Chapter 8 Feedback Controllers 1. On-off Controllers Simple Cheap Used In residential heating and...
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Transcript of Chapter 8 Feedback Controllers 1. On-off Controllers Simple Cheap Used In residential heating and...
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Feedback Controllers
1
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.
Example 1:Example 1: Temperature control of jacketed vessel.
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On-Off Controllers
Synonyms:“two-position” or “bang-bang” controllers.
Controller output has two possible values.
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Practical case (dead band)
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Three Mode (PID) Controller • Proportional• Integral• Derivative
Proportional Control• Define an error signal, e, by e = R - B
whereR= set pointB = measured value of the controlled variable (or equivalent signal from transmitter)
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Since signals are time varying,
e(t) = R(t) - B(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
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Figures 8.4, 8.5in Text
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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) B(t)
Reverse-Acting (Kc > 0)“output increases as input decreases"
cK
%100PB
e(t)K+p=p(t) c
B(t)-R(t)K+p=p(t) c
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• Example 2:Example 2: Flow Control Loop
Assume FT is direct-acting.
1.) Air-to-open (fail close) valve ==> reverse?2.) Air-to-close (fail open) valve ==> direct?
• Consequences of wrong controller action??
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Figure 8.8 in Text
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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"
1 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 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
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Derivative Control Action Ideal derivative action
Used to improve dynamic response of the controlled variable Derivative kick (use db/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
Series Form of PID Control
Historically, it was convenient to construct early analog controllers (both electronic and pneumatic) so that a PI element and a PD element operated in series.
Commercial versions of the series-form controller have a derivative filter that is applied to either the derivative term, as in Eq. 8-12, or to the PD term, as in Eq. 8-15:
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.
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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
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 (8-28)
(pd)- incremental change
1nn
D1n
1kk
Incn ee
te
teKpp
np
ne
t
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PID -Most complicated to tune (Kc, I, D) .-Better performance than PI-No offset-Derivative action may be affected by noise
PI -More complicated to tune (Kc, I) .-Better performance than P-No offset-Most popular FB controller
P -Simplest controller to tune (Kc).-Offset with sustained disturbance or set point change.
Controller Comparison
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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)
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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.
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 reset plus rate:
Most complex
Rapid response
No offset.
Difficult to tune.
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?
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