华科电气自动控制理论12-1

Module 12
Frequency Response and
Nyquist Diagrams •Understand the basic ideas of frequency response method.
•Draw approximate Nyquist diagram of a given TF.
12.1 Introduction of Frequency Response
Why to study Frequency Response?
To see its merits, let’s consider the control of a nuclear reactor, of which the transfer function G(s)is unknown:
Note:Root locus method is not applicable since G(s)is unknown. But,since frequency response(e.g.Bode plot)can be measured through experiments,frequency response method can help us determine stability of the closed-loop system.
)
()()(ωωωj R j C j G =(Frequency-response function)
)
(s G The magnitude and phase of the output signal differ from those of the input.
t r t r ωsin )(0=)
sin()(0ϕω+=t c t c
12.2 Different forms of Frequency Response There are two major styles of demonstrating frequency response: Bode plot and Nyquist diagram:
Frequency response method:A method of system analysis and controller design based on frequency responses of system components.
How to analysis and design?
Draw frequency response plot
Analyze and design
12.3 Nyquist Diagram
∏∏∏∏++++++=
)
e s d s
()p s (s
)
c s b s ()z s (K )s (GH l l 2
k
n
j j 2
i E ∏∏∏∏+++++
+=
)12(
)1()
12(
)1()(2222s s
s T s
s s
s T K s GH nl
l nl
k n
nj j
nj i B ωζω
ωζ
ω
Im
Re
K
Nyquist diagram of Basic elements
ω=0
ω=∞
2
22
2
T
1T
)(Y T
11
)(X ωωωωω+−=+=
22
2)
21()(Y )2
1)(X (=+−ωω
ω=0
ω=∞
T
)(Y 1)(X ωωω==
Im
Re 0
1
=n 2
=n 3
=n ω=0
ω=∞-ω
ωω1
)(0)(−
==Y X
j
0ω=0
ω=∞
1
ω=ωl
M φ
1
,90 2o
l M
ωωϕ
ζ
=⇒==−
Im
Re
ω=0
ω=∞
ω=ωl
2,90
o
l M ωωζϕ=⇒==1
s
e s G τ−=)(ωτωj e
j G −=)(1()()57.3o M rad ϕωωτωτ==−=−（）
j 0－11 幅频与ω无关，相角与ω有关
Case 8:Delay element (延迟环节)
()1()()c t t r t ττ=−−
n n ωωn s
ω9009090n s n ωn 900
∞n 90n 90s e τ−1ωτ
−110∞∞
Plot the Nyquist diagram of open loop T.F.
¾Starting point(ω=0)
¾End point(ω=∞)
¾The direction of the ωvaries
¾crossing with the imaginary axis(Re=0)¾crossing with the real axis(Im=0)
12()i
M M M M ω=⋅ 12()i
ϕωϕϕϕ=+++
241)21(21)(ω
ωωω+−=+=j K j K j GH 2411
ω+⋅=K M )2(1ωφ−−=tg M ωϕ↓↑⇒↓
o o K 9090000M −=−==⋅=∞=ϕω
21611110M ω+⋅ω⋅=)
4(tg 901
ω−−=φ−M ωϕ↓↑⇒↓。