Lab session on week 11 (7th meeting)
1 or 2 more weeks to go…
IT1005
Lab 7 – Quick Check
• Have you all received my reply for lab 7?
– Of course not. I have not finished grading your submissions yet >.<
– I have important research paper deadline >.<
• http://www.cs.mu.oz.au/cp2008/
• Abstract due: 4 April 2008, Paper due: 8 April 2008.
– My future reply should contains:
• Remarks on your M files (look for SH7 tags again)
– For CSTR_input.m, CSTR_matrix.m, and CSTR_soln.m, etc
• Remarks on your Microsoft Word file
– For the other stuffs
• Your marks is stored in the file “Marks.txt” inside the returned zip file!
– I do not think I will be strict mode , the answers are standard for L7!
• Note that my marking scheme is slightly different from the standard one,
as I put emphasize on coding style (indentation, white spaces), efficiency,
things like proper plotting, etc…
L7.Q1 – 4 tanks CSTRs
Q1. A. Transform the non standard set of Linear Equations into standard format:
-(Q+k*V) *CA1 +
0
*CA2 +
0
*CA3 +
0
*CA4 = -Q*CA0
Q *CA1 -(Q+k*V)
*CA2 +
0
*CA3 +
0
*CA4 = 0
0 *CA1 +
Q
*CA2 -(Q+k*V)
*CA3 +
0
*CA4 = 0
0 *CA1 +
0
*CA2 +
*CA3 -(Q+k*V)
Q
*CA4 = 0
Q1. B. Convert the standard format into matrix format. Another straightforward task:
[
-(Q+k*V)
0
0
0
]
[
Q
-(Q+k*V) 0
0
[
0
Q
-(Q+k*V) 0
[
0
0
Q
*
[CA1] = [-Q*CA0
]
]
[CA2]
[0
]
]
[CA3]
[0
]
-(Q+k*V) ]
[CA4]
[0
]
Q1. C. You just need to do:
V = 1; Q = 0.01; CA0 = 10; k = 0.01;
A = [-(Q+k*V) 0 0 0; Q -(Q+k*V) 0 0; 0 Q -(Q+k*V) 0; 0 0 Q -(Q+k*V)];
b = [-Q*CA0; 0; 0; 0];
x = A\b % you should get x = [5; 2.5; 1.25; 0.625], x = inv(A) * b is NOT encouraged!
% Using fsolve for this is also not encouraged!
L7.Q2 – n tanks CSTRs
Q2. A. CSTR_input.m
Answer is very generic, similar to read_input.m in L6.Q2 (Car Simulation)
Q2. B. CSTR_matrix.m
% This is my geek version, do NOT USE THIS VERSION (too confusing for novice)!
function [A b] = CSTR_matrix(Q, CA0, V, k, N)
A = diag(repmat(-(Q+k*V),1,N)) + diag(repmat(Q,1,N-1),-1); % This is a bit crazy :$
b = zeros(N,1); b(1) = -Q*CA0;
Q2. C. CSTR_soln.m
clear; clc; clf; % New trick, but important ! Clear everything before starting our program!
[Q CA0 V k N] = CSTR_input();
[A b] = CSTR_matrix(Q, CA0, V, k, N);
plot(A\b,'o'); % I prefer not to connect the plot with line, but it is ok if you do so.
title('CA of each tank'); xlabel('Tank no k'); ylabel('CA_k'); % Good for CSTR_plot.m
axis([0.5 N+0.5 0 CA0]); % Fix y axis so that it is consistent across 4 plots (same CA0!)
L7.Q2 – Good Plot
Remember:
Plot A against B means that
A is the Y axis, B is the X axis!
Should just stop here (n = 4) for case 1
L7.Q3 and Q4
Q3. Type these at command window:
syms x y; % no need to say syms f1 f2, the next two lines will create f1 and f2 anyway
f1 = x^2 * y^2 - 2*x - 1;
f2 = x^2 - y^2 - 1;
[a b] = solve(f1,f2); % or >> s = solve(f1,f2); a = s.x; b = s.y;
eval(a), eval(b) % convert the symbolic values to numeric values, these are the roots
Q4. Create this function
function F = lab07d(x)
F(1) = sin(x(1)) + x(2)^2 + log(x(3)) - 7;
F(2) = 3*x(1) + 2^(x(2)) + 1 - x(3)^3;
F(3) = x(1) + x(2) + x(3) - 5;
Tips for guessing logically:
1.Plot the functions with some range, see
the region of zero intercepts…
2.Guess from the easiest equation!
e.g. x + y + z = 5
3. log(z)… hm… z should be > 0
At command window (wild guesses will likely give you many imaginary numbers):
fsolve(@lab07d, [1 1 1])  x=0.5991, y=2.3959, z=2.0050
fsolve(@lab07d, [5 -1 1])  x = 5.1004, y = -2.6442, z= 2.5438
Application 5: IVP (revisited)
• Equation
– A statement showing the equality of two expressions usually separated by left
and right signs and joined by an equals sign.
• Differential Equation
– A description of how something continuously changes over time.
• Ordinary Differential Equation
– A relation that contains functions of only one independent variable, and one
or more of its derivatives with respect to that variable.
• Initial Value Problem
– An ODE together with specified value, called the initial condition, of the
unknown function at a given point in the domain of the solution.
Application 5: IVP (revisited)
• We have seen several examples of IVP throughout IT1005
– Spidey Fall example (Lecture 2, Lecture ODE 1)
• How velocity v change over time dt? (dv/dt)
– Depends on gravity minus drag!
• How displacement s change over time dt? (ds/dt)
– Depends on the velocity at that time.
• The Initial Values for v and s at time 0  v(0) = 0; s(0) = 0;
– Car accelerates up an incline example (Lab 6, Q2)
• How velocity v change over time dt? (dv/dt)
– Depends on engine force on some kg car minus friction and gravity factor!
• How displacement s change over time dt? (ds/dt)
– Depends on the velocity at that time.
• The Initial Values for v and s at time 0  v(0) = 0; s(0) = 0;
– And two more for Term Assignment (Q2 and Q3) 
Application 5: IVP (revisited)
• Solving IVP (either 1 ODE or set of coupled ODEs):
– Hard way/Traditional way/Euler method:
• Time is chopped into delta_time, then starting from the initial values for each variable,
simulate its changes over time using the specified differential equation!
• What Colin has shown in Lecture 2 for Spidey Fall is a kind of Euler method.
• What you have written for Lab 6 Question 2 (Car Simulation) is also Euler method.
– Matlab IVP solvers: mostly numerical solutions for ODE.
• Create a derivative function to tell Matlab about how a variable change over time!
function dydt = bla(t,y) % always have at least these two arguments
% explain to Matlab how to derive dy/dt! Can be for coupled ODEs!
dydt = dydt'; % always return a column vector!
• Call one of the ODE solver with certain time span and initial values
[t, y] = ode45(@bla, [tStart tEnd], IVs); % IVs is a column vector for coupled ODEs!
plot(t,y(:,1)); % we can immediately plot the results (also in column vector!)
Term Assignment – Admin
•
This is 30% of your final IT1005 grade... Be very serious with it.
– No plagiarism please!
• Even though you can ‘cross check’ your answers with your friends (we cannot
prevent that), you must give a very strong ‘individual flavor’ in your answers!
• The grader will likely grade number per number, so he will be very curious
if he see similar answers across many students. Do not compromise your 30%!
– Who will grade our term assignment?
• I may not be the one doing the grading! Perhaps all the full time staff… dunno yet.
– Submit your zip file (containing all files that you use to answer the questions)
to IVLE “Term Assignment” folder! NOT to my Gmail!
– Your zip file name should be: yyy-uxxxxxx.zip, NOT according to my style!
– Strict deadline, Saturday, 5 April 08, 5pm
That IVLE folder will auto close by Saturday, 5 April 08, 5.01pm
Be careful with NETWORK CONGESTION around these final minutes…
To avoid that problem, submit early, e.g. Friday, 4 April 08, night.
Term Assignment – Q1
• Question 1: Trapezium rule for finding integration
– A. Naïve one. Explain your results!
– B. More accurate one. Explain your results!
– References:
• help quad
• http://en.wikipedia.org/wiki/Numerical_integration
• http://en.wikipedia.org/wiki/Trapezium_rule
– Revision(s) to the question:
• Symbol ‘a’ changed to ‘c’ inside function f(t)!
• In Q1.B, the rows in column ‘c’ are [0.001 0.5 10.0 100.0] not [0.01 0.5 1.0 10.0]!
• In Q1.B, the range of k is changed from k = 2:n-1 to k = 1:n-1,
but it is ok if you stick with the old one!
Term Assignment - Q2
• Question 2: Zebra Population versus Lion Population
–
–
–
–
–
–
A. IVP, coupled, non-linear ODEs
B. Explain what you see in the graph of part A above.
C. Steady state issue.
D. IVP again, but change the IVs according to part C above. Comment!
E. IVP with different IVs, and different plotting method. Comment!
References:
• Google the term ‘predator prey’ as mentioned in the question.
• help odeXX (depends on the chosen solver)
• http://en.wikipedia.org/wiki/Steady_state
– Revision(s) to the question:
• No change so far…
Term Assignment – Q3
• Question 3: Similar to Q2, Predator-Prey: n = 4 species
– A. IVP again, 4 coupled, non-linear ODEs.
Dr Saif said that we must use ode15s! (See ODE 3 & 4 lecture note)
– B. IVP, same IVs, 1.000 years, 3D plot x1-x2-x3 (x4 is not compulsory),
and explain.
– C. Explain plot in B as best as you can.
– References:
• http://en.wikipedia.org/wiki/Lotka-Volterra_equation (mentioned in the question).
• Google ‘Matlab 3D plot’
• help odeXX (depends on the chosen solver)
– Revision(s) to the question:
• The ODE equations are updated! Read the newest one!
• The coefficient r(3) is changed from 1.53 to 1.43!
Free and Easy Time
• Now, you are free to explore Matlab to:
– Do your Term Assignment (all q1, q2, and q3)
– You should NOT use me as an oracle, e.g.
• I cannot find the bug in my program, can you help me?
• Are my 2-D/3-D plots correct?
• Are my …. bla bla … correct?