Lect. 17 CHE 185 – PID CONTROLx

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15 Νοε 2013 (πριν από 4 χρόνια και 7 μήνες)

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CHE 185


PROCESS
CONTROL AND
DYNAMICS

PID CONTROLLER
FUNDAMENTALS

CLOSED LOOP
COMPONENTS


GENERAL DEFINITIONS


OPEN LOOPS ARE MANUAL CONTROL


FEEDBACK LOOPS ARE CLOSED


EXAMPLE P&ID FOR FEEDBACK
CONTROL LOOP

UTILITY FLOW
T
TC
S/P
DISPLAY
TCV
HEAT
EXCHANGER
CLOSED LOOP
COMPONENTS


GENERAL BLOCK DIAGRAM FOR
FEEDBACK CONTROL LOOP (FIGURE
7.2.1 FROM TEXT)

CLOSED LOOP
COMPONENTS


OVERALL TRANSFER FUNCTION

Y
s
G
s
D
s
G
s
G
s
G
s
E
s
E
s
Y
s
Y
s
Y
s
G
s
Y
s
Y
s
G
s
D
s
G
s
G
s
G
s
Y
s
G
s
Y
s
Y
s
G
s
D
s
G
s
G
s
G
s
Y
s
G
s
G
d
p
c
a
sp
s
sp
s
d
p
c
a
sp
s
d
p
c
a
sp
p
c
(
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(
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(
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(
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(
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s
G
s
G
s
a
s

1
TYPICAL TRANSFER FUNCTIONS
FOR A FEEDBACK LOOP


CONSIDER THE RESPONSE TO A
DISTURBANCE


WITH CONSTANT S/P (
Y
sp
(S) = 0)


REGULATORY CONTROL
OR
DISTURBANCE

REJECTION





THIS REPRESENTS A PROCESS AT


STEADY STATE RESPONDING TO


BACKGROUND DISTURBANCES

Y
s
G
s
G
s
G
s
Y
s
G
s
G
s
G
s
G
s
p
c
a
sp
p
c
a
s
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)


1
TYPICAL TRANSFER FUNCTIONS
FOR A FEEDBACK LOOP


CONSIDER THE SETPOINT RESPONSE


WITH NO DISTURBANCE (D(
s
) = 0)


SETPOINT TRACKING
OR
SERVO
CONTROL







THIS MODEL REPRESENTS THE SYSTEM
RESPONSE TO A S/P ADJUSTMENT

TYPICAL TRANSFER FUNCTIONS
FOR A FEEDBACK LOOP


GENERALIZATIONS REGARDING
THE FORM OF THE TRANSFER
FUNCTIONS


THE NUMERATOR IS THE
PRODUCT OF ALL TRANSFER
FUNCTIONS BETWEEN THE INPUT
AND THE OUTPUT


THE DENOMINATOR IS EQUAL TO
THE NUMERATOR + 1

TYPICAL TRANSFER FUNCTIONS
FOR A FEEDBACK LOOP


CHARACTERISTIC EQUATION




OBTAINED

BY SETTING THE
DENOMINATOR = 0


ROOTS FOR THIS EQUATION WILL BE:


OVERDAMPED LOOP


COMPLEX ROOTS, FOR AN OSCILLATORY
LOOP


AT LEAST ONE REAL POSITIVE ROOT FOR
AN UNSTABLE LOOP

FEEDBACK CONTROL
ANALYSIS


THE LOOP GAIN (
K
c
K
a
K
p
K
s
) SHOULD
BE POSITIVE FOR STABLE
FEEDBACK CONTROL.


AN OPEN
-
LOOP UNSTABLE
PROCESS CAN BE MADE STABLE BY
APPLYING THE PROPER LEVEL OF
FEEDBACK CONTROL.


CHARACTERISTIC
EQUATION EXAMPLE


CONSIDER THE DYNAMIC BEHAVIOR OF A
P
-
ONLY CONTROLLER APPLIED TO A CST
THERMAL MIXER (
K
p
=1;
τ
p
=60
SEC) WHERE
THE TEMPERATURE SENSOR HAS A
τ
s
=20
SEC AND
τ
a

IS ASSUMED SMALL. NOTE
THAT
G
c
(s
)=K
c
.

CHARACTERISTIC EQUATION
EXAMPLE
-

CLOSED LOOP POLES


WHEN
K
c

=0, POLES ARE
-
0.05 AND
-
0.0167
WHICH CORRESPOND TO THE INVERSE
OF
τ
p
AND
τ
s
.


AS
K
c

IS INCREASED FROM ZERO, THE
VALUES OF THE POLES BEGIN TO
APPROACH ONE ANOTHER.


CRITICALLY DAMPED BEHAVIOR OCCURS
WHEN THE POLES ARE EQUAL.


UNDERDAMPED

BEHAVIOR RESULTS
WHEN
K
c

IS INCREASED FURTHER DUE TO
THE IMAGINARY COMPONENTS IN THE
POLES.


PID ALGORITHM
-

POSITION FORM


ISA POSITION FORM FOR PID:






FOR PROPORTIONAL ONLY



c
t
c
K
e
t
c
(
)
(
)


0
DEFINITION OF TERMS


e(t
)
-

THE ERROR FROM
SETPOINT

[
e(t)=
y
sp
-
y
s
].


K
c
-

THE CONTROLLER GAIN IS A TUNING
PARAMETER AND LARGELY DETERMINES
THE CONTROLLER AGGRESSIVENESS.



τ
I
-

THE RESET TIME IS A TUNING
PARAMETER AND DETERMINES THE
AMOUNT OF INTEGRAL ACTION.



τ
D
-

THE DERIVATIVE TIME IS A TUNING
PARAMETER AND DETERMINES THE
AMOUNT OF DERIVATIVE ACTION.


PID
CONTROLLER
TRANSFER
FUNCTION

PID ALGORITHM
-

POSITION FORM


FOR PROPORTIONAL/INTEGRAL:





FOR PROPORTIONAL/DERIVATIVE

c
t
c
K
e
t
e
t
dt
c
I
t
(
)
(
)
(
)










0
0
1

c
t
c
K
e
t
de
t
dt
c
D
(
)
(
)
(
)









0

PID ALGORITHM
-

POSITION FORM


TRANSFER FUNCTION FOR PID
CONTROLLER:




G
s
C
s
E
s
K
e
t
e
t
dt
de
t
dt
c
c
I
D
t
(
)
(
)
(
)
(
)
(
)
(
)











1
0


PID ALGORITHM
-

POSITION FORM


DERIVATIVE KICK:


RESULTS FROM AN ERROR SPIKE (INCREASE IN

(
𝑡
)
𝑡
) WHEN A SETPOINT CHANGE IS INITIATED


CAN BE ELIMINATED BY REPLACING THE
CHANGE IN ERROR WITH A CHANGE IN THE
CONTROLLED VARIABLE


𝑦
𝑠
(
𝑡
)
𝑡

IN THE PID
ALGORITHM


RESULTING EQUATION IS CALLED THE
DERIVATIVE
-
ON
-
MEASUREMENT FORM OF THE
PID ALGORITHM



c
t
c
K
e
t
e
t
dt
dy
t
dt
c
I
D
s
t
(
)
(
)
(
)
(
)











0
0
1


DIGITAL VERSIONS OF
THE PID ALGORITHM


DIGITAL CONTROL SYSTEMS REQUIRE
CONVERSION OF ANALOG SIGNALS TO
DIGITAL SIGNALS FOR PROCESSING.


DIGITAL VERSION OF THE PREVIOUS
EQUATION IN DIGITAL FORMAT BASED ON
A SINGLE TIME INTERVAL,
Δ
t
: YIELDS THE
VELOCITY FORM

OF THE PID ALGORITHM

c
t
t
c
t
K
e
t
e
t
t
t
e
t
t
y
t
y
t
t
y
t
t
c
I
D
s
s
s
(
)
(
)
(
)
(
)
(
)
(
(
)
(
)
(
)































2
2
DIGITAL VERSIONS OF
THE PID ALGORITHM


FOR INTEGRATION OVER A TIME PERIOD,
t
, WHERE
n =
t/
Δ
t
:


c
t
c
t
t
K
e
t
t
e
i
t
t
y
t
y
t
t
c
I
i
n
D
s
s
(
)
(
)
(
)
(
*
)
(
(
)
(
)




























1
DIGITAL VERSIONS OF
THE PID ALGORITHM


PROPORTIONAL KICK


RESULTS FROM THE INITIAL RESPONSE TO A
SETPOINT CHANGE


CAN BE ELIMINATED IN THE VELOCITY
EQUATION BY REPLACING THE ERROR TERM IN
THE ALGORITHM WITH THE SENSOR TERM


c
t
c
t
t
K
y
t
t
y
t
t
e
i
t
t
y
t
y
t
t
c
s
s
I
i
n
D
s
s
(
)
(
)
(
)
(
)
(
*
)
(
(
)
(
)































1
FIRST
ORDER PROCESS WITH A
PI
CONTROLLER EXAMPLE

PI
CONTROLLER APPLIED TO A
SECOND ORDER
PROCESS
EXAMPLE





PROPORTIONAL ACTION


USES A MULTIPLE OF THE ERROR
AS A SIGNAL TO THE CONTROLLER,
CONTROLLER GAIN
,



HAS INVERSE UNITS TO PROCESS
GAIN

K
c
y
c
s



K
y
c
p
s



PROPORTIONAL ACTION
PROPERTIES


CLOSED LOOP TRANSFER
FUNCTION BASE ON A P
-
ONLY CONTROLLER
APPLIED TO A FIRST
ORDER PROCESS.


PROPERTIES OF P
CONTROL


DOES NOT CHANGE ORDER
OF PROCESS


CLOSED LOOP TIME
CONSTANT IS SMALLER
THAN OPEN LOOP
τ
p



DOES NOT ELIMINATE
OFFSET.

P
-
ONLY CONTROL OFFSET

PROPORTIONAL
RESPONSE
ACTION WITH A
PI
CONTROLLER




PROPORTIONAL CONTROL


RESPONSE OF FIRST ORDER
PROCESS TO STEP FUNCTION


OPEN
LOOP
-

NO
CONTROL




CLOSED LOOP
-

PROPORTIONAL
CONTROL



Y
Y
e
s
SP
t
p



(
)
1



Y
Y
K
K
e
s
SP
c
p
t
K
K
p
c
p





1
1
1
(
)

PROPORTIONAL CONTROL


PROPORTIONAL CONTROL MEANS
THE CLOSED SYSTEM RESPONDS
QUICKER THAN THE OPEN SYSTEM
TO A CHANGE.


OFFSET
IS A RESULT OF
PROPORTIONAL CONTROL. AS T
INCREASES, THE RESULT IS:





Y
Y
K
K
OR
Y
Y
K
K
s
SP
c
p
s
SP
c
p




1
1
1
INTEGRAL ACTION



THE PRIMARY BENEFIT OF
INTEGRAL ACTION IS THAT IT
REMOVES OFFSET FROM
SETPOINT
.


IN ADDITION, FOR A PI CONTROLLER
ALL THE STEADY
-
STATE CHANGE IN
THE CONTROLLER OUTPUT
RESULTS FROM INTEGRAL ACTION.


INTEGRAL ACTION


WHERE PROPORTIONAL MODE
GOES TO A NEW STEADY
-
STATE
VALUE WITH OFFSET, INTEGRAL
DOES NOT HAVE A LIMIT IN TIME,
AND PERSISTS AS LONG AS THERE
IS A DIFFERENCE.


INTEGRAL
WORKS ON THE
CONTROLLER
GAIN


INTEGRAL
SLOWS DOWN THE
RESPONSE OF THE CONTROLLER
WHEN PRESENT WITH
PROPORTIONAL

INTEGRAL ACTION


INTEGRAL ADDS AN ORDER TO THE
CONTROL FUNCTION FOR A
CLOSED LOOP


FOR
THE FIRST ORDER PROCESS
WITH PI CONTROL, THE TRANSFER
FUNCTION IS
:




WHERE


AND

G
s
Y
s
Y
s
s
s
p
SP
p
p
(
)
(
)
(
)
`
`
`




1
2
1
2






`
p
I
p
c
p
K
K




`
p
I
p
c
p
K
K

1
2
DERIVATIVE ACTION
PROPERTIES


THE DERIVATIVE MODE RESPONDS
TO THE
SLOPE




THIS MODE AMPLIFIES SUDDEN
CHANGES IN THE CONTROLLER
INPUT SIGNAL
-

INCREASES
CONTROLLER SENSITIVITY


dy
t
dt
s
(
)
DERIVATIVE ACTION
PROPERTIES


DERIVATIVE
MODE CAN
COUNTERACT INTEGRAL MODE TO
SPEED UP THE RESPONSE OF THE
CONTROLLER.


DERIVATIVE
DOES NOT REMOVE
OFFSET


IMPROPER
TUNING CAN RESULT IN
HIGH
-
FREQUENCY VARIATION IN
THE MANIPULATED VARIABLE



7.6

DOES
NOT WORK WELL
WITH NOISY SYSTEMS

DERIVATIVE ACTION
PROPERTIES


PROPERTIES OF DERIVATIVE CONTROL:


DOES NOT CHANGE THE ORDER OF THE
PROCESS


DOES NOT ELIMINATE OFFSET


REDUCES THE OSCILLATORY NATURE OF THE
FEEDBACK RESPONSE


CLOSED LOOP TRANSFER FUNCTION FOR
DERIVATIVE
-
ONLY CONTROL APPLIED TO
A SECOND ORDER PROCESS
.


DERIVATIVE ACTION
RESPONSE FOR
A PID
CONTROLLER


DERIVATIVE ACTION


THE PRIMARY BENEFIT OF
DERIVATIVE ACTION IS THAT IT
REDUCES THE OSCILLATORY
NATURE OF THE CLOSED
-
LOOP
RESPONSE.