Linear approximation

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

Multiple target angle tracking using sensor array outputs

Abstract: The use of the output of an array of sensors to track multiple independently moving targets is reported. The output of each sensor in the array is the sum of signals received from each of the targets. The results of direction-of-arrival estimation by eigenvalue analysis are extended to derive a recursive procedure based on a matrix quadratic equation. The solution of this matrix quadratic equation is used to provide updated target positions. A linear approximation method for estimating the solution of the matrix equation is presented. The algorithm is demonstrated by the simulated tracking of two targets. The main advantage of the algorithm is that a closed-form solution for updating the target angle estimates has been obtained. Also, its application is straightforward, and the data association problem due to uncertainty in the origin of the measurements is avoided. However, it requires the inversion of an N*N as well as other linear operations, so that the computational burden becomes substantial as N becomes very large. >

Computing optical flow from an overconstrained system of linear algebraic equations

Abstract: A method is presented for the recovery of optical flow. The key idea is that the local spatial structure of optical flow, with the exception of surface boundaries, is usually rather coherent and can thus be appropriately approximated by a linear vector field. According to the proposed method, the optical flow components and their first order spatial derivatives are computed at the central points of rather large and overlapping patches which cover the image plane as the solution to a highly overconstrained system of linear algebraic equations. The equations, which are solved through the use of standard least mean square techniques, are derived from the assumptions that the changing image brightness is stationary everywhere over time and that optical flow is, locally, a linear vector field. The method has been tested on many sequences of synthetic and real images and the obtained optical flow has been used to estimate three-dimensional motion parameters with very good results. >

Drone routing with energy function: Formulation and exact algorithm

Abstract: Drone delivery is known as a potential contributor in improving efficiency and alleviating last-mile delivery problems. For this reason, drone routing and scheduling has become a highly active area of research in recent years. Unlike the vehicle routing problem, however, designing drones’ routes is challenging due to multiple operational characteristics including multi-trip operations, recharge planning, and energy consumption calculation. To fill some important gaps in the literature, this paper solves a multi-trip drone routing problem, where drones’ energy consumption is modeled as a nonlinear function of payload and travel distance. We propose adding logical cuts and subgradient cuts in the solution process to tackle the more complex nonlinear (convex) energy function, instead of using the linear approximation method as in the literature, which can fail to detect infeasible routes due to excess energy consumption. We use a 2-index formulation to model the problem and develop a branch-and-cut algorithm for the formulation. Benchmark instances are first generated for this problem. Numerical tests indicate that even though the original model is nonlinear, the proposed approach can solve large problems to optimality. In addition, in multiple instances, the linear approximation model yields routes that under the nonlinear energy model would be energy infeasible. Use of a linear approximation for drone energy leads to differences in energy consumption of about 9% on average compared to the nonlinear energy model.

Robustness Issues of the Best Linear Approximation of a Nonlinear System

Abstract: In many engineering applications, linear models are preferred, even if it is known that the system is disturbed by nonlinear distortions. A large class of nonlinear systems, which are excited with a ldquoGaussianrdquo random excitation, can be represented as a linear system G BLA plus a nonlinear noise source Y S . The nonlinear noise source represents that part of the output that is not captured by the linear approximation. In this paper, it is shown that the best linear approximation G BLA and the power spectrum S Y S of the nonlinear noise source Y S are invariants for a wide class of excitations with a user-specified power spectrum. This shows that the alternative ldquolinear representationrdquo of a nonlinear system is robust, making its use in the daily engineering practice very attractive. This result also opens perspectives to a new generation of dynamic system analyzers that also provide information on the nonlinear behavior of the tested system without increasing the measurement time.