On the Possibility of Constructing a Self-Tuning Control System for Betatron Injection with Stochastic Search

A. A. Vorob'ev, V. M. Razin

In high-current betatrons [1, 2], electron injection is carried out from a high-voltage injector (several hundred kilovolts) using a deflecting capacitor-inflector. In this case, the optimal injection conditions, which ensure the maximum value of the charge of electrons accelerated in one cycle, depend on a large number of additional factors in comparison with a simple betatron. Here, in addition to the factors indicated above, such parameters as the shape and amplitude, as well as the generation phase, of this pulse acquire substantial significance. Thus, the process of optimizing the injection conditions, in the sense of ensuring maximum performance, becomes multiparametric.

\[ Q = Q\left(U_{\text{net}},\, U_i,\, \varphi_i,\, I_i,\, U_{in},\, \varphi_{in}\ldots\right). \tag{1} \]

Here \(Q\) is the charge of electrons captured in the process of betatron acceleration;

\(U_{\text{net}}\) is the voltage of the mains supplying the betatron electromagnet, which affects the magnetic-field intensity at the moment of injection;

\(U_i\) is the voltage (injection energy);

\(\varphi_i\) is the generation phase of the injection-voltage pulse;

\(I_i\) is the emission current of the injector cathode at the moment of injection;

\(U_{in}\) is the voltage on the inflector (pulse amplitude);

\(\varphi_{in}\) is the generation phase of the voltage pulse on the inflector.

In addition to the parameters indicated, there is a substantial influence of such factors as the vacuum depth in the vacuum chamber, the position of the injector and inflector relative to the vacuum chamber and the magnetic field in the working zone, and a number of others, which may undergo changes during operation of the betatron.

Some of the disturbing actions can be compensated or stabilized; the influence of others is insignificant. As applied to optimization of the functional (1), in our view, one may restrict oneself to taking into account three control quantities, such as the injection phase \(\varphi_i\), the inflection phase \(\varphi_{in}\), and the inflection voltage \(U_{in}\), considering \(U_{\text{net}}\) and \(U_i\) to be unregulated. In the absence of stabilization of \(U_{\text{net}}\) and \(U_i\), their influence can be taken into account analytically and compensated by regulation with respect to disturbances. As for the influence of the emission current of the injector cathode, more precisely, the electron current on the conduct-

...the layer adjacent to the wall of the vacuum chamber, then here one can proceed in two ways. One route consists in stabilizing the emission current by known methods. The other method consists in the fact that the weak extremal dependence of the radiation maximum on the emission current is brought to a maximum during the initial adjustment. Random changes in the cathode emission current will not have a substantial direct effect on the conditions for capturing electrons into betatron acceleration. The influence may be indirect, through an effect on the amplitude of the injection voltage. Stabilization of the filament voltage is also necessary under these conditions, since the filament voltage has a very strong effect on the emission characteristics of the cathode; however, the required stabilization is not very high. In addition, it should be borne in mind that stabilizing the filament voltage of the injector cathode is technically considerably simpler than implementing stabilization of the emission current.

Fig. 1. Block diagram of a betatron-radiation optimizer with stochastic search

In the monograph by L. A. Rastrigin \[3\] it is shown that, in problems of optimizing multiparameter systems, in a number of cases a random search with self-learning is expedient. The operating principle of such an optimization system is illustrated in Fig. 1. In this case, generators of random pulse sequences, at definite instants of time specified by a synchronizer, operate randomly, either issuing or not issuing at that instant a pulse into the control system for increments of the coordinates $(\Delta x_1, \Delta x_2 \ldots \Delta x_n)$ of the controlling actions. The probabilistic characteristics of the random generators are readjusted under the action of the control voltage $e_{\mathrm{ctrl}}$ in such a way that, as the voltage increases, the probability of generating a pulse at the synchronization instant increases. After integration, the random search changes of the coordinates $\Delta x_1, \Delta x_2 \ldots \Delta x_n$

act on the object of optimization in the form of control actions \(x_1, x_2 \ldots x_n\). The quality function is characterized in the system by the parameter \(Q\). Further, the system provides devices for measuring the signs of the increments \(\Delta x_i\) and \(\Delta Q\), i.e., for determining \(\operatorname{sign}\Delta x_i\) and \(\operatorname{sign} Q\).

The self-learning algorithm is written as follows:

\[ e_{\mathrm{ctrl}_i}^{(N+1)} = e_{\mathrm{ctrl}_i}^{(N)} + \delta\left(\operatorname{sign}\Delta x_i^{(N)}\right) \left(\operatorname{sign}\Delta Q_N\right), \tag{2} \]

where \(e_{\mathrm{ctrl}_i}^{(N+1)}\) is the voltage controlling the probabilistic properties of the \(i\)-th randomness generator at the \(N+1\)-st search step;

\(\Delta x_i^{(N)}\) is the increment of the \(i\)-th controlled parameter at the \(N\)-th search step;

\(\Delta Q_N\) is the increment of the quality function at the \(N\)-th search step;

\[ \Delta Q_N = Q_N - Q_{N-1}, \]

\(\delta\) is the “speed” parameter of learning.

The product \(\left(\operatorname{sign}\Delta x_i^{(N)}\right)\left(\operatorname{sign}\Delta Q_N\right)\) is formed by the corresponding logic circuit. The values \(e_{\mathrm{ctrl}}\) are formed and accumulated at the outputs of the integrators \(\dfrac{1}{p\tau_2}\), where \(\tau_2\) is the time constant of the storage elements.

Self-learning increases the convergence of the search and the noise immunity of optimization; however, operation of the optimizer in accordance with algorithm (2) leads to predetermination and increased search losses [3].

In our opinion, the technical implementation of the optimization method with random search and self-learning during operation does not cause serious difficulties when applied to betatron installations.

In betatron installations of broad industrial use, it is expedient to apply a double-magnetization scheme for the zero-field sensor in such a way as to ensure the maximum value of the mathematical expectation of the radiation-dose power in the presence of significant fluctuations of the line voltage. In practice, the necessary adjustment is carried out by selecting the proper position of the sliders of the regulating autotransformers [4].

It is clear that this control system may be regarded as a two-parameter object, and a system of automatic optimization with respect to two parameters may be applied to it. The artificial introduction, proposed by the author, of two parameters for regulating the injection phase instead of the generally accepted single parameter makes it possible to produce a system that provides optimization in the presence of random changes of the line voltage over a wide range, as occurs under production conditions.

The role of the integrators \(\dfrac{1}{p\tau_1}\) in the block diagram of Fig. 1 may be performed by reversible dc electric motors controlling the position of the sliders of the regulating autotransformers. In all other respects, the control circuit of Fig. 1 may be technically implemented using the same elements as in [3].

Of considerable interest is a more advanced self-learning algorithm considered in [5]. This algorithm is defined as follows. Let, in the Cartesian space of the optimized parameters \(x_1, x_2 \ldots x_n\), a vector \(\vec W(w_1, w_2 \ldots w_n)\) be specified. On the vector \(\vec W\), a hyperellipsoid of revolution is constructed with foci at the beginning and end

of this vector. The parameters of the ellipsoid of revolution are determined by the system of algebraic equations

\[ |W|=2C, \]

\[ a^2+(n-1)b^2=1, \]

\[ a^2-b^2=c^2, \]

where
\(a\) is the major semiaxis of the ellipsoid;
\(b\) is the minor semiaxis of the ellipsoid;
\(c\) is the distance from the center of the ellipsoid to the focus;
\(n\) is the number of parameters of the system being optimized.

The length of the vector \(\vec W\) determines the degree of probability of the hyperellipsoid, and its direction determines the orientation of the hyperellipsoid. This ellipsoid is called the hyperellipsoid of probabilities.

The random-step vector \(\vec \xi\) is defined as follows. On the surface of the hyperellipsoid of probabilities a random point is selected according to a distribution law uniform over this surface. A straight line is drawn through this point and the beginning of the vector \(\vec W\), and this line determines the direction of the random step. The hyperellipsoid of probabilities determines the \(n\)-dimensional probability distribution law \(P(\vec \xi)\) of the random-step vector in the parameter space of the system being minimized.

The self-learning process is determined by the recurrent formula

\[ \vec W^{(N)}=K\vec W^{(N-1)}-\delta(\Delta Q_{N-1}-d)\Delta x_{N-1}, \tag{3} \]

where
\(\vec W\) is the vector parameter of self-learning;
\(K\) is the coefficient of “forgetting” of previous experience;
\(\delta\) is the coefficient of the “learning rate”;
\(\Delta Q_N=Q_N-Q_{N-1}\) is the increment of the quality function, computed after the \(N\)-th step;
\(\Delta x_N=x_N-x_{N-1}\) is the displacement in parameter space during one, the \(N\)-th, search step;
\(d\) is the coefficient of “skepticism.”

In order that the system not become deterministic, a restriction is introduced for the vector \(W\):

\[ \vec W= \begin{cases} \vec W, & \text{for } |\vec W|<l,\\ l, & \text{for } |\vec W|>l. \end{cases} \tag{4} \]

Optimization of multiparameter systems by the method of statistical search with self-learning makes it possible to increase the convergence of the search and improves its noise immunity; however, hardware implementation of the algorithm in accordance with formula (3) presents certain difficulties. Taking into account the advantages of statistical search in optimization, one should, however, pose and attempt to solve the problem of implementing this algorithm as applied to high-current betatrons, since it is precisely there that the optimization problem is multiparameter; moreover, it seems obvious that computer facilities can find the widest application in this case. Apparently, the most expedient approach would be the development of a specialized control computing device.

In conclusion, we note that systems of this kind may prove promising for automatic regulation of the electron beam during injection and acceleration.

References

1. V. A. Moskalev, M. F. Filippov, A. G. Skorikov, Yu. M. Skvortsov. Izv. vuzov “Fizika”, No. 5, 35–44, 1959.

2. B. V. Okulov, A. E. Erzylyov. Proceedings of the Third Interuniversity Conference on Electron Accelerators, 117–123. Publishing House of Tomsk State University, Tomsk, 1961.

3. L. A. Rastrigin. Random Search in Problems of Optimization of Multiparameter Systems. “Zinatne” Publishing House, Riga, 1965.

4. V. M. Razin. Izv. TPI, vol. 87, 206–215, 1957.

5. L. A. Rastrigin, K. K. Ripa, L. V. Sytenko. In: Self-Learning Automatic Systems, p. 280. “Nauka” Publishing House, Moscow, 1966.