Multivariable Control Systems
ECSE 6460
Fall 2009
Lecture 27: 8 December 2009
Uncertain MIMO Plant
Gp(s)
Frequency Response
Bode Diagram
From: In(1)
From: In(2)
To: Out(1)
0
-20
-40
Magnitude (dB)
-60
-80
0
To: Out(2)
-20
-40
-60
-80
-2
10
0
10
2
10
-2
10
Frequency (rad/sec)
0
10
2
10
Feedback Control Loop
+
K(s)
Gp(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 uncertainty
a la unstructured uncertainty
 a la structured uncertainty


Assessment of robust stability
a la unstructured uncertainty
 a la structured uncertainty


Assessment of robust performance
a la unstructured uncertainty
 a 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
Unstructured
Uncertainty
Wo (s)
+
K(s)
+
-
G (s)
+
+
WP (s)
Robust Performance Assessment
Structured Uncertainty
Ws (s)
+
K(s)
+
-
G (s)
+
WP (s)
+
Robust Performance Assessment
DP
D
N
Robust Performance Assessment
D 0
0 D
P
N
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
D
-1
N
D
In DK iteration, the K step consists of solving H infinity
synthesis for weighted N.
New plant model with weights
-1
D
P
D
K
The new plant model can be formed by including the weights.
Robust Performance Assessment
Unstructured
Uncertainty
Wo (s)
+
G (s)
+
-
+
+
WP (s)
First iteration
(N(j))
2
1
Magnitude (dB)
0
-1
-2
-3
-4
-5
-5
10
0
10
5
10
 (rad/sec)
See unstruct_ex.m
Second iteration
(N(j))
2
1
Magnitude (dB)
0
-1
-2
-3
-4
-5
-5
10
0
10
5
10
 (rad/sec)
See unstruct_ex.m
Robust Performance Assessment
Structured Uncertainty
Ws (s)
+
G (s)
+
-
+
WP (s)
+
First iteration
(N(j))
-1.5
-2
Magnitude (dB)
-2.5
-3
-3.5
-4
-4.5
-5
-5
10
0
10
5
10
 (rad/sec)
See struct_ex.m
Second iteration
(N(j))
-1.5
-2
Magnitude (dB)
-2.5
-3
-3.5
-4
-4.5
-5
-5
10
0
10
5
10
 (rad/sec)
See struct_ex.m
Third iteration
(N(j))
-1.5
-2
Magnitude (dB)
-2.5
-3
-3.5
-4
-4.5
-5
-5
10
0
10
5
10
 (rad/sec)
See struct_ex.m