Linear Dynamic Equations Research Articles

The Karamata integration theorem on time scales and its applications in dynamic and difference equations

We derive a time scale version of the well-known result from the theory of regular variation, namely the Karamata integration theorem. We show an application of this theorem in asymptotic analysis of linear second order dynamic equations. We obtain a classification and asymptotic formulae for all (positive) solutions, which unify, extend, and improve the existing results. In addition, we utilize these results, in combination with a transformation between equations on different time scales, to study the critical double-root case in linear difference equations. This leads to solving open problems posed in the literature.

Autonomous entry guidance using Linear Pseudospectral Model Predictive Control

This paper aims at developing an autonomous entry guidance method that requires no mission-dependent adjustments and will be applicable to a wide range of entry scenarios with worldwide destinations. Firstly, a nonlinear reduced-order entry dynamical system with coupled lateral and longitudinal motions as well as a spherical and rotating Earth is proposed. This system relates the vertical and lateral lift-to-drag ratio profiles to the position and azimuth angle variables with the energy as the independent variable. It has a high computational accuracy even only three differential equations are involved. Secondly, a trajectory planning problem with only two bank reversals is formulated based on this reduced-order system with parameterized control. Trajectory integration prediction and linearization method are applied to transfer the original planning problem into iteratively solving a group of linear dynamical equations. Gauss Pseudospectral Method and Calculus of variations are employed to discrete them so as to derive a series of analytical correction formulas to eliminate the final errors, mathematically, which achieves high accuracy with a small number of points. Moreover, these control parameters include not only the magnitude of bank angle, but also bank reversal points, which will significantly increase its ability to shape the entry trajectory. After the last bank reversal, lateral and longitudinal guidance laws are designed to ensure multiple final constraints. Nominal testing and Monte Carlo simulations on the proposed method and the comparison with the typical predictor–corrector method are carried out. Results demonstrate that, even in highly dispersed environments, this method has wide applicability, strong robustness, and excellent performance. Moreover, its computational efficiency is so high that it sufficiently satisfies the requirement on onboard application.

Dynamics of flexible multibody systems with variable-speed control moment gyroscopes

This paper presents a generic global matrix formulation for the dynamics of flexible multibody systems with variable-speed control moment gyroscopes (VSCMGs). The flexible bodies are assumed to exhibit only small deformation, and they are connected in a tree topology by hinges permitting large rotation and translation. A cluster of VSCMGs is mounted on each body for actuation; it is assumed that the VSCMGs are statically and dynamically balanced, and their rotors are axisymmetric. A minimum set of dynamic equations are derived systematically via a mixed use of Kane's method and Newton–Euler equations. The parameters of each flexible body are augmented to take into account the inertias of the attached VSCMGs. Moreover, a skew-symmetric gyroscopic matrix and three control-input-mapping matrices are defined to represent the passive and the active gyroscopic torques of the VSCMGs in a global matrix manner. Three examples are given to show the usefulness, versatility, and correctness of the proposed formulation. As an additional contribution, also presented are linear dynamic equations for a spacecraft with flexible appendages and embedded VSCMGs.

New Discrete-Time ZNN Models for Least-Squares Solution of Dynamic Linear Equation System With Time-Varying Rank-Deficient Coefficient.

In this brief, a new one-step-ahead numerical differentiation rule called six-instant -cube finite difference (6I CFD) formula is proposed for the first-order derivative approximation with higher precision than existing finite difference formulas (i.e., Euler and Taylor types). Subsequently, by exploiting the proposed 6I CFD formula to discretize the continuous-time Zhang neural network model, two new-type discrete-time ZNN (DTZNN) models, namely, new-type DTZNNK and DTZNNU models, are designed and generalized to compute the least-squares solution of dynamic linear equation system with time-varying rank-deficient coefficient in real time, which is quite different from the existing ZNN-related studies on solving continuous-time and discrete-time (dynamic or static) linear equation systems in the context of full-rank coefficients. Specifically, the corresponding dynamic normal equation system, of which the solution exactly corresponds to the least-squares solution of dynamic linear equation system, is elegantly introduced to solve such a rank-deficient least-squares problem efficiently and accurately. Theoretical analyses show that the maximal steady-state residual errors of the two new-type DTZNN models have an pattern, where denotes the sampling gap. Comparative numerical experimental results further substantiate the superior computational performance of the new-type DTZNN models to solve the rank-deficient least-squares problem of dynamic linear equation systems.

Convex optimisation approach to constrained fuel optimal control of spacecraft in close relative motion

This paper describes an interesting and powerful approach to the constrained fuel-optimal control of spacecraft in close relative motion. The proposed approach is well suited for problems under linear dynamic equations, therefore perfectly fitting to the case of spacecraft flying in close relative motion. If the solution of the optimisation is approximated as a polynomial with respect to the time variable, then the problem can be approached with a technique developed in the control engineering community, known as “Sum Of Squares” (SOS), and the constraints can be reduced to bounds on the polynomials. Such a technique allows rewriting polynomial bounding problems in the form of convex optimisation problems, at the cost of a certain amount of conservatism. The principles of the techniques are explained and some application related to spacecraft flying in close relative motion are shown.

Inexact Cuts in Deterministic and Stochastic Dual Dynamic Programming Applied to Linear Optimization Problems

We introduce an extension of Dual Dynamic Programming (DDP) to solve linear dynamic programming equations. We call this extension IDDP-LP which applies to situations where some or all primal and dual subproblems to be solved along the iterations of the method are solved with a bounded error (inexactly). We provide convergence theorems both in the case when errors are bounded and for asymptotically vanishing errors. We extend the analysis to stochastic linear dynamic programming equations, introducing Inexact Stochastic Dual Dynamic Programming for linear programs (ISDDP-LP), an inexact variant of SDDP applied to linear programs corresponding to the situation where some or all problems to be solved in the forward and backward passes of SDDP are solved approximately. We also provide convergence theorems for ISDDP-LP for bounded and asymptotically vanishing errors. Finally, we present the results of numerical experiments comparing SDDP and ISSDP-LP on a portfolio problem with direct transation costs modelled as a multistage stochastic linear optimization problem. On these experiments, for some values of the noises, ISDDP-LP can converge significantly quicker than SDDP.

A UNIFIED ANALYSIS OF LINEAR QUATERNION DYNAMIC EQUATIONS ON TIME SCALES

Over the last years, considerable attention has been paid to the role of the quaternion differential equations (QDEs) which extend the ordinary differential equations. The theory of QDEs was recently well established and it has wide applications in physics and life science. This paper establishes a systematic frame work for the theory of linear quaternion dynamic equations on time scales (QDETS), which can be applied to wave phenomena modeling, fluid dynamics and filter design. The algebraic structure of the solutions to the QDETS is actually a left- or right- module, not a linear vector space. On the non-commutativity of the quaternion algebra, many concepts and properties of the classical dynamic equations on time scales (DETS) can not be applied. They should be redefined accordingly. Using $q$-determinant, a novel definition of Wronskian is introduced under the framework of quaternions which is different from the standard one in DETS. Liouville's formula for QDETS is also analyzed. Upon these, the solutions to the linear QDETS are established. The Putzer's algorithms to evaluate the fundamental solution matrix for homogeneous QDETS are presented. Furthermore, the variation of constants formula to solve the nonhomogeneous QDETs is given. Some concrete examples are provided to illustrate the feasibility of the proposed algorithms.

Improving The Manoeuvrability of Electric Vehicle with Four-Wheel Drive and Four-Wheel Steering – A Nonlinear Model Vehicle Dynamics Approach

The dynamics motion of a vehicle is inherently a nonlinear dynamics system especially at high speed. Majority of past researches on four-wheel steering (4WS) vehicle adopt easier way of modelling a control system based on vehicle with linear dynamic equation of motion. This paper study on the vehicle dynamics of an electric vehicle with 4WD and 4WS based on nonlinear vehicle dynamic approach. A numerical simulation was performed to analyse the variance of a linear model and nonlinear model during cornering at various constant speed. The results show that during low speed cornering at 10 km/h, the linear and nonlinear model produced similar steady state cornering based on the trajectory and yaw rotational speed. However, the variants of linear and nonlinear started to appear as the vehicle speed increase. By obtaining the steady state cornering speed, another numerical simulation was performed to analyse the characteristics of the 4WD and 4WS electric vehicle. A passive control of the rear wheels’ steer angle was implement in the simulation. The results show that the parallel steering mode decreased the yaw rotational speed which broaden the trajectory of the cornering, while the opposite steering mode increased the yaw rotational speed that led to a tighter trajectory during cornering.

EXISTENCE AND GLOBAL STABILITY OF ALMOST AUTOMORPHIC SOLUTIONS FOR SHUNTING INHIBITORY CELLULAR NEURAL NETWORKS WITH TIME-VARYING DELAYS IN LEAKAGE TERMS ON TIME SCALES

In this paper, shunting inhibitory cellular neural networks(SICNNs) with time-varying delays in leakage terms on time scales are investigated. With the aid of the existence of the exponential dichotomy of linear dynamic equations on time scales, fixed point theorem and the theory of calculus on time scales, we establish some sufficient conditions to ensure the existence and exponential stability of almost automorphic solutions for the model. An example with its numerical simulations is given to illustrate the feasibility and effectiveness of the theoretical findings.

Weighted Pseudo-Almost Periodic Solutions for Shunting Inhibitory Cellular Neural Networks on Time Scales

In this paper, by using the exponential dichotomy of linear dynamic equations on time scales, a fixed point theorem and the theory of calculus on time scales, we obtain sufficient conditions for the existence, uniqueness and global exponential stability of weighted pseudo-almost periodic solutions of a class of shunting inhibitory cellular neural networks with mixed delays on time scales. A numerical example is also presented to illustrate the feasibility of our results.

Nonlocal vibration and pull-in instability analysis of electrostatic carbon-nanotube based NEMS devices

The objective of this paper is to investigate dynamical pull-in behavior of an electrostatic actuated nano-device based on Eringen's nonlocal elasticity theory. The Euler–Bernoulli beam model is used to establish the dynamic equation of motion of the nano-device subjected to both electrostatic and intermolecular forces. The nanobeam is considered with axially immovable ends and the geometrically nonlinearity due to mid-plane stretching is incorporated to the model as well. A new intermolecular attractive force model based on the macroscopic interactions of a circular cross section nanobeam and a flat surface is presented for the carbon nanotube based nano-device. The nonlinear static equation and the linear dynamic equation are treated by the Differential Quadrature Method (DQM) and through a comprehensive survey, the effect of small scale parameter, DC voltage, intermolecular force, residual stress and geometrical nonlinearity on the fundamental natural frequency and pull-in phenomenon are studied. The results are verified with the literature and a good agreement is achieved.

The Ulam Stability of First Order Linear Dynamic Equations on Time Scales

In the present paper, the Ulam stability of the first order linear dynamic equation and its adjoint equation on a time scale is established by using the integrating factor method. Meantime, two examples are provided to illustrate our main results.

A novel time integration formulation for nonlinear dynamic analysis

To solve the linear and nonlinear dynamic equation of motion, a family of explicit higher-order time integration algorithms is presented. The differential transformation method in conjunction with the state–space concept is used. In this study, four members of this family are considered thoroughly. To validate the performance of the new algorithms, comparison studies are accomplished with the well-known time integration schemes. By solving some linear and nonlinear dynamic problems, the merits of the authors' formulations will be clearly demonstrated.

On Different Types of Stability for Linear Delay Dynamic Equations

We provide explicit conditions for uniform stability, global asymptotic stability and uniform exponential stability for dynamic equations with a single delay and a non-negative coefficient. Some examples on nonstandard time scales are also given to show applicability and sharpness of the new results.

A Wiener-Type Dynamic Neural Network Approach to the Modeling of Nonlinear Microwave Devices

A new Wiener-type dynamic neural network (DNN) approach for nonlinear device modeling is proposed in this paper. The proposed analytical formulation of Wiener-type DNN structure consists of a cascade of a simplified linear dynamic part and a nonlinear static part. The simplified linear dynamic equations are obtained by vector fitting enhancing the efficiency of the model. A new formulation and new sensitivity analysis technique is derived to train the proposed Wiener-type DNN with dc, small- and large-signal data. A new gradient-based training algorithm is also formulated to speed up the training of proposed Wiener-type DNN model. The proposed Wiener-type DNN model can be trained to be accurate relative to device data. Furthermore, the proposed Wiener-type DNN provides enhanced convergence properties over existing neural network approaches such as time delay neural network (TDNN) and TDNN with extrapolation. Application examples on modeling GaAs metal-semiconductor-field-effect transistor, GaAs high-electron mobility transistor (HEMT), and a real <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2 \times 50$ </tex-math></inline-formula> gatewidth GaAs pseudomorphic HEMT are presented. The use of Wiener-type DNN model in harmonic balance simulations demonstrates that the Wiener-type DNN is a robust method for developing models for different types of microwave devices. These models could be implemented into circuit simulators conveniently.

Reduced-order modelling of parameter-dependent, linear and nonlinear dynamic partial differential equation models.

In this paper, we develop reduced-order models for dynamic, parameter-dependent, linear and nonlinear partial differential equations using proper orthogonal decomposition (POD). The main challenges are to accurately and efficiently approximate the POD bases for new parameter values and, in the case of nonlinear problems, to efficiently handle the nonlinear terms. We use a Bayesian nonlinear regression approach to learn the snapshots of the solutions and the nonlinearities for new parameter values. Computational efficiency is ensured by using manifold learning to perform the emulation in a low-dimensional space. The accuracy of the method is demonstrated on a linear and a nonlinear example, with comparisons with a global basis approach.

Localization of whistler wave and turbulent spectra in the magnetotail region

AbstractObservations show that whistler waves play a very crucial role both in magnetic reconnection and turbulence. In this context, linear and nonlinear dynamical equations of whistler wave have been derived and further these equations have been solved numerically as well as semianalytically. For application purpose, magnetotail parameters have been chosen. The obtained results reveal that the nonlinearity as well as field perturbation results in the formation of localized structures. These structures contribute to the turbulence and lead to the formation of current sheets in the magnetotail. Finally, the power spectrum has been evaluated and compared with the observed spectra in the magnetotail ion diffusion region as reported.

Oscillation criteria of second order half linear delay dynamic equations on time scales

In this paper, we establish some new oscillation criteria for a non autonomous second order delay dynamic equation $${\left( {r\left( t \right)g\left( {{x^\Delta }\left( t \right)} \right)} \right)^\Delta } + p\left( t \right)f\left( {x\left( {\tau \left( t \right)} \right)} \right) = 0,$$ on a time scale T. Oscillation behavior of this equation is not studied before. Our results not only apply on differential equations when T=ℝ, difference equations when T=ℕ but can be applied on different types of time scales such as when T=qℕ for q > 1 and also improve most previous results. Finally, we give some examples to illustrate our main results.

Atomic spectroscopy in periodic fields

Using the density matrix formalism, we prove the existence of the periodic steady-state for an arbitrary periodically driven system described by linear dynamic equations. The presented derivation simultaneously contains a simple and effective computational algorithm, which automatically guarantees a full account of all frequency components.

Almost Automorphic Functions on the Quantum Time Scale and Applications

We first propose two types of concepts of almost automorphic functions on the quantum time scale. Secondly, we study some basic properties of almost automorphic functions on the quantum time scale. Then, we introduce a transformation between functions defined on the quantum time scale and functions defined on the set of generalized integer numbers; by using this transformation we give equivalent definitions of almost automorphic functions on the quantum time scale; following the idea of the transformation, we also give a concept of almost automorphic functions on more general time scales that can unify the concepts of almost automorphic functions on almost periodic time scales and on the quantum time scale. Finally, as an application of our results, we establish the existence of almost automorphic solutions of linear and semilinear dynamic equations on the quantum time scale.