控制工程3(英文)PPT课件

A necessary condition: All the coefficients of the
polynomial must have the same sign and be nonzero if all
the roots are in left-hand plane (LHP).
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Routh-Hurwitz stability criterion
Hurwitz and Routh published independently a method of investigating the stability of a linear system. The number of roots of q(s) with positive real parts is equal to the number of changes in sign of the first column of the Routh array.
Relative stability: Given that a closed-loop system is stable, we can further characterize the degree of stability. This is referred to as relative stability.
Ch6 The Stability of Linear Feedback Systems
The concept of stability The Routh-Hurwitz stability criterion The relative stability
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6.1 The concept of stability
j1
i1
q
C(s)
Aj
j1ssj
kr1s2B 2kskk C k k2
where q2r n
q
r
C (t) A jesjt
B ke k ktco s(k
1 k2)t
j 1
k 1
r C kB k k kekktsin(
k 1 k 1k2
k
1k2)t,t0
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A necessary and sufficient condition for a feedback system to be stable is that all the poles of the system transfer function have negative real parts.
The issue of ensuring the stability of a closed-loop feedback system is central to control system design. An unstable closedloop system is generally of no practical value.
A stable system is a dynamic system with a bounded output to a bounded input (BIBO).
ຫໍສະໝຸດ Baidu
absolute stability, relative stability
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Absolute stability: We can say that a closed-loop feedback system is either stable or it is not stable. This type of stable/not stable characterization is referred to as absolute stability.
The relative stability of a system can be defined as the property that is measured by the relative real part of each root or pair of roots.
Axis shift and examples
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The characteristic equation is written as
( s ) q ( s ) a n s n a n 1 s n 1 . . . a 1 s a 0 0
a n (s r 1 )(s r 2 ) (s r n ) 0 q(s)ansnan(r1r2...rn)sn 1 an(r1r2r2r3...)sn 2 an( 1 )nr1r2r3...rn
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CASE1 No element in the first column is zero.
CASE2 Zero in the first column while some other elements of row containing a zero in the first column are nonzero.
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Case 3 Consider the characteristic polynomial
The Routh array is
The auxiliary polynomial
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2, j
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Design example： welding control
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6.3 The relative stability
CASE3 Zeros in the first column,and other elements of the row containing the zero are also zero.
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Consider the characteristic polynomial
The Routh array is
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6.2 The Routh-Hurwitz stability criterion
G (s)C R ((ss))b a 0 0 ssm n b a 1 1 ssm n 1 1
bm 1sbm an1san
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m
K(szi)
C(s) q
i1 r
(ssj) (s2 2kk k2)