multivariable control systems ecse 6460
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Multivariable Control SystemsECSE 6460
Fall 2009
Lecture 27: 8 December 2009
Uncertain MIMO Plant
Gp(s)
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)
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
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
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).
Robust Performance Assessment
K(s)+
-
G (s)
Wo (s)
++
Unstructured Uncertainty
+WP (s)
Robust Performance Assessment
K(s)+
-
G (s)
Ws (s)
+
+
Structured Uncertainty
+
WP (s)
Robust Performance Assessment
N
P
Robust Performance Assessment
N
P
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.
Robust Performance Assessment
N DD-1
In DK iteration, the K step consists of solving H infinity synthesis for weighted N.
New plant model with weights
P
K
The new plant model can be formed by including the weights.
D-1
D
Robust Performance Assessment
+
-
G (s)
Wo (s)
++
Unstructured Uncertainty
+WP (s)
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
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
Robust Performance Assessment
+
-
G (s)
Ws (s)
+
+
Structured Uncertainty
+
WP (s)
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
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
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
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