Application of Statistical Methods to the Mathematical Description of Betatrons
Abstract Generated abstract
This paper addresses the problem of describing and optimizing betatron gamma radiation intensity under random fluctuations of input parameters and uncontrolled disturbances. It argues that passive experimental data are insufficient for reliable control-oriented modeling over a broad factor range, and applies planned extremal experiment methods, specifically a central composite rotatable design, to four selected operating factors. The study constructs a second-order regression polynomial for radiation intensity using coded variables, including quadratic and interaction terms, and specifies the experimental levels and variation steps. The adequacy of the resulting betatron model is evaluated using Fisher’s criterion, providing a statistical basis for selecting control algorithms aimed at maintaining maximum radiation intensity.
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PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE
OF THE ORDER OF THE OCTOBER REVOLUTION AND THE ORDER
OF THE RED BANNER OF LABOR
NAMED AFTER S. M. KIROV
Volume 267 1975
APPLICATION OF STATISTICAL METHODS
TO THE MATHEMATICAL DESCRIPTION OF BETATRONS
M. S. Aleinikov, N. S. Bolnykh, V. A. Kochegurov, A. A. Novikov
(Presented by the scientific-technical seminar of the Department of Instruments and Devices of Automation Systems)
In betatrons, which have found wide practical application in industry, there are significant fluctuations in the intensity of bremsstrahlung γ-radiation, which are determined both by input parameters varying according to a random law with a normal distribution [1], and by uncontrolled disturbances. In this connection there arises the problem of finding and maintaining the maximum radiation intensity of betatrons under continuous fluctuations of the input parameters and in the presence of disturbances. For a well-founded choice of a control algorithm, it is necessary to study betatrons as objects of automatic control. Information about the properties of the controlled object is provided by a mathematical description, that is, equations describing the static and dynamic relationships that actually exist between input and output quantities. As shown in [1], for these purposes it is expedient to apply statistical methods. The intensity of γ-radiation depends on such input parameters as the voltage on the electromagnet winding, the injector, and the inflector; the moment at which voltage is applied to the injector (injection phase) and to the inflector (inflection phase), as well as the phase of dumping the accelerated electrons onto the target and a number of other factors. In [2] an equation of multiple regression (without taking into account insignificant relationships and factors) was obtained from the results of a passive experiment. A disadvantage of a passive experiment is that the relationship equation obtained from the material of this experiment does not contain substantially useful information about the process over a large interval of input variables [3]. This is explained by the following reasons:
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The interval of variation of the input and output variables is too narrow, i.e., the data are collected essentially at a single point.
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The initial assumptions on which methods for processing the results of a passive experiment are based are often not satisfied.
In addition, a passive experiment requires the development of special sensors and recording equipment, which does not always meet accuracy requirements and is associated with a significant amount of computation.
To obtain maximum information about the process under study with a minimum number of experiments (taking into account the results of the passive experiment), we shall use methods for planning extremal experiments [3].
We represent the unknown function \(y=f(x_1,x_2,\ldots,x_n)\), as in the processing of the results of a passive experiment, in the form of a second-order regression polynomial:
\[ y=b_0+\sum_{i=1}^{n} b_i x_i+\sum_{i=1}^{n}\sum_{j=1}^{n} b_{ij}x_i x_j+\sum_{i=1}^{n} b_{ii}x_i^2+\cdots \tag{1} \]
In order to estimate all \(k=\dfrac{(n+1)(n+2)}{2}\) coefficients of the quadratic model, it is necessary to choose a design in which each variable is varied at least at three different levels and the matrix of the normal equations is nonsingular. In addition, the design must satisfy a certain optimality criterion; namely, the experimental points in factor space should be placed on the basis of the conditions of orthogonality and rotatability of the design, or else the selected design should be required to be \(D\)-optimal [3, 4].
In practice one should use designs requiring the least expenditure for carrying out the experiment and allowing the coefficients of model (1) to be computed simply. Most often, in practice, central composite orthogonal (CCOD) and rotatable designs (RCD), developed by Box, are used to obtain a mathematical description of the object under study. They are obtained by adding “star” points of the type \((\pm\alpha,0,\ldots,0)\), \((0,\pm\alpha,0,\ldots,0)\), \((0,0,\ldots,0,\pm\alpha)\) and a certain number of central points \(n_0(0,0,\ldots,0)\) to a “core” formed by a full factorial experiment (FFE) of type \(2^n\) or by its regular replica. The number of trials in a composite rotatable design is determined by the formula:
\[ N=2^n+2n+N_0. \tag{2} \]
where \(N_0\) is the number of central points,
\(2^n\) is the FFE,
\(2n\) is the number of star points,
\(n\) is the number of factors.
For the selection of significant factors, screening experiments must be carried out, which in a number of cases require large expenditures. Therefore, in such cases it is important, before planning the main experiment, to use all available a priori information without resorting to experiments. Of great benefit here are the results of processing the data of a passive experiment and the conduct of an a priori ranking of factors based on a survey of qualified specialists operating the accelerator. The degree of agreement between the data of the a priori ranking and the results of processing the data of a passive experiment, which was carried out in one region of the factor space, proved to be relatively close [2].
To obtain a relationship equation of nonlinear form (for the same factors as in [2]) we use the matrix of a rotatable CCD. The variation interval for each parameter was chosen from the condition \(\lambda_{i\min}\ge 4\delta_i\), where \(\delta_i\) is the root-mean-square measurement error. The values of the basic level and of the variation steps for each factor are given in Table 1.
After processing the results of the experiment, an approximating polynomial of the following form was obtained:
Table 1
Experimental plan
| \(\varphi=-120\) msec | \(U_{\mathrm{mag}}=-250\) V | \(U_{\mathrm{inj}}=120\) V | \(U_{\mathrm{inf}}=105\) V | |
|---|---|---|---|---|
| Basic level | \(\varphi=-120\) msec | \(U_{\mathrm{mag}}=-250\) V | \(U_{\mathrm{inj}}=120\) V | \(U_{\mathrm{inf}}=105\) V |
| Variation step | \(\lambda=5\) msec | \(\lambda=\pm 10\) V | \(\lambda=\pm 5\) V | \(\lambda=\pm 5\) V |
Designations of factors
| \(x_0\) | \(x_1\) | \(x_2\) | \(x_3\) | \(x_4\) | \(x_1^2\) | \(x_2^2\) | \(x_3^2\) | \(x_4^2\) | \(x_1x_2\) | \(x_1x_3\) | \(x_1x_4\) | \(x_2x_3\) | \(x_2x_4\) | \(x_3x_4\) | \(y\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| +1 | −1 | −1 | −1 | −1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | 0,02 |
| +1 | +1 | −1 | +1 | −1 | +1 | +1 | +1 | +1 | −1 | +1 | −1 | −1 | +1 | −1 | 0,1 |
| +1 | −1 | −1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | −1 | −1 | −1 | −1 | +1 | 0,02 |
| +1 | −1 | +1 | −1 | +1 | +1 | +1 | +1 | +1 | −1 | +1 | −1 | −1 | +1 | −1 | 0,1 |
| +1 | +1 | +1 | −1 | −1 | +1 | +1 | +1 | +1 | +1 | −1 | −1 | −1 | −1 | +1 | 0,01 |
| +1 | +1 | −1 | −1 | +1 | +1 | +1 | +1 | +1 | −1 | −1 | +1 | +1 | −1 | −1 | 4,2 |
| +1 | −1 | +1 | +1 | −1 | +1 | +1 | +1 | +1 | −1 | −1 | +1 | +1 | −1 | −1 | 3,5 |
| +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | 1,5 |
| +1 | −1 | −1 | −1 | +1 | +1 | +1 | +1 | +1 | +1 | −1 | −1 | +1 | −1 | −1 | 0,014 |
| +1 | +1 | −1 | −1 | −1 | +1 | +1 | +1 | +1 | +1 | −1 | −1 | +1 | +1 | +1 | 2,8 |
| +1 | −1 | +1 | −1 | −1 | +1 | +1 | +1 | +1 | −1 | +1 | +1 | −1 | −1 | +1 | 5,0 |
| +1 | −1 | −1 | +1 | −1 | +1 | +1 | +1 | +1 | +1 | −1 | +1 | −1 | +1 | −1 | 0,65 |
| +1 | +1 | +1 | +1 | −1 | +1 | +1 | +1 | +1 | +1 | +1 | −1 | +1 | −1 | −1 | 0,00 |
| +1 | −1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | −1 | −1 | −1 | +1 | +1 | +1 | 3,0 |
| +1 | +1 | −1 | +1 | +1 | +1 | +1 | +1 | +1 | −1 | +1 | +1 | −1 | −1 | +1 | 5,5 |
| +1 | +1 | +1 | −1 | +1 | +1 | +1 | +1 | +1 | +1 | −1 | +1 | −1 | +1 | −1 | 0,02 |
| 1 | −2 | 0 | 0 | 0 | +4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,01 |
| 1 | +2 | 0 | 0 | 0 | +4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,01 |
| 1 | 0 | −2 | 0 | 0 | 0 | +4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2,3 |
| 1 | 0 | +2 | 0 | 0 | 0 | +4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,03 |
| 1 | 0 | 0 | −2 | 0 | 0 | 0 | +4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4,3 |
| 1 | 0 | 0 | −2 | 0 | 0 | 0 | +4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3,3 |
| 1 | 0 | 0 | 0 | −2 | 0 | 0 | 0 | +4 | 0 | 0 | 0 | 0 | 0 | 0 | 0,06 |
| 1 | 0 | 0 | 0 | +2 | 0 | 0 | 0 | +4 | 0 | 0 | 0 | 0 | 0 | 0 | 2,0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 5,2 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 5,2 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 5,2 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 5,2 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 5,2 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 5,2 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 5,2 |
\[ \begin{aligned} y_p={}&5{,}15-1{,}25x_1^2-0{,}976x_2^2-0{,}32x_3^2-1{,}04x_4^2\\ &-0{,}124x_1x_3-1{,}4x_1x_2+0{,}91x_1x_4+0{,}255x_2x_3\\ &-0{,}67x_2x_4+0{,}585x_3x_4, \end{aligned} \tag{3} \]
where
\[ x_i=\frac{X_i-X_i^*}{\lambda_i}. \tag{4} \]
\(X^*\) — the basic level of the factor,
\(\lambda_i\) — the variation step of the factor,
\(x_i\) is the coded value of the factor.
The adequacy of the obtained betatron model was checked using Fisher’s criterion [3]:
\[ F_{\mathrm{t}} > F_{p}. \]
\[ F = \frac{\sigma_{ag}^{2}}{\sigma_{y}^{2}} = \frac{N_c - 1}{f}; \tag{5} \]
where \(N_c\) is the number of central points,
\(f\) is the number of degrees of freedom.
\(F_{\mathrm{t}}\) is taken for \(k_1=f\) and \(k_2=N_c-1\), usually at the 5% error level.
LITERATURE
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M. S. Aleinikov, V. A. Kocheturov. Izv. TPI, vol. 193, Tomsk.
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V. V. Nalimov, N. A. Chernova, Statistical Methods for Planning Extreme Experiments. Moscow, “Nauka,” 1965.
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Problems of Experimental Design. Edited by G. K. Krug. Moscow, “Nauka,” 1969.