Optimization of longitudinal motion of charged particles taking into account the requirements of sensitivity
DOI:
https://doi.org/10.31548/energiya2019.02.096Abstract
Summary. The statements of problems of structural-parametric optimization of the longitudinal motion of charged particles for the case of relay control are considered. If there are restrictions on the control function, the initial problems of minimax optimization are reduced to problems on the optimal choice of switching points (points at which the control function changes its value). For their numerical solution, an iterative gradient descent procedure is proposed. The area of the initial conditions for the phase coordinates of the particles is preliminarily represented in discrete form, and the gradient of the target function over the switching points is determined using sensitivity functions characterizing the magnitude of the rate of change of the disturbed motion relative to the calculated value of the parameter vector. To solve the problem of dimension and reduce the computation time of the sensitivity functions, an independent variable has been applied in the corresponding Cauchy problem. With this approach, the calculation of the optimal control parameters was carried out taking into account possible deviations of the calculated trajectories on the real operating modes of the system under study. An analysis of the proposed computational scheme and directions for further research on designing a low-sensitivity accelerator system by jointly solving the problem of trajectory optimization and the problem of minimizing the maximum sensitivity throughout the entire range of the functioning of the motion of charged particles is given.
Keywords: automatic control system, parametric optimization, relay control, switching points, gradient, sensitivity functions
Topicality. Solving the practical problems of optimizing the dynamics of charged bunches is associated with many difficulties in the computational plan . These tasks are minimax , the operation is given at a sufficiently large time interval, it is necessary to determine the accelerating and focusing fields according to the corresponding Maxwell equations for the simulation of the trajectories. Therefore, for numerical calculations, the specific tasks of controlling charged particles are parameterized with respect to the fields and apply the method of complication of the
© L.A. Pantaliyenko, 2019
мodel . This approach allows us to reduce the initial control task to finite-dimensional and to determine the optimal regimes in physically implemented structures.
However, the optimum parameters of the structure of the accelerating system are often not suitable for practice. The latter is explained by the fact that in real operating modes of the installation, the calculation parameters always have some spread due to technical reasons and conditions of the environment. In this case, small changes in the design parameters can cause significant deviations of the trajectory of the system from the optimal. In this regard, the relevance of setting tasks for the design of control systems, taking into account the requirements for the sensitivity of their parameters .
Analysis of recent research and publications. One of the directions of studying the dynamic and trajectory properties of charged particle beams is to develop methods of practical stability and structural-parametric optimization in relation to optimal formation of fields of accelerators of charged particles according to the given criteria . Analysis of the stability of parametric systems allows numerical calculation of optimal parameters, depending on the change of trajectories in real modes. Such statements serve as an important part of a complex of tasks for the design of low-sensitivity (insensitive) automatic control systems .
The purpose of the research ─ development of numerical methods for solving structural and parametric optimization of the longitudinal motion of charged particles taking into account the requirements regarding the sensitivity of their parameters.
Materials and methods of research. The methods of the theory of sensitivity, structural-parametric and undifferentiated trajectory optimization are used in the work.
Research results and their discussion. For the equations of longitudinal motion of charged particles, a number of problems of the class of undifferentiated trajectory optimization have been investigated. The case of relay control is considered in the presence of restrictions on the control function. An iterative gradient descent procedure was used for numerical calculation of the optimization problems formulated. In this case, the region of initial conditions by phase coordinates of the particles is pre-submitted in a discrete form, and the gradient of the function of the target by the switch points is determined using the sensitivity functions. In order to solve the problem of dimensionality and reduce the computational time, the replacement of a variable in the Cauchy problem with respect to the sensitivity functions was performed.
Conclusions and perspectives of further research. The formulation of the problems of undifferentiated trajectory optimization for the longitudinal equations of charged particles is considered. The algorithms for solving this class of problems with the help of sensitivity functions are proposed. With this approach, the calculation of optimal control parameters is carried out taking into account possible deviations of the calculated trajectories on the real operating modes of the system, and the definition of sensitivity functions in the process of optimization makes it possible to carry out further design of a low-sensitive accelerating system.
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