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The Universal Framework for Science and Engineering - Part 3: Control systems. Processing of signals.
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Introduction

An universality of this framework has one disadvantage. It is difficult to study it, since it has a large number of facilities. Even author of this software was being frequently discouraged. The best method of studying or using this soft is handling a lot of samples of its applications. This paper contains samples devoted to automatic control and processing of signals.

Background

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 examples of applications to automatic control and processing of signals.

Example 1. Two-channel follow-up system

This example contains simulation of follow-up system of radar (See picture above). It has two channels: channel of azimuth and channel of elevation angle. The geometry of this system is represented on 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 y - axis of Ground frame and the Elevation frame is rotated around z - axis of the Azimuth frame. In result the Elevation frame is the frame of antenna. We also have the Vehicle frame. Its motion is defined by the Motion component. We've chosen such motion that the following graphs looks beautiful. The Measurements component performs measurements parameters of Vehicle frame relatively Elevation frame. These measurements shall be used below for the construction of the control system. We let that antenna performs an angular measurements of axisrving line. These measurements are unambiguously defined by y/d and z/d where y, z, d are y, z - coordinates and distance of the Vehicle expressed in reference frame of antenna. We shall use the following control law:

where α and β are angles of gimbal suspension and a, b, c, f g, h are constants. This control law is in fact standard of the control theory and constants are usually selected by stability and other criterions of this theory. Then we shall construct the control law:

The Delta calculates y/d and z/d:

A data-in of Delta are parameters of Measurements:

The System component solves differential equations of control law. These two equations of second order are decomposed by four ordinary differential equations of first order. It looks like:

Then we add another components for calculation of quaternions of orientation of Azimuth and Elevation components. We also add indication components. Click here to download this situation. A graphical analysis of quantity of control is presented on the following picture:

Light red and red curves correspond to azimuth of Vehicle and antenna orientation. Similarly light blue and blue curves correspond to elevation channel. Thus a control system is constructed. Also we may to install a virtual camera on antenna (Elevation) and observe 3D video of Vehicle motion. We may install camera on the Ground or the Vehicle. Virtual cameras are used at part 2 of this software description. Later we may to install virtual radio transmitter/receiver on the antenna and virtual electronic equipment etc. Click here to download this sample.

Example 2. On-off control element

A lot of control systems use on-off control elements those have the following in - out function

This function have a following math description:

If we have a several sets of photos of a 3D object, we can obtain its 6D position. In this example, this problem is solved by 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 n - the time moment. We shall use a recursive element for simulation of this element. Simple situation of usage of this element is presented below:

The Recursive component contains recursive equations of on-off control element. These equatins looks like:

As result we have following charts:

A red curve is an input and a green one is an output. Click here to download this sample.

Example 3. Continuous filter

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 continuous filter. Result of filtration is presented on the following picture:

A green curve is an input of the filter and a green on is an output. Output looks more smooth. It is shifted by phase. You can process obtained inverse time signal thought this filter to avoid this shift.

Example 4. Digital filter

Digital filter is any electronic filter that works by performing digital math operations on an intermediate form of a signal. We'll consider digital version of third order Butterworth filter. A Z-transform of this filter looks like:

where

The is a cutoff angular frequency. I've been constructed the filter as cascade of recursive elements with transfer functions . As well as continuous filter it performs smoothing.
Click here to download this sample of digital filter.

Example 5. Frequency difference detector

In this sample I've used digital version of standard frequency demodulation of FM signal. The method looks like:

The Diff component performs numerical differentiation. The Abs is a digital rectifier. The Integral is a digital low frequency filter. The input signal is a chirp. Result of demodulation is presented below:

The error of demodulation caused by nonlinearity gain-frequency characteristic of the filter.

Click here to download this sample

Example 6. Phase discriminator

This example is devoted to determination of cosine of differences of phases of two signals. I've used following correlation scheme:

The p1 and p2 contain two harmonic signals x(t) and y(t) having almost equal frequencies. The Multiplication calculates squares and product of these signals. The Integration performs digital calculation the following integrals:

The Integration also performs digital filtering with the following transfer function:

The Post calculates the cosine by the following expression:

As it was expected this discriminator shows the following picture of determination of cosine of differences of phases:

Click here to download this sample

Points of interests

A name "Universal framework" looks like pure advertisement only. However if I say that this software enables me operate with every kind of absorbing layer then it results to shock of my interlocutor. Then I show how can I operate with every kind. "Indeed your software enables you operate with every kind"- reply the interlocutor.