =

Y0 (ω) V (ω) + . H(jω) H(ω)

(76)

Hence, the inverse of the measuring system is applied to the noise-free output signal and the additive noise process individually. Applying the inverse Fourier transform to the expression (76) gives x(t) =

∞

∞

−∞

−∞

1 V (ω) −jωt 1 e dω ∫ X0 (ω)e−jωt dω + ∫ 2π 2π H(jω)

(77)

∞

= x0 (t) +

V (ω) −jωt 1 e dω ∫ 2π H(jω)

(78)

−∞

The integral in the second line may not exist due to the noise ν(t) having significant high-frequency components and H(jω) → 0 for ω → ∞. In any case, the value of this integral term will be very large compared to the value of x0 (t). To this end, a stabilizing component 0 ≤ f (ω, α) ≤ 1 with α > 0 is added as follows: ∞

xα (t) =

Y (ω) 1 f (ω, α)e−jωt dω. ∫ 2π H(jω)

(79)

−∞

The function f (ω, α) is to be chosen such that it suppresses high frequency components of Y (ω)/H(jω) whilst not affecting the spectrum of X0 (ω). Hence, here again the above described compensation bandwidth [0, ωc ] has to be considered. That is, the ideal stabilizer equals f (ω, α) = 1 within the compensation bandwidth and |f (ω, α)| → 0 for ω → ∞ sufficiently fast to render (79) stable. This condition is met by [17], f (ω, α) =

|H(jω)|2 |H(jω)|2 + αQ(ω)

(80)

where |H(jω)|2 = H(jω)H(jω) and Q(ω) a nonnegative function of ω. For a chosen α, the resulting compensation system is then given by Hα (jω) =

H(jω) . |H(jω)|2 + αQ(ω)

(81)

The resulting estimate of the input signal’s spectrum is thus calculated as Xα (ω) = Hα (jω)Y (ω).

Fundamentals of dynamic measurement analysis for LTI systems

� 25

The important task in the Tikhonov regularization method is the selection of a suitable value for α. One approach is to use the expression ϕ(α) =

∞ ∞ 󵄨󵄨 |H(jω)|2 Y (ω) 󵄨󵄨󵄨2 1 1 󵄨 󵄨 󵄨2 󵄨󵄨 dω − Y (ω) ∫ 󵄨󵄨󵄨H(jω)Xα (ω) − Y (ω)󵄨󵄨󵄨 dω = ∫ 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 |H(jω)|2 + αQ(ω) 2π 2π 󵄨 −∞ −∞

(82)

∞

=

1 α2 Q2 (ω)|Y (ω)|2 dω. ∫ 2π [|H(jω)|2 + αQ(ω)]2

(83)

−∞

Thus, it holds that ϕ(0) = 0 and ∞

1 󵄨 󵄨2 󵄨 󵄨2 ϕ(α) ≤ ∫ 󵄨󵄨󵄨Y (ω)󵄨󵄨󵄨 dω = 󵄨󵄨󵄨y(t)󵄨󵄨󵄨 , 2π

(84)

−∞

with ϕ(α) → |y(t)|2 for α → ∞. Hence, the function ϕ(α) is a strictly increasing function of α. Therefore, a suitable value of α can be selected by increasing it until ϕ(α) is sufficiently close to |y(t)|2 . Another approach is the so-called L-curve method [18]. The name of this method stems from the typical shape of the curve one obtains when plotting ‖(h ∗ xα )(t) − y(t)‖ (x axis) against ‖xα ‖ (y axis) in logarithmic scale. This approach takes into account that larger values of α will result in a large error when inserting the estimate xα (t) back in the convolution equation, whereas smaller values of α will result in an insufficient damping of high-frequency deteriorations. Hence, the L-curve method considers the typical regularization trade-off between noise attenuation and reconstruction quality. The heuristic approach for selecting α is to identify the “corner” of the L-curve [18, 19]. Although not imposed explicitly, the quality of several heuristic approaches depend on prior knowledge about x(t) [19].

6.2.2 Wiener–Kolmogorov deconvolution filter The above-described method of Tikhonov regularization does not use a priori information about the input signal explicitly. However, when such information is available it is of advantage to take that into account. The Wiener–Kolmogorov filter method is an approach to this end [20]. Consider the system input signal x(t) and the noise component of the system output v(t) to be uncorrelated stationary stochastic processes. We assume that the spectral densities Sx (f ) and Sv (f ) of x(t) and v(t), respectively, are known and that the measuring system with impulse response h(t) is stationary, too. Note that the knowledge of the spectral density is not equivalent with knowing the actual signal x(t). The corresponding measurement equation is given as y(t) = (h ∗ x)(t) + v(t).

(85)

26 � S. Eichstädt and M. Gruber The Wiener–Kolmogorov deconvolution filter g(t) is designed such that the estimate ̂ = (g ∗ y)(t) x(t)

(86)

minimizes the quadratic error 2

̂ ϵ(t) = (x(t) − x(t)) .

(87)

In the frequency domain, this leads to 󵄨 ̂ )󵄨󵄨󵄨2 E(f ) = 𝔼󵄨󵄨󵄨X(f ) − X(f 󵄨 󵄨󵄨 󵄨2 = 𝔼󵄨󵄨X(f ) − G(f )(H(f )X(f ) + V (f ))󵄨󵄨󵄨

(88) (89)

After some simple reformulation, taking into account that 𝔼{V (f )X ∗ (f )} = 0 because of the assumed independence, one obtains ∗

E(f ) = (1 − G(f )H(f ))(1 − G(f )H(f )) Sx (f ) + G(f )G∗ (f )Sv (f ).

(90)

Minimization of this expression w. r. t. G(f ) then yields the Wiener–Kolmogorov deconvolution filter as G(f ) =

H ∗ (f )Sx (f ) , |H(f )|2 Sx (f ) + Sv (f )

(91)

which is often also written as G(f ) =

H ∗ (f ) |H(f )|2 +

Sv (f ) Sx (f )

.

(92)

It is worth noting that the reciprocal of Sv (f )/Sx (f ) is also known as the signal-to-noise ratio (SNR). That is, the filter g(t) with frequency response G(f ) is regularized based on the expected SNR. When there is zero noise, i. e., Sv (f ) ≡ 0, then G(f ) = 1/H(f ) and no regularization is applied.

6.2.3 Low-pass filter regularization Both approaches to regularizing the deconvolution problem introduced in the previous section have a similar structure. The Tikhonov regularization gives Gα (jω) =

H ∗ (jω) |H(jω)|2 1 = , |H(jω)|2 + αQ(ω) H(jω) |H(jω)|2 + αQ(ω)

and the Wiener–Kolmogorov deconvolution filter can be written as

(93)

Fundamentals of dynamic measurement analysis for LTI systems

G(jω) =

H ∗ (f ) |H(f )|2 +

Sv (f ) Sx (f )

=

|H(jω)|2 1 . 2 H(jω) |H(jω)| + Sv (ω)/Sx (ω)

� 27

(94)

When the second term in the denominator in those expressions is larger than zero, the second factor in both equations resembles a low-pass filter. Hence, both approaches result in a cascade of the reciprocal of the system’s frequency response H(jω) and a certain low-pass filter. The characteristics of the low-pass filter is determined by the choice of the respective regularization parameters. That is, the low-pass filter part for the Tikhonov regularization is determined by α and Q(ω) whereas the low-pass filter part for the Wiener–Kolmogorov deconvolution filter is determined by the signal-to-noise-ratio. As a generalization of this, one may also use any low-pass filter for regularizing the inverse system. The regularization parameter is then the vector of filter parameters, such as filter order and cut-off frequency. The upper bound (69) for the dynamic error (70) could be used to determine an optimal low-pass filter characteristics [21] then. When designing the regularization in this way, one has to take into account the expected spectral content of the measurands to be considered in the application. For instance, in the hydrophone measurements considered in Chapter 4 a certain type and spectral content of input signals exist. This knowledge can be employed to ensure a low regularization error. To visualize the effect of the choice of low-pass filter characteristics, we take a simple example: – The measurement system is modeled by a second-order system with known parameters (gain, damping, resonance frequency) – The input signal (i. e., the measurand) is a bell-shaped pulse with spectral content in the vicinity of the system’s resonance frequency – The output signal (i. e., the indicated signal) shows ringing, because of the system’s resonance frequency – The input estimation is regularized by a low-pass filter with certain cut-off frequency The input and output signals are shown in Figure 8. The output signal shows a strong ringing that is attenuated over time due to the system’s damping. Note that the ringing cannot be compensated for with algebraic operations, i. e., scaling or shifting the observed output signal. Hence, a dynamic compensation system must be designed to remove the ringing. For the compensation of the ringing and other dynamic behavior, the ideal inverse system is accompanied with a low-pass filter. We hold the filter order fix and vary only its cut-off frequency. The filter cut-off frequency determines up to which frequency the filter does not attenuate signal content. That is, a low cut-off frequency may attenuate desired content of the measurand to be estimated. At the same time, a high cut-off frequency may unsufficiently attenuate signal noise in the observed measurement. Figure 9 shows the magnitude of the system’s frequency response, its ideal inverse and of

28 � S. Eichstädt and M. Gruber

Figure 8: Example measurand with corresponding simulated output signal for the considered secondorder system with resonance frequency.

Figure 9: Magnitude frequency response of the considered system, its ideal inverse, and three different regularized deconvolution systems.

the compensation systems for three different choices of low-pass filter cut-off frequency. Note that the compensation system with the low cut-off frequency attenuates frequency content within the vicinity of the systems resonance frequency. This compensation system is thus not able to compensate the effect of ringing. The different compensation systems result in different estimates of the value of the dynamic measurand, of course. In this illustrative, simulated example the measurand has significant spectral content in the vicinity of the resonance frequency. Thus, the compensation system with the low cut-off frequency for the regularizing low-pass filter causes a significant estimation error:

Fundamentals of dynamic measurement analysis for LTI systems

– – –

� 29

high cut-off frequency low-pass filter compensation system: rms estimation error of approximately 0.67e-3 medium cut-off frequency low-pass filter compensation system: rms estimation error of approximately 0.77e-3 low cut-off frequency low-pass filter compensation system: rms estimation error of approximately 8.5e-3

In this example, the low cut-off frequency compensation system has a root mean square (rms) error, which is one magnitude larger than for the other two choices of low-pass filter cut-off frequency. This is also illustrated in Figure 10, which shows the estimation error in the time domain.

Figure 10: Estimation error for the three different compensation systems each applied to the system’s output signal.

Note that we here know the true value of the measurand, because it is a simulated example. We can thus actually calculate the estimation error. In practice, however, this is not the case and one has to estimate the error based on the available knowledge; for instance, by using upper bound function introduced in equation (69).

6.3 Other forms of regularized deconvolution On a more abstract level it can be argued that regularization utilizes some knowledge or assumption about the measurand to be estimated. This assumption (or knowledge) can be represented in various forms. Some of these have been discussed above: In the Wiener–Kolmogorov deconvolution, the assumption is made that the measurand’s

30 � S. Eichstädt and M. Gruber value, the measurement and the noise can be modeled as stationary stochastic processes. This assumption is then used to derive the low-pass filter characteristic of the compensation system. In practice, the Wiener–Kolmogorov deconvolution is sometimes applied using the frequency response function of the measuring system and Fourier transform of the system’s output together with the SNR function instead of actual stochastic processes’ spectral densities. Strictly speaking, this violates the method’s assumptions, but the result is nevertheless often usable. The Tikhonov deconvolution, in particular with the L-curve method to determine the regularization parameter, employs certain assumptions about the statistical properties of the system output and the measurand’s value [19]. In practice, these assumptions are typically not validated when applying this method, which can lead to a significant estimation bias [19]. Application of other parametric low-pass filters utilize assumptions about the spectral content of the measurand’s value for the determination of reasonable parameter values. In principle, this can be implemented as an optimization problem, e. g., using the above introduced upper bound for the spectrum [21]. In practice, one may also choose a rather conservative (i. e., high) low-pass filter cut-off frequency such that the regularization bias effect can be considered negligibly small compared to other sources of uncertainty. A much more explicit approach of considering knowledge about the measurand is to apply a parametric model in the time or frequency domain. For instance, a (multi)sinusoidal function model may be assumed for the measurand: N

x(t) = ∑ ak sin(bk t + ck ). k=1

(95)

Estimation of the value of the measurand can then be implemented as a regression problem: 󵄩 󵄩2 θ̂ = arg min 󵄩󵄩󵄩(h ∗ xθ ) − y󵄩󵄩󵄩 θ∈ℝN

(96)

with θ = (a1 , . . . , aN , b1 , . . . , bN , c1 , . . . , cN ) the vector of model parameters. That is, the model parameters for x(t) are chosen such that the application of the measuring system to x(t) is as close as possible to the observed system output y(t). For a fixed N, this can be considered a regularization because it does not amplify noise, i. e., it renders the illposed estimation problem stable. When the number N of sinusoidal components is also considered as parameter to be optimized, the risk of noise fitting occurs. That is, a high value of N may lead to smaller estimation errors for the observed y(t) just because the resulting model function for x(t) leads to (h ∗ x)(t) mimicking the noisy y(t). In other words, an ideal regularization should consider the model y(t) = y0 (t) + ε(t) with y0 (t) the noise-free ideal output signal and achieve (h ∗ x)(t) ≈ y0 (t).

(97)

Fundamentals of dynamic measurement analysis for LTI systems

� 31

An example of a model-based input estimation, which is not a regularization, is the “differential operator method” for bandpass correction in spectral measurements [22, 23]. The mathematical model for bandpass measurements is ̃ λ̃ − λ)dλ̃ M(λ) = ∫ S(λ)b(

(98)

with b(λ) the so-called “bandpass function.” When replacing the bandpass function by ← 󳨀 its reverse b (λ − λ)̃ = b(λ̃ − λ), equation (98) becomes a convolution and the estimation ← 󳨀 of S(λ) from knowledge about M(λ) and b (λ) becomes a deconvolution problem. The differential operator method for bandpass correction considers a Taylor series expansion of the measurand S(λ) at a certain λk : 1 S(λ) = S(λk ) + (λ − λk )S ′ (λk ) + (λ − λk )2 S ′′ (λk ) + ⋅ ⋅ ⋅ . 2

(99)

Plugging the expression (99) into the measurement equation (98) and calculating the operator inverse, one obtains an expression for S(λk ) as [22]: S(λk ) = θ0 M(λk ) + θ1 M ′ (λk ) + ⋅ ⋅ ⋅ .

(100)

That is, the measurand is expressed in terms of the derivatives of the measurement. Replacing the derivates in (100) with finite differences, one obtains a closed formula for each chosen largest derivative of M. The final formulae for resulting polynomials of order 3 and 5 are given in [22]. This approach to bandpass correction applies a parametric model for the measurand, i. e., the Taylor series expansion, and results in a parametric model for the estimation of its value. However, this method is not a regularized deconvolution. The reason is that due to the use of finite differences in the model, noise approximation increases with decreasing sampling period. That is, the closer the discrete measurement gets to the (continuous) analog measurement, the smaller the region in which the polynomial approximation is applied. In fact, the parameter values increase indefinitely with decreasing sampling period [23]. Another approach to bandpass correction, and to deconvolution in general, is the use of an iterative method such as the so-called Richardson–Lucy method [24, 25]. This method was originally developed individually by Richardson and Lucy for image correction in astronomy. However, the method is also applied in bandpass correction [23] or for deconvolution of certain effects when calculating COVID-19 epidemic curves from available incidence data and non-pharmaceutical inventions; see, for instance, [26] among others. An advantage of the Richardson–Lucy method is that under mild assumptions it guarantees an estimate of the measurand with nonnegative values. This is of importance, for instance, for spectral reconstruction or epidemic cases. Starting with a basically arbitrary initial estimate of the measurand, the Richardson–Lucy method iteratively improves this estimate by repeating the following steps:

32 � S. Eichstädt and M. Gruber 1. 2. 3. 4.

̃ k ) at ̃ k ) from the current estimate S(λ Calculate an estimate of the measurement M(λ all available instances λk . ̃ k ) at all available instances λk . Calculate a correction factor as Q(λk ) = M(λk )/M(λ Calculate the damping term as R(λk ) = (b ∗ Q)(λk ) at all available instances λk . ̃ k ) at all available instances λk . Calculated the updated estimate as S̃ + (λk ) = R(λk )S(λ

A simple initial estimate could be the measurement M(λ) itself. As long as the initial estimate and the convolution kernel b(λ) are nonnegative, the above calculations will ̃ k ) is zero or always result in a nonnegative S.̃ Care has to be taken for λk where M(λ close to zero. One could then set Q(λk ) to zero at those instances. Because the correction factor is calculated from the noisy measurement M, an indefinite repetition of the above updates would lead to noise fitting. Thus, regularization of this iterative deconvolution has to determine a suitable number of iterations, i. e., repetitions. In [23], an approach to determine a suitable number of iterations has been proposed that resembles the L-curve method known from Tikhonov regularization.

Bibliography [1] [2] [3]

[4] [5]

[6]

[7] [8] [9] [10] [11] [12] [13]

A. V. Oppenheim and R. W. Schafer. Discrete-time signal processing: Pearson New International Edition. Pearson Education, Limited, 2013. B. P. Lathi. Linear Systems and Signals. Oxford University Press, 2004. A. C. G. C. Diniz, A. B. S. Oliveira, J. N. de Souza Vianna, and F. J. R. Neves. Dynamic calibration methods for pressure sensors and development of standard devices for dynamic pressure. In XVIII IMEKO WORLD CONGRESS, 2006. E. Amer, M. Wozniak, G. Jönsson, and F. Arrhén. Evaluation of shock tube retrofitted with fast-opening valve for dynamic pressure calibration. Sensors, 21(13):4470, 2021. C. Matthews, F. Pennecchi, S. Eichstädt, A. Malengo, T. Esward, I. M. Smith, C. Elster, A. Knott, F. Arrhén, and A. Lakka. Mathematical modelling to support traceable dynamic calibration of pressure sensors. Metrologia, 51(3):326–338, 2014. H. Füser, S. Eichstädt, K. Baaske, C. Elster, K. Kuhlmann, R. Judaschke, K. Pierz, and M. Bieler. Optoelectronic time-domain characterization of a 100 ghz sampling oscilloscope. Meas. Sci. Technol., 23(2):025201, 2012. W. L. Briggs and V. E. Henson. The DFT: An Owners’ Manual for the Discrete Fourier Transform. Society for Industrial and Applied Mathematics, 1995. V. Strejc. State space theory of discrete linear control. Wiley, Chichester, 1981. G. F. Franklin, J. D. Powell, and M. L. Workman. Digital Control of Dynamic Systems. Addison-Wesley 3rd edition, 1980. S. M. Riad. The deconvolution problem: an overview. Proc. IEEE, 74(1):82–85, 1986. S. Eichstädt, V. Wilkens, A. Dienstfrey, P. Hale, B. Hughes, and C. Jarvis. On challenges in the uncertainty evaluation for time-dependent measurements. Metrologia, 4(53):S125, 2016. B. Ludwig, S. Eichstädt et al. PyDynamic – python package for the analysis of dynamic measurements, 2020. S. Eichstädt, C. Elster, T. J. Esward, and J. P. Hessling. Deconvolution filters for the analysis of dynamic measurement processes: a tutorial. Metrologia, 47(5):522–533, 2010.

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[14] R. Vuerinckx, Y. Rolain, J. Schoukens, and R. Pintelon. Design of stable iir filters in the complex domain by automatic delay selection. IEEE Trans. Signal Process., 44(9):2339–2344, 1996. [15] O. M. Vasilevskyi, P. I. Kulakov, K. V. Ovchynnykov, and V. M. Didych. Evaluation of dynamic measurement uncertainty in the time domain in the application to high speed rotating machinery. Int. J. Metrol. Qual. Eng., 8:25, 2017. [16] A. Dienstfrey and P. D. Hale. Analysis for dynamic metrology. Meas. Sci. Technol., 25(3):035001, 2014. [17] A. N. Tikhonov and V. J. Arsenin. Solution of Ill-posed Problems. Winston, Washington, 1977. [18] P. C. Hansen and D. P. O’Leary. The use of the l-curve in the regularization of discrete ill-posed problems. SIAM J. Sci. Comput., 14(6):1487–1503, 1993. [19] A. Dienstfrey and P. D. Hale. Colored noise and regularization parameter selection for waveform metrology. IEEE Trans. Instrum. Meas., 63(7):1769–1778, 2014. [20] N. Wiener. Extrapolation, Interpolation, and Smoothing of Stationary Time Series. Technology press of MIT and Wiley, New York, 1949. [21] S. Eichstädt, A. Link, Th. Bruns, and C. Elster. On-line dynamic error compensation of accelerometers by uncertainty-optimal filtering. Measurement, 43(5):708–713, 2010. [22] E. R. Woolliams, R. Baribeau, A. Bialek, and M. G. Cox. Spectrometer bandwidth correction for generalized bandpass functions. Metrologia, 48:164–172, 2011. [23] S. Eichstädt, F. Schmähling, G. Wübbeler, K. Anhalt, L. Bünger, U. Krüger, and C. Elster. Comparison of the Richardson–Lucy method and a classical approach for spectrometer bandpass correction. Metrologia, 50(2):107, 2013. [24] W. H. Richardson. Bayesian-based iterative method of image restoration. J. Opt. Soc. Am., 62:55–59, 1972. [25] L. B. Lucy. An iterative technique for the rectification of observed distributions. Astron. J., 79:745–754, 1974. [26] L. Y. Chan, B. Yuan, and M. Convertino. Covid-19 non-pharmaceutical intervention portfolio effectiveness and risk communication predominance. Sci. Rep., 11:10605, 2021.

M. Gruber, S. Eichstädt, and C. Elster

Modeling dynamic measurements in metrology and propagation of uncertainties This chapter provides an overview of methods for evaluating dynamic measurement uncertainty in metrology. The common approach behind most of these methods is the application of the GUM uncertainty framework to discrete linear time-invariant (LTI) systems. To bridge theory and practice, the mathematical background is accompanied with software support through the Python library PyDynamic. The feature that distinguishes a static from a dynamic system is its dependency on the history of its previous states. Along with the history-dependent nature of dynamic measurements, the associated measurement uncertainties of estimated quantities will show dynamic properties as well. It is of fundamental interest to metrology to quantify these dynamic measurement uncertainties. Figure 1 illustrates a typical scenario considered in this chapter. The figure shows the indicated measurements of a simulated dynamic sensor (red), the underlying measurand (black) and an estimate of the measurand based on the indication and the sensor’s uncertain transfer behavior (purple).

Figure 1: Time series plot of simulated sensor indication, measurand and measurand estimate in arbitrary units (a. u.). Colored bands indicate standard uncertainty. Results can be reproduced using Listing 2.1 in Appendix A.2.

https://doi.org/10.1515/9783110713107-002

36 � M. Gruber et al. The goal of this chapter is to provide means for the calculation of the dynamic uncertainty associated with the estimation of a dynamic measurand such as indicated by the scenario of Figure 1. The development of methods for evaluating dynamic uncertainty will be based on the “Guide to the expression of uncertainty in measurement” (GUM [1]). The GUM provides the de-facto standard for uncertainty evaluation in metrology. The chapter is structured as follows: in Section 1 the concepts and notations of the GUM are briefly introduced and the assumptions made about the dynamic measurements are stated. We then proceed by introducing the mathematical concepts used to model the dynamic measurements (Section 2), to calibrate these models (Section 3), to estimate the input from indications (Section 4) and to evaluate the uncertainty associated with these estimates (Section 5). Software support for the presented methods is introduced in Section 6. Finally, the source code complementing the mathematical analysis in the examples shown throughout this chapter is provided in the chapter appendix, Appendix A.

1 GUM uncertainty framework Uncertainty propagation in metrology typically refers to the “Guide to the expression of uncertainty in measurement” (GUM) and its constituent parts and supplements [1–4]. The document GUM-3 [1] covers uncertainty evaluation for scalar quantities depending on multiple input quantities using a first order approximation of the functional relationship between the quantity under study and the input quantities. GUM-7 [2] considers the same scenario, but uses Monte-Carlo simulations to enable an uncertainty evaluation for nonlinear models and distributions associated with the input quantities, and provides the result as a distribution for the measurand. GUM-8 [3] then extends these to vector valued quantities. GUM-6 [4] provides an extensive guidance on the development and use of measurement models. It also covers how dynamic measurement uncertainties can be treated within the existing GUM framework.

1.1 Uncertainty evaluation To evaluate uncertainty, the GUM proposes three stages: formulation of model assumptions, propagation of uncertainty (or of distributions associated with the input quantities) through the model and summarizing the propagation result in terms of an estimate and its associated uncertainty [3]. The formulation stage requires to establish a measurement function f : ℝN → ℝM that relates the sought quantity Y ∈ ℝM to the input quantities X ∈ ℝN , such that Y = f (X).

(1)

This model can range from explicit mathematical equations to involved algorithms. It is necessary to model the knowledge about X by specifying a (joint) probability density

Modeling dynamic measurements in metrology and propagation of uncertainties

� 37

function (PDF) g X or, alternatively, to provide an estimate x of X along with a covariance matrix Ux . These estimates represent the mean of g X for x and the covariance matrix of g X for Ux . The result of the propagation stage is then either a PDF g Y that represents the knowledge about the measurand Y , or an estimate y along with a covariance matrix Uy . The GUM proposes (1) analytical methods, (2) the law of propagation of uncertainty or (3) Monte Carlo methods to accomplish this task. While analytical methods provide an ideal and mathematical representation of g Y , the complexity of the necessary calculations remain manageable only for rather simple measurement models and PDFs. An alternative is the “law of propagation of uncertainty” which propagates uncertainties using a first order approximation of the functional relationship f and is a sufficiently accurate representation of uncertainty in many cases. The resulting estimate y of the measurand Y is obtained by inserting the estimate x of the input quantity X into the functional relationship (Equation (1)) and the associated covariance matrix Uy is calculated from the covariance of the inputs Ux and the sensitivity C of the measurement function f [3] with Ux = Cov(X) 𝜕f1 [ 𝜕X1

[ . C = [ .. [

𝜕fM [ 𝜕X1

(2) ... .. . ...

Uy = Cov(Y )

= CUx C T .

𝜕f1 𝜕XN

] .. ] . ] ]

𝜕fM 𝜕XN

(3)

] (4)

In cases where a propagation of uncertainties is no longer feasible, a Monte Carlo method (MCM) can be used instead to numerically propagate the PDF associated with the input quantities through the functional relationship (Equation (1)), yielding a PDF for the measurand which provides the most comprehensive uncertainty quantification. The MCM draws samples x i with i = 1, 2, . . . , m from the PDF g X and for each sample the model (Equation (1)) is evaluated. The resulting samples yi with i = 1, 2, . . . , m then provide independent samples from the sought PDF g Y for the measurand. The proceeding takes full account of nonlinearities in the measurement model. The MCM can become computationally intensive depending on the chosen number of drawn samples (“runs”) and the complexity of f . The summarizing stage extracts information about the expectation and a covariance matrix (or coverage region) of Y from the calculated PDF. When applying a propagation of distributions, the estimate y and covariance Uy of the output quantity Y are given by the expectation and covariance operator: y = E(Y )

Uy = Cov(Y ).

(5) (6)

38 � M. Gruber et al. When applying a MCM with m runs for the propagation of distributions, the expectation value and covariance matrix can be estimated according to:1 y= Uy =

1 m ∑y m i=1 i

1 m ∑(y − y)(yi − y)T . m − 1 i=1 i

(7) (8)

In this book (if not stated otherwise), uncertainty refers to the standard uncertainty. It is the “uncertainty of the result of a measurement expressed as a standard deviation” [1, Section 2.3.1]. In addition, the term expanded uncertainty refers to “an interval about the result of a measurement that may be expected to encompass a large fraction of the distribution of values that could reasonably be attributed to the measurand” [1, Section 2.3.5]. Typically, a coverage factor relates the expanded uncertainty to the standard uncertainty.

1.2 Assumptions for dynamic measurements Following the approach described in multiple publications [4, 6–8], we consider a realvalued time-dependent measurand ψ(t) ∈ ℝ (with time t ∈ ℝ) that is measured by a sensor with transfer behavior 𝒮 . The sensor converts the measurand to an indicated value γ̃𝒮 (t) based on its evaluation model 𝒮 . Moreover, an analog-digital converter (ADC) discretizes the indicated value in time and value γ𝒮 (tk ) = ADC(γ̃𝒮 (tk )). In the case of equidistant time-steps Δt it is also common to write γ𝒮 [k] = γ𝒮 (tk ) = γ𝒮 (kΔt). The indicated value γ𝒮 (tk ) is influenced by the observed (potentially dynamic) measurand ψ(t) and the transfer behavior of the sensor. The latter effect is typically unwanted and the idea is to apply a reconstruction ℛ to yield a discrete-time estimate of the measurand ψ̂ ℛ (tk ) given knowledge about the sensor 𝒮 used to compensate the influence. The process is visualized in Figure 2 and two distinctive tasks arise in which uncertainty needs to be evaluated: – To estimate the dynamic transfer behavior of 𝒮 (see Section 3) – To estimate the input ψ(t) (see Section 4) To accomplish these tasks, a model for the dynamic transfer behavior of 𝒮 needs to be established, as well as a reconstruction model ℛ derived from 𝒮 . In the following

1 If it is unfeasible to store all results in memory before evaluating Equations (7) and (8), numerically stable variations are proposed in [5] to incrementally evaluate these equations after some (≥ 1) runs. Suitable implementations are also mentioned in Section 6.

Modeling dynamic measurements in metrology and propagation of uncertainties

� 39

Figure 2: Schematic view of the reconstruction task.

Section 2, a class of models applicable to dynamic measurements in metrology is introduced.2

2 Models for dynamic measurements In order to characterize the input-output behavior of dynamical systems, systems of (ordinary) differential equations have proven to be a powerful modeling tool [9–11]. Such equations describe how an incoming signal x affects an internal state z. Additionally, an output equation usually maps the internal state z to the output γ of the dynamical system. This can be expressed by Equation (9) for continuous-time systems and by Equation (10) for discrete-time systems [9]. Small bold letters are used to denote vectors and capital bold letters for matrices.3 ̇ = f ̃(z(t), ψ(t)) z(t) ̃ γ(t) = h(z(t), ψ(t))

(9)

̃ γ[k] = h(z[k], ψ[k])

(10)

z[k + 1] = f ̃(z[k], ψ[k])

These equations are the basis for many system and control engineering approaches [9, 12–14]. The Equations (9) and (10) are generic and allow for fully nonlinear behavior. However, a nonlinear treatment is often not required [15]. Linear time invariant (LTI) systems form a subclass which cover many practical scenarios. Furthermore, LTI systems enjoy analytical properties that allow for efficient numerical treatment. Three

2 Some of the notation and models already presented in Chapter 1 are repeated in order to draw the connection to the GUM uncertainty framework. 3 To avoid naming conflicts for variables with the GUM notation, we use ψ, Ψ, γ, Γ, f ̃, h,̃ C̃ instead of the more common x, X, y, Y , f , g, C.

40 � M. Gruber et al. different ways to represent scalar LTI systems in continuous and discrete time are presented in the following: state-space notation, impulse response, frequency response. Moreover, the concept of an observer is discussed.

2.1 State-space system notation A straightforward simplification of Equation (9) is to assume that f ̃ and h̃ weight the internal state z, z0 ∈ ℝN and input ψ ∈ ℝM with constant matrices A, Ã ∈ ℝN×N , B, B̃ ∈ ℝN×M , C̃ ∈ ℝN×P and D̃ ∈ ℝM×P to update the internal state z and calculate the output γ ∈ ℝP as follows: ̇ = Az(t) + Bψ(t), z(t) ̃ ̃ γ(t) = Cz(t) + Dψ(t).

z(0) = z0

(11)

In the discrete case the state equations are given by ̃ ̃ z[k + 1] = Az[k] + Bψ[k], ̃ ̃ γ[k] = Cz[k] + Dψ[k].

z[0] = z0

(12)

These equations can also be visualized conveniently using the block diagram in Figure 3. Under the assumption that the Nyquist-criterion holds (see Section 4 in Chapter 1), the continuous and discrete forms can be transformed into each other without information loss for equidistant sampling with Δt [16]: Ã = eAΔt Δt

B̃ = ∫ eAτ Bdτ 0

The GUM framework considers only the influence of finite dimensional input quantities X. Because uncertainty in the model description and in the input signal is expected, these input quantities need to represent the model (A,̃ B,̃ C,̃ D,̃ z0 ) and the input signal (ψ). Suitable ways of representing the model could e. g. store the actual entries of all matrices or just a subset of the entries (if some entries are fixed or dependent by construction) as influencing quantities. The input signal could be parameterized using a (piecewise) polynomial approximation in the continuous-time case or the actual signal values in the discrete-time case. Under certain conditions, (a part of) the internal state z of such a state-space system can be estimated by another system by observing the output γ. A system with such an observing capability is called an observer. Special cases of observers are the Luenberger observer and Kalman filter – the latter already provides uncertainty information

Modeling dynamic measurements in metrology and propagation of uncertainties

� 41

Figure 3: Block diagram representation of a linear state-space models Equations (11) and (12) and hint to more generic nonlinear Equations (9) and (10). s−1 represents the integration operation in the continuoustime case and z−1 the delaying operation in the discrete-time case.

about the estimated state which under specific assumptions can be compatible with the GUM [17]. Observability criteria and design guidelines for observers are available in the control engineering literature [9, 16].

2.2 Impulse and step response An LTI system with scalar input ψ and output γ can also be represented in the time domain by stating its response h to a Dirac impulse input or unit step [16]. By using the impulse response as a convolution kernel the input-output relation is: ∞

γ(t) = (ψ ∗ h)(t) = ∫ ψ(t − τ) ⋅ h(τ)dτ, −∞ ∞

γ[k] = (ψ ∗ h)[k] = ∑ ψ[k − i] ⋅ h[i]. i=−∞

(13) (14)

An exemplary impulse response is visualized in Figure 5. To preserve causality, h is typically zero for negative time arguments – otherwise the systems would react to input values before “seeing” them. If h is non-zero only over an interval of finite length it is said to be finite. Two important subgroups of discrete causal impulse responses are covered, as they are frequently used in practical applications in the form of digital filters [10]. A (causal discrete) finite impulse response (FIR) filter can be represented by a coefficient vector b ∈ ℝm+1 : m

γ[k] = ∑ ψ[k − i] ⋅ b[i] i=0

(15)

42 � M. Gruber et al. This approach can represent arbitrary long responses, but it will not achieve true infinite impulse response behavior. A (causal discrete) infinite impulse response (IIR) filter can be represented by coefficient vectors b ∈ ℝm+1 and a ∈ ℝn+1 . Typically, the first element of a is set to 1 without loss of generality (a[0] ≡ 1). m

n

i=0

j=1

γ[k] = ∑ ψ[k − i] ⋅ b[i] − ∑ γ[k − j] ⋅ a[j]

(16)

The infinite filter response is realized via an output recursion. This corresponds to a frequency response that is a rational function, as seen in the next subsection. Uncertainty evaluation of IIR filters according to the GUM will consider the model representation (b, a), the input signal (ψ) and initial conditions as influencing quantities. Typically, these are set to all entries of the coefficient vectors (except a[0]) and a finite length of the discrete input signal. It is also possible to represent an IIR filter as a discrete time state-space model4 by setting à = AIIR , B̃ = BIIR , C̃ = C IIR and D̃ = DIIR in Equation (12) with [8]:

AIIR

BIIR

0 [ [ 0 [ [ . = [ .. [ [ 0 [ [−a[p]

0 [.] [ .. ] ] =[ [ ] [0] [1]

1 0 0 −a[p − 1]

0 1 0 −a[p − 2]

⋅⋅⋅ ⋅⋅⋅

0 0

⋅⋅⋅ ⋅⋅⋅

0 −a[2]

T

C̃ IIR

p = max(m, n)

b[p] − b[0]a[p] [ ] .. [ ] =[ . ] [ b[1] − b[0]a[1] ] ∀j>n a[j] = 0

0 ] 0 ] ] ] ] ] 1 ] ] −a[1]]

(17)

D̃ IIR = [b[0]]

∀i>m b[i] = 0

2.3 Frequency response By transforming the impulse response description introduced in Equations (13) and (14) into the frequency domain, another relevant representation of LTI systems is obtained – the frequency response. Applying a Laplace transformation to Equation (13) and 𝒵 -transformation to Equation (14) transforms the convolution operation into a simple multiplication in the frequency domain. Γ(s) = ℒ{γ(t)}(s) = ℒ{(ψ ∗ h)(t)}(s) = H(s) ⋅ Ψ(s)

4 this is one of infinitely many representations and is called “controller canonical form”.

(18)

Modeling dynamic measurements in metrology and propagation of uncertainties

Γ[z] = 𝒵 {γ[k]}[z] = 𝒵 {(ψ ∗ h)[k]}[z] = H[z] ⋅ Ψ[z]

� 43

(19)

Examples of frequency responses for the continuous- and discrete-time cases are visualized in Figure 4. When considering the Laplace transform along the imaginary axis (i. e. s = jω) ́, the Fourier-transform is obtained. The Fourier approach allows the interpretation of the system behavior intuitively in terms of a spectrum alteration (e. g. high-pass, low-pass, band-pass, band-stop). For the previously mentioned IIR filter Equation (16) (with the exact same b[i] and a[i]) the transfer function H[z] = HIIR [z] is given by [10] −i ∑m+1 i=0 b[i]z

HIIR [z] =

−i ∑n+1 i=0 a[i]z

Gb [z] . Ga [z]

=

(20)

As in the previous two subsections, uncertainty evaluation will consider the model representation (H) and the input signal (Ψ) as influencing quantities. Both aspects could be parameterized using a polynomial approximation in the continuous-frequency case or the actual spectrum values in the frequency-discrete case. This involves transforming the input signal and its uncertainty into the frequency domain.

2.4 Example: accelerometer As an example, we use the accelerometer described in [18] with a continuous-time second order transfer behavior. The selection of this specific model structure is motivated from prior knowledge that the sensor shows resonance around a certain input frequency. Such a behavior can be modeled as a mass-spring-damper system5 G(s) =

S0 ω20

s2 + 2δω0 s + ω20

(21)

,

with assumed parametrization θ̂ = [δ̂

T

ω0̂

= [0.01

2π ⋅ 30 kHz

(0.1δ)̂ 2

[ Uθ = [ [

(22)

S0̂ ]

(0.03ω0̂ )2

T

(23)

1]

2

] ].

(0.01S0̂ ) ]

(24)

This corresponds to a continuous IIR filter with numerator b and denominator a, such that 5 Of course, the parametrization of chosen model structure needs to be identifiable by the available experimental data. For more details on this, please refer to e. g. [19–22].

44 � M. Gruber et al. T

b = [S0̂ ω0̂ 2 ]

(25) 2 T

= [3.55 ⋅ 1010 Hz ] , a = [1.0 = [1.0 0 [ C ba = [2ω0̂ [ 0

U ba =

(26)

2 T

2δ̂ ω0̂

3.77 ⋅ 103 Hz 2ω0̂ S0̂ 2δ̂

2 T

2

3.55 ⋅ 1010 Hz ] ,

(28)

ω0̂ ] 0 ], 0 ]

2ω0̂

C ba U θ C Tba 17

(27)

ω0̂ ]

4

2.41 ⋅ 10 Hz [ = [ 6.11 ⋅ 109 Hz3 17 4 [ 1.15 ⋅ 10 Hz

(29) (30) 9

3

6.11 ⋅ 10 Hz 1.42 ⋅ 105 Hz2 6.11 ⋅ 109 Hz3

17

4

1.15 ⋅ 10 Hz ] 6.11 ⋅ 109 Hz3 ] . 1.15 ⋅ 101 Hz4 ]

(31)

This can be transformed into a discrete time IIR filter at a sample rate of 500 kHz with the MC_cont2discrete method from Listing 2.5 with6 bdiscrete = [6.99 ⋅ 10−2 adiscrete = [1.0

−1.85

8.12 ⋅ 10 [ 8.09 ⋅ 10−7 [ =[ [ 7.78 ⋅ 10−7 −8 [−1.87 ⋅ 10 −7

U ba,discrete

T

(32)

6.97 ⋅ 10−2 ] , T

(33)

0.99] , 8.09 ⋅ 10 8.07 ⋅ 10−7 7.62 ⋅ 10−7 −4.99 ⋅ 10−9 −7

7.78 ⋅ 10 7.62 ⋅ 10−7 2.21 ⋅ 10−6 −5.87 ⋅ 10−7 −7

−1.87 ⋅ 10 −4.99 ⋅ 10−9 ] ] ]. −5.87 ⋅ 10−7 ] 5.80 ⋅ 10−7 ] −8

(34)

The frequency response of the continuous and discrete system are plotted in Figure 4, showing the equivalence of the two representations. Figure 5 shows the impulse response of the discrete time system. All the findings of this example can be reproduced using Listing 2.2 in Appendix A.2.

3 Calibration for dynamic measurements Estimating the transfer behavior of a (dynamic) system involves the selection of a suitable mathematical model and identification of its parameters [23]. Because this establishes a relation between indicated signal and the measurand, it can be considered a calibration in the sense of the “International vocabulary of metrology” (VIM) [24] – provided that the reference signal is obtained by traceable measurement standards. The 6 The discrete coefficients have unit 1 because z = esT .

Modeling dynamic measurements in metrology and propagation of uncertainties

� 45

Figure 4: Frequency response of a continuous-time and corresponding discrete time system with uncertainty. The corresponding impulse response is shown in Figure 5. The small deviation at high frequencies is expected by the chosen analog-discrete conversion method. Results can be reproduced using Listing 2.2.

Figure 5: Impulse response of a discrete time system with uncertainty. The corresponding frequency response is shown in Figure 4. Results can be reproduced using Listing 2.2.

46 � M. Gruber et al. result of this calibration process is a fixed model structure f θ with parameter estimate θ̂ ∈ Θ ⊆ ℝR and corresponding covariance Uθ̂ ∈ ℝR×R . In that general regard, a calibration for dynamic systems is similar to a static calibration. In static calibration, the same measurand is observed in an experiment using two different sensors – one (traceable, ideally more accurate) reference device and the device to be calibrated. Different values of the measurand are applied and the input output characteristic is then fitted to the experimental data. If the same experiment was carried out to evaluate the dynamic characteristics of a sensor, it would likely yield unusable results because the experiment tests only for its response for input signals of 0 Hz (static signal). The distinguishing feature of dynamic measurement calibration therefore is a time-varying measurand that allows to show the dynamics of the assumed transfer behavior.

3.1 Formulation in terms of the GUM framework A calibration can be expressed by means of the GUM uncertainty framework (see Section 1). The (unknown true) scalar measurand ψm (t) is measured simultaneously by two sensors. The reference sensor provides a discrete and finite time series ψr with corresponding uncertainty (quantified by a PDF g r ). The device to be calibrated provides a discrete and finite time series ψc that could come with an indication uncertainty (quantified by a PDF g c ). ψr = [ψr (t0 ), . . . , ψr (tk−1 )]

ψc = [ψc (t0 ), . . . , ψc (tk−1 )]

(35) (36)

A function f cal extracts a parameter estimate for the assumed model structure f θ . An initial estimate θ0 of the parameter that defines f θ is available with PDF g θ0 . This leads to the following formulation using the GUM notation (see Section 1.1) with X = [ψr , ψc , θ0 ],

g X = g r ⋅ g c ⋅ g θ0 , Y = f cal (X), θ̂ = E(Y ),

Uθ̂ = V (Y ).

(formulation) (propagation) (summarizing)

In a similar manner, the input signals could be directly available in the frequency domain (see Section 5), such that Ψr = [Ψr (ω0 ), . . . , Ψr (ωk−1 )]

Ψc = [Ψc (ω0 ), . . . , Ψc (ωk−1 )]

(37) (38)

Modeling dynamic measurements in metrology and propagation of uncertainties

̃r , Ψ ̃ c , θ0 ] X = [Ψ

� 47

(39)

̃ r and Ψ ̃ c being the concatenated real and imaginary parts of Ψr and Ψc respecwith Ψ tively. This is in analogy to Equation (89) to take care of the GUM requirement of realvalued inputs.

3.2 Generic regression task The main estimation step of a calibration can in general be seen as a regression or system identification process [25–27]. A common approach is to reformulate the parameter estimation as a (constrained) optimization problem, i. e. to minimize the difference between modeled and observed system output with respect to some norm: 󵄩 󵄩 θ̂ = arg min 󵄩󵄩󵄩ψ̃ r − f ̃θ (ψ̃ c )󵄩󵄩󵄩 θ∈Θ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(40)

f cal

Here, ψ̃ r|c and f ̃θ denote suitable vectorizations of the assumed model structure and input signals to e. g. allow minimization over multiple observations. With this reformulation, it is possible to apply existing optimization methods, e. g. [28–30].7 To obtain a GUM-compliant uncertainty for the parameters θ,̂ one of the three suitable propagation approaches needs to be chosen (check Section 1). Because of the (often) nonlinear nature of optimization techniques, Monte Carlo approaches are a common choice to implement the propagation stage.

3.3 FIR and IIR filter If the input signals are directly given in the frequency domain (Equation (39)), the coefficients of an FIR or IIR filter can be estimated [18, 31]. The general idea is to augment the existing least squares based fitting method with an uncertainty statement by evaluating them with a MCM (see Section 1), such that 󵄩 󵄩 θ̂ = arg min 󵄩󵄩󵄩Ψr − H θ ⊙ Ψc 󵄩󵄩󵄩 θ∈Θ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(41)

f cal

where ⊙ denotes element-wise multiplication and H θ is the transfer function of the filter evaluated at the same ωi for which the input signals Ψr and Ψc are available: 7 To check the identified parameters, a validation should follow the estimation step. This can be achieved by applying the model to data that has not been used in the estimation process and compare the model’s output with the measured output.

48 � M. Gruber et al. H θ = [Hθ (ω0 ), . . . , Hθ (ωk−1 )]

(42)

For FIR filters, this directly yields a linear expression in θ. In the case of an IIR filter, it is possible to maintain a linear expression in θ by considering that Hθ can be seen as a fraction of numerator and denominator polynomials (see Equation (20)) [18], such that Hθ (ωi ) =

Bθ (ωi ) Aθ (ωi )

(43)

󵄩 󵄩 θ̂ = arg min 󵄩󵄩󵄩Aθ ⊙ Ψr − Bθ ⊙ Ψc 󵄩󵄩󵄩 . θ∈Θ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(44)

f cal

If the L2-norm is chosen, this leads to a least square problem in both cases. Moreover, it is even possible to directly yield a regularized inverse deconvolution filter, which is detailed in Section 4 [18].

3.4 Second-order model System dynamics with decaying oscillations can often be captured using a second order model (SOM). As already detailed in Section 2.4, any SOM can be directly seen as an IIR filter [18]. It is therefore possible to refer back to the calibration methods for IIR filters in Section 3.3 to identify an SOM system. Depending on the application, the resulting discrete IIR filter might already be suitable (e. g. as in Section 3.6). However, if the actual parameters of the SOM are sought, an additional step is required: The mapping from the filter coefficients b, a back to model parameters [δ̂ ω0̂ S0̂ ]T needs to be established with the corresponding uncertainties propagated accordingly.8

3.5 Sine fitting In Chapter 3 a mechanical system is calibrated by analyzing the input-output behavior for monofrequent input signals at different amplitudes. The method directly yields tuples of (ωi , H(jωi ), UH (jωi )) that can be interpreted as a non-parametric representation of the frequency response H with uncertainty UH . If required, suitable parameterized transfer models (e. g. IIR filter, second order model) can be fitted against this curve. The practical implementation with further details is given in Chapter 3 and Section 1.1.

3.6 Example: dynamic calibration of an accelerometer In the following we present the example of the calibration of an accelerometer. The example is based on simulated data using the dynamic model from Section 2.4. The follow8 In Section 2.4 the inverse conversion is given – from model parameters to filter coefficients.

� 49

Modeling dynamic measurements in metrology and propagation of uncertainties

ing values for the model parameters were taken in accordance with [18]: θtrue = [8.3 ⋅ 10−3

29.4 kHz

T

(45)

0.985]

The underlying acceleration is simulated as a chirp signal with linearly increasing frequencies from 0.01 kHz to 40 kHz. Figure 6 visualizes this acceleration, together with the assumed measurements of a reference sensor as well as the measurements of the device under test (DUT). The latter was calculated using the underlying acceleration together with the assumed dynamic model of the DUT. Transforming the reference and DUT signal into the frequency domain, a stabilized IIR filter is fitted against the quotient of both spectra. The true transfer function, the quotient and the fitted IIR are shown in Figure 7 and the obtained coefficients are bfitted = [0.00 Hz afitted = [1.0

T

0.13 Hz2 ]

−1.86 Hz

U ab

2

2.07 ⋅ 10 Hz [−1.96 ⋅ 10−9 Hz3 [ =[ [−2.70 ⋅ 10−9 Hz2 −9 3 [ 3.37 ⋅ 10 Hz −9

(46) 2 T

(47)

0.99 Hz ]

3

−1.96 ⋅ 10 Hz 2.13 ⋅ 10−9 Hz4 2.20 ⋅ 10−9 Hz3 −2.60 ⋅ 10−9 Hz4 −9

2

−2.70 ⋅ 10 Hz 2.20 ⋅ 10−9 Hz3 2.13 ⋅ 10−7 Hz2 −1.95 ⋅ 10−7 Hz3 −9

3

3.37 ⋅ 10 Hz −2.60 ⋅ 10−9 Hz4 ] ] ] (48) −1.95 ⋅ 10−7 Hz3 ] 1.97 ⋅ 10−7 Hz4 ] −9

The results of this example can be reproduced using Listing 2.3.

Figure 6: Time series of true acceleration, together with the output of a reference sensor and the device under test. Results can be reproduced using Listing 2.3.

50 � M. Gruber et al.

Figure 7: Transfer functions of the device under test, (simulated) experimental data and fitted IIR filter. Reproduce using Listing 2.3.

4 Input estimation Looking again at Figure 2, the result of a calibration (Section 3) is a model for the transfer behavior 𝒮 . In practical applications, it is not so much of interest what the sensor indicates (in other words: how the sensor responds to the sensed quantity), but to deduce the value of the physical quantity being measured. This is called input estimation, signal reconstruction or sensor compensation. This can be accomplished by the application of an LTI system which corresponds to the inverse of the dynamic (forward) model (see Figure 2). The inverse model ℛ describes how to obtain an estimate ψ̂ ℛ of the measured quantity ψ(t) from the indicated signal γ𝒮 (see also Figure 2). Some aspects of such a compensation have already been described in Section 6 in Chapter 1. One way of obtaining ℛ could be to re-run the system identification process of a calibration with exchanged input and output. This requires access to the original calibration data and the selection of a suitable model structure for ℛ (which can differ from that of 𝒮 ). We refer to Section 3 for suitable methods. When the original calibration data is not available and only a parameterized model is known, it is possible to obtain ℛ directly from 𝒮 . This seemingly easy task, however, should be undertaken with a degree of precaution as direct (naive) inversion of a dynamic system can lead to unstable or non-causal models. To avoid such effects, the common approach is to perform the inversion using additional assumptions tailored to the

Modeling dynamic measurements in metrology and propagation of uncertainties

� 51

application at hand. For example one could restrict the frequency response of the inverse system above a certain frequency as that would otherwise only amplify the noise. The incorporation of such assumptions into the design of an inverse system is called regularization and is a common procedure for ill-posed problems [32]. In the field of control engineering, model inversion is an important aspect of controller design, as a controller needs to invert (some) of a given system’s behavior in order to deal with or adjust its output [11]. In the following, we will look at some specialized approaches to underline and detail the need for regularization when working with ill-posed model inversion problems. Further examples of regularization in a practical context is provided in [33].

4.1 Finite Impulse Response (FIR) Assume that the sensor’s transfer behavior 𝒮 is given as a frequency response H(jω) at discrete frequencies ω = ωk with k = 1, . . . , k. An FIR filter of order N shall represent the compensation filter ℛ [18, 31]. Although any FIR filter is inherently stable, performing a minimization purely against H −1 will likely not yield a suitable compensation filter due to the delaying nature of FIR filters. A common regularization approach in this case would therefore be to fit the FIR against a delayed version of the inverted frequency response Ĥ −1 (jω) = H −1 (jω)e−jωτ with a user-defined delay τ typically chosen to be a multiple of the inverse sample rate [31]. Note that the regularization term has no effect on the amplitude of Ĥ −1 as |e−jωτ | = 1. The identification task can be expressed as an optimization task (see Equation (40)) where N

2 b̂ = arg min ∑ (Ĥ −1 (jωk ) − Gb (jωk )) b∈ℝN k=1

(49)

with Gb (jω) being the frequency response of an FIR filter with coefficient vector b. Define T −1 Ĥ = [Ĥ −1 (jω0 ), . . . , Ĥ −1 −1(jωk )] , T

Gb = [Gb (jω0 ), . . . , Gb (jωk )] , −1

(50) (51)

R(Ĥ ) −1 H̃ = [ −1 ] , I(Ĥ )

(52)

R(Gb ) G̃ b = [ ]. I(Gb )

(53)

Moreover, define (similar to Equations (100) and (101)):

52 � M. Gruber et al.

Cexp,N

1 [. [ = [ .. [1

e−jω1 Ts .. . e−jωk Ts

... .. . ...

e−jNω1 Ts .. ] ] . ] e−jNωk Ts ]

R(Cexp,N ) MN = [ ] I(Cexp,N )

(54)

(55)

The matrix expression Cexp,N b represents the complex Fourier spectrum of b at the given discrete frequencies ωk . Therefore, the product M N b directly yields the discrete version G̃ b of Gb (jω) with separate real and imaginary parts as desired. Equation (49) can then be reformulated as a (weighted) linear least-squares fit [18], the solution to which can be found in most standard statistics textbooks; see, e. g. [34]. In GUM notation, the estimation of b with weights W can be expressed by −1 X = H̃

(56)

Y =b

= f (X) = arg min (H̃

−1

b∈ℝN

T

− Mb) W −1 (H̃

−1

− Mb).

(57)

The propagation stage requires to state the distribution of X, e. g. −1

X ∼ g X = 𝒩 (H̃ , UH −1 )

(58)

−1

= Cov(X)

(59)

U UH = [ RR,H U IR,H

U RI,H ] U II,H

(60)

UH

−1

= (UH )

The propagation can then be carried out using [35] or MCM.

4.2 Infinite Impulse Response (IIR) Another common approach is to approximate ℛ with an IIR filter. The idea is similar to the already presented procedure for FIR filter. However, the details differ and are G (jω) given in [18, 36]. Let Gb,a (jω) = Gb (jω) be the frequency response of the IIR filter, where a

Gb (jω) and Ga (jω) are defined as in Equation (20). The estimation problem could then be (naively) formulated like N

2 b,̂ â = arg min ∑ (H −1 (jωk ) − Gb,a (jωk )) b,a

i=k

This approach has two issues: (1) the formulation is nonlinear in a and (2) the resulting filter given by b and a can be unstable. While the former can be linearized by a refor-

Modeling dynamic measurements in metrology and propagation of uncertainties

� 53

mulation, the latter requires an iterative regularizing approach that reuses the idea of an introduced delay from Equation (49). N

2 b,̂ â = arg min ∑ (Ga (jωk )H −1 (jωi )e−jωk τk − Gb (jωk )) b,a

k=1

(61)

As proposed by [36], the delay τi is iteratively incremented until a stable solution is found. The update considers the group delay of the fitted filter and a stabilized version of it.9 Using M N and M M as in Equation (55), this can again be reformulated as a GUMcompliant least-squares problem: −1 X = H̃

(62)

b Y =[ ] a = f (X) = arg min (MN a ⊙ H̃ b,a∈ℝN+M

−1

T

− M N b) W −1 (MN a ⊙ H̃

−1

− M N b)

(63)

The propagation stage requires to state the distribution of X, e. g. X ∼ g X = 𝒩 (H̃ , UH −1 )

−1

(64)

−1

−1

= Cov(X)

(65)

U UH = [ RR,H U IR,H

U RI,H ] U II,H

(66)

UH

= (UH )

The propagation can then be carried out using MCM.

4.3 Example: input estimation of an accelerometer In the previous examples (Sections 2.4 and 3.6), we have seen how an acceleration sensor can be modeled as an IIR filter and how the parameters in this model can be estimated by means of a calibration. It is now of interest to compensate the influence of the transfer behavior and estimate the input to the sensor.

9 Alternatively, another regularization method for approximation of inverse system behavior with IIR filters is achieved by decomposing the resulting filter into a minimum-phase system and an all-pass system [33]. While the all-pass system has a constant amplitude response, the minimum-phase system and its inverse are stable. The regularization would therefore be to only use the minimum-phase system as ℛ.

54 � M. Gruber et al. The continuous (Equation (21)) representation of the accelerometer is evaluated at discrete frequencies ωi and an IIR filter is fitted to the inverse behavior.10 A lowpass filter is added to regularize the inverse behavior above ∼60 kHz. The true, fitted, inverted and regularized transfer functions are shown in Figure 8. Application of the regularized inverse behavior to the indicated signal used for calibration in Section 3.6 is compared to the true acceleration and the uncompensated signal in Figure 9. Note, that in this example the effect of the resonance frequency is damped, but not completely compensated. Uncertainty propagation is calculated as described in Section 5. The results of this example can be reproduced using Listing 2.4.

5 Uncertainty propagation for dynamic models To express the uncertainty of processed dynamic measurements by means of the GUM, a signal ψ ∼ g ψ is propagated through a system represented by model parameters θ ∼ g θ to obtain the processed signal γ ∼ g γ .11 To further fulfill the GUM-requirement of a finite set of input variables (see Equation (1)), our focus lies on the uncertainty propagation for discrete-time systems as stated in Section 2. This does not impose major practical drawbacks, as measurement data is typically already discrete-time. For information about uncertainty propagation in continuous systems the reader is referred to the second chapter of [6] and references therein, and for the assignment of uncertainties to input quantities to e. g. the GUM [1, clause 4 and annex F].12

5.1 State-space models Discrete state-space systems with uncorrelated input uncertainties can be dealt with using the GUM’s “law of propagation of uncertainties” [3, 37]. To formulate the task in GUM notation, consider the following input X and model f (see also Equations (1) and (12)):

10 The accelerometer is modeled as a second order system. To obtain a compensation model, a stable IIR filter of similar structure (i. e. b ∈ ℝ1 and a ∈ ℝ3 ) or a higher-order FIR filter (a ∈ ℝ1 2) are suitable choices [18]. 11 E. g. in an input estimation the signal ψ corresponds to the indicated signal of the employed sensor and the processed signal γ equals the estimate of the input signal to the sensor, i. e. the measurand. 12 The GUM [1] considers two types of uncertainty: While type A uncertainty is calculated based on provided observations, type B uncertainty is established by scientific judgment of multiple information source e. g. manufacturer specifications, calibration certificates, previous data or experience.

Modeling dynamic measurements in metrology and propagation of uncertainties

� 55

Figure 8: Frequency responses of the true, calibrated and inverted DUT behavior, as well as the regularized compensation filter. Results can be reproduced using Listing 2.4.

56 � M. Gruber et al.

Figure 9: Time series of the true simulated quantity, the indication of the device under test and the compensated indication. Results can be reproduced using Listing 2.4.

Modeling dynamic measurements in metrology and propagation of uncertainties

z[k] [ ] X = [ψ[k]] [ θ ] Y =[

� 57

(67)

z[k + 1] ] γ[k]

= f (X) = f (z[k], ψ[k], θ) ̃ ̃ ̃ ̃ Ã = A(θ), B̃ = B(θ), C̃ = C(θ), D̃ = D(θ) { { ̃ ̃ ={ z[k + 1] = Az[k] + Bψ[k] { ̃ ̃ γ[k] = Cz[k] + Dψ[k] {

(68)

The propagation according to Equation (4) requires the uncertainty Ux of X and the sensitivities C of f with respect to X. Typically, no correlations between the input signal, model parameters and model state are expected, leading to a sparse Ux . We assume here in addition that X follows a Gaussian distribution, such that the uncertainty Ux and sensitivity C are given by X ∼ g X = 𝒩 (μx , Ux ), U z[k] [ Ux = [ 0 [ 0 Ã C=[ ̃ C

0 U ψ[k] 0

B̃ D̃

Φz ]. Φγ

(69) 0 ] 0 ] = Cov(X), U θ]

(70) (71)

The sub-sensitivities Φz and Φγ are calculated such that13 𝜕z[k + 1] 𝜕θ 𝜕z[k] ̃ = A(θ) ⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝜕θ const.

Φz =

from previous

+

̃ ̃ 𝜕A(θ) 𝜕B(θ) z[k] ψ[k] ⏟⏟⏟⏟⏟⏟⏟ + ⏟⏟⏟⏟⏟⏟⏟ , ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝜕θ known ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝜕θ const.

𝜕γ[k] Φγ = 𝜕θ ̃ = C(θ) ⏟⏟⏟⏟⏟⏟⏟ const.

𝜕z[k] ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝜕θ

from Equation (72)

+

const.

known

̃ ̃ 𝜕C(θ) 𝜕D(θ) z[k] ψ[k] ⏟⏟⏟⏟⏟⏟⏟ + ⏟⏟⏟⏟⏟⏟⏟ . ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝜕θ known ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝜕θ const.

const.

known

(72)

(73)

The application of Equation (4) yields a block matrix

13 Initial values (k = 0) of z[k], 𝜕z[k] and U z[k] can be obtained e. g. by assuming a steady state for the 𝜕θ state-space model. Details are given in [38].

58 � M. Gruber et al.

Uy = [

Uzz Uγz

Uzγ ]. Uγγ

(74)

The lower right block provides the uncertainty information about the output signal γ[k], such that ̃ z[k] C̃ T + DU ̃ ψ[k] D̃ T + Φγ U θ ΦT . Uγγ = CU γ

(75)

5.2 FIR models Practical implementations of impulse response models are typically discrete-time FIR and IIR filters. Uncertainty for these can be propagated using the result of the previous subsection in conjunction with Equation (17) [8]. However, a specialized approach exists for FIR filters that provide computational benefits and operate on blocks of discrete time series. The approach suited for FIR filter makes use of efficient convolution operations and allows to provide full output covariance information. Stating the task in GUM notation uses the following input X and model f (see also Equations (1) and (15)) with T ψ X = [ ] = [ψ[0], . . . , ψ[N], b[0], . . . , b[m]] , b

(76)

T

Y = γ = [γ[0], . . . , γ[k], . . . , γ[N]] = f (X) = f (ψ, b),

∑m i=0 ψ[0 − i] ⋅ b[i] ] [ .. ] [ ] [ . ] [ m [ = [ ∑i=0 ψ[k − i] ⋅ b[i] ] ]. ] [ .. ] [ . ] [ m ψ[N − i] ⋅ b[i] ∑ i=0 ] [

(77)

The propagation stage requires to state the distribution of X. Independent multivariate Gaussian distributions are chosen for the input signal and filter coefficients. X ∼ gX = gψ ⋅ gb

(78)

g ψ = 𝒩 (μψ , Uψ )

(79)

g b = 𝒩 (μb , Ub )

(80)

The propagation for the FIR filter can then be carried out under full consideration of the measurement model and input distribution (without first-order approximations) [7, 39]. This involves the integration of the variance expression [40]

� 59

Modeling dynamic measurements in metrology and propagation of uncertainties

T

Uy = ∫ g X (η)(f (η) − y)(f (η) − y) dη,

(81)

ℝN+m

which yields the elements [uy ]i,j of the covariance matrix U y with T [uy ]i,j = b U ψi ,ψj b + ψTi U b ψj + Tr(U ψi ,ψj U b ) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(82)

first order only

where ψi and U ψi denote corresponding “cutouts” of ψ and Uψ matching the filter length m + 1 and time indices i, j, such that ψi = [ψ[i]

ψ[i − 1]

...

u(ψ[i], ψ[j]) .. . [u(ψ[i − m], ψ[j])

[ U ψi ,ψj = [ [

T

(83)

ψ[i − m]] ... .. . ...

u(ψ[i − m], ψ[j]) ] .. ]. . ] u(ψ[i − m], ψ[j − m])]

(84)

It should be noted, that in contrast to Equation (75), Equation (82) does not describe the uncertainty of the output vector at a single point in time, but the covariance matrix of multiple sequential scalar output values. Equation (82) can be efficiently implemented using two-dimensional convolution operations; see also Section 6.

5.3 IIR models Uncertainty propagation for IIR models refers back to the uncertainty propagation of state-space systems by transforming the IIR filter (e. g.) using Equation (17). The uncertainty propagation is then carried out using the respective state-space model [8]. Alternatively, Monte Carlo methods can be used to evaluate the uncertainty of IIR filter models. Different available methods here optimize the calculations for implementation simplicity, parallel scalability, memory usage or online use. Further information is given in Section 6.

5.4 Frequency response models Let ψ be a real-valued equidistant discrete time signal of length N with complex frequency spectrum Ψ and a complex transfer behavior H – represented by vectors ψ, Ψ and H respectively. For ease of notation, we assume N to be even. T

T

ψ = [ψ[0], . . . , ψ[N − 1]] = [ψ(t0 + 0Δt), . . . , ψ(t0 + (N − 1)Δt)] T

(85) T

−1 0 Ψ = [Ψ[0], . . . , Ψ[N − 1]] = [Ψ( NΔt ), . . . , Ψ( N/2−1 ), Ψ( −N/2 ), . . . , Ψ( NΔt )] NΔt NΔt T

T

−1 0 H = [H[0], . . . , H[N − 1]] = [H( NΔt ), . . . , H( N/2−1 ), H( −N/2 ), . . . , H( NΔt )] NΔt NΔt

(86) (87)

60 � M. Gruber et al. ψ and Ψ are linked by the discrete Fourier transform N−1

Ψ[k] = ∑ ψ[n] exp(−jk n=0

2πn ) N

(88)

Recall that for real valued signals only half the spectrum is required for reconstruction. To take both of these aspects into account, consider the following representation of the signal spectra Ψ and H: ̃ = [R(Ψ[0]), . . . , R(Ψ[N/2 − 1]), I(Ψ[0]), . . . , I(Ψ[N/2 − 1])]T Ψ

(89)

H̃ = [R(Ψ[0]), . . . , R(Ψ[N/2 − 1]), I(Ψ[0]), . . . , I(Ψ[N/2 − 1])]

(90)

T

To evaluate the uncertainty for Equation (19) the following mapping to the GUM notation is used [41] ̃ Ψ X = [ ̃] H

(91)

R(Γ) Y = Γ̃ = [ ] I(Γ) ̃ ⊙ H̃ = f (X) = Ψ ̃ ̃ H[0] ⋅ Ψ[0] ] .. ] . ] ̃ ̃ H[N] ⋅ Ψ[N] ] [

[ =[ [

(92)

The propagation stage requires to state the distribution of X. It is assumed that the spectra of the input signal and transfer behavior are independent,which is reflected by the block diagonal structure of Ux . X ∼ g X = 𝒩 (μx , Ux ) Ux = [

UΨ ̃ 0

C=[

T H̃

0

(93)

0 ] = Cov(X) U H̃

(94)

0 T] ̃ Ψ

(95)

Application of Equation (4) directly yields the uncertainty of the filtered spectrum Γ̃ T ̃ ̃T ̃ Uy = H̃ U Ψ ̃ H + Ψ U H̃ Ψ

(96)

UΨ ̃ and U H̃ are made up of blocks covering the covariances between all real parts (RR), all imaginary parts (II) and the real-imaginary-mixed parts (RI, IR).

Modeling dynamic measurements in metrology and propagation of uncertainties

UΨ ̃ =[

U RR,Ψ U IR,Ψ

U RI,Ψ ] U II,Ψ

U H̃ = [

U RR,H U IR,H

U RI,H ] U II,H

� 61

(97)

5.5 Discrete Fourier transform As detailed by [41], there are more things to consider when working in the frequency domain. Many measurement processes do not directly record frequency domain values and one has to transform the measured data from the time-domain into the frequencydomain (and back, if required). The tool of choice is typically the discrete Fourier transform (DFT, see Equation (88)). To evaluate the uncertainty propagation for the DFT, consider the following mapping to the GUM notation: X =ψ

(98)

̃=[ Y =Ψ

R(Ψ) ] I(Ψ)

C = f (X) = [ cos ] ψ Csin

(99)

with Ccos and Csin transforming Equation (88) into a matrix expression for separate real and imaginary parts in the output.

Ccos

cos(0 ∗ β0 ) [ .. [ =[ . N cos( ∗ β0 ) [ 2

− sin(0 ∗ β0 ) [ .. Csin = [ . [ N − sin( ∗ β0 ) [ 2 n βn = 2π N

... .. . ... ... .. . ...

cos(0 ∗ βN−1 ) ] .. ] . ] cos( N2 ∗ βN−1 )]

− sin(0 ∗ βN−1 ) ] .. ] . ] N − sin( 2 ∗ βN−1 )]

(100)

(101) (102)

To carry out the uncertainty propagation, a multivariate Gaussian distribution is assumed for X. The sensitivity matrix C is composed of the individual sensitivities for the real (Ccos ) and imaginary (Csin ) parts given by X ∼ g X = 𝒩 (μx , Ux ),

Ux = Uψ = Cov(X), C C = [ cos ] . Csin

(103) (104) (105)

The uncertainty is then directly given by Equation (4) which upon further inspection presents the block-matrix structure already mentioned earlier in Equation (97).

62 � M. Gruber et al. Ccos Ux Csin T ] Csin Ux Csin T

C U C T Uy = [ cos x cosT Csin Ux Ccos U = [ RR,Ψ U IR,Ψ

U RI,Ψ ] U II,Ψ

(106) (107)

= UΨ ̃

(108)

5.6 Inverse discrete Fourier transform The inverse discrete Fourier transform (IDFT) is very similar to the DFT and uncertainty evaluation for the IDFT is given in [41]. Formulating it in terms of the GUM notation yields: ̃ X =Ψ

(109)

Y =ψ

= f (X) = [C̃cos

̃ C̃sin ] Ψ

(110)

with C̃cos = W Ccos C̃sin = W Csin 1 [ 1 [ [0 W = [. N [. [. [0

(111) (112) 0 2 ...

... ..

. 0

0 .. ] .] ] ] ] 0] 2]

(113)

To prepare the propagation stage, assume a multivariate Gaussian distribution for X that has a block matrix structure. The sensitivity is (similarly to the DFT) given by the compound of both individual sensitivities. X ∼ g X = 𝒩 (μx , Ux ) U RR,Ψ Ux = U Ψ ̃ =[ U IR,Ψ C = [C̃cos

(114) U RI,Ψ ] = Cov(X) U II,Ψ

C̃sin ]

(115) (116)

Propagation according to Equation (4) and exploiting the block structure yields T T T Uy = C̃cos U RR,Ψ C̃cos + 2C̃cos U RI,Ψ C̃sin + C̃sin U II,Ψ C̃sin

= Uψ

(117) (118)

Modeling dynamic measurements in metrology and propagation of uncertainties

� 63

6 Software support: the PyDynamic library PyDynamic is a “Python library for the analysis of dynamic measurements” [42] with special attention to GUM-compliant uncertainty evaluation. It provides implementations for many of the methods presented in this chapter. Section 3 covered the calibration of dynamic sensors. The estimation of parameters for FIR, IIR and second order models to fit a given frequency response can be accomplished by the following functions from the model_estimation module: – PyDynamic.model_estimation.LSFIR – PyDynamic.model_estimation.LSIIR – PyDynamic.model_estimation.fit_som All three methods propagate uncertainty according to a Monte Carlo scheme. Section 4 dealt with the estimation of an inverse system. The functions LSFIR and LSIIR already mentioned above can be used for this purpose as well by setting the appropriate switch to true (e. g. LSIIR(..., inv=True)). Section 5 covered the propagation of uncertainty through a given system or operation. The propagation schemes for FIR and IIR filters are covered by the uncertainty module – PyDynamic.uncertainty.FIRuncFilter – PyDynamic.uncertainty.IIRuncFilter – PyDynamic.uncertainty.MC – PyDynamic.uncertainty.SMC – PyDynamic.uncertainty.UMC The MC, SMC and UMC methods also provide IIR uncertainty evaluation, but based on different Monte Carlo implementations in contrast to the analytical calculation in IIRuncFilter. MC implements a straight forward Monte Carlo variant of an IIR filter. SMC achieves a memory efficient implementation by only evaluating the filter output one time-step at a time, hence advancing sequentially in time. UMC implements another memory efficient variant by evaluating one (or multiple) time-series at a time from which the mean and standard deviation of the result are updated incrementally. Moreover, the propagation schemes from and to the frequency domain, as well as operations (multiplication, division, conversion of polar and Cartesian complex representations) are provided: – PyDynamic.uncertainty.GUM_DFT – PyDynamic.uncertainty.GUM_iDFT – PyDynamic.uncertainty.DFT_multiply – PyDynamic.uncertainty.AmpPhase2DFT() – PyDynamic.uncertainty.DFT2AmpPhase

64 � M. Gruber et al. Although not mentioned in this chapter, the uncertainty module also provides functions for uncertainty-aware interpolation, discrete wavelet transformation and convolution. Some of these functions are used in the examples of this chapter to which the source code is available in Appendix A. Further documentation and implementation details of PyDynamic are available at https://github.com/PTB-M4D/PyDynamic, https:// pydynamic.readthedocs.io/en/latest/index.html.

Appendix A. Practical examples in Python A.1 Prerequisites to run the code The above code examples are executed using the following packages: – Python 3.11.2 – NumPy 1.24.2 – SciPy 1.10.1 – Matplotlib 3.7.0 – PyDynamic 2.3.2 To set up a programming environment, please refer to the official Python documentation: https://docs.python.org/3/installing/index.html.

A.2 Code implementations The following code blocks (Listings 2.1 to 2.5) only highlight the main calculation steps of each example. The full code for the examples (which includes the required imports and the plot-related code) of this book is published as [43] and can be accessed at the following web addresses: https://doi.org/10.5281/zenodo.7189016. # measurand signal N = 200 dt = 0.05 t = dt * np.arange(N) x = np.full(N, 1.0) x[N//3:2*N//3] = 0.0 ux = np.full_like(x, 0.002) # sensor transfer behavior (second order lowpass with resonance) wc = 8 # / 2pi zeros = np.array([3.5]) * wc poles = np.array([-0.25+1j, -0.25-1j]) * wc b_gain, a_gain = zpk2tf(zeros, poles, 1.0) gain = a_gain[-1] / b_gain[-1] # ensure stationary accuracy

Modeling dynamic measurements in metrology and propagation of uncertainties

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b, a, _ = cont2discrete(zpk2tf(zeros, poles, gain), dt=dt) b = np.squeeze(b) Uba = np.diag(np.hstack((np.full(len(b), 1e-8), np.full(len(a)-1, 1e-8)))) # sensor indication signal state0 = x[0] * lfilter_zi(b, a) y, _ = lfilter(b, a, x, zi=state0) uy = np.full_like(y, 5 / 2**8) # from 8-bit quantization # get inverse filter b_inv, a_inv, uab_inv = MC_inverse_iir(b, a, Uba, dt_input=dt, dt_IIR=dt, tau=1) # obtain estimate of measurand by applying inverse filter init_state = IIR_get_initial_state(b_inv, a_inv, Uab=uab_inv, x0=y[0], U0=uy[0]) x_hat, ux_hat, _ = IIRuncFilter(y, uy, b_inv, a_inv, Uab=uab_inv, kind="diag", state=init_state) Listing 2.1: Simulation and compensation of a dynamic sensor with uncertainty.

# estimated parameter values delta_hat = 0.01 # 1 f0_hat = 30e3 # Hz omega0_hat = 2 * np.pi * f0_hat # Hz S0_hat = 1 # 1 U_theta_hat = np.diag(np.square([0.1 * delta_hat, 0.03 * f0_hat, 0.01 * S0_hat])) # a b C

estimated filter coefficients = np.array([1.0, 2 * delta_hat * omega0_hat, omega0_hat ** 2]) = np.array([S0_hat * omega0_hat ** 2]) = np.array( [ [2 * omega0_hat, 2 * delta_hat, 0], [0, 2 * omega0_hat, 0], [0, 2 * omega0_hat * S0_hat, omega0_hat ** 2], ] ) # sensitivities ab = np.hstack([a[1:], b]) Uab = C @ U_theta_hat @ C.T # discrete filter coefficients fs = 500e3 # Hz b_discrete, a_discrete, Uab_discrete = MC_cont2discrete( b, a, Uab, dt=1.0 / fs, runs=500 ) # get spectra W_conti, H_conti, UH_conti_diag = MC_freq(b, a, Uab) W_discrete, H_discrete, UH_discrete_diag = MC_freq(

66 � M. Gruber et al.

b_discrete, a_discrete, Uab_discrete, dt=1.0 / fs, use_w=W_conti ) # get impulse / step response t_discrete = np.arange(0, 1000 / fs, 1 / fs) t_discrete, h_discrete, uh_discrete_diag = MC_impulse( b_discrete, a_discrete, Uab_discrete, dt=1.0 / fs, use_t=t_discrete ) Listing 2.2: Visualization of the transfer behavior of an acceleration sensor in the frequency and time domain.

# true device under test (DUT) transfer behavior delta_true = 8.3e-3 # 1 f0_true = 29.4e3 # Hz omega0_true = 2 * np.pi * f0_true # Hz S0_true = 0.985 # 1 a = np.array([1.0, 2 * delta_true * omega0_true, omega0_true ** 2]) b = np.array([S0_true * omega0_true ** 2]) # true acceleration fs = 500e3 # Hz t = np.arange(0, 0.01, 1 / fs) acc_true = chirp(t, f0=10, t1=t[-1], f1=40e3, method="linear", phi=-90) # measured reference signal u_ref = 0.05 acc_ref = acc_true + u_ref * np.random.randn(acc_true.size) # measured DUT signal t, acc_dut, xout = lsim((b, a), U=acc_true, T=t) # transfrom to frequency domain f = GUM_DFTfreq(len(t), dt=1 / fs) F_ref, UF_ref = GUM_DFT(acc_ref, u_ref ** 2) F_dut, UF_dut = GUM_DFT(acc_dut, 0.0) H, UH = DFT_deconv(F_ref, F_dut, UF_ref, UF_dut) H = real_imag_2_complex(H)

# calibration (fit IIR to empirical spectrum) # only fit up to 40kHz, as noise is expected above fi = fit_indices = f < 40e3 fifi = np.hstack((fi, fi)) b_discrete, a_discrete, tau, Uab_discrete = LSIIR( H[fi], UH=UH[fifi, :][:, fifi], f=f[fi],

Modeling dynamic measurements in metrology and propagation of uncertainties

� 67

Nb=1, Na=2, Fs=fs, mc_runs=500, verbose=False, ) # calculate true transfer functions of true sensor and DUT (only for visualization) W, H_true = freqs(b, a, worN=2 * np.pi * f) W, H_dut, UH_dut = MC_freq( b_discrete, a_discrete, Uab_discrete, dt=1 / fs, use_w=2 * np.pi * f, return_full_cov=True, ) UH_dut_diag = np.sqrt(np.diag(UH_dut)) UH_dut_abs = np.abs(real_imag_2_complex(UH_dut_diag)) UH_dut_phase = np.angle(real_imag_2_complex(UH_dut_diag)) Listing 2.3: Visualize and perform a calibration for an acceleration sensor based on simulated data.

# true device under test (DUT) transfer behavior ( taken from Section 3.6 ) a_true = np.array([1.00e00, 3.07e03, 3.41e10]) b_true = np.array([3.36e10]) # estimate of DUT parameters ( taken from Section 2.4 ) a = np.array([1.00e00, 3.77e03, 3.55e10]) b = np.array([3.55e10]) Uab = np.array( [ [1.42e05, 6.11e09, 6.11e09], [6.11e09, 1.15e17, 1.15e17], [6.11e09, 1.15e17, 2.41e17], ] ) # estimated DUT inverse behavior fs = 500e3 # Hz b_inv, a_inv, Uab_inv = MC_inverse_iir(b, a, Uab, dt_input=None, dt_IIR=1 / fs) # lowpass regularization b_low, a_low = cheby2(4, 30, 100e3, btype="low", analog=False, output="ba", fs=fs) b_reg, a_reg, Uab_reg = chain_filter_with_lowpass(b_inv, a_inv, Uab_inv, b_low, a_low) # acceleration input t = np.arange(0, 0.01, 1 / fs)

68 � M. Gruber et al.

acc_true = chirp(t, f0=10, t1=t[-1], f1=40e3, method="linear", phi=-90) # measured DUT signal t, acc_dut, _ = lsim((b_true, a_true), U=acc_true, T=t) Uacc_dut = np.full_like( t, 0.2 ) # assumed uncertainty of indiciation, i.e. known from a calibration # compensated DUT signal acc_comp, Uacc_comp, _ = IIRuncFilter( acc_dut, Uacc_dut, b_reg, a_reg, Uab_reg, kind="diag" ) Listing 2.4: Visualize and perform an input estimation for an acceleration sensor.

def MC_cont2discrete(b, a, Uab, dt, runs=500): ab = np.hstack((a[1:], b)) Na_conti = len(a) - 1 results = [] AB = multivariate_normal(ab, Uab, allow_singular=True).rvs(runs) for ab_tmp in AB: a_tmp = np.hstack(([1.0], ab_tmp[:Na_conti])) b_tmp = ab_tmp[Na_conti:] b_discrete, a_discrete, _ = cont2discrete((b_tmp, a_tmp), dt=dt) results.append(np.hstack((a_discrete[1:], np.squeeze(b_discrete)))) Na_discrete = a_discrete.size - 1 ab_discrete = np.mean(results, axis=0) a_discrete = np.hstack(([1.0], ab_discrete[:Na_discrete])) b_discrete = ab_discrete[Na_discrete:] uab_discrete = np.cov(np.array(results).T) return b_discrete, a_discrete, uab_discrete

def MC_step(b, a, Uab, runs=500, dt=None, use_t=None): ab = np.hstack((a[1:], b)) Na = len(a) - 1 t = use_t h_list = [] AB = multivariate_normal(ab, Uab, allow_singular=True).rvs(runs) for ab_tmp in AB: a_tmp = np.hstack(([1.0], ab_tmp[:Na])) b_tmp = ab_tmp[Na:] if dt is None: t, h_tmp = step((b_tmp, a_tmp), T=t) else: t, h_tmp = dstep((b_tmp, a_tmp, dt), t=t) h_tmp = np.squeeze(h_tmp)

Modeling dynamic measurements in metrology and propagation of uncertainties

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h_list.append(h_tmp) h = np.mean(h_list, axis=0) uh = np.cov(np.array(h_list).T) uh_diag = np.sqrt(np.diag(uh)) return t, h, uh_diag

def MC_impulse(b, a, Uab, runs=500, dt=None, use_t=None): ab = np.hstack((a[1:], b)) Na = len(a) - 1 t = use_t h_list = [] AB = multivariate_normal(ab, Uab, allow_singular=True).rvs(runs) for ab_tmp in AB: a_tmp = np.hstack(([1.0], ab_tmp[:Na])) b_tmp = ab_tmp[Na:] if dt is None: t, h_tmp = impulse((b_tmp, a_tmp), T=t) else: t, h_tmp = dimpulse((b_tmp, a_tmp, dt), t=t) h_tmp = np.squeeze(h_tmp) h_list.append(h_tmp) h = np.mean(h_list, axis=0) uh = np.cov(np.array(h_list).T) uh_diag = np.sqrt(np.diag(uh)) return t, h, uh_diag

def MC_freq(b, a, Uab, dt=None, runs=500, use_w=None, return_full_cov=False): W = use_w ab = np.hstack((a[1:], b)) Na = len(a) - 1 H_list = [] AB = multivariate_normal(ab, Uab, allow_singular=True).rvs(runs) for ab_tmp in AB: a_tmp = np.hstack(([1.0], ab_tmp[:Na])) b_tmp = ab_tmp[Na:] if dt is None: W, H_tmp = freqs(b_tmp, a_tmp, worN=W) else: W, H_tmp = freqz(b_tmp, a_tmp, worN=W, fs=2 * np.pi / dt) H_tmp = np.squeeze(H_tmp) H_list.append(H_tmp)

70 � M. Gruber et al.

HH = np.array(H_list) H = np.mean(HH, axis=0) UH = np.cov(np.hstack((np.real(HH), np.imag(HH))).T) if return_full_cov: return W, H, UH else: UH_diag = np.sqrt(np.diag(UH)[: len(W)] + np.diag(UH)[len(W) :]) return W, H, UH_diag

def MC_inverse_iir(b, a, Uab, runs=500, dt_input=None, dt_IIR=1.0, tau=0.0): # this seems not to work currently # generate frequency response with uncertainty via Monte Carlo W, H, UH = MC_freq(b, a, Uab, dt=dt_input, return_full_cov=True) Na = len(a[1:]) Nb = len(b) b_inv, a_inv, tau_inv, Uab_inv = LSIIR( H, UH=UH, Nb=Na+2, Na=Nb, f=W / (2*np.pi), Fs=1.0 / dt_IIR, tau=tau, inv=True, mc_runs=runs ) # gain correction gc = np.sum(a_inv) / np.sum(b_inv) b_inv = gc * b_inv Uab_inv[Na:,Na:] = Uab_inv[Na:,Na:] * gc**2 return b_inv, a_inv, Uab_inv

def chain_filter_with_lowpass(b, a, Uab, b_low, a_low): # define shortcuts Na = len(a) - 1 uaa = block_diag([0.0], Uab[:Na,:Na]) ubb = Uab[Na:,Na:] bc, Ubc = convolve_unc(b, ubb, b_low, None, mode="full") ac, Uac = convolve_unc(a, uaa, a_low, None, mode="full") UC = block_diag(Uac[1:,1:], Ubc) return bc, ac, UC Listing 2.5: Monte-Carlo helper functions used in the examples.

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[29] N. Hansen. The CMA Evolution Strategy: A Comparing Review. In J. A. Lozano, P. Larrañaga, I. Inza and E. Bengoetxea, editors, Towards a New Evolutionary Computation: Advances in the Estimation of Distribution Algorithms. Studies in Fuzziness and Soft Computing, pages 75–102. Springer, Berlin, 2006. [30] J. Nocedal and S. Wright. Numerical Optimization. Springer Series in Operations Research and Financial Engineering. Springer, New York, 2nd edition, 2006. [31] P. J. Parker and R. R. Bitmead. Approximation of Stable and Unstable Systems via Frequency Response Interpolation. IFAC Proc. Vol., 20(5, Part 10):357–362, 1987. [32] R. C. Aster, B. Borchers, and C. H. Thurber. Parameter Estimation and Inverse Problems. Elsevier, 3rd edition, 2018. [33] S. Eichstädt, V. Wilkens, A. Dienstfrey, P. Hale, B. Hughes, and C. Jarvis. On challenges in the uncertainty evaluation for time-dependent measurements. Metrologia, 4(53):S125, 2016. [34] T. Strutz. Data Fitting and Uncertainty (A Practical Introduction to Weighted Least Squares and Beyond), 2010. [35] C. Elster and B. Toman. Bayesian uncertainty analysis for a regression model versus application of GUM Supplement 1 to the least-squares estimate. Metrologia, 48(5):233–240, 2011. [36] R. Vuerinckx, Y. Rolain, J. Schoukens, and R. Pintelon. Design of stable iir filters in the complex domain by automatic delay selection. IEEE Trans. Signal Process., 44(9):2339–2344, 1996. [37] M. Wegener and E. Schnieder. Application of the GUM method for state-space systems in the case of uncorrelated input uncertainties. Meas. Sci. Technol., 24(2):025003, 2013. [38] M. Gruber, T. Dorst, A. Schütze, S. Eichstädt, and C. Elster. Discrete wavelet transform on uncertain data: Efficient online implementation for practical applications. In Advanced Mathematical and Computational Tools in Metrology and Testing XII, volume 90 of Series on Advances in Mathematics for Applied Sciences, pages 249–261, World Scientific, 2021. [39] C. Elster, A. Link, and Th. Bruns. Analysis of dynamic measurements and determination of time-dependent measurement uncertainty using a second-order model. Meas. Sci. Technol., 18(12):3682–3687, 2007. [40] M. G. Cox and B. R. L. Siebert. The use of a Monte Carlo method for evaluating uncertainty and expanded uncertainty. Metrologia, 43(4):S178, 2006. [41] S. Eichstädt and V. Wilkens. GUM2DFT—a software tool for uncertainty evaluation of transient signals in the frequency domain. Meas. Sci. Technol., 27(5):055001, 2016. [42] B. Ludwig, M. Gruber, P. Kruse, S. Eichstädt, M. Weber, M. Sieberer, A. M. Piniella, and Th. Bruns. PTB-M4D/PyDynamic. Zenodo, December 2021. [43] M. Gruber. PTB-M4D/dgsm_code_examples: Initial Code Release. Zenodo, October 2022.

Th. Bruns, L. Klaus, and M. Kobusch

Traceable calibration for the dynamic measurement of mechanical quantities The vast majority of inertial motion sensors nowadays are so-called seismic sensors. This term is owed to the fact that the sensing element makes use of the inertia of a seismic mass, which is coupled to the housing via an elastical structure that can be described as a spring. When the housing is moved and changes its state of motion, the inertia counteracts the connecting spring, and by measuring the deformation of the spring, we obtain a measure of the applied change of motion. The simplest (and most frequently used) approach to model this motion sensor is a single mass-spring system of one degree of freedom, where the measurement equation relates a motion quantity of the (rigid) housing to the resulting elongation of the spring. Figure 1 illustrates the principle. The equation of motion xs of the seismic mass ms in relation to the motion of the base xb is then given by 0 = d(ẋb − ẋs ) + k(xb − xs ) − ms ẍs ,

(1)

Figure 1: Illustration of the motion sensor as a spring-mass-damped system of one degree of freedom.

where the first component on the right-hand side is the (viscous) damping force due to the deformation velocity of the spring, the second term is the elastic force due the deformation itself, and the third term is the inertial reactive force of the seismic mass. Dividing the equation by ms transforms it to an equation of motion quantities instead of forces and scales the coefficients: 0=

d k (ẋ − ẋs ) + (x − xs ) − ẍs . ms b ms b

(2)

With the usual approach of a (complex) monofrequency vibration for all the motions like x(t) = x̂ ⋅ eiωt , https://doi.org/10.1515/9783110713107-003

̇ = iω x̂ ⋅ eiωt , x(t)

̈ = −ω2 x̂ ⋅ eiωt , x(t)

(3)

74 � Th. Bruns et al. one can derive a relation between the elongation of the spring (xs − xb ) and the acceleration of the housing (ẍb ): x̂s − x̂b x̂s − x̂b = = ω2 x̂b x̂̈b

k ms

1

+ iω md − ω2

(4)

.

s

For the assumption of an electrical output proportional to the spring deformation, and with the usual textbook substitution of k/ms = ω20 for the (undamped) resonance frequency and d/ms = γ for the specific damping, this is already the transfer function of an accelerometer usually written as Sqa = ρ ⋅

ω20

S0 1 = 2 γ + iγω − ω 1 + i ω2 ω − 0

ω2 ω20

.

(5)

Here, for example, Sqa models the so-called complex charge sensitivity of an accelerometer, with ρ being the charge conversion factor of a piezoelectric spring, and S0 being the static sensitivity. For the case of different kinds of motion sensors, like geophones or seismometers, which measure velocity instead of acceleration, the approach is the same but the ratio considered in equation (4) has to be related to the base velocity ẋb instead of acceleration ẍb .

1 Calibration of motion sensors Equation (5) provides the idealized transfer function of a linear seismic accelerometer. However, there are three parameters in this equation, which are generally unknown, depending on the individual sensor and to some extent even on environmental conditions. Those are S0 (which describes the asymptotic low frequency sensitivity), γ, and ω0 . In order to derive the actual acceleration from the electrical output of the sensor, those parameters, or more generally, the complex transfer function has to be determined experimentally. This process of calibration can be performed in several ways, which are internationally harmonized via the series of the ISO 16063 standards. These guidelines distinguish between primary calibration by the use of laser interferometry and secondary calibration by the use of a reference standard, which is a calibrated motion sensor itself. In the following sections, both approaches are briefly introduced.

1.1 Primary calibration by laser interferometry Acceleration is the second time derivative of the displacement. Hence, for a primary realization, the obvious approach is to measure displacement, which is a length, and time with high accuracy and resolution. Accurate time measurements in the context of me-

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chanical processes are no exceptional problem with the currently available oscillators in today’s data acquisition systems. Time-dependent (dynamic) length measurements are more challenging. In length metrology, the use of laser interferometry is common practice for decades by now, and it is used in acceleration metrology for almost the same time. A typical setup for primary calibration is depicted in Figure 2. Its basic components are a electrodynamic exciter (shaker), which provides a sinusoidal vibration to the motion sensor to be calibrated, a laser interferometer, which is capable to measure the dynamics of the applied motion, and the motion sensor itself. Not shown are the data acquisition devices to capture the electric signals from the measuring devices. During calibration, a monofrequency sinusoidal vibration is applied to the motion sensor and its output is sampled over time with the data acquisition system. Simultaneously, the motion of the reference surface (of the motion) is measured with the laser interferometer. In order to derive the full complex frequency response, the timing relation between the two channels must be well-defined; the simplest case here is a synchronous sampling system. The sampled motion data of the distinct channels are subsequently analyzed by the sine approximation method, which fits a sine function with a bias of the form x(t) = a ⋅ sin(ωt) + b ⋅ cos(ωt) + c

(6)

Figure 2: Illustration of a primary calibration setup for motion sensors using a laser interferometer. The principle is shown on the left, a realization on the right. The red arrow points at the (tiny) accelerometer.

to the measured data. Amplitude and initial phase of the signal is then given as x̂ = √a2 + b2 ,

a φx = arctan( ). b

Based on that, the model functions for the signals after fitting are

(7)

76 � Th. Bruns et al. ̇ = ωx̂ ⋅ ei(ωt+φx + 2 ) , x(t) π

i(ωt+φq )

q(t) = q̂ ⋅ e

̈ = ω2 x̂ ⋅ e−i(ωt+φx +π) , x(t)

(8) (9)

.

Equation (8) models the velocity or acceleration based on the interferometric data and equation (9) models the electrical response of the motion sensor. Since interferometers sense displacement, the model functions in (8) are scaled and phase-shifted from the original fit to transform the measurand from displacement to vë The measured transfer coefficient for the angular frequency locity (x)̇ or acceleration (x). ω in relation to acceleration is then given as Sqa (ω) =

̂ q(ω) 󵄨 󵄨 iφ (ω) ⋅ ei(φq −φx −π) = 󵄨󵄨󵄨Sqa (ω)󵄨󵄨󵄨 ⋅ e Sqa , ̂ ω2 x(ω)

(10)

which is the experimental result according to equation (5). |Sqa | is called the magnitude and φSqa the phase of the complex sensitivity. More details on the standardized procedure are given in [1].

1.2 Secondary calibration Once primary calibrated motion sensors are available (cf. previous section), it is a rather simple task to compare an unknown sensor, the device under test (DUT), with one that is already traceably calibrated, the reference. This process of direct comparison of two motion sensors is called a secondary calibration [2]. Direct comparison in this context means that both sensors are mounted together on the exciter and both outputs during the vibration excitation are again sampled synchronously. Figure 3 depicts such a socalled back-to-back configuration as a schematic and in a photograph. There are special sensor designs available for such back-to-back mounting, where the seismic system is connected to the top side of the sensor. Also, in secondary calibration, the excitation is usually applied as a single frequency sinusoidal motion. Accordingly, the amplitude and the initial phase of the electrical output can be determined by sine approximation again. Provided that the known complex sensitivity of the reference is Sref , and its output amplitude and initial phase are q̂ ref and φref , respectively, then the reference motion during calibration was aref =

q̂ ref ⋅ ei(ωt+φref ) qref = . Sref Sref

(11)

According to the definition of the complex sensitivity, this provides the necessary information for the calculation of the complex sensitivity of the DUT as SDUT (ω) =

qDUT (ω) ⋅ Sref (ω), qref (ω)

(12)

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Figure 3: Illustration of a secondary calibration setup for motion sensors using the preferred back-to-back configuration. The left image shows the principle, and the right shows a realization. Note that the reference planes of the two seismic systems are in direct contact to each other, because the sensor at the bottom has internally an inverted arrangement of mass and spring.

or in terms of magnitudes and phases |SDUT | = |Sref | ⋅

q̂ DUT , q̂ ref

φDUT = φout − φin − φref .

(13) (14)

The charge output qx is usually derived from a voltage measurement via a dynamically calibrated charge amplifier (cf. Section 2). For this secondary calibration, the standardized methods are documented in [2].

1.3 Disturbing effects of the real world The calibration conditions were idealized in the previous Sections 1.1 and 1.2. This assumes, e. g., a purely uniaxial motion strictly in the direction of the sensitive axis of reference and a device under test with a clean sinusoidal curve over time. In reality, those assumptions fail to a smaller or larger degree and give rise to deviations to be quantified by the uncertainty of measurement. Typical effects influencing the magnitude of sensitivity include:1 – Misalignment between the direction of motion and direction of sensitivity. – Transverse sensitivity of reference or DUT in conjunction with rocking or transverse motion [3]. 1 The lists show typical examples and are not meant to be exhaustive.

78 � Th. Bruns et al. – – – – –

Asymmetric motion measurement by laser interferometry [4]. Electromagnetic interference from magnetic stray fields of the electrodynamic shaker. Relative motion between the reference and the DUT (due to finite mechanical stiffness or non-optimal mounting) [5, 6]. Residual alternating gravitational components. Mass loading sensitivity of the reference in back-to-back calibration.

Effects influencing the phase of the sensitivity include: – Dynamic tilting motion of the shaker armature. – Unknown (group) delay in complementary components like conditioning amplifiers or laser interferometers [7, 8]. – Jitter of the sample clock of either data acquisition channel. The international standards of the ISO 16063 series require that such influencing effects are investigated, analyzed, and quantified in a measurement uncertainty budget. Accordingly, a proper measurement uncertainty evaluation is an elaborate and complex task. However, guidance can be found in the standards themselves as well as the scientific literature (see references). Once determined, such calibration uncertainties are stated in the calibration certificates and should be used in subsequent applications of the sensor as an input to dynamic measurement uncertainty propagation as described in the previous chapter “Modeling dynamic measurements in metrology and propagation of uncertainties”.

1.4 Parameter identification Results from the previously described calibrations describe the behavior of motion sensors in the frequency domain for a selected set of discrete frequencies. Typically, those frequencies are taken from the set of standard frequencies according to the standard ISO 266 [9]. However, for general dynamic measurements it is often necessary to describe the sensor’s response in a continuous frequency range or even in the time domain. An imminent example of such a case are transient shock measurements of all kinds, where no predominant frequency could be identified. An appropriate approach to retrieve traceable measurements in such cases is the use of the model parameter identification; also often referred to as system identification. A prerequisite for the model parameter identification is (of course) an appropriate model of the used electromechanical system, i. e., the sensor or measuring chain in the application. Motion sensors, as considered in this chapter, are typically modeled as systems of masses, elastic springs, and viscous damping elements, which are already indicated in the previous figures.

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A simple linear single mass-spring-damper model like depicted in Figure 1 can be represented by the complex-valued transfer function of the form H(ω) =

Y (ω) = X(ω)

k ms iω md − s

S0 ⋅ k ms

+

ω2

=

ω20

S0 ⋅ ω20

+ iωγ − ω2

,

(15)

where the base sensitivity S0 , the seismic mass ms , the spring stiffness k, and the damping coefficient d are the physical parameters. However, only three of four parameters are actually mathematically independent, which results in the right-hand side of equation (15) with the parameter set {S0 , ω0 , γ}. Here, ω0 is the undamped resonance frequency and γ is the specific damping. Use cases and approaches for the identification process based on this model can be found in a large number in the literature (e. g., [10, 11]) and even made it into an international standard [12]. However, the simplicity of the model calls for an extension to improve the accuracy in some applications. Typically, this happens when the used frequency range comes close to or even covers resonances of the measurement setup. Then additional effects like the elasticity of the mounting or an internal structure of the sensor have to be considered [6, 13, 14]. From the mathematical point of view, the parameter identification is an optimization problem, which in the framework of this book which is handled in the previous chapter. From the metrological point of view, it has to be noted that the uncertainty of measurement is handled by a propagation of uncertainty from the calibration results to an uncertainty of the parameters. The methods for this transfer are introduced in the previous chapter.

2 Conditioning amplifiers Although the electromechanical conversion process of modeled motion sensors can be derived in a straightforward way, the measured data usually do not follow this description directly. This is due to the simple fact that the sensor is connected to the sampling system via a conditioning amplifier. These are devices that convert between different electrical quantities often with the additional option to adjust a gain factor and/or the frequency band of transmission. Typical types of conditioning amplifiers are listed in the subsequent table. For the purpose of measurements, these devices are designed to behave linearly in terms of input quantity value to output voltage. Hence, they can amplifier type

conversion from → to

sensor type

voltage amplifier charge amplifier bridge amplifier

voltage → voltage (gain and bandwidth only) charge → voltage resistance ratio → voltage

voltage output servoelectric piezoelectric piezoresistive strain gage

80 � Th. Bruns et al.

Figure 4: Graphical representation of the complex transfer function of conditioning amplifiers in the frequency domain, a voltage amplifier on the left and a charge amplifier on the right. In both cases, the top shows the phase response while the bottom shows the magnitude (gain) response.

be considered as LTI systems. Their transfer characteristics can usually be modeled as a combination of a high-pass and a low-pass filter [15, 16]. Typical frequency-domain transfer functions are depicted in Figure 4. While classical (analogue) electronic devices typically implement conversion stages with filter characteristics of second or third order, there are more and more devices on the market with intermediate digital processing, where high-order filters with sometimes tens of parameters are realized. While the specs of these latter devices may look excellent, their benefit in terms of input prediction processing and other metrological measurement data analysis still needs to be proven. With respect to modeling, the increased number of unknown parameters (and unknown number of parameters, in fact) poses a huge challenge for sure. Recently, comprehensive guidelines [17] were developed and published concerning the proper dynamic calibration of various types of conditioning amplifier. These describe the experimental setups, the calibration processes, the measurement uncertainty contributions, and common pitfalls. Since they are openly available, the details should not be discussed here. In preparation of this guideline, extensive investigations [7] took place, which repeatedly identified the impedance of the connected measurement chain as a potentially detrimental source of influence. This means two things: 1. The calibration data of a conditioning amplifier are only valid for a given measurement, if the source impedance of the sensor is close enough to the source impedance used during calibration of the amplifier. What “close enough” means in a particular case depends strongly on the individual type of conditioning amplifier and specifically on its complex input impedance [18]. 2. The input impedance of the data acquisition system has to be high in comparison to the output (source) impedance of the conditioning amplifier. The ratio of deviation caused by a too low input impedance is approximately given by the ratio of the two impedances of sampling system and amplifier. If both impedances are known, the influence can be corrected. If, however, the calibration is handled properly in all respect, the result is a complex frequency response function given by samples in the frequency domain, equivalent to the characterization of the motion sensor. The frequency response function of the com-

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pleted measuring chain consisting of sensor and conditioning amplifier is accordingly given by their product as H(ω)chain = Samp (ω) ⋅ Ssens (ω).

(16)

3 Dynamic force calibration The vast majority of force sensor calibrations are static calibrations only. For most static force calibrations, different force levels in tension or compression are applied for a certain time. The output of the sensor under test is read, and then the force load is changed to the next level [19]. Afterwards, a sensitivity can be calculated as a one-number value or a quadratic approximation, depending on the required level of measurement uncertainty. Many applications differ significantly from these calibration conditions. In case of rapidly changing force levels, a static calibration is not sufficient. Albeit not available universally, the dynamic calibration of force sensors was included into normative documents recently in a general way [20] and for dedicated applications [21]. The force sensors used for dynamic measurements typically incorporate one of two measurement principles: piezoelectric elements or strain gauges. The former type applies piezoceramics or piezocrystals, which emit or absorb electric charge if being compressed or decompressed [22]. These sensors are only suitable for dynamic measurements. Due to charge leakage, they cannot measure true static forces. The output signal of the conditioning electronics, the charge amplifier, is nearly always equipped with a high-pass filter to avoid a drifting output signal. As result of this, piezoelectric sensors are well suited for the measurement of dynamic forces in presence of a dominant static load, which then is not present in the measured signal. The latter strain-gauge based sensors are suitable for both static and dynamic measurements. These sensors have one or more structural elements with reduced stiffness. Strain gauges are applied on these places to sense the force-dependent deformation at these more compliant structural parts. Strain gauges are resistive elements that change their resistance proportionally to the applied elongation [23]. Multiple strain gauges can be connected in a Wheatstone bridge circuit. If a known voltage is applied, the small changes in resistance lead to an—also small—voltage, which can be conditioned in bridge amplifiers. Temperature and offset compensation can be realized by cleverly adding resistors in the bridge circuit. A wide range of strain gauge sensors from precision sensors, over sensors dedicated to dynamic measurements, to mass-produced load cells used in scales are available. More details on the measurement principle and the mechanical design of strain gauge sensors can be found in [24, 25]. Other force measurement principles exist but have no significance for traceable dynamic measurements. More details to the various measurement principles can be found in [24].

82 � Th. Bruns et al. Dynamic force applications In material testing, cyclic fatigue testing is carried out to demonstrate the safety of structures or to test the strength of materials when being exposed to cyclic loads. If these tests are carried out in fatigue testing machines, the applied load is often measured by means of force sensors. In the automotive industry, the safety of new vehicle developments is analyzed by means of crash tests. To analyze the force of the impact, the duration and the peak force during a crash test is measured using sensor-equipped walls. Often, the sensors used for crash applications are based on the piezoelectric measurement principle. Standards on instrumentation for crash tests in the automotive industry like ISO 6487 [26] or SAE J211/1 [27] already specify requirements for force sensors used in such tests.

3.1 State of the art in dynamic force calibration The dynamic calibration of force sensors is still a topic of scientific research. Although the challenges of traceable dynamic measurements are well known for a long time, up to now, no commonly accepted solutions are available. The general suitability of a force sensor for dynamic measurements can be roughly estimated on the basis of its resonance frequency, which is typically given by the manufacturer in the sensor’s data sheet, although it is unknown how the manufacturer derived the information (calculation, simulation, or measurement) and it is also not known how accurate the given data is. For reliable knowledge, the sensor needs to be calibrated. This always means that not only properties of the device under test are determined, but also the associated measurement uncertainty (obtained with the given calibration device) is estimated. The uncertainty allows to classify the degree of confidence of the calibration results.

3.1.1 Sine force excitation: electrodynamic shakers, prestressed frames with hydraulic exciters The dynamic calibration of force sensors began in the field of material testing machines [28, 29]. The excitation form used in material testing machines and in the dynamic calibration of force sensors is a mono-frequency sinusoidal force. Two different setups are typically used for this kind of calibration. An electrodynamic shaker can be used to generate oscillations in a wide range of frequency and for different types of waveforms. A mass-loaded force sensor mounted on the shaker armature experiences a force when accelerated. The dynamic force F(t) is dependent on the acceleration a(t) and on the mass m mounted on top of the force sensor according to Newton’s second law giving

Traceable calibration for the dynamic measurement of mechanical quantities

F(t) = m ⋅ a(t).

� 83

(17)

A calibration setup based on a electrodynamic shaker is shown in principle and as a photo in Figure 5.

Figure 5: Dynamic force calibration setup based on an electrodynamic shaker. The calibration principle is illustrated on the left; a photograph of a calibration setup is shown on the right [30].

This type of dynamic force calibration setup provides a primary calibration (i. e., a calibration traced to other quantities than force) in a wide frequency range (usually ranging from a few hertz to a few kilohertz). The acceleration can be measured by an accelerometer placed on the load mass or by means of a laser interferometer (as shown in Figure 5). The force magnitudes are limited to the capabilities of the shakers used and go up to about 10 kN. Such types of dynamic force calibration setups were established in several national metrology institutes (NMIs) worldwide [31–34]. The most simple dynamic characterization of the sensor obtained with such type of calibration setup is the sensor sensitivity S at different excitation frequencies ω. For this purpose, the measured force Fsens and the reference force input Fref (derived according to equation (17)) are compared giving the sensor’s transfer function S(ω) =

Fsens (ω) Fout (ω) = . Fref (ω) m ⋅ aref (ω)

(18)

This complex transfer function is usually given as magnitude response and phase response, respectively. Due to parasitic mechanical influences like tilting of the shaker, the reference acceleration might be not evenly distributed. Such possible influences need to be taken

84 � Th. Bruns et al. into account in case of a calibration [35, 36]. Calibrations like these are offered as a service from PTB [37]. International comparison measurements showed good agreement, but also issues due to parasitic motions [38]. Investigations of uniaxial force sensors are carried out on such setups, but also research on the calibration of multicomponent sensors was carried out [39–42]. In addition to calibration setups based on electrodynamic shakers, there are also calibration devices that operate with hydraulic excitation. Hydraulic exciters can generate much higher dynamic forces (in the range of 100 kN) but only up to frequencies of about 100 Hz. To generate dynamic forces by the above-mentioned principles based on Newton’s second law, large masses need to be coupled to the force sensor [43]. To generate higher force levels, the hydraulic exciter and the force sensor to be calibrated can be mounted and prestressed within a machine frame. This setup is very similar to material testing machines, which operate with such a frame, too. If two force sensors are placed in a series arrangement in this frame, a calibration can be carried out by comparison (secondary calibration). One of the sensors will be the reference and the other sensor under test is compared to this reference [43, 44]. 3.1.2 Shock force: Colliding masses The limited force magnitudes of electrodynamic shakers and the very limited frequency range of hydraulic exciters led to the development of shock force calibration setups in the early 2000s [45]. Shock force calibration devices use the same principle of traceability of the dynamic force as the shaker-based calibration devices (see equation (17)) and, therefore, also provide primary calibrations. The use of shock forces has some advantages and disadvantages compared to sinusoidal calibrations. Force magnitude: The forces generated in shock force calibration devices can be significantly larger than that of shaker-based setups. While electrodynamic shakers are limited to about 10 kN, shock force calibration devices for forces up to 250 kN and 1 MN exist [46, 47]. Frequency range, waveform control: Electrodynamic shakers can excite frequencies up to several kilohertz (while hydraulic exciters are limited to a few hundred hertz) and allow the operator to choose the waveform (single frequency, multifrequency sine, random noise excitation). In a shock force calibration device, the generated waveform is predominantly defined by the stiffness of the contact surface pairing as well as the stiffness and mass distribution of the sensor under test and the connected components. It is possible to control the contact properties to some extend (e. g., by softening and, therefore, stretching the shock pulse), but the possibilities are very limited compared to a shaker excitation.

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To date, two different types of shock force calibration devices have been established at different NMIs: The first type is a horizontal setup based on two movable masses. The first mass is accelerated to a certain velocity and then impacts onto the force sensor, which is mounted on the second mass. The (de) acceleration of the masses during the brief time of contact can be measured by accelerometers or laser interferometers. From the measurements on both masses, the impact force and the reaction force can be derived independently according to Newton’s second law (cf. equation (17)). Examples for this design are the setups at PTB [46, 48] or at the Shanghai Institute of Quality Inspection and Technical Research [49]. An illustration of a setup and a photo is displayed in Figure 6.

Figure 6: Shock force calibration setup. On the left, an illustration of a horizontal design, and on the right the experimental setup [30].

The second type is a vertical design that uses a falling mass, which impacts onto the force sensor mounted at the bottom. This design was implemented at NIM in China and NIS in Egypt [47, 50]. The height of the release position of the mass can be varied to adjust the force magnitude. The measurement principle is the same as of the aforementioned horizontal design. The dynamic force generated by the falling mass body is given by the product of the mass and time-dependent acceleration during impact.

3.2 Model-based dynamic force calibration A significant challenge of the dynamic calibration of force sensors (the same applies also to torque sensors) is the fact that they are always mechanically coupled to their environment on both sides. In contrast to, e. g., accelerometers, this environment influences the dynamic behavior of the sensor and always needs to be taken into account. If the sensor is placed into a different environment (e. g., in a calibration device for the calibration) or the mechanical properties of the application change, the dynamic behavior of the sensor may change, too. If a transfer function is derived like described in Section 3.1.1, it is only valid in later applications if coupled masses and their connection stiffnesses are the same as during calibration.

86 � Th. Bruns et al. A solution for the aforementioned issue of the influence of the mechanical environment can be a model-based calibration. In this case, the sensor is described by an appropriate mechanical model. During calibration, the model parameters for this model are identified; see above. The model usually chosen for force sensors resembles the mechanical design. The most simple model is linear and time-invariant (LTI) and consists of two rigid masses (base mass mB , top/head mass mH ) connected by a spring k and a damper d in parallel. Figure 7 illustrates the model in a calibration setup.

Figure 7: Illustration of the model of a force sensor mounted on a shaker, as shown in Figure 5. The sensor is described by two masses connected by a spring and a damper.

This model leads to an ordinary differential equation system as follows: M ẍ + Dẋ + Kx = L,

(19)

with the mass matrix M, damping matrix D, stiffness matrix K, and load vector L. The excitations are found in the displacement vector x and its derivative equivalents ẋ and x.̈ Based on the mechanical design of the force sensor and of the measuring device, different degrees of complexity of the sensor model and of the calibration device might be required. The most simple approach is the aforementioned four parameter model of the sensor. Sometimes it might be necessary to extend the sensor model to account for compliant mounting bolts of the sensor, or other elasticities. In other cases, the mounting of the mass bodies might be so compliant that an additional spring element is necessary. In Figure 8, different models for the shaker-based sinusoidal calibration are shown with the known parameters of the measuring device and the parameters to be identified during the calibration. The same is given in Figure 9 for the shock force calibration.

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Figure 8: Models used for parameter identification using sinusoidal excitation.

Figure 9: Models used for parameter identification using shock force excitation [30].

The model should always be chosen as simple (i. e., with as few parameters) as possible, an “overfitting” should be avoided. More parameters do not mean a more precise result. On the contrary, if there are too many parameters to identify, the identification becomes more difficult. This principle is also known as “Occam’s razor.” Additionally, if there are too many “stiff” parameters (i. e., parameters with only little influence on the behavior of a system), the identification of all parameters will be less stable. More details can be found in [51, Chapter 5].

88 � Th. Bruns et al. The model parameter identification for calibration setups with sinusoidal excitation is carried out in the frequency domain. Based on the chosen model, the matrices of the equation systems as given in equation (19) will have different sizes and will require the motion quantities at different positions to be measured. Although the excitation is required to be measured as displacement, velocity, and acceleration, in case of the sinusoidal excitation, these quantities can be derived from each other by simple calculation. In case of a sinusoidal excitation, these different motion quantities can be expressed as follows: x(t) = x̂ eiωt ,

̇ = iω x̂ eiωt x(t) 2

iωt

̈ = −ω x̂ e x(t)

̇ = iω x(t) or x(t) or

respectively,

(20)

2

̈ = −ω x(t). x(t)

From the equation system (19), different transfer functions can be defined. This of course is dependent on the chosen model. With the most simple model assuming a rigidly coupled sensor under test to both exciter and mass body and, therefore, with only degree of freedom, transfer functions like the one presented in Section 1.4 will be the result. In practice, at least a nonrigid mounting of the mass body is often required to be incorporated in the model [37]. In general, the more complicated the model is, the more measuring positions need to be acquired for the model parameter estimation. Due to technical restrictions, it is not always possible to measure the acceleration directly below and above the sensor. In these cases, it can be helpful to assume the output of the sensor UF to be proportional to its compression or tension giving UF (t) = ρ(xH (t) − xB (t))

(21)

with an unknown proportionality constant ρ, which adds as a parameter to identify. The derived transfer functions then can be used for the parameter identification using measurement data. The estimation of the parameter vector θ (consisting of all parameters to be identified) is carried out by minimizing the squared cost function K(ω, θ), which is the difference of the model function G and the corresponding measurement data Y giving K(ω, θ) = (G(ω, θ) − Y (ω)) ⋅ W

(22)

with an (optional) weighting matrix W . The parameter identification for shock force calibration can be carried out in the frequency domain or in the time domain. As the shock pulses are typically very short, the frequency resolution is low after carrying out a discrete Fourier transform. This resolution is directly proportional to the measuring time. Generally, it is advantageous to carry out the parameter identification in the time domain if the excitation signals

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cannot be chosen and/or the frequency domain representation is not optimal because of the frequency resolution or leakage. More details on advantages and disadvantages of the time or frequency domain identification can be found in [51, Chapter 3]. If the model parameters of a sensor were identified, they can be used to calculate or model the behavior of the sensor in a later application. By principle, this requires a similar knowledge about the components that are coupled in this specific application to the sensor. It can require a lot of experiments or simulations to derive this information. If the sensor will be used in frequently changing measurement setups, a proper determination of the properties of the mechanical environment might be not feasible for the user.

3.3 In situ calibration without modeling To overcome the issue of the demanding modeling of a dynamic force application, alternative methods—which also have their limitations—have been developed. Here, we give some examples of such approaches and their limitations. Fatigue testing of material probes is carried out to determine the material properties. For this purpose, a load is periodically applied to a specimen in a so-called fatigue testing machine. Such a machine consists of a stable frame, an exciter (hydraulic, electromagnetic, etc.), and clamping mechanisms to mount the specimen. It is important to correctly adjust the applied dynamic force during fatigue testing to be able to correctly estimate the capabilities of the specimen under testing. For this purpose, force sensors are installed in the machine to measure the applied force. The results from those tests are then used to estimate the lifetime of designed components. Due to the dynamic excitation, the effective sensitivity of the force sensor can differ from the static calibration value. The international standard ISO 4965 [21, 52] addresses this issue. The approaches given in this standard differ from those of the model-based calibration previously presented. Sheet 1 of the standard (ISO 4965-1) describes two approaches to tackle the deviation of the force sensor’s output at dynamic excitations: 1. One replica test-piece will be equipped with strain gauges to measure the actual force. A static calibration is used to determine the sensitivity of the test piece. In a second step, dynamic measurements are carried out. The differences between the output of the replica test piece and that of the sensor are calculated and compensated for in subsequent tests. This approach is only valid for specimens of the same stiffness or compliance as the replica test piece. 2. A compliance envelope is calculated with two force sensors. One has a lower compliance than the hardest specimen in use, and one has a higher compliance than the softest specimen in use. The maximum deviations are determined in dynamic tests and a compliance envelope is defined, in which these deviations are smaller than 1 %. Only within the compliance envelope, an operation is permitted.

90 � Th. Bruns et al. The second sheet of the standard (ISO 4695-2 [21]) ensures that the conditioning electronics and the data acquisition and indication comply with the requirements. An approach to dynamically calibrate force sensors in situ was proposed by [53]; additional investigations of possible influences showed a successful implementation [54]. The idea is to use an impact hammer as a transportable force reference. Impact hammers are hammers equipped with a force sensor at the tip. If this force sensor is calibrated dynamically, it can be carried around and be used to calibrate a force sensor in situ. Figure 10 illustrates the calibration principle and shows an example measurement.

Figure 10: Dynamic calibration of a force sensor with a calibrated impact hammer, principle on the left, application on the right [54].

Advantages of the approach are that the impact hammer and its conditioning electronics can be calibrated as a measurement chain and do not need to be assessed individually. The impact hammer’s excitation bandwidth defines the calibration bandwidth. The procedure is well suited for applications, at which one end of the force sensor is accessible for the hammer impact, like crash walls or the measurement of wave impacts [55]. The impulse that can be applied to the system to be calibrated is limited by the impact hammer.

4 Dynamic torque calibration Torque measurement is in many aspects similar to force measurement. The torque M equals the cross-product of a location vector r and a vectorial force F giving

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M = r × F,

(23)

more commonly denoted in scalar units as a force F acting on a lever arm r giving M = r ⋅ F.

(24)

Torque as a rotatory quantity is measured in many rotating shafts. When analyzing the applications, a distinction not only between static and dynamic torque loads is necessary, but also between rotating and nonrotating conditions. The sensors are either reaction torque sensors, which are not suited for rotation, or shaft- and flange-type sensors designed to operate also in rotating conditions. The measurement principle of torque sensors is similar to force sensors (cf. Section 3). Reaction torque sensors are mostly based on the piezoelectric [22] or on the strain gauge measurement principle. Most of the shaft-type and flange-type sensors are based on strain gauges; different mechanical designs for different applications [56]. For dynamic applications, the manufacturers try to develop sensors with high torsional stiffness. Other measurement principles also exist, but do not have high importance for dynamic measuring applications at the moment. More information on the different measuring principles can be found in [25]. Sensors designed for rotating conditions need to transmit the measurement signals from the rotating measuring element to some kind of receiver. For this purpose, passive methods like slip rings have been available for many decades. Noncontact methods consist of two sets of two coils each (one rotating, one stationary); one set of coils transfers energy to the rotating components, while the other set receives the measurement signal. In the past, the transfer was passive (using transformers) or the torque signal was transferred as a frequency modulated analogue signal. Today, the signal is often not only conditioned on the rotating shaft, but also digitized. A digital torque data stream is then transmitted to the stator. As an example, let us consider engine test benches. The efficiency of internal combustion engines can be determined by measuring the output torque M output and the rotational speed n. To determine the efficiency, the mechanical output power P is derived as follows: P = n ⋅ M,

(25)

and can be compared with the consumed energy, i. e., the amount of fuel used in a certain time. Due to its operation principle with many combustions in a short time, the output torque of an internal combustion engine (ICE) changes rapidly over time. This dynamic torque change can lead to problems if not taken into account. The dynamic components can be significantly higher than the mean torque and, therefore, overload the torque sensors until they may fail mechanically. Nevertheless, the state of the art for calibrations of torque sensors used in engine test benches are still only static calibrations. The sensors are often mechanically oversized to account for the unknown

92 � Th. Bruns et al. amount of superimposed dynamic torque components, and are partly only calibrated for lower torque levels than the higher nominal load of the sensor, which increases the safety margin for mechanical failure. In many industrial production processes, screw connections are fastened with so-called impact and impulse wrenches. These wrenches apply short torque impulses to fasten the screw connection, generated by mechanical impact (impact wrenches) and by hydraulic pulses (impulse wrenches). Due to the inertia of the tool, only a small amount (the shorter the pulse, the less) of the torque pulse is absorbed by the operator, making these tools ergonomic. All measuring fastening tools in production processes need to be calibrated according to ISO 9000. Up to now, no dynamic calibration procedures for reference torque sensors used for the calibration of these fastening tools exist. Instead, statically calibrated reference sensors need to be used [57, Appendix D].

4.1 Rotation and dynamic torque Both torque measurements in engine test benches and impact wrenches deal with dynamic torque signals. While the sensor used for calibrating torque tools does not need to be rotatable, the in-line measurement at the drive shaft in engine test benches requires sensors designed for rotation. The output power can also be measured without a rotating torque sensor by measuring the reaction torque of the brake. The dynamic calibration of torque sensors is still under research. First proposals for calibration devices were made by PTB in the early 2000s [58]. First measurement results and concepts for the traceability were presented later [59, 60]. Recently, a new dynamic torque calibration device was presented [61]. Meanwhile, other institutes also worked on the dynamic calibration of torque sensors, e. g., the Brazilian NMI INMETRO [62, 63], the Chinese CIMM [64], or Japan’s NMIJ [65]. The last mentioned approach from NMIJ is unique and different from the others; it is based on the principle of a Kibble balance. The formerly mentioned dynamic torque calibration devices are using the same measurement principle based on Newton’s second law. The dynamic torque M(t) is defined ̈ giving by the product of the mass moment of inertia J and the angular acceleration φ(t) ̈ M(t) = J ⋅ φ(t).

(26)

This approach is similar to the dynamic calibration of force sensors but transferred to rotational quantities. The mechanical design of a dynamic torque calibration setup consists of a rotational exciter (typically an electric motor of some sort) to which the sensor under test is connected. On the other side of the sensor, a mass moment of inertia body in mounted. Typically, this kind of setup requires additional coupling elements and bearings to connect the components and to avoid parasitic forces and bending moments acting on the sensor. The calibration devices can be designed as a horizontal or vertical arrangement of the components. An example of a horizontal design is shown in Figure 11.

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Figure 11: Dynamic torque calibration device at PTB [61].

4.2 Model-based calibration for torque measurements Due to the similarities of the mechanical design and the measurement principle (cf. Section 4), the model of a torque sensor can be made in analogy to that of a force sensor. The model assumptions of a linear and time-invariant behavior are the same, too. A typical sensor model therefore consists of two mass moments of inertia elements (MMOI) connected by a linear torsional spring and a torsional damper. A schematic of a calibration device with a sensor as a model is depicted in Figure 12. In order to be able to identify the model parameters, the calibration device needs to be modeled as well. The model parameters of the calibration device need to be determined prior to the model parameter identification. This model leads to a system of ordinary differential equations in analogy to (19) for the quantity force giving J φ̈ + Dφ̇ + Kφ = L,

(27)

with the mass moment of inertia matrix J, damping matrix D, stiffness matrix K, and load vector L. The angular excitations are found in the angular displacement vector φ and its derivative equivalents φ̇ and φ.̈ The model chosen for the model-based calibration at PTB is also shown in Figure 12. In the practical application, the model parameter identification for torque sensors is more complex than for force sensors. This is due to several reasons: Angle measurement Instead of the angular acceleration, a measured angle position is often used to calculate the angular acceleration. The position of the angle measurement can typically not be chosen as freely as an acceleration measurement position. Model parameters of the calibration device For the determination of the model parameters of the calibration device, additional experiments need to be set up. Whereas in case of dynamic force calibrations the mass value can be easily determined by weighing, the determination of the mass moment of inertia used in dynamic torque calibrations is much more challenging and requires dedicated experiments.

94 � Th. Bruns et al.

Figure 12: Illustration of components of the dynamic torque calibration device (left) and the mechanical model (right) [66].

Limited options to modify the measurement device The model parameter identification becomes much more robust if the masses (in case of force calibration) or mass moments of inertia (torque calibration) can be modified. In this case, a wellknown “detuning” can be carried out. This can be implemented much more easily for translatory quantities than for rotatory ones. The model-based calibration approach of torque sensors, which was published in [66], required a rather complex model. This was because of the mechanical design of the calibration device and the limited options of the placement of the angle measuring devices. To be able to identify the model parameters, not only the two measured rotational quantities above and below the sensors were used to calculate two transfer functions, but also the output of the sensor was used for this purpose assuming it to be proportional to its torsion giving UM (t) = ρ(φH (t) − φB (t)) = ρ ΔφHB (t),

(28)

with ρ as an unknown scaling factor. This scaling factor is an additional unknown parameter in the identification. The two derived transfer functions included all the un-

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known parameters of the device under a test. These functions and their corresponding measurement quantities are shown in Figure 13.

Figure 13: Derived transfer functions and corresponding measurement quantities [66].

With the model parameter identification, the mechanical properties of a sensor under test will be identified. To do this properly, influences due to signal conditioning and data acquisition need to be compensated. This can be done by calibrating the data acquisition electronics and the signal conditioning amplifier separately. In contrast to force sensors, which are typically passive sensors without integrated electronics, this separation is difficult with torque sensors. The increasing use of integrated electronics in the sensors requires a dynamic calibration of this electronics, which is not intended by design by the manufacturers and would require direct access to the bridge amplifier inputs. Strain gauge sensors are often sealed in order to reduce influences due to environmental conditions, especially humidity. Moreover, the bridge amplifier inputs are very sensitive to electromagnetic disturbances and, therefore, need to be shielded. This issue cannot be solved without changing the design of the sensors. A dynamic calibration of the electronics during or prior to the assembly of the sensor might help in this regard, but will not account for changes during the lifetime of the sensor.

96 � Th. Bruns et al.

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V. Wilkens and M. Weber

Traceable measurements with dynamically calibrated hydrophones 1 Introduction ultrasonics Ultrasonic waves comprise more than 20000 mechanical stress or pressure cycles per second. This is a highly dynamic physical process per definition, and dynamic properties need to be quantified during measurements. The dynamics of the field quantities are still much slower than in optics, which offers the direct access to time resolved detection, a prerequisite for the many technical pulse-echo and time-of-flight measurement implementations. Ultrasound in the kHz frequency range is largely applied in industry, for instance for material cleaning and processing and in sonochemistry, whereas nondestructive testing and the prominent medical applications mostly use even higher frequencies in the MHz range. Such medical applications comprise the nowadays widespread use of sonography for imaging and diagnostics, as well as therapeutic applications like physiotherapy and high intensity therapeutic ultrasound, for instance, for tumor ablation. Propagation of MHz-ultrasonic waves through tissue enables highly resolved imaging of opaque media as well as treatment of deeply located parts inside the body without the need for surgical opening.

2 Ultrasound exposimetry To ensure both, the safety for the patient as well as the efficiency of treatments, the acoustic output of ultrasonic medical equipment needs to be determined before devices can be marketed and subsequently be used on patients. Modern ultrasound scanners offer a lot of modalities to not only determine anatomic images from inside the body but to also provide spatially resolved information on movements, flow velocities, tissue composition, elastic parameters, and others. A variety of ultrasonic signals with different working frequencies, acoustic pressure amplitudes, pulse lengths, repetition rates, and spatial scanning patterns is applied to support those modalities. In general, sonography is considered a particularly safe imaging modality. However, a potential for thermal and mechanical hazards exists if the natural tolerances of human tissue against sonication are exceeded, and some diagnostic modalities would produce stronger effects in tissue than others. For instance, the determination of blood flow in the umbilical cord requires longer pulse lengths and higher repetition rates than usual brightness mode imaging of the anatomy of organs, and exposes a small region of tissue for the time of the measurement instead of spatially scanning larger volumes. As the actual acoustic output of a sonography device will depend on a large number of system settings, it is https://doi.org/10.1515/9783110713107-004

100 � V. Wilkens and M. Weber required that the system provides information on the monitor for the user. The acoustic output display according to IEC 60601-2-37:2007 uses mechanical and thermal indices for this purpose [1]. Along with these numbers the sonographer should also consider existing guidelines for exposure duration [2]. In particular, the examination time should be kept short if high thermal index values are displayed, for instance, by freezing the image which deactivates the acoustic output and allows the tissue to cool down before exposure might be activated again for further assessments.

3 Basic ultrasonic exposure quantities Acoustic output declarations for medical ultrasonic equipment either in form of index data for diagnostic machines or in other forms to specify the acoustic fields of therapeutic systems, are all based on acoustic field measurements where the treatment head emits into water as a standard propagation medium. The most important physical quantities to be measured are: 1. the time averaged emitted acoustic power P, 2. the time dependent acoustic pressure waveform p(t) including peak compressional pc and peak rarefactional pressure pr and acoustic working frequency f , 3. the local temporal average ultrasonic intensity I, 4. geometrical dimensions of the acoustic beams. The time averaged overall acoustic power emitted from an ultrasonic transducer (1.) is a relevant safety parameter mainly regarding the thermal effect of ultrasound in tissue. The preferred measurement method uses radiation force balances [3]. The ultrasonic beam is directed toward an appropriately designed absorber, which is mechanically connected with a sensitive balance. The radiation force is determined during switching on and off of the ultrasonic source from the change in weight measured with the balance. For this type of measurement, it is actually not necessary to resolve the high ultrasonic frequency in the detection process. For all the other basic quantities (2. to 4.), measurements providing temporal and spatial resolution of the acoustic field quantities are required. The ultrasonic waves emitted into water are characterized through scanning with piezoelectric local pressure receivers, so called hydrophones [4]. From the measured pressure-time waveforms, direct pressure parameters are derived that are relevant for mechanical hazard. Further parameters like local intensity parameters and beam dimensions all relevant for the local heating hazard are also derived from the hydrophone-based acoustic pressure measurements [5].

4 Ultrasonic hydrophones Hydrophones for the measurement of medical ultrasonic fields use piezoelectric receiving elements to convert the ultrasonic pressure into signal voltage. While piezoceramics

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are still in use, most hydrophones for broadband detection in the MHz frequency range are based on PVDF (polyvinylidenefluoride) films, which offers lower acoustic impedence mismatch with the sound propagation medium water. Different types of PVDF hydrophones like needle-type, capsule-type, and membrane type offer different advantages for specific measurement conditions and tasks. While needle- and capsule-type hydrophones offer most flexibility and practical advantages for large scale water tank scanning applications, membrane hydrophones provide more homogeneous frequency responses, and are often used as reference receivers. Figure 1 shows an example of a membrane hydrophone with differential electrode design and an integrated preamplifier that can be used for ultrasound frequencies up to 100 MHz and even beyond.

Figure 1: Example of an ultrasonic membrane hydrophone (PTB RS072) of differential electrode design [6]. The small sensing element in the center of the ring is formed through spot-poling at the overlap of the front (golden) and the rear (chrome) electrode and has a diameter of 0.2 mm. The larger ground electrodes are for noise reduction. A preamplifier is located inside the outer ring.

To address large detection bandwidths and small effective receiving areas— these should be small compared to the acoustic wavelength to keep the spatial averaging of the acoustic pressure reasonable—it is necessary to use short passive electrical paths. Thus, a preamplifier and signal cable driver electronics are often located close to the sensing element inside the hydrophone system. The frequency dependent transfer function of such a hydrophone assembly, usually termed hydrophone sensitivity, relies on several mechanical and electronical design choices and material parameters. Therefore, to enable precise and reliable acoustic pressure measurements, ultrasonic hydrophones need to be calibrated individually, and recalibrated periodically after appropriate periods of time.

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5 Hydrophone calibration and voltage-to-pressure transfer functions Hydrophone calibration services are offered by a number of national metrology institutes and by other calibration laboratories, for instance, run by hydrophone manufacturers [7–14]. Some metrology institutes hold a primary realization of the measurand of ultrasound pressure as well. Primary calibration provides direct traceability to SI unit quantities, while secondary calibration methods allow the comparison of two hydrophones within a substitution arrangement. Different calibration techniques using a variety of ultrasonic excitation pressure waveforms comprising different pressure amplitude ranges and frequency compositions have been proposed and applied over the past decades. Such excitations include single-frequency tonebursts and swept frequency tonebursts in the kPa amplitude range of quasilinear acoustics as well as higher amplitude burst [7] and impulse excitations including nonlinear propagation [15]. Guidance on the application of different techniques is provided in IEC 62127-2 [16]. Traditionally and for technical reasons, hydrophone calibrations have been performed mostly as magnitude or modulus calibrations only, in the past. However, calibration methods were further developed through the years to meet the demands of calibrations at higher frequencies and with reduced uncertainties and to provide calibration data comprising both modulus and phase [17–21]. Complex-valued calibration data are particularly important for deconvolution applications when characterizing broadband ultrasonic fields [21–27]. A calibration result for the broadband membrane hydrophone (PTB RS072) depicted in Figure 1 obtained by a primary calibration technique [20] is shown in Figure 2. A relatively flat response can be observed between 1 MHz to 40 MHz and an increase of sensitivity up to the thickness mode resonance of the sensing element located at about 70 MHz followed by a high frequency roll off. The thickness mode resonance frequency depends on the thickness of the sensing element, which is 11 µm in this example. For an unbacked hydrophone like this, it appears at the frequency corresponding to half an acoustic wavelength in the material fitting to the element thickness. For hydrophone designs including an absorbing backing material, the resonance is at a quarter of the wavelength, e. g., the resonance frequency is lower for the same PVDF foil thickness. Other hydrophone designs like needle-type hydrophones usually provide more sensitivity variations than membrane hydrophones, particularly in the low MHz frequency range due to the stronger impact of radial resonances and diffraction effects of such probes.

6 Hydrophone measurements—signal deconvolution For acoustic pressure waveform measurements, the output voltage uL (t) of a hydrophone is acquired using an oscilloscope. For a pure, single frequency waveform,

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Figure 2: Broadband calibration data of a hydrophone depicted as logarithmic sensitivity level or modulus of the sensitivity (upper plot) and corresponding phase response; uncertainty ranges are shown for a coverage factor of k = 2 for 95 %.

the voltage-time signal represents the pressure-time waveform at the position of the sensitive element in the acoustic field. The actual pressure data p(t) can be determined by scaling the voltage signal with the modulus sensitivity of the hydrophone |M L | as read from the calibration data at the frequency of the waveform, the acoustic working frequency fawf : p(t) =

uL (t) |M L (fawf )|

(1)

If the hydrophone would provide the same sensitivity for all frequencies, the same conversion method could be applied to waveforms comprising more than one frequency

104 � V. Wilkens and M. Weber as well. But as outlined before, ultrasonic detection always suffers more or less from specific internal resonance effects of sensing elements as well as from outer reflection and diffraction of the acoustic waves impinging on the receivers. Thus, transfer functions are affected by these phenomena and will show variation with frequency rather than being constant. Nevertheless, for the vast majority of hydrophone measurements the conversion by scaling has been performed by scaling so far for all types of waveforms. Corrections may then have been addressed by empirical methods or the imperfections were attributed by increased uncertainty estimates. The former IEC TS 61220 [28] published in 1993 provided such an empirical method to remove the signal overshoot of nonlinearly distorted waveforms to avoid large overestimation of peak compressional pressures. In 2007, the succeeding (and 2022 updated) standard IEC 62127-1 [4] already replaced this method with instead recommending waveform deconvolution according to p(t) = ℱ −1 [

U L (f ) ] M L (f )

(2)

where ℱ −1 denotes the inverse Fourier transform and U L (f ) is the Fourier transform result of uL (t), for specified measurement situations where the so-called narrowband approximation of equation (1) would not hold. Recognizing wider availability of broadband and complex-valued calibration data over the years, the second edition of this standard [4] changes the recommendation for deconvolution toward a requirement. Only in specified cases with low impact of nonlinear propagation or sufficient flatness of the hydrophone frequency response, the narrowband approximation can still be applied. However, such a requirement needed to be accompanied by more detailed description of the deconvolution method as a prerequisite for broader application by many users. To further support the method and its standardization, a calculation scheme including stepby-step guidance and examples were developed as a deconvolution tutorial available through online publication [29]. In addition, the determination of output parameter uncertainties when using waveform deconvolution needed specific attention beforehand [30–32], since commonly used methods possible with the narrowband approximation, e. g., when applying just a single hydrophone sensitivity factor, cannot directly be followed in the same way with this method.

7 Tutorial for waveform deconvolution A tutorial was developed and published, to help users with the practical application of waveform deconvolution. It uses freeware software tools only and comprises all necessary procedures for deconvolution and associated uncertainty estimation. Figure 3 shows the schematic workflow of the method. Some details of the various steps are discussed in the following subclauses.

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Figure 3: Schematic workflow of signal deconvolution for an ultrasonic hydrophone measurement.

The tutorial provides all mathematical procedures and functions and makes use of the PyDynamics package [33]. The script can be run online, or on a local computer after download according to the instructions provided at the beginning. Hydrophone calibration data are provided for a set of four hydrophones of different types as examples. To keep differences caused by spatial averaging effects small, hydrophones offering small effective element diameters in the range from 0.11 mm to 0.14 mm were chosen. Ultrasonic waveforms of an ultrasonic diagnostic device were measured by each of the hydrophones. The device was used in two different modes. The first mode used was the motion (M) mode which produces waveforms that are similar to the common brightness (B) mode, but without scanning the ultrasonic beam. This mode uses a short pulse with only a few cycles. The other mode was the pulse Doppler (pD) mode, which is used to investigate the flow of liquids. For the Doppler mode, a burst signal with several cycles is produced. Both modes were operated at two different frequencies, approximately 3 MHz and approximately 7 MHz. The signals at the lower acoustic work frequency were emitted from a convex array, and the signals at the higher acoustic work frequency from a linear array. Settings were chosen for the acoustic work frequency, the output power, the focal length, the region of interest, and the pulse repetition frequency, which produced the maximum spatial-peak temporal-average intensity of the diagnostic machine with the respective transducer for the particular mode. The waveforms were acquired at the position of the spatial-peak temporal-average intensity. Altogether, the data sets comprise 16 different combinations, e. g., measurement situations that directly can be investigated. Of course, the functions of the tutorial can also be applied to own hydrophone and waveform data.

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7.1 Input data As can be seen in Figure 2, the tutorial needs data input for hydrophone sensitivity and for measured voltage signals including information on associated uncertainties. The complex-valued hydrophone calibration data were determined in the frequency range from 0.1 MHz to 100 MHz by means of primary [20] or secondary calibration [34]. The hydrophone sensitivity level, e. g., the modulus of the transfer function in logarithmic scales is depicted in Figure 4 including calibration uncertainties for the four example hydrophones.

Figure 4: Hydrophone sensitivity level for four hydrophones of different type.

In general, the combination of sampling interval and number of points acquired determines the frequency interval of the spectrum of the measured signal according to the equation: Δf =

1 Δt × N

(3)

where Δf , Δt, and N denote the frequency increment, the time increment, and the number of data points, respectively. Frequently, it is found that this frequency increment of the measured data differs from that supplied between consecutive points in the hydrophone calibration data. To ensure that calibration data is available at the appropriate frequency points, it may be necessary to interpolate values at the required spacing from the calibration data. For the deconvolution calculation according to equation (2), the data sets of the voltage spectrum U L (f ) and the sensitivity M L (f ) need to cover the same frequency ranges

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from zero to the Nyquist frequency of the hydrophone measurement. For instance, for a typical waveform sample rate of 500 MS/s, data is needed up to 250 MHz. Therefore, extrapolation of M L (f ) may be used to cover frequencies f outside the calibration range available if necessary. A convenient way is to start with extrapolating the modulus data first and to adjust the phase data in a second step using equation (4) below. This will assure consistent complex-valued response extrapolation, including the phase value being zero at zero frequency as is necessary for any real-valued time domain impulse response [24]. The modulus extrapolation should follow the trend indicated by the calibration range limit region, for instance, if some thickness mode resonance behaviour can be observed at the high frequency end a decreasing behavior with frequency may be estimated from that. Of course, the sensitivity cannot become zero. This would induce singularities in the deconvolution because some noise can be expected in the voltage spectrum even at frequencies not transduced by the hydrophone due to zero sensitivity. Several investigations have shown that rather simple extrapolation schemes work well for most applications, e. g., lead to reasonable deconvolved waveform results. For membrane hydrophones, using the calibration data point modulus at the lowest frequency as constant value down to zero and using the calibration data point modulus at the highest frequency as constant value above the calibration range may be sufficient [24, 26, 27, 35] and can be applied as standard procedure. The tutorial follows this standard procedure for all hydrophones. However, if the preamplifier is known to cause a significant roll-off at low frequencies, either from manufacturers specifications or from electrical measurements, such sensitivity decrease should be accounted for in obtaining ML (f ). For needle type hydrophones, a decreasing sensitivity toward low frequencies may be expected due to the impact of diffraction at the sensor tip. Here, decreasing sensitivity modulus functions show good results [23]. If available, more comprehensive hydrophone frequency response modeling may be used for the purpose of extrapolation. Relatively simple secondary impulse calibration methods [34, 36] can also be applied by the hydrophone user to extrapolate sensitivity data reasonably, as can measurements of similar units of the same design. An alternative approach to measuring hydrophone phase response is the computation of phase response data from modulus response data. A minimum phase system can be assumed for some hydrophones, which allows an estimation of the frequency dependent phase response using, e. g., Hilbert-transform methods or iterative phase signal reconstruction techniques [19, 37, 38]. A straightforward easy-to-implement method of calculating the discrete phase response data arg M(fm ) in rad from modulus response data for minimum phase systems is provided by the Bode gain-phase relation formulation [39], which can be written for problems with discrete and uniform frequency intervals as arg M(fm ) =

2fm N/2 ln(|M(fn )|/|M(fm )|) Δf ∑ π n=0 fn2 − fm2

(4)

108 � V. Wilkens and M. Weber where n and m run from 0 to N/2, n ≠ m, N denotes the number of time signal data points (resulting in N/2 + 1 frequency domain data points in the single-sided spectrum), fn and fm denote discrete frequencies, and Δf is the constant frequency increment, e. g., Δf = fn+1 − fn , respectively. The methods of phase calculation from modulus responses may also be helpful when used in combination with measured hydrophone phase data to correct the phase calibration data for linear-with-frequency terms by this comparison. Furthermore, it may be used to adequately extrapolate the measured data below and above the experimentally available frequency ranges by assuming reasonable modulus data for that frequency ranges first, and then calculating the suitable phase response extrapolation with equation (4). The tutorial provides two alternatives: Either using the hydrophone phase response data as determined through calibration, or using equation (4) to determine phase data from sensitivity modulus data. Figure 5 shows the modulus of the inter and extrapolated calibration data of a needle-type hydrophone. The example described herein refers to data set 4 of the tutorial. Extrapolation was performed for this example below 1 MHz to 0 MHz and beyond 50 MHz up to 250 MHz. The corresponding phase response data determined from those modulus data using equation (4) is depicted in Figure 6. The tutorial offers the calculation of phase uncertainties from the modulus uncertainties as well if the phase response calculation option is chosen. Both, modulus and phase response uncertainties are then used in the further uncertainty determination for the deconvolved waveforms.

Figure 5: Modulus of a needle-type hydrophone transfer function as determined by calibration and inter and extrapolations.

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Figure 6: Phase of a needle-type hydrophone transfer function as determined from modulus data shown in Figure 5 using the Bode formula.

The hydrophone measurement voltage data are imported in the tutorial as single column list of values representing uL (t) at equal steps in time. To each data point of the measurement, an uncertainty should be assigned. In a simple (but typical) case, the uncertainty may be estimated for each point by means of the quantization uncertainty of the instrument as an upper bound. Furthermore, the additional contribution of a possible scaling error should be taken into account. The quantization error may be assumed to have a rectangular distribution. It is calculated by means of the full range of measurement and by the number of bits of the AD converter 2Δa = Ufullrange /2n . This distribution can be transferred into a standard deviation 2Δa : ux = Δa /√3 that will be used in later uncertainty calculations. For other distributions, the value needs to be corrected. The conversion factors can be found in the literature [40]. The uncertainty vector is calculated from this value. If more information is known about the uncertainty at each time step—derived, for instance, from a separate voltage calibration measurement of the oscilloscope, then this vector can be altered and all the information should be used for further calculation. The preferred way is to determine the uncertainty experimentally. The hydrophone is connected to the oscilloscope (as in the waveform measurements). The oscilloscope has the same settings (voltage resolution, sample rate, number of averages, and so on) as during the acquisition of the waveform. But this time, no ultrasonic signal is transmitted, so that only the noise of the hydrophone (together with the noise of the oscilloscope) is present. From this data, the standard deviation from the mean value (voltage offset) can be calculated and be used as the uncertainty of the signal voltages. In addition, the uncertainty of the voltage scale itself should be determined by a separate calibration of the oscilloscope.

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7.2 Deconvolution of waveforms Deconvolution is performed using the function from the GUM2DFT package [30, 33] supplied with PyDynamic. The Fourier-transformed time-domain voltage signal of the hydrophone measurement is the numerator and the hydrophone frequency response is the denominator in equation (2). The spectrum of the measured signal corrected for the frequency weighting of the hydrophone is then transformed from the frequency domain to the time domain through inverse Fourier transformation. Figure 7 shows as example the deconvolved pressure waveform for 3 MHz M-mode operation measured with a needle-type hydrophone compared with the waveform obtained by scaling according to equation (1).

Figure 7: Deconvolved pressure waveform for 3 MHz M-mode operation measured with a needle-type hydrophone compared with waveform obtained by scaling.

For the deconvolution, hydrophone calibration modulus data were used from 1 MHz to 50 MHz with constant value extrapolation below and beyond this range (Figure 5) and phase response data were determined from modulus data (Figure 6). Due to the rather strong variations of the sensitivity for this needle-type hydrophone large deviations are obtained between the deconvolved and the scaled pressure waveform. Typical signal overshoot of the peak compressional part of the waveform is removed and the peak compressional pressure obtained by deconvolution is much more realistic and comparable to the corresponding result obtained with the other hydrophones offering more homogeneous frequency responses. The peak rarefactional pressure is also affected, but

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here the deconvolution leads to an increase. Again, the deconvolved result is in closer agreement with the corresponding results of the other hydrophones.

7.3 Regularization filtering To avoid excessive additional high frequency noise in the deconvolved waveforms, regularization filtering may be necessary. Such noise can arise from low hydrophone sensitivity (at high frequencies) and from the Gibbs phenomenon. Appropriate suppression may be achieved by low-pass filtering of the pressure pulse spectrum prior to the back transformation to the time domain: p(t) = ℱ −1 (LP(f )

U L (f ) ) M L (f )

(5)

On the other hand, any additional low-pass filtering causes additional effective limitations of the detection bandwidth and, in particular, peak compressional parts of nonlinearly distorted ultrasonic waveforms incorporating high frequency components may possibly be cut. Different numerical low-pass filters LP(f ) can be used for regularization. A standard low-pass filter, a low-pass filter with critical damping and a Bessel filter were implemented in the tutorial, as well as the option of not filtering. Other filter customized functions may be added, if needed.

7.4 Regularization uncertainty estimation In order to estimate the additional uncertainty that is introduced by the regularization filtering, the upper-bound function method described in [31] is applied. By means of this method, the impact of the induced systematic error is investigated quantitatively. This is done by considering a continuous upper-bound function in the frequency domain for the pressure modulus spectrum, using an approach with a simple basic function, which has one parameter only. Prior knowledge is included (such as the monotonous decrease of the spectral components with increasing frequency—as can be expected for nonlinearly distorted acoustic pulse waveforms—and the typical noise at high frequencies that differs from the contribution of the signal). For the uncertainty estimation, information is required about the signal at all frequencies, even if this information cannot be obtained by measurements. For that reason, it is necessary to carry out a signal approximation. In the publication of Eichstaedt and Wilkens [31], a wavelet-like approach was used. This is rather a simple approximation, for which no additional knowledge is necessary. On the other hand, the model is rather conservative and may overestimate the uncertainty. Better models for the approximation might be chosen. Figure 8

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Figure 8: Deconvolved pressure waveform for 3 MHz M-mode operation measured with a needle-type hydrophone including regularization and uncertainty (k = 2).

shows the same example as Figure 7 but as final result including the regularization filtering using the standard low-pass filter with −3 dB cut-off frequency set to 100 MHz, as well as the overall uncertainty of the pressure waveform determined with the tutorial script.

7.5 Reference data sets To enable a comparison of the waveforms obtained by deconvolution using the tutorial with the results of other implementations, a set of results was stored for reference in the results folder for download. Furthermore, the example parameters used for those evaluations are summed up in a table at the end of the script for all 16 hydrophone waveform combinations along with the ultrasonic parameter results like peak compressional, peak rarefactional pressure, and pulse-pressure-squared integral ppsi, which is the basis for local intensity parameters, all including the associated uncertainties (k = 1). Table 1 shows the data of the first 5 of the 16 examples. The uncertainties include the—often conservative—estimation of the contribution from regularization. This compilation of results is meant to support quick comparison when varying the calculation parameters like regularization bandwidth or calibration data ranges extrapolated as well as validation purposes for own deconvolution implementations.

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Table 1: Part of measurement examples, evaluation settings, and acoustic output parameter results stored with the deconvolution tutorial as reference data. Number (i)

1

2

3

4

5

Mode Frequency Hydrophon

M 3 MHz Membrane MH44 1 MHz to 60 MHz yes

M 3 MHz Membrane MH46 1 MHz to 100 MHz yes

M 3 MHz Capsule 1704 0.1 MHz to 50 MHz no

M 3 MHz Needle 1434 1 MHz to 50 MHz yes

pD 3 MHz Membrane MH44 1 MHz to 60 MHz yes

yes low-pass 100 MHz 5.62 ± 0.12 2.63 ± 0.12 1.637 ± 0.016

yes low-pass 100 MHz 5.02 ± 0.13 2.51 ± 0.13 1.572 ± 0.016

no low-pass 100 MHz 5.30 ± 0.10 2.57 ± 0.10 1.600 ± 0.060

yes low-pass 100 MHz 5.33 ± 0.11 2.54 ± 0.11 1.639 ± 0.025

yes low-pass 120 MHz 3.38 ± 0.08 1.10 ± 0.08 2.618 ± 0.017

Calibration range amplitude calibration only Bode equation filter type cut-off fc pc /MPa pr /MPa ppsi/kPa2 s

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IEC. IEC 60601-2-37:2007 Medical electrical equipment - Part 2-37: Particular requirements for the basic safety and essential performance of ultrasonic medical diagnostic and monitoring equipment. International Standard, 2007. [2] G. R. Harris, C. C. Church, D. Dalecki, M. C. Ziskin, and J. E. Bagley. Comparison of thermal safety practice guidelines for diagnostic ultrasound exposures. Ultrasound Med. Biol., 42:345–357, 2016. [3] IEC. IEC 61161:2013 ultrasonics – power measurement – radiation force balances and performance requirements. International Standard, 2013. [4] IEC. IEC 62127-1:2022 Ultrasonics – hydrophones – part 1: Measurement and characterization of medical ultrasonic fields. International Standard, 2022. [5] IEC. IEC 62359 Ultrasonics – field characterization – test methods for the determination of thermal and mechanical indices related to medical diagnostic ultrasonic fields. International Standard, 2010. [6] V. Wilkens and W. Molkenstruck. Broadband PVDF membrane hydrophone for comparisons of hydrophone calibration methods up to 140 MHz. IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 54(9):1784–1791, 2007. [7] D. R. Bacon. Primary calibration of ultrasonic hydrophones using optical interferometry. IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 29(2):152–161, 1982. [8] C. Koch and W. Molkenstruck. Primary calibration of hydrophones with extended frequency range 1 to 70 MHz using optical interferometry. IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 46(5):1303–1314, 1999. [9] G. Ludwig and K. Brendel. Calibration of hydrophones based on reciprocity and time delay spectrometry. IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 35(2):168–174, 1988. [10] Y. Matsuda, M. Yoshioka, and T. Uchida. Primary calibration of hydrophones up to 40 MHz in ultrasonic far-field using optical interferometry. Proc. Symp. Ultrason. Electron., 34:403–404, 2013. [11] R. C. Preston, S. P. Robinson, B. Zeqiri, T. Esward, P. N. Gélat, and N. D. Lee. Primary calibration of membrane hydrophones in the frequency range 0.5 MHz to 60 MHz. Metrologia, 36:331–343, 1999. [12] E. G. Radulescu, J. Woicik, P. Lewin, and A. Nowicki. Calibration of ultrasonic hydrophone probes up to 100 MHz using time gating frequency analysis and finite amplitude waves. Ultrasonics, 41(4):247–254, 2003.

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[13] R. A. Smith and D. R. Bacon. A multiple-frequency hydrophone calibration technique. J. Acoust. Soc. Am., 87(5):2231–2243, 1990. [14] P. Yang, G. Xing, and L. He. Calibration of high-frequency hydrophone up to 40 MHz by heterodyne interferometer. Ultrasonics, 54(1):402–407, 2014. [15] M. Weber and V. Wilkens. A comparison of different calibration techniques for hydrophones used in medical ultrasonic field measurement. IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 68:1919–1929, 2021. [16] IEC. IEC 62127-2 Ultrasonics – hydrophones – part 2: Calibration for ultrasonic fields up to 40 MHz. International Standard, 2007. [17] M. P. Cooling and V. F. Humphrey. A nonlinear propagation model-based phase calibration technique for membrane hydrophones. IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 55(1):84–93, 2008. [18] C. Koch. Amplitude and phase calibration of hydrophones by heterodyne and time-gated time-delay spectrometry. IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 50(3):344–348, 2004. [19] K. A. Wear, P. M. Gammell, S. Maruvada, Y. Liu, and G. R. Harris. Time-delay spectrometry measurement of magnitude and phase of hydrophone response. IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 58(11):2325–2333, 2011. [20] M. Weber and V. Wilkens. Using a heterodyne vibrometer in combination with pulse excitation for primary calibration of ultrasonic hydrophones in amplitude and phase. Metrologia, 54(4):432–444, 2017. [21] V. Wilkens and C. Koch. Amplitude and phase calibration of hydrophones up to 70 MHz using broadband pulse excitation and an optical reference hydrophone. J. Acoust. Soc. Am., 115(6):2892–2903, 2004. [22] A. M. Hurrell. Voltage to pressure conversion: are you getting ’phased’ by the problem? J. Phys. Conf. Ser., 1(1):57, 2004. [23] A. M. Hurrell and S. Rajagopal. The practicalities of obtaining and using hydrophone calibration data to derive pressure waveforms. IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 64(1):126–140, 2017. [24] K. A. Wear, P. M. Gammell, S. Maruvada, Y. Liu, and G. R. Harris. Improved measurement of acoustic output using complex deconvolution of hydrophone sensitivity. IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 61(1):62–75, 2014. [25] K. A. Wear, Y. Liu, P. M. Gammell, S. Maruvada, and G. R. Harris. Correction for frequency-dependent hydrophone response to nonlinear pressure waves using complex deconvolution and rarefactional filtering: application with fiber optic hydrophones. IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 62(1):152–164, 2015. [26] M. Weber and V. Wilkens. HIFU waveform measurement at clinical amplitude levels: primary hydrophone calibration, waveform deconvolution and uncertainty estimation. In IUS 2017 IEEE International Ultrasonics Symposium, Washington DC, IEEE, 2017. [27] V. Wilkens, S. Sonntag, and O. Georg. Robust spot-poled membrane hydrophones for measurement of large amplitude pressure waveforms generated by high intensity therapeutic ultrasonic transducers. J. Acoust. Soc. Am., 139(3):1319–1332, 2016. [28] IEC. IEC 61220 Ultrasonics – fields – guidance for the measurement and characterization of ultrasonic fields generated by medical ultrasonic equipment using hydrophones in the frequency range 0,5 MHz to 15 MHz (meanwhile withdrawn technical specification). Technical Specification, 1993. [29] M. Weber and V. Wilkens and S. Eichstädt. Tutorial for the deconvolution of hydrophone measurement data. https://doi.org/10.5281/zenodo.4012242 2020. [30] S. Eichstädt, V. Wilkens, A. Dienstfrey, P. Hale, B. Hughes, and C. Jarvis. On challenges in the uncertainty evaluation for time-dependent measurements. Metrologia, 4(53):S125, 2016. [31] S. Eichstädt and V. Wilkens. Evaluation of uncertainty for regularized deconvolution: A case study in hydrophone measurements. J. Acoust. Soc. Am., 141(6):4155–4167, 2017. [32] S. Eichstädt, N. Makarava, and C. Elster. On the evaluation of uncertainties for state estimation with the Kalman filter. Meas. Sci. Technol., 27(12):125009, 2016.

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[33] B. Ludwig, S. Eichstädt et al. PyDynamic – python package for the analysis of dynamic measurements, 2020. [34] V. Wilkens, S. Sonntag, and M. Weber. Secondary complex-valued hydrophone calibration up to 100 MHz using broadband pulse excitation and a reference membrane hydrophone. In 2019 IEEE International Ultrasonics Symposium (IUS), pages 1846–1849. IEEE, Glasgow, 2019. [35] Y. Chiba and M. Yoshioka. Effectiveness evaluation of extrapolation to frequency response of hydrophone sensitivity for measuring instantaneous acoustic pressure of diagnostic ultrasound. Jpn. J. Appl. Phys., 60(SD):SDDE14-1–SDDE14-7, 2021. [36] S. Howard. Calibration of reflectance-based fiber-optic hydrophones. In Proceedings of the IEEE Ultrasonics Symposium, 2016. [37] P. E. Bloomfield and P. A. Lewin. Determination of ultrasound hydrophone phase from Fourier-Hilbert transformed 1 to 40 MHz time delay spectrometry amplitude. IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 61(4):622–672, 2014. [38] Th. F. Quateri and A. V. Oppenheim. Iterative techniques for minimum phase signal reconstruction from phase of magnitude. IEEE Trans. Acoust. Speech Signal Process., ASSP–29(6):1187–1193, 1981. [39] J. Bechhoefer. Kramers–Kronig, Bode, and the meaning of zero. Am. J. Phys., 79(10):1053–1059, 2011. [40] BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, and OIML. Guide to the expression of uncertainty in measurement, 2008.

S. Eichstädt

Dynamic measurement analysis and the internet of things 1 Introduction With the rise of the Internet of Things (IoT), as in Industry 4.0 or Smart City, the world of measurement is changing rapidly. For instance, sensor networks are preferred over single sensors, since data analysis for huge data sets in almost real-time has become feasible and low-cost sensors are easily available. The role of mathematical modeling is changing, because of the increasingly often application of purely data-driven methods, based on machine learning. This also leads to a different role of calibration information and the communication of measurement data [1, 2]. Usually, the sensors applied in the IoT are measuring continuously irrespective of how the measured values are used. Thus, it has to be ensured that the sensor behavior is well known for a wide range of measurement situations. This includes situations where the measurand, i. e., the sensor input signal, changes rapidly over time. The performance of the sensor in such scenarios needs to be known in order to assess the reliability and quality of the reported measurement values. The way these measured values are acquired and used, however, may depend on the scenario. For instance, in some scenarios measured values are reported only if a certain event was detected, e. g., if a given threshold for the sensor output signal was reached. In other scenarios, measured values may be provided upon request by another instance in the network, e. g., in a client-server subscription based environment like OPC-UA. This chapter addresses the link between the developments for dynamic measurements and requirements from the IoT. Data analysis in the IoT typically relies on methods from machine learning, because of the complexity of the sensor network and the amount of data acquired. Uncertainty evaluation for machine learning is an important topic and considered in several research activities. However, this is only possible when the uncertainty associated with the machine learning input values is available. To this end, this chapter focuses on the uncertainty for data preprocessing as the initial step in machine learning for IoT.

2 Data analysis for machine learning pre-processing In situations where IoT devices are used, measurements are usually time-dependent, i. e., dynamic, because measurements are taken continuously and no efforts are taken to ensure a steady state of the measured processes. Examples are air quality monitoring, https://doi.org/10.1515/9783110713107-005

118 � S. Eichstädt traffic surveillance, production control, or mobile health measurements. Thus, signal processing methods are regularly applied for data preprocessing in IoT scenarios. For instance, the discrete Fourier transform is often applied to extract magnitude and phase values from a measurement of vibration, which are then used in a subsequent machine learning method as features. Other examples for preprocessing are: synchronizing the time axis of sensors, because sensor data is acquired from different sources that have different time keeping; interpolation of sensor data to accommodate for missing values or nonequidistant sampling; low-pass filtering to reduce noise or other unwanted highfrequency components in the measured data. Another reason for the application of data preprocessing is the reduction of data dimensionality. This may be necessary simply due to storage or data transfer bandwidth limitations. Moreover, it reduces the complexity of the machine learning required to learn the relevant aspects from the data by letting it focus on a preselected representation of the data. In this way, one can also steer the machine learning in the right direction. Thus, data preprocessing is a very relevant step in machine learning. The above kinds of preprocessing are usually applied before the actual machinelearning; see Figure 1. For instance, the interpolated data may be Fourier transformed to extract the frequency components with the highest absolute value, i. e., magnitudes. These features are then fed into a machine learning method for classification [3]. Uncertainty-aware machine learning (UA-ML) means that the machine learning method takes the uncertainties associated with its input values into account. For example, the classification method may use the uncertainties associated with the frequency components as a weighting in the selection of suitable features. Hence, uncertainty-aware machine learning requires the propagation of uncertainties from the sensor data through the preprocessing steps. Most of the uncertainty evaluation methods needed for the machine learning preprocessing are similar to tasks in the analysis of dynamic measurements: digital filtering, Fourier transform, Wavelet transform, interpolation, etc. Thus, the toolbox of methods for dynamic measurements provides an excellent starting point to enable uncertainty-aware machine learning (UA-ML).

Figure 1: (Placeholder image) Typical workflow in machine learning with data preprocessing including use of signal processing methods as first step.

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Evaluation of uncertainty for such data processing pipelines needs to be real-time capable, robust, and easy to implement in order to fit into IoT scenarios. That is, similar to the fact that sensors are becoming more intelligent, the analysis of the sensor data needs to become automated and intelligent, too. One reason is the rapidly increasing amount of data that needs to be processed in IoT environments. For IoT, it is not one individual measurement for which a model, uncertainty budget definition, and uncertainty evaluation are to be carried out. Instead, data streams from several (hundreds of) sensors need to be analyzed online. This can be achieved only by automated, flexible, and intelligent approaches and implementations.

3 Calibrated measuring instruments in the Internet of Things Measuring instruments in the IoT usually need to: provide their measured values via a digital interface, ideally over a network interface; work reliably under a wide range of conditions and ideally, detect and report adversarial conditions; report on their health status upon request; work automatically without significant human interaction, ideally containing some sort of self-control and adaption to changing conditions. Sensors that have these properties are also called “smart sensors” [3–5]. Thus, these are measuring devices that contain some sort of preprocessing. Due to the rise of IoT and related applications, smart sensors are increasingly requested from sensor manufacturers. Their preprocessing may range from a simple analogue-to-digital conversion up to a complex compensation of influencing effects measured by sensing elements integrated into the “smart sensor.” Calibration traceable to the international system of units (SI), a harmonized treatment of measurement uncertainties and the application of industry standards and guidelines are the main components of a metrological infrastructure that enables globalized production and international trade. This holds true as well for the IoT. The calibration and verification of smart sensors, though, is very challenging, because to the user these devices usually remain a “black box.” Moreover, common standards for the calibration or treatment of measurement uncertainty for the integrated preprocessing are not available yet. The approaches and experiences from dynamic measurement analysis in metrology provide a good starting point to address some of these challenges. The concept of the IoT relies on a versatile and flexible combination of measuring instruments, the automated acquisition and processing of the measured data, and the application of intelligent algorithms to derive conclusions or decisions. Moreover, due to the increasing availability of low-cost measuring devices, sensor networks in the IoT are much larger than is usually the case for traditional measurement applications. As a consequence of the size and complexity of the networks, data analysis is typically datadriven and, in particular, machine learning methods are often applied. In contrast to

120 � S. Eichstädt mathematical models that rely on a physical understanding of the measured process, machine learning can be applied directly based on the sensors’ output data. Thus, the need for calibration is not as obvious as for “traditional” measurements. However, calibrated measuring instruments in the IoT offer several benefits: – Calibrated sensors can serve as reference devices in the network to assess and improve data quality; – Calibration of the sensor enables the estimation of the measurand, i. e., its input signal. In this way, a replacement of the sensor by another calibrated one does not affect the machine learning; – Calibration enables traceability to the international system of units of measurement (SI), and thus, the comparability between different sites and countries; – Calibrated sensors improve the explainability of the obtained output from the machine learning. Moreover, there is an increasing use of digital twins, in which a model of a physical object is updated based on an evolving set of data [6], and software-based sensors as a fusion of models with sensor data. In the IoT, digital twins provide insights into the behavior of the system being observed. Calibration information is the first step to build reliable digital twins. Software-based sensors, or simply “soft sensors,” are aggregating data from a set of sensors. The outcome may be considered a “measurement” provided a calibration has been carried out such that a unit of measurement can be associated with the values of the soft sensor. The simplest form of a soft sensor is a smart sensor that uses measurement information about its surroundings in the internal data preprocessing. Thus, calibration information in IoT play an important role. The use of this information varies with the place where in the IoT architecture the sensor data is processed. The simplest form of IoT is a direct communication between the individual devices with a central unit, usually considered to be in the “cloud,” i. e., a computer somewhere in the internet. Sometimes, the preprocessing of data is carried out before data is sent to the cloud. This is also called processing at the “edge” and devices that integrate such preprocessing are called “edge devices.” For larger and more complex IoT architectures, it may be infeasible to communicate all data to the cloud. In these cases, preprocessing at the edge, soft sensors, and other approaches are combined into what is called the “fog” also considered as a layer between the edge and the cloud [7]. Figure 2 illustrates the role of calibration in the different layers in the IoT. Calibration of the individual sensors provides traceability to the SI as well as the basis for the evaluation of measurement uncertainty. For homogeneous sensor networks, i. e., with sensors measuring the same quantity or even the same measurand, novel concepts such as cocalibration may be considered. For example, one of the sensors could be considered as reference and the parameters of the LTI model of the other sensors adapted accordingly. This could also be used to detect sensor drift and automatically adapt the model parameters online. For heterogeneous sensor networks, i. e., with sensors measuring different quantities, traceability to the SI simplifies the replacement of

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Figure 2: The role of calibration in the different layers in the IoT: calibration of the individual sensors provides information for the data preprocessing at the edge or in the fog layer. The uncertainties associated with the preprocessed data and the general calibration information is used for the assignment of quality of data metrics in the cloud layer.

sensors and improves the interpretability and explainability of data. Software-based sensors aggregate data from individual sensors into a measurement result using a mathematical model or data-driven machine learning. At the edge layer, measurement uncertainties and other calibration information can be used in the data preprocessing. Examples are uncertainty-aware feature selection, plausibility assessment, and anomaly detection. The propagation of uncertainties through these processing steps is necessary to enable its use in the fog and cloud layers. In the cloud layer, calibration data and measurement uncertainties associated with the data streams form the basis for an assessment of data quality. This enables the selection of high-quality data as reference, and it improves the trustworthiness of data-driven conclusions and decision-making. The application of metrological principles for the IoT leads to several calibration and traceability challenges. For example, the establishment of metrology in IoT means that calibration possibilities have to be extended to sensors with a purely digital output signal. This requires, among other things, new concepts for the generation of time stamps for the signals from sensors. This is particularly important for dynamic, i. e.,

122 � S. Eichstädt frequency-dependent, calibration. The reason is that the reliable calibration of the phase change in the sensor signal is an important element for time-dependent quantities as outlined in Chapter 3. In typical IoT applications, sensors provide digital, timedependent output signals, and have internal signal processing capabilities. This makes the calibration of the sensor phase response difficult, because the internal time measurement of the sensor is no longer managed by the calibration system, which requires new concepts for the calibration of such sensors. To this end, the authors in [8] developed a microcontroller board, which can accommodate one or more sensors, provides options for connection to external, traceable timers, and allows online preprocessing of the measured data. With the integration of traceable timers, the time axis is traceable to the SI and the sensor can be dynamically calibrated with conventional approaches, including its phase response. With the appropriate hardware with proper calibration as described in [8, 9], a sensor can then be expanded to allow an easy integration into IoT. In addition to the traced back time stamps of the sensor signals, evaluation methods can be implemented on-board of the microcontroller. These methods could provide the determination of the measurement uncertainty for each measured value. In this way, basic principles of measurement value treatment can be integrated directly on the extended sensor or applied in an edge computing approach close to the sensor, i. e., at the edge. For this purpose, the applied methods of data and signal processing must be extended by methods for propagating measurement uncertainties as described in Chapter 2. The basic mathematical groundwork for huge variety of methods is already available and basic implementations in Python are also available as part of the PyDynamic software package. Important for the practical use is the modular structure of the implementation in order to be applicable to many application areas in a flexible way using so-called agents as outline below. Provision of calibration information in the IoT also includes a machine-readable data structure, e. g., in terms of a digital calibration certificate (DCC) [10]. A DCC should at least contain the information required by the ISO 17025. This information can be used in the fog or cloud layer to monitor the overall status of measurement instrumentation in the network. In some industrial areas, such as in the pharmaceutical sector, this is a requirement by regulation authorities. In addition, a DCC could also include a validated expression of the calibrated dynamic system model or a validated inverse model for deconvolution. The combination of the above described components (calibration of digital sensor; DCC integration into the sensor; extension of sensor to provide a measurement result rather than measured values) lead to what can be called “smart traceability.” This concept uses hardware and software elements that can be moved easily between the individual smart sensor, the edge, the fog, or the cloud—depending on the available computing power and application. With smart traceability present in the IoT, novel concepts for using semantic information can be applied. The authors in [1] describe an approach that utilizes the information from smart sensors and other sources to obtain a semantic description of the sensor network. This information is provided in a machine-readable way

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such that an algorithm can make use of it in the subsequent data analysis. For example, the expression of the calibration dynamic system model can be used to automatically design a deconvolution filter with associated uncertainty evaluation. Again, this can be implemented using so-called agents as outlined below.

4 From calibration data to a digital twin Beyond the mere extension of the calibration equipment, approaches for the treatment of data preprocessing methods integrated in the sensor must be developed. As a rule, these methods will not be visible to the calibration laboratory. Formally, knowledge of the methods used in the sensor is not necessary to perform the calibration. In practice, however, the methods can lead to strong nonlinear effects. For example, an algorithm for avoiding jumps in the sensor signal by smoothing can lead to a nonlinear behavior of the sensor in a shock calibration. The previously linearly reacting sensor thus becomes a nonlinear measuring instrument due to the software used and the result of the calibration may no longer reflect the real behavior of the measuring instrument in practice. As a result, calibration regulations need to be revised. However, this will first require extensive basic research and the definition of approved types of methods for data preprocessing. It must be ensured that the methods and algorithms integrated in the measuring instrument cannot lead to misjudgments regarding the measuring capability and precision as a result of the calibration. A mere disclosure of the algorithms is probably not enough for this purpose, since the complexity of corresponding procedures is increasing rapidly. In particular, if methods of machine learning should be integrated into measuring instruments, a pure analysis of the algorithm will probably not provide reliable statements for the uncertainty of the calibration result. However, without a correct calibration with associated measurement uncertainty, the later measurement result is not traceable to SI units. Alternatively, targeted approaches could be developed, which implement recognized methods of measurement data evaluation including the determination of measurement uncertainties close to the sensor. This could result in smart measuring devices that combine the traceability to SI units achieved by calibration with intelligent data preprocessing in the measuring device. This means, e. g., that the calibration results are also implemented directly in the measuring instrument in the form of methods for compensating dynamic effects of the sensor. The result would be the principle of “smart traceability” already mentioned above. In this way, the customers’ desire for intelligent sensors can be met by the manufacturers, and at the same time the application of methods for evaluating measured data recognized in metrology can be simplified.

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5 Modeling of dynamic measurements in IoT In Chapter 1, mathematical models for individual measuring instruments as well as for complex measuring systems are discussed. These kinds of models for the individual sensors can also be applied in IoT scenarios. For instance, a low-pass filter or second-order model may be applicable for a MEMS acceleration sensor similarly to a piezo-type accelerometer. The digital technologies embedded in the (smart) sensors, the edge or somewhere else in the IoT network allow for a more efficient provision and automated use of such models, provided they are given in a machine-readable way [1]. That is, the IoT can make use of the sensor model in a more efficient way. The aspect of the machinereadability of the model is important for the IoT, because of the typically large number of sensors being in place. With hundreds or even thousands of sensors in place, a manual application of a model, data analysis and integration cannot be carried out reasonably. The often used state-space models for measuring systems, consisting of several sensors, as discussed in Chapter 1, is usually not used in IoT scenarios, though. The reason is that such models assume knowledge of a physical interconnection of the system, which is usually not given in IoT. For instance, a state-space model of a measuring system observing the position of a moving vehicle may contain measurements of inertia, velocity, and acceleration. The model is then derived based on the known physical relationship between these quantities to estimate the position of the vehicle. In a typical IoT approach to this problem, though, usually data-driven models are used instead. These models require large amounts of data and derive a relationship between the measured values in the system and of the target value. In this way, even large and complex systems can be handled efficiently. Provided the information about the sensors, their interconnection and the target are provided in an appropriate machine-readable way, the estimation of the model can even be carried out automatically. With an existing data-driven model in place, also new insights into the physical relationship in a sensor network can be achieved. For instance, a data analysis as in [5] of a sensor network estimating the health status of a machine part may result in the finding that the change in sound characteristics of the working machine part is more relevant for the target value than the actual physical components such as electrical power consumption or movement of the part. With the new possibilities for making use of sensor models and for working with sensor networks, also the way in which models and methods are implemented change. One aspect is the above-mentioned machine-readability of information. The other aspect is the implementation of the actual methods, e. g., for uncertainty evaluation. The methods described in the previous chapters are all implemented with a manual application in mind. That is, it is assumed that a human acquires the sensor data, runs the software, checks and saves the results manually. This also includes the interpretation of software interfaces, the potential adjustment of variables and other fixes. In an IoT scenario, this is not a practical approach. Instead, the software implementing the methods must have clear and largely standardized interfaces. In addition, the data analysis

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methods must be able to work with data streams. This requires implementation of methods in such a way that they can either work sequentially or with a moving window of data. This also requires that the methods need to run efficient enough to ensure that the results are calculated at least as quickly as new data arrives. Sequential uncertainty evaluation methods have been developed for some scenarios, such as Fourier transform [11], Wavelet transform [12], and digital filtering [13]. It can be expected that more work in this direction will follow in the near future.

6 Conclusions and outlook The analysis of dynamic measurements in metrology has focused on the increase of precision and accuracy of sensors by improving the underlying mathematical model and its proper calibration. This is in line with the general metrological approach to improve understanding and working of measuring instruments. The sensors in the IoT, though, are often low-cost devices, such as MEMS sensors. For these types of measuring instruments, an improvement of precision and accuracy via calibrations with smaller uncertainties is not an appropriate consideration. Nevertheless, the models and methods from dynamic metrology are highly relevant for the IoT, too. Understanding dynamic effects, modeling of complex systems and the development of uncertainty evaluation methods will be the basis for the emerging field of sensor network metrology. These methods, however, will need to be further developed to meet the novel requirements that differ from the laboratory practice. In addition, completely new developments are needed, such as, e. g., in situ and cocalibration in IoT, and semantic description of measurands, conditions, and measuring instruments in accordance with metrology standards.

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Index ADC see analog-to-digital conversion algebraic equation 2 amplifier – bridge 79 – charge 79 – conditioning 79 analog-to-digital conversion 3 anti-alias filter 14 back-to-back configuration 76 bandpass correction 30 bandpass function 30 bandwidth 9 bilinear transform 15 Bode gain-phase relation 107 bound function 21 calibration 44, see also dynamic calibration – bandwidth 90 – dynamic 4 – uncertainty 77 – guideline 80 charge sensitivity 74 cloud 120 combustion engine 91 compensation – bandwidth 17 – dynamic compensation 16 – filter 19 compensation system 16, 17, 27 convolution 7 damping 73 deconvolution 16, 102, 104, 110, 111, see also input estimation – filter 18 – frequency domain 17 differential operator method 30 digital compensation filter 19 digital output 80 digital twin 120 discrete-time 3 – Fourier transform 13 dynamic 1 – behavior 2 – ringing 2 – calibration 82 https://doi.org/10.1515/9783110713107-006

– measurand 1 – measurement 1 – measuring system 1 – quantity 1 – system 1 dynamic error 4, 16, 21, 22 dynamic error correction 16 dynamic force calibration 81 dynamic input estimation 16 dynamic measurement uncertainty see uncertainty dynamic metrology 3 dynamic model 78, 86, 94, see transfer behavior dynamic torque 90, 92, 93 dynamic uncertainty 3, 23 dynamic uncertainty component 4, 23 edge device 120 error bound 21 extrapolation 108, 111 falling mass calibration 85 fatigue testing 89 filter – FIR 18 – IIR 18 FIR see transfer behavior FIR filter 18 fog 120 Fourier domain – convolution 7 – deconvolution 107 Fourier transform 7, 104, 110 – inverse 7 – one-sided 13 frequency domain estimation 17 frequency response 20 frequency response function 7 – amplitude 108 – extrapolation 108 – interpolation 108 – magnitude 107 – modulus 107, 108 – phase 107, 108 frequency warping 15 GUM see uncertainty

128 � Index

Hilbert transform 107 hydrophone 101 – calibration 102 – calibration data 106 – sensitivity 102, 103, 106, 111 – thickness mode resonance 102 IIR see transfer behavior IIR filter 18 ill-posed problem 16 impact hammer 90 impulse invariance method 14 impulse response 5, 20 Industrie 4.0 117 input estimation 50 Internet of Things 117 inverse filter 18 inverse model see input estimation inverse problem 16 inverse system 20 IoT 117 L-curve method 25 Laplace transform 9, 12 laser interferometry 74 linear time-invariant system 4 low-pass filter 111 lowpass filter cut-off 27 LTI see linear time-invariant system, see transfer behavior machine learning 117 – uncertainty-aware 118 mass moment of inertia 92 mass-spring-damper model 73 measurement – dynamic 99 measurement process 1 minimum phase system 107 model see transfer behavior – amplifier 79 model-free calibration 89 Monte Carlo see uncertainty motion sensor 73 narrowband approximation 104 Nyquist frequency 14 ODE see ordinary differential equation

OPC-UA 117 ordinary differential equation 1 piezoelectric 81 pre-processing 118 pre-warping 15 primary calibration 74, 75 PyDynamic 35, 63, 64, 105, 110 quantity – dynamic 99 regularization 16, 23, 27, 111 – filter 111 – uncertainty estimation 111 regularization error 27 resonance frequency 8 rotation torque 92 secondary calibration 76 seismic sensor 73 shock excitation 84 shock force 88 SI 117 signal 1 – analog 3 – continuous-time 3 – digital 3 – discrete-time 3 – input 1 – multivariate 1 – output 1 – time-dependent 1 signal processing 117 signal-to-noise ratio 26 sine approximation method 75 sinusoidal excitation 75, 87 SNR see signal-to-noise ratio soft sensor 120 state-space system 11 static 2 – measurement 2 – measuring system 2 – system 2 step response 5, 20 strain gauges 81 Tikhonov 23 time-dependent measurement 1

Index

traceable – dynamic measurement 3 transfer behavior 8, see also system – finite impulse response 41 – frequency response 42, see also Fourier, see also frequency response – impulse response 41, see also impulse response – infinite impulse response 42 – linear time invariant 39 – state-space 40 – step response 41, see also step response transfer function 10 – poles 10, 19 – roots 10 – zeros 10, 19 ultrasonic waves 99 ultrasound 99 – exposimetry 100 – hydrophone 101

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– peak compressional pressure 100, 112 – peak rarefactional pressure 100, 112 – power 100 – pressure 100 – temporal average ultrasonic intensity 100, 112 uncertainty – dynamic measurement 38 – evaluation see uncertainty – framework 36 – propagation 36, 47 uncertainty estimation 104, 109 unit step signal 5 upper bound 21 wavelet 111 Wiener deconvolution 25 Wiener filter 25 z-domain 18