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Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 26: 4 December 2009
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Multivariable Control SystemsECSE 6460
Fall 2009
Lecture 26: 4 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
Menu for the day
Representing the uncertaintya la unstructured uncertaintya la structured uncertainty
Assessment of robust stabilitya la unstructured uncertaintya la structured uncertainty
Assessment of robust stabilitya la unstructured uncertaintya la structured uncertainty
Unstructured Uncertainty
G (s)
Wo (s) (s)
G (s)
Wi (s) (s)
++
++
Output uncertainty
Input uncertainty
Output Uncertainty
G (s)
Wo (s) (s)
++
Output uncertainty
We need to select a weight such that:
Input Uncertainty
We need to select a weight such that:
G (s)
Wi (s) (s)
++
Input uncertainty
Weight Wo(s)
10-5
100
105
10-3
10-2
10-1
100
101
Output Uncertainty
Refer to uncert1.m
Structured Uncertainty
Structured Uncertainty
G (s)
Ws (s)
++
relax to
Relaxation from real to complex uncertainty
Assessment of Robustness
In this topic, we first design a controller for the nominal system, then we assess the robustness properties of the system.
Two control designs are explored, decentralized inverse-based, and H-infinity synthesis.
Gp(s)K(s)+
-
Decentralized Inverse Based
Based on nominal plant model
Refer to decent.m
Nominal Performance
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.5
1
1.5
System: TnomOutput: Out(2)Time (sec): 0.451Amplitude: 0.9
To:
Out
(2)
Nominal Close Loop Plant Step Response
Time (sec)
Am
plitu
de 0
0.5
1
1.5 System: TnomOutput: Out(1)Time (sec): 0.556Amplitude: 0.901
To:
Out
(1)
H synthesis for nominal plant
Use S/T synthesis with:
Refer to hinf_ex.m
Nominal Performance
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
System: TnomOutput: Out(2)Time (sec): 0.465Amplitude: 0.901
To:
Out
(2)
Nominal Close Loop Plant Step Response
Time (sec)
Am
plitu
de 0
0.2
0.4
0.6
0.8
1
System: TnomOutput: Out(1)Time (sec): 0.465Amplitude: 0.901
To:
Out
(1)
Robust Stability Assessment
K(s)+
-
G (s)
Wo (s) (s)
++
Unstructured Uncertainty
Robust Stability Assessment
K(s)
-
G (s)
Wo (s)
++
Robust Stability Assessment
10-5
100
105
10-4
10-3
10-2
10-1
100
101
102
max
(T) (blue) vs 1/|Wo| (red)
Refer to decent.m
Robust Stability Assessment
10-5
100
105
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
max
(T) (blue) vs 1/|Wo| (red)
Refer to hinf_ex.m
Robust Stability Assessment
K(s)+
-
G (s)
Ws (s)
++
Structured Uncertainty
Robust Stability Assessment
K(s)+
-
G (s)
Ws (s)
++
Structured Uncertainty
Structured Singular Value
10-5
100
105
-90
-80
-70
-60
-50
-40
-30
-20
-10M
agni
tude
(dB
)
(M(j))
(rad/sec)
Peak at 0.130
For decentralized controller
Perturbed system responses
-0.5
0
0.5
1
1.5T
o: O
ut(1
)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.5
1
1.5
To:
Out
(2)
Perturbed Close Loop Plant Step Responses
Time (sec)
Am
plitu
de
10-5
100
105
-110
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10M
agni
tude
(dB
)
(M(j))
(rad/sec)
Structured Singular Value
Peak at 0.150
Perturbed system responses
-0.5
0
0.5
1
1.5
To:
Out
(1)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
To:
Out
(2)
Perturbed Close Loop Plant Step Responses
Time (sec)
Am
plitu
de
Nominal Performance
10-5
100
105
10-6
10-5
10-4
10-3
10-2
10-1
100
101
max
(S) (blue) vs 1/|Wp| (red)
Refer to hinf_ex.m
Robust Performance Assessment
K(s)+
-
G (s)
Wo (s)
++
Unstructured Uncertainty
+WP (s)
Robust Performance Assessment
K(s)+
-
G (s)
Wo (s)
++
Unstructured Uncertainty
+WP (s)
Robust Performance Assessment
N
P
Robust Performance Assessment
10-5
100
105
-30
-25
-20
-15
-10
-5
0
5
10
Mag
nitu
de (
dB)
(N(j))
(rad/sec)
Robust Performance Assessment
K(s)+
-
G (s)
Ws (s)
+
+
Structured Uncertainty
+
WP (s)
Robust Performance Assessment
K(s)+
-
G (s)
Ws (s)
+
+
Structured Uncertainty
+
WP (s)
Robust Performance Assessment
N
P
Robust Performance Assessment
10-5
100
105
-60
-50
-40
-30
-20
-10
0
10
Mag
nitu
de (
dB)
(N(j)) Structured vs Unstructured
(rad/sec)
