10. closed-loop dynamics - 2013
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Min-Sen Chiu
Department of Chemical and Biomolecular Engineering
National University of Singapore
CN3121 Process Dynamics and Control
10. Dynamic Behavior of Closed-loop Control Systems
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Dynamic Behavior of Closed-Loop
Control Systems
Learning Objectives
Become familiar with the major elements in thefeedback control system
Evaluate the dynamic behavior of processesoperated under feedback control
Develop closed-loop transfer functions
10
DynamicsofClosed
-LoopControlSystems
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10
DynamicsofClosed
-LoopControlSystems
The combination of the process and the feedback
controller is called the closed-loop system.
Variables of a closed-loop system:1. Inputs set-point and disturbance variables.
2. Output - controlled variable.
The analysis of closed-loop systems can be difficultdue to the presence of feedback.
Two useful tools:
1. Block diagram2. Closed-loop transfer function
Block diagrams can provide quantitative information if
each block is represented by a transfer function.
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Process
Approximate dynamic model of the stirred-tank blending
system is available:
1 21 2 (11-1) 1 1
K KX s X s W s
s s
11 2
1
, , and (11-2)wV x
K Kw w w
where
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DynamicsofClosed
-LoopControlSystems
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Composition Sensor
Assume a first-order transfer function:
(11-3)
1
m m
m
X s K
X s s
10
DynamicsofClosed
-LoopControlSystems
Usually m
.
Compared to the (slow) process dynamics, sensor
dynamics is considered as negligible (fast) dynamics;
thus its transfer function can be further simplified as
a steady-state gain Km.
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Controller
Suppose that a PI controller is used, its
transfer function is
1
1 (11-4)
cI
P sK
E s s
, E(s) - Laplace transforms of the controller outputand the error signal e(t) P s p t
and e(t) - electrical signals (units of mA)
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DynamicsofClosed
-LoopControlSystems
Set point expressed as an
electrical current signal
Set point expressed as the
actual physical variable
p t
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The error signal is )()(~)( '' txtxte msp (11-5)
Transform
)()(~)( '' sXsXsE msp (11-6)
)('~ txsp is internal set-point related to the actual set-point
by sensor gain Km :)(' tx sp
)()(~ '' txKtx spmsp
m
sp
spK
sXsX
)()(
~
'
'
(11-7)
(11-8)
Eqs. 11-4, 11-6 and 11-8 are shown in the controller
block diagram.
Transform
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ynamicsofClosed
-LoopControlSystems
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Current-to-Pressure (I/P) Transducer
Usually has linear characteristics and negligible (fast)dynamics, thus the transfer function merely consists of
a steady-state gain KIP.
(11-9)t
IP
P s
KP s
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D
ynamicsofClosed
-LoopControlSystems
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2
(11-10) 1
v
t v
W s K
P s s
Assume that the valve can be modeled as
Control Valve
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D
ynamicsofClosed
-LoopControlSystems
[psi]
The valve dynamics is generally nonlinear =>
approximated by a linear 1st-order model in the vicinity
of the nominal operating condition
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Combining the block diagrams for individual components
obtains the overall block diagram of the feedback control
system as follows:10
D
ynamicsofClosed
-LoopControlSystems
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Block Diagram
Basic elements in a block diagram:
Arrowindicates flow of information, e.g. p = Ge = G(r - c).
Circlerepresents algebraic relation of the input arrows,
e.g. e = r c.
Blockrepresents the relevant dynamics (by transfer
function model) between the input and output.
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D
ynamicsofClosed
-LoopControlSystems
Gr +
-
p
p
e
c
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Some basic rules
1. Y = A B C
2. Y = G1G2A G1 G2A Y
G2 G1A Y
G1G2A Y
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D
ynamicsofClosed
-LoopControlSystems
A +
-
+
-
B C
YA - B
B -
-
+
+
C A
Y- B - C
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3. Y = G1(A B)
G1A
B
Y+-
G1
G1
YA
B
+
-
4. Y = (G1+G2)A
G1
G2
A Y+
+
G1+G2A Y
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D
ynamicsofClosed
-LoopControlSystems
G2 G1/G2A+
+
Y
G2A G1A
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D
ynamicsofClosed
-LoopControlSystems
This is a general diagram that can be used to represent
a wide variety of practical control problems.
Gp Effect of manipulated variable on the controlled variable
Gd Effect of load variable on the controlled variable
Block Diagram for Closed-loop System
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Standard symbols
G = transfer function
(subscripts c, v, p, d, m = controller, valve, process,
disturbance, and measurement respectively)
Y - process output D - disturbance or load variable
Ym - measured output Ysp - set-point
sp - internal set-point E - error
P - controller output U - manipulated variable
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D
ynamicsofClosed-LoopControlSystems
Note
Each variable in the figure is the Laplace transform of a deviationvariable.
For simplicity, the primes and s have been omitted; thus Ymeans Y(s).
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D
ynamicsofClosed-LoopControlSystems
Closed-Loop Transfer Functions
The objective is to find the transfer functions between
the inputs (Ysp and D) and the output (Y) of the closed-loop or feedback control system.
For the process input,
)()~
( YGYKGGYYGGEGGPGU mspmcvmspcvcvv
Process output is obtained as
DGUGY dp
From the above two equations,
DGYGYKGGGY dmspmcvp )(
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Rearranging,
DGGGG
GYGGGG
KGGG
Ymcvp
dsp
mcvp
mcvp
11
Effect of Yspon Y Effect o f D on Y
(11-30)
Eq. 11-30 illustrates the important role of Laplace Transformin the analysis of feedback control system.
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D
ynamicsofClosed-LoopControlSystems
Ysp Y+
Gd
1+GcGvGpGmD
+
GcGvGpKm
1+GcG
vG
pG
m
Closed-loop dynamics can now be
given in an open-loop manner
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Two Control Problems
1. Servo problem (set-point change)
This is equivalent to Ysp 0 (set-point change) and D = 0(disturbance change is zero), from the last equation
mcvp
mcvp
sp GGGG
KGGG
Y
Y
1
(11-26)
(closed-loop transfer function for set point change)
2. Regulator problem (disturbance change)
mcvp
d
GGGG
G
D
Y
1(11-29)
In this case, D 0 and Ysp = 0 (constant set-point):
(closed-loop transfer function for load change)
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ynamicsofClosed-LoopControlSystems
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Remarks
1. Closed-loop transfer functions (eqs. 11-26 and 11-29)
depend on dynamics of process, measurement device,
controller and control valve.
3. Overall transfer function = (product of transfer functions inthe forward path)/(1 + product of all transfer functions in the
feedback loop).
2. Denominator for both transfer functions, eqs. 11-26 and 11-29,
is the same => (1 + product of all the transfer functions in the
feedback loop), i.e. (1+GpGvGcGm).
4. For simultaneous changes in disturbance and set-point (i.e.,
D 0 and Ysp 0), eq 11-30 holds => overall response is the
sum of the individual response.
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D
ynamicsofClosed-LoopControlSystems
Ysp
Y+
Gd
1+GcGvGpGmD
+GcGvGpKm
1+GcGvGpGm
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1
c v p m
sp c v p m
G G G K Y
Y G G G G
Negative feedback
1d
c v p m
GY
D G G G G
Forward path from Ysp to Y
Forward path from D to Y
Product of all transfer
functions in the loop
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D
ynamicsofClosed-LoopControlSystems
Remark 3
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1
c v p m
sp c v p m
G G G K Y
Y G G G G
Negative feedback
1
p
c v p m
GY
D G G G G
Forward path from Ysp to Y
Forward path from D to Y
Product of all transfer
functions in the loop
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D
ynamicsofClosed-LoopControlSystems
Consider this feedback system,
+
-
Gc
Gm
Y+ Gp
D
+
GvYsp Km
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Analysis and Design Problems
Analysis Given particular Gp, Gv, Gc, Gm,
- Is the closed-loop system stable?
- Speed of response? oscillatory
response ?
Design Given Gp, Gv, Gm, and Gd, design Gc toachieve requirements like:
- Closed-loop stability;
- Closed-loop dynamics are sufficiently fast andsmooth (without excessive oscillations);
- Y/Ysp has an (ideal) gain of?? and Y/D has an
(ideal) gain of??
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ynamicsofClosed-LoopControlSy
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Effect of Proportional Control on Closed-Loop Response
Consider 1st-order process and P-controller.
Then1
s
KG
p
p 1
s
KG dd
cc KG
For simplicity, let andvv KG mm KG
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ynamicsofClosed-LoopControlSy
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From eq. 11-30,
DKKKKs
KY
KKKKs
KKKKY
mpvc
dsp
mpvc
mpvc
11
Ds
KY
s
Ksp
11 1
2
1
1
mpvc KKKK
11
mpvc
mpvc
KKKK
KKKKK
11
mpvc
d
KKKK
KK
12
Decreases with increasing KcAlways < , i.e., CL response is
faster than OL response
1 unless Kc =
Always < 1
0 unless Kc =
Always < Kd
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ynamicsofClosed-LoopControlSy
stems
Note:
CL system is 1st order with time
constant 1. For both TFs, time
constant is the same, but gain is
different
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Servo problem
Step change of magnitude M in set-point, i.e., Ysp = M/s
and D = 0.
Then from the last eq.s
M
s
KY
11
1
Inverting, )1()( 1/
1
teMKty
mpvc KKKK
M
1
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ynamicsofClosed-LoopControlSy
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Less than the desired
value M.
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Offset = current set-point (final value of the response)
mpvcmpvc
mpvc
KKKK
M
KKKK
KKKMKMMKM
111=
As t , output response neverreaches new set-point.The discrepancy is called (steady-state) offset.
Offset decreases with increasing Kc.
This is the characteristic of P-control.
Theoretically, offset when Kc .
Can it happen in practice? If not, why?
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ynamicsofClosed-LoopControlSy
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Regulator problem
In this case, D = M/s Ysp = 0 and Ds
KY
11
2
Inverting, )1()( 1/
2
t
eMKty
0
M D(t)
y(t)
no control (Kc=0)KdM
Time0
K2Mwith control
offset
Offset = current set-point (final value of the response)
mpvc
d
KKKK
MKMK
10 2
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ynamicsofClosed-LoopControlSy
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Kc offset
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Proportional-Integral Control for Disturbance Change
In this case, )1
1(
s
KG
I
cc
)1()1()
11(
11
1
sKKKKss
sK
ss
KKKKs
K
D
Y
ImpvcI
Id
I
mpvc
d
Rearrange
12 332
3
3
ss
sK
D
Y
where
I
mpvc
mpvc
mpvc
I
mpvc
Id
KKKK
KKKK
KKKKKKKK
KK
15.0,,
333
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ynamicsofClosed-LoopControlSy
stems
Gain = ?
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Remarks
1. Integral action eliminates offset, ysp() y() = 0
= 1, Kp = 1, Kv= 1
I
= 0.25 Kc= 3.5
y(t) y(t)
2. For Kc or I response speeds up
3. For Kc response more oscillatory (unexpected)
5. Note In general, closed loop response becomes more oscillatory
as Kc. The anomalous result above is due to neglected valveand measurement dynamics. When these two dynamics are
included, the TF is no longer 2nd-order.
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ynamicsofClosed-LoopControlSy
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4. ForI response more oscillatory (expected)
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Proportional-Integral Control for Set Point Change
(1 1 / ) / ( 1)
1 (1 1 / ) / ( 1)
c p v m I
sp c p v m I
K K K K s sY
Y K K K K s s
For this case,
Standard form
12
1
33
22
3
ss
s
Y
Y I
sp
3, 3 as defined earlier
For a unit step change in Ysp
Gain = ?
Offset = ?
2nd order dynamics
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ynamicsofClosed-LoopControlSy
stems
Kc= 1, Kp= 1
Kv= 1, Km= 1
I= 1, = 1
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Effect of Measurement Lag
As before let
1
sKG pp 1
sKG dd
cc KG 1 vv KG,,,
Assume significant measurement lag:1
1
sG
m
m
Resulting feedback control system:
1
s
KG dd
Kc1
sKG
p
p
1
1
sG
m
m
+ +
+
-
Ysp Y
D
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ynamicsofClosed-LoopControlSy
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Ym
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Consider set point change,
12
)1(
)1)(1(1
1
55
22
5
5
ss
sK
ss
KKs
KK
Y
Y m
m
pc
pc
sp
5 ,1
c p
p c
K KK
K K ,15
cp
m
KK
cpm
m
KK
1
1
2
5
where and
Response may be oscillatory depending on the choice of
, m, Kp and Kc. One possibilityis as follows:
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ynamicsofClosed-LoopControlSy
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Measurement lag produces poorer transients
Remarks
Offset results from the use of P controller
Kp = 1
=1Kc = 8
Ysp = 1/sy(t)
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ynamicsofClosed-LoopControlSy
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S
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When we place chemical process in a closed-loop with
sensors, transducers, valves and controllers, we have a more
complicated system than the original process. Nevertheless,
our analysis shows that a closed-loop system can be written
as one single transfer function.
Thus we recognize that a block diagram provides a
convenient representation for analyzing control systems.
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ynamicsofClosed-LoopControlSy
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Summary
We understand the key features of P and PI control using
simple first-order processes.
We notice that measurement dynamics can cause
deterioration in the control system performance.
Further reading: Chapter 10.1 to 10.3, SEM