Piecewise linear function

About: Piecewise linear function is a research topic. Over the lifetime, 8133 publications have been published within this topic receiving 161444 citations.

The Focus-Center-Limit Cycle Bifurcation in Symmetric 3D Piecewise Linear Systems

Abstract: The birth of limit cycles in 3D (three-dimensional) piecewise linear systems for the relevant case of symmetrical oscillators is considered. A technique already used by the authors in planar systems is extended to cope with 3D systems, where a greater complexity is involved.Under some given nondegeneracy conditions, the corresponding theorem characterizing the bifurcation is stated. In terms of the deviation from the critical value of the bifurcation parameter, expressions in the form of power series for the period, amplitude, and the characteristic multipliers of the bifurcating limit cycle are also obtained.The results are applied to accurately predict the birth of symmetrical periodic oscillations in a 3D electronic circuit genealogically related to the classical Van der Pol oscillator.

Polyhedral Tools for Control

Abstract: Polyhedral operations play a central role in constrained control. One of the most fundamental operations is that of projection, required both by addition and multiplication. This thesis investigates projection and its relation to multi-parametric linear optimisation for the types of problems that are of particular interest to the control community. The first part of the thesis introduces an algorithm for the projection of polytopes in halfspace form, called Equality Set Projection (ESP). ESP has the desirable property of output sensitivity for non-degenerate polytopes. That is, a linear number of linear programs are needed per output facet of the projection. It is demonstrated that ESP is particularly well suited to control problems and comparative simulations are given, which greatly favour ESP. Part two is an investigation into the multi-parametric linear program (mpLP). The mpLP has received a lot of attention in the control literature as certain model predictive control problems can be posed as mpLPs and thereby pre-solved, eliminating the need for online optimisation. The structure of the solution to the mpLP is studied and an approach is pre- sented that eliminates degeneracy. This approach causes the control input to be continuous, preventing chattering, which is a significant problem in control with a linear cost. Four new enumeration methods are presented that have benefits for various control problems and comparative simulations demonstrate that they outperform existing codes. The third part studies the relationship between projection and multi-parametric linear programs. It is shown that projections can be posed as mpLPs and mpLPs as projections, demonstrating the fundamental nature of both of these problems. The output of a multi-parametric linear program that has been solved for the MPC control inputs offline is a piecewise linear controller defined over a union of polyhedra. The online work is then to determine which region the current measured state is in and apply the appropriate linear control law. This final part introduces a new method of searching for the appropriate region by posing the problem as a nearest neighbour search. This search can be done in logarithmic time and we demonstrate speed increases from 20Hz to 20kHz for a large example system.

Adaptive design of a global opacity transfer function for direct volume rendering of ultrasound data

Abstract: While there are a couple of transfer function design approaches for CT and MRI (magnetic resonance imaging) data, direct volume rendering of ultrasound data still relies on manual adjustment of an inflexible piecewise linear opacity transfer function (OTF) on a trial-and-error basis. The main challenge of automatically designing an OTF for visualization of sonographic data is the low signal-to-noise ratio in combination with real time data acquisition at frame rates up to 25 volumes per second. In this paper, we present an efficient solution of this task. Our approach is based on the evaluation of tube cores, i.e., collections of voxels gathered by traversing the volume in rendering directions. We use information about the probable position of an interface between tissues of different echogenicity to adaptively design an OTF in a multiplicative way. We show the appropriateness of our approach by examples, deliberately on data sets of moderate quality arising frequently in clinical settings.