Modeling and Optimizing the Turn of a Loaded Elastic Rod Controlled by an Electric Drive
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ROL IN SYSTEMS WITH DISTRIBUTED PARAMETERS
Modeling and Optimizing the Turn of a Loaded Elastic Rod Controlled by an Electric Drive G. V. Kostin Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, 119526 Russia e-mail: [email protected] Received February 2, 2020; revised February 28, 2020; accepted March 30, 2020
Abstract—In this study, we consider a flat rotation of an electric rectilinear rod controlled by an electric drive, loaded at the free end by a point mass. The problem is to optimally control the voltage in the motor’s winding, which transfers the system to the final state with the damping of the elastic vibrations. A generalized formulation of the equations of the rod’s state and a method for approximating unknown displacements, momentum density, and bending moment are proposed. The procedure for minimizing the objective functional and regularizing the numerical error is described. DOI: 10.1134/S1064230720040103
INTRODUCTION Methods for optimizing the movement of systems with distributed parameters are being actively developed in control theory. The mathematical base for solving optimal control problems for dynamical systems, which are described by linear partial differential equations (PDEs), in the case of convex objective functionals, was laid in the studies of the Lions scientific school [1]. The controllability of systems with hyperbolic-type PDEs was considered by other researchers [2, 3]. An introduction to the control theory of distributed oscillations with a detailed review of the literature may be found in [4]. Oscillations in network models are studied, for example, in [5–7]. The practical need for reliable modeling of such systems has led to the development of special approaches to solving direct and inverse problems of dynamics. Two important directions in the study of controllability for spatially distributed processes can be identified. The first approach uses the so-called late discretization, in which the law of the optimal control is directly derived for the original model with PDEs. The input action as a function of time and, possibly, as a function of spatial coordinates is approximated by an element from a finite-dimensional subspace of the initial infinite-dimensional functional space only if necessary, at the last stage. Optimization in this case may be based on the spectral theory of linear operators [8, 9]. The variable separation method and decomposition based on the Fourier method are used [10], which for some of the simplest systems of mathematical physics allow constructing a quasi-optimal limited action that brings the system to a specified state in a finite time period. Another approach to constrained optimal control problems in the form of hyperbolictype PDEs is associated with the selection method [11]. Even for the simplest system that describes the movement of a closed string, obtaining an explicit form of control and proving its optimality require the use of a delicate mathematical apparatus [12]. The analysis of controllability and, in p
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