Page 33 - Mechatronic Systems Modelling and Simulation with HDLs

22 2 PRINCIPLES OF MODELLING AND SIMULATION

The first stage of signal analysis is the establishment of a suitable test signal,
which is triggered by the system. Possibilities here are step functions, rectan-
gular pulses, triangular pulses and many more. An inspection in the frequency
range facilitates investigations into whether the system to be identified is suffi-
ciently excited over the spectrum of interest. Measurements are generally only
made at discrete time points, so that a sampling interval must also be determined.
Furthermore, a measurement time must be specified, the lower limit of which is
determined by the point at which sufficient data is available for identification. The
progressive nature of a real system imposes an upper limit on the measurement
time. Then, signal processing procedures may also be used, such as averaging,
root mean square calculation, or Fourier analysis, correlation analysis, and spec-
tral analysis. So, for example, statistically dispersive noise signal components can
be disposed of by averaging similar measurements which, however, multiplies the
measurement time.
In stage two, determining the model approach, we can choose between prefabri-
cated and customised structures, see for example Ljung [234]. The former may, for
example, consist of canonical models in the state space and lead to a ‘black-box’
parameterisation, i.e. model structure and parameters have no physical significance,
but rather serve merely as a vehicle for reflecting the observed behaviour. Cus-
tomised equation system structures, on the other hand, are based upon a physical
modelling of the system, so that the identified parameters also possess a physi-
cal significance. In any case, however, all available information about the system
should be fed into this. This applies in particular to the faults that are virtually
always present, which in most cases rule out an exact solution.
The identification typically rests upon minimising the discrepancy between mea-
surement and simulated behaviour or a functional of this. Various quality criteria
are used for this, one of which is selected in the third stage. Criteria are particularly
frequently selected that assess a quadratic function of the measurement error.
To conclude the identification, numerical procedures are used in order to min-
imise the quality criteria selected in the third stage. These procedures are performed
for all model structures proposed in the second stage, so that not only are the param-
eters in question determined in this stage, the quality of the individual structures
in relation to one another are also established. This facilitates a selection of the
model structure.
In the simplest case we can, as in Kramer and Neculau [206], quote the following
equation for the system under investigation:

y k = a · x k + n k (2.1)

where x k denotes an input quantity, y k an output quantity, n k a disturbance vari-
able in relation to the measurement and a is the parameter to be estimated. This
relationship should be modelled on the basis of the following approach:

ˆ y k = ˆa · x k (2.2)