Toward Analytical Homogenized Relaxation Modulus for Fibrous Composite Material with Reduced Order Homogenization Method

In this manuscript,we propose an analytical equivalent linear viscoelastic constitutive model for fiber-reinforced composites,bypassing general computational homogenization.The method is based on the reduced-order homogenization(ROH)approach.The ROH method typically involves solving multiple finite element problems under periodic conditions to evaluate elastic strain and eigenstrain influence functions in an‘off-line’stage,which offers substantial cost savings compared to direct computational homogenization methods.Due to the unique structure of the fibrous unit cell,“off-line”stage calculation can be eliminated by influence functions obtained analytically.Introducing the standard solid model to the ROH method enables the creation of a comprehensive analytical homogeneous viscoelastic constitutive model.This method treats fibrous composite materials as homogeneous,anisotropic viscoelastic materials,significantly reducing computational time due to its analytical nature.This approach also enables precise determination of a homogenized anisotropic relaxation modulus and accurate capture of various viscoelastic responses under different loading conditions.Three sets of numerical examples,including unit cell tests,three-point beam bending tests,and torsion tests,are given to demonstrate the predictive performance of the homogenized viscoelastic model.Furthermore,the model is validated against experimental measurements,confirming its accuracy and reliability.

Meta-Auto-Decoder:a Meta-Learning-Based Reduced Order Model for Solving Parametric Partial Differential Equations

Many important problems in science and engineering require solving the so-called parametric partial differential equations(PDEs),i.e.,PDEs with different physical parameters,boundary conditions,shapes of computational domains,etc.Typical reduced order modeling techniques accelerate the solution of the parametric PDEs by projecting them onto a linear trial manifold constructed in the ofline stage.These methods often need a predefined mesh as well as a series of precomputed solution snapshots,and may struggle to balance between the efficiency and accuracy due to the limitation of the linear ansatz.Utilizing the nonlinear representation of neural networks(NNs),we propose the Meta-Auto-Decoder(MAD)to construct a nonlinear trial manifold,whose best possible performance is measured theoretically by the decoder width.Based on the meta-learning concept,the trial manifold can be learned in a mesh-free and unsupervised way during the pre-training stage.Fast adaptation to new(possibly heterogeneous)PDE parameters is enabled by searching on this trial manifold,and optionally fine-tuning the trial manifold at the same time.Extensive numerical experiments show that the MAD method exhibits a faster convergence speed without losing the accuracy than other deep learning-based methods.

Adding-Point Strategy for Reduced-Order Hypersonic Aerothermodynamics Modeling Based on Fuzzy Clustering

Reduced order models（ROMs） based on the snapshots on the CFD high-fidelity simulations have been paid great attention recently due to their capability of capturing the features of the complex geometries and flow configurations. To improve the efficiency and precision of the ROMs, it is indispensable to add extra sampling points to the initial snapshots, since the number of sampling points to achieve an adequately accurate ROM is generally unknown in prior, but a large number of initial sampling points reduces the parsimony of the ROMs. A fuzzy-clustering-based adding-point strategy is proposed and the fuzzy clustering acts an indicator of the region in which the precision of ROMs is relatively low. The proposed method is applied to construct the ROMs for the benchmark mathematical examples and a numerical example of hypersonic aerothermodynamics prediction for a typical control surface. The proposed method can achieve a 34.5% improvement on the efficiency than the estimated mean squared error prediction algorithm and shows same-level prediction accuracy.

The Reduced-order Design of Robust Adaptive Backstepping Controller

For a class of systems with unmodeled dynamics, robust adaptive stabilization problemis considered in this paper. Firstly, by a series of coordinate changes, the original system is re-parameterized. Then, by introducing a reduced-order observer, an error system is obtained. Basedon the system, a reduced-order adaptive backstepping controller design scheme is given. It is provedthat all the signals in the adaptive control system are globally uniformly bounded, and the regulationerror converges to zero asymptotically. Due to the order deduction of the controller, the design schemein this paper has more practical values. A simulation example further demonstrates the e?ciency ofthe control scheme.

Observer based projective reduced-order synchronization of different chaotic systems

The projective reduced-order synchronization of two different chaotic systems with different orders is investigated based on the observer design in this paper.According to the observer theory,the reduced-order observer is designed.The projective synchronization can be realized by choosing the transition matrix of the observer as a diagonal matrix.Further,the synchronization between hyperchaotic Chen system(fourth order)and Rssler system(third order)is taken as the example to demonstrate the effectiveness of the proposed observer.Numerical simulations confirm the effectiveness of the method.

Distributed Reduced-order Optimal Fusion Kalman Filters for Stochastic Singular Systems

Based on the optimal fusion algorithm weighted by matrices in the linear minimum variance (LMV) sense, a distributed full-order optimal fusion Kalman filter (DFFKF) is given for discrete-time stochastic singular systems with multiple sensors, which involves the inverse of a high-dimension matrix to compute matrix weights. To reduce the computational burden, a distributed reduced-order fusion Kalman filter (DRFKF) is presented, which involves in parallel the inverses of two relatively low-dimension matrices to compute matrix weights. A simulation example shows the effectiveness.

Generalized reduced-order synchronization of chaotic system based on fast slide mode

A new kind of generalized reduced-order synchronization of different chaotic systems is proposed in this paper. It is shown that dynamical evolution of third-order oscillator can be synchronized with the canonical projection of a fourth-order chaotic system generated through nonsingular states transformation from a cell neural net chaotic system. In this sense, it is said that generalized synchronization is achieved in reduced-order. The synchronization discussed here expands the scope of reduced-order synchronization studied in relevant literatures. In this way, we can achieve generalized reduced-order synchronization between many famous chaotic systems such as the second-order Drifting system and the third-order Lorenz system by designing a fast slide mode controller. Simulation results are provided to verify the operation of the designed synchronization.

Reduced-order Kalman filtering for state constrained linear systems

This paper aims at solving the state filtering problem for linear systems with state constraints. Three classes of typical state constraints, i.e., linear equality, quadratic equality and inequality, are discussed. By using the linear relationships among different state variables, a reduced-order Kalman filter is derived for the system with linear equality constraints. Afterwards, such a solution is applied to the cases of the quadratic equality constraint and inequality constraints and the two constrained state filtering problems are transformed into two relative constrained optimization problems. Then they are solved by the Lagrangian multiplier and linear matrix inequality techniques, respectively. Finally, two simple tracking examples are provided to illustrate the effectiveness of the reduced-order filters.

DESIGNING REDUCED-ORDER CONTROLLERS OF MIXEDSENSITIVITY PROBLEM FOR FLIGHT CONTROL SYSTEMS

Based on linear matrix inequalities (LMI), the design method of reduced order controllers of mixed sensitivity problem is studied for flight control systems. It is shown that there exists a controller with order not greater than the difference between the generalized plant order and the number of independent control variables, if the mixed sensitivity problem is solvable for strict regular flight control plants. The proof is constructive, and an approach to design such a controller can be obtained in terms of a pair of feasible solution to the well known 3 LMI. Finally, an example of mixed sensitivity problem for a flight control system is given to demonstrate practice of the approach.

A reduced-order extrapolation algorithm based on CNLSMFE formulation and POD technique for two-dimensional Sobolev equations

A reduced-order extrapolation algorithm based on Crank-Nicolson least-squares mixed finite element （CNLSMFE） formulation and proper orthogonal decomposition （POD） technique for two-dimensional （2D） Sobolev equations is established. The error estimates of the reduced-order CNLSMFE solutions and the implementation for the reduced-order extrapolation algorithm are provided. A numerical example is used to show that the results of numerical computations are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order extrapolation algorithm is feasible and efficient for seeking numerical solutions to 2D Sobolev equations.

Computationally efficient direction finding using polynomial rooting with reduced-order and real-valued computations

The root multiple signal classification（root-MUSIC） algorithm is one of the most important techniques for direction of arrival（DOA） estimation. Using a uniform linear array（ULA） composed of M sensors, this method usually estimates L signal DOAs by finding roots that lie closest to the unit circle of a（2M-1）-order polynomial, where L 〈 M. A novel efficient root-MUSIC-based method for direction estimation is presented, in which the order of polynomial is efficiently reduced to 2L. Compared with the unitary root-MUSIC（U-root-MUSIC） approach which involves real-valued computations only in the subspace decomposition stage, both tasks of subspace decomposition and polynomial rooting are implemented with real-valued computations in the new technique,which hence shows a significant efficiency advantage over most state-of-the-art techniques. Numerical simulations are conducted to verify the correctness and efficiency of the new estimator.

Reduced-order finite element method based on POD for fractional Tricomi-type equation

The reduced-order finite element method （FEM） based on a proper orthogo- nal decomposition （POD） theory is applied to the time fractional Tricomi-type equation. The present method is an improvement on the general FEM. It can significantly save mem- ory space and effectively relieve the computing load due to its reconstruction of POD basis functions. Furthermore, the reduced-order finite element （FE） scheme is shown to be un- conditionally stable, and error estimation is derived in detail. Two numerical examples are presented to show the feasibility and effectiveness of the method for time fractional differential equations （FDEs）.

Nonlinear Control of an Induction Motor Using a Reduced-Order Extended Sliding Mode Observer for Rotor Flux and Speed Sensorless Estimation

This article proposes an innovative strategy to the problem of non-linear estimation of states for electrical machine systems. This method allows the estimation of variables that are difficult to access or that are simply impossible to measure. Thus, as compared with a full-order sliding mode observer, in order to reduce the execution time of the estimation, a reduced-order discrete-time Extended sliding mode observer is proposed for on-line estimation of rotor flux, speed and rotor resistance in an induction motor using a robust feedback linearization control. Simulations results on Matlab-Simulink environment for a 1.8 kW induction motor are presented to prove the effectiveness and high robustness of the proposed nonlinear control and observer against modeling uncertainty and measurement noise.

Research of influence of reduced-order boundary on accuracy and solution of interior points

The flow field with a high order scheme is usually calculated so as to solve complex flow problems and describe the flow structure accurately. However, there are two problems, i.e., the reduced-order boundary is inevitable and the order of the scheme at the discontinuous shock wave contained in the flow field as the supersonic flow field is low. It is questionable whether the reduced-order boundary and the low-order scheme at the shock wave have an effect on the numerical solution and accuracy of the flow field inside. In this paper, according to the actual situation of the direct numerical simulation of the flow field, two model equations with the exact solutions are solved, which are steady and unsteady, respectively, to study the question with a high order scheme at the interior of the domain and the reduced-order method at the boundary and center of the domain. Comparing with the exact solutions, it is found that the effect of reduced-order exists and cannot be ignored. In addition, the other two model equations with the exact solutions, which are often used in fluid mechanics, are also studied with the same process for the reduced-order problem.

The Design of Reduced-order Observer for Systems with Monotone Nonlinearities

与单调非线性为系统关于存在和完整顺序的观察员的设计方法基于讨论，一个还原剂顺序观察员设计方法在线性矩阵不平等(LMI ) 有的假设下面被开发积极明确的矩阵答案和还原剂顺序观察员获得矩阵被 LMI 的解决方案计算。由线性转变，不包含系统产量的衍生物的信息的一个还原剂顺序观察员被提供。一个模型被模仿，一些结论基于还原剂顺序观察员的结果的比较被得出到完整顺序的观察员的。模拟证明这份报纸开发的设计方法有好性能。

Reduced-order proper orthogonal decomposition extrapolating finite volume element format for two-dimensional hyperbolic equations

This paper is concerned with establishing a reduced-order extrapolating fi- nite volume element （FVE） format based on proper orthogonal decomposition （POD） for two-dimensional （2D） hyperbolic equations. For this purpose, a semi discrete variational format relative time and a fully discrete FVE format for the 2D hyperbolic equations are built, and a set of snapshots from the very few FVE solutions are extracted on the first very short time interval. Then, the POD basis from the snapshots is formulated, and the reduced-order POD extrapolating FVE format containing very few degrees of freedom but holding sufficiently high accuracy is built. Next, the error estimates of the reduced-order solutions and the algorithm procedure for solving the reduced-order for- mat are furnished. Finally, a numerical example is shown to confirm the correctness of theoretical conclusions. This means that the format is efficient and feasible to solve the 2D hyperbolic equations.

Fourth-Order Predictive Modelling: II. 4th-BERRU-PM Methodology for Combining Measurements with Computations to Obtain Best-Estimate Results with Reduced Uncertainties

This work presents a comprehensive fourth-order predictive modeling (PM) methodology that uses the MaxEnt principle to incorporate fourth-order moments (means, covariances, skewness, kurtosis) of model parameters, computed and measured model responses, as well as fourth (and higher) order sensitivities of computed model responses to model parameters. This new methodology is designated by the acronym 4th-BERRU-PM, which stands for “fourth-order best-estimate results with reduced uncertainties.” The results predicted by the 4th-BERRU-PM incorporates, as particular cases, the results previously predicted by the second-order predictive modeling methodology 2nd-BERRU-PM, and vastly generalizes the results produced by extant data assimilation and data adjustment procedures.

Sliding Mode Control Design via Reduced Order Model Approach

This paper presents a design of continuous-time sliding mode control for the higher order systems via reduced order model. It is shown that a continuous-time sliding mode control designed for the reduced order model gives similar performance for thc higher order system. The method is illustrated by numerical examples. The paper also introduces a technique for design of a sliding surface such that the system satisfies a cost-optimality condition when on the sliding surface.

Reduced order observer based identification of base isolated buildings

The objective of this study is to identify system parameters from the recorded response of base isolated buildings,such as USC hospital building,during the 1994 Northridge earthquake.Full state measurements are not available for identification.Additionally,the response is nonlinear due to the yielding of the lead-rubber bearings.Two new approaches are presented in this paper to solve the aforementioned problems.First,a reduced order observer is used to estimate the unmeasured states.Second,a least squares technique with time segments is developed to identify the piece-wise linear system properties.The observer is used to estimate the initial conditions needed for the time segmented identification.A series of equivalent linear system parameters are identified in different time segments.It is shown that the change in system parameters,such as frequencies and damping ratios,due to nonlinear behavior of the lead-rubber bearings,are reliably estimated using the presented technique.It is shown that the response was reduced due to yielding of the lead-rubber bearings and period lengthening.

Stable Model Order Reduction Method for Fractional-Order Systems Based on Unsymmetric Lanczos Algorithm

This study explores a stable model order reduction method for fractional-order systems. Using the unsymmetric Lanczos algorithm, the reduced order system with a certain number of matched moments is generated. To obtain a stable reduced order system, the stable model order reduction procedure is discussed. By the revised operation on the tridiagonal matrix produced by the unsymmetric Lanczos algorithm, we propose a reduced order modeling method for a fractional-order system to achieve a satisfactory fitting effect with the original system by the matched moments in the frequency domain. Besides, the bound function of the order reduction error is offered. Two numerical examples are presented to illustrate the effectiveness of the proposed method.