Efficient model reduction of interconnects via double gramians approximation

A new mixed method for relative error model order reduction is proposed. In the proposed method the frequency domain balanced stochastic truncation method is improved by applying the generalized singular perturbation method to the frequency domain balanced system in the reduction procedure. The frequency domain balanced stochastic truncation method, which was proposed in [15] and [17] by the author, is based on two recently developed methods, namely frequency domain balanced truncation within a desired frequency bound and inner-outer factorization techniques. The proposed method in this paper is a carry over of the frequency-domain balanced stochastic truncation and is of interest for practical model order reduction because in this context it shows to keep the accuracy of the approximation as high as possible without sacrificing the computational efficiency and important system properties. It is shown that some important properties of the frequency domain stochastic balanced reduction technique are extended to the proposed reduction method by using the concept and properties of the reciprocal systems. Numerical results show the accuracy, simplicity and flexibility enhancement of the method.

This paper presents a framework for model order reduction based on Balanced Truncation (BT) method and Artificial Neural Networks (ANN). As social, architectural and technical models are witnessing rapid complexity increase, it has become essential to find solutions to obtain a lower complex version of models while still preserving the model integrity. The main idea of the proposed framework is to reduce uncertainty by cascading two different Model Order Reduction (MOR) approaches, the BT technique and the Jordan Recurrent Neural Network (JRNN). The reduced model obtained by the JRNN network is estimated using the Instrument Variable (IV) estimation approach. Finally, the BT reduced model and the estimated JRNN reduced model are cascaded to obtain the final reduced order version of the original plant. The cascading procedure takes the powerfulness of both techniques and present them in a single system capable to compensate for possible errors and uncertainties. Simulation results prove the efficiency of the proposed framework in terms of the deduction strength and the integrity level.

Parameterization theory of balanced truncation method for any evenly distributed RC interconnect circuits and transmission lines

Considering the use of dynamical systems in practical applications, often only limited regions in the time or frequency domain are of interest. Therefore, it usually pays off to compute local approximations of the used dynamical systems in the frequency and time domain. In this paper, we consider a structure-preserving extension of the frequency- and time-limited balanced truncation methods to second-order dynamical systems. We give a full overview about the first-order limited balanced truncation methods and extend those to second-order systems by using the different second-order balanced truncation formulas from the literature. Also, we present numerical methods for solving the arising large-scale sparse matrix equations and give numerical modifications to deal with the problematic case of second-order systems. The results are then illustrated on three numerical examples.

Most of the physical structures may be described in terms of mathematical models. The mathematical methods of system modelling also lead to a thorough explanation of the mechanism in the form of mathematical equations, which are often difficult to use for both analysis and controller synthesis. Consequently, it is useful and very important to determine the likelihood of different calculations of the same type, but in a lower-order representation, which can be assumed to correctly represent almost all the basic features of the system under examination. In this article, the proposed method is based on the modified balanced truncation method (BTM) by which the steady-state value from the problem of the BTM has been circumvented. This weakness has been eliminated by using a modified BTM to narrow the deviations by incorporating a gain factor into the response transfer matrix of the reduced-order model (ROM) to adjust the steady-state value of the ROM, without affecting the variations in dynamical behaviour as compared to the original system. To illustrate the proposed method, a real-time application model has been reduced where the ROM retains all the essential characteristics of the original system. In order to analyse the effectiveness, accuracy and validation with the other existing reduction methods, two standard numerical test systems have been taken from the literature and been tested also.

Studies a class of balanced truncation algorithms applicable to relative/multiplicative model reduction. These algorithms seek to balance the controllability gramian of a given transfer function and the observability gramian of its right inverse. For this reason, the algorithms are referred to as inverse-weighted balanced truncation (IWBT) algorithms. It is shown that by using IWBT algorithms one can derive relative and multiplicative L/sub /spl infin// error bounds that are known to hold for other reduction algorithms. It is also shown that the balanced stochastic truncation (BST) method is actually one special, but an optimal version of the IWBT algorithms. As such, the authors' result also serves to establish the fact that the available error bounds pertaining to BST algorithms actually hold for IWBT algorithms.

Balanced truncation is a standard method for model reduction, possessing many desirable properties such as preservation of model stability and a priori error bounds. Balanced truncation is conducted using controllability and observability Gramians. Generalized Gramians can be found by solving a set of linear matrix inequalities. In this paper, we show that these linear matrix inequalities can be extended so that the number of decision variables are at least doubled, leading to the concept of extended Gramians. Herein it is shown that the desirable properties of balanced truncation also hold with extended Gramians. The extended Gramians are especially useful for improving error bounds and for models possessing additional structural constraints, as is demonstrated by means of examples herein.

Flexible structures occur in a large field of applications such as aerospace, biomedical, military applications. It involves many control problems ranging from vibration control to transient shape control. One of the most important applications of flexible structures is in adaptive optics. In this paper, a reducedorder controller design based on the internal model principle is proposed for deformable mirrors in adaptive optics. The finitedimensional model of the two-dimensional partial differential equation of the deformable mirror is obtained by applying the Galerkin approximation. Then, the balanced truncation method is applied to the controller of the high order finite-dimensional model of the deformable mirror. Simulation results show the success of the proposed method.

This paper studies stability- and symmetry-preserving H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> optimal model reduction problems of linear systems, which include linear gradient systems as a special case. The problem is formulated as a nonlinear optimization problem on the product manifold of the manifold of symmetric positive-definite matrices and two Euclidean spaces. To solve the problem by using the trust-region method, the gradient and Hessian of the objective function are derived. Furthermore, it is shown that if we restrict our systems to gradient systems, the gradient and Hessian can be obtained more efficiently. More concretely, by symmetry, we can reduce linear matrix equations to be solved. In addition, by a simple example, we show that the solutions to our problem and a similar problem in some literature works are not unique, and the solution sets of both problems do not contain each other in general. Also, it is revealed that the attained optimal values do not coincide. Numerical experiments show that the proposed method gives a reduced system with the same structure with the original system although the balanced truncation method does not.

Offshore structures like jackets are exposed to various kinds of external loading conditions and examination of broader range of modes and natural frequencies are required for their dynamic analysis. In this study, four model reduction techniques — two modal truncation methods and two balanced truncation methods — are introduced and applied to a jacket structure. The reduction results are compared with those of the conventional Paz condensation methods. Four new methods show better fidelity, especially in high frequency ranges than the conventional methods.

In this paper, we develop a method for solving the problem of minimizing the $H^2$ error norm between the transfer functions of original and reduced systems on the set of stable matrices and two Euclidean spaces. That is, we develop a method for identifying the optimal reduced system from all stable linear systems. However, it is difficult to develop an algorithm for solving this problem, because the set of stable matrices is highly non-convex. To overcome this issue, we show that the problem can be transformed into a tractable Riemannian optimization on the product manifold of the set of skew-symmetric matrices, the manifold of the symmetric positive-definite matrices, and two Euclidean spaces. The stability of the reduced systems constructed using the optimal solutions to our problem is preserved. To solve the reduced problem, the Riemannian gradient and Hessian are derived and a Riemannian trust-region method is developed. The initial point in the proposed approach is selected using the output from the balanced truncation (BT) method. Numerical experiments demonstrate that our method considerably improves the results given by BT in the sense of the $H^2$ norm, and also provides reduced systems that are globally near-optimal solutions to the problem of minimizing the $H^{\infty}$ error norm. Moreover, we show that our method provides a better reduced model than BT from the viewpoint of the frequency response.

The traditional centralized control strategy to vibration suppression of large-scale thin plate structures may increase the design difficulty of the controller. In this article, a decentralized vibration active control method is proposed to suppress the vibration of the thin plate structure with smart constrained layer damping treatment. First, the dynamics model of the smart constrained layer damping plate is established based on the finite element method, and the characteristics of viscoelastic materials with temperature and frequency are described by Golla-Hughes-McTavish damping model. Subsequently, a decentralized subsystem control model is obtained from the balanced model reduction method and complex mode truncation method. The modal test proves that the theoretical model is accurate. Then, the particular emphasis is placed on the stability and vibration attenuation of a decentralized system, which is composed of multiple subsystems. The local state feedback stabilization, using interaction of local state feedback and output feedback, is introduced to achieve system stability. To solve the practical problem of local state feedback, a decentralized controller with an observer is developed by adopting the pole placement method. Finally, the numerical simulation and hardware-in-the-loop experiment under different excitation are performed to investigate the effectiveness of decentralized vibration active control. The results demonstrate that the decentralized controller can effectively suppress the vibration, especially under mixed periodic signal and Gauss white noise signal.

We propose an alternative approach to the Balanced Model Truncation method (standard method). This approach reduces substantially the order of minimum-phase inverse filters for equalizing room acoustics. This method is based on a property of the filter z transform function, which modifies the corresponding FIR coefficients before the application of the standard technique to the modified FIR coefficients filter version. In the standard technique, the Hankel singular values plot is the chief guide for a user for the selection of a reduced filter order. Results for minimum-phase inverse filter corresponding to partial equalization of measured acoustic impulse response show the superiority of the proposed method over the standard technique, in terms of reduced filters order selection.

Vibration and dissipation in vibro-acoustic systems can be assessed using frequency response analysis. Evaluating a frequency sweep on a full-order model can be very costly, so model order reduction methods are employed to compute cheap-to-evaluate surrogates. This work compares structure-preserving model reduction methods based on rational interpolation and balanced truncation with a specific focus on their applicability to vibro-acoustic systems. Such models typically exhibit a second-order structure and their material properties as well as their excitation may be depending on the driving frequency. We demonstrate the effectiveness of all considered methods in terms of their accuracy and computational cost by applying them to numerical models of vibro-acoustic systems depicting structural vibration, sound transmission, acoustic scattering, and poroelastic problems. The results of the experiments are extensively discussed to derive guidelines for the choice of model reduction methods based on the problem setting.

The problem of model reduction of linear systems with certain interconnection structure is considered in this paper. To preserve the interconnection structure between subsystems in the reduction, special care needs to be taken. This problem is important and timely because of the recent focus on complex networked systems in control engineering. Two different model reduction methods are introduced and compared in this paper. Both methods are extensions to the well‐known balanced truncation method. Compared with earlier work in the area these methods use a more general linear fractional transformation framework, and utilize linear matrix inequalities. Furthermore, new approximation error bounds that reduce to classical bounds in special cases are derived. The so‐called structured Hankel singular values are used in the methods, and indicate how important states in the subsystems are with respect to a chosen input–output map for the entire interconnected system. It is shown how these structured Hankel singular values can be used to select an approximation order. Finally, the two methods are applied to a model of a mechanical device. Copyright © 2008 John Wiley &amp; Sons, Ltd.