These videos and handouts are supplemental documents of paper X. Li, Z. Huang. An Inverted Classroom Approach to Educate MATLAB in Chemical Process
|
|
- Agatha Norman
- 5 years ago
- Views:
Transcription
1 These videos and handouts are supplemental documents of paper X. Li, Z. Huang. An Inverted Classroom Approach to Educate MATLAB in Chemical Process Control, Education for Chemical Engineers, 9, -, 7. The copyright of these materials belongs to Dr. Huang and Villanova University, PA, USA. Any use of these materials should be in compliance with U.S. Copyright law, including the TEACH Act. These materials are only allowed for academic purpose, NOT for commercial usage.
2 CHE 43: Chemical Process Control, Spring 6, by Zuyi (Jacky) Huang, Villanova University MATLAB Training - Laplace Transform and Transfer Function Objective o Basic commands for Laplace transform and inverse Laplace transform to solve ODEs o Represent ODEs using transfer functions in Simulink o Prediction from transfer functions for first-order, second-order, lead-lag, higher-order and other complicated models Introduction of the commands for Laplace transform and inverse Laplace transform o Laplace transform and inverse Laplace transform Use command laplace and ilaplace to perform Laplace transform and inverse Laplace transform Example : get the inverse Laplace transform for the following terms and get the time profiles for them. -s + e ) H( (4s + )(3s + ) ) H ( s( s + 4s + 4) s( s + ) + s 3) H 3( ( s + s + ) Solution: General procedure: type the Laplace term using symbolic variable s, and then use the command to get the time t function. -s + e ) H( (4s + )(3s + ) >> syms s %Define s as a symbolic variable for inverse Laplace-Transform >> H (+ exp(-*)/(4*s+)/(3*s+); >> h_t ilaplace(h) You should get h_t exp(-t/4) - exp(-t/3) + heaviside(t - )*(exp(/ - t/4) - exp(/3 - t/3)) >> t :.:; >> h_t exp(-t/4)-exp(-t/3)+heaviside(t-).*(exp(/-t/4) - exp(/3 - t/3)); >> plot(t,h_t,'linewidth',); >> xlabel('time (','Fontsize',) >> ylabel('h','fontsize',).5 This line is used to highlight the dot. You don t need to type it in MATLAB..5 h Time ( Note: you can save all these commands into a document, i.e., Main_example_.m, so that you just need to type Main_example_ in the command window again to get the simulation results. Here is
3 CHE 43: Chemical Process Control, Spring 6, by Zuyi (Jacky) Huang, Villanova University how you create a new script with the name of Main_example_.m: click New Script -> input all commands into the new script, and make sure that >> is not input in the new script -> press Ctrl + s on the keyboard to save the new script as Main_example_ -> type Main_example_ in the command window to run the simulation and plot the result Window for setting a new script: Save all the commands in a new script (i.e., Main_example_) Run the script in the command window: ) H ( s( s + 4s + 4) s( s + ) >> syms s %Define s as a symbolic variable for inverse Laplace- Transform >> H /s/(s + )^ >> h_t ilaplace(h) You should get h_t /4 - (t*exp(-*t))/ - exp(-*t)/4 >> t :.:; >> h_t /4 - (t.*exp(-*t))/ - exp(-*t)/4; >> plot(t,h_t,'linewidth',); >> xlabel('time (','Fontsize',) >> ylabel('h','fontsize',)
4 CHE 43: Chemical Process Control, Spring 6, by Zuyi (Jacky) Huang, Villanova University Save the program in a script named Main_example_ The profile of the output should be as follow: 3) + s H 3( ( s + s + ) >> syms s %Define s as a symbolic variable for inverse Laplace- Transform >> H3 (s+)/(s^ + s + ); >> h_t3 ilaplace(h3) You should get h_t3 exp(-t/)*(cos((3^(/)*t)/) + 3^(/)*sin((3^(/)*t)/)) >> t :.:; >> h_t3 exp(-t/).*(cos((3^(/)*t)/)+3^(/)*sin((3^(/)*t)/)); >> plot(t,h_t3,'linewidth',); >> xlabel('time (','Fontsize',) >> ylabel('h3','fontsize',) Save the program in a script named Main_example_3 The profile of the output should be as follow: 3
5 CHE 43: Chemical Process Control, Spring 6, by Zuyi (Jacky) Huang, Villanova University Example 3: get the time profile of the change of a chemical concentration (i.e., y, in the unit of nm) due to the change of the complicated inlet flowrates (shown in the right hand side of the equation) from the following differential equation: t d y dy y e d + step( t - 5) + sin3t dt dt ò - t dy t with y() and ( ) dt Step : The Laplace transform of the left-hand side: s Y( + 3sY( + Y ( ( s + 3s + ) Y( Step : Get the Laplace transform of the right-hand side of the equation: >> syms tau t >> laplace(*int(exp(-tau),,t)+*heaviside(t-5)+sin(3*t)) Note: () the Laplace operation is defined as a symbolic operation, so you have to define tau and t as symbolic variables; () you can find the expressions for other functions in the following website The Laplace transform of the right-hand side turns out to be: (*exp(-5*)/s - /(s + ) + 3/(s^ + 9) + /s Step 3: get the Laplace transform of y(t) by dividing the coefficient term (s + 3s + ) on the Laplace transform of the right-hand side >> syms s; >> Y_s ((*exp(-5*)/s - /(s + )+3/(s^ + 9) + / / (s^ +3*s+ ) Step 4: apply inverse Laplace transform on Y( to get the time profile of y(t) >> y_t ilaplace(y_ The time profile of y is y_t (3*exp(-t))/ - (9*cos(3*t))/3 - (6*exp(-*t))/3 - (7*sin(3*t))/3 - *t*exp(-t) + *heaviside(t - 5)*(exp( - *t)/ - exp(5 - t) + /) + Step 5: plot the profile of y over time >> t :.:; >> y_t (3*exp(-t))/ - (9*cos(3*t))/3 - (6*exp(-*t))/3 - (7*sin(3*t))/3 - *t.*exp(-t) + *heaviside(t - 5).*(exp( - *t)/ - exp(5 - t) + /) + ;% this is one line in MATLAB >> plot(t,y_t,'linewidth',) >> xlabel('time (', 'fontsize', ); >> ylabel('y (nm)', 'fontsize', ); 4
6 CHE 43: Chemical Process Control, Spring 6, by Zuyi (Jacky) Huang, Villanova University.5 y (nm) Time ( Solve ODEs using transfer functions in Simulink o Represent transfer function in Simulink d y dy Example 4: We redo the example: y e d + step( t - 5) + sin3t dt dt ò - t t in Simulink. d y dy Step : re-write the model as y u( t), as we can get the input functions on dt dt the left-hand side of the equation in Simulink. Step : get the transfer function to link the input u to the output y: s Y( + 3sY( + Y ( U( ( s + 3s + ) Y( U( > Y( U ( s + 3s + Step 3: input the transfer function in Simulink, get the functions for s + 3s + t òe - t dt, step( t - 5), sin3t as the input of the transfer function, and simulate the model in Simulink to get the output profile for y o You can find the module for transfer function from the Continuous category. t 5
7 CHE 43: Chemical Process Control, Spring 6, by Zuyi (Jacky) Huang, Villanova University o Double-click to input the coefficient of s terms (from the highest order to the lowest) for both the numerator and denominator For the transfer function, the coefficients for the numerator and s + 3s + denominator are [] and [ 3 ] respectively. o Here is the procedure to add t ò - e t dt, the first term of the input signals on the right-hand side of the equation, into the Simulink model: get the exp(u) from the Math Operation category, and set the input u as t. Add the clock from the Sources category to get the variable t, which then times - to get t. Integrator is added from the Continuous category. Set the initial value of the Integrator to zero. o Add step( t -5), sin 3t from the Sources category into the Simulink model. Set the final value, the initial value, and the step time of the step signal step(t-5) as,, and 5 respectively. For the sin3t, set the Amplitude as and the Frequency as 3. o Add the workspace for the time t and the output y_t. Make sure that you should change the variable names to t and y_t with the Save format as Array. Otherwise, you can plot the profile of y_t versus t. 6
8 CHE 43: Chemical Process Control, Spring 6, by Zuyi (Jacky) Huang, Villanova University o Set the Stop time as minutes, and the solver as ode5s. Run the simulation and plot the y profile. We plot the profile of the output y with the one obtained from the Laplace transform approach in the same figure. Here are the steps you should use to plot the output profiles from the Laplace transform (i.e., t VS y_t) and the Simulink model (i.e., t VS y_t). ) Run the Simulink model to get the profiles for t and y_t. ) Run the following commands to get the profiles. 3) Save the figure in the *.jpg format and insert into the WORD document. >> syms tau t >> laplace(*int(exp(-tau),,t)+*heaviside(t-5)+sin(3*t)) >> sys s;%define s as a symbolic variable for inverse Laplace-Transform >> Y_s ((*exp(-5*)/s - /(s + )+3/(s^ + 9) + / / (s^ +3*s+ ) >> y_t ilaplace(y_ >> t :.:; % Specify the time >> y_t (3*exp(-t))/ - (9*cos(3*t))/3 - (6*exp(-*t))/3 - (7*sin(3*t))/3 - *t.*exp(-t) + *heaviside(t - 5).*(exp( - *t)/ - exp(5 - t) + /) + ; % this one line >> plot(t,y_t,'linewidth',) >> hold on; >> plot(t,y_t,'r+', 'linewidth',) >> xlabel('time (Minute', 'fontsize', ); >> ylabel('y (nm)', 'fontsize', ); >> legend ('Laplace Transform approach','simulink simulation approach') Here are the profiles you should get: 7
9 CHE 43: Chemical Process Control, Spring 6, by Zuyi (Jacky) Huang, Villanova University Prediction from transfer functions for first-order, second-order, lead-lag, higher-order and other complicated models Model prediction for first order plus time delay systems Example 5: Liquid storage system with a long transmission line can be described by a first order plus time delay model:! " # $h $&! "( ) h where R v is the valve coefficient. min/m, A is the cross-section area of the tank 4 m, q i is in-let flow rate. The steady state value for q i and h are 4 m 3 /min and.4 m respectively. Determine the time profile of h..8.6 q in (m 3 /min).4. Solution: 5 5 Time (minute 8
10 CHE 43: Chemical Process Control, Spring 6, by Zuyi (Jacky) Huang, Villanova University The deviation ODE model is! " # $h $&! "( ) h where q i is q i q i () as shown in the following figure.4.3. q in ' (m 3 /min). 5 5 Time (minute After performing Laplace transform on the deviation ODE, we can get the following transfer function:, -! ".! " #. + )(.) Now we can use Simulink to get the profile of h upon the change in q i : Step : Instead of decomposing the q i as the summation of two step functions, use the Signal Builder in the Sources category to get the profile of q i. o Double click Signal Builder to pick the pulse signal, change the time Range to minutes. The default time unit in the Signal Builder is second. We don t have to change it to minute here, although the unit should be minute. 9
11 CHE 43: Chemical Process Control, Spring 6, by Zuyi (Jacky) Huang, Villanova University o Set the duration and amplitude for the pulse signal with the following procedure: ) move the mouse around the top line of pulse and left-click it ; ) after selecting the top line, double left-click the line to set the duration and amplitude of the pulse. This line should be in BOLD format for editing The beginning time for the pulse: the nd minute The ending time for the pulse: the 6 th minute The amplitude of the pulse is set to.4 m 3 /min o Check the profile from the scope or plot the output profile of the Signal Builder. In the scope, you should get the following figure: Step : Generate the transfer function in the Simulink and link the input (i.e., the output of the Signal Builder) to the transfer function. 5 o The transfer function should be 6 ;.: ; : ;.89:
12 CHE 43: Chemical Process Control, Spring 6, by Zuyi (Jacky) Huang, Villanova University o Add the output of the transfer function, i.e., h, with the initial value of h, i.e., h ().4 m. After this, we can get h (t) h () + h (t). o Save the output h in the workspace. o Add the time signal from the sources category and save the time in the workspace Step 3: Plot the result of h over time by using the following commands to plot the output profile. >> plot(t,h,'linewidth',) >> xlabel('time (minute','fontsize', ) >> ylabel('h (m)','fontsize', ) Model prediction for second order systems Example 6: The temperature in a heat-exchanger, represented by the variable y, can be described by the following equation in which u is the electrical heating power. K is a coefficient that can be manipulated by changing the cooling methods. The steady state value for y and u are 3 o C and W respectively. Predict the profile of y when the heating power u increases from to 5 W at time zero for K equal to, 4, and 8. Simulate the model for 5 minutes. Solution: The ODE can be re-written in the deviation variable form: $ >? $& > $? + 4? B $& The transfer function from the deviation ODE is: / 4 / 4 Y' ( u'( u'( u'( 4 s + Ks + K s + s + t s + tzs The procedure to implement the model in Simulink: Step : For K, input the transfer function and step input in Simulink. Make sure that the input, u (t), is a step with a zero initial value and a 3 W final value.
13 CHE 43: Chemical Process Control, Spring 6, by Zuyi (Jacky) Huang, Villanova University Step : Add the output y with its initial value, i.e., y y + 3, and save it into the workspace y ; add the time t signal into the Simulink model. Step 3: Apply the same approach to K equal to 4 and get profile y. Similarly, get the profile y 3 for K equal to 8. You can copy the module for y for y and y 3. You can put the modules for three outputs together in the same Simulink model. Note: to copy the entire module, select the module, press Ctrl on the keyboard and at the same time use the mouse to draw the module.
14 CHE 43: Chemical Process Control, Spring 6, by Zuyi (Jacky) Huang, Villanova University Step 4: Plot the three profiles, i.e., y, y, and y 3, together. >> plot(t,y,'r','linewidth',) >> hold on; >> plot(t,y,'b','linewidth',) >> hold on; >> plot(t,y3,'k','linewidth',) >> xlabel('time (minute','fontsize',) >> ylabel('y (^oc)','fontsize',) >> legend('k ','K 4','K 8','fontsize',) Model prediction for lead-lag systems (including inverse response system Example 7: Lead-lag systems can be presented by the following transfer function K(+ t 3 Y( U( (+ t (+ t where t > 3t. Assume that t 4, t, and K, plot the output profiles for t 3 6, 4,, and -4 for a unit step change in the input u. Assume the initial value of y is zero. Simulate the model for 3 minutes. Solution: Step : Get the transfer function for t 3 6, add the time signal t in the Simulink model, add the unit step input, and save the output of the transfer function y in the workspace; Step : Repeat Step for t 3 4,, and -4, and save the output to y, y 3 and y 4. 3
15 CHE 43: Chemical Process Control, Spring 6, by Zuyi (Jacky) Huang, Villanova University Step 3: Plot all the output profiles together in the same figure using the following commands: >> plot(t,y,'r','linewidth',) >> hold on; >> plot(t,y,'b','linewidth',) >> hold on; >> plot(t,y3,'k','linewidth',) >> hold on; >> plot(t,y4,'k','linewidth',) >> xlabel('time (minute','fontsize',) >> ylabel('y','fontsize',) >> legend('\tau_3 6','\tau_3 4','\tau_3 ','\tau_3 ','fontsize',) Example 8: A change in the steam pressure to the reboiler will lead to inverse response in the reboiler level. In Process (G ), a steam pressure increase leads to more vapor flow which can cause a rapid spillover of liquid from some trays to the reboiler (fast effect). In Process (G ), removing significant amount of the liquid mixture from the reboiler is through increased boiling, that is, more steam will lead to a lower liquid level in the reboiler over time (slow effect). U( + + Y( 4
16 CHE 43: Chemical Process Control, Spring 6, by Zuyi (Jacky) Huang, Villanova University Assume K 3, t, K -6, t. Predict the change of liquid level (i.e., y) due to a step change of psi in the steam pressure (i.e., u) for 8 minutes. Assume the system is at steadystate at the beginning. Solution: Step : build the two first-order models and add them together, add the step change with psi as the final value and zero as the initial value, add the time signal, and save the time t and the output y into the Workspace. Step : run the simulation for 8 minutes, and plot the output y profile by the following commands: >> plot(t,y,'r','linewidth',) >> xlabel('time (minute','fontsize',) >> ylabel('y (mm)','fontsize',) y (mm) - - Model prediction for model simplification Time (minute Example 9 The reboiler in a distillation column is heated up by steam which is supplied from a tank heated by electrical power. To design a controller to control the liquid level in the tank, you need to get the transfer function for the controlled objects in the loop and then simplify the transfer function to the first order plus time delay model (as most available controller design methods were developed for first order plus time delay system). q: electrical power G Q: heating power G V: steam pressure ( -s + ).5s + 3 s G + 3 5s + Controller Heating element Steam creator Reboilor L: liquid level The transfer function from q to L is: ( -s + ) L ( q( (.5s + ) (3s + ) (5s + ) 5
17 CHE 43: Chemical Process Control, Spring 6, by Zuyi (Jacky) Huang, Villanova University The simplified model via Taylor series expansion approach is: -4.5s e L( q( 5s + The simplified model via the Skogestad s half rule is: -3s e L3( q( 6.5s + Compare the output profiles for these three transfer functions when the electrical power has a step change of 5 W at time zero. Simulate the model for 5 minutes Solution: Step : Input the model L in the Simulink, the time signal t, and record the output as L in the Workspace; Step : Input the model L and L 3 in the Simulink o Both models have a delay term. o Add the delay term Transport Delay from Continuous category, and set the delay time 4.5 for L and 3 for L 3. 6
18 CHE 43: Chemical Process Control, Spring 6, by Zuyi (Jacky) Huang, Villanova University Step 3: Simulate the model for 5 minutes and plot the output profiles for the three models together. >> plot(t,l,'r','linewidth',) >> hold on; >> plot(t,l,'b','linewidth',) >> hold on; >> plot(t,l3,'k','linewidth',) >> xlabel('time (minute','fontsize',) >> ylabel('l (mm)','fontsize',) >> legend('l model','l model','l3 model') 8 6 L model L model L3 model L (mm) Time (minute 7
MAT 275 Laboratory 7 Laplace Transform and the Symbolic Math Toolbox
Laplace Transform and the Symbolic Math Toolbox 1 MAT 275 Laboratory 7 Laplace Transform and the Symbolic Math Toolbox In this laboratory session we will learn how to 1. Use the Symbolic Math Toolbox 2.
More informationECE 3793 Matlab Project 3 Solution
ECE 3793 Matlab Project 3 Solution Spring 27 Dr. Havlicek. (a) In text problem 9.22(d), we are given X(s) = s + 2 s 2 + 7s + 2 4 < Re {s} < 3. The following Matlab statements determine the partial fraction
More informationECE 3793 Matlab Project 3
ECE 3793 Matlab Project 3 Spring 2017 Dr. Havlicek DUE: 04/25/2017, 11:59 PM What to Turn In: Make one file that contains your solution for this assignment. It can be an MS WORD file or a PDF file. Make
More informationES205 Analysis and Design of Engineering Systems: Lab 1: An Introductory Tutorial: Getting Started with SIMULINK
ES205 Analysis and Design of Engineering Systems: Lab 1: An Introductory Tutorial: Getting Started with SIMULINK What is SIMULINK? SIMULINK is a software package for modeling, simulating, and analyzing
More informationSolving Differential Equations Using MATLAB
Solving Differential Equations Using MATLAB Abraham Asfaw aasfaw.student@manhattan.edu November 28, 2011 1 Introduction In this lecture, we will follow up on lecture 2 with a discussion of solutions to
More informationHomework 1 Solutions
18-9 Signals and Systems Profs. Byron Yu and Pulkit Grover Fall 18 Homework 1 Solutions Part One 1. (8 points) Consider the DT signal given by the algorithm: x[] = 1 x[1] = x[n] = x[n 1] x[n ] (a) Plot
More informationLab 1: Dynamic Simulation Using Simulink and Matlab
Lab 1: Dynamic Simulation Using Simulink and Matlab Objectives In this lab you will learn how to use a program called Simulink to simulate dynamic systems. Simulink runs under Matlab and uses block diagrams
More informationChapter 1: Introduction
Chapter 1: Introduction Definition: A differential equation is an equation involving the derivative of a function. If the function depends on a single variable, then only ordinary derivatives appear and
More informationCourse roadmap. ME451: Control Systems. Example of Laplace transform. Lecture 2 Laplace transform. Laplace transform
ME45: Control Systems Lecture 2 Prof. Jongeun Choi Department of Mechanical Engineering Michigan State University Modeling Transfer function Models for systems electrical mechanical electromechanical Block
More informationEE -213 BASIC CIRCUIT ANALYSIS LAB MANUAL
EE -213 BASIC CIRCUIT ANALYSIS LAB MANUAL EE 213 Fall 2009 LABORATORY #1 INTRODUCTION TO MATLAB INTRODUCTION The purpose of this laboratory is to introduce you to Matlab and to illustrate some of its circuit
More informationChemical Engineering 436 Laplace Transforms (1)
Chemical Engineering 436 Laplace Transforms () Why Laplace Transforms?? ) Converts differential equations to algebraic equations- facilitates combination of multiple components in a system to get the total
More informationEE 4314 Lab 1 Handout Control Systems Simulation with MATLAB and SIMULINK Spring Lab Information
EE 4314 Lab 1 Handout Control Systems Simulation with MATLAB and SIMULINK Spring 2013 1. Lab Information This is a take-home lab assignment. There is no experiment for this lab. You will study the tutorial
More informationCHAPTER 6 STATE SPACE: FREQUENCY RESPONSE, TIME DOMAIN
CHAPTER 6 STATE SPACE: FREQUENCY RESPONSE, TIME DOMAIN 6. Introduction Frequency Response This chapter will begin with the state space form of the equations of motion. We will use Laplace transforms to
More informationLaplace Transforms Chapter 3
Laplace Transforms Important analytical method for solving linear ordinary differential equations. - Application to nonlinear ODEs? Must linearize first. Laplace transforms play a key role in important
More informationYTÜ Mechanical Engineering Department
YTÜ Mechanical Engineering Department Lecture of Special Laboratory of Machine Theory, System Dynamics and Control Division Coupled Tank 1 Level Control with using Feedforward PI Controller Lab Date: Lab
More informationUsing MATLAB with the Convolution Method
ECE 350 Linear Systems I MATLAB Tutorial #5 Using MATLAB with the Convolution Method A linear system with input, x(t), and output, y(t), can be described in terms of its impulse response, h(t). x(t) h(t)
More informationAMS 27L LAB #6 Winter 2009
AMS 27L LAB #6 Winter 2009 Symbolically Solving Differential Equations Objectives: 1. To learn about the MATLAB Symbolic Solver 2. To expand knowledge of solutions to Diff-EQs 1 Symbolically Solving Differential
More informationProfessor Fearing EE C128 / ME C134 Problem Set 7 Solution Fall 2010 Jansen Sheng and Wenjie Chen, UC Berkeley
Professor Fearing EE C8 / ME C34 Problem Set 7 Solution Fall Jansen Sheng and Wenjie Chen, UC Berkeley. 35 pts Lag compensation. For open loop plant Gs ss+5s+8 a Find compensator gain Ds k such that the
More informationLaboratory handout 5 Mode shapes and resonance
laboratory handouts, me 34 82 Laboratory handout 5 Mode shapes and resonance In this handout, material and assignments marked as optional can be skipped when preparing for the lab, but may provide a useful
More informationSystem Identification of RTD Dynamics (Tom Co, 11/26/2006; 11/25/2007, 11/15/2009)
System Identification of RTD Dynamics (Tom Co, 11/26/26; 11/25/27, 11/15/29) Laboratory Objective: To obtain a mathematical model of RTD dynamics General procedures for mathematical modeling of dynamic
More informationBITS-Pilani Dubai, International Academic City, Dubai Second Semester. Academic Year
BITS-Pilani Dubai, International Academic City, Dubai Second Semester. Academic Year 2007-2008 Evaluation Com anent: Com rehensive Examination Closed Book CHE UC441/11NSTR UC 45'1 PROCESS CONTROL Date:
More information(t) ), ( ) where t is time, T is the reaction time, u n is the position of the nth car, and the "sensitivity coefficient" λ may depend on un 1(
Page 1 A Model of Traffic Flow Everyone has had the experience of sitting in a traffic jam, or of seeing cars bunch up on a road for no apparent good reason MATLAB and SIMULINK are good tools for studying
More informationMAE143A Signals & Systems - Homework 5, Winter 2013 due by the end of class Tuesday February 12, 2013.
MAE43A Signals & Systems - Homework 5, Winter 23 due by the end of class Tuesday February 2, 23. If left under my door, then straight to the recycling bin with it. This week s homework will be a refresher
More informationMATLAB Ordinary Differential Equation (ODE) solver for a simple example 1. Introduction
MATLAB Ordinary Differential Equation (ODE) solver for a simple example 1. Introduction Differential equations are a convenient way to express mathematically a change of a dependent variable (e.g. concentration
More informationSolution to Homework Assignment 1
ECE602 Fall 2008 Homework Solution September 2, 2008 Solution to Homework Assignment. Consider the two-input two-output system described by D (p)y (t)+d 2 (p)y 2 (t) N (p)u (t)+n 2 (p)u 2 (t) D 2 (p)y
More informationBasic Procedures for Common Problems
Basic Procedures for Common Problems ECHE 550, Fall 2002 Steady State Multivariable Modeling and Control 1 Determine what variables are available to manipulate (inputs, u) and what variables are available
More informationMECH : a Primer for Matlab s ode suite of functions
Objectives MECH 4-563: a Primer for Matlab s ode suite of functions. Review the fundamentals of initial value problems and why numerical integration methods are needed.. Introduce the ode suite of numerical
More informationLAB 1: MATLAB - Introduction to Programming. Objective:
LAB 1: MATLAB - Introduction to Programming Objective: The objective of this laboratory is to review how to use MATLAB as a programming tool and to review a classic analytical solution to a steady-state
More informationLesson 19: Process Characteristics- 1 st Order Lag & Dead-Time Processes
1 Lesson 19: Process Characteristics- 1 st Order Lag & Dead-Time Processes ET 438a Automatic Control Systems Technology 2 Learning Objectives After this series of presentations you will be able to: Describe
More informationSimulink Modeling Tutorial
Simulink Modeling Tutorial Train system Free body diagram and Newton's law Model Construction Running the Model Obtaining MATLAB Model In Simulink, it is very straightforward to represent a physical system
More informationApplication Note. U. Heat of Formation of Ethyl Alcohol and Dimethyl Ether. Introduction
Application Note U. Introduction The molecular builder (Molecular Builder) is part of the MEDEA standard suite of building tools. This tutorial provides an overview of the Molecular Builder s basic functionality.
More informationLaboratory 1. Solving differential equations with nonzero initial conditions
Laboratory 1 Solving differential equations with nonzero initial conditions 1. Purpose of the exercise: - learning symbolic and numerical methods of differential equations solving with MATLAB - using Simulink
More informationLab-Report Control Engineering. Proportional Control of a Liquid Level System
Lab-Report Control Engineering Proportional Control of a Liquid Level System Name: Dirk Becker Course: BEng 2 Group: A Student No.: 9801351 Date: 10/April/1999 1. Contents 1. CONTENTS... 2 2. INTRODUCTION...
More information1 Overview of Simulink. 2 State-space equations
Modelling and simulation of engineering systems Simulink Exercise 1 - translational mechanical systems Dr. M. Turner (mct6@sun.engg.le.ac.uk 1 Overview of Simulink Simulink is a package which runs in the
More informationRotary Motion Servo Plant: SRV02. Rotary Experiment #01: Modeling. SRV02 Modeling using QuaRC. Student Manual
Rotary Motion Servo Plant: SRV02 Rotary Experiment #01: Modeling SRV02 Modeling using QuaRC Student Manual SRV02 Modeling Laboratory Student Manual Table of Contents 1. INTRODUCTION...1 2. PREREQUISITES...1
More informationRead through A Graphing Checklist for Physics Graduate Students before submitting any plots in this module. See the course website for the link.
Module C2: MatLab The Physics and Astronomy Department has several copies of MATLAB installed on computers that can be accessed by students. MATLAB is a highly developed software package that allows the
More informationEE -213 BASIC CIRCUIT ANALYSIS LAB MANUAL
EE -213 BASIC CIRCUIT ANALYSIS LAB MANUAL EE 213 Spring 2008 LABORATORY #1 INTRODUCTION TO MATLAB INTRODUCTION The purpose of this laboratory is to introduce you to Matlab and to illustrate some of its
More informationCHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION. Professor Dae Ryook Yang
CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION Professor Dae Ryook Yang Spring 2018 Dept. of Chemical and Biological Engineering 5-1 Road Map of the Lecture V Laplace Transform and Transfer
More informationD(s) G(s) A control system design definition
R E Compensation D(s) U Plant G(s) Y Figure 7. A control system design definition x x x 2 x 2 U 2 s s 7 2 Y Figure 7.2 A block diagram representing Eq. (7.) in control form z U 2 s z Y 4 z 2 s z 2 3 Figure
More informationEE Experiment 11 The Laplace Transform and Control System Characteristics
EE216:11 1 EE 216 - Experiment 11 The Laplace Transform and Control System Characteristics Objectives: To illustrate computer usage in determining inverse Laplace transforms. Also to determine useful signal
More informationLaplace Transforms. Chapter 3. Pierre Simon Laplace Born: 23 March 1749 in Beaumont-en-Auge, Normandy, France Died: 5 March 1827 in Paris, France
Pierre Simon Laplace Born: 23 March 1749 in Beaumont-en-Auge, Normandy, France Died: 5 March 1827 in Paris, France Laplace Transforms Dr. M. A. A. Shoukat Choudhury 1 Laplace Transforms Important analytical
More informationProcess Control Exercise 2
Process Control Exercise 2 1 Distillation Case Study Distillation is a method of separating liquid mixtures by means of partial evaporation. The volatility α of a compound determines how enriched the liquid
More informationClass 27: Block Diagrams
Class 7: Block Diagrams Dynamic Behavior and Stability of Closed-Loop Control Systems We no ant to consider the dynamic behavior of processes that are operated using feedback control. The combination of
More informationYTÜ Mechanical Engineering Department
YTÜ Mechanical Engineering Department Lecture of Special Laboratory of Machine Theory, System Dynamics and Control Division Coupled Tank 1 Level Control with using Feedforward PI Controller Lab Report
More information( ) ( = ) = ( ) ( ) ( )
( ) Vρ C st s T t 0 wc Ti s T s Q s (8) K T ( s) Q ( s) + Ti ( s) (0) τs+ τs+ V ρ K and τ wc w T (s)g (s)q (s) + G (s)t(s) i G and G are transfer functions and independent of the inputs, Q and T i. Note
More informationLaboratory handouts, ME 340
Laboratory handouts, ME 340 This document contains summary theory, solved exercises, prelab assignments, lab instructions, and report assignments for Lab 4. 2014-2016 Harry Dankowicz, unless otherwise
More informationMAT 311 Midterm #1 Show your work! 1. The existence and uniqueness theorem says that, given a point (x 0, y 0 ) the ODE. y = (1 x 2 y 2 ) 1/3
MAT 3 Midterm # Show your work!. The existence and uniqueness theorem says that, given a point (x 0, y 0 ) the ODE y = ( x 2 y 2 ) /3 has a unique (local) solution with initial condition y(x 0 ) = y 0
More informationFinite Element Modules for Enhancing Undergraduate Transport Courses: Application to Fuel Cell Fundamentals
Finite Element Modules for Enhancing Undergraduate Transport Courses: Application to Fuel Cell Fundamentals Originally published in 007 American Society for Engineering Education Conference Proceedings
More informationCHAPTER 10: STABILITY &TUNING
When I complete this chapter, I want to be able to do the following. Determine the stability of a process without control Determine the stability of a closed-loop feedback control system Use these approaches
More informationSimulink Tutorial 1 CPE562
Simulink Tutorial 1 CPE562 Week 1 Introduction to Simulink Familiarization with Simulink blocks Sources: Constants Sinks: Display Operations: Sum, Product, Add, Divide. Mathematical operations involving
More informationCONTROL SYSTEMS LABORATORY ECE311 LAB 1: The Magnetic Ball Suspension System: Modelling and Simulation Using Matlab
CONTROL SYSTEMS LABORATORY ECE311 LAB 1: The Magnetic Ball Suspension System: Modelling and Simulation Using Matlab 1 Introduction and Purpose The purpose of this experiment is to familiarize you with
More informationTime Response of Systems
Chapter 0 Time Response of Systems 0. Some Standard Time Responses Let us try to get some impulse time responses just by inspection: Poles F (s) f(t) s-plane Time response p =0 s p =0,p 2 =0 s 2 t p =
More informationMATLAB for Chemical Engineering
MATLAB for Chemical Engineering Dr. M. Subramanian Associate Professor Department of Chemical Engineering Sri Sivasubramaniya Nadar College of Engineering OMR, Chennai 603110 msubbu.in[at]gmail.com 16
More informationRotary Motion Servo Plant: SRV02. Rotary Experiment #11: 1-DOF Torsion. 1-DOF Torsion Position Control using QuaRC. Student Manual
Rotary Motion Servo Plant: SRV02 Rotary Experiment #11: 1-DOF Torsion 1-DOF Torsion Position Control using QuaRC Student Manual Table of Contents 1. INTRODUCTION...1 2. PREREQUISITES...1 3. OVERVIEW OF
More informationMultivariable Control Laboratory experiment 2 The Quadruple Tank 1
Multivariable Control Laboratory experiment 2 The Quadruple Tank 1 Department of Automatic Control Lund Institute of Technology 1. Introduction The aim of this laboratory exercise is to study some different
More informationExperiment 81 - Design of a Feedback Control System
Experiment 81 - Design of a Feedback Control System 201139030 (Group 44) ELEC273 May 9, 2016 Abstract This report discussed the establishment of open-loop system using FOPDT medel which is usually used
More informationWe begin exploring Euler s method by looking at direction fields. Consider the direction field below.
Emma Reid- MA 66, Lesson 9 (SU 17) Euler s Method (.7) So far in this course, we have seen some very special types of first order ODEs. We ve seen methods to solve linear, separable, homogeneous, Bernoulli,
More informationSolutions to Homework 3
Solutions to Homework 3 Section 3.4, Repeated Roots; Reduction of Order Q 1). Find the general solution to 2y + y = 0. Answer: The charactertic equation : r 2 2r + 1 = 0, solving it we get r = 1 as a repeated
More informationLaboratory 11 Control Systems Laboratory ECE3557. State Feedback Controller for Position Control of a Flexible Joint
Laboratory 11 State Feedback Controller for Position Control of a Flexible Joint 11.1 Objective The objective of this laboratory is to design a full state feedback controller for endpoint position control
More informationMathematical Modelling Using SimScape (Electrical Systems)
Experiment Three Mathematical Modelling Using SimScape (Electrical Systems) Control Systems Laboratory Dr. Zaer Abo Hammour Dr. Zaer Abo Hammour Control Systems Laboratory 1. Model and simulate MultiDomain
More information2 Solving Ordinary Differential Equations Using MATLAB
Penn State Erie, The Behrend College School of Engineering E E 383 Signals and Control Lab Spring 2008 Lab 3 System Responses January 31, 2008 Due: February 7, 2008 Number of Lab Periods: 1 1 Objective
More informationProfessor Fearing EE C128 / ME C134 Problem Set 10 Solution Fall 2010 Jansen Sheng and Wenjie Chen, UC Berkeley
Professor Fearing EE C28 / ME C34 Problem Set Solution Fall 2 Jansen Sheng and Wenjie Chen, UC Berkeley. (5 pts) Final Value Given the following continuous time (CT) system ẋ = Ax+Bu = 5 9 7 x+ u(t), y
More informationLab Experiment 2: Performance of First order and second order systems
Lab Experiment 2: Performance of First order and second order systems Objective: The objective of this exercise will be to study the performance characteristics of first and second order systems using
More informationLecture 14 - Using the MATLAB Control System Toolbox and Simulink Friday, February 8, 2013
Today s Objectives ENGR 105: Feedback Control Design Winter 2013 Lecture 14 - Using the MATLAB Control System Toolbox and Simulink Friday, February 8, 2013 1. introduce the MATLAB Control System Toolbox
More informationExperiment # 5 5. Coupled Water Tanks
Experiment # 5 5. Coupled Water Tanks 5.. Objectives The Coupled-Tank plant is a Two-Tank module consisting of a pump with a water basin and two tanks. The two tanks are mounted on the front plate such
More informationSolving a RLC Circuit using Convolution with DERIVE for Windows
Solving a RLC Circuit using Convolution with DERIVE for Windows Michel Beaudin École de technologie supérieure, rue Notre-Dame Ouest Montréal (Québec) Canada, H3C K3 mbeaudin@seg.etsmtl.ca - Introduction
More informationSolutions. Chapter 1. Problem 1.1. Solution. Simulate free response of damped harmonic oscillator
Chapter Solutions Problem. Simulate free response of damped harmonic oscillator ẍ + ζẋ + x = for different values of damping ration ζ and initial conditions. Plot the response x(t) and ẋ(t) versus t and
More informationRepresenting Polynomials
Lab 4 Representing Polynomials A polynomial of nth degree looks like: a n s n +a n 1 a n 1 +...+a 2 s 2 +a 1 s+a 0 The coefficients a n, a n-1,, a 2, a 1, a 0 are the coefficients of decreasing powers
More informationCorrection for Simple System Example and Notes on Laplace Transforms / Deviation Variables ECHE 550 Fall 2002
Correction for Simple Sytem Example and Note on Laplace Tranform / Deviation Variable ECHE 55 Fall 22 Conider a tank draining from an initial height of h o at time t =. With no flow into the tank (F in
More informationPROCESS CONTROL (IT62) SEMESTER: VI BRANCH: INSTRUMENTATION TECHNOLOGY
PROCESS CONTROL (IT62) SEMESTER: VI BRANCH: INSTRUMENTATION TECHNOLOGY by, Dr. Mallikarjun S. Holi Professor & Head Department of Biomedical Engineering Bapuji Institute of Engineering & Technology Davangere-577004
More information( 1) EE 313 Linear Signals & Systems (Fall 2018) Solution Set for Homework #10 on Laplace Transforms
EE 33 Linear Signal & Sytem (Fall 08) Solution Set for Homework #0 on Laplace Tranform By: Mr. Houhang Salimian & Prof. Brian L. Evan Problem. a) xt () = ut () ut ( ) From lecture Lut { ()} = and { } t
More informationDESIGN AND CONTROL OF BUTYL ACRYLATE REACTIVE DISTILLATION COLUMN SYSTEM. I-Lung Chien and Kai-Luen Zeng
DESIGN AND CONTROL OF BUTYL ACRYLATE REACTIVE DISTILLATION COLUMN SYSTEM I-Lung Chien and Kai-Luen Zeng Department of Chemical Engineering, National Taiwan University of Science and Technology, Taipei
More informationMatlab Sheet 3 - Solution. Plotting in Matlab
f(x) Faculty of Engineering Spring 217 Mechanical Engineering Department Matlab Sheet 3 - Solution Plotting in Matlab 1. a. Estimate the roots of the following equation by plotting the equation. x 3 3x
More informationIntroduction to Process Control
Introduction to Process Control For more visit :- www.mpgirnari.in By: M. P. Girnari (SSEC, Bhavnagar) For more visit:- www.mpgirnari.in 1 Contents: Introduction Process control Dynamics Stability The
More informationMatlab Instruction Primer; Chem 691, Spring 2016
1 Matlab Instruction Primer; Chem 691, Spring 2016 This version dated February 10, 2017 CONTENTS I. Help: To obtain information about any instruction in Matlab 1 II. Scripting 1 III. Loops, determine an
More informationProcess Control and Instrumentation Prof. A. K. Jana Department of Chemical Engineering Indian Institute of Technology, Kharagpur
Process Control and Instrumentation Prof. A. K. Jana Department of Chemical Engineering Indian Institute of Technology, Kharagpur Lecture - 8 Dynamic Behavior of Chemical Processes (Contd.) (Refer Slide
More informationDr. Ian R. Manchester
Dr Ian R. Manchester Week Content Notes 1 Introduction 2 Frequency Domain Modelling 3 Transient Performance and the s-plane 4 Block Diagrams 5 Feedback System Characteristics Assign 1 Due 6 Root Locus
More informationProcess Control, 3P4 Assignment 5
Process Control, 3P4 Assignment 5 Kevin Dunn, kevin.dunn@mcmaster.ca Due date: 12 March 2014 This assignment is due on Wednesday, 12 March 2014. Late hand-ins are not allowed. Since it is posted mainly
More informationGetting started with BatchReactor Example : Simulation of the Chlorotoluene chlorination
Getting started with BatchReactor Example : Simulation of the Chlorotoluene chlorination 2011 ProSim S.A. All rights reserved. Introduction This document presents the different steps to follow in order
More informationOKLAHOMA STATE UNIVERSITY
OKLAHOMA STATE UNIVERSITY ECEN 4413 - Automatic Control Systems Matlab Lecture 1 Introduction and Control Basics Presented by Moayed Daneshyari 1 What is Matlab? Invented by Cleve Moler in late 1970s to
More informationProcess Control, 3P4 Assignment 6
Process Control, 3P4 Assignment 6 Kevin Dunn, kevin.dunn@mcmaster.ca Due date: 28 March 204 This assignment gives you practice with cascade control and feedforward control. Question [0 = 6 + 4] The outlet
More informationGraded and supplementary homework, Math 2584, Section 4, Fall 2017
Graded and supplementary homework, Math 2584, Section 4, Fall 2017 (AB 1) (a) Is y = cos(2x) a solution to the differential equation d2 y + 4y = 0? dx2 (b) Is y = e 2x a solution to the differential equation
More informationEE/ME/AE324: Dynamical Systems. Chapter 4: Block Diagrams
EE/ME/AE324: Dynamical Systems Chapter 4: Block Diagrams and Computer Simulation Block Diagrams A block diagram is an interconnection of: Blocks representing math operations Wires representing signals
More informationMAT 275 Laboratory 4 MATLAB solvers for First-Order IVP
MAT 275 Laboratory 4 MATLAB solvers for First-Order IVP In this laboratory session we will learn how to. Use MATLAB solvers for solving scalar IVP 2. Use MATLAB solvers for solving higher order ODEs and
More informationWILEY. Differential Equations with MATLAB (Third Edition) Brian R. Hunt Ronald L. Lipsman John E. Osborn Jonathan M. Rosenberg
Differential Equations with MATLAB (Third Edition) Updated for MATLAB 2011b (7.13), Simulink 7.8, and Symbolic Math Toolbox 5.7 Brian R. Hunt Ronald L. Lipsman John E. Osborn Jonathan M. Rosenberg All
More informationMatlab Controller Design. 1. Control system toolbox 2. Functions for model analysis 3. Linear system simulation 4. Biochemical reactor linearization
Matlab Controller Design. Control system toolbox 2. Functions for model analysis 3. Linear system simulation 4. Biochemical reactor linearization Control System Toolbox Provides algorithms and tools for
More informationSolutions for Tutorial 10 Stability Analysis
Solutions for Tutorial 1 Stability Analysis 1.1 In this question, you will analyze the series of three isothermal CSTR s show in Figure 1.1. The model for each reactor is the same at presented in Textbook
More informationPressure Swing Distillation with Aspen Plus V8.0
Pressure Swing Distillation with Aspen Plus V8.0 1. Lesson Objectives Aspen Plus property analysis RadFrac distillation modeling Design Specs NQ Curves Tear streams Understand and overcome azeotrope Select
More informationLab 13: Ordinary Differential Equations
EGR 53L - Fall 2009 Lab 13: Ordinary Differential Equations 13.1 Introduction This lab is aimed at introducing techniques for solving initial-value problems involving ordinary differential equations using
More information37. f(t) sin 2t cos 2t 38. f(t) cos 2 t. 39. f(t) sin(4t 5) 40.
28 CHAPTER 7 THE LAPLACE TRANSFORM EXERCISES 7 In Problems 8 use Definition 7 to find {f(t)} 2 3 4 5 6 7 8 9 f (t),, f (t) 4,, f (t) t,, f (t) 2t,, f (t) sin t,, f (t), cos t, t t t 2 t 2 t t t t t t t
More informationSection 6.4 DEs with Discontinuous Forcing Functions
Section 6.4 DEs with Discontinuous Forcing Functions Key terms/ideas: Discontinuous forcing function in nd order linear IVPs Application of Laplace transforms Comparison to viewing the problem s solution
More informationEEE 480 LAB EXPERIMENTS. K. Tsakalis. November 25, 2002
EEE 480 LAB EXPERIMENTS K. Tsakalis November 25, 2002 1. Introduction The following set of experiments aims to supplement the EEE 480 classroom instruction by providing a more detailed and hands-on experience
More informationDYNAMIC SIMULATOR-BASED APC DESIGN FOR A NAPHTHA REDISTILLATION COLUMN
HUNGARIAN JOURNAL OF INDUSTRY AND CHEMISTRY Vol. 45(1) pp. 17 22 (2017) hjic.mk.uni-pannon.hu DOI: 10.1515/hjic-2017-0004 DYNAMIC SIMULATOR-BASED APC DESIGN FOR A NAPHTHA REDISTILLATION COLUMN LÁSZLÓ SZABÓ,
More informationProcess Unit Control System Design
Process Unit Control System Design 1. Introduction 2. Influence of process design 3. Control degrees of freedom 4. Selection of control system variables 5. Process safety Introduction Control system requirements»
More informationLesson 14: Van der Pol Circuit and ode23s
Lesson 4: Van der Pol Circuit and ode3s 4. Applied Problem. A series LRC circuit when coupled via mutual inductance with a triode circuit can generate a sequence of pulsing currents that have very rapid
More informationMASS TRANSFER AND GAS ABSORPTION EQUIPMENT
MASS TRANSFER AND GAS ABSORPTION EQUIPMENT Mark J McCready University of Notre Dame Indiana, USA TOPICS Review of heat transfer and heat exchangers Some fundamental aspects of mass transfer Analysis of
More informationLecture 2. Today s Goal. Linearization Around a Trajectory, cont. Linearization Around a Trajectory. Linearization, cont d.
Lecture Today s Goal Linearization Stability definitions Simulation in Matlab/Simulink To be able to linearize, bot around euilibria and trajectories eplain definitions of stability ceck local stability
More informationCHEE 319 Tutorial 3 Solutions. 1. Using partial fraction expansions, find the causal function f whose Laplace transform. F (s) F (s) = C 1 s + C 2
CHEE 39 Tutorial 3 Solutions. Using partial fraction expansions, find the causal function f whose Laplace transform is given by: F (s) 0 f(t)e st dt (.) F (s) = s(s+) ; Solution: Note that the polynomial
More informationAnalysis and Design of Control Systems in the Time Domain
Chapter 6 Analysis and Design of Control Systems in the Time Domain 6. Concepts of feedback control Given a system, we can classify it as an open loop or a closed loop depends on the usage of the feedback.
More informationPractice problems from old exams for math 132 William H. Meeks III
Practice problems from old exams for math 32 William H. Meeks III Disclaimer: Your instructor covers far more materials that we can possibly fit into a four/five questions exams. These practice tests are
More information