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Unformatted text preview: P D I Chapter 8 Feedback Controllers: Week 1 Figure 8.1 Schematic diagram for a stirredtank blending
system. “Normal”, Kc < 0
“Reverse”, Kc > 0
p change wrt ym
not wrt e = error 1 Although Eq. 81 indicates that the set point can be timevarying,
in many process control problems it is kept constant for long
periods of time. Basic Control Modes
Next we consider the three basic control modes starting with the
simplest mode, proportional control. For proportional control, the controller output is proportional to
the error signal, Chapter 8 Chapter 8 Proportional Control
Control
In feedback control, the objective is to reduce the error signal to
zero where
e ( t ) = ysp ( t ) − ym ( t )
and e (t ) (81) = error signal ysp ( t ) = set point 2 (ysp and ym must have same units) p ( t ) = p + Kce ( t ) (82) where:
p ( t ) = controller output
p = bias (steadystate) value
K c = controller gain (usually dimensionless) ym ( t ) = measured value of the controlled variable
(or equivalent signal from the sensor/transmitter)
3 4 1 The key concepts behind proportional control are the following: Chapter 8 Chapter 8 1. The controller gain can be adjusted to make the controller
output changes as sensitive as desired to deviations between
set point and controlled variable; In order to derive the transfer function for an ideal proportional
controller (without saturation limits), define a deviation variable
p′ ( t ) as
p′ ( t ) p ( t ) − p
(84) Chapter 8 Chapter 8 (85) The transfer function for proportionalonly control:
= Kc 6 For integral control action, the controller output depends on the
integral of the error signal over time,
p (t ) = p + p′ ( t ) = K c e ( t ) E (s) Some controllers have a proportional band setting instead of a
controller gain. The proportional band PB (in %) is defined as
Th
PB
%)
100%
100
PB
or K c =
(83)
Kc
PB Integral Control Then Eq. 82 can be written as P′ ( s ) For proportional controllers, bias p can be adjusted, a procedure
referred to as manual reset. For “PB”, Kc must be dimensionless, e.g. p, e in mA, each 420 mA range
Example: if Kc = 2, a p change is twice the e change. Output p would
change by 100% for an e change of 50% so PB = 50%. 5 Gc = 2. the sign of Kc can be chosed to make the controller output
sign
be chosed to make the controller output
increase (or decrease) as the error signal increases. (86) An inherent disadvantage of proportionalonly control is that a
steadystate error usually occurs after a setpoint change or a
sustained disturbance.
7 1
τI ∫0 e ( t *)dt *
t (87) where τ I , an adjustable parameter referred to as the integral time
or reset time, has units of time.
Integral control action is widely used because it provides an
important practical advantage, the elimination of offset.
Consequently, integral control action is normally used in
conjunction with proportional control as the proportionalintegral
(PI) controller:
⎛
1
p ( t ) = p + Kc ⎜ e ( t ) +
τI
⎝ ⎞ ∫0 e ( t *) dt * ⎟
⎠
t (88)
8 2 The corresponding transfer function for the PI controller in
Eq. 88 is given by
P′ ( s ) ⎛
⎛ τI s +1 ⎞
1⎞
= Kc ⎜1 +
⎟ = Kc ⎜
⎟
E (s)
⎝ τI s ⎠
⎝ τI s ⎠ • Further buildup of the integral term while the controller is
saturated (“pegged at a limiting value by the controller”) is
referred to as reset windup or integral windup. (89) Chapter 8 Chapter 8 Gc = • When a sustained error occurs, the integral term becomes
quite large and the controller output eventually saturates. Some commercial controllers are calibrated in terms of 1/ τ I
(repeats per minute) rather than τ I (minutes, or minutes per
repeat).
Reset Windup
• An inherent disadvantage of integral control action is a
phenomenon known as reset windup or integral windup. Derivative Control
The function of derivative control action is to “anticipate” the
future behavior of the error signal by considering its rate of
change. An alternative description is that derivative action seeks
to minimize the rate of change of the error de/dt.
• The anticipatory strategy used by the experienced operator can
be incorporated in automatic controllers by making the
controller output proportional to the rate of change of the error
signal or the controlled variable. • Recall that the integral mode causes the controller output to
change as long as e(t*) ≠ 0 in Eq. 88.
9 10 • Thus, for ideal derivative action,
p (t ) = p + τD de ( t )
dt • For analog controllers, the PD controller transfer function in
(811) can be approximated by
(810)
Gc = For example, an ideal PD controller has the transfer function:
Gc = P′ ( s )
E (s) = K c (1 + τ D s ) Chapter 8 Chapter 8 where τ D , the derivative time, has units of time. (811) • By providing anticipatory control action, the derivative mode
providing anticipatory control action the derivative mode
tends to stabilize the controlled process. P′ ( s )
E (s) ⎛
τDs ⎞
= K c ⎜1 +
⎟
ατ D s + 1 ⎠
⎝ (812) where the constant α typically has a value between 0.05 and
0.2, with 0.1 being a common choice.
• In Eq. 812 the derivative term includes a derivative mode
filter (also called a derivative filter) that reduces the sensitivity
of the control calculations to highfrequency noise in the
measurement. • Unfortunately, the ideal proportionalderivative control
algorithm in Eq. 810 is physically unrealizable because it
cannot be implemented exactly.
11 12 3 The corresponding transfer function is: ProportionalIntegralDerivative (PID) Control
Now we consider the combination of the proportional, integral,
and derivative control modes as a PID controller. Gc = • Next, we consider the three most common forms. Chapter 8 Chapter 8 • Many variations of PID control are used in practice. Parallel Form of PID Control
The parallel form of the PID control algorithm (without a
derivative filter) is given by
⎡
1
p ( t ) = p + K c ⎢e ( t ) +
τI
⎣ ∫0 e ( t *) dt * + τ D
t de ( t ) ⎤
⎥
dt ⎦ (813) P′ ( s ) ⎡
⎤
1
= K c ⎢1 +
+ τDs⎥
E (s)
⎣ τI s
⎦ (814) Series Form of PID Control
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 seriesform controller have a
derivative filter that is applied to either the derivative term, as in
derivative filter that is applied to either the derivative term, as in
Eq. 812, or to the PD term, as in Eq. 815:
P′ ( s ) ⎛ τ s + 1 ⎞⎛ τ D s + 1 ⎞
= Kc ⎜ I
⎟⎜
⎟
E (s)
⎝ τ I s ⎠⎝ ατ D s + 1 ⎠ (815) 13 14 • This sudden change is undesirable and can be avoided by basing
the derivative action on the measurement, ym, rather than on the
error signal, e, but then the controller has two inputs: e and ym. Expanded Form of PID Control
In addition to the wellknown series and parallel forms, the
expanded form of PID control in Eq. 816 is sometimes used:
de ( t ) 0 dt Chapter 8 • We illustrate the elimination of derivative kick by considering
the parallel form of PID control in Eq. 813. (816) Chapter 8 t p ( t ) = p + K c e ( t ) + K I ∫ e ( t *) dt * + K D Features of PID Controllers
Elimination of Derivative and Proportional Kick
• One disadvantage of the previous PID controllers is that a
disadvantage of the previous PID controllers is that
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.
15 • Replacing de/dt by –dym/dt gives
⎡
1
p ( t ) = p + K c ⎢e ( t ) +
τI
⎣ ∫0 e ( t *) dt * −τ D
t dym ( t ) ⎤
⎥
dt ⎦ (817) Reverse or Direct Action
• The controller gain can be made either negative or positive. 16 4 • For proportional control, when Kc > 0, the controller output p(t)
increases as its input signal ym(t) decreases, as can be seen by
combining Eqs. 82 and 81:
(822) • This controller is an example of a reverseacting controller. Figure 8.11 Reverse
and directacting
proportional
controllers. (a) reverse
acting (Kc > 0. (b)
direct acting (Kc < 0) Chapter 8 Chapter 8 p ( t ) − p = K c ⎡ ysp ( t ) − ym ( t ) ⎤
⎣
⎦ • When Kc < 0, the controller is said to be direct acting because
the controller output p increases as the ym “input” increases.
• Equations 82 through 816 describe how controllers perform
during the automatic mode of operation.
• However, in certain situations the plant operator may decide to
override the automatic mode and adjust the controller output
manually = controller is “on manual”. 17 18 • Example: Flow Control Loop Automatic and Manual Control Modes Chapter 8 Chapter 8 • Assume FT is directacting, what kind of P
controller (direct or reverse acting) should be used
for the two kinds of valves ?
th
ki
1. Airtoopen (fail close) valve ==> ?
2. Airtoclose (fail open) valve ==> ?
19 Automatic Mode
Controller output, p(t), depends on e(t), controller
yp
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 startup
plant shutdown
emergencies
• Percentage of controllers "on manual” ??
(30% in 2001, Honeywell survey)
20 5 Example: Liquid Level Control
• Control valves are airtoopen
• Level transmitters are direct acting Chapter 8 Chapter 8 OnOff 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. Question: Type of controller action?
21 22 OnOff Controllers (continued) Practical case (dead band) = no change unless “large” change in e Chapter 8 Chapter 8 Synonyms:
“twoposition” or “bangbang” controllers. Controller output has two possible values.
23 24 6 P PID Controller D I • Ideal controller Chapter 8 • Transfer function (ideal)
t
⎡
1
de ⎤
p(t ) = p + K c ⎢e(t ) + ∫ e(t ′)dt ′ + τ D ⎥
τI 0
dt ⎦
⎣ ⎛
⎞
P′(s)
1
= K c ⎜1 +
+τ Ds ⎟
E(s)
τIs
⎝
⎠ • Transfer function (actual)
⎛ τ s +1⎞ ⎛ τ Ds +1 ⎞
P′(s)
= Kc ⎜ I
⎟⎜
⎟
E(s)
⎝ τ I s ⎠ ⎝ ατ D s + 1 ⎠ “Normal”, Kc < 0
“Reverse”, Kc > 0
p change wrt ym
not wrt e = error lead / lag units α = small number (0.05 to 0.20, 0.05 used in CHE461OSU)
25 Typical Response of Feedback Control Systems
Consider response of a controlled system after a
sustained disturbance occurs (e.g., step change in
the disturbance variable) Controller Comparison
 Simplest controller to tune (Kc).
with sustained disturbance or setpoint
 Offset with sustained disturbance or setpoint
change. PI  More complicated to tune (Kc, τI) .
Better performance than P
No offset
Most popular FB controller PID  Most complicated to tune (Kc, τI, τD) .
Better performance than PI
No offset
Derivative action may be affected by noise Chapter 8 Chapter 8 P 26 y Figure 8.12. Typical process responses with feedback control.
27 28 7 ...
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