Instrumentation & Measurement Magazine 23-2 - 68

Fig. 9. Reconstruction in the case when only the effect of the physical quantity can be measured by sensor. Also the effect of signal paths in the physical system
needs to be taken into account. (a) Physical system can be described by a model with known parameters. (b) Signal path in the physical system can be described by
a model having unknown, but measurable parameters. (c) Quantity to be measured is an unmeasurable internal state of a physical system.

Case 2: Secondary Effect of a Physical
Quantity Can be Directly Measured by a
Sensor
A more complicated task is to reconstruct the signal if the
physical quantity to be observed cannot be directly measured
by a sensor, only its effect. In this case, the distorting effect in
the physical system from the signal to be observed to the one
that can be measured with sensor needs to be identified, either from the a priori knowledge of physics, or from system
identification measurements. If this correspondence can be
described by an invertible model, having known parameters
(Fig. 9a), the effect of the two systems (physical and measurement system) can be combined and described together. With
this combination, the problem is reduced to a previous one
(Fig. 5). Life gets more complicated if the effect of the physical
system in Fig. 9a cannot be described by a model with known
68	

parameters (even not with time varying parameters), because
the parameters are unknown physical quantities (e.g., temperature dependence of transfer function). These parameters are
time varying, but their fluctuation is not known. However, we
might have the chance to measure them (Fig. 9b). In that case,
three different distortion and disturbance sources need to be
taken into account during the inverse operation: distortion
and disturbance of the physical system, that of the measurement system measuring the secondary effect of the signal to
be observed, and finally, the distortion and disturbance of the
measurement systems measuring the physical quantities that
influence the model parameters of the physical system.
The most complicated case investigated in this paper is
when the physical quantity to be observed is an unmeasurable
internal sate of a physical system. In this case, observation theory might help to determine the internal state. Observers copy

IEEE Instrumentation & Measurement Magazine	

April 2020



Instrumentation & Measurement Magazine 23-2

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