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

QUI ANDRA’ IL COLLEGAMENTO CON IL GLOSSARIO 22/08/2019 Process Instrumentation and Control - Prof M. Miccio

FIRST PRINCIPLE MODELS 22/08/2019 Process Instrumentation and Control - Prof M. Miccio

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:0098-1354, Pages:136-148 22/08/2019 Process Instrumentation and Control - Prof M. Miccio

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] 22/08/2019 Process Instrumentation and Control - Prof M. Miccio

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: §10.1 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” 22/08/2019 Process Instrumentation and Control - Prof M. Miccio

DYNAMIC SYSTEMS WITH INPUT / OUTPUT REPRESENTATION 22/08/2019 Process Instrumentation and Control - Prof M. Miccio

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

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) 22/08/2019 Process Instrumentation and Control - Prof M. Miccio

DYNAMIC SYSTEMS WITH INPUT / STATE / OUTPUT REPRESENTATION 22/08/2019 Process Instrumentation and Control - Prof M. Miccio

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 22/08/2019 Process Instrumentation and Control - Prof M. Miccio

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: §10.1 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” 22/08/2019 Process Instrumentation and Control - Prof M. Miccio

VARIABLES: DEFINITION FARE IL COLLEGAMENTO CON IL GLOSSARIO VARIABLES: DEFINITION 22/08/2019 Process Instrumentation and Control - Prof M. Miccio

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. 22/08/2019 Process Instrumentation and Control - Prof M. Miccio

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

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

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

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: § 6.3 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” 22/08/2019 Process Instrumentation and Control - Prof M. Miccio

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. 22/08/2019 Process Instrumentation and Control - Prof M. Miccio

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 22/08/2019 Process Instrumentation and Control - Prof M. Miccio