Piecewise

About: Piecewise is a research topic. Over the lifetime, 21064 publications have been published within this topic receiving 432096 citations. The topic is also known as: piecewise-defined function & hybrid function.

A piecewise trajectory optimization model for connected automated vehicles: Exact optimization algorithm and queue propagation analysis

Abstract: This paper formulates a simplified traffic smoothing model for guiding movements of connected automated vehicles on a general one-lane highway segment. Adapted from the shooting heuristic proposed by Zhou et al. (2017) and Ma et al. (2017), this model confines each vehicle’s trajectory as a piecewise quadratic function with no more than five pieces and lets all trajectories in the same platoon share identical acceleration and deceleration rates. Similar to the shooting heuristic, the proposed simplified model is able to control the overall smoothness of a platoon of connected automated vehicles and approximately optimize traffic performance in terms of fuel efficiency and driving comfort. While the shooting heuristic relies on numerical meta-heuristic algorithms that cannot ensure solution optimality, we discover a set of elegant theoretical properties for the general objective function and the associated constraints in the proposed simplified model, and consequentially propose an efficient analytical algorithm for solving this problem to the exact optimum. Interestingly, this exact algorithm has intuitive physical interpretations, i.e., stretching the transitional parts of the trajectories (i.e., parts with acceleration and deceleration adjustments) as far as they reach the upstream end of the investigated segment, and then balancing the acceleration and deceleration magnitudes as close as possible. This analytical exact model can be considered as a core module to a range of general trajectory optimization problems at various infrastructure settings. Numerical examples reveal that this exact algorithm has much more efficient computational performance and the same or better solution quality compared with the previously proposed shooting heuristic. These examples also illustrate how to apply this model to CAV control problems on signalized segments and at non-stop intersections. Further, we study a homogeneous special case of this model and analytically formulate the relationship between queue propagation and trajectory smoothing. One counter-intuitive finding is that trajectory smoothing may not always cause longer queue propagation but instead may mitigate queue propagation with appropriate settings. This theoretical finding has valuable implications to joint optimization of queuing management and traffic smoothing in complex transportation networks.

Feedforward Controller of Ill-Conditioned Hysteresis Using Singularity-Free Prandtl–Ishlinskii Model

Abstract: Piezoelectric, magnetostrictive, and shape memory alloy actuators are gaining importance in high-frequency precision applications constrained by space. Their intrinsic hysteretic behavior makes control difficult. The Prandtl-Ishlinskii (PI) operator can model hysteresis well, albeit a major inadequacy: the inverse operator does not exist when the hysteretic curve gradient is not positive definite, i.e., ill condition occurs when slope is negative. An inevitable tradeoff between modeling accuracy and inversion stability exists. The hysteretic modeling improves with increasing number of play operators. But as the piecewise continuous interval of each operator reduces, the model tends to be ill-conditioned, especially at the turning points. Similar ill-conditioned situation arises when these actuators move heavy loads or operate at high frequency. This paper proposes an extended PI operator to map hysteresis to a domain where inversion is well behaved. The inverse weights are then evaluated to determine the inverse hysteresis model for the feedforward controller. For illustration purpose, a piezoelectric actuator is used.

Non-parametric Bayesian Estimation of a Spatial Poisson Intensity

Abstract: A method introduced by Arjas & Gasbarra (1994) and later modified by Arjas & Heikkinen (1997) for the non-parametric Bayesian estimation of an intensity on the real line is generalized to cover spatial processes. The method is based on a model approximation where the approximating intensities have the structure of a piecewise constant function. Random step functions on the plane are generated using Voronoi tessellations of random point patterns. Smoothing between nearby intensity values is applied by means of a Markov random field prior in the spirit of Bayesian image analysis. The performance of the method is illustrated in examples with both real and simulated data.

Event-triggered consensus control for second-order multi-agent systems

Abstract: In this study, an event-triggered control strategy is proposed to achieve consensus in a multi-agent system under a directed topology. The proposed control strategy utilises a piecewise continuous control law and an event-triggering function for each agent. The control law only updates at discrete event instants computed using an event-triggering function, which depends on the states of the agents at the current and outdated event instant. This control approach is first applied to a first-order system and is further extended to a second-order system. Simulation examples are presented to illustrate the efficiency of the proposed control strategy.

An inexact successive quadratic approximation method for L-1 regularized optimization

Abstract: We study a Newton-like method for the minimization of an objective function $$\phi $$? that is the sum of a smooth function and an $$\ell _1$$l1 regularization term. This method, which is sometimes referred to in the literature as a proximal Newton method, computes a step by minimizing a piecewise quadratic model $$q_k$$qk of the objective function $$\phi $$?. In order to make this approach efficient in practice, it is imperative to perform this inner minimization inexactly. In this paper, we give inexactness conditions that guarantee global convergence and that can be used to control the local rate of convergence of the iteration. Our inexactness conditions are based on a semi-smooth function that represents a (continuous) measure of the optimality conditions of the problem, and that embodies the soft-thresholding iteration. We give careful consideration to the algorithm employed for the inner minimization, and report numerical results on two test sets originating in machine learning.