Friday, October 4, 2013
Simulink:
MATLAB is based on the vector algebra; even the scalars are
treated as 1x1 matrix. Because of the inherent support of MATLAB for matrices
the MATLAB programmers are encouraged to employ matrix algebra instead of
iterative loops for speedier calculations in MATLAB.
Starting Simulink:
Simulink is started from the MATLAB command prompt by entering the
following command: Simulink Alternatively, you can click on the "Simulink
Library Browser" button at the top of the MATLAB command window as shown
below:
The Simulink Library Browser window should now appear on the
screen. Most of the blocks needed for modeling basic systems can be
There are two major classes of elements in Simulink: blocks and
lines. Blocks are used to
generate, modify, combine, output, and display signals. Lines are
used to transfer signals from one block to another.
Blocks
The subfolders underneath the "Simulink" folder indicate
the general classes of blocks available for us to use:
• Continuous: Linear,
continuous-time system elements (integrators, transfer functions, state-space
models, etc.)
• Discrete: Linear, discrete-time system elements
(integrators, transfer functions, state- space models, etc.)
• Functions & Tables:
User-defined functions and tables for interpolating function values
• Math: Mathematical
operators (sum, gain, dot product, etc.)
• Nonlinear: Nonlinear
operators (coulomb/viscous friction, switches, relays, etc.)
• Signals & Systems:
Blocks for controlling/monitoring signal(s) and for creating subsystems
• Sinks: Used to output or
display signals (displays, scopes, graphs, etc.)
• Sources: Used to generate various signals (step, ramp,
sinusoidal, 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.
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. One exception to this is that a line can tap
off of another line. This sends the
original signal to each of two (or more) destination blocks, as shown below:
Lines can never inject a signal into another line; lines must be
combined through the use of a block such as a summing junction. A signal can be
either a scalar signal or a vector signal. For Single-Input, Single-Output
systems, scalar signals are generally used. For Multi-Input, Multi-Output
systems, vector signals are often used, consisting of two or more scalar
signals. The lines used to transmit scalar and vector signals are identical.
The type of signal carried by a line is determined by the blocks on either end
of the line.
Building a System
To demonstrate how a system is represented using Simulink, we will
build the block diagram for a simple model consisting of a sinusoidal input multiplied
by a constant gain, which is shown below:
This model will consist of three blocks: Sine Wave, Gain, and Scope. The Sine Wave is a Source Block from which a
sinusoidal input signal originates. This
signal is transferred through a line in the direction indicated by the arrow to
the Gain Math Block. The Gain block
modifies its input signal (multiplies it by a constant value) and outputs a new
signal through a line to the Scope block.
The Scope is a Sink Block used to display a signal (much like an
oscilloscope). We begin building our system by bringing up a new model window
in which to create the block diagram.
This is done by clicking on the "New Model" button in the
toolbar of the Simulink Library Browser (looks like a blank page).
Building the system model is then accomplished through a series of
steps:
1.
The necessary blocks are gathered from the Library Browser and placed in the
model window.
2.
The parameters of the blocks are then modified to correspond with the system we
are modeling.
3.
Finally, the blocks are connected with lines to complete the model. Each of
these steps will be explained in detail using our example system. Once a system is built, simulations are run
to analyze its behavior.
Gathering Blocks
Each
of the blocks we will use in our example model will be taken from the Simulink
Library Browser. To place the Sine Wave
block into the model window, follow these steps:
2.
Scroll down until you see the "Sine Wave" block. Clicking on this will display a short
explanation of what that block does in the space below the folder list:
3. To insert a Sine Wave block into your model
window, click on it in the Library Browser and drag the block into your
workspace. The same method can be used to place the Gain and Scope blocks in
the model window. The "Gain"
block can be found in the "Math" subfolder and the "Scope"
block is located in the "Sink" subfolder. Arrange the three blocks in the work space
(done by selecting and dragging an individual block to a new location) so that
they look similar to the following:
Modifying the Blocks
Simulink
allows us to modify the blocks in our model so that they accurately reflect the
characteristics of the system we are analyzing.
For example, we can modify the Sine wave block by double-clicking on
it. Doing so will cause the following
window to appear:
This
window allows us to adjust the amplitude, frequency, and phase shift of the sinusoidal
input. The "Sample time" value
indicates the time interval between successive readings of the signal. Setting this value to 0 indicates the signal
is sampled continuously. Let us assume that our system's sinusoidal input has:
•
Amplitude = 2
•
Frequency = pi
•
Phase = pi/2
Enter these values into the appropriate fields (leave
the "Sample time" set to 0) and click "OK" to accept them
and exit the window. Note that the
frequency and phase for our system contain 'pi’ (3.1415...). These values can be entered into Simulink
just as they have been shown. Next, we modify the Gain block by double-clicking
on it in the model window. The following
window will then appear:
Note
that Simulink gives a brief explanation of the block's function in the top portion
of this window. In the case of the Gain
block, the signal input to the block (u) is multiplied by a constant (k) to
create the block's output signal (y).
Changing the "Gain" parameter in this window changes the value
of k. For our system, we will let k = 5.
Enter this value in the "Gain" field, and click "OK"
to close the window. The Scope block simply plots its input signal as a
function of time, and thus there are no system parameters that we can change
for it. We will look at the Scope block in
more detail after we have run our simulation.
Connecting the Blocks
For
a block diagram to accurately reflect the system we are modeling, the Simulink
blocks must be properly connected. In
our example system, the signal output by the Sine Wave block is transmitted to
the Gain block. The Gain block amplifies
this signal and outputs its new value to the Scope block, which graphs the
signal as a function of time. Thus, we
need to draw lines from the output of the Sine Wave block to the input of the
Gain block, and from the output of the Gain block to the input of the Scope
block.
Lines
are drawn by dragging the mouse from where a signal starts (output terminal of
a block) to where it ends (input terminal of another block). When drawing lines, it is important to make
sure that the signal reaches each of its intended terminals. Simulink will turn the mouse pointer into a
crosshair when it is close enough to an output terminal to begin drawing a
line, and the pointer will change into a double crosshair when it is close
enough to snap to an input terminal. A
signal is properly connected if its arrowhead is filled in. If the arrowhead is open, it means the signal
is not connected to both blocks. To fix
an open signal, you can treat the open arrowhead as an output terminal and
continue drawing the line to an input terminal in the same manner as explained
before.
Open
Signal
When drawing lines, you do not need to worry about the path you
follow. The lines will route themselves
automatically. Once blocks are
connected, they can be repositioned for a neater appearance. This is done by clicking on and dragging each
block to its desired location (signals will stay properly connected and will re-route
themselves).
In
some models, it will be necessary to branch a signal so that it is transmitted
to two or more different input terminals.
This is done by first placing the mouse cursor at the location where the
signal is to branch. Then, using either
the CTRL key in conjunction with the left mouse button or just the right mouse
button, drag the new line to its intended destination. This method was used to construct the branch
in the Sine Wave output signal shown below:
The
routing of lines and the location of branches can be changed by dragging them
to their desired new position. To delete
an incorrectly drawn line, simply click on it to select it, and hit the DELETE
key.
Running Simulations
Now
that our model has been constructed, we are ready to simulate the system. To do this, go to the Simulation menu and
click on Start, or just click on the "Start/Pause Simulation" button
in the model window toolbar (looks like the "Play" button on a
VCR). Because our example is a
relatively simple model, its simulation runs almost instantaneously. With more complicated systems, however, you
will be able to see the progress of the simulation by observing its running
time in the the lower box of the model window.
Double-click the Scope block to view the output of the Gain block for
the simulation as a function of time. Once
the Scope window appears, click the "Auto scale" button in its
toolbar (looks like a pair of binoculars) to scale the graph to better fit the
window. Having done this, you should see
the following:
Note that the output of our system appears as a cosine curve with a period of 2 seconds and amplitude equal to 10. Does this result agree with the system parameters we set? Its amplitude makes sense when we consider that the amplitude of the input signal was 2 and the constant gain of the system was 5 (2 x 5 = 10). The output's period should be the same as that of the input signal, and this value is a function of the frequency we entered for the Sine Wave block (which was set equal to pi). Finally, the output's shape as a cosine curve is due to the phase value of pi/2 we set for the input (sine and cosine graphs differ by a phase shift of pi/2). What if we were to modify the gain of the system to be 0.5? How would this affect the output of the Gain block as observed by the Scope? Make this change by double-clicking on the Gain block and changing the gain value to 0.5. Then, re-run the simulation and view the Scope (the Scope graph will not change unless the simulation is re-run, even though the gain value has been modified).
Note
that the only difference between this output and the one from our original
system is the amplitude of the cosine curve.
In the second case, the amplitude is equal to 1, or 1/10th of 10, which
is a result of the gain value being 1/10th as large as it originally was.
