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This lecture is from Process Control course. Some key points for this lecture are: Frequency Response Analysis, Closed Loop Behavior, Closed Loop Stability, Good Disturbance, Set Point Tracking, Satisfactory, Degree of Robustness, Process Variations, Model Uncertainty, Low Sensitivity
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Frequency response concepts and techniques play an important role in control system design and analysis.
In general, a feedback control system should satisfy the following design objectives:
Controller Design Using Frequency Response Criteria Advantages of FR Analysis:
( )
( ) ( )( ) ( )
( ) ( ) ( ) ( )
( )
1 2 1 1 2 1 1 2 1
π π
Dynamic Behavior of Closed- Loop Control Systems
(^) 4-20 mA
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
OL c v p m OL M c M v p m
( )
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
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
=
∴ =
= = ∠ = → ∞ → ∠ → −
( )
( ) (^) ( ) ( ) ( )
( ) (^) ( ) ( ) ( ) ( ) (^) ( ) ( ) ( ) ( ) (^) ( ) ( ) ( )
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
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
− →∞ →∞ →∞ − +
= (^) + + ⇒ =
= = − − − + (^) − + − +
⋅ = (^) ⋅ + ⋅ + ≈ →