Optimal Estimates Research Articles

Rotor Dynamics of Aircraft Gas Turbine Engines

Ever since Sir Frank Whittle built the first jet engine for aircraft propulsion, rotor dynamics became the key issue in design and continues to be most vexing problem for the designers. The rotors are gradually replaced by drum construction and the bearing supports have become very flexible owing to thin engine casings mounted on the wings. The compressor and turbine rotors are no more coupled serially through couplings as in steam turbines but are mounted one over other as spools. Now, there are several critical speeds of multi spools excited by unbalances and misalignment. The disks mounted on the rotors carrying the blades have also become very flexible with several critical speeds excited by different per rev excitations from flow path and subjected to severe resonances during start up and shut down operations. Besides, these mounted parts have become globally elastic but locally plastic structures and their lifing has become an important design problem. This paper traces the developments taken place leading to optimum designs and life estimation of gas turbine rotating parts.

Slip ratio adaptive control for distributed drive electric vehicle based on wheel speed

Aiming at the problems of poor control effect and poor robustness for distributed drive electric vehicles driving on complex and changeable roads, this paper proposes a slip ratio adaptive control method based on wheel speed. Firstly, an optimal slip ratio estimator considering axle load transfer was developed by analyzing the single-wheel dynamics of the vehicle and based on the Burckhardt tire model. Secondly, a cubic polynomial function is used to fit and optimize the optimal slip ratio-peak adhesion coefficient curve, which effectively improves the accuracy of obtaining the optimal slip ratio. Then, a conditional integral sliding mode controller based on wheel speed is designed in order to enhance the transient characteristics of the sliding mode control, to avoid integral saturation, and to improve the control effect of the sliding ratio controller in the low speed start-up phase. Finally, three typical working conditions, namely single road, flat road, and slip road, were established for simulation and experimental validation. The results show that the method in this paper is able to estimate the optimal slip ratio of the road in real time, realize the wheel slip ratio adaptive control, and improve the dynamic performance and lateral stability of distributed drive electric vehicle.

A modified weak Galerkin finite element method for parabolic equations on anisotropic meshes

In this paper, we study a modified weak Galerkin finite element method (MWG-FEM) on anisotropic triangular meshes for a class of parabolic equations. Different from conventional weak Galerkin methods, the MWG-FEM replaces the boundary functions by the average of interior functions, which possesses flexibility in the approximation functions and mesh generation. Moreover, MWG-FEM is compatible with anisotropic meshes and suitable for solving problems with anisotropic property. By using the anisotropic interpolation and projection operators, the optimal order error estimate in L2 norm is derived. Numerical experiments are performed to demonstrate the stability and efficiency of the method.

Real Time Estimate and Computational Model of Fighter Aircraft Fuel Gauge System

An onboard program for the fighter aircraft fuel gauge system is developed using a computational model resulting in to an optimal estimate of fuel gauge system. A fighter aircraft has several fuel tanks and the fuel quantity of the fuel tanks is required to be gauged reliably taking into account the various dynamic effects of flight maneuvers like aircraft pitch angle, roll angle and acceleration parameters for aircraft CG management. Real time estimation and computational modeling of fuel gauging system is realised by using the least square estimation numerical method with Microcontroller based Electronic unit. It was seen that the fluctuations in the fuel quantity estimation under the dynamic flight conditions are reduced and fuel gauging accuracy is improved.

Optimal Linear MMSE Estimation Under Correlation Uncertainty in Restricted Bayesian Framework

A restricted Bayes approach is proposed for linear estimation of a scalar random parameter based on a scalar observation under uncertainty regarding the correlation between the parameter and the observation. In particular, the optimal linear estimator that minimizes the average mean-squared error (MSE) is derived under a constraint on the worst-case MSE by considering possible values of the correlation coefficient and its probability distribution. A closed-form expression is derived for the optimal linear estimator in the proposed restricted Bayesian framework by considering a generic statistical characterization of the correlation coefficient. Performance of the proposed estimator is evaluated via numerical examples and its benefits are illustrated in various scenarios. The proposed framework is also extended to the case of vector-valued observation and the properties of the optimal linear estimator are characterized.

L2 error estimates for a family of cubic finite volume methods on triangular meshes

This paper considers the error estimates of a family of Lagrange cubic finite volume methods for solving two dimensional elliptic boundary value problems over triangular meshes. By introducing a novel map of finite volume method, we propose a family of cubic finite volume method schemes with four independent map coefficients. It is detailedly proved that the proposed finite volume method schemes is exactly equivalent to the existing traditional finite volume method schemes of the same boundary value problems. A geometric constraint requirement on the triangular meshes of primary partitions is imposed to guarantee the uniform ellipticity of discrete elliptic bilinear forms resulted from the proposed schemes. The geometric requirements of the primary partitions could be greatly attenuated when the map coefficients are appropriately designed. Once the uniform ellipticity of the discrete elliptic bilinear forms is established, the proposed schemes have the unique solvability, and the optimal H1-norm error estimates without any restriction on dual elements of dual partitions of the finite volume methods, which is consistent with the results of literatures. With the help of the introduced map, the proposed schemes have the optimal L2-norm error estimates with a mesh restriction for the dual elements of their dual partitions significantly weaker than the existing theoretical results, in which the Aubin-Nitsche duality argument and the orthogonal conditions based on the dual partitions are employed. Finally, we validate our theoretical results by numerical experiments.

A Deterministic Least Squares Approach for Simultaneous Input and State Estimation

This paper considers a deterministic estimation problem to find the input and state of a linear dynamical system which minimise a weighted integral squared error between the resulting output and the measured output. A completion of squares approach is used to find the unique optimum in terms of the solution of a Riccati differential equation. The optimal estimate is obtained from a two-stage procedure that is reminiscent of the Kalman filter. The first stage is an end-of-interval estimator for the finite horizon which may be solved in real time as the horizon length increases. The second stage computes the unique optimum over a fixed horizon by a backwards integration over the horizon. A related tracking problem is solved in an analogous manner. Making use of the solution to both the estimation and tracking problems a constrained estimation problem is solved which shows that the Riccati equation solution has a least squares interpretation that is analogous to the meaning of the covariance matrix in stochastic filtering. The paper shows that the estimation and tracking problems considered here include the Kalman filter and the linear quadratic regulator as special cases. The infinite horizon case is also considered for both the estimation and tracking problems. Stability and convergence conditions are provided and the optimal solutions are shown to take the form of left inverses of the original system.

Secure Estimation With Privacy Protection.

In this article, we focus on the state estimation problems for a system with protecting user privacy. Regarding whether the user has conducted a sensitive action in the system as a kind of privacy, we propose a privacy-preserving mechanism (PPM) to prevent its action results from being disclosed or inferred. For such a system with the PPM, we first obtain the optimal estimator (OE). Subject to the inoperability of the OE in practice, we turn to designing a computationally efficient suboptimal estimator (SE) as an alternative. Then, we prove that this SE can remain stable while satisfying the user's requirements on both privacy protection and estimation performance. By solving a privacy-preserving optimization problem, a set of guidelines is established to customize a tradeoff between privacy and performance according to the user's demand. Finally, illustrated examples are used to illustrate the main theoretical results.

Unconditional energy stability and temporal convergence of first-order numerical scheme for the square phase-field crystal model

This paper presents a sixth-order nonlinear parabolic problem of the square phase-field crystal model. We first demonstrate the time-discrete backward Euler scheme with mass conservation and energy stability. Then, we prove the unconditionally optimal error estimates for the time-discrete backward Euler scheme. In the end, we present 2D and 3D numerical simulations to confirm the theoretical results.

Two-grid algorithms based on FEM for nonlinear time-fractional wave equations with variable coefficient

In this paper, a fully discrete finite element scheme is established with L1 approximation for solving the nonlinear time-fractional wave equation with a variable coefficient. The stability of the scheme is analyzed and the optimal error estimates are derived. To improve the computational efficiency, we propose a two-grid algorithm based on the fully discrete finite element scheme. With the proposed technique, a small-scale nonlinear problem is calculated by iteration in a coarse-grid space with mesh size H, and then a linearized problem is solved in a fine-grid space with mesh size h, H≫h. This technique not only maintains the numerical precision, but also saves a lot of computing time. The stability and optimal convergence order are considered. Using the Taylor expansion, the second two-grid algorithm is constructed to improve the optimal convergence order. The similar properties are discussed in detail. Numerical experiments are provided to illustrate the theoretical analysis and test the performance of the proposed algorithms.

Linearized fast time-stepping schemes for time–space fractional Schrödinger equations

This paper proposes a linearized fast high-order time-stepping scheme to solve the time–space fractional Schrödinger equations. The time approximation is done by using the fast L2-1σ formula on nonuniform meshes. The spatial discretization is done by using Fourier spectral methods. Optimal error estimates of the numerical method are obtained unconditionally. Such unconditional convergence results are proved directly under the assumption f∈C1(R) by the fractional Sobolev inequalities and boundedness of the numerical solutions. In contrast, the previous unconditionally convergence results are usually proved by using the error splitting argument under the assumption f∈C3(R) and the help of some additional systems. Numerical experiments are given to confirm the theoretical results.

Direction finding of electromagnetic radiation sources using the volumetric phased arrays consisting of the small number of elements

Problem statement. This article proposes a method to measure the bearing of electromagnetic radiation sources in the conditions of passive interference with multipath propagation and in the conditions of using "rough" (small number of elements) statistics taking into account particular design features of the direction finder located on a mobile object. The technical design of an adaptive direction finder (ADF) with a small number of channels M = 7 and “short spatial samples” m = 2 (bases) is substantiated, which leads to reducing hardware and software costs. Target. Determine the method of direction finding of electromagnetic radiation sources by volumetric phased arrays consisting of the small number of elements, which allow for the presence of passive interference with multipath radiation of electromagnetic radiation sources, and include spatial digital filters (DF) of a low order r = 2. Results. The problem of measuring the bearing of electromagnetic radiation sources under the conditions of correlated multipath passive interference and limited hardware and computing resources was solved. Direction finder includes a 7-element monopulse direction finder, which was built according to a sequential scheme at the nodes of a volumetric hexagonal grid with an equivalent spatial quantization step of 0.5l, forming programmable small-sized bases with a size of no more than l using the method of multiplexing (HF switching). Practical significance. The relevance of the proposed approach is substantiated due to the need to notch "multipath components", parasitic reflections from local objects surrounding the adaptive direction finder, and reduce the instrumental error in bearing measurement. A new approach to the construction of a volumetric phased array of an adaptive direction finder is proposed. Novelty. The synthesis of the optimal estimate is based on the principle of “whitewashing” passive correlated interference and the maximum likelihood method, which guarantees the achievement of potential direction finding accuracy in the conditions of using rough statistics that reduce hardware and software costs.

Optimal out‐of‐sample forecast evaluation under stationarity

AbstractIt is a common practice to split a time series into an in‐sample and pseudo‐out‐of‐sample segments and estimate the out‐of‐sample loss for a given statistical model by evaluating forecasting performance over the pseudo‐out‐of‐sample segment. We propose an alternative estimator of the out‐of‐sample loss, which, contrary to the conventional wisdom, utilizes criteria measured both in‐ and out‐of‐sample via a carefully constructed system of affine weights. We prove that, provided that the time series is stationary, the proposed estimator is the best linear unbiased estimator of the out‐of‐sample loss and outperforms the conventional estimator in terms of sampling variability. Application of the optimal estimator to Diebold–Mariano type tests of predictive ability leads to a substantial power gain without increasing finite sample size distortions. An extensive evaluation on real‐world time series from the M4 forecasting competition confirms superiority of the proposed estimator and also demonstrates substantial robustness to violations of the underlying assumption of stationarity.

Analysis of the MAC scheme for the three dimensional Stokes problem

In this paper, we propose a marker and cell method (MAC) with a proper quadrature scheme over non-uniform cuboid grids by using a staggered finite volume element method (FVEM) for 3D Stokes equations. The stability of the proposed MAC scheme is proven under Petrov-Galerkin method. By establishing a connection between MAC and FVEM, we rigorously prove the superconvergence property and the optimal order L2 error estimate. Finally, numerical results verify our theoretical findings.

Approximate message passing from random initialization with applications to Z2 synchronization

This paper is concerned with the problem of reconstructing an unknown rank-one matrix with prior structural information from noisy observations. While computing the Bayes optimal estimator is intractable in general due to the requirement of computing high-dimensional integrations/summations, Approximate Message Passing (AMP) emerges as an efficient first-order method to approximate the Bayes optimal estimator. However, the theoretical underpinnings of AMP remain largely unavailable when it starts from random initialization, a scheme of critical practical utility. Focusing on a prototypical model called [Formula: see text] synchronization, we characterize the finite-sample dynamics of AMP from random initialization, uncovering its rapid global convergence. Our theory-which is nonasymptotic in nature-in this model unveils the non-necessity of a careful initialization for the success of AMP.

A Space-Time Finite Element Method for the Fractional Ginzburg–Landau Equation

A fully discrete space-time finite element method for the fractional Ginzburg–Landau equation is developed, in which the discontinuous Galerkin finite element scheme is adopted in the temporal direction and the Galerkin finite element scheme is used in the spatial orientation. By taking advantage of the valuable properties of Radau numerical integration and Lagrange interpolation polynomials at the Radau points of each time subinterval In, the well-posedness of the discrete solution is proven. Moreover, the optimal order error estimate in L∞(L2) is also considered in detail. Some numerical examples are provided to evaluate the validity and effectiveness of the theoretical analysis.

Linearized transformed [formula omitted] finite element methods for semi-linear time-fractional parabolic problems

A linearized Galerkin finite element method is presented for numerically solving the semi-linear time-fractional parabolic problems, whose solutions always display a initial weak singularity. The transformed L1 scheme based on a change of variable is used to approximate Caputo derivatives and the finite element approximations to the spatial variables. By the temporal-spatial error splitting argument, unconditionally optimal error estimates of the proposed schemes are proved. Finally, several numerical experiments are given to demonstrate our theoretical results.

Unconditionally optimal error estimate of the Crank–Nicolson extrapolation Galerkin finite element method for Kuramoto–Tsuzuki equation

Unconditionally optimal error estimate of the Crank–Nicolson extrapolation Galerkin finite element method for Kuramoto–Tsuzuki equation

Optimal Error Estimates for Chebyshev Approximations of Functions with Endpoint Singularities in Fractional Spaces

Optimal Error Estimates for Chebyshev Approximations of Functions with Endpoint Singularities in Fractional Spaces

Stability and temporal error analysis for SAV schemes for electrohydrodynamic model with variable density

In this paper, we construct and analyze first- and second-order schemes based on scalar auxiliary variable (SAV) approach for the electrohydrodynamic (EHD) model with variable density. These schemes only require solving a sequence of linear differential equations plus a linear well-posed algebraic equation at each time step, and are unconditionally energy stable. We carry out a rigorous error analysis for the first-order semi-discrete SAV scheme in two-dimensional case and derive the maximum principle, the optimal error estimates and the regularity estimates. Numerical experiments verify the accuracy and stability of the presented schemes.