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Transfer Functions
What is transfer function?-is an algebraic expression for dynamic relation between a selected input and output of the process model-can only be derived for a linear model because Laplace Transform can be applied only to linear equations.
What is Laplace transforms?
-method to convert the ODE (representing the dynamicbehaviour of process output variables) into algebraic equations-integral transforms that transforms a function of time [ f(t) ] into a function of complex variables s, F(s)
• Consider the tank heating system shown in Figure 1. A liquid enters
the tank with a flowrate Fi (ft3/min), and a temperature Ti, where it is heated with steam (having a flowrate Q, lb/min). Let F and T be the flowrate and temperature of the stream leaving the tank. The tank is considered to be well stirred, which implies that the temperature of the effluent is equal to the temperature of the liquid in the tank.
• The control objective of this heater is to keep the effluent temperature T at a desired value. Draw a P& ID and block diagram for this control system.
Unit 1: Process Control Loop
Process control loop
I/PProcess
Sensor
Transmitter
Controller
Transducer
Control valve
4-20 mA1-5 Vdc
PIDFuzzy logic
4-20 mA3-15 psig
dP cellCapacitanceRadar, SonicMagneticResistanceIR/Laser
PressureFlowLevelTemperaturepH
LinearEqual percentage
© Abdul Aziz Ishak, Universiti Teknologi MARA Malaysia (2009)
Unit 1: Process Control Loop
Process
Sensor
Transmitter
Controller
Transducer
Control valve
PressureFlowLevelTemperaturepH
dP cellCapacitanceRadar, SonicMagneticResistanceIR/Laser
4-20 mA1-5 VdcField/profibus
PIDFuzzy logic
4-20 mA3-15 psig
LinearEqual percentage
CONTROLLER
CONTROLVALVE
PROCESS
TRANSMITTER
PV
SP
Process control loop
© Abdul Aziz Ishak, Universiti Teknologi MARA Malaysia (2009)
*SP = set point*PV = process value
Unit 1: Process Control Loop
CO
NTR
OLL
ER
CO
NTR
OL
VALV
E
PR
OC
ES
S
PVSP
Simulation mode
Process control loop: The Block Diagram
© Abdul Aziz Ishak, Universiti Teknologi MARA Malaysia (2009)
TRANSMITTER
• Convenient representation of a linear, dynamic model.• A transfer function (TF) relates one input and one output:
systemx t y tX s Y s
The following terminology is used:x
input
forcing function
“cause”
y
output
response
“effect”
Cha
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4Transfer Functions
The TF model enables us to determine the output response to any change in an input.
Type of forcing function: Step, Impulse, Ramp, SinusoidalType of response :Step, Impulse, Ramp, Sinusoidal
Credit: Process Systems Analysis and Control, Coughanowr and Leblanc
Definition of the transfer function:Let G(s) denote the transfer function between an input, x, and an output, y. Then, by definition
where:
Y sG s
X s
Y s y t
X s x t
LL
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4
Figure 2.3 Stirred-tank heating process with constant holdup, V.
Page 27
Cha
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4Example: Stirred Tank Heating System
Transfer Functions for a Process
Equation (1) is the energy balance of the stirred-tank heating system, assuming constant liquid holdup and flow rates:
(2-36)idTV C wC T T Qdt
Suppose the process is at steady state:
0 (2)iwC T T Q
Subtract (2) from (1):
(3)i idTV C wC T T T T Q Qdt
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4
(1)
But,
(4)idTV C wC T T Qdt
where the “deviation variables” are
, ,i i iT T T T T T Q Q Q
0 (5)iV C sT s T wC T s T s Q s
Take L of (4):
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4
At the initial steady state, T′(0) = 0.Because T’(0) = T-Ts= Ts- Ts =0
Rearrange (5) to solve for
1 (6)1 1 i
KT s Q s T ss s
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4
where
1 and VKwC w
0 (5)iV C sT s T wC T s T s Q s
(s)T(s)G(s)Q(s)(s)=GT i 21
1 (6)1 1 i
KT s Q s T ss s
(s)T(s)G(s)Q(s)(s)=GT i 21
1 (6)1 1 i
KT s Q s T ss s
K (gain) – it describes how far the output will travel with the change of the input.
Gain, K = Δ y Δ x*If a process has a large K, then a small change in the input will cause the output to move a large amount.
If a process has a small K, the same input change will move the output a small amount
50= 100 20
0.05= 1 20
MV
Credit: controlguru.com
(time constant) – describes how fast the output moves in response to a change in the input
Large values of mean a slow process responseSmall values of mean a fast response
*The time constant must be positive and it must have units of time
MV
Credit: controlguru.com
Cha
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4Order of transfer function
)(1
)( sXsKsY
• General first order transfer function
• General second order transfer function
)(12
)(22
sXss
KsY
)(1
)(0
sXs
KesYst
• First-order-plus-dead-time (FOPDT)
aka first-order lag and single exponential stage
LAPLACE TRANSFORM • integral transforms that transforms a function of time, f(t)
into a function of complex variables s, F(s)
We often abbreviate this to
In short,
The step / unit step function, S(t)
f(t) = S(t)
*Laplace transform only work in the range of 0<t<∞
So, S(t)=0 is not taken as t<0
Thus, S(t)= 1 when t>0.
L S(t) = L {1} =
First Order Derivative
Higher Order Derivative
• a new variable, is introduced.
Eg 1.
Eg 2.
Exponential Functions
Ramp FunctionsEg 1.
What is Response with Time Delay/Dead Time?
X(t) Y(t)
ΔX= Xf – X0ΔY= Yf – Y0
X(t) Y(t)
• In a process when MV changes, the PV only responds after a certain time has elapsed.
• This elapsed time is known as the dead time/time delay. • Similarly, the response of the PV is delayed when the MV changes
back
X0 Y0
X(t)=MVY(t)=PV
XfYf
X(t)=MVY(t)=PV
X(t)
Y(t)
• A typical example here is a belt conveyor, where there is a certain time delay before a change in the chute feed rate is recorded at the measurement location.
• Another example is when someone tell you a joke/explanation, only after a while you crack a laugh/ understand.
Do you get it what it meant by Response with Time Delay/Dead Time??
• The relationship between process value x and manipulating variable y is as follows:
Gain, K = Δ y Δ x
Response with time delay
to=Time delay/dead time
X(t)
Y(t)
t=0 t=t0
• All first order systems forced by a step function ( a type MV function) will have a response of this same shape.
Step response for first order system
ΔY= Yf – Y0 = ???
To calculate the gain and time constant from the graph
xyK
Gain,
Time constant, – value of t which the response is 63.2% complete
= ?? 10
1. A sensor measures and transmits the current value of the process variable, PV, back to the controller 2. Controller error at current time t is computed as set point minus measured process variable, or e(t) = SP – PV3. The controller uses this e(t) in a control algorithm to compute a new controller output signal, CO4. The CO signal is sent to the final control element (e.g. valve, pump, heater, fan) causing it to change5. The change in the final control element (FCE) causes a change in a manipulated variable6. The change in the manipulated variable (e.g. flow rate of liquid or gas) causes a change in the PV
The goal of the controller is to make e(t) = 0 in spite of unplanned and unmeasured disturbances. Since e(t) = SP – PV, this is the same as saying a controller seeks to make PV = SP.
Transfer Functions for a Transmitter
KT = transmitter gain
H(s)Process variable
PV(s)
Transmitter output
C(s)
1K
)()()( T
ssPVsCsH
T
T = transmitter time constant
For proportional control, the controller output is proportional to the error signal,
(8-2)cp t p K e t
where: controller output
bias (steady-state) valuecontroller gain (usually dimensionless)c
p t
pK
Proportional Control
is controller output when the error is zero
-The proportionality is given by the controller gain, Kc-The controller gain determines how much the output from the controller changes for a given change in error
p
Transfer Functions for a Controller
Transfer function
LaplaceTransform
2. Introduce deviation variable, into
1
1
1.
3.
How to obtain Transfer Function for a Proportional Controller?
For process with self-regulation,
Examples of proportional processes are:- mechanical gearing without slip- mechanical transmission by lever- transistor (collector current Ic
follows the base current IB with virtually no delay)
46
Integral ControlFor integral control action, the controller output depends on the integral of the error signal over time,
0
1 * * (8-7)τ
t
Ip t p e t dt
where , an adjustable parameter referred to as the integral time or reset time, has units of time.
τI
• Integral control action is normally used in conjunction with proportional control as the proportional-integral (PI) controller :
0
1 * * (8-8)τ
tc
Ip t p K e t e t dt
Function of integral term:While the proportional term considers the current size of e(t) only at the time of the controller calculation;The integral term considers the history of the error/how long or how far the PV has been (away) from the SP over time
E(t) continually changes size and sign as time passes
Integral of each shaded portion has the same sign as the error.
The total integral sum grows as long as e(t) is positive andshrinks when it is negative.
It calculates accumulation of error over time.
τI• The PI controller has two parameters, Kc and
Transfer function
11.
2. Introduce deviation variable: p’(t) = into 1
3. p’(t) =
How to obtain Transfer Function for a Proportional-Integral Controller?
Proportional-Integral processes usually found in aircraftand liquid level in tank. Rowing a kayak, canoe, sampan byyour hand roughly mimicks the PI controller
Derivative Control
• The function of derivative control gives the controller the capability to anticipate where the process is heading by calculating the derivative error
• Thus, for ideal derivative action,
τ (8-10)D
de tp t p
dt
where , the derivative time, has units of time.τD
Transfer function
• controller output is equal to the nominal value p as long as the error is constant (that is, as long as de/dt = 0).
• Derivative action is based on how rapidly the error is changing, not the magnitude of the error or how long the error has persisted. It is based on the slope of the error
versus time curve at any instant in time.
• Therefore, a rapidly changing error signal will induce a large derivative response. “Noisy” error signals cause significant problems for derivative action because of the rapidly changing slope of the error caused by noise.
• Derivative control should be avoided in these situations unless the error signal can be filtered to remove the noise.
For example, suppose that a reactor temperature increases by 10 oc in a short period of time, say, 3 min. This clearly is a more rapid increase in temperature than a 10 oc rise in 30 min, and it could indicate a potential runaway situation for an exothermic reaction. If the reactor were under manual control, an experienced plant operator would anticipate the consequences and quickly take appropriate corrective action to reduce the temperature. Such a response would not be obtainable from the proportional and integral control modes discussed so far. Note that a proportional controller reacts to a deviation in temperature only, making no distinction as to the time period over which the deviation develops. Integral control action is also ineffective for a sudden deviation in temperature, because the corrective action depends on the duration of the deviation
Transfer function for Valve
Proportional-Integral-Derivative (PID) Control
Now we consider the combination of the proportional, integral, and derivative control modes as a PID controller.
Form of PID Control
The form of the PID control algorithm is given by
0
1 * * τ (8-13)τ
tc D
I
de tp t p K e t e t dt
dt
The corresponding transfer function is:
Transfer function