The document discusses using MATLAB and Simulink to design and simulate control systems. It provides instructions on how to: 1) Use the sisotool in MATLAB to design a cascade compensator and determine the root locus, gain where it intersects a 0.6 damping ratio line, and where it intersects the jω-axis. 2) Build models in Simulink by adding blocks like sources, sinks, transfers functions, gains and connecting them. 3) Modify block parameters and simulate the model response in the scope to analyze system performance.

1. CONTROL SYSTEMS
THEORY
MATLAB & SIMULINK
CHAPTER 9
STB 35103

2. Objectives

To use the MATLAB to design a cascade
compensator

To use SIMULINK to simulate the response
of a system

3. SISOTOOL

You are going to learn how to design or
improve a control system using MATLAB
control system toolbox Root Locus Design
Graphical User Interface (GUI).

There are two design tools in the MATLAB
that you can use for root locus:


sisotool
rltool

4. SISOTOOL

Example 1

In the figure below, Gc(s) is a proportional
controller, K,

Using sisotool determine the following:



The root locus for this system.
The exact point and gain where the locus crosses the
0.6 damping ratio line.
The exact point and gain where the locus crosses the
jω-axis

5. SISOTOOL
to make the plant model and start the
SISO design tool, at the MATLAB prompt
type
>>Gp = tf(1,[1 7 10 0])
>>sisotool


To add a damping ratio line,

Left click + design constrain + new
Constraint type: damping ratio
 Damping ratio > 0.6


6. SISOTOOL
Move the closed loop poles until it
intersect the damping ratio.
 Get the value of gain, K, and the
coordinate of the intersect.


7. SIMULINK

To start the SIMULINK at the MATLAB
prompt type
>>simulink

8. SIMULINK



To see the content of the blockset, click on
the "+" sign at the beginning of each toolbox.
To start a model click on the NEW FILE ICON
as shown in the screenshot above.
Alternately, you may use keystrokes CTRL+N.
A new window will appear on the screen. You
will be constructing your model in this
window. Also in this window the constructed
model is simulated. A screenshot of a typical
working (model) window is shown below:

9. SIMULINK

workspace

10. SIMULINK

Two basic elements in SIMULINK

Blocks


used to generate, modify, combine, output, and
display signals
Lines

Lines are used to transfer signals from one block to
another.

11. SIMULINK

Blocks







There are several general classes of blocks:Sources: Used
to generate various signals
Sinks: Used to output or display signals
Discrete: Linear, discrete-time system elements (transfer
functions, state-space models, etc.)
Linear: Linear, continuous-time system elements and
connections (summing junctions, gains, etc.)
Nonlinear: Nonlinear operators (arbitrary functions,
saturation, delay, etc.)
Connections: Multiplex, Demultiplex, System Macros, etc.
Blocks have zero to several input terminals and zero to
several output terminals. Unused input terminals are
indicated by a small open triangle. Unused output terminals
are indicated by a small triangular point. The block shown
below has an unused input terminal on the left and an
unused output terminal on the right.

12. SIMULINK

Lines

Lines transmit signals in the direction indicated
by the arrow. Lines must always transmit
signals from the output terminal of one block
to the input terminal of another block. On
exception to this is a line can tap off of another
line, splitting the signal to each of two
destination blocks, as shown below

13. SIMULINK

Modifying blocks

A block can be modified by double-clicking on
it.
[1] → 1
[ 1 1] → s + 1
[ 1 2 1] → s 2 + 2s + 1
[ 1 2 0] → s 2 + 2 s + 0

14. SIMULINK

Example

Produce the output signal for the system
below.

15. SIMULINK

Gathering Blocks

Follow the steps below to collect the necessary
blocks:

Create a new model (New from the File menu or
Ctrl-N). You will get a blank model window.

16. SIMULINK
Double-click on the Sources
icon in the main Simulink
window.
 This opens the Sources
window which contains the
Sources Block Library.
Sources are used to generate
signals.
 Drag the Step block from the
sources window into the left
side of your model window.


17. SIMULINK
Select “Math operations” and drag “gain”, and “sum”
to the workspace.
 Select “Continuous” and drag two “Transfer Fcn” to
the workspace.
 Select “Sink” drag “scope” to the workspace.


18. SIMULINK

Modify blocks




Follow these steps to properly modify the blocks in
your model.Double-click your Sum block. Since you
will want the second input to be subtracted, enter
+- into the list of signs field. Close the dialog box.
Double-click your Gain block. Change the gain to
2.5 and close the dialog box.
Double-click the leftmost Transfer Function block.
Change the numerator to [1 2] and the
denominator to [1 0]. Close the dialog box.
Double-click the rightmost Transfer Function block.
Leave the numerator [1], but change the
denominator to [1 2 4]. Close the dialog box. Your
model should appear as:

19. SIMULINK

You workspace should look like below

20. SIMULINK

Change the name of the first Transfer Function block by
clicking on the words "Transfer Fcn". A box and an
editing cursor will appear on the block's name as shown
below. Use the keyboard (the mouse is also useful) to
delete the existing name and type in the new name, "PI
Controller". Click anywhere outside the name box to
finish editing.

21. SIMULINK

Connecting Blocks with Lines

Now that the blocks are properly laid out, you
will now connect them together. Follow these
steps.

Drag the mouse from the output terminal of the Step block
to the upper (positive) input of the Sum block. Let go of
the mouse button only when the mouse is right on the
input terminal. Do not worry about the path you follow
while dragging, the line will route itself. You should see the
following.

22. SIMULINK

Your complete diagram should look like this

23. SIMULINK

Simulation

Now that the model is complete, you can simulate the
model. Select Start from the Simulation menu to run
the simulation. Double-click on the Scope block to view
its output. Hit the autoscale button (binoculars) and you
should see the following.