Оn the design of low-sensitivity accelerating-focusing systems using metods of practical stability

Abstract

The mathematical models of problems which are included in the complex of problems of designing of low-sensitive accelerating-focusing systems are considered. With this approach, the calculation of optimal control parameters is made taking into account possible deviations of their calculated values in real modes. To account for the sensitivity requirements, algorithms of practical stability for parametric systems in the space of sensitivity functions were applied. In the framework of the formulated problems, the equations of motion of particles in an accelerating voltage field in the absence of Coulomb interaction forces are investigated. The formulation of tasks for calculating the structure of an accelerator with optimal beam characteristics in dynamics in real modes is given. For the case of relay control, the original optimization tasks are reduced to the tasks of optimal selection of switching points in the presence of sensitivity requirements.

Keywords: mathematical model, sensitivity, practical stability, parametric system, dynamic constraints

Topicality. By solving optimization problems with the help of parametrization methods, problems of designing various complex systems are caused . In this case, the original control problem is reduced to a finite-dimensional nonlinear programming problem with respect to the parameters characterizing the transient process, the control functions and the structure of the control object.

In practice, due to various types of physical and technological reasons, the values of real parameters always differ from the estimated (optimal), which leads to a change in the characteristics and quality of the system's operation. If the magnitude of the deviation of the characteristics will be quite large, then the system under study becomes generally inoperable. In this connection, the task of designing low-sensitivity (insensitive) control systems and calculating guaranteed sensitivity becomes relevant .

It turns out that when designing real systems of sensitivity problems, they are closely associated with the formulation of problems of practical stability of parametric systems. The latter allows us to apply the practical stability algorithms in the corresponding function space . The specification of this approach is the tasks included in the task of designing low-sensitive accelerating-focusing systems for the equation of motion of particles in an accelerating voltage field without taking into account the forces of the Coulomb interaction.

Analysis of recent research and publications. One of the directions of the development of the theory of stability is the numerical solution of stability problems on a finite period of time . Unlike other statements , the analysis of the stability of parametric systems  can significantly expand the range of problems to be studied and solve them numerically. In addition, such formations are an important component of the design of low-sensitivity control systems.

The purpose of the research is to develop numerical methods for solving the problems of designing low-sensitivity accelerating-focusing systems by practical stability methods.

Materials and methods of research. The paper uses methods of stability theory, sensitivity and structural-parametric optimization.

Results of the research and their discussion. Let us consider one of the models of the equations of motion of charged particles of the form

 Here the first two equations characterize the longitudinal motion, and the last two are transverse. − the amplitude of the longitudinal component of the stress of the accelerating field; , ,  − spatial coordinates of particles.  − a function that further provides focusing on radial coordinates;  −  wavelength of high-frequency field;  − the length of the accelerator.

         Sometimes, instead of an independent variable , a new dimensionless coordinate   is introduced. Then equations (1) acquire the form

In  the equation of motion in planes , , they coincide, so instead of (2) it is expedient to simulate such a system of equations:

         We formulate a sequence of tasks for calculating the structure of an accelerator with optimal beam characteristics in dynamics, taking into account real operating conditions.

         Task 1. Determine the parameters of the accelerator structure so that the capture of particles in the process of acceleration by phase and energy was maximal after given phase-energy spread of the beam at the end of the accelerator.

         Task 2. Given the radial vibrations for a particle beam, determine the structure of the accelerating-focusing system in order to obtain a minimum spread over the phase coordinates at the output and fulfill the requirements (restrictions) for the functions of sensitivity in the dynamics.

         Task 3. For a beam of trajectories for a given initial set of capture, find parameters of the accelerating-focusing system that minimize the sensitivity of the state vector to the interval of functioning.

One of the optimization tasks will be to minimize the maximum deviation of the particles by energy from the given:

For the case of relay control (4), the optimal selection of switching points      is reduced to the problem. This approach allows for the design of a low-sensitive accelerator system by solving the problem (1) and minimizing the maximum sensitivity

         If the initial conditions are selected from a given set , then the switching points must be chosen in such a way as to minimize the functional  and perform sensitivity limitations on features, such as appearance

          The given problems belong to the class of problems of undifferentiated optimization , and restrictions (4) are taken into account by stability algorithms .

Conclusions and perspectives of further research. The formulation of tasks of low-sensitive accelerating-focusing systems is formulated. For solving problems with limited sensitivity, algorithms for practical stability of parametric systems are proposed. This approach can also be implemented for other types of accelerator models and optimal control.