Model reduction of bilinear systems described by input–output difference equation

A class of single input single output bilinear systems described by their input–output difference equation is considered. A simple expression for the Volterra kernels of the system is derived in terms of the coefficients of difference equation. An algorithm, based on the singular value decomposition of a generalized Hankel matrix, is also developed. The algorithm is then used to find a reduced-order bilinear state-space model. The Hankel approach will be extensively studied under different data length cases and different orders of the state-space models. A numerical example is presented to illustrate the effectiveness of the proposed algorithm.     

An Algorithm for the Generalized Singular Value Decomposition on Massively Parallel Computers

A new algorithm for computing the generalized singular value decomposition that diagonalizes two matrices is introduced. First, two existing algorithms are studied, one due to Paige and the other to Han and Veselic. The former requires only orthogonal plane transformations, resulting in two matrices with parallel rows. The latter employs nonsingular (not necessarily orthogonal) transformations to directly diagonalize the two data matrices. For many applications, it is preferable to be given both the diagonal matrices and the transformation matrices explicitly. Our algorithm has the advantages that the nonsingular transformation is readily available, computation of transformations is simple, and the convergence test is efficient in a parallel computing environment. We present implementation results, obtained on a Connection Machine 2, to compare our new algorithm with that of Paige.     

基于规范变量分析的子空间辨识方法及应用

现代控制理论的研究如滤波、预测、控制等大多建立在系统的状态空间模型形式上,而传统辨识方法需要预先参数化,并且计算比较复杂。本文深入研究和推广一种基于规范变量分析（CVA）的子空间辨识方法,并引入一种新的Akaike信息判据来获得系统阶次,求解过程简单,结果精确。由于此法直接由数据确定系统状态,避免了预先参数化;在计算上,主要依赖于奇异值分解（SVD）,也不会遇到与传统方法有关的数值困难。仿真研究结果表明,引入新的系统阶次求解方法后能很精确的辨识出系统模型,具有十分广阔的应用前景。     

A PMF-based subspace method for continuous-time model identification. Application to a multivariable winding process

This paper presents a methodology for system identification of continuous-time state-space models from finite sampled input-output signals. The estimation problem of the consecutive time-derivatives and integrals of the input-output signals is considered. The appropriate frequency characteristcs of a linear filtering based on the Poisson moment functionals in regards to the derivative or integral estimation problem is shown. The proposed method combines therefore the Poisson moment functionals technique with subspace based state-space system identification methods. The developed algorithm is based on a generalized singular value decomposition to compensate the noise colouring caused by the linear prefiltering of the input-output data. Rules of thumb are presented to choose the design parameters and new regards to the selection of the Poisson filter cut-off frequency are introduced. Finally, the proposed method is applied to a multivariable winding processes. The experimental results emphasize the applicability of the developed methodology.     

An improved algorithm for the computation of structural invariants of a system pencil and related geometric aspects

In this paper we propose a new recursive algorithm for computing the staircase form of a matrix pencil, and implicitly its Kronecker structure. The algorithm compares favorably to existing ones in terms of elegance, versatility, and complexity. In particular, the algorithm without any modification yields the structural invariants associated with a generalized state-space system and its system pencil. Two related geometric aspects are also discussed: we show that an appropriate choice of a set of nested spaces related to the pencil leads directly to the staircase form; we extend the notion of deflating subspace to the singular pencil case.     

An efficient algorithm for the tensor product model transformation

The tensor-product (TP) model transformation was proposed recently as a numerical and automatically executable method which is capable of transforming linear parameter varying (LPV) state-space models into the higher order singular value decomposition (HOSVD) based canonical form of polytopic models. The crucial disadvantage of the TP model transformation is that its computational load explodes with the density of discretization and the dimensionality of the parameter vector of the parameter-varying state-space model. In this paper we propose a new algorithm that leads to considerable reduction of the computation in the TP model transformation. The main idea behind the modified algorithm is to minimize the number of discretized points to acquire as much information as possible. The modified TP model transformation can readily be executed on a regular computer efficiently and concisely, especially in higher dimensional cases when the original TP model transformation fails. The paper also presents numerical examples to show the effectiveness of the new algorithm.     

Computation of transfer-function matrices of generalized state-space models

A reliable computational algorithm is proposed for the evaluation of the transfer-function matrices of linear multivariable systems described by generalized state-Space models. The algorithm is based on a new, efficient and numerically stable procedure for computing irreducible state-space realizations of generalized state-space models. For each input-output channel, the proposed algorithm computes an irreducible realization using this procedure. Then, the parameters of the corresponding transfer function (gain, poles, zeros) are determined from the resulting irreducible model.     

Chained aggregation of singular systems

Extends the notion of aggregation to singular systems (sE-A). Necessary and sufficient conditions for complete aggregability are given which reduce in the state-space case, E=I, to Aoki's conditions. A geometric quotient-space analysis is given which uses a new result on the pencil induced in the quotient space by a regular matrix pencil. A generalized chained aggregation algorithm is used to put the system into a Hessenberg form which provides a reduced-order model for the output dynamics. It is shown that, as in the state-space situation, chained aggregation corresponds to removal of the unobservable subspace. It is also pointed out that the usual state-space chained aggregation algorithm is very efficient and numerically unstable. Specializing the singular system chained aggregation algorithm to the state-space case yields a better and more streamlined approach     

The generalized center problem of resonant infinity for a polynomial differential system

This paper explores the problems of generalized center conditions and integrability of resonant infinity for a complex polynomial differential system. The method is based on converting resonant infinity into an elementary singular point by a homeomorphic transformation. The calculation of generalized singular point quantities is an effective way of finding necessary conditions for integrable systems. A new recursive algorithm for computing generalized singular point quantities at the origin of the transformed system is derived. At the same time, a necessary and sufficient condition for resonant infinity to be a generalized complex center is presented. As an application of the new recursive algorithm, the generalized center conditions for resonant infinity for a class of cubic systems are discussed. To the best of our knowledge, this is the first time that the generalized center problem has been considered for p:−q resonant infinity.     

核不相关鉴别分析以及它在字符识别中的应用

核不相关鉴别分析是在线性不相关鉴别分析的基础上发展起来的．然而,由于核函数的运用,计算核不相关矢量集变得更加复杂．为了解决这个问题,提出一种解决核不相关鉴别分析的有效算法．该算法巧妙地利用了矩阵的分解,然后在一个矩阵对上进行广义奇异值分解．与此同时,提出了几个相关的定理．最重要的是,提出的算法能克服核不相关鉴别分析中矩阵的奇异问题．在某种意义上,提出的算法拓宽了已有的算法,即从线性问题到非线性问题．最后,用手写数字字符识别实验来验证提出的算法是可行和有效的．     

协同过滤系统的矩阵稀疏性问题的研究

应用奇异值算法得到一个无缺失的矩阵,引进了一种增强的、基于参数的Pearson相关系统算法来提高相关性算法的准确性。提出一个基于奇异值分解和增强Pearson系数的“HybridSVD”算法,用MovieLens数据集来评价该算法,并和其他经典的传统算法做了比较。实验结果证明,“HybridSVD”算法比其他传统算法能更好地处理协同过滤中的稀疏性问题。     

Reliable state-space upper bounds for the peak structured singular value

A state-space method for computing upper bounds for the peak of the structured singular value over frequency for both real and complex uncertainties is presented. These bounds are based on the positivity and Popov criteria for one-sided, sector-bounded and for norm-bounded, block-structured linear uncertainty. These criteria are restated and used to derive upper bounds for the peak structured singular value by equating the feasibility of a linear matrix inequality which involves a plant state-space realization to the strict positive realness of a transfer function. Numerical examples are given to illustrate these upper bounds. © 1998 John Wiley & Sons, Ltd.     

彩色图像四元数频域奇异值分解水印算法

将四元数傅里叶变换与四元数奇异值分解技术相结合并引入到对彩色图像的水印处理,提出一种基于四元数频域奇异值分解的彩色图像盲水印算法.首先对彩色载体图像进行分块并采用四元数傅里叶变换(quaternion Fourier transform,QFT)得到其频域矩阵,然后对频域矩阵中的单位小块进行四元数奇异值分解(quaternion singular value decomposition,QSVD),得到实系数奇异值,使用奇偶量化调制法将水印信号嵌入到单位小块的最大奇异值中.仿真实验结果表明,嵌入的水印分布在空域图像各彩色分量中,在不可见性以及鲁棒性的比较中优于传统的彩色图像亮度域以及独立多通道处理方法.     

Truncated balanced realization of a stable non-minimal state-space system

In this paper we present a numerically reliable algorithm to compute the balanced realization of a stable state-space system that may be arbitrarily close to being unobservable and/or uncontrollable. The resulting realization, which is known to be a good approximation of the original system, must be minimal and therefore may contain a reduced number of states. Depending on the choice of partitioning of the Hankel singular values, this algorithm can be used either as a form of minimal realization or of model reduction. This illustrates that in finite precision arithmetic these two procedures are closely related. In addition to real matrix multiplication, the algorithm only requires the solution of two Lyapunov equations and one singular value decomposition of an upper-triangular matrix.     

A parallel method for computing the generalized singular value decomposition

We describe a new parallel algorithm for computing the generalized singular value decomposition of two n × n matrices, one of which is nonsingular. Our procedure requires O(n) time and one triangular array of O(n²) processors.     

Subspace based system identification with periodic excitation signals

An identification algorithm for use with data generated by periodic inputs is presented. The algorithm is based on the geometrical properties of the resulting periodic output signal and a state-space model is derived from the signal subspace of a Hankel matrix by means of a singular value decomposition. It is shown that 2n + 1 noise-free output measurements are required to identify an nth order system. The algorithm is demonstrated to be consistent when the output measurements are corrupted by zero mean noise characterized by decaying covariances. The computational complexity of the algorithm is several orders of magnitude lower than standard methods.     

基于多分辨率奇异值分解的图像融合

提出一种新的基于多分辨率奇异值分解（ MSVD）图像融合算法。算法对源图像进行MSVD处理,使其分解为互不相关的平滑和细节分量,并对平滑分量进行多层次的分解与处理。类似于小波变换,多分辨率奇异值分解的基本思想是在平滑分量的每一层上用奇异值分解（ SVD）来取代滤波,最终利用融合规则对图像进行MSVD融合。利用5种评价算子来评价算法,得到的融合效果很好。与基于小波分解的算法相比,算法计算简单、实时性突出,对复杂、高像素图像处理更简单方便。     

A method for computing the partial singular value decomposition

A method for computing the partial singular value decomposition of a matrix is described. The method is appropriate to problems where the matrix is known to be of low rank and only the principal singular vectors are of interest. The technique is simple, easy to implement in integer arithmetic, and places modest memory requirements. The convergence properties of the algorithm are investigated analytically and by simulation.     

Minimum phase robustness for uncertain state-space systems

The minimum phase robustness of an uncertain state-space system with affine parametric uncertainties in the state-space matrices is studied. A tolerable margin in terms of the structured singular value is given for uncertain parameters to guarantee the minimum phase property of the system. Based on the linear fractional transformation methodology, the matrix sizes involved in the computation of structured singular value are reduced significantly to improve computational burden. The approach can be applied to the proper or strictly proper linear uncertain systems.     

Model order reduction via real Schur-form decomposition

A new algorithm is presented for computing Moore's reduced-order transfer-function matrix without calculating the balancing transformation, which tends to be ill-conditioned, especially when the original system is non-minimal or when it has very nearly uncontrollable or unobservable modes. The algorithm is based on finding the eigenspaces associated with large eigenvalues of the cross-gramian matrix W_co using the real Schur-form decomposition. The algorithm does not require a minimal model to start with. The state-space realization obtained by this method is related to the balanced realization by a non-singular matrix. An example is presented to illustrate the proposed algorithm