Introduction to Matlab 7 Lesson IV

1
Introduction to Matlab 7Lesson IV

• Marco Lattuada
• Swiss Federal Institute of Technology - ETH
• Institut für Chemie und Bioingenieurwissenschaften
  
• ETH Hönggerberg/ HCI F135 Zürich (Switzerland)
• E-mail lattuada_at_chem.ethz.ch
• http//www.morbidelli-group.ethz.ch/education/inde
  x

2
Example of Function

• The M-File the "mysum" function (saved in
  mysum.m)

3
Exercise

• Create a new function called 'my_second_function'
• Scope of the function
• Compute the function f(x) 2x2-exp(x)3
• Parameters of the function
• Input a generic array x with n components
• Output an array n?1 with the values of the
  function in x
• Evaluate the function for 40 x values ranging
  from -5 to 5
• Plot f(x) versus x

4
Exercise

• Create the function 'bisection_root' that
• Computes the zero of a function f(x) by the
  'Bisection' method
• The objective function is f(x) 2x2-exp(x)3
• The interval within the search for the zero is
  carried out is given to the function as input
  parameter (xmin, xmax)
• The function returns as output the zero (x0) and
  the corresponding value of the function (f(x0))
• Iterate the search for the zero until the search
  interval is less than 10-4
• Provide to the function the desired precision e
• Add a preliminary check on the supplied
  uncertainty interval
• If f(xmin)f(xmax) gt 0 return an error (use the
  command error)

• Compute x0
• Compute f(x0)
• Change the interval x1, x2
• Iterate 1-gt3 until (x2-x1) lt e

5
How to Pass a Function to a Function
How can I use the function 'bisection_root' to
compute the root of a generic function fobj(x)?
6
Simulation of a Batch Reactor

• Hp T const V const
• Variables CA, CB , CC
• Equations

R1 A ? 2B with R1 k1A R2 B ? C with R2 k2B

• Create the function 'batch_reactor' which
• Receives the time t and the column vector C with
  the concentration of the three components
• Returns a vector dCdt with the derivatives dCi/dt
• k1 and k2 are fixed values in the function

7
Integration of Systems of ODEs

• Function call
• t, dydt ode15s (_at_fcn_dydt, t_grid, y_init)
• The function ' fcn_dydt' must have the following
  header
• function dydt fcn_dydt (t, y)

Let us use the built in Matlab function ode15s
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Simulation of a Batch Reactor

• Hp T const V const
• Variables CA, CB , CC
• Equations

R1 A ? 2B with R1 k1A R2 B ? C with R2 k2B

• Modify the function 'batch_reactor' so that
• It receives the time t and the column vector C of
  the concentration of the three components
• It returns a vector dCdt with the derivatives
  dCi/dt
• It receives k1 and k2 as input parameters

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Integration of Systems of ODEs
The new function header does not correspond to
that required for the subroutine ode15s function
dydt fcn_dydt (t, y)
Alternative call 1 t_out,C ode15s(_at_(t,
C) batch_reactor(t, C, k),t_in,C_init)
Alternative call 2 t_out,C
ode15s(_at_batch_reactor,t_in,C_init,,k)
Alternative call 3 global k t_out,C
ode15s(_at_batch_reactor2,t_in,C_init)
batch_reactor_an _at_(t, C) batch_reactor(t, C, k)
The value of k becomes part of the "body" of
batch_reactor_an
Function "handle"
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Anonymous Functions
11
Input-output functions

• To enter date from the keyboard during the
  execution of a program, use the command input
• Variable_nameinput(Message)
• Stops the execution of the program until
  something is typed in. The value that is typed in
  is stored in the variable variable-name
• To retrieve data from a file, there are different
  possibilities
• Load loads vectors and matrices from a binary
  file
• Dataload(file_name.ext)
• Will store the values in the file file_name.ext
  in Data. Remember the values in the file need to
  be stored in matrix form (some number of rows and
  columns everywhere) and the file has to be ASCII

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Read from - and write to - a file

• fidfopen(filename.ext,w) creates a file
  named filename.ext
• or it overwrites it if already existing.
• Instread of w, the permission can be
  r(read), ora (append)
• A fread(fid) similar to load, reads data
  and stores them in A
• A fscanf(fid, format) reads formatted data
  (specified by format) and stores it in A
• tl fgetl(fid) reads one line from the file and
  stores it in tl
• fwrite(fid, A, precision) writes matrix A as a
  column vector with specified precision
• fprintf(fid,14.10e\n,a(1,1)) writes a(1,1) in
  the file with specified format (in this case,
  exponential notation)
• Fclose(fid) closes file

File identifier
Permission
13
Exercise

• Create the function 'my_derivative' that
• Computes sin(x) in x
• Computes the derivative of sin(x) in x
• Uses a step Dx 1e-8 to compute the derivative
• Compare the result produced by the function with
  the analytical solution of the derivative
• Provide the step Dx as an input parameter
• Select a value of Dx and run the function for x
  00.1pi
• Select x 1 and run the function of decreasing
  Dx
• Compare previous result with the analytical
  solution
• For which Dx the derivative is best approximated?
• Repeat points 5 to 7 for the function F(x) a
  exp(-b x) c x2 where (a, b, c) (2, 3, -4)

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My Derivative