Linear approximation

About: Linear approximation is a research topic. Over the lifetime, 3901 publications have been published within this topic receiving 74764 citations.

Loss-Minimizing Control of PMSM With the Use of Polynomial Approximations

Abstract: Normally, lookup-table-based methods are being utilized for loss-minimizing control of permanent magnet synchronous motors (PMSMs). But numerous repetitive experiments are required to make a lookup table, and the program size becomes bulky. In this paper, analytic methods for finding the loss-minimizing solution are studied. Since the solution lies either in the interior or on the voltage limit boundary, two different cases are dealt with separately. In both cases, fourth-order polynomials are derived. To obtain approximate solutions, methods of order reduction and linear approximation are utilized. The accuracies are good enough for practical use. These approximate solutions are fused into a proposed loss-minimizing algorithm and implemented in an inverter digital signal processor. Experiments were done with a real PMSM developed for a sport utility fuel cell electric vehicle. The analytically derived minima were justified by experimental evidences, and the dynamic performances over a wide range of speed were shown to be satisfactory.

On the dynamics of a system of rigid bodies

Abstract: The existent treatises on rational mechanics do not pay attention to the dynamics of a system of rigid bodies, confining their interest to the general theorems of mechanics, the theory of rotation of a rigid body about a fixed point, some problems of analytical mechanics and the theory of stability. At the same time, the problems whose formulation is reduced precisely to the dynamics of a system of rigid bodies are quite common nowadays, at least during the last 30 years. An attempt to solve such problems on the basis of Lagrange’s second method leads, as a rule, to cumbersome differential equations with a large number of terms; the mechanical interpretation of those is not at all easy. Even computers are then of little help. A solution of the problem of motion of a system of rigid bodies in terms of a linear approximation is a superposition of harmonic motions, and is, therefore, sufficiently transparent. However, in order to get the important qualitative characteristics of the motion, it is necessary to take into account the non-linear terms of initial differential equations and to integrate these equations over a time interval considerably longer than the period of partial oscillations of the associated linear system. However, such integration leads to a rapid accumulation of errors because the actual calculations by the method of finite differences involve a large number of steps.

Efficient point coverage in wireless sensor networks

Abstract: We study minimum-cost sensor placement on a bounded 3D sensing field, R, which comprises a number of discrete points that may or may not be grid points. Suppose we have l types of sensors available with different sensing ranges and different costs. We want to find, given an integer σ ≥ 1, a selection of sensors and a subset of points to place these sensors such that every point in R is covered by at least σ sensors and the total cost of the sensors is minimum. This problem is known to be NP-hard. Let ki denote the maximum number of points that can be covered by a sensor of the ith type. We present in this paper a polynomial-time approximation algorithm for this problem with a proven approximation ratio \({\gamma = \sum_{i=1}^{\ell} k_{i} -\sigma+1}\). In applications where the distance of any two points has a fixed positive lower bound, each ki is a constant, and so we have a polynomial-time approximation algorithms with a constant guarantee. While γ may be large, we note that it is only a worst-case upper bound. In practice the actual approximation ratio is small, even on randomly generated points that do not have a fixed positive minimum distance between them. We provide a number of numerical results for comparing approximation solutions and optimal solutions, and show that the actual approximation ratios in these examples are all less than 3, even though γ is substantially larger.

Analytic Solutions to the Dynamic Programming Subproblem in Hybrid Vehicle Energy Management

Abstract: The computationally demanding dynamic programming (DP) algorithm is frequently used in academic research to solve the energy management problem of a hybrid electric vehicle (HEV). This paper is exclusively focused on how the computational demand of such a computation can be reduced. The main idea is to use a local approximation of the gridded cost-to-go and derive an analytic solution for the optimal torque split decision at each point in the time and state grid. Thereby, it is not necessary to quantize the torque split and identify the optimal decision by interpolating in the cost-to-go. Two different approximations of the cost-to-go are considered in this paper: 1) a local linear approximation and 2) a quadratic spline approximation. The results indicate that computation time can be reduced by orders of magnitude with only a slight degradation in simulated fuel economy. Furthermore, with a spline approximated cost-to-go, it is also possible to significantly reduce the memory storage requirements. A parallel plug-in HEV is considered in this paper, but the method is also applicable to an HEV.