|
Academic Open Internet Journal |
Volume 13, 2004 |
Linearization output parameters for electron beam accelerator with neural network
Sotir Sotirov* - ssotirov@btu.bg, Andrey Nenov** - andrey@nenov.com
*Burgas University "prof. Asen Zlatarov" - Burgas , Bulgaria
** Wholesale
Wealth Systems, Scottsdale,
AZ, USA
www.wholesalewealthsystems.com
Abstract:
The accelerators of the elementary pieces are of particular importance for the research of the matter structure – the structure of the elementary pieces, the nature of the nuclear power, etc. This paper approaches a simple, quick and easy way of implementing algorithm for a linearization of electron beam current using neural network.
Introduction:
The accelerators of the elementary pieces are of particular importance for the research of the matter structure – the structure of the elementary pieces, the nature of the nuclear power, etc. As a rule, accelerators with energy not more than several tens megaelectron-volts are used for applied purposes. The industrial accelerators of electrons, used mostly in the processes of radiation-chemical modification and sterilization of materials, are provided in most of the cases with safety systems for monitoring the level of radiation and the electric parameters controlling the beam. The analyses of the matter meet the requirements for strict accuracy and authenticity of the received results. It is often difficult to measure directly all working characteristics because of the specific working zone. Scientists have problem to measure real electron beam data and compare it with data from standard sensors. Indirect methods are applied quite often to evaluate the effect of the electron beams, as well as the quantities of energy the materials have absorbed.
Aurora IV electron accelerator used at Burgas University has got maximum energy of 750 keV, 60 mA, and initial window dimensions 2000 x 180 mm. Because of the nature of the electron beam its angle of descent at the two sides of the window is quite big (fig.2). Due to this fact the quantity of the received energy is different from the one received at the zone where the beam descends perpendicular. The uneven distribution of the descending beam (energy) is due to the impossibility of 100% focusing on a definite point without affecting the adjacent zones. This problem causes inaccuracy when reading the absorbed doses and it is therefore basic for the analysis of the investigated materials. The received parameters of the processed patterns are characterized by great inaccuracy depending on the zones of radiation they have been in. The accelerator is provided with a precise dosimetric control system, as well as with a system for controlling the provided energy – milliamperemetre, voltmetre, and controlling the maximum diversion and focus of the beam.
Solution of the problem.
To solve the above mentioned problem our team has developed an indicator for measuring the intensity and a computer system for analysis of the provided energy that reads directly the energy supplied to the material
The indicator consists of a wide wafer with chess-board fields, which is laid under the radiating window of the accelerator. The descending beams describe and fall consecutively on every single metal element of the indicator while the microprocessing system reads every level of the energy.
 |
Fig.1. Initial form of the signal
With the help of the indicator our initial research has shown that it is almost impossible to achieve good focus of the beam without automatic regulation system. All received graphics have had nonlinear form – fig.1
In order to receive best accuracy of the synchronization and the control of the electron beam we have developed a model with a neural network for adjustment of the initial beam.
It has been found out that the following types of initial characteristics of the beam are appropriate for the research needs of the laboratory:
À) Triangle – the focusing of the beam has got a linear form of decreasing when moving away from the focusing point
I = Icentral/d
Icentral is the intensity of the beam in the focusing center and d is the distance to the measured point
B) Rectangular – there is a sharp difference in the intensity in and out of the focusing point. Rectangular form, however, can be achieved only in the idealized case is – the standard form of this solution is trapezium-sized.
A better decision for the research needs would be the solution of the rectangular form, but it is difficult to be achieved. Because of that we have developed a system for linearization output parameters of the beam on the basis of neural network
To linearize the information of the indicator additional processing on the basis of neural network is done.
 |
Fig.2 Visualization of the beam currents
When moving the beam starts to re-cover the indicator to a greater extent and as a result the received signal has got the form shown at (Fig.2).
Levenberg-Marquard Algorithm [3] is a variation of the Newton method and it is used to minimize the functions which are sums of nonlinear functions.
If the error vector is indicated as
(1)
and the parameters vector is
(2)
In Levenberg-Marquard Algorithm Taylor’s line is described as
(3)
equals zero and therefore Newton’s method is
defined with the equation
(4)
If the sensitivity index of one neural network is F(x) and stands for
(5)
where 
(6)
and 
(7)
if we accept that F(x) is the sum of quadratic functions:
(8)
then the j element of the gradient will be
(9)
the gradient could be written in matrix form
(10)
where 
(11)
is Jacobi’s matrix
then we can find out Hesianov’s matrix. The elements k,j from Hesianov’s matrix can be
(13)
Hesianov’s matrix can be expressed in matrix form
(14)
where 
(15)
If we accept that S(x) is small then we receive the
following 
(16)
Replacing (15) and (10) in (5) we come to the equation of Gauss-Newton method
(17)
To avoid the
problem with the inverting of the matrix received by Gauss-Newton method we
apply the equation of Levenberg-Marquard method, namely 
(18)
where I is a separate matrix and m is a coefficient. If we replace (1) and (2)
in (11) we come to Jacobi’s matrix of multilayer neural network
(19)
Jacobi’s matrix
can be calculated on the basis of Backpropagation algorithm
with little changes. In the standard Backpropagation algorithm
the change of the error is calculated 
(20)
For the Jacobi’s matrix elements we have to calculate
(21)
In the standard algorithm Backpropagation derivative
(22)
where the first member of the right side is defined as sensitivity:
(23)
and is calculated from the last to the first layer.
In Levenberg- Marquard algorithm the so called Marquard sensitivity is used:
(24)
where h=(q-1)SM+k
Now we can calculate the elements of Jacobi’s matrix
(25)
when xi is a weigh coefficient, and if xi is a displacement then
(26)
Marquard sensitivity in the latter is
(27)
when the entrance Pq is applied to the neural network then at the exit we come to àÌq, Levenberg- Marquard algorithm begins with
(28)
where
(29)
Then the distribution of the calculation of Maquard sensitivity will be
(30)
The weight coefficients Wni,j
and the displacement bni. can be re-calculated like this
Realization
The neural network consists of three layers and the number of the neurons in the hidden layer is six. Indirectly every layer characterizes a particular object (process) in the system for controlling the electron beam.
The neural network is pre-trained in a medium of MATLAB. The medium is a common way for mathematical solutions and it gives opportunities for analogue analyses in other similar laboratories and circles.
The change in the error of training is shown in Fig.4 and the schedule of training of the neural network is shown further below.
 |
Fig.4 Mean square error
Conclusion
The developed system for control of the electron beam is an important solution for the specialists working with accelerators of small pieces. It allows a significant improvement of the control characteristics and parameters of the investigated materials.
In addition this system and approach can be successfully applied to decisions in the field of monitors and televisions due to the similar working principle of the electron accelerators. Still more, the focus of the beams and the artificial intellect are in the basis of these technologies.
References
[1] A.G.Nenov, S.N.Sotirov, "PC - based method for measuring of beam of electrons" - 2nd Symposium on Applied Electromagnetism of trans black Sea Region, 27-29 June 2000, Greece
[2] S.N.Sotirov, A.G. Nenov, P. Ilieva, "PC-based system for measurement output parameters electron accelerator" -14th International Conference “Systems for Automation of Engineering and Research” (SAER'2000) 18-19 September 2000 - St. Konstantin, BULGARIA
[3] M.T.Hagan, H.B.Demuth, M.Beale, “Neural Network Design”, PWS Publishing Company, Boston, 1996
Technical
College - Bourgas,
All rights reserved, © March, 2000