MATLAB: Intro and infrastructure

Outline

• Infrastructure

  • Tips about the MATLAB program

  • Tips about MATLAB code

  • Some is relevant to later lectures, some is just nice to know

• Time ODEs

  • Mechanical systems as a base

  • Hand-rolled Euler, ode45, linear state-space implementations

  • Simulink basics & block diagram equivalents

• Systems modeling

  • More mechanical system examples

  • Simscape libraries: Mechanical, Electrical, Thermal

  • Fluids and Pneumatic (if I can figure them out)

• Composing system models

Basic tips

Function documentation

• Quick help from help <function>

  • Arguments, inputs, outputs

• Detailed help from doc <function>

  • Same as the webpage by default

  • Offline access by going to Home/Preferences/Help/Documentation Location

Up arrow

• Use the up arrow in the MATLAB command window to execute the thing you did
  last

• Start typing, and that behavior will be filtered

PATH and startup.m

• %userprofile%/Documents/MATLAB is on your "MATLAB PATH"

  • A "PATH", in any context, is just "where your program globally looks for functions"

  • MATLAB's built-in functions and libraries are on MATLAB's PATH

  • Any function you put to this kind of directory is call-able no matter where you are

• %userprofile%/Documents/MATLAB/startup.m runs whenever MATLAB starts

  • Configure parts of MATLAB not visibile in Preferences

  • Example: Dock figure window by default

  set(groot, "DefaultFigureWindowStyle", "docked");

  • Example: Default figure font size of 14

  set(groot, "DefaultAxesFontSize", 14);

Value output

• Probably you use disp to print values

  • Fine if you're just printing a number

  • disp(['The value of pi is ', num2str(pi, 2), 'to two decimal places'])

• Instead we can use fprintf(fmt, ...)

  • fprintf("The value of pi is %.2f to two decimal places\\n", pi)

• Most common to use

  • %d for integers

  • %f for decimals

  • %e for exponental

  • %s for strings

Vectorization

Operating on entire arrays "at once" is better than using a for loop.

• Many functions in MATLAB are vectorized

  • Implicitly applying to entire arrays (sin, sum, exp, etc.)

  • Mathematically equivalent to doing a for-loop in MATLAB and doing elements
    one-at-a-time

  • There's still a for-loop under the hood, but it's in a lower-level
    (faster) language

  • Can be stupidly faster in some cases ~ orders of magnitude is a reasonable
    expectation

• When each element of the array doesn't depend on the others, vectorization is
  possible

• You can't vectorize numerical simulation (except by explicitly solving the
  system)

1D indexing

• Let vector v be a row or column vector (doesn't matter)

• v(n) takes the n-th element of v

• v(end) gets the last element of v

2D indexing

• Ordered (row, col)

• row and col are as flexible as they were before

  • end to mean the end

  • a:b to create a continuous slice

  • [a b c d] to create a non-continuous slice

Linear & logical indexing

• To be perfectly consistent, the 1D examples should have used 1, n or n, 1
  to specify "the first column"

  • Lack of this is not a special case of indexing

• What if you just use one index anyway?

  • A(4) seems to work

  • Can you go off the end? Sure! A(15) gives a result

  • The single index treats the matrix as a big stack of columns

• find returns linear indices

  • fives = find(A == 5) is all of the linear inidces where A is 5

  • A(fives) = 0 makes all of those zero (gets rid of all of the fives)

• Equivalently, make an array of the same dimensions of logical (true/false)
  values

  • fives = (A == 5)

  • A(fives) = 0

  • Even more briefly A(A == 5) = 0

• Don't try to write up a list of linear indices or ones and zeros yourself; you should get from find, sub2ind, or a logical operation

• A(:) turns anything into a column (useful if you want to garauntee
  orientation)

Broadcast dimensions

• Two kinds of math: Matrix operations and element-wise

  • * is matrix multiplication, .* is element-by-element

  • Same for /, ^ and their element-wise counterparts ./, .^

• When you try to use element-wise math on mismatched arrays, what happens?

  • If one of the dimensions is 1, it gets copied

  • (N, 1) .* (1, M) -> (N, M)

  • Scalar multiplication is a special case---think of the scalar as getting
    copied

Operating along a dimension

• sum and similar functions say that their second argument makes the funciton
  operate "along" a dimension

  • Good mnemonic is that this "along" variable is the dimension removed by
    the operation

  • Example: sum(A, 2) removes 2, which is the column, "squashing" the array
    down.

Performance claims

• As far as you know, when you write code, MATLAB just runs it

• Actually, MATLAB's interpreter attempts to do optimization (JIT compilation)

• Actually actually, the operating system is scheduling thousands of threads

• Actually (last time for now), the CPU is trying to branch predict

• Don't say it's faster without good benchmarking!

  • Significant difference

  • Long-enough runtime

  • Persistant across attempts

Debugging

• Type "keyboard" or click the red dot next to the line

• Inspect variables to make sure they turn out how you expect

• Test calculations

Example: Fourier Series

• The sine series for a square wave has given coefficients $b_{n}$.
  Compute the sine series $f(t)$ to 100 terms between -2 and 2 seconds using a

  • double for-loop (solve to f_t_a)

  • single for-loop---sine evaluated for all time (solve to f_t_b)

  • single vectorized operation (solve to f_t_c)

$$ b_{n} = \frac{4}{\pi n}\quad \text{for} \quad n = 1,3,5,\cdots $$
$$ f(t) = \sum_{n} b_{n} \sin(n \pi t) $$

Solution: Double for-loop

Solution: Single for-loop

Solution: No for-loop

Timing

• tic before and toc after displays the time

• profile on before profile viewer after breaks down the time in detail

Example: Curve stitching

• A machine error introduces jumps into a dataset

• All element-to-element jumps greater than 0.5 are erroneous

• Remove these jumps

  • Need to remove their effect on all subsequent elements

Hint
diff returns the difference between consecutive elements,
cumsum takes the cumulative sum of its argument

Solution: Curve stitching

Data structures

Structures

• You can create a variable that has variables inside it

• struct.a creates a variable struct that has a field a

Why timeseries?

• When you output Simulink data, you'll have the option to use an array or
  structure

• In interpreting physical signals, you want two things to be true

  • Data never makes sense in isolation from time

  • Data and time cannot have the same operations done to them

• The Simulink array output has data and time in one matrix

  • If adding signals, hope and pray they have the same time vector

  • Hope you don't accidentally add one time vector to another

What timeseries?

• A Timeseries is a structure with fields Data and Time

  • ts.Data to get the data, ts.Time to get time

  • The MECE department says you should use arrays for simulink output but
    they are wrong

  • Array output treats time and data the same, and hides the time-dependence

  • You can plot a timeseries and MATLAB figures out the axes

  • synchronize to make the times for two timeseries the same, allowing math
    on the elements, or resample for just one

Timetables

• A timeseries but with serveral fields synchronized to a time column

• This time column has to have units

  • If it doesn't already come from Simulink, you have to use seconds, hours,
    milliseconds, etc.

Conclusion

• More proficiency with basics

• Know how to instrument code (benchmark, profile, debug)

• Know that vectorizing is a thing

• Can use structures, timeseries, and timetables

• Notes will go on the wiki sometime this week

Remarks

• In a 3-D array, the third index is the "page"

• There are more options to speed up code

  • Column and row-major ordering differ

  • Contiguous and non-contiguous slicing differ

  • Not likely to be worth it (unless you're doing a lot of operations)

• The first argument of fpritnf may be a file descriptor, which is a number
  representing an open file

  • fprintf(fid, ...)

  • Use fid = fopen("path_to_file") to print to a file you open

  • Use fid = 1 to print to the console

• compose works like fprintf but on arrays of strings

References/Further reading

• Format specifiers

• Matrix Indexing in MATLAB

• Vectorization

• Array memory layout