Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 December 2009.
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Transcript of Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 December 2009.
![Page 1: Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 December 2009.](https://reader036.fdocuments.net/reader036/viewer/2022081513/56649e4e5503460f94b44b4a/html5/thumbnails/1.jpg)
Multivariable Control SystemsECSE 6460
Fall 2009
Lecture 27: 8 December 2009
![Page 2: Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 December 2009.](https://reader036.fdocuments.net/reader036/viewer/2022081513/56649e4e5503460f94b44b4a/html5/thumbnails/2.jpg)
Uncertain MIMO Plant
Gp(s)
![Page 3: Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 December 2009.](https://reader036.fdocuments.net/reader036/viewer/2022081513/56649e4e5503460f94b44b4a/html5/thumbnails/3.jpg)
Frequency Response
-80
-60
-40
-20
0
From: In(1)
To:
Out
(1)
10-2
100
102
-80
-60
-40
-20
0
To:
Out
(2)
From: In(2)
10-2
100
102
Bode Diagram
Frequency (rad/sec)
Mag
nitu
de (
dB)
![Page 4: Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 December 2009.](https://reader036.fdocuments.net/reader036/viewer/2022081513/56649e4e5503460f94b44b4a/html5/thumbnails/4.jpg)
Feedback Control Loop
Gp(s)K(s)+
-
• practically zero steady state error for step reference• rise time less than 1 second• overshoot less than 5%• robustly for any value of c
![Page 5: Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 December 2009.](https://reader036.fdocuments.net/reader036/viewer/2022081513/56649e4e5503460f94b44b4a/html5/thumbnails/5.jpg)
From previous lecture
Representing the uncertaintya la unstructured uncertaintya la structured uncertainty
Assessment of robust stabilitya la unstructured uncertaintya la structured uncertainty
Assessment of robust performancea la unstructured uncertaintya la structured uncertainty
![Page 6: Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 December 2009.](https://reader036.fdocuments.net/reader036/viewer/2022081513/56649e4e5503460f94b44b4a/html5/thumbnails/6.jpg)
Today’s class
How to design a controller that minimizes the structured singular value.
This is a sub-optimal solution, but works well in practice.
The technique is called DK iteration (see dksyn function in MATLAB).
![Page 7: Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 December 2009.](https://reader036.fdocuments.net/reader036/viewer/2022081513/56649e4e5503460f94b44b4a/html5/thumbnails/7.jpg)
Robust Performance Assessment
K(s)+
-
G (s)
Wo (s)
++
Unstructured Uncertainty
+WP (s)
![Page 8: Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 December 2009.](https://reader036.fdocuments.net/reader036/viewer/2022081513/56649e4e5503460f94b44b4a/html5/thumbnails/8.jpg)
Robust Performance Assessment
K(s)+
-
G (s)
Ws (s)
+
+
Structured Uncertainty
+
WP (s)
![Page 9: Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 December 2009.](https://reader036.fdocuments.net/reader036/viewer/2022081513/56649e4e5503460f94b44b4a/html5/thumbnails/9.jpg)
Robust Performance Assessment
N
P
![Page 10: Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 December 2009.](https://reader036.fdocuments.net/reader036/viewer/2022081513/56649e4e5503460f94b44b4a/html5/thumbnails/10.jpg)
Robust Performance Assessment
N
P
![Page 11: Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 December 2009.](https://reader036.fdocuments.net/reader036/viewer/2022081513/56649e4e5503460f94b44b4a/html5/thumbnails/11.jpg)
N as a ‘function’ of K
P
K
Note that, from previous classes, we know how to design K so as to minimize the H-infinity norm of N.
![Page 12: Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 December 2009.](https://reader036.fdocuments.net/reader036/viewer/2022081513/56649e4e5503460f94b44b4a/html5/thumbnails/12.jpg)
Robust Performance Assessment
N DD-1
In DK iteration, the K step consists of solving H infinity synthesis for weighted N.
![Page 13: Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 December 2009.](https://reader036.fdocuments.net/reader036/viewer/2022081513/56649e4e5503460f94b44b4a/html5/thumbnails/13.jpg)
New plant model with weights
P
K
The new plant model can be formed by including the weights.
D-1
D
![Page 14: Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 December 2009.](https://reader036.fdocuments.net/reader036/viewer/2022081513/56649e4e5503460f94b44b4a/html5/thumbnails/14.jpg)
![Page 15: Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 December 2009.](https://reader036.fdocuments.net/reader036/viewer/2022081513/56649e4e5503460f94b44b4a/html5/thumbnails/15.jpg)
Robust Performance Assessment
+
-
G (s)
Wo (s)
++
Unstructured Uncertainty
+WP (s)
![Page 16: Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 December 2009.](https://reader036.fdocuments.net/reader036/viewer/2022081513/56649e4e5503460f94b44b4a/html5/thumbnails/16.jpg)
First iteration
10-5
100
105
-5
-4
-3
-2
-1
0
1
2M
agni
tude
(dB
)
(N(j))
(rad/sec)
See unstruct_ex.m
![Page 17: Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 December 2009.](https://reader036.fdocuments.net/reader036/viewer/2022081513/56649e4e5503460f94b44b4a/html5/thumbnails/17.jpg)
Second iteration
(N(j))
(rad/sec)
10-5
100
105
-5
-4
-3
-2
-1
0
1
2
Mag
nitu
de (
dB)
See unstruct_ex.m
![Page 18: Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 December 2009.](https://reader036.fdocuments.net/reader036/viewer/2022081513/56649e4e5503460f94b44b4a/html5/thumbnails/18.jpg)
![Page 19: Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 December 2009.](https://reader036.fdocuments.net/reader036/viewer/2022081513/56649e4e5503460f94b44b4a/html5/thumbnails/19.jpg)
Robust Performance Assessment
+
-
G (s)
Ws (s)
+
+
Structured Uncertainty
+
WP (s)
![Page 20: Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 December 2009.](https://reader036.fdocuments.net/reader036/viewer/2022081513/56649e4e5503460f94b44b4a/html5/thumbnails/20.jpg)
First iteration
10-5
100
105
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5M
agni
tude
(dB
)
(N(j))
(rad/sec)
See struct_ex.m
![Page 21: Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 December 2009.](https://reader036.fdocuments.net/reader036/viewer/2022081513/56649e4e5503460f94b44b4a/html5/thumbnails/21.jpg)
Second iteration
10-5
100
105
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5M
agni
tude
(dB
)
(N(j))
(rad/sec)
See struct_ex.m
![Page 22: Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 December 2009.](https://reader036.fdocuments.net/reader036/viewer/2022081513/56649e4e5503460f94b44b4a/html5/thumbnails/22.jpg)
Third iteration
10-5
100
105
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5M
agni
tude
(dB
)
(N(j))
(rad/sec)
See struct_ex.m