Continuous Approximation Method Research Articles

Robust [formula omitted] stabilization for systems with uncertain input time-delay

This paper presents a robust H∞ stabilization approach for actively controlled systems that include uncertain input time-delay. The approach comprises two essential steps: (1) converting a controlled system with uncertain time-delay to have only deterministic time-delay through a time-scale transformation; and (2) applying the Chebyshev spectral continuous time approximation (CHsCTA) method and defining appropriate extension matrices to transform the system so there are no time-delays. A robust H∞ control design is implemented on the transformed system, which facilitates the derivation of a robust H∞ controller. An iterative algorithm is proposed for computing the feedback gain matrix, which overcomes the challenges posed by the nonlinear coupling terms when solving the matrix inequalities. Case studies validate the effectiveness of the proposed control algorithm, which, through comparison with existing control algorithms, is shown to be less conservative.

Jointly Active/Passive Beamforming Optimization for Intelligent-Reflecting Surface-Assisted Cognitive-IoT Networks

To overcome challenges such as limited energy availability for terminal devices, constrained network coverage, and suboptimal spectrum resource utilization, with the overarching objective of establishing a sustainable and efficient interconnection infrastructure, we introduce an innovative Intelligent Reflective Surface (IRS) technology. This cutting-edge IRS technology is employed to architect a wireless and energy-efficient cognitive secure communication network assisted by IRS. To further optimize the overall energy harvesting of this network, we present a cognitive secure resource allocation scheme, aiming to maximize the system’s total collected energy. This scheme carefully considers various constraints, including transmission power constraints for cognitive base stations, power constraints for jammer devices, interference limitations for all primary users, minimum security rate constraints for all cognitive Internet of Things (IoT) devices, and phase shift constraints for IRS. We establish a comprehensive hybrid cognitive secure resource allocation model, encompassing joint cognitive transmission beam design, jammer device transmission beam design, and phase shift design. Given the non-convex nature of the formulated problem and the intricate coupling relationships among variables, we devise an effective block coordinate descent (BCD) iterative algorithm. The realization of joint cognitive/jammer base station transmission beam design and phase shift design employs sophisticated techniques such as continuous convex approximation methods and semi-definite programming. Simulation results underscore the superior performance of the proposed scheme compared to existing resource allocation approaches, particularly in terms of total harvested energy and other critical metrics.

Variational Wasserstein Barycenters with C-cyclical Monotonicity Regularization

Wasserstein barycenter, built on the theory of Optimal Transport (OT), provides a powerful framework to aggregate probability distributions, and it has increasingly attracted great attention within the machine learning community. However, it is often intractable to precisely compute, especially for high dimensional and continuous settings. To alleviate this problem, we develop a novel regularization by using the fact that c-cyclical monotonicity is often necessary and sufficient conditions for optimality in OT problems, and incorporate it into the dual formulation of Wasserstein barycenters. For efficient computations, we adopt a variational distribution as the approximation of the true continuous barycenter, so as to frame the Wasserstein barycenters problem as an optimization problem with respect to variational parameters. Upon those ideas, we propose a novel end-to-end continuous approximation method, namely Variational Wasserstein Barycenters with c-Cyclical Monotonicity Regularization (VWB-CMR), given sample access to the input distributions. We show theoretical convergence analysis and demonstrate the superior performance of VWB-CMR on synthetic data and real applications of subset posterior aggregation.

A New Distributed Robust Power Control for Two-Layer Cooperative Communication Networks in Smart Grids with Reduced Utility Costs

The packet loss during transmission of load control commands can lead to regulation errors in the smart grid and increase the cost of utility agencies due to the purchase of additional automatic generation control (AGC) services. In this paper, a two-layer cooperative communication network between the utility company and relays is presented. The utility company rents the relay to assist with the downlink transmission to improve the reliability of communication and reduce the data transmission cost due to packet loss. Furthermore, the uncertainty of channel gain is considered, and a two-tier game model is established. A distributed robust power control algorithm based on the continuous convex approximation method is proposed to obtain the optimal relay power allocation and price. Through the simulation analysis of the proposed scheme and the two comparison schemes, the cost of the utility company was reduced by 6% and 21%, and the standard deviation of income value between the relays was reduced by 40% and 48%, respectively.

Security performance analysis of active intelligent reflective surface assisted wireless communication

As a new communication technology, Intelligent Reflecting Surface (IRS) can intelligently reconfigure the wireless propagation environment by integrating many passive/active reflective elements on the plane. According to the characteristics that IRS can adjust the propagation channel intelligently, this paper applies IRS to wireless security communication, and studies how to make the security rate reach the optimal security capacity from the perspective of optimization technology. In this paper, two schemes of passive/active IRS are considered, and the corresponding safety rate maximization algorithm is proposed. In view of the nonconvexity of the objective function, on the one hand, in the passive IRS scheme, the Dinkelbach method is used to transform the objective function into a form that is easy to handle, and the original problem is transformed into a convex problem through the continuous convex approximation method; On the other hand, under the active IRS scheme, aiming at the complexity of the original problem, the first order Taylor expansion is used to obtain the lower bound of the optimization problem, and a minimax optimization algorithm is proposed. Finally, the performance of the proposed algorithm is verified by simulation. The simulation results show that the algorithm designed with active IRS has better security rate than the algorithm designed with passive IRS under the same parameter settings.

A Fairness Index Based on Rate Variance for Downlink Non-Orthogonal Multiple Access System

Aiming at the resource allocation problem of a non-orthogonal multiple access (NOMA) system, a fairness index based on sample variance of users’ transmission rates is proposed, which has a fixed range and high sensitivity. Based on the proposed fairness index, the fairness-constrained power allocation problem in NOMA system is studied; the problem is decoupled into the intra cluster power allocation problem and the inter cluster power allocation problem. The nonconvex optimization problem is solved by the continuous convex approximation (SCA) method, and an intra and inter cluster power iterative allocation algorithm with fairness constrained is proposed to maximize the total throughput. Simulation results show that the proposed algorithm can take into account intra cluster, inter cluster, and system fairness, and maximize the system throughput on the premise of fairness.

Designing transit-oriented multi-modal transportation systems considering travelers’ choices

One goal of future transit-oriented transportation systems is to promote door-to-door mobility for travelers by integrating different public transportation modes into a whole. We propose a mathematical design framework for such a transit-oriented multi-modal transportation system from a societal aspect considering three categories of public transportation modes, i.e., general on-demand modes, local on-demand modes, and fixed-schedule modes. A system-state equilibrium is brought up to describe travelers’ rational travel choices and their reverse effects on agency service levels using the continuous approximation method, and a centralized system designer then manages to achieve a system-beneficial outcome with the minimum cost. To solve the design problem, we prove that the transportation system reaches a unique equilibrium when decision variables are given. By this discovery, we construct a global search framework based on the DIRECT algorithm to solve the optimal design. In analyzing the problem property, we rigorously prove that in the designed systems, the bus service as a fixed-schedule mode is absent from the design scheme under the cases with sufficiently low demands, and the design problem thus reduces to a one-dimensional line search. The ride-hailing service as a general on-demand mode is similarly proved to be excluded when the demand is sufficiently high. In this context, the approximate design parameters of the bus service and the total system cost are developed analytically. The local on-demand mode, bike-sharing service, as an option of bus feeders, is proved to be efficient under a realistic setting. Extensive numerical examples provide evidence verifying the preceding analyses and indicating the behavior of travelers and agencies. Further sensitivity tests show that the subway is favorable for intensive demands and autonomous vehicles may promote the ride-hailing industry. For completeness, an immediate application of the proposed framework in generalized cases validates the model reliability.

Multiple-Trigger Catastrophe Bond Pricing Model and Its Simulation Using Numerical Methods

Investor interest in single-trigger catastrophe bonds (STCB) has the potential to decline in the future. It is triggered by the increasing trend of global catastrophe loss and intensity every year, which increases the probability that a claim of STCB will occur. To increase investor interest again, the issuance of multiple-trigger catastrophe bonds (MTCB) can be one solution. However, to issue MTCB, its pricing is more complex because it involves more factors than STCB. Therefore, this study aims to design a simple MTCB pricing model. The claim trigger indices used are actual loss and fatality. Then, a nonhomogeneous compound Poisson process is used to model actual losses and fatalities aggregate to consider catastrophe intensity. In addition, this study proposes numerical methods, namely the continuous distribution approximation method and the Nuel recursive method, to facilitate the application of the model. Finally, an analysis of the effect of catastrophe intensity and other factors on MTCB prices is also presented. This study is expected to help special-purpose vehicles as MTCB issuers in MTCB pricing.

Design of a two-echelon last-mile delivery model

Due to high congestion in cities and growing demand for last-mile delivery services, several companies have been implementing two-echelon distribution strategies over the past few years. Notably, the installation of urban transshipment points has gained increasing attention, used by logistics operators to transfer goods from large freight trucks to smaller and more agile vehicles for last-mile delivery. Nevertheless, the main challenge is how to decide the number and location of these facilities under the presence of demand uncertainty. In this paper, we develop a two-stage stochastic program to design two-echelon last-mile delivery networks under demand uncertainty. This approach decomposes the problem into strategic decisions (facility location) and operational decisions (daily distribution of goods). To address large-scale instances, we solve the model through the sample average approximation (SAA) technique and estimate the optimal routing costs (of the SAA counterpart) using a continuous approximation method. Using a real-world case study with more than 1300 customers from New York City, our results provide several managerial insights regarding the mix of transportation modes, facility location, and the impact of allowing the outsourcing of customer demand. We provide extensive validation of the two-stage stochastic program results through a simulation-based approach and the calculation of the value of the stochastic solutions. • We develop a two-stage stochastic program to design two-echelon last-mile delivery networks under demand uncertainty. • We use sample average approximation and continuum approximation of route cost to address large-scale instances. • For a real-world case study, we provide managerial insights regarding the mix of transportation modes, facility locations, and the benefits of outsourcing deliveries. • We extensively validate our results using a simulation-based approach and assess the value of accounting for stochasticity.

Spatial analysis of public transportation infrastructure in Santiago, Chile, using the continuous approximation method

Santiago, the capital city of Chile, has seven million inhabitants in an area of 850 km2. This city has a metro network with seven lines extending 140 kilometers and transports approximately 2.6 million people daily. The bus system has undergone significant transformations over the last three decades. The most relevant change having been Transantiago, the public transportation system implemented in 2007 for Santiago, Child, which combines the use of Metro and buses (BRT). Metropolitan Mobility Network (called Red) is the latest version of the public transportation plan. This paper aims to analyze the current subway infrastructure using the continuous approximation method for Santiago, Chile. We previously proposed a macroscopic methodology to identify the needs for an adequate level of service in urban mobility and transportation, and we applied it to Santiago’s Metro network. Our work focuses on functionality and demand distribution. Santiago’s demand varies spatially in volume and extension throughout the city. Using the latest origin-destination survey from 2012, we deduct the critical components in this current network structure. It is worth mentioning that the Metro design bases its network on a ring-radial structure. With our macroscopic model applied to Santiago, Chile, we have detected infrastructure needs in the current transit network. The supply of infrastructure should increase for two reasons: first, to achieve balanced cost levels between users and the agency and second, to reduce subway occupations. The optimal model outcomes for Santiago define the optimal network in which the system requires five rings and ten end-to-end longitudinal lines (20 radial routes), including lower levels of occupation. The obtained results are a good preliminary solution, considering the subway infrastructure supply could be sub-estimated in the public transportation plan.

Periodic Response and Stability of a Maglev System with Delayed Feedback Control Under Aerodynamic Lift

In this research, the periodic response and stability of a nonlinear maglev system under the combined effects of steady and unsteady aerodynamic lifts is investigated, considering time delay in the feedback control loop. First, a nonlinear maglev system with a single levitation point that accounts for the nonlinearity of the electromagnetic force, time delay in the feedback control loop, and effect of aerodynamic lift is established. Then the periodic solutions of the maglev system with aerodynamic lift and time delays are obtained by an incremental harmonic balance analysis, in which the explicit time-delay action matrices used indicate that the effect of time delay on the response of the maglev system is periodic. The stability of the periodic solutions based on a finite difference continuous time approximation method and Floquet theory is studied, from which the critical time delay is obtained. Also, the relationship between the periodic vibration amplitude and the time delay is examined, along with the steady aerodynamic lift coefficient, and frequency of the unsteady aerodynamic lift, as well as the variation of critical delay with respect to the position feedback and velocity feedback with the control gain parameters. In addition, the stability boundary for the simultaneous time-delayed position and velocity feedback is obtained.

Enhancement of Deposition Process Controlling in Electron Beam Metal Wire Deposition Method

The urgency of improving control systems for electron beam metal wire deposition method is proved in terms of creating methods for controlling the geometric parameters of deposited layer. It is shown that to solve this problem in electron beam metal wire deposition method, it is advisable to use reflected electron signal detection when scanning the deposited layer. Using a mathematical model based on the application of continuous loss approximation and Monte-Carlo method, the probe characteristics recorded during cylindrical layer scanning are investigated. Experimental verification of the results obtained was carried out using an electron-beam technological installation. The effect of the distance between the layer and electron collector on the recorded signals is investigated. The possibility of using this method for simultaneous control of deposited layer height and width is shown.

Modeling public transportation networks for a circular city: the role of urban subcenters and mobility density

The concentration of both employment and services in a specific area of a town generates positive effects, but also impacts (congestion, transit issues, and others). Urban subcenters seek to approach economic activities to residents in peripheral urban spaces. The objective of this research is to evaluate the contribution to the mobility of implementing urban subcenters in a city. The model has a total cost function (users and agency costs) on a circular city (ring and radial routes) formulated using the continuous approximation method. The model solution addresses with mathematical optimization. The model evaluates a BRT network applied to scenarios of urban subcenters. The results of the modeling show that the implementation of subcenters obtains savings of 3.5% in rush hour. Thus, this strategy of urban planning generates improvements in the functioning of a public transportation system. Moreover, the maximum benefits are obtained in medium-sized subcenters in comparison to the CBD, which allows balancing user and agency costs. Therefore, the outcomes may be better with an urban pattern with subcenters, and a transit scheme adapted to the demand needs.

High-order Mass-lumped Schemes for Nonlinear Degenerate Elliptic Equations

We present and analyze a numerical framework for the approximation of nonlinear degenerate elliptic equations of the Stefan or porous medium types. This framework is based on piecewise constant approximations for the functions, which we show are essentially necessary to obtain convergence and error estimates. Convergence is established without regularity assumption on the solution. A detailed analysis is then performed to understand the design properties that enable a scheme, despite these piecewise constant approximations and the degeneracy of the model, to satisfy high-order error estimates if the solution is piecewise smooth. Numerical tests, based on continuous and discontinuous approximation methods, are provided on a variety of one- and two-dimensional problems, showing the influence on the convergence rate of the nature of the degeneracy and of the design choices.

АНАЛІЗ НЕПЕРЕРВНИХ МЕТОДІВ ІНТЕРПОЛЯЦІЇ ТА АПРОКСИМАЦІЇ ПЛОСКИХ КРИВИХ

The article provides an overview of continuous interpolation and approximation methods that are designed for discretely presented source data. It has been established that a common drawback is the lack of the possibility of local adjustment of the result of geometric modeling without re-computing the entire solution to the problem. To fulfill the conditions for the interpolation curve to pass through predetermined points, it is necessary to solve a system of linear equations. An increase in the size of the corresponding determinant leads to an increase in the error of the solution. It should be noted that, depending on the initial information and the purpose of the research, methods of continuous or discrete interpolation and approximation can be used. This article analyzes directly continuous methods of discrete interpolation and approximation. If we consider splines, then the parameters that control the shape of its individual segments are determined by the points that are the source data and determine the discretely presented curve. Spline methods are studied in the works of Yu.S. Zavyalova, N.P. Korneychuk and D. Rogers, of which it is known that the most important characteristics of a spline are its degree and defect. The degree of spline is determined by the highest degree of segment from the segments that make up the spline. A spline defect is the smallest derivative in which a discontinuity occurs, of all derivatives at the edges of segments that are spline components. In applied geometry, second-order curves are most often used due to the developed geometric apparatus and the high adaptability of their application. The limitations for their use in the interpolation process is the obligatory arrangement of the source data in accordance with the shape defined by the second-order curve. Although, in the process of approximation, this restriction is less significant, but the reproduction of the curve should be performed with a certain, predetermined error. Key words: approximation, interpolation, plane curves, Bezier curves, splines.

Optimal traffic signal control under dynamic user equilibrium and link constraints in a general network

In this study, an optimal traffic signal control framework is proposed for finding the signal control settings that minimize the total travel time in a road network with traffic lights. A novel aspect of this framework is its integration of the continuous-time double queue traffic flow model in a signal controlled traffic network to capture queue spillbacks in continuous-time. Furthermore, drivers’ long term responses to changes in traffic signal control settings are captured by their route choices following Wardrop’s first principle, which results in the dynamic user equilibrium state. Two signal control strategies, the fixed-timing control and the adaptive signal control, are considered. A continuous approximation method for the signal control is applied to eliminate integer variables and enhance the computational efficiency. A heuristic genetic algorithm based solution procedure is proposed to solve the proposed nonlinear programming problem with time-varying delay terms. Numerical tests are conducted in two testing networks and the results show that adaptive control with drivers taking into account signal timing on their route travel times performs best, and in some cases nearly as well as the benchmark performance derived from system optimal control without equilibrium constraints. The results also show that the advantage of adaptive over fixed-time signal control is more pronounced under UE than SO routing behavior.

Exploring paradigm shift impacts in urban mobility: Autonomous Vehicles and Smart Cities

Urban mobility is a dynamic system that has had a (slow) natural evolution. Scientists and engineers are currently developing new mobility technologies. A progressive paradigm shift will change everything from the fuel type to the way of driving vehicles. Vehicles will progressively become autonomous and will communicate and cooperate with each other. In the long run, profound changes are expected in mobility as a service. Furthermore, urban areas will have a higher level of development, and cities will likely turn into Smart Cities in which the vehicles will interact with the urban infrastructure. The main objective of this paper is to explore the macroscopic effects of mobility interaction in a radial-circular urban road system for current and future cities (Smart Cities). In the literature, there is documentation of the direct effects of autonomous vehicles, but some indirect effects will cause undesirable impacts such as an increase in demand and more congestion, which change the demand behavior and the urban structure. Finally, this paper exhibits the results of direct and indirect effects calculated through analytical tools (Continuous Approximation Method). In fact, our research shows that if demand increases by about 50%, the current scenario could have the same total cost as the future scenario with autonomous vehicles. Moreover, if the city radius increases by about 33% and the subjective value of time decreases about 20%, the benefits of the autonomous cars will be compensated. Therefore, the paper proves that autonomous vehicles could encourage the urban sprawl in the long run. Finally, Administrations should define transport strategies and policies to control these externalities, because autonomous driving could deteriorate mobility even worse than it is now.

Hunting stability analysis of high-speed train bogie under the frame lateral vibration active control

ABSTRACTIn this paper, we study a multi-objective optimal design of three different frame vibration control configurations and compare their performances in improving the lateral stability of a high-speed train bogie. The existence of the time-delay in the control system and its impact on the bogie hunting stability are also investigated. The continuous time approximation method is used to approximate the time-delay system dynamics and then the root locus curves of the system before and after applying control are depicted. The analysis results show that the three control cases could improve the bogie hunting stability effectively. But the root locus of low- frequency hunting mode of bogie which determinates the system critical speed is different, thus affecting the system stability with the increasing of speed. Based on the stability analysis at different bogie dynamics parameters, the robustness of the control case (1) is the strongest. However, the case (2) is more suitable for the dynamic performance requirements of bogie. For the case (1), the time-delay over 10 ms may lead to instability of the control system which will affect the bogie hunting stability seriously. For the case (2) and (3), the increasing time-delay reduces the hunting stability gradually over the high-speed range. At a certain speed, such as 200 km/h, an appropriate time-delay is favourable to the bogie hunting stability. The mechanism is proposed according to the root locus analysis of time-delay system. At last, the nonlinear bifurcation characteristics of the bogie control system are studied by the numerical integration methods to verify the effects of these active control configurations and the delay on the bogie hunting stability.

Modeling Vehicle Miles Traveled on Local Roads Using Classification Roadway Spatial Structure

This paper models the relationship between vehicle miles traveled (VMT) on local and collector roads with an objective to predict local road VMT by using collector road VMT. Through a continuous approximation method typically used for vehicle routing, it first analytically reveals this relationship mainly as a function of roadway density ratios between multiple roadway classifications. This structural relationship suggests regression equations using density ratios or logarithmic values of them as the explanatory variables. The use of regression equations enables to account for varying spatial distributions of roadways and demand through parameter calibration. The proposed regression equations are proved good fits through computer simulation using distinct community road network topologies. In addition, practical data from Hennepin County of Minnesota, U.S.A. that encompasses Minneapolis indicates that our developed regression equations can work well.

A continuous approximation method for dynamic pricing problem under costly price modifications

This paper presents a heuristic method to solve a dynamic pricing problem under costly price modifications. This is an extremely difficult nonlinear problem that has been solved only for a few special instances. Here we provide a new approach that involves an approximate reformulation of the problem, which can subsequently be solved in closed-form using elementary calculus techniques. Numerical results show that the approach is quite accurate; approximating the optimal revenue with errors usually much less than 1%. Moreover, the accuracy rapidly improves as the optimal number of price changes increases, which are precisely the cases conventional approaches would fail.