Analysis and Synthesis of Spatially Multidimensional Distributed Systems with a Descriptor Structure of the Mathematical Model

Abstract

The problem of plasma stabilization in a magnetic field is considered. The plant is described by the hydrodynamics model of a weightless potential ideally conducting fluid in a cylindrical coordinate system. The deviation of the perturbed surface, the velocity of deflection of the perturbed surface, the acceleration potential are the functions of state. The acceleration potential on the outer generatrix of cylinder surface is used as the control. The state and control functions depending on the radial and axial coordinates are presented as series with respect to orthonormal systems of functions. Via the spectral method of the distributed systems analysis and synthesis the theoretical positions for the transition from the initial mathematical model described by PDE system to an infinite descriptor system are done. The components of state and control vectors of the descriptor system are the time-varying amplitudes of the spatial harmonics of the series. Expressions for the calculation of the operational differentiation matrices, the operational matrices of factors depending on spatial coordinates depended vectors, the matrices of boundary conditions are given. The matrices of the descriptor plant representation are obtained. The questions of existence, uniqueness, and convergence of the solution of the received descriptor system are investigated. For the plant represented by the descriptor system, a control low described by the differential system is constructed. The solution of the problem is based on the variational approach. The H2 optimization criterion is used. The analysis and synthesis of the closed-loop system via Matlab is done. The results of the analysis indicate that there is no overshoot and oscillation in the system. Calculations based on a limited number of equations show that with the increase in the number of differential and algebraic equations and, respectively, the spatial modes of the expansion of the control functions and the state of the process, the obtained result tends to a certain limit.