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Commit 0e6f0ff7 authored by zhiyupan's avatar zhiyupan
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add v10 and DC teil2

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%% Cell type:markdown id:a2b19476-4bf6-4dba-b506-3d530bf60bd2 tags:
# <span style='color:OrangeRed'>V10 KALMAN FILTER</span>
%% Cell type:markdown id:68f373f1-ea17-4b52-86ed-b4c3afaf5b21 tags:
<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
Gegeben ist einen Wechselstrommotor als zeitdiskretes System.
Es wird nun ein Kalman-Filter zur Schätzung des Zustandsvektors
x(k) verwendet. Als Messung liegt der Motorwellenwinkel als ein verrauschtes Sensorsignal
vor. Ferner ist der Eingang des Motors eine verrauschte Eingangsspannung,
die nicht mit dem Messrauschen korreliert ist. Ein quivalentes zeidiskretes System des
Wechselstrommotors wird mittels gest rter Zustandsgleichungen wie folgt beschrieben:
%% Cell type:code id:402f525f-8a64-4584-8d05-dbd5e42e5516 tags:
``` octave
Ad = [1 0.09516; 0 0.9048]
Bd = [0.004837; 0.09516]
Cd = [1 0]
Dd = 0
```
%% Output
Ad =
1.00000 0.09516
0.00000 0.90480
Bd =
0.0048370
0.0951600
Cd =
1 0
Dd = 0
%% Cell type:markdown id:8629e5a2-ae35-418d-8a65-7e511440917e tags:
<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
Die Varianz des Prozessrauschens ist mit Qk = 1,0 und die Varianz des Messrauschens
mit Rk = 0,45 angegeben — beide sind zeitinvariant. Der initiale Schätzfehler ist mittelwertfrei
und Gauß-verteilt, wobei dessen initiale Kovarianz durch einen Matrix Sigma beschrieben wird.
Der Motor rotiert initial nicht und der Anfangswert des Motorwellenwinkels
beträgt pi/4.
%% Cell type:code id:12fc81f8-de9e-4a3d-9210-2fdc15f0338c tags:
``` octave
Qk = 1.0
Rk = 0.45
Sigma = [100 0; 0 10]
xo = [pi/4; 0]
```
%% Output
Qk = 1
Rk = 0.45000
Sigma =
100 0
0 10
xo =
0.78540
0.00000
%% Cell type:markdown id:5292b94c-4bbe-4127-8c8a-55f20706ba40 tags:
<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
a) Überprüfen Sie zunächst, ob das System beobachtar ist
%% Cell type:markdown id:4f60313c-d4d0-4fe8-892f-1a9e96774d0a tags:
<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
Die Beobachtbarkeit l sst sich wie folgt prüfen durch:
%% Cell type:code id:a579b6da-ea06-469d-bdad-e438d840fd37 tags:
``` octave
So = [Cd; Cd*Ad]
```
%% Output
So =
1.00000 0.00000
1.00000 0.09516
%% Cell type:markdown id:d4fac333-e43d-49c7-aaec-6168b86bd6a6 tags:
<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
Das System ist beobachtar wenn der Rang ist 2, dass heißt wenn der Determinant nicht null ist:
%% Cell type:code id:0d90c253-316f-47ed-9f7b-561ff0ed4868 tags:
``` octave
det(So)
```
%% Output
ans = 0.095160
%% Cell type:markdown id:a6ed8d3e-8d76-451c-8480-7182a8723ab2 tags:
<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
b) Berechnen Sie den Anfangswert der Kalman-Verstärkung und den Wert der Zustandsschätzung
nach dem ersten Korrekturschritt.
%% Cell type:code id:828217c8-bd90-42da-8b36-9e1f05727a24 tags:
``` octave
Ko = (Sigma*Cd')*inv(Cd*Sigma*Cd'+Rk)
```
%% Output
Ko =
0.99552
0.00000
%% Cell type:markdown id:8a4be558-f51c-415e-96f7-922b8e417422 tags:
<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
Der Wert der Zustandsschätzung nach dem ersten Korrekturschritt lautet:
%% Cell type:code id:058ad783-d553-4bcc-873c-a6f521c16975 tags:
``` octave
yo = pi/4;
xc = xo+Ko*(yo-Cd*xo)
```
%% Output
xc =
0.78540
0.00000
%% Cell type:markdown id:ebc1ad68-243b-41eb-909d-0520bc17630d tags:
<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
c) Berechnen Sie die Kovarianzmatrix des Sch tzfehlers nach dem ersten Korrekturschritt.
%% Cell type:code id:6bce8c32-255e-4236-8fb7-fb5e7bcdac6e tags:
``` octave
Sigma2 = (eye(2)-Ko*Cd)*Sigma
```
%% Output
Sigma2 =
0.44798 0.00000
0.00000 10.00000
%% Cell type:markdown id:4ed1cc00-cac1-4390-b64e-6dac83a0f0aa tags:
<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
d) Berechnen Sie den Anfangswert der Kalman-Verstärkung und den Wert der Zustandsschätzung
nach dem ersten Korrekturschritt für den Fall, dass die Kovarianz
des Schätzfehlers vernachlässigbar ist.
%% Cell type:markdown id:05bfa508-a9f3-4fc7-9978-a8c83f8580e5 tags:
<div style="font-family: 'times'; font-size: 13pt; text-align: justify">
In diesem Fall ist die Kalman-Verstärkung = 0
%% Cell type:code id:4234449c-03a4-493c-9e00-ef44ec741442 tags:
``` octave
```
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