Linear approximation

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

Linear Decision Functions, with Application to Pattern Recognition

Abstract: Many pattern recognition machines may be considered to consist of two principal parts, a receptor and a categorizer. The receptor makes certain measurements on the unknown pattern to be recognized; the categorizer determines from these measurements the particular allowable pattern class to which the unknown pattern belongs. This paper is concerned with the study of a particular class of categorizers, the linear decision function. The optimum linear decision function is the best linear approximation to the optimum decision function in the following sense: 1) "Optimum" is taken to mean minimum loss (which includes minimum error systems). 2) "Linear" is taken to mean that each pair of pattern classes is separated by one and only one hyperplane in the measurement space. This class of categorizers is of practical interest for two reasons: 1) It can be empirically designed without making any assumptions whatsoever about either the distribution of the receptor measurements or the a priori probabilities of occurrence of the pattern classes, providing an appropriate pattern source is available. 2) Its implementation is quite simple and inexpensive. Various properties of linear decision functions are discussed. One such property is that a linear decision function is guaranteed to perform at least as well as a minimum distance categorizer. Procedures are then developed for the estimation (or design) of the optimum linear decision function based upon an appropriate sampling from the pattern classes to be categorized.

A Novel Contour Error Estimation for Position Loop-Based Cross-Coupled Control

Abstract: Reduction of contour error is the main control objective in contour-following applications. A common approach to this objective is to design a controller based on the contour error directly. In this case, the contour error estimation is a key factor in the contour-following operation. Contour error can be approximated by the linear distance from the actual position to the tangent line or plane at the desired position. This approach suffers from a significant error due to linear approximation. A novel approach to contour error calculation of an arbitrary smooth path is proposed in this paper. The proposed method is based on coordinate transformation and circular approximation. In this method, the contour error is represented by the coordinate of the actual position with respect to a specific virtual coordinate frame. The method is incorporated in a position loop-based cross-coupled control structure. An equivalent robust control system is used to establish stability of the closed-loop system. Experimental results demonstrate the efficiency and performance of the proposed contour error estimation algorithm and the motion control strategy.

On time scaling for nonlinear systems: Application to linearization

Abstract: In this note, we propose a method to analyze systems in a time scale which is varied depending on the state such as dt/d\tau = s(x) (where t and τ are the actual time scale and that of new one, respectively, and s(x) is the function which we call time scaling function). Analysis of the system in the new time scale τ enables us to investigate the intrinsic structure of the system. A linearization problem in the new time scale is formulated as wide-sense feedback equivalence and is solved. It is also shown that the time scaling function which makes the system linear is derived as the solution of differential equations.