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
S/P
DISPLAY
TC
TCV
T
UTILITY FLOW
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 )  Gd ( s ) D( s )  G p ( s )Gc ( s )Ga ( s ) E ( s )
E ( s )  Ysp ( s )  Ys ( s )  Ysp ( s )  Gs ( s )Y ( s )
Y ( s )  Gd ( s ) D( s )  G p ( s )Gc ( s )Ga ( s )[Ysp ( s )  Gs ( s )Y ( s )]
Gd ( s ) D( s )  G p ( s )Gc ( s )Ga ( s )Ysp ( s )
Y ( s) 
G p ( s )Gc ( s )Ga ( s )Gs ( s ) 1
TYPICAL TRANSFER FUNCTIONS
FOR A FEEDBACK LOOP
• CONSIDER THE RESPONSE TO A
DISTURBANCE
• WITH CONSTANT S/P (Ysp(S) = 0)
• REGULATORY CONTROL OR
DISTURBANCE REJECTION
G p ( s )Gc ( s )Ga ( s )Ysp ( s )
Y ( s) 
G p ( s )Gc ( s )Ga ( s )Gs ( s ) 1
• THIS REPRESENTS A PROCESS AT
• STEADY STATE RESPONDING TO
• BACKGROUND DISTURBANCES
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 (KcKaKpKs) 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 (Kp=1; τp=60 SEC) WHERE
THE TEMPERATURE SENSOR HAS A τs=20
SEC AND τa IS ASSUMED SMALL. NOTE
THAT Gc(s)=Kc.
CHARACTERISTIC EQUATION
EXAMPLE- CLOSED LOOP POLES
• WHEN Kc =0, POLES ARE -0.05 AND -0.0167
WHICH CORRESPOND TO THE INVERSE
OF τp AND τs.
• AS Kc 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 Kc 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 )  c0  K c e( t )
DEFINITION OF TERMS
• e(t) - THE ERROR FROM SETPOINT
[e(t)=ysp-ys].
• Kc - THE CONTROLLER GAIN IS A TUNING
PARAMETER AND LARGELY DETERMINES
THE CONTROLLER AGGRESSIVENESS.
• τI - THE RESET TIME IS A TUNING
PARAMETER AND DETERMINES THE
AMOUNT OF INTEGRAL ACTION.
• τD - 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:


1 t
c( t )  c0  Kc e(t )   e( t )dt 
0



I
• FOR PROPORTIONAL/DERIVATIVE
de( t ) 

c( t )  c0  Kc e( t )   D
dt 

PID ALGORITHM POSITION FORM
• TRANSFER FUNCTION FOR PID
CONTROLLER:

C( s )
1
Gc ( s ) 
 Kc e( t ) 
E( s)
I

de( t ) 
0 e(t )dt   D dt 
t
PID ALGORITHM POSITION FORM
• DERIVATIVE KICK:
– RESULTS FROM AN ERROR SPIKE (INCREASE IN
𝑑𝑒(𝑡)
) WHEN A SETPOINT CHANGE IS INITIATED
𝑑𝑡
– CAN BE ELIMINATED BY REPLACING THE
CHANGE IN ERROR WITH A CHANGE IN THE
−𝑑𝑦𝑠 (𝑡)
CONTROLLED VARIABLE
IN THE PID
𝑑𝑡
ALGORITHM
– RESULTING EQUATION IS CALLED THE
DERIVATIVE-ON-MEASUREMENT FORM OF THE
PID ALGORITHM

1
c(t )  c0  Kc e(t ) 
I

dys (t ) 
0 e(t )dt   D dt 
t
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, Δt: YIELDS THE
VELOCITY FORM OF THE PID ALGORITHM
t


e
(
t
)

e
(
t


t
)

e
(
t
)


I
c( t  t )  c( t )  K c 

  D ( y ( t )  2 y ( t  t )  y ( t  2t )
s
s
 t s

DIGITAL VERSIONS OF
THE PID ALGORITHM
• FOR INTEGRATION OVER A TIME PERIOD,
t, WHERE n = t/Δt:
t n


e( t )    e( i * t ) 
I i 1

c( t )  c( t  t )  K c 
 D

 t ( ys ( t )  ys ( t  t )
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
t n


 y s ( t  t )  y s ( t )    e( i * t )
I i 1

c( t )  c( t  t )  K c 
 D


(
y
(
t
)

y
(
t


t
)
s
 t s

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,
c
Kc 
ys
• HAS INVERSE UNITS TO PROCESS
GAIN
ys
Kp 
c
PROPORTIONAL ACTION
PROPERTIES
• CLOSED LOOP TRANSFER
FUNCTION BASE ON A PONLY 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 τp
– 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
t
p
Ys  YSP (1  e )
• CLOSED LOOP - PROPORTIONAL
CONTROL
t
p
YSP
K c K p 1
Ys 
(1  e
)
Kc K p  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:
YSP
Ys
1
Ys 
OR

Kc K p  1
YSP Kc K p  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:
Y ( s)
1
Gp ( s) 

2
YSP ( s)  `p s  2 ` `p s  1
• WHERE
 `p 
 I p
Kc K p
AND
1
I
 `p 
2  p Kc K p
DERIVATIVE ACTION
PROPERTIES
• THE DERIVATIVE MODE RESPONDS
TO THE SLOPE
dy s ( t )
dt
• THIS MODE AMPLIFIES SUDDEN
CHANGES IN THE CONTROLLER
INPUT SIGNAL - INCREASES
CONTROLLER SENSITIVITY
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.