Differential Matrix Research Articles

Differential quadrature discrete time transfer matrix method for vibration mechanics

Transfer matrix method is a practical technology for vibration analysis of engineering mechanics. In this article, a new version of mixed differential quadrature discrete time transfer matrix method is presented for simulating vibrations. First, ordinary differential equations of the substructure or the element of a mechanical system are determined by classical mechanics rules or the finite element method. These ordinary differential equations can be transformed to a set of algebraic equations at some discrete time points by the application of a differential quadrature method. After extending the state vector of the transfer matrix method, new transfer equations and transfer matrices of the substructures (or elements) of a mechanical system are developed finally. Riccati transformation is used to improve the computational convergence of the method. Several numerical examples show that the proposed method can be regarded as an efficient tool for transient response analysis of vibration systems.

Mapped Chebyshev pseudospectral methods for optimal trajectory planning of differentially flat hypersonic vehicle systems

This paper presents a novel trajectory planning algorithm for hypersonic glide vehicles based on differentially flatness theory and mapped Chebyshev pseudospectral method (MCPM). Firstly, the flatness property of a hypersonic glide vehicle dynamics system is extended explicitly by introducing a pseudo control input along the opposite direction of aerodynamics drag. In this way, all the states and controls of the system can be represented by flat outputs and their derivatives, and the trajectory planning problem is formulated into the flat output space with differential constraints entirely eliminated. Secondly, the optimization problem in the flat output space is converted into a nonlinear programming one via the mapped Chebyshev pseudospectral method. The MCPM employs the conformal mapping principle to discretize the flat outputs at the time interval, in terms of which, the closely clustered Chebyshev points near the boundary are stretched and the ill-conditions of the differential matrix is improved. The Barycentric Lagrange interpolation algorithm is then utilized to interpolate the flat output functions, which is seized of better approximation properties than polynomial interpolation. Finally, the transformed nonlinear programming problem are solved by sequence quadratic program technique. The states and controls are then generated by the expressions of the flatness outputs. At the end of this paper, numerical simulations are performed to evaluate the proposed approaches, and comparisons are made with traditional methods. Simulation results demonstrate that the proposed differentially flatness based mapped Chebyshev pseudospectral method possessed better computing efficiency than traditional ones without losing the merits of high precision.

GPU acceleration of splitting schemes applied to differential matrix equations

We consider differential Lyapunov and Riccati equations, and generalized versions thereof. Such equations arise in many different areas and are especially important within the field of optimal control. In order to approximate their solution, one may use several different kinds of numerical methods. Of these, splitting schemes are often a very competitive choice. In this article, we investigate the use of graphical processing units (GPUs) to parallelize such schemes and thereby further increase their effectiveness. According to our numerical experiments, large speed-ups are often observed for sufficiently large matrices. We also provide a comparison between different splitting strategies, demonstrating that splitting the equations into a moderate number of subproblems is generally optimal.

Differential Linear Matrix Inequalities Optimization

This letter proposes a new method to solve convex programming problems with constraints expressed by differential linear matrix inequalities (DLMIs). Initially, feasible solutions of interest are characterized and a general numerical method, based on the well known outer linearization technique, is proposed and discussed from theoretical and numerical viewpoints. Feasible solutions are written as a truncated series of a given set of time valued continuous functions with symmetric matrix coefficients to be determined. The numerical method encompasses the piecewise linear solution usually adopted in the literature with lower computational burden. In the sequel, several sampled-data control design problems whose optimality conditions can be expressed in this mathematical framework are provided. They are solved in order to put in evidence the most important aspects of the proposed method as well as to evaluate and compare numerical efficiency and limitations. Moreover, it is shown that DLMIs are particularly well adapted to cope with this class of optimal control design problems.

THU-099-Collagen is not just collagen: Differential matrix expression induced by TgF-beta and PDGFs

THU-099-Collagen is not just collagen: Differential matrix expression induced by TgF-beta and PDGFs

Delay‐dependent conditions for finite time stability of linear time‐varying systems with delay

AbstractThis paper investigates the problem of finite time stability of linear time‐varying system with delay. By constructing an augmented time‐varying Lyapunov functional and using the Wirtinger‐type inequality deductively, delay‐dependent finite time stability conditions are derived and presented in terms of differential linear matrix inequalities (DLMIs). Then, the DLMIs are transformed into a series of recursive linear matrix inequalities (RLMIs) by discretizing the time interval into equally spaced time distances, and an algorithm is given to solve the RLMIs. Examples illustrate the feasibility and effectiveness of the proposed method.

Method of interpreting Mueller matrix of anisotropic medium.

The differential Mueller matrix is an important concept for analyzing the polarization properties of an optically homogeneous anisotropic sample, both nondepolarizing and depolarizing. In this work, we present a new method of interpreting Mueller matrix of anisotropic medium based on the relationships that exist between the components of a differential Mueller matrix and the polar components of the corresponding macroscopic Mueller matrix, and the necessary conditions are determined that guarantee the physical realizability of the generating matrices. Finally, a group of the experimental data of a sample from the literature with some known polarization properties was used to demonstrate the analysis. The work is helpful for obtaining new insights or new interpretations of the measured Mueller matrix of the medium.

Mathematical descriptions of heat-mass-exchange processes in construction industry at control automation

The paper covers matters arising in building mathematical model of processes at thermal treatment of construction materials. On the basis of analysis of heat energy and moisture flows in intermittent steam chamber and continuous tunnel drying chamber, analytic and structure models of heat-mass-exchange processes in processing vessels are drawn. The structural model of heat-mass-exchange processes allowed to evaluate the relationship of heat energy and moisture flows at heat treatment processes for gypsum and reinforced-concrete articles. The resulting system of interrelated differential equations is based on a structural model. Analytical studies showed that the considering heat treatment units are characterized by non-stationary, non-linear, stochastic and distributed technological parameters. An experimental study of technological devices has shown that in a limited time range, the processes of heat-mass-exchange can be characterized by a system of linear differential equations with constant coefficients with sufficient accuracy for practice. Acceptable allowances and simplifying assumptions at analytical description of heat-mass-exchange processes in processing vessels are considered. As a result of performed research, various mathematical forms (differential equation system, matrix and operator forms) of mathematical models of processing vessels are obtained. The built mathematical models may be applied for constructing the processing vessels with preset dynamic properties, as well for control-system designing by those vessels.

Leader-following consensus of multi-agent systems via event-based impulsive control

Leader-following consensus of multi-agent systems with general linear models is investigated via event-based impulsive control approach. Event-based impulsive controller and state-dependent triggering function are designed. Impulsive instants are determined by certain triggering events, that is, impulsive effect performs only at the instants when events occur, and so as controller update. Sufficient conditions on leader-following consensus are proposed by using stability theory of impulsive differential equations, matrix theory, and inequality technique. Zeno-behavior is also excluded for the concerned closed-loop system. Besides, a technique that used the state information of agents only at event instants is proposed to avoid continuous communication among agents. Finally, an example is presented to illustrate the effectiveness of the obtained theoretical results.

An Alternative PML Implementation for Arbitrary Media

An efficient implementation of the complex frequency-shifted perfectly matched layer (PML) is proposed to terminate the finite-difference time domain (FDTD) lattices. The proposal is based on the matrix exponential algorithm originally used for modeling frequency dispersive media. As the equations of describing $E (D)$ fields, $H (B)$ fields, and auxiliary variables associated with PML are formulated as a compact first-order differential matrix form, no convolution operations, variable substitutions, or equations rearrangements are needed in its mathematical derivations. Via the concept of generalized material-independent PML, this new implementation can also be used to truncate arbitrary media. Furthermore, since its iteration forms are similar to those of the convolutional PML (CPML) that has been widely used in FDTD simulations, the proposal can be directly applied to existing codes with only minor changes. Meanwhile, it can achieve better absorption effect than CPML. Through a numerical example in terms of the printed F antenna and corresponding analyses of performance and efficiency, we find that the proposal works well as absorbing boundary condition.

On finite time stability with guaranteed cost control of uncertain linear systems

This paper deals with the design of a robust state feedback control law for a class of uncertain linear time varying systems. Uncertainties are assumed to be time varying, in one-block norm bounded form. The proposed state feedback control law guarantees finite time stability and satisfies a given bound for an integral quadratic cost function. The contribution of this paper is to provide a sufficient condition in terms of differential linear matrix inequalities for the existence and the construction of the proposed robust control law. In particular, the construction of the feedback control law is brought back to a feasibility problem which can be solved inside the convex optimization framework. The effectiveness of the proposed approach is shown by means of the results obtained on a numerical and a physical example.

Differential linear matrix inequality in optimal sampled-data control

This paper addresses the design of optimal sampled-data output feedback full order controllers for linear continuous-time invariant systems. First, H2 and H∞ performance indices are determined and expressed through differential linear matrix inequalities (DLMIs). Second, in each scenario, the optimal sampled-data controller of the aforementioned class is determined from a convex programming problem expressed by DLMIs. Theoretical achievements are based on the direct application of the celebrated Bellman’s Principle of Optimality expressed in terms of the dynamic programming equation associated to the time interval corresponding to two successive sampling instants. The design problems to be dealt with are convex and are solved by converting all constraints to LMI and adopting a piecewise linear solution. The possibility of aperiodic sampling is considered and briefly discussed. Finally, academic examples are solved for illustration.

Nonlinear differential equations linear approximation unstructured additive uncertainty estimation in computational structural analysis

This paper presents a nonlinear differential equations system linear approximation unstructured additive uncertainty estimation technique. Additive uncertainty linear algebraic and differential equations matrix parameters are estimated using nonlinear and linear differential equations systems time responses through linear programming. The results of proposed approach application are presented.

Finite horizon robustness analysis of LTV systems using integral quadratic constraints

The goal of this paper is to assess the robustness of an uncertain linear time-varying (LTV) system on a finite time horizon. The uncertain system is modeled as an interconnection of a known LTV system and a perturbation. The input/output behavior of the perturbation is described by time-domain, integral quadratic constraints (IQCs). Typical notions of robustness, e.g. nominal stability and gain/phase margins, can be insufficient for finite-horizon analysis. Instead, this paper focuses on robust induced gains and bounds on the reachable set of states. Sufficient conditions to compute robust performance bounds are formulated using dissipation inequalities and IQCs. The analysis conditions are provided in two equivalent forms as Riccati differential equations and differential linear matrix inequalities, and an algorithm is developed leveraging both forms.

Prediction of 3D grinding temperature field based on meshless method considering infinite element

A three-dimensional numerical model to calculate the grinding temperature field distribution is presented. The finite block method, which is developed from meshless method, is used to deal with the stationary and the transient heat conduction problems in this paper. The influences of workpiece feed velocity, cooling coefficient, and the depth of cut on temperature distribution are considered. The model with temperature-dependent thermal conductivity and specific heat is presented. The Lagrange partial differential matrix from the heat transfer governing equation is obtained by using Lagrange series and mapping technique. The grinding wheel-workpiece contact area is assumed as a moving distributed square heat source. The Laplace transformation method and Durbin’s inverse technique are employed in the transient heat conduction analysis. The results of the developed model are compared with others’ finite element method solutions and analytical solutions where a good agreement is demonstrated. And the finite block method was proved a better convergence and accuracy than finite element method by comparing the ABAQUS results. In addition, the three-dimensional infinite element is introduced to perform the thermal analysis, and there is a great of advantages in the simulation of large boundary problems.

Thermo-mechanical performance of layered transversely isotropic media around a cylindrical/tubular heat source

We develop a new numerical model based on a precise integration method to investigate the coupled thermo-mechanical performance of layered transversely isotropic media around a cylindrical/tubular heat source. To obtain the relational matrices of the extended precise integration method, we first convert the governing equations of the problem into ordinary differential matrix equations through the Laplace–Hankel transform. Then, the cylindrical heat source is divided into a series of plane heat sources, and the plane temperature load term is added to the state vector between layer elements. By combining the layer elements, we build a layered transversely isotropic numerical model containing a cylindrical heat source in the transformed domain. Finally, we solve the model in the transformed domain and obtain the solution of the problem in the real domain through the Laplace–Hankel transform inversion. The accuracy of this method is verified by comparing the solutions with the results of the analytical method and the finite element method. Then, we study the influence of the anisotropy of thermal parameters, the embedded depth, the length/radius ratio, the type of heat source and the stratification of the medium on the thermo-mechanical coupled performance.

Design Method of Three-Rotation and Translation Spatial Full Compliant Mechanism

According to the kinematics analysis of 4-RRCR spatial parallel mechanism with three rotation and translation(3R1T), establish differential Jacobian matrix relationship with input and output of four degree of freedom(DOF) spatial parallel mechanism, and used to design space of 3R1T full compliance mechanism. In the design, the method of solving the model of by method of Isotropic Material with Penalization(SIMP) is optimality criteria (OC) method. Through static simulation analysis, the compliance mechanism of obtained by flexible hinge replacement method and by the topology optimization method are respectively in HyperWorks®. The simulation analysis results show us the 3R1T full compliance mechanism design by topology optimization method has same motion characteristics as the traditional parallel mechanism, and the compliance mechanism design by topology optimization method global stiffness and material utilization are better than the compliance design by flexible hinge replacement method.

Infrared spectroscopy of propene in solid para-hydrogen and helium droplets: The role of matrix shifts in the analysis of anharmonic resonances

The infrared spectrum of propene in the CH stretching region is complicated by anharmonic resonance polyads associated with the coupling of CH stretch fundamentals to overtones and combinations of CHn bends and CC stretches. We report the spectra of propene isolated in both helium nanodroplets (HENDI) and solid para-hydrogen (p-H2). Spectral assignments and anharmonic polyad memberships are obtained with a VPT2+K effective Hamiltonian. In the 2800–3120 cm−1 region, the average differential matrix shift in going from HENDI to p–H2 is ∼4.4 cm−1 to the red, with a standard deviation of 1.9 cm−1. Moreover, the choice of matrix environment influences the positions and intensity ratios of transitions within the resonance polyads. Two-state interaction models are used to confirm that differential matrix shifts less than 10 cm−1 are sufficient to account for the observed differences.

Mueller matrix spectroscopy of fano resonance in plasmonic oligomers

Fano resonance in plasmonic oligomers originating from the interference of a spectrally broad superradiant mode and a discrete subradiant mode is under intensive recent investigations due to numerous potential applications. In this regard, development of experimental means to understand and control the complex Fano interference process and to modulate the resulting asymmetric Fano spectral line shape is highly sought after. Here we present a polarization Mueller matrix measurement and inverse analysis approach for quantitative understanding and interpretation of the complex interference process that lead to Fano resonance in symmetry broken plasmonic oligomers. The spectral Mueller matrices of the plasmonic oligomers were recorded using a custom designed dark-field Mueller matrix spectroscopy system. These were subsequently analyzed using differential Mueller matrix decomposition technique to yield the quantitative sample polarimetry characteristics, namely, polarization diattenuation (d) and linear retardance ({\delta}) parameters. The unique signature of the interference of the superradiant dipolar plasmon mode and the subradiant quadrupolar mode of the symmetry broken plasmonic oligomers manifested as rapid spectral variation of the diattenuation and the linear retardance parameters across the Fano spectral dip. The polarization information contained in the Mueller matrix was further utilized to desirably control the Fano spectral line shape. The experimental Mueller matrix analysis was complemented with finite element based numerical simulations, which enabled quantitative understanding of the interference of the superradiant and the subradiant plasmon modes and its link with the polarization diattenuation and retardance parameters.

Weighted generalized Korn inequalities on John domains

The goal of this work is to show that the generalized Korn inequality that replaces the symmetric part of the differential matrix in the classical Korn inequality by its trace‐free part is valid over John domains and weighted Sobolev spaces. The weights considered are nonnegative powers of the distance to the boundary.