Lecture Control system design: The stability of linear feedback systems include all of the following content: The concept of stability, the Routh – Hurwitz stability criterion, the stability of state variable systems, system stability using control design software.
Trang 1Nguyễn Công Phương
CONTROL SYSTEM DESIGN
The Stability
of Linear Feedback Systems
Trang 2I Introduction
II Mathematical Models of Systems
III State Variable Models
IV Feedback Control System Characteristics
V The Performance of Feedback Control Systems
VI The Stability of Linear Feedback Systems
VII The Root Locus Method
VIII.Frequency Response Methods
IX Stability in the Frequency Domain
X The Design of Feedback Control Systems
XI The Design of State Variable Feedback Systems
XII Robust Control Systems
XIII.Digital Control Systems
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Trang 3The Stability
of Linear Feedback Systems
1 The Concept of Stability
2 The Routh – Hurwitz Stability Criterion
3 The Stability of State Variable Systems
4 System Stability Using Control Design
Software
Trang 4The Concept of Stability (1)
• Stability is of the utmost importance.
• A close – loop feedback system that is unstable
is of little value.
• A stable system is a dynamic system with a
bounded response to a bounded input.
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Trang 5The Concept of Stability (2)
http://www.ctc.org.uk/cyclists-library/bikes-and-other-cycles/cycle-styles/city-bike
Trang 6-1 0 1
-10 0 10
-1 0 1
-1 0 1
0 1 2
0 0.5 1
Trang 7The Stability
of Linear Feedback Systems
1 The Concept of Stability
2 The Routh – Hurwitz Stability Criterion
3 The Stability of State Variable Systems
4 System Stability Using Control Design
Software
Trang 8n n n
a a b
n n n
Trang 9The Routh – Hurwitz Stability
Criterion (2)
1 No element in the 1 st column is
zero.
2 There is a zero in the 1 st column,
but some other elements of the row containing the zero in the 1 st
column are nonzero.
3 There is a zero in the 1 st column,
and the other elements of the row containing the zero are also zero.
4 Repeated roots of the characteristic
equation on the jω – axis.
Trang 10s an1 an3 an5 2
n
s bn1 bn3 bn5 3
The system is stable if a2, a1 & a0 are all positive or all negative
Trang 11The Routh – Hurwitz Stability
n
s bn1 bn3 bn5 3
b
1
1 50 1
50,
48 0 48
0
48 0 48
Trang 120 0
0 0
a a
a a a a a a
0
0 0
0 0
a a
a a a a a
Trang 13The Routh – Hurwitz Stability
Trang 14One sign change one root with positive real part
the system is unstable for all values of K
Trang 15The Routh – Hurwitz Stability
Trang 17The Routh – Hurwitz Stability
( 6) 6
0 0
K
b Ka
Trang 18The Stability
of Linear Feedback Systems
1 The Concept of Stability
2 The Routh – Hurwitz Stability Criterion
3 The Stability of State Variable Systems
4 System Stability Using Control Design
Software
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Trang 19The Stability of State Variable
Trang 20u dt
Trang 21The Stability
of Linear Feedback Systems
1 The Concept of Stability
2 The Routh – Hurwitz Stability Criterion
3 The Stability of State Variable Systems
4 System Stability Using Control Design
Software
Trang 22( )
Trang 23System Stability Using Control
Trang 24sites.google.com/site/ncpdhbkhn 24
System Stability Using Control
Design Software (3)
Ex 3
Given a characteristic equation q(s) = s4 + 8s3 + 17s2 + (K + 10)s + aK = 0.
Find a & K such that the system is stable.
0
(126 )( 10)
8 0