From Nondynamic Errors to Dynamic Errors - A Structural Deployment

Errors represent a special kind of signals. There are many types of errors. The most prominent ones are systematic and / or random errors, as well as constant and / or time dependent errors. Different types may overlap. The modelling procedure of error quantities is an important issue. There are processes of interest, error sources, error processes, error superpositions, and error compensations, a veritable error chain. The review at hand develops and interprets some structures and properties of this chain. Introduction Errors have been discussed continuously, with more or less power of persuasion and success. At present, intentions to neglect errors are fashionable. However, there seems to be a revival of the error to come, since concepts and structures are still debatable. Error analysis, error description, and error correction are no easy tasks. In some fields, in production processes for example, an error is just an ad hoc information about a particular property value: Some report is due and decisions can be taken. However, there are processes, for example continuous, realtime, on-line measurement processes, where time dependent acquisition results are error prone. One has to reconstruct the original values or trajectories of quantity of interest in real time. Further on, one may want to separate the errors from the erroneous quantities by whatever means. The latter procedure is notably malicious due to time dependent errors and dynamic processes. The following sections provide an introduction to a deeper and more systematic understanding of this tricky area. In principle, all fields of Science and Technology are concerned, but not all provide the necessary background for an appropriate handling. In this respect, Signal and System Theory (SST), a field, which is famous for consistent logical statements and for mathematically rigorous relations, delivers valuable tools to analyse, describe, and solve the diverse challenges. Up to now, errors in Metrology are seldom addressed by Signal and System Theory (SST). The main reasons are their demanding theoretic capabilities, which seem not to account for specific needs in daily practice. This is a misjudgement. Quite the contrary, the definition of relations between quantities as the description of processes of interest always utilise the very same concepts and tools. This is important, since the measurement and observation chain normally consists of many interacting subprocesses, which are quite diverse indeed. Therefore, the following treatment of error quantities and error processes has to take a broad view. 1 To whom any correspondence should be addressed. XXII World Congress of the International Measurement Confederation (IMEKO 2018) IOP Conf. Series: Journal of Physics: Conf. Series 1065 (2018) 212020 IOP Publishing doi:10.1088/1742-6596/1065/21/212020 2 1. Quantities intended to be acquired and processes of interest Important: No process P of interest is measured or observed, only its defined and specified quantities are! At an early stage of any metrological procedure, it is important to properly define the quantities intended to be measured and possibly observed. The model of a particular quantity (signal) with its properties (structure plus parameters) is theoretically and / or empirically identified (calibrated) either from scratch or more likely from measurement, based on a hypothesis of the model structure. For an acquisition of quantities of a process P, we need a measurement process M, which includes a sensor process S and a reconstruction process RS. Defined abstract error signals come into existence within systems and are visible within system blocks of Signal Relation Graphs (SRG). This may be surprising, since we know that signals are often disturbed by their environment and that vice versa signals may disturb their environment. But we have to be aware that mathematical models do not map the real world as such. They just describe signals and relations between signals. Therefore, a disturbance of a signal is considered a relation between signals and thus a model of a disturbed system. In other words, all procedures concerning signals occur in systems. 2. Ideal measurement procedures The idealistic hypothesis assumes that defined output quantities y(t) of process P with actual properties are acquired by an ideal (nominal) measurement process, denoted by MN. This assumed ideal measurement process MN serves as a reference and as a benchmark for further definitions, discussions, and decisions, which always take place on the model level and not on the real-world level. An ideal (nominal) measurement process MN is described by the constant transfer identity matrix GMN = I, which means that MN ˆ(t) (t) (t) = = y G y Iy . For this fictitious, ideal situation, this relation is remodelled, in order to include the total measurement error ey(t), which fictitiously is zero: ˆ(t) (t) (t) − = = y y y e o . 3. Almost ideal measurement procedures Since no ideal measurement process M exists, the question comes up, what a almost ideal measurement process M is, whose measurement errors ey(t) will almost equal zero. Such a definition does not just depend on the properties of the measurement process M, but on the quantities y(t) of process P, intended to be measured and in addition on the requirements of the measurement task. Therefore, any measurement procedure by means of an almost ideal measurement process M is a compromise. 4. Errors in nonideal, dynamic systems As a first step, an extension of the so-called ideal system is made, in order to enable nonideal structures as well as descriptions of the different error signal types. Therefore, a general nonideal system P is assumed to have two inputs, the input signal vector u(t) and the disturbance signal vector v(t), and, in addition, two outputs, the output signal vector y(t) and the loading signal vector z(t), which influences precedent systems. Internal subsystems, for example nonlinear and / or dynamic subsystems, are embedded, in order to enable the description of error signal vectors e(t). This concept is unique and valid for all further systems. XXII World Congress of the International Measurement Confederation (IMEKO 2018) IOP Conf. Series: Journal of Physics: Conf. Series 1065 (2018) 212020 IOP Publishing doi:10.1088/1742-6596/1065/21/212020 3 For this extension, the conventional State Description (SD) of a dynamic system includes one additional column for the additional, disturbing input signal vector v(t), and one additional row for the additional, loading output signal vector z(t). A vector-matrix differential equation describes the possible internal dynamic relations. This structure serves almost all applicational needs. Nonlinear effects are includable.