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Commit 5f54ea3f authored by zhiyupan's avatar zhiyupan
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replace exam example/V6.ipynb

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function [num den] = myss2tf2(A,B,C)
n2 = -A(1,1)*B(2)*C(2)+A(1,2)*B(2)*C(1)+A(2,1)*B(1)*C(2)-A(2,2)*B(1)*C(1);
n1 = B(1)*C(1)+B(2)*C(2);
d1 = -A(1,1)-A(2,2);
d2 = A(1,1)*A(2,2)-A(1,2)*A(2,1);
num = [n1 n2];
den = [1 d1 d2];
%% Cell type:markdown id:1f0f7673-ab1a-413f-b73c-0a248ba39ed9 tags:
# <span style='color:OrangeRed'>V6 NORMALFORMEN IN ZUSTANDRAUMDARSTELLUNG</span>
%% Cell type:markdown id:9653081e-df65-4ccd-9204-e11c542a469f tags: %% Cell type:markdown id:9653081e-df65-4ccd-9204-e11c542a469f tags:
<div style="font-family: 'times'; font-size: 13pt; text-align: justify"> <div style="font-family: 'times'; font-size: 13pt; text-align: justify">
Gegeben ist die Prinzipskizze einer Wasserturbine zur Stromerzeugung: Gegeben ist die Prinzipskizze einer Wasserturbine zur Stromerzeugung:
%% Cell type:markdown id:880aa4ff-2004-4875-affc-5759aba8104e tags: %% Cell type:markdown id:880aa4ff-2004-4875-affc-5759aba8104e tags:
<img src="Bilder/Turbine.png" alt="drawing" width="600" height="300"/> <img src="Turbine.png" alt="drawing" width="600" height="300"/>
%% Cell type:markdown id:a27e62db-897e-41e8-9b56-fd8c6b48f3bf tags: %% Cell type:markdown id:a27e62db-897e-41e8-9b56-fd8c6b48f3bf tags:
<div style="font-family: 'times'; font-size: 13pt; text-align: justify"> <div style="font-family: 'times'; font-size: 13pt; text-align: justify">
Die Wasserturbine sei in dieser Aufgabe die zu betrachtende Regelstrecke, die als linear Die Wasserturbine sei in dieser Aufgabe die zu betrachtende Regelstrecke, die als linear
angenommen wird. Damit lässt sich das System der Wasserturbine gemäß des nachfolgenden angenommen wird. Damit lässt sich das System der Wasserturbine gemäß des nachfolgenden
Wirkungsplans darstellen: Wirkungsplans darstellen:
%% Cell type:markdown id:add7e04a-83e6-4685-b845-47137f44a388 tags: %% Cell type:markdown id:add7e04a-83e6-4685-b845-47137f44a388 tags:
<img src="Bilder/TurbineSchema.png" alt="drawing" width="600" height="300"/> <img src="TurbineSchema.png" alt="drawing" width="600" height="300"/>
%% Cell type:code id:e050a273-a079-40b9-ac0c-194e5e7f12e8 tags: %% Cell type:code id:e050a273-a079-40b9-ac0c-194e5e7f12e8 tags:
``` octave ``` octave
% Necessary to use control toolbox % Necessary to use control toolbox
pkg load control pkg load control
clear all clear all
``` ```
%% Cell type:markdown id:1943a7dc-b837-4623-8792-a1775026155d tags: %% Cell type:markdown id:1943a7dc-b837-4623-8792-a1775026155d tags:
<div style="font-family: 'times'; font-size: 13pt; text-align: justify"> <div style="font-family: 'times'; font-size: 13pt; text-align: justify">
Geben Sie die Zustandsraumdarstellung des Systems Geben Sie die Zustandsraumdarstellung des Systems
%% Cell type:code id:ef3d3be5-73cd-48d2-a773-d177b638230a tags: %% Cell type:code id:ef3d3be5-73cd-48d2-a773-d177b638230a tags:
``` octave ``` octave
Kb = 500 Kb = 500
Kt = 104 Kt = 104
Kv = 2 Kv = 2
J = 100 J = 100
``` ```
%% Output %% Output
Kb = 500 Kb = 500
Kt = 104 Kt = 104
Kv = 2 Kv = 2
J = 100 J = 100
%% Cell type:code id:402f525f-8a64-4584-8d05-dbd5e42e5516 tags: %% Cell type:code id:402f525f-8a64-4584-8d05-dbd5e42e5516 tags:
``` octave ``` octave
A = [0 -Kv;Kt/J -Kb/J] A = [0 -Kv;Kt/J -Kb/J]
B = [1; 0] B = [1; 0]
C = [0 1] C = [0 1]
D = 0 D = 0
``` ```
%% Output %% Output
A = A =
0.00000 -2.00000 0.00000 -2.00000
1.04000 -5.00000 1.04000 -5.00000
B = B =
1 1
0 0
C = C =
0 1 0 1
D = 0 D = 0
%% Cell type:markdown id:8629e5a2-ae35-418d-8a65-7e511440917e tags: %% Cell type:markdown id:8629e5a2-ae35-418d-8a65-7e511440917e tags:
<div style="font-family: 'times'; font-size: 13pt; text-align: justify"> <div style="font-family: 'times'; font-size: 13pt; text-align: justify">
Dann berechnen wir die Eigenwerte Dann berechnen wir die Eigenwerte
%% Cell type:code id:1ed03ed5-3913-4107-95c9-f48e52bb50ee tags: %% Cell type:code id:1ed03ed5-3913-4107-95c9-f48e52bb50ee tags:
``` octave ``` octave
p = eigs(A) p = eigs(A)
``` ```
%% Output %% Output
p = p =
-4.54206 -4.54206
-0.45794 -0.45794
%% Cell type:markdown id:0353e887-ebe2-44d0-b7fb-a191622cabff tags: %% Cell type:markdown id:0353e887-ebe2-44d0-b7fb-a191622cabff tags:
<div style="font-family: 'times'; font-size: 13pt; text-align: justify"> <div style="font-family: 'times'; font-size: 13pt; text-align: justify">
Dann wenden wir das Caley-Hamilton-Theorem an Dann wenden wir das Caley-Hamilton-Theorem an
%% Cell type:code id:0d90c253-316f-47ed-9f7b-561ff0ed4868 tags: %% Cell type:code id:0d90c253-316f-47ed-9f7b-561ff0ed4868 tags:
``` octave ``` octave
Ts = 0.1; Ts = 0.1;
Ac = [1 p(1);1 p(2)] Ac = [1 p(1);1 p(2)]
Bc = [expm(p(1)*Ts);expm(p(2)*Ts)] Bc = [expm(p(1)*Ts);expm(p(2)*Ts)]
alfa = inv(Ac)*Bc alfa = inv(Ac)*Bc
``` ```
%% Output %% Output
Ac = Ac =
1.00000 -4.54206 1.00000 -4.54206
1.00000 -0.45794 1.00000 -0.45794
Bc = Bc =
0.63495 0.63495
0.95524 0.95524
alfa = alfa =
0.991151 0.991151
0.078422 0.078422
%% Cell type:markdown id:d4fac333-e43d-49c7-aaec-6168b86bd6a6 tags: %% Cell type:markdown id:d4fac333-e43d-49c7-aaec-6168b86bd6a6 tags:
<div style="font-family: 'times'; font-size: 13pt; text-align: justify"> <div style="font-family: 'times'; font-size: 13pt; text-align: justify">
und dann: und dann:
%% Cell type:code id:828217c8-bd90-42da-8b36-9e1f05727a24 tags: %% Cell type:code id:828217c8-bd90-42da-8b36-9e1f05727a24 tags:
``` octave ``` octave
Fi = alfa(1)*eye(2)+alfa(2)*A Fi = alfa(1)*eye(2)+alfa(2)*A
``` ```
%% Output %% Output
Fi = Fi =
0.991151 -0.156845 0.991151 -0.156845
0.081559 0.599039 0.081559 0.599039
%% Cell type:markdown id:a6ed8d3e-8d76-451c-8480-7182a8723ab2 tags: %% Cell type:markdown id:a6ed8d3e-8d76-451c-8480-7182a8723ab2 tags:
<div style="font-family: 'times'; font-size: 13pt; text-align: justify"> <div style="font-family: 'times'; font-size: 13pt; text-align: justify">
Wir Konnen vergleichen unsere Berechnungen mit Octave: Wir Konnen vergleichen unsere Berechnungen mit Octave:
%% Cell type:code id:058ad783-d553-4bcc-873c-a6f521c16975 tags: %% Cell type:code id:058ad783-d553-4bcc-873c-a6f521c16975 tags:
``` octave ``` octave
Fio = expm(A*Ts) Fio = expm(A*Ts)
``` ```
%% Output %% Output
Fio = Fio =
0.991151 -0.156845 0.991151 -0.156845
0.081559 0.599039 0.081559 0.599039
%% Cell type:markdown id:63fb7dec-c4e2-4e53-b6f9-bea0649fd521 tags: %% Cell type:markdown id:63fb7dec-c4e2-4e53-b6f9-bea0649fd521 tags:
<div style="font-family: 'times'; font-size: 13pt; text-align: justify"> <div style="font-family: 'times'; font-size: 13pt; text-align: justify">
Die andere Matrix können wir so berechnen: Die andere Matrix können wir so berechnen:
%% Cell type:code id:0c2d94f8-536c-421a-98ae-69771c0e6ca2 tags: %% Cell type:code id:0c2d94f8-536c-421a-98ae-69771c0e6ca2 tags:
``` octave ``` octave
Gi = (Fi-eye(2))*inv(A)*B Gi = (Fi-eye(2))*inv(A)*B
``` ```
%% Output %% Output
Gi = Gi =
0.0996930 0.0996930
0.0044243 0.0044243
%% Cell type:markdown id:8a4be558-f51c-415e-96f7-922b8e417422 tags: %% Cell type:markdown id:8a4be558-f51c-415e-96f7-922b8e417422 tags:
<div style="font-family: 'times'; font-size: 13pt; text-align: justify"> <div style="font-family: 'times'; font-size: 13pt; text-align: justify">
Zu vergleichen: Zu vergleichen:
%% Cell type:code id:69e7fd06-f71a-48ec-af34-d2e406f9067c tags: %% Cell type:code id:69e7fd06-f71a-48ec-af34-d2e406f9067c tags:
``` octave ``` octave
sys = ss(A,B,C,D) sys = ss(A,B,C,D)
sysd = c2d(sys,Ts,'zoh') sysd = c2d(sys,Ts,'zoh')
``` ```
%% Output %% Output
sys.a = sys.a =
x1 x2 x1 x2
x1 0 -2 x1 0 -2
x2 1.04 -5 x2 1.04 -5
sys.b = sys.b =
u1 u1
x1 1 x1 1
x2 0 x2 0
sys.c = sys.c =
x1 x2 x1 x2
y1 0 1 y1 0 1
sys.d = sys.d =
u1 u1
y1 0 y1 0
Continuous-time model. Continuous-time model.
sysd.a = sysd.a =
x1 x2 x1 x2
x1 0.9912 -0.1568 x1 0.9912 -0.1568
x2 0.08156 0.599 x2 0.08156 0.599
sysd.b = sysd.b =
u1 u1
x1 0.09969 x1 0.09969
x2 0.004424 x2 0.004424
sysd.c = sysd.c =
x1 x2 x1 x2
y1 0 1 y1 0 1
sysd.d = sysd.d =
u1 u1
y1 0 y1 0
Sampling time: 0.1 s Sampling time: 0.1 s
Discrete-time model. Discrete-time model.
%% Cell type:markdown id:4fb92a07-3450-4c37-bef4-7e3734cc8c20 tags: %% Cell type:markdown id:4fb92a07-3450-4c37-bef4-7e3734cc8c20 tags:
<div style="font-family: 'times'; font-size: 13pt; text-align: justify"> <div style="font-family: 'times'; font-size: 13pt; text-align: justify">
Jetzt können wir die Übertragungsfunktion in z-domän berechnen: Jetzt können wir die Übertragungsfunktion in z-domän berechnen:
%% Cell type:code id:cbf41502-116d-40b5-8fec-39faf2062a03 tags: %% Cell type:code id:cbf41502-116d-40b5-8fec-39faf2062a03 tags:
``` octave ``` octave
[num den] = myss2tf2(Fi,Gi,C) pkg load signal
[num den] = ss2tf(Fi,Gi,C,D)
Gz = tf(num,den,Ts) Gz = tf(num,den,Ts)
``` ```
%% Output %% Output
num = num =
0.0044243 0.0037458 0.0000000 0.0044243 0.0037458
den = den =
1.00000 -1.59019 0.60653 1.00000 -1.59019 0.60653
Transfer function 'Gz' from input 'u1' to output ... Transfer function 'Gz' from input 'u1' to output ...
0.004424 z + 0.003746 0.004424 z + 0.003746
y1: --------------------- y1: ---------------------
z^2 - 1.59 z + 0.6065 z^2 - 1.59 z + 0.6065
Sampling time: 0.1 s Sampling time: 0.1 s
Discrete-time model. Discrete-time model.
%% Cell type:code id:6ebd4c8d-4218-41b7-aa86-902033541283 tags: %% Cell type:code id:6ebd4c8d-4218-41b7-aa86-902033541283 tags:
``` octave ``` octave
``` ```
......
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