For

System Dynamics & Control

By Dr. Hong Zhang

Start Matlab

Octave◦ http://www.gnu.org/software/octave/◦ Very Similar commands◦ Can run most M-files◦ No built-in Simulink package◦ Pure command line

Free Matlab Clones

Scilab◦ http://www.scilab.

org/◦ Some commands

are different◦ Built-in Xcos to

clone Simulink◦ Some Graphic

interface

Free Matlab Clones

Given a transfer function a2s2 + a1s + a0

b2s2 + b1s + b0

We can define it in Matlab asnum = [a2, a1, a0];

den = [b2, b1, b0];

sys = tf(num, den);

Transfer Function

Unit step responsestep(sys)

Unit impulse responseimpulse(sys)

Arbitrary input responset = tstart: tinterval : tfinish;

u = f(t); % u is a function of t, e.g. ramp is u=t;lsim(sys, u, t)

Plot Transient Response

Just bring the output to a variable. E.g.y1 = step(sys);y2 = impule(sys);y3 = lsim(sys, u, t);

Then we can use the variable. E.g. plot(t,y1, t, y2)

plot(t, u, t, y3)

Response As a Variable

[r, p, k] = residue(num, den);Where

r: rootp: polek: constant

If there are complex terms, we can add the two conjugate ones together to get a 2nd order real term.

Partial Fractional Expansion

Click the Simulink icon in Matlab window

Start Simulink

Simulink library browserSimulink modeling window

Matlab main window

Find, drag and drop following blocks to the window◦ Simulink Continuous Transfer Function◦ Sources Step◦ Sinks ScopeYou will get

Transfer Function

Building Blocks

OutputInput

Except sources and sinks, every block should have an input and an output.

Double click the Transfer function block.

Modify Transfer Function

Change Numerator to [1], denominator to [1 3 2]

Link the blocks by drag the output to input Double click Scope to show Scope window Click Ctrl+T or SimulationStart or button

Run Simulation

Change the spring constant and damping ratio, then you can have different response.

Modify System

[1 2 12][1 2 1]Hint: Hit the binocular to auto-scale the plot.

Replace the source with a Sine wave with frequency =3

Sinusoidal Response

Hint: Double click the block name to change it.

Hint: ◦ Hold Ctrl and click to tap an output line◦ Right click a block and select Format to flip or rotate a block

Flowchart with Feedback

Mass-Spring-Damper Modeling

€

m˙ ̇ x + c ˙ x + kx = f ( t)

€

˙ ̇ x = 1m [ f (t)− c ˙ x − kx]

Rewrite

as

Assumem=2kgc=3NSec/mk=3N/mf(t)=1(t)N