Continuous Approximation Research Articles

Super Twisting Based Lyapunov Redesign for Uncertain Linear Delay Systems

We present a new continuous Lyapunov redesign (LR) methodology for the robust stabilization of a class of uncertain time-delay systems that is based on the so-called super twisting algorithm. The main feature of the proposed approach is that allows one to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">simultaneously</i> adjust the chattering effect and achieve asymptotic stabilization of the uncertain system, which is lost when continuous approximation of the unit control is considered. At the basis of the super twisting based LR methodology is a well-known class of Lyapunov–Krasovskii functionals. The particular form of the time derivative of this class of functionals allows one to define a delay-free sliding manifold on which some class of smooth uncertainties are compensated.

Well-Balanced Unstaggered Central Scheme Based on the Continuous Approximation of the Bottom Topography

A key difficulty of the conventional unstaggered central schemes for the shallow water equations (SWEs) is the well-balanced property that may be missed when the computational domain contains wet-dry fronts. To avoid the numerical difficulty caused by the nonconservative product, we construct a linear piecewise continuous bottom topography. We propose a new discretization of the source term on the staggered cells, and a novel “backward” step based on the water surface elevation. The core of this paper is that, we construct a map between the water surface elevation and the cell average of the free surface on the staggered cells to discretize the source term for maintaining the stationary solutions. The positivity-preserving property is obtained by using the “draining” time-step technique. A number of classical problems of the SWEs can be solved reasonably.

Stretch‐induced wrinkling analysis of thin sheets with splines

SummaryNumerical simulation of stretch‐induced wrinkling in thin elastic sheets is a challenging problem due to a vanishing bending stiffness and the coexistence of superabundant equilibrium solutions. In this work, we present a computational framework to capture the morphological evolution of stretch‐induced wrinkles. The application of modified Föppl‐von Kármán plate model results in a fourth‐order partial differential equation. The convergence of finite‐element solutions necessitates ‐continuous approximations. Herein, Powell‐Sabin B‐splines, which are based on triangles, are utilized for both the approximation of the field variables and the description of the geometry. To trace the wrinkling behavior in thin sheets, a path‐following technique using asymptotic numerical method is considered. The advantage of this method is an adaptive step length, which works incredibly well near the bifurcation points and allows for the computation of the post‐bifurcation diagrams with a quite small perturbation. The versatility and accuracy of the developed computational approach are assessed in three case studies, featuring wrinkling in highly stretched rectangular and annular thin sheets.

Stability analysis of a digital hierarchical steering controller of autonomous vehicles with multiple time delays

This study investigates the lane-keeping control of autonomous vehicles with an emphasis on the digital delayed nature of the system. The vehicle dynamics are represented using a kinematic bicycle model, and a hierarchical lane-keeping controller is introduced with multiple delays in the feedback loop. An extension of the semidiscretization method is presented, in order to perform the stability analysis of the digitally controlled vehicle with multiple discrete time delays. The differences between the continuous approximation and the exact consideration of discrete time delays are highlighted. We show that in certain cases, neglecting the effects of quantization can lead to significant inaccuracies, especially when tuning the lower-level controller. The results are verified using a series of small-scale laboratory experiments.

Optimal Solutions for Joint Beamforming and Antenna Selection: From Branch and Bound to Graph Neural Imitation Learning

This work revisits the joint beamforming (BF) and antenna selection (AS) problem, as well as its robust beamforming (RBF) version under imperfect channel state information (CSI). Such problems arise due to various reasons, e.g., the costly nature of the radio frequency (RF) chains and energy/resource-saving considerations. The joint (R)BF&AS problem is a mixed integer and nonlinear program, and thus finding <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">optimal solutions</i> is often costly, if not outright impossible. The vast majority of the prior works tackled these problems using techniques such as continuous approximations, greedy methods, and supervised machine learning—yet these approaches do not ensure optimality or even feasibility of the solutions. The main contribution of this work is threefold. First, an effective <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">branch and bound</i> (B&B) framework for solving the problems of interest is proposed. Leveraging existing BF and RBF solvers, it is shown that the B&B framework guarantees global optimality of the considered problems. Second, to expedite the potentially costly B&B algorithm, a machine learning (ML)-based scheme is proposed to help skip intermediate states of the B&B search tree. The learning model features a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">graph neural network</i> (GNN)-based design that is resilient to a commonly encountered challenge in wireless communications, namely, the change of problem size (e.g., the number of users) across the training and test stages. Third, comprehensive performance characterizations are presented, showing that the GNN-based method retains the global optimality of B&B with provably reduced complexity, under reasonable conditions. Numerical simulations also show that the ML-based acceleration can often achieve an order-of-magnitude speedup relative to B&B.

Stochastic and continuous Petri nets approximation of Markovian model

Stochastic and continuous Petri nets approximation of Markovian model

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.

Exploring alternative bus operations strategies for adaptive and efficient transit systems in the face of disruptions

Disruptions in public transit systems can have significant impacts on agency and passenger needs. Therefore, it is crucial to implement new transit policies to address these disruptions and to ensure efficient and reliable transit services. This study proposes a framework to assess alternative bus operating strategies to adapt with different disruptions to public transit systems. Conventional all-stop systems, visiting all the stops along a bus corridor, are compared against three alternative operational schemes: Skip-stop, express-local schemes with stop-skipping designs, and on-demand service with a fixed route but flexible stopping patterns, are compared to find the most efficient bus operating service under various circumstances. We developed an optimization model based on the total generalized system cost for each operating strategy using continuous approximation techniques and extended the previous models by comparing a wider range of alternative services and model flexibility to evaluate the optimum system in response to different disruptions. Different factors, such as the level of demand, demand patterns, and sensitivity to various components of transit trips for passengers and the operator, including crowding, denied boarding costs, and fleet constraints, are considered. We found that, given a different disruption scenario, demand, and travel patterns, the most efficient service can vary significantly between on-demand, all-stop, skip-stop, and express-local services. As such, it is suggested that the service scheme be chosen more adaptively by employing such frameworks.

Robust Hierarchical Tracking Control of Vehicle-Manipulator Systems

For vehicle-manipulator systems (VMSs) to perform precise operations, a robust tracking control framework is required. Moreover, to exploit the platform's redundancy, it is desirable that the framework allows several tasks to be completed simultaneously. This work presents a robust hierarchical tracking controller for floating-base robots. In addition to providing uniform global asymptotic stability (UGAS) in free motion and allowing a total task dimension greater than the number of degrees of freedom (DoFs), the proposed controller includes a sliding mode effect and is proved to achieve UGAS even in the presence of bounded disturbances. Moreover, it is proved that uniform global ultimate boundedness (UGUB) is obtained if a continuous approximation of the sliding mode term is used. The stability results are proved mathematically, and are validated through simulations of an articulated intervention autonomous underwater vehicle (AIAUV).

Modelling and optimisation of single-cycle time for mobile rack AS/RS

ABSTRACT This paper deals with mobile-rack automated storage/retrieval systems (AS/RS) that are a particular physical configuration of unit-deep AS/RS, i.e. the class of systems where each cell is designed to contain a single load. There are two main contributions in this paper: the first one is the analytical modelling of the average single-cycle time for mobile-rack AS/RS. The approach adopted for modelling is based on a continuous approximation, which is not recent, however, the approach proposed here for the transition from the discrete to the continuous models is different from those used previously in the literature. In a certain way, the multi-aisle and unit-load configurations can be seen as special cases of the mobile-rack configuration. This allows to deduce analytical models for the average single-cycle time for these two configurations from the one developed for mobile rack. The second main contribution of this paper is to find the dimensions of the AS/RS that minimise the average single-cycle time. The optimisation has been studied formally for the multi-aisle and unit-load configurations, and numerically for the mobile-rack configuration.

Solutions and reachable sets of hybrid dynamical systems: Semicontinuous dependence on initial conditions, time, and perturbations

The sequential compactness afforded hybrid systems under mild regularity constraints guarantee outer/upper semicontinuous dependence of solutions on initial conditions and perturbations. For reachable sets of hybrid systems, this property leads to upper semicontinuous dependence with respect to initial conditions, time, and perturbations. Motivated by these results, we define a counterpart to sequential compactness and show that it leads to lower semicontinuous dependence of solutions on initial conditions and perturbations. In the sequel, it is shown that under appropriate assumptions, reachable sets of systems possessing this novel property depend lower semicontinuously on initial conditions, time, and perturbations. When those assumptions fail, continuous approximations of reachable sets turn out to be still possible. Necessary and sufficient conditions for the introduced property are given by geometric constraints, regularity assumptions, and tangentiality conditions.

A hybrid pressure approximation in the control volume finite element method for multiphase flow and transport in heterogeneous porous media

We present a new hybrid pressure formulation applied to the control volume finite element (CVFE) method to model multiphase flow and transport in highly heterogeneous porous media. The formulation effectively captures sharp saturation changes in the presence of discontinuous material properties by employing a discontinuous pressure approximation at material interfaces. The heterogeneous porous medium is divided into sub-domains within which material properties are uniform or smoothly varying. By construction, the resultant control volume dual mesh is restricted within a sub-domain. The artificial mass leakage across material property boundaries observed in classical CVFE methods is therefore circumvented. The approach applies the robust continuous pressure approximation in the rest of the computational domain; the discontinuous approximation is applied only at the sub-domain boundaries. The discontinuous parameters necessary to achieve mass conservative solutions, locally and globally, are described. We demonstrate the accuracy and efficiency of the new approach by comparison with the classical continuous CVFE method on various examples of heterogeneous domains as well as establishing the convergence of the numerical method. The proposed hybrid formulation significantly outperforms the accuracy and efficiency of classical CVFE methods that use the same order of approximation for modeling multiphase flow in heterogeneous porous media.

Generalization of a class of uniformly optimized [formula omitted]-step hybrid block method for solving two-point boundary value problems

This study aims to develop, analyze and implement an efficient method for approximating two-point boundary value problems of ordinary differential equations. The method contains six and twelve implicit formulas, respectively, for the one-step and two-step schemes. The continuous approximations, using the shifted Chebyshev polynomial as the basis function, were obtained via evaluations at three different points on the selected one-step method, including two optimized hybrid points. Evaluations were carried out on six different points on the selected two-step method, including four generalized optimized hybrid points. Qualitative analysis of the method proves the proposed methods are consistent, zero-stable, convergent and have a larger region of absolute stability. The quantitative analysis shows that the methods compared favorably well and established some superiority strength with the existing methods.

Optimal design of pure battery electric bus system on the grid network

Electrifying bus fleets is a practical way to develop sustainable transit systems. This study develops a continuous approximation model to provide strategic-level insights. Considering conventional depot charging mode, the proposed model optimises the design parameters of a pure electric bus network such as stop spacing, headway, and batteries’ expected charging range. Numerical cases are presented over a wider range of key design parameters. Both analytical and numerical simulations are conducted to compare the results between the proposed system and the diesel bus system. Substituting electric buses for traditional diesel could save CO2 emissions from 47% to 52% and up to 8% of the total cost can be saved with electric buses, and deploying electric bus systems would bring an around 7% increase in capital cost but save 23% operation cost. On a real-world grid network in Xi’an, China, deploying electric buses can save the system cost by 7.7%.

The elastic deformation process of bimetallic objects

In industrial problems linear elasticity simulation has been known to be successfully solved by the Finite Element Methods (FEMs). However, a mathematical model as well as a numerical solution program that can be applied to the deformation simulation of bimetallic materials are still the challenges in this research area. In this paper, we investigate a mathematical model for the linear elastic deformation of bimetallic structures. By reducing the linear elasticity problem to its weak form, we apply the continuous Galerkin approximation to obtain the discrete problem. The existence and the uniqueness of the solution of the weak problem are clearly demonstrated, ensuring the well-posedness of the problem. In addition, the numerical simulations with bimetallic materials have been built on Freefem[Formula: see text] software that illustrate the efficiency and reliability of the numerical method.

Uniqueness for the Fractional Calderón Problem with Quasilocal Perturbations

We study the fractional Schrödinger equation with quasilocal perturbations and show that the qualitative unique continuation and Runge approximation properties hold in the assumption of sufficient decay. Quantitative versions of both results are also obtained via a propagation of smallness analysis for the Caffarelli--Silvestre extension. The results are then used to show uniqueness in the inverse problem of retrieving a quasilocal perturbation from Dirichlet-to-Neumann (DN) data under suitable geometric assumptions. Our work generalizes recent results regarding the locally perturbed fractional Calderón problem.

Performance enhancement in the cell-free massive MIMO SWIPT system with active eavesdropping

This paper investigates the secure transmission for simultaneous wireless information and power transfer (SWIPT) in the cell-free massive multiple-input multiple-output (MIMO) system. To develop green communication, legitimate users harvest energy by the hybrid time switching (TS) and power splitting (PS) strategy in the downlink phase, and the harvested energy can provide power to send uplink pilot sequences for the next time slot. By in-built batteries, the active eavesdropper can send the same pilots with the wiretapped user, which results in undesirable correlations between the channel estimates. Under these scenarios, we derive the closed-form expressions of average harvested energy and achievable rates, and propose an iterative power control (PC) scheme based on max–min fairness algorithm with energy and secrecy constraints (MMF-ESC). This scheme can ensure the uniform good services for all users preserving the distributed architecture advantage of cell-free networks, while meeting the requirements of energy harvested by legitimate users and network security against active eavesdroppers. Besides, continuous approximation, bisection and path tracking are jointly applied to cope with the high-complexity and non-convex optimization. Numerical results demonstrate that MMF-ESC PC scheme can effectively increase the achievable rate and the average harvested energy of each user, and decrease the eavesdropping rate below the threshold. Moreover, the results also reveal that PS strategy is superior in harvesting energy in terms of more stringent network requirements for average achievable rates or security.

Reconfigurable Intelligent Surface-Aided Cell-Free Massive MIMO with Low-Resolution ADCs: Secrecy Performance Analysis and Optimization

The secure communication in reconfigurable intelligent surface-aided cell-free massive MIMO system is investigated with low-resolution ADCs and with the existence of an active eavesdropper. Specifically, an aggregated channel estimation approach is applied to decrease the overhead required to estimate the channels. Using the available imperfect channel state information (CSI), the conjugate beamforming and random beamforming are applied at the APs and the RIS for downlink data transmission, respectively. The closed-form expression of the achievable secrecy rate is acquired to appraise the achievable secrecy performance using only the channel statistics. With the achievable analytical results, the impacts of the quantization bit of ADCs, channel estimation error, the number of RIS elements, and the number of the APs can be unveiled. Aiming to maximize the minimum achievable rate of all legitimate users subject to security constraints, the power control optimization scheme is first formulated. To tackle this nonconvex property of the proposed optimization problem, a path-following algorithm is then utilized to solve the initial problem with continuous approximations and iterative optimization. Numerical results are presented to verify the achieved results along with availability of the presented power allocation approach.

Variable stepsize construction of a two-step optimized hybrid block method with relative stability

Abstract Several numerical techniques for solving initial value problems arise in physical and natural sciences. In many cases, these problems require numerical treatment to achieve the required solution. However, in today’s modern era, numerical algorithms must be cost-effective with suitable convergence and stability features. At least the fifth-order convergent two-step optimized hybrid block method recently proposed in the literature is formulated in this research work with its variable stepsize approach for numerically solving first- and higher-order initial-value problems in ordinary differential equations. It has been constructed using a continuous approximation achieved through interpolation and collocation techniques at two intra-step points chosen by optimizing the local truncation errors of the main formulae. The theoretical analysis, including order stars for the relative stability, is considered. Both fixed and variable stepsize approaches are presented to observe the superiority of the latter approach. When tested on challenging differential systems, the method gives better accuracy, as revealed by the efficiency plots and the error distribution tables, including the machine time measured in seconds.

An iterative constrained least squares method for continuous piecewise linear approximation

In this work, a novel iterative least squares method is proposed to approximate nonlinear functions with continuous piecewise linear (CPWL) functions, where the continuity is guaranteed by constrained least squares. A typical least-squares-based method for CPWL fitting consists of two major steps: to perform least squares for a fixed set of breakpoints (i.e. for a specific partition of function domain), and to update breakpoints to reduce the overall fitting error. The proposed method aims to improve the existing CPWL method, the one with canonical representation, by modifying both major steps. Instead of performing ordinary least squares with canonical representation, partitioned least squares and constrained least squares are successively performed to reduce the computational complexity. For the update of breakpoints, an iterative procedure is proposed, where gradient descent with momentum method is employed to improve the convergence characteristics. The advantages of proposed method are illustrated using illustrative examples.