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Control System Design Based on
Frequency Response Analysis
Frequency response concepts and techniques play an important role in control system design and analysis.
Closed-Loop Behavior
In general, a feedback control system should satisfy the following design objectives:
- Closed-loop stability
- Good disturbance rejection (without excessive control action)
- Fast set-point tracking (without excessive control action)
- A satisfactory degree of robustness to process variations and model uncertainty
- Low sensitivity to measurement noise Docsity.com
Chapter 14
Controller Design Using Frequency Response Criteria Advantages of FR Analysis:
- Applicable to dynamic model of any order (including non-polynomials).
- Designer can specify desired closed-loop response characteristics.
- Information on stability and sensitivity/robustness is provided. Disadvantage: The approach tends to be iterative and hence time- consuming -- interactive computer graphics desirable (MATLAB)
Chapter 14
( )
( ) ( )( ) ( )
( ) ( ) ( ) ( )
( )
1 2 1 1 2 1 1 2 1
will increase by 2 clockwise from each zero
inside the contour, but decrease by 2 for eac
Comformal Mapping
h pole
inside the c
s z s z
F s
s p
F s z s z s p
F s
s F
s z s z s
F s
s
p
π π
ontour,
⇒ (2 -1) encirclements
Dynamic Behavior of Closed- Loop Control Systems
Chapter 11
(^) 4-20 mA
Application of Z-P=N Theorem to
Stability
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
(1) Consider the characteristic equation of the closed-loop
control system:
(2) Let 1 1 , where
and.
(3) Select a contou
OL c v p m OL M c M v p m
G s G s G s G s G s
F s G s B s G s
B s G s G s G s G s G s
( )
r that goes completely around the entire
right-half of -plane.
(4) Plot and determine -.
s
F s N = Z P
Nyquist Stability Criterion
( )
If is the number of times the Nyquist plot encircles the point ( 1, 0) in the complex plane in the clockwise direction, and is the number of open-loop poles of that lie
in the RHP, then
OL
N
P G s Z N P
−
= + ( )
is the number of unstable roots of 1 + GOL s = 0.
Example
3
3
/ 8 1 0
- contour: and 0 / 8 1 0 / 8 0 0 270
OL^ c
OL c
OL c OL OL OL
G s K s P
C s j G j K j G K G j G G j
=
∴ =
= < < ∞
= = ∠ = → ∞ → ∠ → −
Important Conclusion
The C+ contour, i.e., Nyquist plot,
is the only one we need to
determine the system stability!
Ultimate Gain and Ultimate
Frequency
( )
( ) (^) ( ) ( ) ( )
( ) (^) ( ) ( ) ( ) ( ) (^) ( ) ( ) ( ) ( ) (^) ( ) ( ) ( )
3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
OL^ c s j c c
c cu c u
G j j K
s s s
K K
j
K
K
K
ω
ω
ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω
=
= − + = ^
Example 2
( ) (^) ( )( )
( ) ( )
( ) ( ) ( )
( ) (^) ( )( )
2 (^2 2 2 2 2 )
2 2
1 5 1 0
(a) contour: (^1 5 ) (^1 5 6 1 5 36 1 5 )
(b) contour:
lim lim 1 5 1 lim 5 0
(c) contour: cojugate of.
OL^ c
OL c^ c c
R OL j^ j c^ j c j R R R
G s K P s s
C
G j K^ K j K j C G R e K^ K e R e R e R C C
φ φ φ φ
− →∞ →∞ →∞ − +
= (^) + + ⇒ =
= = − − − + (^) − + − +
⋅ = (^) ⋅ + ⋅ + ≈ →
