State estimation with Kalman Filter
State estimation with Kalman Filter
State estimation with Kalman Filter
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F. Haugen: Kompendium for Kyb. 2 ved Høgskolen i Oslo 110<br />
Real system (process)<br />
Process noise<br />
(disturbances)<br />
w(k)<br />
(Commonly no connection)<br />
Measurement<br />
noise<br />
v(k)<br />
u(k)<br />
Known inputs<br />
(control variables<br />
and disturbances)<br />
Process<br />
x(k)<br />
x(k+1) = f[x(k),u(k)] + Gw(k)<br />
(Commonly no connection)<br />
<strong>State</strong> variable<br />
(unknown value)<br />
Sensor<br />
y(k) = g[x(k),u(k)]<br />
+ Hw(k) + v(k)<br />
Measurement<br />
variable<br />
y(k)<br />
(Commonly no connection)<br />
<strong>Kalman</strong> <strong>Filter</strong><br />
f()<br />
System<br />
function<br />
x p (k+1)<br />
1/z<br />
Corrected<br />
estimate<br />
x c (k)<br />
Unit<br />
delay<br />
x c (k)<br />
x p (k)<br />
Predicted<br />
estimate<br />
Applied state<br />
estimate<br />
Ke(k)<br />
g()<br />
K<br />
y p (k)<br />
Measurement<br />
function<br />
<strong>Kalman</strong> gain<br />
e(k)<br />
Innovation<br />
variable<br />
or ”process”<br />
Feedback<br />
correction<br />
of estimate<br />
Figure 8.1: The <strong>Kalman</strong> <strong>Filter</strong> algorithm (8.35) — (8.38) represented by a block<br />
diagram<br />
(8.35) — (8.38) can be represented by the block diagram shown in Figure<br />
8.1.<br />
The <strong>Kalman</strong> <strong>Filter</strong> gain is a time-varying gain matrix. It is given by the<br />
algorithm presented below. In the expressions below the following matrices<br />
are used:<br />
• Auto-covariance matrix (for lag zero) of the <strong>estimation</strong> error of the<br />
corrected estimate:<br />
P c = R ex c (0) = E n(x − m xc )(x − m xc ) T o (8.39)<br />
1 Therefore, I have underlined the formula.
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