Page 270 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

SYSTEM IDENTIFICATION 259

separatelyidentifiablefromthelevelmeasurementsbecausetheseparameters
always occur in the combination /R 1 . Thus, in this case we can treat /R 1 as
one identifiable parameter. Sometimes, it might be necessary to temporarily
use additional (often expensive) sensors so as to enable the estimation of all
relevant parameters. As soon as the system identification is satisfactorily
accomplished, these additional sensors can be removed from the system.
Often, the acquired data need preprocessing before the parameter esti-
mation and evaluation take place. Reasons for doing so are, for instance:

. If the bandwidth of the noise is larger than the bandwidth of
interest, filtering can be applied to suppress the noise in the unim-
portant frequency ranges.
. If a linearized model is strived for, the unimportant offsets should
be removed (offset correction, baseline removal). Often, this is done
by subtraction of the average from the signal.
. Sudden peaks (spikes) in the data are probably caused by disturb-
ances such as mechanical shocks and electrical inferences due to
insufficient shielding. These peaks should be removed.

In order to prevent overfitting, it might be useful to split the data
according to two time intervals. The data in the first interval is used
for parameter estimation. The second interval is used for model evalu-
ation. Cross-evaluation might also be useful.

Example 8.2 Experimental data from the hydraulic system
Figure 8.3 shows data obtained from the hydraulic system depicted in
Figure 8.2. The data is obtained using two level sensors that measure
the levels h 1 and h 2 . The sample period is ¼ 5 (s). The standard
deviation of the sensor noise is about v ¼ 0:04 (cm).
The measured levels in Figure 8.3 correspond to the free response of
the system obtained with zero input and with an initial condition in
which both tanks are completely filled, i.e. h 1 (0) ¼ h 2 (0) ¼ 25 (cm).
Such an experiment is useful if it is envisaged that in the application
this kind of level swings can occur.

8.1.3 Parameter estimation

Suppose that all unknown parameters are gathered in one parameter
vector a. The discrete system equation is denoted by f(x, w, a). The