Control systems. Processing of signals
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The universality of this framework has one disadvantage. It is difficult to study it, since it has a large number of facilities. Even the author of this software was frequently discouraged. The best method of studying or using this software is handling a lot of samples of its applications. This article contains samples devoted to automatic control and processing of signals.
Automatic control and processing of signals is, in general, a pure math. In principle, it is possible to use MatCAD for it. However, the purpose of this framework is not only calculation. Calculations should interoperate with other facilities of this software, for example, 3D graphics. So, I'll show application examples of automatic control and the processing of signals.
Now I'm engaged by a very interesting job, so I've little time for writing a profound description of the framework. Also, I cannot find assistants. A person who can catch the problem usually has their own excellent work. However, the framework is being intensively developed and I would like developers to know about its new updates. Therefore, instead of a profound description, I put in brief remarks about changes to the framework.
Now the framework can test a set of situations. Testing code belongs to the TestCategory.csproj and AstronomyGL_Test.csproj projects. The user can put a set of *.cfa files into a directory or database and the framework can test all of them.
Editors of properties in previous versions can be opened in separate windows only. Now you can find them on the desktop.
Now charts are placed directly on the desktop. Right mouse-clicking on the
caption (for example,
Chart) shows the old layout in a separate
Also, a new version of the software contains DockManager developed by Weifen Luo. I've found that open file operation is very exhausting work. So, the new version enables us to drag a *.cfa file to the desktop. The result is the same as file open.
The framework uses drag and drop in other situations, as well. For example, if we put the series component on the desktop and drag a *.gra file onto it, then whatever is contained in the file series will be loaded to the component:
The new version of the formula editor contains a new operation, . Let us consider usage of this operation. For example, we have the following picture:
Init component has following property editor:
Init has a single formula and accepts Derivation
order 3. This means that any client of
Init may use
derivations of the formula by order up to 3.
Der is a client of
Init. Its property editor is exhibited below:
We have two formulas with one input variable,
x. This variable
is, in fact, Formula 1 of
Init. The variable is
Init. So, we can calculate such formula derivations
that require derivations of
x of orders that do not exceed 3.
The present day version of the framework contains lots of new components: magnetic field, sensor of 3D field, inertial navigation system, etc. Although these features do not belong to pure automatic control theory, they are related to concrete automatic control systems. For example, magnetic field and field sensor components can be used for simulations of spacecraft orientation control systems.
Now there exists a website devoted to the framework. I and my colleagues are more engineers and scientists than software developers or architects. A lot of our work does not correspond to The Code Project. If you wish to know this work, you can visit the framework site. This article contains a new chapter devoted to transfer functions and the Dirac Delta function.
This example contains a simulation of a follow-up system of radar. See the picture above. It has two channels: the channel of azimuth and the channel of elevation angle. The geometry of this system is represented in the following picture:
In this situation, an antenna is installed on gimbal suspension with two degrees of freedom: azimuth and channel of elevation angle. Let us construct the situation.
We have the
Ground frame. We've installed the
Azimuth frame on it and the
Elevation frame on the
Azimuth frame. The
Azimuth frame is
rotated around the y-axis of the
Ground frame and the
Elevation frame is rotated around z-axis of the
Azimuth frame. In result, the
Elevation frame is the
frame of the antenna.
We also have the
Vehicle frame. Its motion is defined by the
Motion component. We've chosen such motion that the following
graphs look beautiful. The
Measurements component performs
measurement parameters of the
Vehicle frame relative to the
Elevation frame. These measurements shall be used below for the
construction of the control system. We let that antenna perform an angular
measurement of the
axisrving line. These measurements are
unambiguously defined by
d are the y-coordinates,
z-coordinates and the distance of the
Vehicle expressed in the
reference frame of the antenna. We shall use the following control law:
...where and are angles of gimbal suspension and
h are constants. This control law is, in fact, standard of the control theory. Constants
are usually selected by stability
and other criteria of this theory. Then we shall construct the control law:
The data-in of
Delta are parameters of
System component solves differential equations of the
control law. These two equations of second order are decomposed by four ordinary
differential equations of first order. It looks like this:
Then we add another component to calculate orientation quaternions of the
Elevation components. We
also add indication components. Click
here to download this situation. A graphical analysis of control quantity is
presented in the following picture:
Light red and dark red curves correspond to the azimuth of
Vehicle and antenna orientation. Similarly, light blue and dark
blue curves correspond to elevation channel. Thus, a control system is
constructed. Also, we may install a virtual camera on the antenna
Elevation) and observe 3D video of
Vehicle motion. At
I installed a camera on the
Ground or the
Virtual cameras are used at part
2 of this software description. Later, we may to install a virtual radio
transmitter/receiver on the antenna and virtual electronic equipment, etc. Click
here to download this sample.
A lot of control systems use on-off control elements that have the following in-out function:
This function has the following math description:
If we have several sets of photos of a 3D object, we can obtain its 6D position. In this example, the problem is solved in the following way. From the photos and virtual cameras, we obtain the contours of the object as it is presented in the following picture:
...where are input and output on the nth time moment. We shall use a recursive element for simulation of this element. A simple situation of usage of this element is presented below:
Recursive component contains recursive equations of the
on-off control element. These equations look like this:
As a result, we have the following charts:
A red curve is an input and a green one is an output. Click here to download this sample.
Usually, continuous filters are described by systems of ordinary differential equations. Since we have considered systems of ordinary differential equations above, our description will be very brief. You should click here to download a sample of a continuous filter. The result of filtration is presented in the following picture:
A red curve is an input of the filter and a green one is an output. The output looks more smooth. It is shifted by phase. You can process obtained inverse time signals through this filter to avoid the shift.
A digital filter is any electronic filter that works by performing digital math operations on an intermediate form of a signal. We'll consider a digital version of the third order Butterworth filter. A Z-transform of this filter looks like this:
The is a cut-off angular frequency. I've constructed the filter as a cascade of recursive elements with transfer functions . As well as continuous filtering, it performs smoothing. Click here to download this sample digital filter.
Diff component performs numerical differentiation. The
Abs component is a digital rectifier.
Integral component is a digital low frequency filter. The input
signal is a chirp. The result
of demodulation is presented below:
The error of demodulation is caused by the non-linearity gain frequency characteristic of the filter. Click here to download this sample.
This example is devoted to determining the differences cosine of two signals' phases. I've used the following correlation scheme:
p2 contain two harmonic signals, x(t)
and y(t), having almost equal frequencies. The
calculates the squares and product of these signals. The
Integration performs a digital calculation of the following
Integration also performs digital filtering with the
following transfer function:
Post calculates the cosine by the following expression:
As was expected, this discriminator shows the following determination picture of the differences cosine of the phases:
A transfer function is the ratio of the output of a system to the input of a system in the Laplace domain, considering its initial conditions to be zero. If we have an input function of X(s) and an output function of Y(s), we define the transfer function H(s) to be:
Readers who have read a circuit theory book will recognize the transfer function as being the Laplace transform of a circuit's impulse response. The new version of the framework contains the new component . Its properties editor is presented in the following picture:
Using the formula editor, we can enter the transfer function. This editor also exhibits transients:
It also represents frequency characteristics using both linear...
...and logarithmic scales:
An example of its usage is presented below:
Input generates a harmonic input signal. The
Transform transforms it. The red curve is input and the blue curve
is the result of the transformation. Click
here to download this sample.
As has been said above, this task requires the Dirac Delta function. The formula editor supports it. The presence of the Delta function at the right part of the ordinary differential equation shows that the resulting function is not continuous. It is presented in the following chart:
The framework can operate with Dirac Delta functions. For example, the following formula...
...is allowed. You can use these formulae as the input of transformation functions, as presented in the following picture:
Also, the Dirac Delta function can be used in ordinary differential equations. Click here to download samples with Dirac Delta function.
The name "Universal Framework" looks like pure advertisement only. However, if I say that this software enables me to operate with every kind of absorbing layer, then it results in the shock of my interlocutor. When I show how I can operate with every kind, "Indeed, your software enables you operate with every kind," replies the interlocutor.