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13 to 07 February 2014
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Readings and preparation for class; video and audio files
- Main reference: Marlin textbook, Chapter 3, Chapter 4, Chapter 5, and Appendix B
- Alternative reference: Seborg textbook, Chapters 2, 3, 4 and 5
| Date
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Class number
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Video and audio files
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Main concepts
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Reading (other)
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Handout
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| 13 January
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02A
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Video (361M)
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Audio (42M)
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- Why we need process models
- What is "steady state"
- General balance equations
- Derived the tank height model
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None
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| 15 January
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02B
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Video (364M)
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Audio (43M)
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- Use of the integrating factor
- Analytical solution of the tank height model
- MATLAB solution of ODEs
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None
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| 17 January
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02C
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Video (227M)
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Audio (42M)
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- Coffee cooling example
- What is the Laplace transform
- Using Laplace to solve the coffee cooling problem
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- An amazing description of where the Laplace transform comes from: Video 1 and continues with video 2
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Table of Laplace transforms (from Seborg)
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| 20 January
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03A
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Video (367M)
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Audio (43M)
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- Review the coffee example
- Initial value and final value theorems
- Using deviation variables
- Solve coffee example in deviation form
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| 22 January
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03B
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Video (305M)
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Audio (42M)
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- Reviewed deviation variables, IVT and FVT
- The tank height problem in deviation form
- Deriving a transfer function for the tank height for varying inlet flow
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None
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None
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| 24 January
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03C
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Video (317M)
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Audio (42M)
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- Confirming understanding of a transfer function (TF)
- How to find system's output, for a given input
- Chaining up TFs for systems in series
- Learning about "pure gain" systems
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None
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None
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| 27 January
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04A
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Video (294M)
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Audio (41M)
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- combining transfer functions in series
- linearization of nonlinear functions
- basic block diagrams
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None
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None
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| 29 January
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04B
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Video (271M)
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Audio (43M)
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- linearization
- block diagrams for series and parallel systems
- time delays
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None
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None
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| 31 January
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04C
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Video (322 M)
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Audio (44 M)
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- time delay derivation and interpretation
- first order system theory
- composite input example
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See code below for class 04C
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None
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| 03 February
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05A
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Video (287 M)
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Audio (42 M)
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- review of all work covered in weeks 1 to 4
- assignment 3, question 3
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None
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Developing process models
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| 05 February
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05B
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Video (267 M)
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Audio (45 M)
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- composite inputs
- formal introduction to second order systems
- question period
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Types of damping
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None
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| 07 February
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05C
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Video (267 M)
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Audio (45 M)
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- wrapped up second order systems
- re-discussed the objective of control systems
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None
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* The book by Seborg et al. is easily available new or second hand, as it was the prescribed textbook in 2013 (Marlin's book was prescribed in 2012). I will make reference to the chapters from Seborg on the website as well.
Test your understanding before and after class with these resources from Dr. Thomas Marlin. This website also contains full Powerpoint slides for each chapter from his textbook. Use this as a resource to get a different teaching perspective on the same topic. It's quite OK if someone else's approach is a "better fit" for you than my approach.
Computer code: class 02B, 15 January
In a file called cstr_height.m:
function d_depnt__d_indep = cstr_height(indep, depnt)
% Dynamic balance for the CSTR height
% indep: the independent ODE variable, such as time or length or the reactor
% depnt: a VECTOR of dependent variables
%
% Returns:
%
% d(depnt)
% ---------- = a vector of ODEs
% d(indep)
% Assign some variables for convenience of notation: one row per DEPENDENT variable
h = depnt(1);
% Constant and other equations
A = 0.5; % m^2
F_i = 0.8; % m^3/min
R = 15; % min/m^2
F_o = h/R; % m^3/min
% Output from this ODE function must be a COLUMN vector, with n rows
% n = how many ODEs in this system?
n = numel(depnt); d_depnt__d_indep = zeros(n,1);
% Specify every element in the vector below: 1, 2, ... n
d_depnt__d_indep(1) = 1/A * (F_i - F_o);
The call the above model file from the "driver"; you can call this file anything, e.g. ODE_driver.m:
% The independent variable always requires an initial and final value:
indep_start = 0.0; % s
indep_final = 50.0; % s
% Set initial condition(s): for integrating variables (dependent variables)
h_depnt_zero = 4.0; % i.e. h(t=0) = 3.0
IC = [h_depnt_zero];
% Integrate the ODE(s):
[indep, depnt] = ode45(@c02B_linear, [indep_start, indep_final], IC);
% Plot the results:
clf;
plot(indep, depnt(:,1))
grid('on')
hold('on')
xlabel('Time [min]')
ylabel('Tank height')
legend('h')
title('Tank height with time')
% Does it match the analytical equation?
height = 12 - 8.*exp(-indep/7.5);
hold on
plot(indep, height, 'r.')
Computer code: class 04C, 31 January
t = 0:0.01:18;
y_regular = 3.*(1-exp(-t./4));
step_function = ones(size(t));
step_function(t<2) = 0;
y_delayed = 3.*(1-exp(-(t-2)./4)) .* step_function;
y_combined = y_regular + y_delayed;
% Show the plots
f = figure;
set(f,'Position',[440 79 841 638])
plot(t, y_regular,'b.-')
hold on
%plot(t, step_function,'k.-') % if you want to visualize the step function
plot(t, y_delayed,'r.-')
plot(t, y_combined,'m.-')
legend('Response from 1 tank', 'Response from delayed tank', ...
'Sum of the two responses', 'location','NorthWest')
grid on
