Dynamic models - 2014

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Class date(s): 13 to 07 February 2014
Download video: Link (plays in Google Chrome) [361 M]

Download video: Link (plays in Google Chrome) [364 M]

Download video: Link (plays in Google Chrome) [227 M]

Download video: Link (plays in Google Chrome) [367 M]

Download video: Link (plays in Google Chrome) [305 M]

Download video: Link (plays in Google Chrome) [317 M]

Download video: Link (plays in Google Chrome) [294 M]

Download video: Link (plays in Google Chrome) [271 M]

Download video: Link (plays in Google Chrome) [322 M]

Download video: Link (plays in Google Chrome) [287 M]

Download video: Link (plays in Google Chrome) [267 M]

Download video: Link (plays in Google Chrome) [215 M]

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 Class number Video and audio files Main concepts Reading (other) Handout
13 January 02A Video (361M) Audio (42M)
  • Why we need process models
  • What is "steady state"
  • General balance equations
  • Derived the tank height model
None
15 January 02B Video (364M) Audio (43M)
  • Use of the integrating factor
  • Analytical solution of the tank height model
  • MATLAB solution of ODEs
None
17 January 02C Video (227M) Audio (42M)
  • Coffee cooling example
  • What is the Laplace transform
  • Using Laplace to solve the coffee cooling problem
An amazing description of where the Laplace transform comes from: Video 1 and continues with video 2
Table of Laplace transforms (from Seborg)
20 January 03A Video (367M) Audio (43M)
  • Review the coffee example
  • Initial value and final value theorems
  • Using deviation variables
  • Solve coffee example in deviation form
22 January 03B Video (305M) Audio (42M)
  • 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
None None
24 January 03C Video (317M) Audio (42M)
  • 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
None None
27 January 04A Video (294M) Audio (41M)
  • combining transfer functions in series
  • linearization of nonlinear functions
  • basic block diagrams
None None
29 January 04B Video (271M) Audio (43M)
  • linearization
  • block diagrams for series and parallel systems
  • time delays
None None
31 January 04C Video (322 M) Audio (44 M)
  • time delay derivation and interpretation
  • first order system theory
  • composite input example
See code below for class 04C None
03 February 05A Video (287 M) Audio (42 M)
  • review of all work covered in weeks 1 to 4
  • assignment 3, question 3
None Developing process models
05 February 05B Video (267 M) Audio (45 M)
  • composite inputs
  • formal introduction to second order systems
  • question period

Types of damping

None
07 February 05C Video (267 M) Audio (45 M)
  • wrapped up second order systems
  • re-discussed the objective of control systems
None

* 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