Second-order Taylor Expansion Research Articles

Efficient Constant Envelope Precoding for Massive MU-MIMO Downlink via Majorization-Minimization Method.

The practical implementation of massive multi-user multi-input-multi-output (MU-MIMO) downlink communication systems power amplifiers that are energy efficient; otherwise, the power consumption of the base station (BS) will be prohibitive. Constant envelope (CE) precoding is gaining increasing interest for its capability to utilize low-cost, high-efficiency nonlinear radio frequency amplifiers. Our work focuses on the topic of CE precoding in massive MU-MIMO systems and presents an efficient CE precoding algorithm. This algorithm uses an alternating minimization (AltMin) framework to optimize the CE precoded signal and precoding factor, aiming to minimize the difference between the received signal and the transmit symbol. For the optimization of the CE precoded signal, we provide a powerful approach that integrates the majorization-minimization (MM) method and the fast iterative shrinkage-thresholding (FISTA) method. This algorithm combines the characteristics of the massive MU-MIMO channel with the second-order Taylor expansion to construct the surrogate function in the MM method, in which minimizing this surrogate function is the worst-case of the system. Specifically, we expand the suggested CE precoding algorithm to involve the discrete constant envelope (DCE) precoding case. In addition, we thoroughly examine the exact property, convergence, and computational complexity of the proposed algorithm. Simulation results demonstrate that the proposed CE precoding algorithm can achievean uncoded biterror rate (BER) performance gain of roughly 1dB compared to the existing CE precoding algorithm and has an acceptable computational complexity. This performance advantage also exists when it comes to DCE precoding.

Risk sensitive linear approximations

We propose a linear approximation to the solution of DSGE models that is sensitive to the effects of risk. If variables remain close to the approximation point in expectation, a second-order Taylor expansion to the equilibrium conditions reduces to a fixed-point problem characterized by a system of linear equations that depends on the second-order moments of the variables. The latter can be solved recursively using standard linear rational expectation methods. The resulting approximation captures the effects of risk in models with constant volatility, stochastic volatility, and GARCH effects through the intercept and/or the slopes of the decision rules. Relative to alternative approximations, our method yields approximation errors that are up to two orders of magnitude smaller, all while preserving a linear structure in the state variables. Finally, we show how to accommodate asymmetric effects from non-normal shocks within our linear approximation using information from a third-order Taylor expansion.

Qubit Count Reduction by Orthogonally Constrained Orbital Optimization for Variational Quantum Excited-State Solvers.

We propose a state-averaged orbital optimization scheme for improving the accuracy of excited states of the electronic structure Hamiltonian for use on near-term quantum computers. Instead of parameterizing the orbital rotation operator in the conventional fashion as an exponential of an antihermitian matrix, we parameterize the orbital rotation as a general partial unitary matrix. Whereas conventional orbital optimization methods minimize the state-averaged energy using successive Newton steps of the second-order Taylor expansion of the energy, the method presented here optimizes the state-averaged energy using an orthogonally constrained gradient projection method that does not require any expansion approximations. Through extensive benchmarking of the method on various small molecular systems, we find that the method is capable of producing more accurate results than fixed basis FCI while simultaneously using fewer qubits. In particular, we show that for H2, the method is capable of matching the accuracy of FCI in the cc-pVTZ basis (56 qubits) while only using 14 qubits.

Kernel-based diffusion approximated Markov decision processes for autonomous navigation and control on unstructured terrains

We propose a diffusion approximation method to the continuous-state Markov decision processes that can be utilized to address autonomous navigation and control in unstructured off-road environments. In contrast to most decision-theoretic planning frameworks that assume fully known state transition models, we design a method that eliminates such a strong assumption that is often extremely difficult to engineer in reality. We first take the second-order Taylor expansion of the value function. The Bellman optimality equation is then approximated by a partial differential equation, which only relies on the first and second moments of the transition model. By combining the kernel representation of the value function, we design an efficient policy iteration algorithm whose policy evaluation step can be represented as a linear system of equations characterized by a finite set of supporting states. We first validate the proposed method through extensive simulations in 2D obstacle avoidance and 2.5D terrain navigation problems. The results show that the proposed approach leads to a much superior performance over several baselines. We then develop a system that integrates our decision-making framework with onboard perception and conduct real-world experiments in both cluttered indoor and unstructured outdoor environments. The results from the physical systems further demonstrate the applicability of our method in challenging real-world environments.

Low-Order Automatic Domain Splitting Approach for Nonlinear Uncertainty Mapping

This paper introduces a novel method for the automatic detection and handling of nonlinearities in a generic transformation. A nonlinearity index that exploits second-order Taylor expansions and polynomial bounding techniques is first introduced to estimate the Jacobian variation of a nonlinear transformation. This index is then embedded into a low-order automatic domain splitting algorithm that accurately describes the mapping of an initial uncertainty set through a generic nonlinear transformation by splitting the domain whenever nonlinearities grow above a predefined threshold. The algorithm is illustrated in the critical case of orbital uncertainty propagation, and it is coupled with a tailored merging process that limits the growth of the domains in time by recombining them when nonlinearities decrease. The low-order automatic domain splitting algorithm is then combined with Gaussian mixture models to accurately describe the propagation of a probability density function.

An AC-Feasible Linear Model in Distribution Networks With Energy Storage

With the increasing deployment of distributed energy resources (DERs), dispatching DERs subject to operational constraints in distribution networks draws much attention. One challenge is the non-convexities in 1) system-wide AC power flow constraints and 2) the individual complementarity constraint of energy storage. To resolve this challenge, this paper studies an AC-feasible linear model in distribution networks with energy storage, including its formulation, analysis, and applications. First, an AC-feasible linear model is formulated as a set of linear constraints on controllable DERs and uncontrollable power demand by 1) converting the non-convex system-wide constraints into linear constraints based on the Brouwer's fixed-point theorem and the second-order Taylor expansion, and 2) replacing the non-convex individual complementarity constraint of energy storage with one properly designed linear constraint. Furthermore, to analyze the power demand level at which the proposed linear model can provide a solution, this paper proposes an examination-based projection method under the Monte Carlo framework to handle projections of thousands of dimensions from linear constraints over time periods. Finally, the potential applications of our AC-feasible linear model are discussed. Numerical experiments are conducted in the IEEE 33-bus and 136-bus test systems to demonstrate the effectiveness of the proposed methods.

A novel refined Caputo kernel and constitutive concepts for semi-exact nonlinear dynamic and creep analyses of suddenly pressurized hollow fractional-order visco-hyperelastic cylinders

Some reports have confirmed that discretizing the mass, damping, and stiffness characteristics of a continuum can result in remarkable deviations from the experimental results; as all these properties are intertwined and coupled within a single material. The viscoelastic constitutive laws with fractional-order time derivatives can model this integrity much more accurately, due to their capability of using a set of an infinite number of orders. Here, an innovative constitutive model with a nonlinear hyperelastic memory element that treats simultaneously the fractional-order viscoelasticity and Mooney-Rivlin hyperelasticity of the continuum is proposed to track the suppression of the short-term dynamic stresses and displacements and the redistribution of the long-term creep stress and deformations of the suddenly pressurized thick cylinders. While the spatial variations of the quantities of the incompressible cylinder are treated by an exact analytical approach, the resulting time-dependent system of equations is converted by a novel numerical solution scheme to an integrodifferential system that contains integer-order partial derivatives and a time-growing number of integral terms. An enhancement to Caputo's singular-kernel integral definition of the fractional-order derivatives is proposed here to replace these derivatives with equivalent integrals whose integer derivatives are computed based on second-order Taylor's expansion. The integer-order and fractional-order time derivatives are then computed numerically using the second-order Runge-Kutta and accumulation-based trapezoidal techniques, respectively. Finally, comparisons are made among the dissipative pattern and time variations of the dynamic/vibration stresses and large-deformations of the hyperelastic and fractional-order visco-hyperelastic cylinders in addition to the assessment of the effects of the constitutive parameters of the fractional-order model. Results show that while the larger magnitudes of both the nonlinear viscosity coefficients and fractional-order time derivatives diminish the resulting deformations and stresses, they respectively lead to smaller and larger frequencies of oscillation.

Weld line structured light stripe center extraction approach based on data dimensionality reduction

In order to improve the accuracy and speed of extracting the center of the line structured light stripe when using the vision measurement technology to get the 3D information of the weld sample, this paper proposes a center extraction approach based on the idea of data dimensionality reduction. First, use the weighted gray centroid approach to get the initial center point of the light stripe. Then, build a 7×7 template centered on the pixel block where the initial point is located and performs singular value decomposition on the 49×3 matrix within the template to get the normal direction of the initial point. Finally, calculate the stripe's exact center points through second-order Taylor expansion and smooth repair by cubic spline interpolation of breakpoints. The comparison experimental results show that this approach can effectively reduce the influence of the noise outside the light stripe on the center extraction, reduce the extraction time while ensuring the measurement accuracy, which can lay a good foundation for the point cloud data processing and surface reconstruction.

A hybrid bias correction algorithm for estimating straight line parameters

In the common robust weighted total least squares estimator of the errors-in-variables model for straight line fitting, a bias exists in the estimated variance component used to reweight the weight matrix of observations when outliers are present. Additionally, when the second-order Taylor expansion of the functional model is not considered, the estimator is often computed with bias. A hybrid bias correction algorithm is proposed to reduce the two biases simultaneously and achieve unbiased results, which is demonstrated by three examples. The results show that the proposed algorithm can realize the bias correction of parameters and variance component estimated by robust weighted total least squares (RWTLS) ,and obtain the minimum variance component estimates. The new method is more suitable for correlation and/or outliers case and can detect outliers that cannot be detected by RWTLS.

A multifidelity approach to robust orbit determination

This paper presents an algorithm for the preprocessing of observation data aimed at improving the robustness of orbit determination tools. Two objectives are fulfilled: obtain a refined solution to the initial orbit determination problem and detect possible outliers in the processed measurements. The uncertainty on the initial estimate is propagated forward in time and progressively reduced by exploiting sensor data available in said propagation window. Differential algebra techniques and a novel automatic domain splitting algorithm for second-order Taylor expansions are used to efficiently propagate uncertainties over time. A multifidelity approach is employed to minimize the computational effort while retaining the accuracy of the propagated estimate. At each observation epoch, a polynomial map is obtained by projecting the propagated states onto the observable space. Domains that do no overlap with the actual measurement are pruned thus reducing the uncertainty to be further propagated. Measurement outliers are also detected in this step. The refined estimate and retained observations are then used to improve the robustness of batch orbit determination tools. The effectiveness of the algorithm is demonstrated for a geostationary transfer orbit object using synthetic and real observation data from the TAROT network.

A Robust Symbol-Level Precoding Method for Multibeam Satellite Systems

Compared to interference mitigating precoding, interference exploiting symbol-level precoding (SLP) requires less transmit power to guarantee the quality of service in a multibeam satellite system. However, due to the large roundtrip time (RTT), it is impractical to obtain real-time channel state information on the satellite side. The random channel state information (CSI) phase error of outdated CSI could cause serious performance deterioration of SLP. To compensate the CSI phase error, we propose an outdated CSI-based robust SLP (RSLP) method, which optimizes the transmit power under outage probability constraints. The central limit theorem (CLT) and second-order Taylor expansion are used to relax the outage probability constraints into convex ones. In addition, because only outdated CSI-based robust block-level precoding exists, we present two comparative RSLP methods accordingly. Without violating outrage probability constraints, the proposed RSLP method requires much lower transmit power than comparative RSLP and existing robust block-level precoding methods. The complexity of the proposed RSLP method is also lower than that of two comparative RSLP methods.

Influence of Random Errors in Element Positions on Performance of Antenna Arrays Considering Mutual Coupling Effect

In engineering, random errors in radiating element positions caused by the manufacturing and assembly process could lead to degradation of the radiation performance of an antenna array. In order to accurately evaluate the deterioration of the antenna performance in the design phase, a statistical analysis method considering the mutual coupling effect between array elements is developed herein. The mutual coupling effect is described in terms of modal coupling and is governed by the modal coupling equilibrium equation. The statistical characteristics such as the mean and variance of the gain pattern are obtained based on the second-order Taylor expansion, and the expansion coefficients are obtained by the adjoint method. The correctness and effectiveness of the proposed method are verified through a comparison with the standard Monte Carlo method.

An overview of a posteriori error estimation and post-processing methods for nonlinear eigenvalue problems

In this article, we present an overview of different a posteriori error analysis and post-processing methods proposed in the context of nonlinear eigenvalue problems, e.g. arising in electronic structure calculations for the calculation of the ground state and compare them. We provide two equivalent error reconstructions based either on a second-order Taylor expansion of the minimized energy, or a first-order expansion of the nonlinear eigenvalue equation. We then show how several a posteriori error estimations as well as post-processing methods can be formulated as specific applications of the derived reconstructed errors, and we compare their range of applicability as well as numerical cost and precision.

Improved characterization for AC-feasible power transfer regions of virtual power plants

Aggregating distributed energy resources as virtual power plants (VPPs) to participate in transmission-level operations is a promising direction for future grids. However, it is challenging to decide the AC-feasible power transfer region of such a VPP at its point of common coupling with the transmission. This paper proposes an improved method to characterize this region. The proposed method constructs linear constraints to inner-approximate the AC-feasible power transfer regions, by applying Brouwer’s fixed-point theorem to the second-order Taylor expansion of the nonlinear Dist-flow equations. An exploration strategy is proposed to further improve our characterization flexibility. Finally, a transmission-level operation problem with VPP participation is formulated and solved through big-M linearization. The proposed methods are verified by numerical experiments on the IEEE 33-bus and 136-bus test systems.

Novel Discrete-Time Recurrent Neural Network for Robot Manipulator: A Direct Discretization Technical Route.

Controlling and processing of time-variant problem is universal in the fields of engineering and science, and the discrete-time recurrent neural network (RNN) model has been proven as an effective method for handling a variety of discrete time-variant problems. However, such model usually originates from the discretization research of continuous time-variant problem, and there is little research on the direct discretization method. To address the aforementioned problem, this article introduces a novel discrete-time RNN model for solving the discrete time-variant problem in a pioneering manner. Specifically, a discrete time-variant nonlinear system, which originates from the mathematical modeling of serial robot manipulator, is presented as a target problem. For solving the problem, first, the technique of second-order Taylor expansion is used to deal with the discrete time-variant nonlinear system, and the novel discrete-time RNN model is proposed subsequently. Second, the theoretical analyses are investigated and developed, which shows the convergence and precision of the proposed discrete-time RNN model. Furthermore, three distinct numerical experiments verify the excellent performance of the proposed discrete-time RNN model. In addition, a robot manipulator example further verifies the effectiveness and practicability of the proposed novel discrete-time RNN model.

Finite deformation analysis of electro-active shells

In this contribution, a novel nonlinear formulation for the finite deformation analysis of shell-like structures made of dielectric polymers and subject to electrical loading is developed. The kinematics of mechanical deformation is described via a seven-parameter shell model. Moreover, a second-order Taylor expansion, containing three parameters, is proposed to approximate the electrical potential through the thickness of the shell. The combined mechanical and electrical parameters constitute a ten-parameter electro-shell model. After developing the variational formulation of the problem, a nonlinear finite element formulation for the numerical solution of the coupled electromechanical problem is also developed. The formulation is further enriched by the enhanced assumed strain method to alleviate locking in thin shells. Finally, the applicability of the proposed formulation in several benchmark examples is demonstrated. In particular, it is shown that the formulation can capture the results available in the literature with high accuracy.

A direct discretization recurrent neurodynamics method for time-variant nonlinear optimization with redundant robot manipulators

Discrete time-variant nonlinear optimization (DTVNO) problems are commonly encountered in various scientific researches and engineering application fields. Nowadays, many discrete-time recurrent neurodynamics (DTRN) methods have been proposed for solving the DTVNO problems. However, these traditional DTRN methods currently employ an indirect technical route in which the discrete-time derivation process requires to interconvert with continuous-time derivation process. In order to break through this traditional research method, we develop a novel DTRN method based on the inspiring direct discrete technique for solving the DTVNO problem more concisely and efficiently. To be specific, firstly, considering that the DTVNO problem emerging in the discrete-time tracing control of robot manipulator, we further abstract and summarize the mathematical definition of DTVNO problem, and then we define the corresponding error function. Secondly, based on the second-order Taylor expansion, we can directly obtain the DTRN method for solving the DTVNO problem, which no longer requires the derivation process in the continuous-time environment. Whereafter, such a DTRN method is theoretically analyzed and its convergence is demonstrated. Furthermore, numerical experiments confirm the effectiveness and superiority of the DTRN method. In addition, the application experiments of the robot manipulators are presented to further demonstrate the superior performance of the DTRN method.

An effective nonlinear interval sequential quadratic programming method for uncertain inverse problems

An effective nonlinear interval sequential quadratic programming method is proposed to provide an efficient tool for uncertain inverse problems. Assisted by the ideology of sequential quadratic programming and dimension-reduction analysis theory, the interval inverse problem is transformed into several interval arithmetic and deterministic optimizations, which could enhance computational efficiency without losing much accuracy. The novelty of the proposed method lies in two main aspects. First, an alternate updating strategy is proposed to identify the radii and midpoints of the interval inputs in each cycle, which could reduce the number of iterative steps. Second, the standard quadratic models are constructed based on the dimension-reduction analysis results, rather than the second-order Taylor expansion. Therefore, the interval arithmetic can be applied to efficiently calculate the interval response, which avoids the inner optimization. Moreover, a novel iterative mechanism is developed to accelerate the convergence rate of the proposed method. Finally, two numerical examples and an engineering application are adopted to verify its feasibility, accuracy and efficiency.

Random walks on hyperbolic spaces: second order expansion of the rate function at the drift

Let (X,d) be a separable geodesic Gromov-hyperbolic space, o∈X a basepoint and μ a countably supported non-elementary probability measure on Isom(X). Denote by z n the random walk on X driven by the probability measure μ. Supposing that μ has a finite exponential moment, we give a second-order Taylor expansion of the large deviation rate function of the sequence 1 nd(z n ,o) and show that the corresponding coefficient is expressed by the variance in the central limit theorem satisfied by the sequence d(z n ,o). This provides a positive answer to a question raised in [6]. The proof relies on the study of the Laplace transform of d(z n ,o) at the origin using a martingale decomposition first introduced by Benoist–Quint together with an exponential submartingale transform and large deviation estimates for the quadratic variation process of certain martingales.

Multi-objective voltage/VAR control for integrated port energy system considering multi-network integration

Energy integration and logistics electrification are very promising solutions to reduce port pollution and improve operational efficiency. However, the access of renewable energy and electrified logistics loads may lead to voltage fluctuation and violation issues. Therefore, this paper proposes a multi-objective voltage/VAR control (VVC) method considering multi-network integration. This method can coordinate multiple devices of electricity, heat, gas, and logistics in multiple timescales to minimize voltage deviation and operating costs. To achieve the targets, a VVC model covering multi-network power flow calculation is established, which makes the control results more accurate. The second-order cone relaxation and Taylor expansion are used to deal with the nonlinear part of the power flow calculation, so that the scheduling model can be solved by commercial solvers. Multi-scenario stochastic optimization is also used to offset the uncertainties in renewable energy, load, and temperature. The case results show that the proposed method can fully use the dynamic properties of the integrated port energy system (IPES) and logistics system, improve the port operation economy, and reduce voltage deviation.