1. UNIVERSITÀ DEGLI STUDI DI SALERNO
Bachelor Degree in Chemical Engineering
Course: Process Instrumentation and Control
(Strumentazione e Controllo dei Processi Chimici)
Input-Output and Input-State-Output
Mathematical Models
Rev. 3.2 – April 9, 2019

2. QUI ANDRA’ IL COLLEGAMENTO CON IL GLOSSARIO
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Process Instrumentation and Control - Prof M. Miccio

3. FIRST PRINCIPLE MODELS
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Process Instrumentation and Control - Prof M. Miccio

4. FIRST PRINCIPLE MODELS
DEFINITION
First Principle Models (FPMs) are those based on fundamental engineering, physics and chemistry principles, in contrast, for example, to empirical mathematical or statistical correlations between input and output variables derived from plant or other data
 Pantelides CC, Renfro JG, 2013, The online use of first-principles models in process operations: Review, current status and future needs, Computers & Chemical Engineering, Vol:51, ISSN: , Pages:
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Process Instrumentation and Control - Prof M. Miccio

5. PARTICULAR CASES for process engineering:
GENERAL BALANCE LAW
Conservation Law of an Entity
[IN] – [OUT] + [GEN] = [ACC]
PARTICULAR CASES for process engineering:
Steady-State:
[IN] – [OUT] + [GEN] = 0
NB: GEN > 0  formation GEN < 0  disappearance
Absence of the term GEN (e.g. in the absence of chemical reactions):
[IN] – [OUT] = [ACC]
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Process Instrumentation and Control - Prof M. Miccio

6. REFERENCE EXAMPLE: THE WATER OPEN TANK
Process Diagram Block Diagramm
h(t)
Dynamic
Model
Input / Output
FLOWRATE = DRIVING FORCE/RESISTANCE
GENERAL CASE :
Torricelli’s Law :
LINEAR CASE :
see:
§ Stephanopoulos, “Chemical process control: an Introduction to theory and practice”
22/08/2019
Process Instrumentation and Control - Prof M. Miccio

7. DYNAMIC SYSTEMS WITH INPUT / OUTPUT REPRESENTATION
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Process Instrumentation and Control - Prof M. Miccio

8. INPUT/OUTPUT REPRESENTATION (scalar case)
see:
Ch.1 - Stephanopoulos,
“Chemical process control: an Introduction to theory and practice”
Input output
where the scalar variables are:
y≠y(t)
i≠i(t)
Steady-state
Mathematical
Model i y
where:
f(t) = Forcing Function
y(t) = Dynamic Response
Dynamic
Mathematical
Model
DEFINITION
Forcing Function:
Variable that represents an input to the model and that is a function of time; as such, it stimulates or "disturbs" the system, causing its
dynamic response
f(t)
y(t)
Dynamic
Mathematical
Model
in Laplace domain
22/08/2019
Process Instrumentation and Control - Prof M. Miccio

9. SISO and MIMO DYNAMIC MODELS
SISO MODEL:
SINGLE INPUT – SINGLE OUTPUT
see:
Ch.1 - Stephanopoulos,
“Chemical process control: an Introduction to theory and practice”
DYNAMIC
MATHEMATICAL
MODEL
f(t)
y(t)
MIMO MODEL:
MULTIPLE INPUT – MULTIPLE OUTPUT
Vector representation
DYNAMIC
MATHEMATICAL
MODEL
where:
y(t)
f(t)
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Process Instrumentation and Control - Prof M. Miccio

10. DYNAMIC SYSTEMS WITH INPUT / STATE / OUTPUT REPRESENTATION
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Process Instrumentation and Control - Prof M. Miccio

11. INPUT / STATE / OUTPUT REPRESENTATION
scalar
case
MATHEMATICAL MODEL
Input state output
The state is an intermediate quantity between input and output.
The state represents the "internal" functioning of the system, and provides its knowledge.
vectorial
case
 From Giua & Seatzu “Analisi dei sistemi dinamici”, 2006
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Process Instrumentation and Control - Prof M. Miccio

12. REFERENCE EXAMPLE: THE WATER OPEN TANK
Process Diagram Block Diagramm
h(t)
Dynamic
Model
Input / Output
FLOWRATE = DRIVING FORCE/RESISTANCE
GENERAL CASE :
Torricelli’s Law :
LINEAR CASE :
see:
§ Stephanopoulos, “Chemical process control: an Introduction to theory and practice”
22/08/2019
Process Instrumentation and Control - Prof M. Miccio

13. VARIABLES: DEFINITION
FARE IL COLLEGAMENTO CON IL GLOSSARIO
VARIABLES: DEFINITION
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Process Instrumentation and Control - Prof M. Miccio

14. Process Instrumentation and Control - Prof M. Miccio
VARIABLES: 1st CLASSIFICATION depending on the meaning assumed in the dynamic models
INPUT
They supply the model with values of quantities coming from outside or on which
the designer acts to make the output variables assume the desired trend.
OUTPUT
They represent the values calculated for unknown functions or the quantities supplied to the external environment and of interest to the designer.
STATE
They characterize the dynamic behavior of the system
They store the past history of system inputs
They focus on themselves the knowledge of the past and present of the system
They may in whole or in part coincide with the output variables
SUGGESTIONS:
The state is often identified by the variables that appear inside the I.C. of the dynamic model.
In general, the choice of state variables is not unique.
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Process Instrumentation and Control - Prof M. Miccio

15. Process Instrumentation and Control - Prof M. Miccio
VARIABLES: 2nd CLASSIFICATION depending on the properties and the role they have in the process control
The input variables are classifiable as:
Manipulated or Adjustable Variables (“Variabili di Controllo” in the nomenclature of Magnani) (VM, i): they are the variables that can be adjusted towards the proper values by the process control system (controller, computer and the human operator)
Disturbances (D, d): they are the variables that cannot be determined either by an operator or a control system. They can be Measured and Unmeasured
Process
INPUT VM
(i)
MEASURED
UNMEASURED
DISTURBANCES (d)
see:
Ch. 2 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”
22/08/2019
Process Instrumentation and Control - Prof M. Miccio

16. Contrasting attribute
VARIABLES: 2nd CLASSIFICATION depending on the properties and the role they have in the process control
The Output Variables can be also classified into the following two categories:
The Controlled Variables ( VC, y ) are the output variables for which design and operating specifications have to be fulfilled, and are those submitted to the AUTOMATIC CONTROL
Model attribute
Contrasting attribute
Measured
Unmeasured
Controlled
Uncontrolled
Process
OUTPUT VARIABLES
(VC, y)
CONTROLLED
UNCONTROLLED
see:
Ch. 2 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”
22/08/2019
Process Instrumentation and Control - Prof M. Miccio

17. MIMO System PROCESSING SYSTEM EXTERNAL DISTURBANCES see:
MEASURED (d)
UNMEASURED (d’)
MANIPULATED
VARIABLES
(m)
MEASURED
OUTPUTS
(y)
UNMEASURED OUTPUTS (z)
see:
Ch.2 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”
22/08/2019
Process Instrumentation and Control - Prof M. Miccio

18. Process Instrumentation and Control - Prof M. Miccio
DEVIATION VARIABLE
The deviation variable is defined as a new variable
(indifferently for Input, State, Output)
calculated as the difference between the current value and the steady state value.
E.g., for the output:
see:
§ Stephanopoulos, “Chemical process control: an Introduction to theory and practice”
22/08/2019
Process Instrumentation and Control - Prof M. Miccio

19. THE ADVANTAGES OF USING THE DEVIATION VARIABLES
PROBLEM!
When we pass to the Laplace domain, the derivation theorem introduces constant values that are not equal to zero value. 
SOLUTION!
We introduce the deviation variables as a change of variables putting the initial condition as null.
When we pass to the Laplace domain in the successive step, the derivation theorem generates constant values equal to zero value.
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Process Instrumentation and Control - Prof M. Miccio

20. Process Instrumentation and Control - Prof M. Miccio
SYSTEMS DYNAMIC
Variation in the input variables of a process induce changes in the internal state of a system
The system response (how and when) depends on:
from the nature of the changed input;
from the intrinsic nature of the process.
The study of the dynamic response of the process to variations in inputs allows us to obtain useful information on its nature. The reverse is also true: knowing the nature of a process it is possible to predict its dynamic behavior (response) to changes in individual inputs.
Systems dynamics is the discipline that studies the dynamic behavior of systems by providing the necessary tools to predict their response following changes in one or more inputs.
The design of a control system cannot ignore a good knowledge of the dynamics of the process in question.
Need for a model of the process
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Process Instrumentation and Control - Prof M. Miccio