Post on 22-Dec-2015
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On-off Controllers
• Simple• Cheap• Used In residential heating and domestic refrigerators• Limited use in process control due to continuous
cycling of controlled variable excessive wear on control valve.
ExamplesExamples•Batch process control (PLC = programmable logic controller)•Solenoid in home heating unit•Sprinkler systems•Cruise control?
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On-Off Controllers
Synonyms:“two-position” or “bang-bang” controllers.
Controller output has two possible values.
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e = error = set point – measured variable
Three Mode (PID) Controller • Proportional• Integral• Derivative
Proportional Control• Define an error signal, e, by e = Ysp – Ym
whereYsp = set pointYm = measured value of the controlled variable (or equivalent signal from transmitter)
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Since signals are time varying,
e(t) = Ysp(t) - Ym (t)
n.b. Watch units!!
• For proportional control: where, p(t) = controller output = bias value (adjustable) Kc = controller gain (dimensionless, adjustable)
p-p=p e(t)K+p=p(t) c
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p
Proportional Band, PB
Reverse or Direct Acting Controller Kc can be made positive or negative Recall for proportional FB control:
or
Direct-Acting (Kc < 0)“output increases as input increases"
p(t) Ym(t)
Reverse-Acting (Kc > 0)“output increases as input decreases"
cK
%100PB
e(t)K+p=p(t) c
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)t(Y)t(YKp)t(p mspc
• Example 2:Example 2: Flow Control Loop
Assume FT is direct-acting. Select sign of Kc so that KcKv > 0
1.) Air-to-open (fail close) valve ==> ?2.) Air-to-close (fail open) valve ==> ?
• Consequences of wrong controller action??
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Transfer Function for Proportional Control:Let
Then controller input/output relation can written as
Take Laplace transform of each side,
or
INTEGRAL CONTROL ACTIONSynonyms: "reset", "floating control"
I reset time (or integral time) - adjustable
p-p(t)(t)p
e(t)K(t)p c
E(s)K(s)P c
cKE(s)
(s)P
s
1
E(s)
(s)P td)t(e
1p)t(p
I
t
0I
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t
0Ic td)t(e
1)t(eKp)t(p
Proportional-Integral (PI) Control
• Response to unit step change in e:
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integral provides memory of emost popular controller
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• Integral action eliminates steady-state error (i.e., offset) Why??? e 0 p is changing with time until e = 0, where p reaches steady state.
s
11K
E(s)
(s)P
Ic
• Transfer function for PI control
Derivative Control Action Ideal derivative action
Used to improve dynamic response of the controlled variable Derivative kick (use -dym/dt ) Use alone?
Some controllers are calibrated in 1/I
("repeats per minute") instead of I .
p
dt
dep)t(p D
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For PI controllers, is not adjustable.
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Proportional-Integral-Derivative (PID) Control
Now we consider the combination of the proportional, integral, and derivative control modes as a PID controller.
• Many variations of PID control are used in practice.
• Next, we consider the three most common forms.
Parallel Form of PID Control
The parallel form of the PID control algorithm (without a derivative filter) is given by
0
1* * τ (8-13)
τ
tc D
I
de tp t p K e t e t dt
dt
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The corresponding transfer function is:
11 τ (8-14)
τc DI
P sK s
E s s
τ 1 τ 1(8-15)
τ ατ 1I D
cI D
P s s sK
E s s s
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Expanded Form of PID Control
In addition to the well-known series and parallel forms, the expanded form of PID control in Eq. 8-16 is sometimes used:
0
* * (8-16)t
c I Dde t
p t p K e t K e t dt Kdt
Features of PID Controllers
Elimination of Derivative and Proportional Kick
• One disadvantage of the previous PID controllers is that a sudden change in set point (and hence the error, e) will cause the derivative term momentarily to become very large and thus provide a derivative kick to the final control element.
Automatic and Manual Control Modes• Automatic Mode
Controller output, p(t), depends on e(t), controller constants, and type of controller used. ( PI vs. PID etc.)
Manual Mode Controller output, p(t), is adjusted manually. Manual Mode is very useful when unusual conditions exist:
plant start-upplant shut-downemergencies
• Percentage of controllers "on manual” ?? (30% in 2001, Honeywell survey)
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Digital PID Controller
finite difference approximation
where,= the sampling period (the time between successive samples of the controlled variable)= controller output at the nth sampling instant, n=1,2,…= error at the nth sampling unit
velocity form - see Equation (8-19)(pn)- incremental change
1
11
nD
n c n k n nkI
DI
tp p K e e e e
t
np
ne
t
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Typical Response of Feedback Control SystemsConsider response of a controlled system after a sustained disturbance occurs (e.g., step change in disturbance variable); y > 0 is off-spec.
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No control (Kc=0)Increasing K
C
Time
y
0
Figure 8.13 Proportional control: effect of Controller gain
Increasing D
Time
y
0
Figure 8.15 PID control: effect of derivative time
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integral action ~ /c IK
Increasing I
Time (a)
y
0
Increasing KC
Time (b)
y
0
Figure 8.14 Proportional-integral control: (a) effect of integral time, (b) effect of controller gain
Summary of the Characteristics of the Most Commonly Used Controller Modes
1. Two Position:Inexpensive.Extremely simple.
2. Proportional:Simple.Inherently stable when properly tuned.Easy to tune.Experiences offset at steady state. (OK for level control)
3. Proportional plus integral:No offset.Better dynamic response than reset alone.Possibilities exist for instability due to lag introduced.
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4. Proportional plus derivative:
Stable.
Less offset than proportional alone (use of
higher gain possible).
Reduces lags, i.e., more rapid response.
5. Proportional plus integral plus derivative:
Most complex
Rapid response
No offset.
Best control if properly tuned.
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Example 3:Example 3: Liquid Level Control• Control valves are air-to-open• Level transmitters are direct acting
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Question:Question:1. Type of controller action? Select Kc so that
0c v pK K K
(a) air-to-open valve: sign of Kv?(b) sign of process gain?
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