System Reduction Method Research Articles

The method of determining optimal control of the thermoelastic state of piece-homogeneous body using a stationary temperature field

This paper proposes a new highly effective method for determining the optimal control of the stress-strain state of spatially multi-connected composite bodies using a stationary temperature field. The proposed method is considered based on the example of a stationary axisymmetric thermoelastic problem for a space with a spherical inclusion and cavity. The proposed method is based on the generalized Fourier method and reduces the original problem to an equivalent problem of optimal control, in which the state of the object is determined by an infinite system of linear algebraic equations, the right-hand side of which parametrically depends on the control. At the same time, the functional of the cost of the initial problem is transformed into a quadratic functional, which depends on the state of the equivalent system and parametrically on the control. The limitation on the temperature distribution is replaced by the value of the control norm in the space of square summable sequences. In fact, this paper considers for the first time the problem of optimal control of an infinite system of linear algebraic equations and develops a method for its solution. The proposed method is based on presenting the solutions of infinite systems in a parametric form, which makes it possible to reduce equivalent problem to the problem of conditional extremum of a quadratic functional, which explicitly depends on the control. A further solution to this problem A further solution to this problem is found by the Lagrange method using the spectral decomposition of the quadratic functional matrix. found by the Lagrange method using the spectral decomposition of the quadratic functional matrix. The method developed in this paper is strictly justified. For all infinite systems, the Fredholm property of their operators is proved. As an important result necessary for substantiation, for the first time, an estimate from below of the module of the multi-parameter determinant of the resolving system of the boundary value problem of conjugation – space with a spherical inclusion – was obtained when solving it using the Fourier method. The theorem that establishes the conditions for the existence and uniqueness of the solution of equivalent problem or optimal control problem without restrictions in the space of square summable sequences is proved. The numerical algorithm is based on a reduction method for infinite systems of linear algebraic equations. Estimates of the practical accuracy of the numerical algorithm demonstrated the stability of the method and sufficiently high accuracy even with close location of the boundary surfaces. Graphs showing the optimal temperature distribution for various geometric parameters of the problem and their analysis are provided. The proposed method extends to boundary value problems with different geometries.

A model reduction method for parametric dynamical systems defined on complex geometries

Dynamic mode decomposition (DMD) describes the dynamical system in an equation-free manner and can be used for the prediction and control. It is an efficient data-driven method for the complex systems. In this paper, we extend DMD to the parameterized problems and propose a model reduction method based on DMD to improve the computation efficiency. This method is an offline-online mechanism. In the offline phase, we need to generate the snapshots data by solving the parameterized equations for each parameter in the training set and perform the singular value decomposition (SVD) to get the reduced operator matrices, which would lead to the substantial computation. The weighted & interpolated nearest-neighbors algorithm (wiNN) is adopted to construct the efficient surrogate models of the reduced operator matrices (including the reduced Koopman operator matrix and SVD-modes). In the online phase, for each parameter, we only need to perform the operations based on the low-dimensional matrices to get the parameter DMD solution. Moreover, we choose the least squares radial basis function finite difference method for the spatial discretization. This can make our method more applicable to the parameterized problems defined on complex geometries. At last, the reaction-diffusion, the incompressible miscible flooding and the incompressible Navier-Stokes models defined on the complex geometries are presented to illustrate the effectiveness of the proposed method.

High-order Krylov subspace model order reduction methods for bilinear time-delay systems

Model order reduction methods via high-order Krylov subspace for bilinear time-delay systems are developed in this paper. The proposed methods are based on the expansion of the Taylor series or Laguerre series. The obtained reduced systems can not only match certain expansion coefficients but also preserve the structure of the original system. We also briefly discuss the two-sided projection reduction method. To address the implementation of our approach, we utilize the high-order block Arnoldi algorithm to generate projection matrices and employ the genetic algorithm to optimize parameter selection during the reduction process. Finally, we validate the performance of the proposed reduction methods through numerical results.

Movement Tracking and False Positive Reduction Method for Microwave Colonoscopy Systems

Movement Tracking and False Positive Reduction Method for Microwave Colonoscopy Systems

Model Reduction of Cooperative Systems Using Separable Energy Functions

In this paper, we develop a model reduction method for cooperative nonlinear systems via their variational systems. Inspired by scalable stability analysis of cooperative systems based on separable functions, we provide novel L1 differential observability and L∞ differential reachability functions and study them by using vector-valued functions. These vector-valued functions characterize not-so-important state variables. By truncating them, model reduction is performed, which preserves cooperativity and stability properties.

Structural static reanalysis using flexibility disassembly perturbation and system reduction

Rapid static reanalysis technology is an important tool commonly used in structural optimization design and reliability analysis. The existing static reanalysis methods are not efficient enough in solving high-rank large modification problems. To overcome this drawback, this paper proposes a new method for structural static reanalysis based on flexibility disassembly perturbation and system reduction. Firstly, a generalized flexibility disassembly perturbation technique is developed to directly compute the flexibility matrix of the modified structure. Subsequently, a new system reduction method is proposed to quickly approximate the exact solution of the modified static displacement. Even for the global large modification problem, the proposed method only requires two basis vectors to obtain very accurate calculation results. Two numerical models of statically indeterminate structures are used to verify the effectiveness and progressiveness of the method. The results show that the proposed reanalysis method has higher computational accuracy than the existing combined approximation (CA) and preconditioned conjugate gradient (PCG) methods, especially for the high-rank large modification case. A steel reticulated shell structure is further used to illustrate the application of the proposed reanalysis method in structural reliability analysis. It can be found that the proposed method significantly saves the calculation time of structural reliability analysis.

Robust ∞ model reduction for the continuous fractional‐order two‐dimensional Roesser system: The 0 < ε ≤ 1 case

This study analyzes the model reduction methods and the robust model reduction methods for the continuous fractional‐order (FO) two‐dimensional (2D) Roesser system with the FO ranging from 0 to 1. A sufficient LMI‐based condition is given by Projection Lemma, the FO‐stable‐region‐analyze method and matrix analysis, guaranteeing that the error system is stable and that its transfer function's ‐norm is bounded. Then, the approach to design the reduced‐model system of the original continuous FO 2D Roesser model is given. The robust model reduction methods for continuous FO 2D Roesser systems with polytopic uncertainties are proposed based on the results above. In the end, two numerical examples are used to demonstrate the effectiveness of the results.

Model reduction for nonlinearizable dynamics via delay-embedded spectral submanifolds

Delay embedding is a commonly employed technique in a wide range of data-driven model reduction methods for dynamical systems, including the dynamic mode decomposition, the Hankel alternative view of the Koopman decomposition (HAVOK), nearest-neighbor predictions and the reduction to spectral submanifolds (SSMs). In developing these applications, multiple authors have observed that delay embedding appears to separate the data into modes, whose orientations depend only on the spectrum of the sampled system. In this work, we make this observation precise by proving that the eigenvectors of the delay-embedded linearized system at a fixed point are determined solely by the corresponding eigenvalues, even for multi-dimensional observables. This implies that the tangent space of a delay-embedded invariant manifold can be predicted a priori using an estimate of the eigenvalues. We apply our results to three datasets to identify multimodal SSMs and analyse their nonlinear modal interactions. While SSMs are the focus of our study, these results generalize to any delay-embedded invariant manifold tangent to a set of eigenvectors at a fixed point. Therefore, we expect this theory to be applicable to a number of data-driven model reduction methods.

A novel image noise reduction method for composite multistable stochastic resonance systems

In the field of digital signal processing, image denoising is an more and more significant research direction. For the traditional noise reduction theory, noise is considered to be harmful, and the image quality can be improved by analyzing noise characteristics and filtering noise. The appearance of stochastic resonance theory proves that noise can be used to enhance signal, which brings new inspiration to image processing. The classical bistable stochastic resonance model has the problems of high potential barrier and easy saturation, which is not conducive to the improvement of image denoising effect. In this paper, a novel type of stochastic resonance potential well model is quoted, which solves the above shortcomings of the bistable stochastic resonance model, and then combines it with the Gaussian model to propose a composite multistable stochastic resonance model. The dynamic principle of the model in signal detection is described, and the influence of system parameters on image noise reduction is analyzed. The whale optimization algorithm is used to optimize the model parameters, and an adaptive compound multistable stochastic resonance system is established to process pictures and measured radar images under different noise backgrounds. The simulation experiment and engineering application show that the model proposed in this paper solves the problem of high potential barrier and easy saturation of the bistable model, and has better image noise reduction ability compared with Wiener filter, median filter, classical bistable stochastic resonance system and novel type of stochastic resonance potential well system.

Parallel model order reduction methods based on structured matrix analysis for discrete-time systems with parametric uncertainty

This paper explores two novel parallel parametric model order reduction methods for discrete-time systems with parametric uncertainty. The proposed methods allow that the parametric dependence is non-affine. With the shift-transformation matrix of the discrete orthogonal polynomials, including Charlier polynomials and Krawtchouk polynomials, the expansion coefficients of state variable in discrete orthogonal polynomials space can be obtained by solving the system of linear algebraic equations (SLAE). By applying the structured matrix analysis of the SLAE, two parallel strategies based on the equivalent transformation of block bi-diagonal matrices and the block discrete Fourier transform of block ϵ-circulant matrices are proposed to solve the SLAE. Then, the projection matrix is constructed to reduce discrete-time parametric systems. Moreover, we analyze the properties of the proposed parallel methods, including the invariable coefficients, the invertibility, and the error analysis. Finally, the effectiveness of the proposed methods is illustrated by the numerical experiments.

H2model order reduction of bilinear systems via linear matrix inequality approach

AbstractThis paper proposes an H2‐optimal model order reduction (MOR) method for bilinear systems based on the linear matrix inequality (LMI) approach. In this method, to reduce the computational complexity, at first, a reduced middle‐order approximation of the system is derived based on common bilinear MOR methods. Next, the H2norm of the error system is minimized to obtain the reduced‐order bilinear model. Generalized Lyapunov equations are added to the optimization problem as LMI constraints to guarantee the specification of type II Gramians of the bilinear system to improve accuracy. Besides, two stability conditions are included to the optimization problem as its constraints to preserve stability of reduced‐order bilinear model. One of advantages of the proposed method is the need for only one of the Gramians of controllability or observability. Since the proposed H2‐optimal MOR problem is a polynomial matrix inequality (PMI) problem, an iterative method is used to convert the PMI to the LMI problems and solve the optimization problem. Three bilinear test systems are considered to show the proposed method's efficiency, while its performance is compared with some classical methods. Results show that the proposed methods lead to a more accurate reduced‐order model than other MOR methods.

Model reduction of discrete time-delay systems based on Charlier polynomials and high-order Krylov subspaces

In this paper, we present an efficient model reduction method for discrete time-delay systems based on the expansions of systems under Charlier polynomials. Making full use of the properties of Charlier polynomials and the structure of discrete time-delay systems, the projection space built by state variables is embedded in a high-order Krylov subspace. Further, a high-order Krylov subspace method is developed to generate discrete time-delay reduced systems. The proposed method is independent of the choice of inputs since the high-order Krylov subspace sequence does not involve the expansion coefficients of inputs. Besides, theoretical analysis shows that the resulting discrete time-delay reduced system characterizes the property of invariable coefficients. Finally, two numerical examples demonstrate the effectiveness of the proposed method.

Neural network-based regression assisted PAPR reduction method for OFDM systems

Neural network-based regression assisted PAPR reduction method for OFDM systems

Controller Reduction for Nonlinear Systems by Generalized Differential Balancing

In this article, we aim at developing computationally tractable methods for nonlinear model/controller reduction. Recently, model reduction by generalized differential (GD) balancing has been proposed for nonlinear systems with constant input-vector fields and linear output functions. First, we study incremental properties in the GD balancing framework. Next, based on these analyses, we provide GD linear quadratic Gaussian (LQG) balancing and GD <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> -balancing as controller reduction methods for nonlinear systems by focusing on linear feedback and observer gains. Especially for GD <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> -balancing, we clarify when the closed-loop system consisting of the full-order system and a reduced-order controller is exponentially stable. All provided methods for controller reduction can be relaxed to linear matrix inequalities (LMIs).

Parallel model order reduction based on block discrete Fourier transform and Krylov subspace for parametric systems

This paper explores a time-domain parallel parametric model order reduction (PMOR) method for parametric systems based on the block discrete Fourier transform (DFT) and Krylov subspace. The proposed method is suitable for parametric systems with non-affine parametric dependence. With Taylor expansion, the expansion coefficients of the state variable are first obtained. Then, we show that the subspace spanned by the expansion coefficients belongs to a Krylov subspace. To speed up the PMOR process, a parallel strategy based on the block DFT and the structured matrices is proposed to compute the matrices involved in the Krylov subspace. This can avoid directly computing the inverse of the large-scale matrix. After that, the reduced parametric systems are constructed with the projection matrix obtained by the Arnoldi algorithm and orthonormalisation. Furthermore, we analyse the invertibility and the error estimations to guarantee the feasibility of the proposed PMOR method. Finally, the numerical experiments are given to demonstrate the effectiveness of the proposed method.

EMI Reduction Method for Wireless Power Transfer Systems With High Power Transfer Efficiency Using Frequency Split Phenomena

This article describes a method to reduce electromagnetic interference (EMI) components at odd harmonic frequencies of a wireless power transfer (WPT) system. In a WPT system, a leakage magnetic field is generated, which may adversely affect the human body or other electronic devices. The previous reactive shielding method could reduce the magnetic field, but it has the disadvantage of reducing the power transfer efficiency (PTE). By using frequency split phenomena, the reactive shielding method can reduce the EMI components with increasing PTE. In this article, a reactive shielding method that can reduce the magnetic field while increasing PTE is proposed. Specifically, the method for designing resonant system of transmitting, receiving, and shielding for the WPT system is proposed. In addition, a method for obtaining a coupling coefficient between a shielding coil and a transmitting coil for designing structure in which magnetic field reduction can be maximized is also proposed. The theoretical analysis based on the impedance of each part is strongly correlated with the simulation and measurement result. As a result of the experiment, the PTE is increased by about 2.7%, and EMI components at odd harmonic frequency can be reduced by up to 6.81 dB in a 50 W-WPT system.

Optimized baseband Nyquist pulse-based PAPR reduction method for SEFDM systems

Optimized baseband Nyquist pulse-based PAPR reduction method for SEFDM systems

Two-sided dimension reduction for linear control systems of [formula omitted]th-order form

In this article, a two-sided structure-preserving dimension reduction method for linear control systems of kth-order form is discussed. We first recall the Petrov–Galerkin framework of dimension reduction for the kth linear systems. Then the moments of systems at a given frequency point s0 are discussed. According to the recurrence form of the moments, we obtain the input Krylov subspace and the projection matrix V. Via dually designing output Krylov subspace, we obtain the output Krylov subspace basis matrix W. Hence, the two-sided dimension reduction method is obtained via combination V with W. Afterwards one can obtain the result that the number of moments matched is twice as much as the column dimension of the projection matrix. Finally, one numerical experiment is performed to verify the proposed dimension reduction method.

Passivity preserving model reduction via spectral factorization

We present a novel model-order reduction (MOR) method for linear time-invariant systems that preserves passivity and is thus suited for structure-preserving MOR for port-Hamiltonian (pH) systems. Our algorithm exploits the well-known spectral factorization of the Popov function by a solution of the Kalman–Yakubovich–Popov (KYP) inequality. It performs MOR directly on the spectral factor inheriting the original system’s sparsity enabling MOR in a large-scale context. Our analysis reveals that the spectral factorization corresponding to the minimal solution of an associated algebraic Riccati equation is preferable from a model reduction perspective and benefits pH-preserving MOR methods such as a modified version of the iterative rational Krylov algorithm (IRKA). Numerical examples demonstrate that our approach can produce high-fidelity reduced-order models close to (unstructured) H2-optimal reduced-order models.

Riemannian optimization model order reduction method for general linear port-Hamiltonian systems

Abstract This paper presents a Riemannian optimal model order reduction method for general linear stable port-Hamiltonian systems based on the Riemannian trust-region method. We consider the $\mathcal{H}_2$ optimal model order reduction problem of the general linear port-Hamiltonian systems. The problem is formulated as an optimization problem on the product manifold, which is composed of the set of skew symmetric matrices, the manifold of the positive definite matrices, the manifold of the positive semidefinite matrices with fixed rank and the Euclidean space. To solve the optimal problem, the Riemannian geometry of the product manifold is given. Moreover, the Riemannian gradient and the Riemannian Hessian of the objective function are derived. Furthermore, we propose the Riemannian trust-region method for the optimization problem and introduce the truncated conjugate gradient method to solve the trust-region subproblem. Finally, the numerical experiments illustrate the efficiency of the proposed method.