First-order Taylor Series Expansion Research Articles

Role of the Observability Gramian in Parameter Estimation: Application to Nonchaotic and Chaotic Systems via the Forward Sensitivity Method

Data assimilation in chaotic regimes is challenging, and among the challenging aspects is placement of observations to induce convexity of the cost function in the space of control. This problem is examined by using Saltzman’s spectral model of convection that admits both chaotic and nonchaotic regimes and is controlled by two parameters—Rayleigh and Prandtl numbers. The problem is simplified by stripping the seven-variable constraint to a three-variable constraint. Since emphasis is placed on observation positioning to avoid cost-function flatness, forecast sensitivity to controls is needed. Four-dimensional variational assimilation (4D-Var) is silent on this issue of observation placement while Forecast Sensitivity Method (FSM) delivers sensitivities used in placement. With knowledge of the temporal forecast sensitivity matrix V, derivatives of the forecast variables to controls, the cost function can be expressed as a function of the observability Gramian VTV using first-order Taylor series expansion. The goal is to locate observations at places that force the Gramian positive definite. Further, locations are chosen such that the condition number of VTV is small and this guarantees convexity in the vicinity of the cost function minimum. Four numerical experiments are executed, and results are compared with the structure of the cost function independently determined though arduous computation over a wide range of the two nondimensional numbers. The results are especially good based on reduction in cost function value and comparison with cost function structure.

Investigations of moiré artifacts induced by flux fluctuations in x-ray dark-field imaging

X-ray dark-field imaging using a grating interferometer has shown potential benefits for a variety of applications in recent years. X-ray dark-field image is commonly retrieved by using discrete Fourier transform from the acquired phase-stepping data. The retrieval process assumes a constant phase step size and a constant flux for each stepped grating position. However, stepping errors and flux fluctuations inevitably occur due to external vibrations and/or thermal drift during data acquisition. Previous studies have shown that those influences introduce errors in the acquired phase-stepping data, which cause obvious moiré artifacts in the retrieved refraction image. This work investigates moiré artifacts in x-ray dark-field imaging as a result of flux fluctuations. For the retrieved mean intensity, amplitude, visibility and dark-field images, the dependence of moiré artifacts on flux fluctuation factors is theoretically derived respectively by using a first-order Taylor series expansion. Results of synchrotron radiation experiments verify the validity of the derived analytical formulas. The spatial frequency characteristics of moiré artifacts are analyzed and compared to those induced by phase-stepping errors. It illustrates that moiré artifacts can be estimated by a weighted mean of flux fluctuation factors, with the weighting factors dependent on the moiré phase and different greatly for each retrieved image. Furthermore, moiré artifacts can even be affected by object’s features not displayed in the particular contrast. These results can be used to interpret images correctly, identify sources of moiré artifacts, and develop dedicated algorithms to remove moiré artifacts in the retrieved multi-contrast images.

A Nonlinear Optimal Control Method for Attitude Stabilization of Micro-Satellites

Attitude control and stabilization of micro-satellites is a nontrivial problem due to the highly nonlinear and multivariable structure of the satellites’ state-space model. In this paper, a novel nonlinear optimal (H-infinity) control approach is developed for this control problem. The dynamic model of the satellite’s attitude dynamics undergoes first approximate linearization around a temporary operating point which is updated at each iteration of the control algorithm. The linearization process relies on first-order Taylor series expansion and on the computation of the Jacobian matrices of the state-space model of the satellite’s attitude dynamics. For the approximately linearized description of the satellite’s attitude a stabilizing H-infinity feedback controller is designed. To compute the controller’s feedback gains, an algebraic Riccati equation is solved at each time-step of the control method. The stability properties of the control scheme are proven through Lyapunov analysis. It is also demonstrated that the control method retains the advantages of linear optimal control that is fast and accurate tracking of the reference setpoints under moderate variations of the control inputs.

Variability response functions for statically indeterminate structures

The response variability of statically indeterminate beam structures caused by uncertain material properties modeled by random fields is studied in this paper. By solving the governing equations of the statically indeterminate structure, the response bending moment along the length of the beam, M(x), is expressed as a function of its (random) zero-moment location. This approach, combined with a second-order Taylor series expansion of the random zero-moment location, leads to novel Variability Response Function-based integral expressions for the variance of the response bending moment, Var[M(x)]. Variability Response Functions have numerous desirable attributes including the capability to perform a full sensitivity analysis of the response variability with respect to the spectral characteristics of the random field modeling the uncertain material/system properties and establishing realizable upper bounds of the response variability. The proposed expressions are fundamentally different from all currently existing expressions that are based inherently on first-order Taylor series expansions, as they involve: (i) two distinct integrals in the expression for Var[M(x)] and, (ii) an expression for the mean value ɛ[M(x)] that depends on the stochastic field modeling the uncertain material properties. In general, the proposed second-order Taylor series-based formulation is expected to provide more accurate results. Extensive numerical examples are provided where the accuracy of the results obtained using the proposed formulation is validated using Monte Carlo simulations involving stochastic fields that follow truncated Gaussian and shifted lognormal probability distribution functions. These Monte Carlo simulation results indicate that the proposed Variability Response Functions are probability-distribution-independent.

The controlling factors of ecosystem water use efficiency in maize fields under drip and border irrigation systems in Northwest China

The ecosystem water use efficiency (WUEe) is the critical link between water and carbon cycling in the terrestrial ecosystem. The evaluations of the biophysical factors influencing WUEe could reveal mechanisms of controlling WUEe. In order to save water in agriculture, drip irrigation (DI) has been promoted to replace border irrigation (BI) in northwest China. Different irrigation methods will cause differences in evapotranspiration (ETa), gross primary productivity (GPP), and WUEe. Further research is needed to quantify the contribution of each factor to the differences in ETa, GPP, and WUEe. To separate the different factors controlling ETa, GPP, and WUEe variability between DI and border BI, the 4-year data under DI and BI measured by eddy covariance systems were collected and analyzed. To explore the differences in ETa, GPP, and WUEe under different irrigation methods, Penman-Monteith (P-M) model and stomatal conductance-photosynthetic transpiration coupling (SMPT-SB) model complemented by perturbation analysis were employed. The results showed that the 4-year mean evapotranspiration (ETa) decreased by 8% under DI compared with the value under BI, and the mean gross primary productivity (GPP) under DI was 5% higher than the BI. The WUEe under DI was 14% higher than that under BI. The P-M model and the SMPT-SB model agreed well with ETa and GPP measured on daily time scales, providing confidence in their ability to separate the biological and physical controls of ETa, GPP, and WUEe. Based on the two models, the first-order Taylor series expansions of the total ETa, GPP, and WUEe derivatives were applied to the P-M model and SMPT-SB model and compared with measured differences in ETa, GPP, and WUEe. More than 60% of the difference in ETa caused by the two irrigation methods was attributed to variations in the coupled surface resistance, and drip irrigation mainly achieves water saving by reducing soil evaporation. The difference in maize growth due to irrigation methods was the main reason for the differences in GPP. The differences in vapor pressure deficit and the coupled surface resistance caused by irrigation methods were the main factors causing the difference in WUEe. It could be surmised that drip irrigation increased WUEe in maize fields by changing surface resistance, which mainly reduced soil evaporation.

Uncertain Quasi-static and Nonlinear Dynamic Analysis of Viscoelastic Dielectric Elastomer with Interval Parameters

Dielectric elastomers as a soft active material have been widely used in the field of artificial muscle actuator, acoustic actuator, loudspeaker, active control of vibration, soft robots and membrane resonators. Compared with traditional materials, there are many unknown uncertainties in the properties of the DE actuators. In this work, a viscoelastic dynamic model of dielectric elastomer is proposed with considering the uncertainties in material parameters, external mechanical load and voltage. By introducing the interval perturbation method and first-order Taylor series expansion method, the creep analysis, relaxation analysis and dynamic analysis of the dielectric elastomer with interval uncertain parameters are implemented. The effectivity of the proposed interval method is verified by the Monte Carlo simulation. This uncertain prediction method could be used in the design of active control systems with dielectric elastomers as actuators or sensors in the future.

Energy-Efficient Hybrid Beamforming for Multilayer RIS-Assisted Secure Integrated Terrestrial-Aerial Networks

The integration of aerial platforms to provide ubiquitous coverage and connectivity for densely deployed terrestrial networks is expected to be a reality in the emerging sixth-generation networks. Energy-effificient and secure transmission designs are two important components for integrated terrestrial-aerial networks (ITAN). Inlight of the potential of reconfigurable intelligent surface (RIS) for significantly reducing the system power consumption and boosting information security, this paper proposes a multi-layer RIS-assisted secure ITAN architecture to defend against simultaneous jamming and eavesdropping attacks, and investigates energy-efficient hybrid beamforming for it. Specifically, with the availability of imperfect angular channel state information (CSI), we propose a block coordinate descent (BCD) framework for the joint optimization of the user’s received decoder, the terrestrial and aerial digital precoder, and the multi-layer RIS analog precoder to maximize the system energy efficiency (EE) performance. For the design of the received decoder, a heuristic beamforming scheme is proposed to convert the worst-case design problem into a min-max one and facilitate the developing a closed-form solution. For the design of the digital precoder, we propose an iterative sequential convex approximation approach via capitalizing the auxiliary variables and first-order Taylor series expansion. Finally, a monotonic vertex-update algorithm with a penalty convex-concave procedure (P-CCP) is proposed to obtain the analog precoder with satisfactory performance. Numerical results show the superiority and effectiveness of the proposed optimization framework and architecture over various benchmark schemes.

UNCERTAINTY MODELING FOR POINT CLOUD-BASED AUTOMATIC INDOOR SCENE RECONSTRUCTION BY STRICT ERROR PROPAGATION ANALYSIS

Abstract. Accurate digital representation of indoor facilities is a key component for the generation of building twins. 3D indoor scenes are often reconstructed from 3D point clouds obtained by various measurement techniques, which usually show different accuracy characteristics. During the reconstruction process, the uncertainties of data and intermediate products propagate into the accuracy of the vectorized model. Although point clouds-based 3D building modeling has been a hot topic of research for at least two decades, a thorough analysis of error propagation for this problem from a geodetic point of view is still underrepresented. In this contribution, we propose an analytical approach to estimate the uncertainty of 3D modeling results using the analytic approach based on first-order Taylor-series expansion. A general model for the input data is established and the uncertainty expressions of all computed products are symbolically derived. We estimate the uncertainty of 3D data fitting, followed by the derivation of vectorized building parameters and their covariance matrices. The results of the theoretical approaches are tested on real data presenting an indoor scene. The practical example is illustrated, thoroughly analysed, and quantified.

A nonlinear optimal control approach for shipboard AC/DC microgrids

Shipboard AC/DC microgrids are used for power supply and electric propulsion in vessels. An indicative form of such a microgrid comprises diesel engines or gas turbines that provide power for the rotation of synchronous or asynchronous generators. Next, the AC output voltage of the generators is turned into DC voltage with the use of AC to DC converters and is distributed through DC voltage buses to the ship’s compartments. Besides, with the use of DC to AC inverters voltage excitation is provided to synchronous or asynchronous motors which can be used in turn for the vessel’s propulsion. The dynamic model of the considered shipboard AC/DC microgrid, being initially expressed in a nonlinear and multivariable state-space form, undergoes approximate linearization around a temporary operating point that is recomputed at each time-step of the control method. The linearization relies on first-order Taylor series expansion and on the computation of the associated Jacobian matrices. For the linearized state-space model of the shipboard AC/DC microgrid a stabilizing optimal (H-infinity) feedback controller is designed. This controller stands for the solution to the nonlinear optimal control problem of the AC/DC microgrid under model uncertainty and external perturbations. To compute the controller’s feedback gains an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The global stability properties of the control method are proven through Lyapunov analysis. Finally, to implement state estimation-based control of the shipboard AC/DC microgrid, without the need to measure its entire state vector, the H-infinity Kalman Filter is used as a robust state estimator. The article’s method provides one of the few algorithmically simple and computationally efficient solutions for the nonlinear optimal control problem of shipboard microgrids.

A nonlinear optimal control approach for the truck and N-trailer robotic system

Truck and N-trailer mobile robots find use in freight transportation, urban transportation, mining as well as in agriculture. The article proposes a nonlinear optimal (H-infinity) control approach for the truck and N-trailer robotic system. The method has been successfully tested so far on the control problem of several types of robotic vehicles and here it is shown that it can also provide an optimal solution to the control problem of the underactuated truck and N-trailer mobile robot. To implement this control scheme, the state-space description of the kinematic model of the truck and N-trailer robotic system undergoes first approximate linearization around a temporary operating point, through first-order Taylor series expansion and through the computation of the associated Jacobian matrices. Next, an optimal (H-infinity) feedback controller is designed. To select the feedback gains of the optimal (H-infinity) controller an algebraic Riccati equation is solved at each time-step of the control method. The global stability properties of the control loop are proven through Lyapunov analysis. Finally, to implement state estimation-based feedback control, the H-infinity Kalman Filter is used as a robust state estimator.

Robust state/output feedback linearization of direct drive robot manipulators: A controllability and observability analysis

In this research, a robust feedback linearization technique is analysed for robot manipulators control. A complete first-order Taylor series expansion is used to linearize the robot dynamics which takes into account initial conditions and the Taylor-series remainder. A modified PD control law with Taylor-series compensation is used to guarantee robust reference tracking. Whilst classic feedback linearization controllers guarantee asymptotic convergence to zero, the proposed approach shows that, for real applications, if the linearized robot dynamics is stable then the nonlinear robot states are also stable and remain bounded. This premise is assessed via Lyapunov stability theory under a controllability and observability analysis; and hence, exponential convergence to a bounded set is concluded. Experiments are carried out using a 1-degree of freedom robot and a 4-degree of freedom exoskeleton robot to validate the proposed approach.

Spacecraft attitude control: Application of fine trajectory linearization control

One of the simplest and most popular methods to design a model-based controller for a nonlinear dynamic system is to apply linear control theories to the linearized model of the nonlinear system. For linearization, first-order Taylor series expansion is a straightforward and widely used method. Although the related linearization method is fairly simple, unfortunately, it causes an error in the linearized model, especially in highly nonlinear systems. Different strategies have been proposed to increase the robustness of linearization-based control methods against the linearization error. All of the methods somehow look at the linearization error as uncertainty or disturbance, which has to be rejected. Generally speaking, the lack of the linearization technique reduces the quality of control. One area in which quality and simplicity of control can be necessary is spacecraft attitude control. For the purpose of spacecraft attitude control, this paper improves a well-known trajectory linearization control (TLC) method as a simple and practical nonlinear trajectory tracking control method using an error-less, fine linearization technique. The modified TLC is then used to develop a control law for spacecraft attitude control with only three tunable control parameters and an approximation of the spacecraft inertia matrix. Finally, due to the elimination of uncertainty caused by the linearization error, in a simple way, the novel attitude controller leads to better performance in comparison with the traditional method. The superiority of the new approach is illustrated in numerical simulations.

A nonlinear optimal control approach for underactuated power-line inspection robots

AbstractThe article proposes a nonlinear optimal (H-infinity) control approach for a type of underactuated power-line inspection robots. To implement this control scheme, the state-space model of the power-line inspection robots undergoes first approximate linearization around a temporary operating point, through first-order Taylor series expansion and through the computation of the associated Jacobian matrices. To select the feedback gains of the controller an algebraic Riccati equation is solved at each time step of the control method. The global stability properties of the control loop are proven through Lyapunov analysis. The significance of the article’s results is outlined in the following: (i) the proposed control method is suitable for treating underactuated robotic systems and in general nonlinear dynamical systems with control inputs gain matrices which are in a nonquadratic form, (ii) by achieving stabilization of the power-line inspection robots in underactuation conditions the proposed control method ensures the reliable functioning of these robotic systems in the case of actuators’ failures or enables the complete removal of certain actuators and the reduction of the weight of these robotic systems, (iii) the proposed control method offers a solution to the nonlinear optimal control problem which is of proven global stability while also remaining computationally tractable, (iv) the proposed nonlinear optimal control method retains the advantages of linear optimal control that is fast and accurate tracking of reference setpoints under moderate variations of the control inputs, and (v) by minimizing the amount of energy that is dispersed by the actuators of the power-line inspection robots the proposed control method improves the autonomy and operational capacity of such robotic systems.

Evidence-Theory-Based Kinematic Uncertainty Analysis of a Dual Crane System With Epistemic Uncertainty

Abstract An evidence-theory-based interval perturbation method (ETIPM) and an evidence-theory-based subinterval perturbation method (ETSPM) are presented for the kinematic uncertainty analysis of a dual cranes system (DCS) with epistemic uncertainty. A multiple evidence variable (MEV) model that consists of evidence variables with focal elements (FEs) and basic probability assignments (BPAs) is constructed. Based on the evidence theory, an evidence-based kinematic equilibrium equation with the MEV model is equivalently transformed to several interval equations. In the ETIPM, the bounds of the luffing angular vector (LAV) with respect to every joint FE are calculated by integrating the first-order Taylor series expansion and interval algorithm. The bounds of the expectation and variance of the LAV and corresponding BPAs are calculated by using the evidence-based uncertainty quantification (UQ) method. In the ETSPM, the subinterval perturbation method (SIPM) is introduced to decompose original FE into several small subintervals. By comparing results yielded by the ETIPM and ETSPM with those by the evidence theory-based Monte Carlo method (ETMCM), numerical examples show that the accuracy and computational time of the ETSPM are higher than those of the ETIPM, and the accuracy of the ETIPM and ETSPM can be significantly improved with the increase of the number of FEs and subintervals.

Generalized correntropy induced metric based total least squares for sparse system identification

The total least squares (TLS) method has been successfully applied to system identification in the errors-in-variables (EIV) model, which can efficiently describe systems where input–output pairs are contaminated by noise. In this paper, we propose a new gradient-descent TLS filtering algorithm based on the generalized correntropy induced metric (GCIM), called as GCIM-TLS, for sparse system identification. By introducing GCIM as a penalty term to the TLS problem, we can achieve improved accuracy of sparse system identification. We also characterize the convergence behaviour analytically for GCIM-TLS. To reduce computational complexity, we use the first-order Taylor series expansion and further derive a simplified version of GCIM-TLS. Simulation results verify the effectiveness of our proposed algorithms in sparse system identification.

Nonlinear Optimal Control for the Synchronization of Distributed Marine-turbine Power Generation Units

Power generation from marine-turbines offers a vast potential for raising the contribution of renewable energy sources to the electricity grid. Typically, marine-turbine power units comprise marine-turbines that receive mechanical excitation from sea currents or tidal streams and synchronous or asynchronous generators that are finally connected to the electricity grid. Due to connection to the same transmission lines, interactions between the excitations of the individual generators appear. To achieve stabilization of the generators’ functioning, as well as synchronization to the grid’s frequency the generators’ control has to compensate for such interactions. The article proposes a nonlinear optimal control approach for distributed marine-turbine power units that use synchronous generators. First, the dynamic model of the distributed power system undergoes approximate linearization around a temporary operating point which is re-computed at each time-step of the control algorithm. The linearization relies on first-order Taylor series expansion and on the computation of the associated Jacobian matrices. For the approximately linearized model of the distributed power system a stabilizing H-infinity (optimal) feedback controller is designed. To compute the controller’s feedback gains an algebraic Riccati equation is repetitively solved at each time-step of the control method. The global stability properties of the control scheme are proven through Lyapunov analysis. The proposed control method retains the advantages of optimal control, that is fast and accurate tracking of the reference setpoints, under moderate variations of the control inputs. Among the advantages of the proposed control method one can note: (i) the presented nonlinear optimal control method has improved performance when compared against other nonlinear control schemes that one can consider for the dynamic model of the distributed marine power system (actually other control methods are neither of proven optimality, nor of proven global stability), (ii) avoids complicated state-space transformations and the related appearance of singularities in the computation of the control inputs which are applied to the nonlinear model (iii) minimizes the consumption of energy by the control system of the distributed marine-turbine generators, thus also reducing the functioning cost of such power units.

Development of positioning error measurement system based on geometric optics for long linear stage

The positioning error of the moving stage has a significant influence on the machining accuracy of a machine tool. Therefore, the positioning error must be measured accurately. Laser interferometer systems are widely used for positioning error calibration due to their accuracy, but these systems also have many shortcomings in standard procedure and measurement methods. In this study, a positioning error measurement system for a long linear stage by using geometric optics rather than interference principles is proposed. However, as the stage moves, geometric errors apart from the positioning error are produced. Accordingly, the proposed measurement system is designed to not only measure the positioning error but also simultaneously measure the multi-degree-of-freedom (MDOF) geometric errors of the linear stage. The proposed measurement system was characterized numerically by using the commercial software Zemax; the mathematical model of the errors was constructed by using a skew-ray tracing method, a homogeneous transformation matrix, and a first-order Taylor series expansion. The feasibility and effectiveness of the proposed measurement system were experimentally verified by using a laboratory-built prototype. The experimental results demonstrate that compared with conventional laser interferometers, the proposed measurement system has similar accuracy and can simultaneously measure the six-degree-of-freedom (6-DOF) geometric errors of a long linear stage.

A new type of WENO scheme in SPH for compressible flows with discontinuities

In the past few decades, the weighted essentially non-oscillatory (WENO) scheme has been widely used in mesh-based methods and has achieved great success, but its application to the smoothed particle hydrodynamics (SPH) is still limited. In this paper, a simple and accurate implementation of the WENO scheme to the SPH method is proposed for solving compressible flows with discontinuities and small-scale structures. In the proposed scheme, several equal spacing points along the line joining two interacting particles are selected to constitute candidate stencils. However, due to the Lagrangian characteristic of SPH, some points may be lost. To solve this problem, we first search the particles closest to these missing points, and then the first-order Taylor series expansion is used to obtain the corresponding primitive variables of these points. After that, through the WENO strategy, the left and right states at the interface between two interacting particles are reconstructed. Finally, the inter-particle interactions are determined by using Roe’s approximate Riemann solver. Several numerical tests show that the proposed WENO-SPH method is robust and able to accurately capture shockwaves, and benefiting from the low-dissipation property, it also has a good performance in resolving small-scale structures in flows.

Income and factor substitution: an investigation on the Solow growth model under the constant elasticity of substitution

PurposeThe purpose of this study is to examine whether the elasticity of substitution (ES) varies between developed and developing countries.Design/methodology/approachThe author derives the growth regressions from the Solow model under the constant elasticity of substitution production function by using the first-order Taylor series expansion and estimate them for each country group classified based on time-varying behavior of income per worker using the data-driven algorithm.FindingsThe ES is not unitary and varies among country groups. Developed countries generally have a higher ES than developing countries.Originality/valueFor the first time, the author uses the first-order Taylor series expansion to linearize the steady-state value of income per worker, as the author considers this approach to be relatively more straight-forward and tractable. Furthermore, the author estimates the equations using both cross-section and panel data techniques and employs the data-driven algorithm proposed by Phillips and Sul (2007) to classify countries.

Uncertainty Propagation Through An Empirical Model of Cutting Forces in End Milling

Abstract Empirical mathematical models of cutting forces in machining processes use experimentally determined input parameters to make predictions. A general method for propagation of input parameter uncertainties through such predictive models is developed. Sources of uncertainty are identified and classified. First, a classical uncertainty procedure is employed to estimate uncertainties associated with the data reduction equation using a first-order Taylor series expansion. Small values of input parameter uncertainties justify this local linearization. Coverage factors required to estimate confidence intervals are computed based on appropriate underlying statistical distributions. A root sum of squares method yields the overall expanded uncertainty in force predictions. A popular model used for predicting cutting forces in end milling is selected to demonstrate the procedure, but the demonstrated approach is general. The analysis is applied to experimental data. Force predictions are quoted along with a confidence interval attached to them. An alternative analysis based on Monte Carlo simulations is also presented. This procedure yields different insights compared with the classical uncertainty analysis and complements it. Monte Carlo simulation provides combined uncertainties directly without sensitivity calculations. Classical uncertainty analysis reveals the impacts of random effects and systematic effects separately. This information can prompt the user to improve the experimental setup if the impact of systematic effects is observed to be comparatively large. The method of quoting an estimate of the uncertainty in force predictions presented in this paper will permit users to assess the suitability of given empirical force prediction models in specific applications.