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EXTENSIONS OF THE KALMAN FILTER 305
Step 2.3 is only required if we are interested in the smoothing covariance
matrix.
Example 8.19 Estimation of a transient of an electrical RC circuit
Figure 8.15 shows an electric circuit consisting of a capacitor con-
nected by means of a switch to a resistor. The resistor represents the
input impedance of an AD converter that measures the voltage z. The
voltage across the capacitor is x.At t ¼ 0 the switch closes, giving rise
to a measured voltage z ¼ x þ v (v is regarded as sensor noise). The system
x
obeys the following state equation _ x ¼ x/(RC) with RC ¼ 10 (ms).
v(t) t=0
x(t) C z(t) R
Figure 8.15 The measurement of a transient in an electrical RC network
20 state (thick) and measurements 20 filtered state (thick) and measurements
10 10
0 0
–10 –10
–20 –20
0 5 10 0 5 10
t (µs) t (µs)
20 smoothed state (thick) and measurements 5 smooth error (thick) and filter error
10
0 0
–10
–20 –5
0 5 10 0 5 10
t (µs) t (µs)
Figure 8.16 Estimation of a transient by means of filtering and by smoothing
