Proper Coordinate Transformation Research Articles

Distributed GNE Seeking Strategy for Second-Integrator Multiplayer Systems Over Directed Topologies.

This article studies the generalized Nash equilibrium (GNE) seeking problem of second-integrator multiplayer systems. In particular, each player is endowed with an individual payoff function with respect to collective decision variables, and simultaneously, a coupling inequality constraint and a set constraint are imposed to each player. The players communicate with their local neighbors over a directed topology. To begin with, a distributed-observer-based seeking strategy is synthesized by leveraging a proper composite variable. It is first demonstrated using nonsmooth analysis that the established distributed observer enables each player to accurately estimate the decision variables of others in terms of a strongly connected topology condition. Upon this basis, all the decision variables are then shown to converge to the expected GNE asymptotically borrowing from convex theory. In addition, three extension results are also given under the built GNE seeking framework. First, under the postulation that the velocity information is unavailable, a velocity-free distributed GNE seeking strategy is synthesized for second-integrator systems by implementing a proper auxiliary dynamics. Second, we consider nonlinear Euler-Lagrange systems with unknown inertia parameters and synthesize an improved distributed GNE seeking strategy resorting to an adaptation technique. Third, we focus on integrator chain systems and synthesize a modified distributed GNE seeking strategy using a new composite variable based on a proper coordinate transformation. For three extension cases, we all show in detail the achievement of the GNE seeking objective. Finally, a practical example is simulated to confirm the built GNE seeking results.

Parameter-dependent controller design of stochastic strict feedforward nonlinear systems

In this paper, the exponentially stable problem is discussed for a class of stochastic strict feedforward nonlinear systems. Firstly, by employing the proper coordinate transformation and the novel parameter-dependent controller, the initial stochastic strict feedforward nonlinear system is converted to an equivalent system. After that, the implementable parameter-dependent controller, which is adopted to handle the nonlinearities, can be achieved by reasonably selecting the designed parameter. Finally, by means of stochastic Lyapunov stability theory, it is rigorously verified that the proposed parameter-dependent state-feedback controller and parameter-dependent output-feedback controller can guarantee that the stochastic closed-loop system is the fourth moment exponentially stable. Simulation results demonstrate the efficiency of the proposed parameter-dependent controller.

Formulation and verification of 3D in-situ stress estimation based on differential-direction drilling

A three-dimensional stress field underground has six independent components. They are among the key factors that affect ground stability. Estimation of the magnitudes and orientations of the 3D in-situ stresses is of practical importance in underground engineering projects. At present the only method that is capable of measuring the complete 3D stresses is the overcoring method, which is however expensive and cannot be applied to a deep well in petroleum engineering due to lack of access. This paper presents an alternative method for estimating the complete 3D stresses based on back-analysis of convergence measured in multiple boreholes . This method requires differential-direction drilling and measurements of diametrical borehole convergence in three non-parallel planes. It has also taken into consideration of the field conditions in petroleum engineering and analyzed the effects of pore pressure , mud pressure and permeability of the rock mass and well wall. Overall, a total of five mathematical models (one in linear elastic and four in poro-elastic) are developed for the anticipated field scenarios. For each model, comprehensive formulations correlating the measured convergence with the in-situ stresses are presented. Data compatibility in different directions of individual boreholes is achieved by proper coordinate transformation. The method has been verified using simulated data and the results match well the original input.

A time-stepping method for multibody systems with frictional impacts based on a return map and boundary layer theory

This work presents a new numerical integration method for determining dynamics of a class of multibody systems involving impact and friction. Specifically, these systems are subject to a set of equality constraints and can exhibit single frictional impact events. Such events are associated to significant numerical stiffness, appearing in the equations of motion. The new method is a time-stepping scheme, involving proper incorporation of a novel return mapping into an augmented Lagrangian formulation, developed recently for systems with bilateral constraints only. Namely, when an impact is detected during a time step, this map is applied at the end of the step in order to bring the system position back to the configuration manifold with the allowable motions. The construction of this map is based on the concept of Jacobi fields on non-flat manifolds. Moreover, once an impact event is detected, the post-impact state is determined by employing a combination of analytical and numerical tools. First, a proper coordinate transformation is performed, bringing the system into a new set of coordinates, which are suitable for describing the impact dynamics. In these coordinates, the dominant dynamics is described by a system of three equations of motion only, which are valid during the short contact interval. In addition, these equations are geometrically discretized by using appropriate cubic splines on the configuration manifold. In this way, the inherent numerical stiffness of the class of systems examined is properly addressed, since it is restricted to a space with a much smaller dimension and a much shorter time scale. Finally, the accuracy and efficiency of the new method is demonstrated by applying it to a selected set of mechanical examples.

Free and Forced Vibration of Coupled Beam Systems Resting on Variable Viscoelastic Foundations

This paper presents a modified variational method for free and forced vibration analysis of coupled beam systems resting on various viscoelastic foundations. Non-uniform as well as uniform curved and straight Timoshenko beam components are considered in the coupled beam system. Using proper coordinate transformations, interactions among the beam components of the coupled beam system are accommodated by combining Lagrange multiplier method and least-square weighted residual method. Interface potential energy for various boundary conditions including the elastic ones is simultaneously formulated. Thus, the proposed method allows flexible choice of the admissible functions, regardless of the boundary conditions. Based on the proposed energy method, Winkler, Pasternak or even variable foundations distributed in a parabolic or sinusoidal manner can be easily introduced into the coupled beam systems. Two kinds of damping, namely the proportional and viscous damping, are also employed to model the energy dissipation of the viscoelastic foundations. Corresponding finite element (FE) simulations are performed where possible and good agreement is observed. Thus, great efficiency and accuracy of the present approach are demonstrated for free, steady-state and transient vibration of the coupled beam systems. The influences of the parameters of the variable viscoelastic foundations on the dynamic properties of the coupled beam system are also examined.

Fuzzy adaptive automatic train operation control with protection constraints: A residual nonlinearity approximation-based approach

In this study, we present fuzzy adaptive control based on residual nonlinearity approximation in the presence of protection constraints for the target trajectory tracking problem observed in automatic train operation. Herein, protection constraints refer to a condition wherein the speed and position of a controlled train are not allowed to surpass the boundaries imposed by automatic train protection and moving authority. By defining proper coordinate transformation, the protection constraints are converted to an error-prescribed performance control problem that facilitates operational efficiency by reducing the margin with respect to target trajectories. Based on the prescribed performance control methodology, we present an improved scheme using fuzzy residual nonlinearity approximation and establish the uniformly ultimately boundedness (UUB) property. A novel feature therein is that the ultimate boundary of the proposed scheme is simultaneously characterized by the prescribed performance functions and control parameters, with rigorous and analytically mathematical expressions; while pioneering the prescribed performance control methodology, the ultimate boundary is characterized solely by the prescribed performance functions. To verify the effectiveness and advantages of the proposed scheme, the controllers are applied to the automatic train operation on the Beijing Yizhuang line, which contains 13 operational intervals. Finally, comparative and simulation results are presented to validate the proposed method.

Quasi‐composite rotating formation control of second‐order multi‐agent systems

In this study, quasi-composite rotating formation control for second-order multi-agent systems over undirected communication topology is considered. To solve this problem, a novel distributed control protocol based on the local information is proposed. Next, the origin closed-loop system is changed into an equivalent one by taking a proper coordinate transformation. Under some mild conditions, it is proved that the quasi-composite rotating formation can be reached by analysing the matrix eigenvalues of the closed-loop system. Finally, the simulation results are provided to demonstrate the effectiveness of the theoretical results.

Stability and Hopf bifurcation analysis of a complex-valued Wilson–Cowan neural network with time delay

Stability and Hopf bifurcation of a class of delayed complex-valued neural networks are investigated in this paper. First, using proper translations and coordinate transformations, we decompose the activation functions and connection weights into their real and imaginary parts, so as to construct an equivalent real-valued system. Then, the sufficient conditions for Hopf bifurcation and its directions are provided through normal form theory and central manifold theorem. In the end, some numerical simulations are given to illustrate the correctness of the results.

Multi-scale asymptotic analysis and computation of the elliptic eigenvalue problems in curvilinear coordinates

A novel second-order two-scale asymptotic method is presented for the eigenvalue problems of the second-order elliptic operator in the general composite domain. The eigenvalue equation is firstly reformulated in curvilinear coordinates with periodic configuration using proper coordinate transformation, and by applying the asymptotic expansion technique, the eigenfunctions of the system are expanded to the second-order terms. Using the argument of the so-called “corrector equations”, the eigenvalues are expressed in terms of the homogenized eigenfunctions and the cell functions are defined in the representative cell domain. The feature of the proposed model is that some homogenized material coefficients and all the microscopic cell functions are dependent on the macroscopic coordinates. Various reduced expressions of the eigenfunctions and eigenvalues are discussed under specific coordinate transformations, and the conditions that the cell functions and homogenized coefficients are decoupled from the macroscopic coordinates are elaborated. The finite element algorithm is developed and three numerical experiments are carried out, which demonstrate the effectiveness of our proposed method in simulating and predicting the vibration behavior of the composite structures. It is also indicated that the second-order correctors are of necessity to capture the locally oscillating behavior within a periodicity of the eigenfunctions. By the coordinate transformation, the asymptotic analysis method can be generalized to more general composite domain with quasi-periodic and non-periodic configurations.

MODELING OF LIQUID DROP HEATING AND EVAPORATION: THE EFFECT OF DROP SHRINKING

The process of heat and mass transfer from a spherical mono-component liquid drop evaporating into a gas environment is investigated, relaxing the commonly adopted quasi-steady approximation and accounting for the inherent unsteadiness caused by the sudden immersion of a liquid drop in a gaseous environment. The drop radius shrinking due to evaporation settles a moving boundary problem, which is transposed to a fixed boundary one by a proper coordinate transformation. The heat and evaporation rates and the drop diameter evolution are quantified by numerical solution of the species and energy conservation equations and the overall mass and energy balances over the drop for different species (water, n-octane, n-dodecane, ethanol).

Three-dimensional wave-induced current model equations and radiation stresses

After the approach by Mellor (2003, 2008), the present paper reports on a repeated effort to derive the equations for three-dimensional wave-induced current. Via the vertical momentum equation and a proper coordinate transformation, the phase-averaged wave dynamic pressure is well treated, and a continuous and depth-dependent radiation stress tensor, rather than the controversial delta Dirac function at the surface shown in Mellor (2008), is provided. Besides, a phase-averaged vertical momentum flux over a sloping bottom is introduced. All the inconsistencies in Mellor (2003, 2008), pointed out by Ardhuin et al. (2008) and Bennis and Ardhuin (2011), are overcome in the presently revised equations. In a test case with a sloping sea bed, as shown in Ardhuin et al. (2008), the wave-driving forces derived in the present equations are in good balance, and no spurious vertical circulation occurs outside the surf zone, indicating that Airy’s wave theory and the approach of Mellor (2003, 2008) are applicable for the derivation of the wave-induced current model.

Asymptotic computation for transient heat conduction performance of periodic porous materials in curvilinear coordinates by the second‐order two‐scale method

A novel second‐order two‐scale (SOTS) analysis method is developed for predicting the transient heat conduction performance of porous materials with periodic configurations in curvilinear coordinates. Under proper coordinate transformations, some non‐periodic porous structures in Cartesian coordinates can be transformed into periodic structures in general curvilinear coordinates, which is our particular interest in this study. The SOTS asymptotic expansion formulas for computing the temperature field of transient heat conduction problem in curvilinear coordinates are constructed, some coordinate transformations are discussed, and the related SOTS formulas are given. The feature of this asymptotic model is that each of the cell functions defined in the periodic cell domain is associated with the macroscopic coordinates and the homogenized material coefficients varies continuously in the macroscopic domain behaving like the functional gradient material. Finally, the corresponding SOTS finite element algorithms are brought forward, and some numerical examples are given in detail. The numerical results demonstrate that the SOTS method proposed in this paper is valid to predict transient heat conduction performance of porous materials with periodicity in curvilinear coordinates. By proper coordinate transformations, the SOTS asymptotic analysis method can be extended to more general non‐periodic porous structures in Cartesian coordinates. Copyright © 2017 John Wiley & Sons, Ltd.

NN‐Based Output‐Feedback Control for Stochastic Nonlinear Systems with Unknown Control Directions

AbstractThis paper addresses the neural network‐based output‐feedback control problem for a class of stochastic nonlinear systems with unknown control directions. The restrictions on the drift and diffusion terms are removed and the conditions on unknown control directions are relaxed. By introducing a proper coordinate transformation, and combining dynamic surface control (DSC) technique with radial basis function neural network (RBF NN) approximation approach, we construct an adaptive output‐feedback controller to guarantee the closed‐loop system to be mean square semi‐globally uniformly ultimately bounded (M‐SGUUB). A simulation example demonstrates the effectiveness of the proposed scheme.

Research on coordinate transformation design of a cylinderical acoustic cloak with pentamode materials

The pentamode material, similar to fluid in physical properties, serves as a useful way for the physical implementation of the anisotropic fluid. Based on the similarity, a method to design cloak with the pentamode materials has been put forward by Norris. To analyze the effect factors and rules of the stealth performance of the cloak, the present article is focused on the studying of the coordinate transformation equation of the pentamode cloak design of Norris. Cloaks with different materials parameters distribution can be achieved by adjusting coordinate transformation equations. There are four kinds of the distribution of pentamode cloak material parameters: the density equation being constant, the modulus equation being constant, the density equation being, power equation and the modulus equation being power equation. The average visibility is considered as the standard of stealth effect and is calculated with different coordinate transformation equations by using the software COMSOL. The average visibility is used to analyze the relationship between stealth effect and coordinate transformation equations. The relationship between the coordinate transformation equation and the route of acoustic wave transmission, the relationship between the materials of obstacle and the stealth effect, and the relationship between the route of acoustic wave transmission and the stealth effect are studied. Two results are achieved by comparing these relationships mentioned above. The first is that the stealth effect of a cloak with aluminum obstacle is worse than one with water obstacle. The reason lies in the impedance mismatch between the aluminum and the cloak material. The second result shows that the coordinate transformation equation is related to the distribution of material parameters and the route of acoustic wave transmission and it can affect the scattering property of the cloak. When the route of acoustic wave transmission is close to inner surface of cloak, the stealth effect is relatively poor, while when the route of acoustic wave transmission is close to outer surface of cloak, the stealth effect is relatively well. The reason is that when the route of acoustic wave transmission is close to inner surface of cloak, the acoustic wave affects the obstacle which leads to the enhancement of the scattering of obstacle. So when designing the cloak, not only the physical realization of the cloak material but also the distributed situation of the route of acoustic wave transmission should be considered. And the route of acoustic wave transmission is decided by the coordinate transformation equation. Therefore the stealth performance can be improved by applying proper coordinate transformation equation.

The Control and Synchronization of a Rotational Relativistic Chaotic System With Parameter Uncertainties and External Disturbance

This paper investigates the control and synchronization of a rotational relativistic chaotic system with parameter uncertainties and external disturbance. By using the proper coordinate transformation, some novel criteria for control or synchronization are proposed via a single input. Numerical simulations are given to show the robustness and efficiency of the proposed approach.

Nonlinear Voltage Control of Photovoltaic Z-Source Inverter DC-Link Based on Zero Dynamic Design

Based on the DC-link voltage-to-shoot-through duty ratio control problem of non-minimum phase in Z-source inverter, the photovoltaic system has been taken as an example, according to the nonlinear characteristics and switching state equation of Z-source inverter, the affine nonlinear model was set, based on the differential geometric, the proper coordinate transformation was constructed. In order to simplify the controller, and ensure the internal dynamic system stability, the zero-dynamic design was used for controlling DC-link voltage of Z-source inverter. Simulation results show that when the change of the output power of photovoltaic cells occurs, this method has better dynamic characteristics.

Efficiency Investigation of Conventional Satellite Initial Acquisition Control with the Consideration of Orbital Motion

In practical, there exist several occasions that satellitesmay temporarily lose their control over body rate: The high tip-off rate after the separation from launch vehicle, the tumbling caused by avionic reset during daily operation and accidental attitude loss during orbit correction. Unfortunately, uncontrolled body rate and attitude usually prevent on-board optical attitude sensors from locking targets, such as Sun, Earth or stars. This report provides a comprehensive analysis of such a process from initial rate reduction to terminal orientation from the perspective of Lyapunov stability.In addition, with proper coordinate transformation, the B-dot control efficiency is found to be orbit inclination angle dependent. Numerical analysis is performed where the conclusion suggests that the average B-dot control efficiency improves as the inclination angle increases.

Coordinate Transform Methods Used in Dynamic Simulation of Satellite Separation

Direction cosine matrix, Euler angles and quaternions are the common methods for translating vector equations into scalar equations. These methods used in separation simulation lack systematic discussion and comparison. This article is trying to present a proper coordinate transformation method for dynamic analysis of satellite separation. 321 and 123 rotations of Euler angles are proposed to construct the dynamic equations, which is different from simulation of missiles by the rotation of 321 and 231. Both Euler angles and quaternions are adopted to model the separation process of a small satellite that uses the helical compression springs mechanism. Feasibility and practicability of the approach are proved by comparing the simulation results, which are solved in MATLAB and ADAMS software platforms. It is concluded that the method of quaternions is more accurate and efficient in dynamic simulation of satellite separation.

Regional Pole Placement via Static Output Feedback Based on Coordinate Transformation

This paper focuses on the regional pole placement via static output feedback. Under proper state coordinate transformation with a free matrix variable, the static output feedback gain may be obtained by solving a linear matrix inequality (LMI). The LMI is feasible only if the poles of a dummy control system are in the given LMI region. The free matrix variable can regulate the dummy system as a state feedback gain matrix. So once the free variable is determined, the static output feedback gain matrix may be obtained by an LMI-based method. The main computations do not concern any reduction or enlargement of matrix inequalities. Numerical examples show the effectiveness of the proposed algorithm.

Performance improvement of network based RTK GPS positioning in Taiwan

Upon the availability of fast and stable Internet access and the mature virtual reference station (VRS) technique, the network based real time kinematic RTK GPS technique, which overcomes the limitations of traditional RTK (single reference station) positioning, has become an important tool of survey engineering. Since 2004, a regional real time GPS network with 86 stations for expanding the effective range of cadastral measurement has been set up in Taiwan. Users can get VRS observations from the control centre via wireless Internet and acquire high quality positioning. In an experiment we conducted, we find that after applying proper coordinate transformation to the coordinates measured by network based RTK GPS positioning, three-dimensional coordinates with centimetre level accuracy can be achieved, with results for the horizontal direction generally better than the vertical direction.