Model reduction of fractional impedance spectra for time–frequency analysis of batteries, fuel cells, and supercapacitors

Joint time–frequency analysis is an emerging method for interpreting the underlying physics in fuel cells,batteries,and supercapacitors.To increase the reliability of time–frequency analysis,a theoretical correlation between frequency-domain stationary analysis and time-domain transient analysis is urgently required.The present work formularizes a thorough model reduction of fractional impedance spectra for electrochemical energy devices involving not only the model reduction from fractional-order models to integer-order models and from high-to low-order RC circuits but also insight into the evolution of the characteristic time constants during the whole reduction process.The following work has been carried out:(i)the model-reduction theory is addressed for typical Warburg elements and RC circuits based on the continued fraction expansion theory and the response error minimization technique,respectively;(ii)the order effect on the model reduction of typical Warburg elements is quantitatively evaluated by time–frequency analysis;(iii)the results of time–frequency analysis are confirmed to be useful to determine the reduction order in terms of the kinetic information needed to be captured;and(iv)the results of time–frequency analysis are validated for the model reduction of fractional impedance spectra for lithium-ion batteries,supercapacitors,and solid oxide fuel cells.In turn,the numerical validation has demonstrated the powerful function of the joint time–frequency analysis.The thorough model reduction of fractional impedance spectra addressed in the present work not only clarifies the relationship between time-domain transient analysis and frequency-domain stationary analysis but also enhances the reliability of the joint time–frequency analysis for electrochemical energy devices.

An osculatory rational interpolation method of model reduction in linear system

A theorem for osculatory rational interpolation was shown to establish a new criterion of interpolation. On the basis of this conclusion a practical algorithm was presented to get a reduction model of the linear systems. Some numerical examples were given to explain the result in this paper.

Static output feedback stabilization for second-order singular systems using model reduction methods

In this paper,the static output feedback stabilization for large-scale unstable second-order singular systems is investigated.First,the upper bound of all unstable eigenvalues of second-order singular systems is derived.Then,by using the argument principle,a computable stability criterion is proposed to check the stability of secondorder singular systems.Furthermore,by applying model reduction methods to original systems,a static output feedback design algorithm for stabilizing second-order singular systems is presented.A simulation example is provided to illustrate the effectiveness of the design algorithm.

Statistical detection of structural damage based on model reduction

This paper proposes a statistical method for damage detection based on the finite element （FE） model reduction technique that utilizes measured modal data with a limited number of sensors. A deterministic damage detection process is formulated based on the model reduction technique. The probabilistie process is integrated into the deterministic damage detection process using a perturbation technique, resulting in a statistical structural damage detection method. This is achieved by deriving the first- and second-order partial derivatives of uncertain parameters, such as elasticity of the damaged member, with respect to the measurement noise, which allows expectation and covariance matrix of the uncertain parameters to be calculated. Besides the theoretical development, this paper reports numerical verification of the proposed method using a portal frame example and Monte Carlo simulation.

Model reduction and active control of flexible beam using internal balance technique

The internal balance technique is effective for the model reduction in flexible structures, especially the ones with dense frequencies. However, due to the difficulty in extracting the internal balance modal coordinates from the physical sensor readings, research on this topic has been mostly theoretical so far, and little has been done in experiments or engineering applications. This paper studies the internal balance method theoretically as well as experimentally and designs an active controller based on the reduction model. The research works on a digital signal processor （DSP） TMS320F2812- based experiment system with a flexible beam and proposes an approximate approach to access the internal balance modal coordinates. The simulation and test results have shown that the proposed approach is feasible and effective, and the designed controller is successful in restraining the beam vibration.

Stable Routh-Padé-type approximation in model reduction of interval systems

An interval Pade-type approximation is introduced and then Routh-Pade-type method （IRPTM） is presented to model reduction in interval systems. The denominator in reduced model is obtained from the stable Routh table, and its numerator is constructed by the interval Pade-type definition. Compared to the existing Routh-Pade method, IRPTM does not need to solve linear interval equations theoretical analysis shows that IRPTM has example is given to illustrate our method. Hence, we do not have to compute smaller computational cost than that interval division in the process. Moreover, of Routh-Pade method. A typical numerical

Model reduction using the genetic algorithmand routh approxi mations

A new method of model reduction combining the genetic algorithm（GA） with the Routh approximation method is presented. It is suggested that a high-order system can be approximated by a low-order model with a time delay. The denominator parameters of the reduced-order model are determined by the Routh approximation method, then the numerator parameters and time delay are identified by the GAL. The reduced-order models obtained by the proposed method will always be stable if the original system is stable and produce a good approximation to the original system in both the frequency domain and time domain. Two numerical examples show that the method is cornputationally simple and efficient.

RobustL_1 model reduction for stochastic time-delay systems

This paper investigates the problem of robust L1 model reduction for continuous-time uncertain stochastic time-delay systems. For a given mean-square stable system, our purpose is to construct reduced-order systems, such that the error system between these two models is mean-square asymptotically stable and has a guaranteed L1 (also called peak-to-peak) performance. The peak-to-peak gain criterion is first established for stochastic time-delay systems, and the corresponding model reduction problem is solved by using projection lemma. Sufficient conditions are obtained for the existence of admissible reduced-order models in terms of linear matrix inequalities (LMIs) plus matrix inverse constraints. Since these obtained conditions are not expressed as strict LMIs, the cone complementarity linearization (CCL) method is exploited to cast them into nonlinear minimization problems subject to LMI constraints, which can be readily solved by standard numerical software. In addition, the development of reduced-order models with special structures, such as the delay-free model, is also presented. The efficiency of the proposed methods is demonstrated via a numerical example.

A matrix Padé-type-Routh model reduction method for multivariable linear systems

In the present paper, the formulae for matrix Padé-type approximation were improved. The mixed model reduction method of matrix Padé-type-Routh for the multivariable linear systems was presented. A well-known example was given to illustrate that the mixed method is efficient.

Model Reduction via Reducibility Matrix

In this work, a new model reduction technique is introduced. The proposed technique is derived using the matrix reducibility concept. The eigenvalues of the reduced model are preserved; that is, the reduced model eigenvalues are a subset of the full order model eigenvalues. This preservation of the eigenvalues makes the mathematical model closer to the physical model. Finally, the outcomes of this method are fully illustrated using simulations of two numeric examples.

Model Reduction Methods for High-Rise Buildings with Active Mass Damper Control Systems

To improve the performance of an active mass damper control system,the controller should be designed based on a reduced-order model. An improved method based on balanced truncation method was proposed to reduce the dimension of high-rise buildings,and was compared with other widely used reduction methods by using a framework with ten floors. This optimized method has improvement of reduction process and choice of the order. Based on the reduced-order model obtained by the improved method and pole-assignment algorithm,a controller was designed. Finally,a comparative analysis of structural responses,transfer functions,and poles was conducted on an actual high-rise building. The results show the effectiveness of the improved method.

Advanced System-Level Model Reduction Method for Multi-Converter DC Power Systems

For dynamic stability analysis and instability mechanism understanding of multi-converter medium voltage DC power systems with droop-based double-loop control,an advanced system-level model reduction method is proposed.With this method,mathematical relationships of control parameters(e.g.,current and voltage control parameters)between the system and its equivalent reduced-order model are established.First,open-loop and closed-loop equivalent reduced-order models of current control loop considering dynamic interaction among converters are established.An instability mechanism(e.g.,unreasonable current control parameters)of the system can be revealed intuitively.Theoretical guidance for adjustment of current control parameters can also be given.Then,considering dynamic interaction of current control among converters,open-loop and closed-loop equivalent reduced-order models of voltage control loop are established.Oscillation frequency and damping factor of DC bus voltage in a wide oscillation frequency range(e.g.,10–50 Hz)can be evaluated accurately.More importantly,accuracy of advanced system-level model reduction method is not compromised,even for MVDC power systems with inconsistent control parameters and different number of converters.Finally,experiments in RT-BOX hardware-in-the-loop experimental platform are conducted to validate the advanced system-level model reduction method.

Projection-Based Dimensional Reduction of Adaptively Refined Nonlinear Models

Adaptive mesh refinement (AMR) is fairly practiced in the context of high-dimensional, mesh-based computational models. However, it is in its infancy in that of low-dimensional, generalized-coordinate-based computational models such as projection-based reduced-order models. This paper presents a complete framework for projection-based model order reduction (PMOR) of nonlinear problems in the presence of AMR that builds on elements from existing methods and augments them with critical new contributions. In particular, it proposes an analytical algorithm for computing a pseudo-meshless inner product between adapted solution snapshots for the purpose of clustering and PMOR. It exploits hyperreduction—specifically, the energy-conserving sampling and weighting hyperreduction method—to deliver for nonlinear and/or parametric problems the desired computational gains. Most importantly, the proposed framework for PMOR in the presence of AMR capitalizes on the concept of state-local reduced-order bases to make the most of the notion of a supermesh, while achieving computational tractability. Its features are illustrated with CFD applications grounded in AMR and its significance is demonstrated by the reported wall-clock speedup factors.

EMTP-type Program Realization of Krylov Subspace Based Model Reduction Methods for Large-Scale Active Distribution Network

Effective model reduction methods are required to deal with new challenges in active distribution network simulations that are on a large scale and have complicated structures.In the development of advanced electromagnetic transient simulation programs,automated model reduction plays an important role.This paper proposes an automated realization algorithm for the Krylov subspace based model reduction methods of an active distribution network with which the reduced model can be automatically established according to a given threshold of reduction error.The combined state-space nodal analysis framework is employed to apply the automated model reduction algorithm in popular EMTP-type simulation programs.Simulations are performed using PSCAD and a self-developed program to show the feasibility and validity of the proposed methods.

MODEL REDUCTION AND ACTIVE CONTROL FOR A FLEXIBLE PLATE

The internal balance technique is effective for model reduction in flexible structures, especially those with dense frequencies. However, due to the difficulty in extracting the internal balance modal coordinates from the physical sensor readings, research so far on this topic has been mostly theoretic and little on experiment or engineering applications. This paper, by working on a DSP TMS320F2812-based experiment system with a flexible plate and bringing forward an approximating approach to accessing the internal balance modal coordinates, studies the internal balance method theoretically as well as experimentally, and further designs an active controller based on the reduced model. Simulation and test results have proven the proposed approximating approach feasible and effective, and the designed controller successful in restraining the plate vibration.

Hilbert-Schmidt-Hankel norm model reduction for matrix second-order linear systems

This paper considers the optimal model reduction problem of matrix second-order linear systems in the sense of Hilbert-Schmidt-Hankel norm, with the reduced order systems preserving the structure of the original systems. The expressions of the error function and its gradient are derived. Two numerical examples are given to illustrate the presented model reduction technique.

Model Reduction and Controller Design for a Nonlinear Heat Conduction Problem Using Finite Element Method

The mathematical models for dynamic distributed parameter systems are given by systems of partial differential equations. With nonlinear material properties, the corresponding finite element (FE) models are large systems of nonlinear ordinary differential equations. However, in most cases, the actual dynamics of interest involve only a few of the variables, for which model reduction strategies based on system theoretical concepts can be immensely useful. This paper considers the problem of controlling a three dimensional profile on nontrivial geometries. Dynamic model is obtained by discretizing the domain using FE method. A nonlinear control law is proposed which transfers any arbitrary initial temperature profile to another arbitrary desired one. The large dynamic model is reduced using proper orthogonal decomposition (POD). Finally, the stability of the control law is proved through Lyapunov analysis. Results of numerical implementation are presented and possible further extensions are identified.

Correlation analysis of PCB and comparison of test-analysis model reduction methods

The validity of correlation analysis between finite element model（FEM） and modal test data is strongly affected by three factors, i.e., quality of excitation and measurement points in modal test,FEM reduction methods, and correlation check techniques. A new criterion based on modified mode participation（MMP） for choosing the best excitation point is presented. Comparison between this new criterion and mode participation（MP） criterion is made by using Case 1 with a simple printed circuit board（PCB）. The result indicates that this new criterion produces better results. In Case 2, 35 measurement points are selected to perform modal test and correlation analysis while 9 selected in Case 3.System equivalent reduction expansion process（SEREP）, modal assurance criteria（MAC）, coordinate modal assurance criteria（CoMAC）, pseudo orthogonality check（POC） and coordinate orthogonality check（CORTHOG） are used to show the error introduced by modal test in Cases 2 and 3. Case 2 shows that additional errors which cannot be identified by using CoMAC can be found by using CORTHOG.In both Cases 2 and 3, Guyan reduction, improved reduced system（IRS） method, SEREP and Hybrid reduction are compared for accuracy and robustness. The results suggest that the quality of the reduction process is problem dependent. However, the IRS method is an improvement over the Guyan reduction, and the Hybrid reduction is an improvement over the SEREP reduction.

Development of stability-preserving time-limited model reduction framework for 2-D and 1-D models with error bound

Due to their complex structure,2-D models are challenging to work with;additionally,simulation,analysis,design,and control get increasingly difficult as the order of the model grows.Moreover,in particular time intervals,Gawronski and Juang’s time-limited model reduction schemes produce an unstable reduced-order model for the 2-D and 1-D models.Researchers revealed some stability preservation solutions to address this key flaw which ensure the stability of 1-D reduced-order systems;nevertheless,these strategies result in large approximation errors.However,to the best of the authors’knowledge,there is no literature available for the stability preserving time-limited-interval Gramian-based model reduction framework for the 2-D discrete-time systems.In this article,2-D models are decomposed into two separate sub-models(i.e.,two cascaded 1-D models)using the condition of minimal rank-decomposition.Model reduction procedures are conducted on these obtained two 1-D sub-models using limited-time Gramian.The suggested methodology works for both 2-D and 1-D models.Moreover,the suggested methodology gives the stability of the reduced model as well as a priori error-bound expressions for the 2-D and 1-D models.Numerical results and comparisons between existing and suggested methodologies are provided to demonstrate the effectiveness of the suggested methodology.

An Improved Model Reduction Method on AIMs for N-S Equations Using Multilevel Finite Element Method and Hierarchical Basis

A numerical method is proposed to approach the Approximate Inertial Man-ifolds(AIMs)in unsteady incompressible Navier-Stokes equations,using multilevel fi-nite element method with hierarchical basis functions.Following AIMS,the unknown variables,velocity and pressure in the governing equations,are divided into two com-ponents,namely low modes and high modes.Then,the couplings between low modes and high modes,which are not accounted by standard Galerkin method,are consid-ered by AIMs,to improve the accuracy of the numerical results.Further,the multilevel finite element method with hierarchical basis functions is introduced to approach low modes and high modes in an efficient way.As an example,the flow around airfoil NACA0012 at different angles of attack has been simulated by the method presented,and the comparisons show that there is a good agreement between the present method and experimental results.In particular,the proposed method takes less computing time than the traditional method.As a conclusion,the present method is efficient in numer-ical analysis of fluid dynamics,especially in computing time.