1. UNIVERSITÁ DEGLI STUDI DI SALERNO FACOLTÀ DI INGEGNERIA
Bachelor Degree in Chemical Engineering
Course:
Process Instrumentation and Control
(Strumentazione e Controllo dei Processi Chimici)
REFERENCE LINEAR DYNAMIC SYSTEMS
FIRST-ORDER PLUS DEAD TIME MODEL
Rev. 3.5 – May 17, 2019

2. First-Order Plus Dead Time (FOPDT) MODEL
Complex or unknown
process
f(t)
y(t)
Dead time
First order process
f(t)
w(t)=f(t-td)
y(t)
 This block representation provides a "frozen" picture of the dynamical system at a given time t
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Process Instrumentation and Control - Prof. M. Miccio

3. Modeling Dynamic Process Behavior by FOPDT
The best way to understand process data is through modeling
Modeling means fitting a First Order Plus Dead Time (FOPDT) dynamic process model to the data set:
where:
y(t) is the measured process variable
u(t) is an input variable (e.g., the controller output signal in “manual mode”)
The important parameters that result are:
Steady State Process Gain, KP
Overall Process Time Constant, P
Apparent Dead Time, P or td
The FOPDT model is low order and linear: so it can only approximate the behavior of real processes
adapted from:
Cooper D. (2008), "Practical Process Control using Loop-Pro Software", PDF textbook
20/09/2019
Process Instrumentation and Control - Prof. M. Miccio

4. First Order Plus Dead Time (FOPDT) MODEL
Complex or unknown
process
f(t)
y(t)
L[y(t)]
20/09/2019
Process Instrumentation and Control - Prof. M. Miccio

5. First Order Plus Dead Time (FOPDT) MODEL in the Laplace domain
 The FOPDT model is low order and linear: so it can only approximate the behavior of real processes
 Cooper D. (2008), "Practical Process Control using Loop-Pro Software", PDF textbook 
20/09/2019
Process Instrumentation and Control - Prof. M. Miccio

6. Method of Process Reaction Curve (STEP TEST)
Modeling process data through fitting a First Order Plus Dead Time (FOPDT) dynamic process model
see:
§ 7.4 of Magnani, Ferretti e Rocco (2007)
§ 16.5 of Stephanopoulos, “Chemical Process Control: an Introduction to Theory and Practice”
Ch. 3 “Modeling Process Dynamics - A Graphical Analysis of Step Test Data” in Cooper D. (2008), "Practical Process Control using Loop-Pro Software", PDF textbook
20/09/2019
Process Instrumentation and Control - Prof. M. Miccio

7. Two cases:
Self-regulating process
Non self-regulating process

8. Step Test Data and Dynamic Process Modeling
Process (real or simulated) starts at steady state
An input variable (e.g., the “controller output” in manual mode) is stepped to new value
The measured process variable is recorded and allowed to complete response

9. Meaning of Process Reaction Curve
Cohen e Coon (1953) used an approximate model and estimated values of parameters K, td, τ, as shown in figure:
see:
par di Stephanopoulos, “Chemical process control: an Introduction to theory and practice”
20/09/2019
Process Instrumentation and Control - Prof. M. Miccio

10. Two cases:
Self-regulating process
Non self-regulating process

11. PROCESS REACTION CURVE for Non-Self Regulating Processes
The dynamic response to the input step change is a straight line
The slope of the response line is Kp*
Example of a dynamic response to a step input change for a tank with an output flowrate withdrawn by a pump
20/09/2019
Process Instrumentation and Control - Prof. M. Miccio

12. Integrating FOPDT model
It is applied when the dynamic response does not reach a new value at steady state.
 an integrating FOPDT model is adopted
FOPDT integrating model has two parameters:
Steady State Process Gain, KP*
Apparent Dead Time, P or td
The two parameters are determined as follows:
Kp* as the angular coefficient of the response line.
td as time elapsed between the instant of the step and the first successive instant in which the dynamic response occurs.
20/09/2019
Process Instrumentation and Control - Prof. M. Miccio

13. EVALUATION CRITERION FOR FOPDT MODEL GOODNESS-OF-FIT
METHOD of the SUM OF SQUARED ERRORS (SSE)
Measured or experimental value
Predicted value at the same time with the FOPDT model
20/09/2019
Process Instrumentation and Control - Prof. M. Miccio

14. FOPDT MODEL ProcedurE for the CALCULATION OF PARAMETERS
SOFTWARE PROCEDURE
GRAPHICAL PROCEDURE
ANALYTICAL PROCEDURE
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Process Instrumentation and Control - Prof. M. Miccio

15. FOPDT MODEL ProcedurE for the CALCULATION OF PARAMETERS
SOFTWARE PROCEDURE
GRAPHICAL PROCEDURE
ANALYTICAL PROCEDURE
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Process Instrumentation and Control - Prof. M. Miccio

16. Process Instrumentation and Control - Prof. M. Miccio
“Automatic” approximation to a first-order system and parameters estimation with the software LOOP-PRO
Available for response to the step test
for self-regulating system
for NON-self-regulating system
for NON-inflected response
for inflected response
It requieres a file to record data of the response curve
It will be used for the students' Project Work
Download example files from:
20/09/2019
Process Instrumentation and Control - Prof. M. Miccio

17. FOPDT MODEL ProcedurE for the CALCULATION OF PARAMETERS
SOFTWARE PROCEDURE
GRAPHICAL PROCEDURE
ANALYTICAL PROCEDURE
20/09/2019
Process Instrumentation and Control - Prof. M. Miccio

18. Process Gain from Step Test Data
KP describes how much the measured process variable, y(t), changes in response to changes in the controller output, u(t)
A step test starts and ends at steady state, so KP can be computed from the plot data:
where:
u(t) and y(t) represent the total change from initial to final steady state
Δ = (final value - initial value)
A large process gain means the process will show a big response to each control action
 Usually, KP is not dimensionless

19. KP for Gravity-Drained Tanks
EXAMPLE:
KP for Gravity-Drained Tanks
Steady state process gain has:
size (0.1)
sign (+0.1)
units (m/%)

20. Time Constant and Dead Time from Step Test Data
They are computed from the plot of the measured process variable, y(t)
The calculation procedure may vary depending on
the y(t) plot presents or doesn’t present an inflection point
A software is used to minimize the model fitting error
Calculation is graphically carried out directly on the plot
Calculation is carried out by analytically using some data points on the plot 21

21. Process Instrumentation and Control - Prof. M. Miccio
TIME CONSTANT and DEAD TIME from data of the step test Case N. 1 The curve responde does not show inflection point to the step test
20/09/2019
Process Instrumentation and Control - Prof. M. Miccio

22. “Manual” approximation to the first-order lag Only graphical procedure
 The curve response is monotonic but it does not present INFLECTION POINT.
Characteristic times of the FOPDT model are determined by the plot of the tangent line to the initial point of the curve response to the step test.
20/09/2019
Process Instrumentation and Control - Prof. M. Miccio

23. Process Instrumentation and Control - Prof. M. Miccio
TIME CONSTANT and DEAD TIME from data of the step test Case N. 2 The curve response shows an inflection point to the step test (“sigmoidal” curve)
20/09/2019
Process Instrumentation and Control - Prof. M. Miccio

24.  The response curve shows an INFLECTION POINT.
“Manual” approximation to the first-order lag Graphic procedure
 The response curve shows an INFLECTION POINT.
Characteristic times of the FOPDT model are directly evaluated by the plot of the tangent line to the inflection point on the response curve to the step test.
Fig. taken from Magnani, Ferretti e Rocco (2007)
Controllability ratio: td/τp
20/09/2019
Process Instrumentation and Control - Prof. M. Miccio

25. MAXIMUM RESPONSE VELOCITY
Let us define another parameter:
MAXIMUM VELOCITY RESPONSE:
NOTE:
RR is not dimensionless
Δ = (final value – initial value)
see:
par. 7.4 di Magnani, Ferretti e Rocco (2007)
20/09/2019
Process Instrumentation and Control - Prof. M. Miccio

26. FOPDT MODEL ProcedurE for the CALCULATION OF PARAMETERS
SOFTWARE PROCEDURE
GRAPHICAL PROCEDURE
ANALYTICAL PROCEDURE
20/09/2019
Process Instrumentation and Control - Prof. M. Miccio

27. Process Instrumentation and Control - Prof. M. Miccio
“Manual” approximation to the first-order lag Analytical procedure based on the plot
 The response curve is monotonic but does not show INFLECTION POINT.
Characteristic times of the FOPDT model are estimated by graphic constructions adn analytical calculations on the response curve to the step test.
20/09/2019
Process Instrumentation and Control - Prof. M. Miccio

28. P for Gravity-Drained Tanks
Locate where the measured process variable first shows a clear initial response to the step change
call this time tYstart
From plot:
tYstart = 9.6 min

29. P for Gravity-Drained Tanks
Locate where the measured process variable reaches y63.2, or where y(t) reaches 63.2% of its total final change
Label time t63.2 as the point in time where y63.2 occurs

30. P for Gravity-Drained Tanks
y(t) starts at 1.93 m and shows a total change y = 0.95 m
y63.2 =
1.93 m (y) =
1.93 m (0.95 m) =
2.53 m
y(t) passes through 2.53 m at t63.2 = 11.2 min

31. P for Gravity-Drained Tanks
The time constant is the time difference between tYstart and t63.2
Time constant must be positive and have units of time
From plot:
P = t63.2  tYstart =
11.2 min  9.6 min =
1.6 min

32. Apparent Dead-Time from Step Test Data
P is the time from when the controller output step is made until when the measured process variable first responds
Apparent dead time, P, is the sum of these effects:
transportation lag, or the time it takes for material to travel from one point to another
sample or instrument lag, or the time it takes to collect analyze or process a measured variable sample
higher order processes naturally appear slow to respond
Notes:
Dead time must be positive and have units of time
Tight control in increasingly difficult as P  P
For important loops, work to avoid unnecessary dead time

33. P for Gravity-Drained Tanks
Dead-Time
P =
tYstart  tUstep =
9.6 min  9.2 min =
0.4 min

34. Process Instrumentation and Control - Prof. M. Miccio
“Manual” approximation to the first-order lag Analytical procedure using areas
HYPOTHESES:
Monotonic sigmoidal response to the step test
Initial and final equilibrium conditions
y, u y
NOTE:
in Magnani, Ferretti e Rocco (2007)
t  td
T  t A0
Dy0 A1 u
Du0 t td tP
PROCEDURE:
Measurement (graphic or by means of integration) of the area A0
Measurement (graphic or by means of integration) ofthe area A1
Calculation of Kp= Dy0/ Du0
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Process Instrumentation and Control - Prof. M. Miccio

35. Process Instrumentation and Control - Prof. M. Miccio
“Manual” approximation to the first-order lag Analytical procedure using areas
If we approximate the response curve to a “real” first-order plus dead time, for which:
the following approximate formulas can be calculated:
td + tP is evaluated from A0 and Dy0. Afterwards, tP is calculated from A1
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Process Instrumentation and Control - Prof. M. Miccio