Transcript Slide 1

Proportional Control and Disturbance Changes
Chapter 11
From Fig. 11.16 and Eq. 11-29 the closed-loop transfer function
for disturbance changes with proportional control is
K p /  τs  1
H  s

Q1  s  1  KOL /  τs  1
(11-53)
H  s
K2

Q1  s  τ1s  1
(11-54)
Rearranging gives
where τ1 is defined in (11-39) and K2 is given by
K2 
Kp
1  KOL
(11-55)
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• A comparison of (11-54) and (11-37) indicates that both closedloop transfer functions are first-order and have the same time
constant.
Chapter 11
• However, the steady-state gains, K1 and K2, are different.
• From Eq. 11-54 it follows that the closed-loop response to a
step change in disturbance of magnitude M is given by

h  t   K2 M 1  et / τ1

(11-56)
   0
The offset can be determined from Eq. 11-56. Now hsp
since we are considering disturbance changes and h     K 2 M
for a step change of magnitude M.
Thus,
offset  0  h      K 2 M  
K pM
1  KOL
(11-57)
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Chapter 11
Figure 11.19 Load responses for Example 11.3.
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Chapter 11
PI Control and Disturbance Changes
For PI control, Gc  s   Kc 1  1/ τ I s  . The closed-loop transfer
function for disturbance changes can then be derived from Fig.
11.16:
K p /  τs  1
H  s

(11-58)
Q1  s  1  KOL 1  1/ τ I s  /  τs  1
Clearing terms in the denominator gives
K pτ I s
H  s

Q1  s  τ I s  τs  1  KOL τ I s
(11-59)
Further rearrangement allows the denominator to be placed in the
standard form for a second-order transfer function:
H  s
K3 s
 2 2
Q1  s  τ3 s  2ζ3τ3s  1
(11-60)
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Chapter 11
where
K3  τ I / K c K v K m
(11-61)
1  1  KOL
ζ3  
2  KOL
(11-62)
 τI

 τ
τ3  ττ I / KOL
(11-63)
For a unit step change in disturbance, Q1  s   1/ s , and (11-59)
becomes
K3
H  s  2 2
(11-64)
τ3 s  2ζ 3τ3 s  1
For 0  ζ 3  1 , the response is a damped oscillation that can be
described by
h  t  
eζ 3t / τ3 sin  1  ζ 32 t / τ3 


1  ζ 32
K3
τ3
(11-65)
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Chapter 11
PI Control of an Integrating Process
Consider the liquid-level control system shown in Fig. 11.22. This
system differs from the previous example in two ways:
Chapter 11
1. the exit line contains a pump and
2. the manipulated variable is the exit flow rate rather than an
inlet flow rate.
In Section 5.3 we saw that a tank with a pump in the exit stream
can act as an integrator with respect to flow rate changes because
H  s
1
 Gp  s   
Q3  s 
As
(11-66)
H  s
1
 Gd  s  
Q1  s 
As
(11-67)
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Chapter 11
Figure 11.22 Liquid-level control system with pump in exit line.
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Chapter 11
If the level transmitter and control valve in Eq. 11.22 have
negligible dynamics, the Gm(s) = Km and Gv(s) = Kv. For PI
control, Gc  s   Kc 1  1/ τ I s  . Substituting these expressions
into the closed-loop transfer function for disturbance changes
H  s
Gd

Q1  s  1  GcGvG pGm
(11-68)
and rearranging gives
H  s
K4 s
 2 2
Q1  s  τ 4 s  2ζ 4 τ 4 s  1
(11-69)
K4  τ / Kc Kv K m
(11-70)
τ 4  τ I / KOL
(11-71)
ζ 4  0.5 KOL τ I
(11-72)
where
And KOL = KcKvKpKm with Kp = - 1/A.
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Stability of Closed-Loop Control Systems
Chapter 11
Example 11.4
Consider the feedback control system shown in Fig. 11.8 with
the following transfer functions:
Gc  K c
1
G p  Gd 
5s  1
1
Gv 
2s  1
1
Gm 
s 1
(11-73)
(11-74)
Show that the closed-loop system produces unstable responses if
controller gain Kc is too large.
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Chapter 11
Figure 11.23. Effect of controller gains on closed-loop
response to a unit step change in set point (example 11.1).
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