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.

Biologically Inspired Spiking Neurons: Piecewise Linear Models and Digital Implementation

Abstract: There has been a strong push recently to examine biological scale simulations of neuromorphic algorithms to achieve stronger inference capabilities. This paper presents a set of piecewise linear spiking neuron models, which can reproduce different behaviors, similar to the biological neuron, both for a single neuron as well as a network of neurons. The proposed models are investigated, in terms of digital implementation feasibility and costs, targeting large scale hardware implementation. Hardware synthesis and physical implementations on FPGA show that the proposed models can produce precise neural behaviors with higher performance and considerably lower implementation costs compared with the original model. Accordingly, a compact structure of the models which can be trained with supervised and unsupervised learning algorithms has been developed. Using this structure and based on a spike rate coding, a character recognition case study has been implemented and tested.

Nonparametric Estimation with Nonlinear Budget Sets

Abstract: Choice models with nonlinear budget sets provide a precise way of accounting for the nonlinear tax structures present in many applications. In this paper we propose a nonparametric approach to estimation of these models. The basic idea is to think of the choice, in our case hours of labor supply, as being a function of the entire budget set. Then we can do nonparametric regression where the variable in the regression is the budget set. We reduce the dimensionality of this problem by exploiting structure implied by utility maximization with piecewise linear convex budget sets. This structure leads to estimators where the number of segments can differ across observations and does not affect accuracy. We give consistency and asymptotic normality results for these estimators. The usefulness of the estimator is demonstrated in an empirical example, where we find it has a large impact on estimated effects of the Swedish tax reform.

Conewise Linear Systems: Non-Zenoness and Observability

Abstract: Conewise linear systems are dynamical systems in which the state space is partitioned into a finite number of nonoverlapping polyhedral cones on each of which the dynamics of the system is described by a linear differential equation. This class of dynamical systems represents a large number of piecewise linear systems, most notably, linear complementarity systems with the P-property and their generalizations to affine variational systems, which have many applications in engineering systems and dynamic optimization. The challenges of dealing with this type of hybrid system are due to two major characteristics: mode switchings are triggered by state evolution, and states are constrained in each mode. In this paper, we first establish the absence of Zeno states in such a system. Based on this fundamental result, we then investigate and relate several state observability notions: short-time and $T$-time (or finite-time) local/global observability. For the short-time observability notions, constructive, finitely verifiable algebraic (both sufficient and necessary) conditions are derived. Due to their long-time mode-transitional behavior, which is very difficult to predict, only partial results are obtained for the $T$-time observable states. Nevertheless, we completely resolve the $T$-time local observability for the bimodal conewise linear system, for finite $T$, and provide numerical examples to illustrate the difficulty associated with the long-time observability.

Periodic motions and grazing in a harmonically forced, piecewise, linear oscillator with impacts

Abstract: In this paper, an idealized, piecewise linear system is presented to model the vibration of gear transmission systems. Periodic motions in a generalized, piecewise linear oscillator with perfectly plastic impacts are predicted analytically. The analytical predictions of periodic motion are based on the mapping structures, and the generic mappings based on the discontinuous boundaries are developed. This method for the analytical prediction of the periodic motions in non-smooth dynamic systems can give all possible periodic motions based on the adequate mapping structures. The stability and bifurcation conditions for specified periodic motions are obtained. The periodic motions and grazing motion are demonstrated. This model is applicable to prediction of periodic motion in nonlinear dynamics of gear transmission systems.