Theory and Practice on Analysis and Design of Linear Control Systems Part II

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Theory and Practice on Analysis and Design of
Linear Control Systems Part II-
Instructor
Prof. Masayuki Fujita
6 / 2, 6 / 16 , 6 / 23, 6 / 30, 7 / 14,
7 / 21
Schedule
(No class) 6 / 9 ACC, 7 / 7 MSC
Textbook
1) K. J. Astrom and R. M. Murray, Feedback
Systems, Princeton University Press (2008)
Reference Books
2
You can download pdf files of books 1) and 3)
1) Feedback Systems
Chapter 11 http//www.fl.ctrl.titech.ac.jp/co
urse/SC/FS_cp11.pdf
Chapter 12 http//www.fl.ctrl.titech.ac.jp/co
urse/SC/FS_cp12.pdf
Web Information on the lecture will be
continuously uploaded
http//www.fl.ctrl.titech.ac.jp/course/SC/09SC/09S
C.html
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11.1 Sensitivity Functions
Basic Feedback Loop




Fig. 11.1
Measurement noise
Process output
Load disturbance
Measured signal
Feedback block
Control variable
Process
Reference signal
Feedforward block
( )
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Control System Transfer Functions
?



Fig. 11.1
5
Gang of Four



Fig. 11.1
Gang of Four


complementary sensitivity function
load sensitivity function


noise sensitivity function
sensitivity function
..(11.3)
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General Representation



Fig. 11.1
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Example 11.1
Fig. 11.1
Process
Stable ?
Controller
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Gang of Four
Complementary Sensitivity Function
Load Sensitivity Function
can be unstable
Noise Sensitivity Function
Sensitivity Function
unstable
(a) Step response
(b) Frequency response
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Internal Stability
Fig. 11.1
Internal stability
All of the Gang of Four are stable.
complementary sensitivity function
load sensitivity function


noise sensitivity function
sensitivity function


..(11.3)
Well-posed
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Youla Parameterization (stable process)
...(12.8)
All stabilizing controllers
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Youla Parameterization (unstable process)
Stable
Coprime Factorization
Stable
Fig.12.8(b)
All stabilizing controllers
Stable
...(12.9)
Gang of Four
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-Break -
American Control Conference (ACC) 2009
St. Louis, Missouri, USA, June 10 - 12, 2009
http//a2c2.org/conferences/acc2009/
Plenary lecture
Donald P. Eckman Award
Asu Ozdaglar (MIT)
"Learning and Dynamics in Networks"
Panos Antsaklis (Univ. of Notre Dame)
"From Hybrid to Networked Cyber-Physical Systems"
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Jun. 17th , 1st
11.1 Sensitivity Functions (12.2 Youla
Parameterization )
Keywords
Basic Feedback Loop, Gang of Four
11.3 Performance Specifications
(12.3 Performance in the Presence of Uncertainty)
Keywords
Sensitivity function, Complementary Sensitivity
function, Load Sensitivity function, Noise
Sensitivity function
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11.3 Performance Specifications
Time Responses ( 5.3)

• Overshoot
• Steady-state value
• Rise time
• Settling time

Step response
Fig. 5.9
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2nd-order system
Natural angular frequency
Damping factor
Im
Relationship between poles and time responses
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Disturbance Attenuation Open Loop vs. Closed
Loop
Open Loop
Closed Loop (feedback)
Fig. 11.1
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resonant peak
peak freq.
sens. crossover
sens. bandwidth
Fig. 11.6(a)
the shortest distance from the Nyquist
curve to the critical point (-1)
Large ? Poor
maximum sensitivity
Im
stability margin
Re
Fig. 11.6(b)
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Process Variations ( 12.3)



Fig. 11.1
?
Exercise
The response to load disturbances is insensitive
to process variations for frequencies where
is small.
?
Exercise
Sensitivity functions around the nominal values
tell us how tracking responses vary as a function
of process perturbations.
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Complementary sensitivity function
resonant peak
bandwidth
peak frequency
Robust Stability is characterized by the
complementary sensitivity function ? 2nd Lecture
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Load Sensitivity Function
with integral action
...(11.8, 11.9)
for small
for large



Fig. 11.1
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Noise Sensitivity Function
...(11.10)
for small
for large



Fig. 11.1
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Jun. 17th , 1st
11.1 Sensitivity Functions (12.2 Youla
Parameterization )
Keywords
Basic Feedback Loop, Gang of Four
11.3 Performance Specifications
(12.3 Performance in the Presence of Uncertainty)
Keywords
Sensitivity function, Complementary Sensitivity
function, Load Sensitivity function, Noise
Sensitivity function
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Jun. 24th , 2nd
12.1 Modeling Uncertainty
Modeling Uncertainty, Parametric
Uncertainty, Unmodeled Dynamics
Keyword
12.2 Stability in the Presence of Uncertainty
Keyword
Complementary sensitivity function