President UniversityErwin SitompulSMI 1/1 Dr.-Ing. Erwin Sitompul President University Lecture 1 System Modeling and Identification

President UniversityErwin SitompulSMI 1/2 Textbook: “Process Modelling, Identification, and Control”, Jan Mikles, Miroslav Fikar, Springer, Syllabus: Chapter 1:Introduction Chapter 2:Mathematical Modeling of Processes Chapter 3:Analysis of Process Models Chapter 4:Dynamical Behavior of Processes Chapter 5:Discrete Time Process Models Chapter 6:Process Identification Textbook and Syllabus

President UniversityErwin SitompulSMI 1/3 Control Chapter 1Introduction Control is the purposeful influence on an object (process) to ensure the fulfillment of a required objectives. The objectives can be to satisfy the safety and optimal operation of the technology, the product specifications under constraints of disturbance, process stability, and other technical related matters. Control systems in the whole consist of technical devices and human factor. Control systems must satisfy: Disturbance attenuation Stability guarantee Optimal process operation

President UniversityErwin SitompulSMI 1/4 Control Chapter 1Introduction There are two main methods of control: Feedback control, where the information about process output is used to calculate the control (manipulated) signal  process output is fed back to process input Feedforward control, where the effect of control is not compared with the desired result Practical control experience confirms the importance of assumptions about dynamical behavior of processes. This behavior is described using mathematical models of processes, which can be constructed from a physical or chemical nature of processes or can be abstract.

President UniversityErwin SitompulSMI 1/5 Chapter 1Introduction The purpose of this course is to learn how to model a process, which may have one of these objectives: Synthesis: Modeling as the fundamentals to influence a process through a controller Analysis: Modeling as the fundamentals to a deep understanding of the process and further to optimization of the process Simulation: Modeling as the fundamentals to emulative calculation under a given boundary condition Process is the entire activities where matter and/or energy are stored, transported, and converted; whereas information is stored, transported, converted, created, or destroyed. System is a part of a process, which is defined by the user, and has an interconnection with the environment regarding the flow of matter, energy, and information. Process, System, Model

President UniversityErwin SitompulSMI 1/6 Chapter 1Introduction Process, System, Model Model is an appropriate description of the flows of a system to its environment, using physical, chemical nature of the system, or using abstract mathematical equations.

President UniversityErwin SitompulSMI 1/7 Process Chapter 1An Example of Process Control A simple heat exchanger T i, q i w T o, q o VT Inlet Outlet V: volume of liquid in heat exchanger [m 3 ] q: Volume flow rate [m 3 /s] T: Temperature [K] w: Heat input [W] Assumptions: Ideal mixing No heat loss Constant heating rate Exchanger has no heat capacity T = T o

President UniversityErwin SitompulSMI 1/8 Steady-State Chapter 1An Example of Process Control A simple heat exchanger T i, q i w T o, q o VT Inlet Outlet Input variables: Inlet temperature T i Heat input w Output variables: Outlet temperature T The process is said to be in steady-state if the input and output variables remain constant in time. The heat balance in the steady-state is of the form: q: Volume flow rate [m 3 /s] ρ: Liquid specific density [kg/m 3 ] c p : Liquid specific heat capacity [J/(kgK)]

President UniversityErwin SitompulSMI 1/9 Process Control Chapter 1An Example of Process Control A simple heat exchanger T i, q i w T o, q o VT Inlet Outlet Control of the heat exchanger in this case means to influence the process so that T will be kept close to T w. This influence is realized with changes in w, which is called manipulated variable. A thermometer must be placed on the outlet of the exchanger and we may choose between manual control or automatic control. Input variables: Inlet temperature T i Heat input w Output variables: Outlet temperature T

President UniversityErwin SitompulSMI 1/10 Dynamical Properties of the Process Chapter 1An Example of Process Control In the case that the control is realized automatically, the knowledge about process response to changes of input variables is required. This is the knowledge about dynamical properties of the process, which is the description of the process in unsteady-state. The heat balance for the heat transfer process in a very short time interval Δt is given by:

President UniversityErwin SitompulSMI 1/11 Assuming q i = q o and T = T o, Dynamical Properties of the Process Chapter 1An Example of Process Control The heat balance in the steady-state may be derived from the last equation, in the case that dT/dt = 0.

President UniversityErwin SitompulSMI 1/12 In case of choosing an automatic control, the control device performs the control actions which is described in a control law. The task of a control device is to minimize the difference between T w and T, which is defined as control error. Suppose that we choose a controller that will change the heat input proportionally to the control error, the control law can be given as: Feedback Process Control Chapter 1An Example of Process Control We speak about proportional control, and P is called the proportional gain.

President UniversityErwin SitompulSMI 1/13 Chapter 1An Example of Process Control Feedback Control of the Heat Exchanger The scheme The block diagram

President UniversityErwin SitompulSMI 1/14 Chapter 2Mathematical Modeling of Processes General Principles of Modeling A system is expressed through mathematical descriptions. These descriptions are called “mathematical models.” The behavior of the system with regard to certain inputs can be characterized by using the mathematical model. Mathematical models can be divided into three groups, depending on how they are obtained: Theoretical model, developed using physical, chemical principles/laws Empirical model, obtained from mathematical analysis of measurement data of the process/ system or through experience Empirical-theoretical model, obtained from a combination of theoretical and empirical modeling approach

President UniversityErwin SitompulSMI 1/15 Theoretical models are derived from the so called “balance equation of conserved quantity” that may include: Mass balance equation Energy balance equation Entropy balance equation Enthalpy balance equation Charge balance equation Heat balance equation Impulse balance equation The balance equations in an unsteady-state are used to obtain the dynamical model, which is expressed using differential equations. In most cases, ordinary differential equations are chosen to keep the mathematical model simple. Chapter 2Mathematical Modeling of Processes General Principles of Modeling

President UniversityErwin SitompulSMI 1/16 Balance equation in integral form: Chapter 2Mathematical Modeling of Processes General Principles of Modeling Balance equation can be written in differential form: The variable m in the equations above can be mass, energy, entropy,..., impulse.

President UniversityErwin SitompulSMI 1/17 Mass balance in an unsteady-state is given by the law of mass conservation: Chapter 2Mathematical Modeling of Processes General Principles of Modeling ρ, ρ i : Specific densities [kg/m 3 ] V: Volume [m 3 ] q, q i : Volume flow rates [m 3 /s] m: Number of inlets n: Number of outlets In Out

President UniversityErwin SitompulSMI 1/18 Energy balance follows the general law of energy conservation: Chapter 2Mathematical Modeling of Processes General Principles of Modeling ρ, ρ i : Specific densities [kg/m 3 ] V: Volume [m 3 ] q, q i : Volume flow rates [m 3 /s] c p, c p,i : Specific heat capacities [J/(kgK) T, T i : Temperatures [K] Q: Heat per unit time [W] m: Number of inlets n: Number of outlets s: Number of heat sources and consumptions

President UniversityErwin SitompulSMI 1/19 Let us examine a liquid storage system shown below: Chapter 2Examples of Dynamic Mathematical Models Single-Tank System The mass balance for this process yields: qiqi qo qo V h ρ: Specific densities [kg/m 3 ] V: Volume [m 3 ] q i, q o : Volume flow rates [m 3 /s] A: Cross-sectional area of the tank [m 2 ] h: Height of liquid in the tank [m] With A and ρ assumed to be constant,

President UniversityErwin SitompulSMI 1/20 v1v1 Applying the law of mechanical energy conservation to the liquid near the outlet: Chapter 2Examples of Dynamic Mathematical Models Single-Tank System v 1 : Outlet flow velocity [m/s] a 1 : Cross-sectional area of the outlet pipe [m 2 ] qiqi qo qo V h

President UniversityErwin SitompulSMI 1/21 Inserting q 0 = v 1 a 1 into the mass balance equation of the system: Chapter 2Examples of Dynamic Mathematical Models Single-Tank System or The initial condition (i.e., the initial height of the liquid) can be arbitrary, h(0) = h 0. The tank will be in steady-state if dh/dt = 0. For a constant inlet flow rate q i, the steady-state liquid height h s is given by:

President UniversityErwin SitompulSMI 1/22 Chapter 2Examples of Dynamic Mathematical Models Simulation of Single-Tank System The dynamic mathematical model of the single-tank system will now be simulated using Matlab Simulink. a 1 = 20 cm 2 = 2010 –4 m 2 = 210 –3 m 2 A = 2500 cm 2 = 250010 –4 m 2 = 0.25 m 2 Manual calculation of steady-state liquid height yields: g = 9.8 m/s 2 q i = 5 liters/s = 510 –3 m 3 /s t sim = 200 s

President UniversityErwin SitompulSMI 1/23 After construction, the Matlab Simulink block diagram is given as: Chapter 2Examples of Dynamic Mathematical Models Simulation of Single-Tank System

President UniversityErwin SitompulSMI 1/24 The simulation result, from transient until steady-state can be observed by clicking the Scope. Chapter 2Examples of Dynamic Mathematical Models Simulation of Single-Tank System

President UniversityErwin SitompulSMI 1/25 Homework 1: Interacting Tank-in-Series System v1v1 qiqi qo qo h1h1 Chapter 2Examples of Dynamic Mathematical Models Derive a dynamic mathematical model for the interacting tank-in-series system as shown below. h2h2 v2v2 q1 q1 a1 a1 a2 a2

President UniversityErwin SitompulSMI 1/26 Homework 1: Triangular-Prism-Shaped Tank Chapter 2Examples of Dynamic Mathematical Models Derive a dynamic mathematical model for the tank with the form of a triangular prism as shown below. NEW v q i1 qo qo a q i2 h max h ρ: Specific densities [kg/m 3 ] V: Volume of liquid in the tank [m 3 ] h: Height of liquid in the tank [m] h max : Height of the tank [m] A: Cross-sectional area of the liquid surface [m 2 ] A max : Cross-sectional area of the tank at the top [m 2 ] q i1, q i2 : Volume flow rates of inlets [m 3 /s] q o : Volume flow rates of outlet[m 3 /s] v: Outlet flow velocity [m/s] a 1 : Cross-sectional area of the outlet pipe [m 2 ]