Transcript No Slide Title

Basic Concepts
 Block diagram representation of control
systems
 Transfer functions
 Analysis of block diagrams
 P, PI and PID controllers ( Continuous and
discrete forms)
 Stability of feedback control systems
Basic Block Diagram
INPUT VARIABLE
SYSTEM
OUTPUT VARIABLE
PID:Process Instrumentation Diagram
temperature
transmitter
TT
101
TC
101
Feed
electronic
transmission
line
temperature
controller
pneumatic
line
Thermocouple
sensor
Automatic Control
Valve cooling water
inlet
Product
Manual
Valve
cooling water
out
Block Diagram of Feedback Control System
CONTROLLER
OUTPUT
ACTUATOR
ERROR
SETPOINT
Tset

CONTROLLER
SIGNAL TO
VALVE
Tm (MEASURED VARIABLE)
CONTROLLER
VARIABLE
MANIPULATED
VARIABLE
PROCESS
CONTROL
VALVE
TANK
WATER
FLOW
TRANSMITTER
TEMPERATURE
Laplace Transform
F s 

o
f t e  st dt
L f t   F s 
L1 F s   f t 
Common Signals
Name
Unit step function
Unit impulse function(Dirac
delta function)
Ramp function
Sine function
Function, f(t)
u t   0 t  0
ut   1 t  0
 t   , t  0
 t   0 t  0
r t   0 t  0
r t   kt t  0
xt   sin wt
Laplace transform, F(s)
1
s
1
k
s2

s2  2
Properties of Laplace Transform
Property
L f1 t   f 2 t 
Linearity Property
Description
Lk f1 t 
where K

L f1 t   L f 2 t 
 k L f1 t 
 constant
L f t     e s L f t 
Time Delay
 e s F s 
Differentiation
Integration
Final Value Theorem
li m f t 
 
 d f t  
L
  s F s   f o 
 dt 
t
1
L   f t  dt  F s 
o
 s
 li m s F s 

hand side exists.
provided the limit on the left
Transfer functions
a1
dx(t )
 a 2 x (t )  b1u (t ),
dt
x (0)  0
(BC.3)
a1 s x s   x0   a2 xs   b1 u s 
 b1

 u s 
xs   
a
s

a
2 
 1
K

 u s 
s  1
(BC.4)
G s  
(BC.5)
K
 s1
Proportional Control
mt   K c et   m
ms   K c  es 
et   setpoint measurement
Proportional Integral Control

1
mt   K c et  
I


 et  dt


1 
ms   K c 1 
 es 
 Is
Proportional Integral Derivative
(PID) Control

1
mt   K c et  
I

m( s )  K c (1 
1
Is
 e dt   D
de 

dt 
  D s )e( s )
 Ds
m( s )  K c (1 

)e( s )
 I s s  1
1
Common Transfer Functions
Second-Order Transfer Functions

2
d 2x
dt
2
 2 
xs  
dx
 x  Ku,
dt
x(0)  0, x' (0)  0

 s  2 s  1
2 2
 u s 
First-Order Plus Dead-Time
(FOPDT)
s
Ke
y s  
 s 1
Stability of Systems
y s  
K
xs 
s  p 
xt    t ,
y s  
xs   1
(BC.19)
(BC.20)
K
s p
y t   Ke pt
p  a  ib
(BC.22)
Location of pole
Imaginary Axis
y  Ke pt
 Ke at cosbtisinbt
p  a  ib
 re if
r  a2  b2
r
φ
b
a
Real Axis
Generalization
G s  


N s 
Ds 
N s 
s  p1  s  p2  ....... s  pn 
Kn
K1
K2

 ....... 
s  p1 s  p 2
s  pn
yt   K1 e
p1t
 ...  K n e
pnt
Stability and Pole Location
"For a transfer function to be stable, all its poles must lie to the left of the imaginary axis in the complex plane, i.e. in the left half plane (LHP)".
Imaginary Axis


Sustained Oscillations

Oscillatory growth
Oscillatory Decay
Exponential growth


Expnential Decay

Oscillatory Decay

Sustained Oscillations

Oscillatory growth
Real Axis
Stability of Closed Loop Systems
CONTROLLER ACTUATOR
y set +

-
e(s)
Gc s 
m(s)
q(s)
Gv s 
PROCESS OUTPUT
y(s)
G p s 
G (s)
MEASURED VARIABLE
TRANSMITTER
GT s 
ym s 
y
Gc Gv G p
1  G p Gc Gv GT
y set
1  G pGcGvGT  0
Example: Third order process
1
G p s  
s  1 s  2 s  3
Gc s   K c, Gv s   GT s   1
y
y set

Kc
s  1 s  2 s  3  K c
s  1 s  2 s  3  K c  0
Root Locus
Table BC.4 Table of roots of the character’s equation for various valves ofK c
Kc
0.1
0.2
0.39
0.6
1.0
10.0
20.0
30.0
60.0
100.0
root1
-3.0467
-3.0880
-3.1564
-3.2212
-3.3247
-4.3089
-4.8371
-5.2145
-6.0000
-6.7134
root2
root3
-1.8990
-1.7909
-1.4218 - 0.0542i
-1.3894 - 0.3442i
-1.3376 - 0.5623i
-0.8455 - 1.7316i
-0.5814 - 2.2443i
-0.3928 - 2.5980i
0 - 3.3166i
0.3567 - 3.9575i
-1.0544
-1.1211
-1.4218 + 0.0542i
-1.3894 + 0.3442i
-1.3376 + 0.5623i
-0.8455 + 1.7316i
-0.5814 + 2.2443i
-0.3928 + 2.5980i
0 + 3.3166i
0.3567 + 3.9575I
Root Locus Graph
6
4
2
0
Imag Axis
-2
-4
-6
-6
-4
-2
0
Real Axis
2
4
6
CONTROLLER TUNING
m( s )  K c (1 
1
Is

1
 D s ) e( s )
(s  1)
Overshoot
Rise time, Tr
Overshoot, A
1.4
B
New set point
0
2
3
4
5
2
12
12
 Period 4
0.2
1

RiseTime
ISE   e 2 dt
Decay Ratio  B / A
0
 e
 2
e(t), Error in control
0.4
 2 s2  2   s  1
Penod
0.8
0.6
1
Decay Ratio  e
1.2
1
G s  
6
12
Ziegler Nichols Tuning
Ke  s
G s  
s  1
Table BC.5 Ziegler-Nichols tuning correlation
P roportion
al P 
P roportion
al IntegralPI 
P rop- Integral- DerivativePID
KI
1  
 
K  
0.9   
 
K  
1.2   
 
K  
I
D
3.33
2.00
.5