This document describes how to perform common tasks in 6.302 using Octave, Scilab, and Matlab. The first two software packages are free alternatives to Matlab, and their use is encouraged.

Each software package employs different methods to create, examine, print, and save transfer functions. For the purposes of this tutorial, we will use the following transfer function as an example:

`L(s) = 3e4 * (0.05s + 1)^2 / ((s+1)^3 * (0.01s + 1))`

GNU Octave is a numerical computation software package very similar
to Matlab. In fact, the syntax is almost exactly the same. For help,
type `help `

. For an excellent
tutorial on control-related functions, type
`function_name``DEMOcontrol`

.

For the purposes of 6.302, there exist two ways to create transfer
functions in Octave. The first is `tf2sys`

, which takes as
arguments the coefficients of the numerator and denominator of the
transfer function. With the example transfer function, you would type:

`L = tf2sys(3e4 * [0.0025 0.1 1], [0.01 1.03 3.03 3.01 1]);`

Another command is `zp2sys`

, which takes as arguments
the zeros, poles, and leading coefficient of the transfer function.
Consequently, you would type:

`L = zp2sys([-20 -20], [-1 -1 -1 -100], 3e4);`

You will notice that if you merely type `L`

, the output
isn't very readable. Use `sysout(L, "tf")`

to display the
function in polynomial form and `sysout(L, "zp")`

for
zero-pole form.

For root locus, use `rlocus(L)`

.

L = tf2sys(3e4 * [0.0025 0.1 1], [0.01 1.03 3.03 3.01 1]); title("Root locus of L"); rlocus(L);

For Bode, use `bode(L)`

.

L = tf2sys(3e4 * [0.0025 0.1 1], [0.01 1.03 3.03 3.01 1]); bode(L);

For Nyquist, use `nyquist(L)`

.

L = tf2sys(3e4 * [0.0025 0.1 1], [0.01 1.03 3.03 3.01 1]); nyquist(L, logspace(-1, 1, 100)); title("Nyquist plot of L"); xlabel("Real axis"); ylabel("Imag axis"); replot;

For Nichols, use `nichols(L)`

.

L = tf2sys(3e4 * [0.0025 0.1 1], [0.01 1.03 3.03 3.01 1]); nichols(L); title("Nichols plot of L"); replot;

As in Matlab, the commands `title`

, `xlabel`

,
and `ylabel`

assign strings to the graphs. Often after
these commands, `replot`

is necessary to activate the changes.

However, `bode`

does not work after a `replot`

command; if you find yourself in a situation where a `replot`

is necessary, plot the data from `bode`

manually using
`plot`

. (See the online help for details.)

Octave uses Gnuplot to plot its graphs, so the saving and printing process is not as straightforward as clicking a button.

To save a graph in PNG format (which can be read by almost any graphics program), use the following commands:

gset terminal png; gset output "filename.png"; replot;

Again, with Bode plots, just use `bode(L)`

again instead
of `replot`

.

To print a graph, use the following commands:

gset terminal postscript; gset output "|lpr"; replot;

After saving or printing, reset the environment with the command
`gset terminal x11`

, or else any subsequent graphs will be
saved/printed without being displayed onscreen.

Scilab is a free software package similar to Maple. Its syntax is considerably different from Matlab, but Scilab can perform symbolic manipulation, unlike Matlab. It includes excellent demos, as found by pressing the "Demos" button on the toolbar.

This is perhaps the easiest task of all. Just type the following:

s = poly(0, "s"); L = syslin('c', 3e4 * (0.05*s + 1)^2 / ((s+1)^3 * (0.01*s + 1)));

The `poly`

command simply defines a polynomial with the
symbol "s." Don't forget it! Simply type `L`

to see the
transfer function you've just defined.

For root locus, use `evans(L)`

.

s = poly(0, "s"); L = syslin('c', 3e4 * (0.05*s + 1)^2 / ((s+1)^3 * (0.01*s + 1))); evans(L, 2.6);

For Bode, use `bode(L)`

.

s = poly(0, "s"); L = syslin('c', 3e4 * (0.05*s + 1)^2 / ((s+1)^3 * (0.01*s + 1))); bode(L);

For Nyquist, use `nyquist(L)`

.

s = poly(0, "s"); L = syslin('c', 3e4 * (0.05*s + 1)^2 / ((s+1)^3 * (0.01*s + 1))); nyquist(L);

For Nichols, use `black(L)`

.

s = poly(0, "s"); L = syslin('c', 3e4 * (0.05*s + 1)^2 / ((s+1)^3 * (0.01*s + 1))); black(L);

Unfortunately, there doesn't seem to be any easy way to replace
pre-existing titles and axis labels. If you want true customization,
consider using `plot2d`

manually.

Just use the menu items on the graph screens.

Matlab is one of the de facto standards of scientific computing. Though it is very powerful, it is unfortunately also very expensive. This is why the availability of packages like Octave and Scilab is so desirable. Nevertheless, the following Matlab material is provided for reference and comparison.

Like Scilab, Matlab makes this relatively easy. Just type the following:

s = tf ('s'); L = 3e4 * (0.05*s + 1)^2 / ((s+1)^3 * (0.01*s + 1));

For root locus, use `rlocus(L)`

.

s = tf ('s'); L = 3e4 * (0.05*s + 1)^2 / ((s+1)^3 * (0.01*s + 1)); rlocus(L);

For bode, with margins displayed, use `margin(L)`

.

s = tf ('s'); L = 3e4 * (0.05*s + 1)^2 / ((s+1)^3 * (0.01*s + 1)); margin(L);

For Nyquist, use `nyquist(L)`

.

s = tf ('s'); L = 3e4 * (0.05*s + 1)^2 / ((s+1)^3 * (0.01*s + 1)); nyquist(L);

For Nichols, you must calculate the magnitude and phase separately
with `nichols(L)`

, then explicitly graph it (probably as
a semilog plot).

s = tf ('s'); L = 3e4 * (0.05*s + 1)^2 / ((s+1)^3 * (0.01*s + 1)); [m,p,w] = nichols(L); semilogy (p(:,:), m(:,:)); grid; title ('Gain/phase plot'); xlabel ('open loop phase (deg)'); ylabel ('open loop magnitude');

You can also show the Nichols Chart on a dB axis:

s = tf ('s'); L = 3e4 * (0.05*s + 1)^2 / ((s+1)^3 * (0.01*s + 1)); [m,p,w] = nichols(L); plot (p(:,:), 20*log10 (m(:,:))); ngrid; title ('Gain/phase plot'); xlabel ('Open loop Phase (deg)'); ylabel ('Open loop Magnitude (dB)');

Again like Scilab, Matlab provides a menu interface that makes it easy to save and print graphs as needed.

Written by Charlie Kehoe, Eric Syu, and Dr. Kent Lundberg.