Multivariable Control Systems ECSE 6460

1
Multivariable Control SystemsECSE 6460

• Fall 2009
• Lecture 27 8 December 2009

2
Uncertain MIMO Plant
Gp(s)
3
Frequency Response
4
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

5
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

6
Todays 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).

7
Robust Performance Assessment
Unstructured Uncertainty
Wo (s)


K(s)
G (s)

WP (s)

-
8
Robust Performance Assessment
Structured Uncertainty
Ws (s)


K(s)
G (s)
WP (s)


-
9
Robust Performance Assessment
DP
D
N
10
Robust Performance Assessment
N
11
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.
12
Robust Performance Assessment
N
D
D
-1
In DK iteration, the K step consists of solving H
infinity synthesis for weighted N.
13
New plant model with weights
D
-1
D
P
K
The new plant model can be formed by including
the weights.
14
Unstructured Uncertainty
15
Robust Performance Assessment
Unstructured Uncertainty
Wo (s)


G (s)

WP (s)

-
16
First iteration
See unstruct_ex.m
17
Second iteration
See unstruct_ex.m
18
Structured Uncertainty
19
Robust Performance Assessment
Structured Uncertainty
Ws (s)


G (s)
WP (s)


-
20
First iteration
See struct_ex.m
21
Second iteration
See struct_ex.m
22
Third iteration
See struct_ex.m