A Method for Constructing a Betatron Model

V. V. Zakharov, L. M. Rogachevskaya

(Presented by the seminar of the AU sector of the Research Institute of Nuclear Physics and Electronics)

The portable betatron PMB-6, manufactured serially at the Research Institute of Nuclear Physics of TPI, is regulated by means of two parameters: the current and the injection phase. In order to make maximum use of the capabilities of the betatron, it is necessary to equip it with a small optimizer whose connection would ensure the maximum value of the intensity of $\gamma$-radiation, considered as a function of the controlled variables $x_1$ and $x_2$ (the current and the injection phase, respectively). In order, from several possible variants of optimizers, to choose the best one as applied to the PMB-6, one should compare the efficiency of the optimizer algorithms by means of simulation on a computer. First of all, the question arises of constructing a betatron model intended for these purposes.

Preliminary studies showed that the PMB-6 betatron model is rather complex and cannot be adequately described by a polynomial of the second degree over the entire range of variation of the variables $x_1$ and $x_2$. The use of polynomials of higher degrees, however, is associated with substantial computational difficulties [1]. In the present article, for constructing a betatron model, one heuristic method is proposed, based on the idea of penalty functions for approximation to the boundary of a region, used in [2] for solving a nonlinear programming problem. This idea is also applied in [3] for constructing functions of many variables used for comparing methods of extremum search.

As applied to the construction of models of objects of small dimension (with the number of controlled variables of the order of 2–3), the method consists in the following. Suppose it is necessary to construct a function $F(x_1,x_2)$ whose level lines reproduce the outlines of the level lines of the object (obtained from experiment). We construct some region $T=\{x/f_j(x_1,x_2,a_j)>0,\ j=\overline{1,m}\}$, using basic functions $f_j(x_1,x_2,a_j)$ of the simplest form. By changing the components of the vectors $a$, we achieve approximate coincidence of the outlines of the region $T$ with some level line $F(x_1,x_2)=\mathrm{const}$ and fix the parameters $a_{ji}$ $(i=\overline{1,n})$.

We choose the modeling function in the form

\[ \widetilde{F}(x_1,x_2)=\sum_{j=1}^{m} d_j \ln f_j(x_1,x_2,a_{ji}). \tag{1} \]

Function (1) imposes a penalty for approaching the boundary of the region \(T\) and repeats the outlines of the region near the boundaries. As the example of constructing a betatron model given below shows, by a further selection of the parameters \(a_{ji}\) one can obtain a significant similarity of the outlines of the level lines \(F=\mathrm{const}\) and \(\tilde F=\mathrm{const}\).

In Figs. 1 and 2 the level lines are given for one realization of the radiation intensity \(I(x_1,x_2,t)\), a random function of the arguments \(x_1\) and \(x_2\), at the instants of time \(t_1\), \(t_2\), respectively. The dependences shown in Figs. 1 and 2 were taken from different betatrons.

Fig. 1

Fig. 2

Comparing these figures, we note that the absolute maximum drifts approximately parallel to the \(ox_1\) axis and is located close to the right-hand part of the zero level lines. There is a local maximum (the maxima in Figs. 1–2 are marked by crosses). The right-hand part of the zero level lines changes only slightly with time. These features of the random variation of the intensity were taken into account in the modeling.

As elements of the modeling functions we take six basic functions

\[ \begin{aligned} f_1&=a_{10}-a_{11}x_1-a_{12}x_2,\\ f_2&=a_{20}-a_{21}x_1,\\ f_3&=-a_{30}+a_{31}x_1+a_{32}x_2,\\ f_4&=-a_{40}+a_{41}x_1,\\ f_5&=a_{50}(x_1-a_{51})^2+a_{52}(x_2-a_{53})^2+a_{54},\\ f_6&=a_{60}(x_1-a_{61})^2+a_{62}(x_2-a_{63})^2+a_{64}, \end{aligned} \]

where \(a_{ji}>0,\ m=6,\ n=2\).

We choose the betatron model in the form

\[ \tilde I(x_1,x_2)=\sum_{j=1}^{6} a_j \ln f_j(x_1,x_2)+a_7 . \tag{2} \]

We impose the following restriction on the functions \(f_j(x_1,x_2)\) \((j=\overline{1,4})\): the straight lines \(f_j(x_1,x_2)=0\) must coincide with the previously selected straight lines on the plane (in Figs. 1–4 they are shown by dash-dotted lines). Owing to the logarithmic functions, the level lines of the modeling function (2) will repeat the outlines of the boundary of the region \(T=\{x:f_j(x_1,x_2)>0,\ j=\overline{1,4}\}\). Taking into account the behavior of the level lines \(I(x_1,x_2,t_i)\) outside this region, we introduce the basis functions \(f_5\) and \(f_6\)—elliptic paraboloids. The paraboloids \(f_5, f_6\) are placed so that the coordinates of the center of the paraboloid \(f_5\) are located

Fig. 3

Fig. 4

in the region of the minimum, and the coordinates of the paraboloid \(f_6\) are located near the local maximum of the modeled function \(I(x_1,x_2,t_1)\), shown in Fig. 1. In order that the model \(\tilde I(x_1,x_2)\) be defined over the entire domain of values \(x_1, x_2\), we extend it beyond the boundary \(f_3(x_1,x_2)=0\) as follows. Put \(f_3(x_1,x_2)=\delta/(1+\delta-f_3(x_1,x_2))\), if \(f_3(x_1,x_2)<\delta\) \((\delta>0)\). For the same purpose we additionally define \(\tilde I(x_1,x_2)=0\), if \(f_1(x_1,x_2)<0\). Varying on the BESM-4 computer the parameters \(\delta_{ji}\) and visually estimating the degree of closeness of the obtained model and the realization of the random intensity function \(I(x_1,x_2,t_1)\) (Fig. 1), we obtain as a satisfactory model (Fig. 3):

\[ \begin{aligned} \tilde I(x_1,x_2)={}&2\ln(10-0.008x_1-1.773x_2)+4\ln(-30+0.057x_1+5.19x_2)+{}\\ &+\ln(2-0.0112x_1)+2\ln\{0.01(x_1-60)^2+10(x_2-3.1)^2+1\}-{}\\ &-0.6\ln\{0.1(x_1-146)^2+10(x_2-3.6)^2+1\}+0.6\ln(-1+0.025x_1)+9 . \end{aligned} \tag{3} \]

In an analogous manner, for the random function \(I(x_1,x_2,t_2)\) (Fig. 2) we obtain the model (Fig. 4)

\[ \begin{aligned} \tilde I(x_1,x_2)={}&2\ln(10-0.0012\,x_1-1.85\,x_2)+4\ln(-30-0.012\,x_1+7.65\,x_2)+{}\\ &+\ln(2-0.003\,x_1)+0.01\ln\{0.01(x_1-60)^2+10(x_2-3.1)^2+1\}-{}\\ &-0.02\ln\{0.1(x_1-146)^2+10(x_2-36)^2+1\}+0.6\ln(-1+0.025\,x_1)+14. \end{aligned} \tag{4} \]

It is important to emphasize that the form of the dependence \(\tilde I(x_1,x_2)\) is preserved for different betatrons: expressions (3) and (4) differ only by constant coefficients. This circumstance makes it possible to speak of function (2) as a model of the PMB-6 betatron.

In conclusion, we note that there is no point in further detailing the form of (3), (4), since in an experimental verification of optimizers (or their algorithms) it is convenient to regard the models (3), (4) as certain random realizations \(I(x_1,x_2,t)\). Then, in the process of an experimental search for the maximum, one can obtain a whole spectrum of models determined by realizations of the parameters \(a_{ji}\), which are modeled as normally distributed random variables.

REFERENCES

1. V. V. Nalimov, N. A. Chernova. Statistical Methods for Planning Experiments. Moscow, “Nauka,” 1965.

2. C. W. Carrol. The CRS Technique for Optimising Nonlinear Restrained Systems. Operations Res., v. 9, No. 2, 1961.

3. V. V. Zakharov. A system of test functions for experimentation with methods of searching for an extremum. “Avtomatika i vychislitel'naya tekhnika,” 1968, No. 4.