A simple algorithm on minimal balanced realization for transferfunction matrices

Given a transfer function matrix, it is shown that its minimal balanced realization can be obtained directly from the singular values and the singular vectors of a constant matrix constructed from the coefficients of the least common denominator polynomial. The algorithm is based on the close relationship of controllability and the observability gramians to the singular value decomposition of the associated infinite block Hankel matrix     

奇异系统的一个实现方法

对于线性时不变奇异系统,给出传递矩阵构造它的最小有限维实现。所提算法的关键是将传递矩阵变成正常的,因而可以享用对正常系统已有实现结果的一些长处。     

广义系统的最小实现问题

本文利用传递函数对偶的概念,给出了非真有理分式阵的实现算法和最小实现算法,提出了广义系统的最小实现定理和最小实现之间的强等价定理。     

HANKEL矩阵的奇异值分析和系统结构及实现的数值算法

本文通过奇异分解分析了系统结构的定量性质和在数字计算机上计算结构参数及实现时所遇到的数值问题,并指出了结构定量性质和数值稳定性间的关系。本文给出了由HANKEL矩阵奇异值分解所得的几个典型最小实现,提出了一个借助奇异值分解由HANKEL矩阵直接计算结构参数的可靠的数值算法。     

An elementary derivation of Rosenbrock's minimal realization algorithm

Most minimal realization procedures utilize the Hankel matrix or properties of the controllability matrix. Recently a useful algorithm for obtaining minimal realizations has been developed by Rosenbrock using the system theory developed by him. This note shows that the algorithm can be simply developed using the properties of similarity transformation and matrix reduction procedures.     

广义快子系统的H∞最优模型降阶

In this paper, H∞ optimal model reduction for singular fast subsystems will be investigated. First, error system is established to measure the error magnitude between the original and reduced systems, and it is demonstrated that the new feature for model reduction of singular systems is to make H_ norm of the error system finite and minimal. The necessary and sufficient condition is derived for the existence of the H∞ suboptimal model reduction problem. Next, we give an exact and practicable algorithm to get the parameters of the reduced subsystems by applying the matrix theory. Meanwhile, the reduced system may be also impulsive. The advantages of the proposed algorithm are that it is more flexible in a straight-forward way without much extra computation, and the order of the reduced systems is as minimal as possible. Finally, one illustrative example is given to illustrate the effectiveness of the proposed model reduction approach.     

Principal component analysis in linear systems: Controllability, observability, and model reduction

Kalman's minimal realization theory involves geometric objects (controllable, unobservable subspaces) which are subject to structural instability. Specifically, arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces. This situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization. Structural instability associated with geometric theories is not unique to control; it arises in the theory of linear equations as well. In this setting, the computational problems have been studied for decades and excellent tools have been developed for coping with the situation. One of the main goals of this paper is to call attention to principal component analysis (Hotelling, 1933), and an algorithm (Golub and Reinsch, 1970) for computing the singular value decompositon of a matrix. Together they form a powerful tool for coping with structural instability in dynamic systems. As developed in this paper, principal component analysis is a technique for analyzing signals. (Singular value decomposition provides the computational machinery.) For this reason, Kalman's minimal realization theory is recast in terms of responses to injected signals. Application of the signal analysis to controllability and observability leads to a coordinate system in which the "internally balanced" model has special properties. For asymptotically stable systems, this yields working approximations ofX_{c}, X_{bar{o}}, the controllable and unobservable subspaces. It is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces.     

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     

Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms

An algorithm is presented in this paper for computing state-space balancing transformations directly from a state-space realization. The algorithm requires no "squaring up" or unnecessary matrix products. Various algorithmic aspects are discussed in detail. A key feature of the algorithm is the determination of a contragredient transformation through computing the singular value decomposition of a certain product of matrices without explicitly forming the product. Other contragredient transformation applications are also described. It is further shown that a similar approach may be taken, involving the generalized singular value decomposition, to the classical simultaneous diagonalization problem. These SVD-based simultaneous diagonalization algorithms provide a computational alternative to existing methods for solving certain classes of symmetric positive definite generalized eigenvalue problems.     

二维广义系统的实现方法

给出两种二维广义系统的实现方法，第一种方法是通过两次一维广义实现来完成一种特殊的广义Ｒｏｅｓｓｅｒ模型实现；第二种方法根据两个二维广义系统实现与它们的传递函数阵乘积的实现关系，把待实现的二维传函函数阵与写成容易实现的传递函数矩阵乘积的形式而得到原系统的实现。     

求多输入多输出有理传递函数的最小实现的一种算法

本文叙述了多输入多输出有理传递两数到状态方程的最小实现的方法,在所述的算法中用SVD分解求得最小实现的维数,再求解一系列相当于系数矩阵为上三角阵的线性方程组,得出最小实现的状态方程系数矩阵A、B、C。     

基于系统矩阵实Schur分解的集结法模型降阶

通过有序实Schur分解将系统矩阵变成分块对角阵,得到一种数值稳定的集结法模型降 阶,并给出降阶的L∞-误差界.降价系统保留了原系统的主导极点且为最小实现.     

求解非线性最小二乘问题的实用型方法

１．引言对于非线性最小二乘问题其中,为残差向量且,这里是指通常意义下的范数,即二范数．目标函数的梯度和Ｈｅｓｓｅ矩阵为其中　矩阵, 求解非线性最小二乘问题（１．１）的最基本方法是Ｇａｕｓｓ－Ｎｅｗｔｏｎ法,迭代格式为其中ｄｋ为线性方程组的解,这． 当人为满秩矩阵时,线性方程组（１．５）有唯一解,即并且有如下不等式：其中　是矩阵　的最小特征值．当　人接近奇异时,因此有可能存在着 ｄｋ,使得,即某一步迭代的步长太大,导致 Ｇａｕｓｓ－Ｎｅｗｔｏｎ法迭代失败． 另外,当　为奇异矩阵时,线性方程组（１．５）…     

A stochastic realization algorithm via block LQ decomposition in Hilbert space

A stochastic realization problem of a stationary stochastic process is re-visited, and a new stochastically balanced realization algorithm is derived in a Hilbert space generated by second-order stationary processes. The present algorithm computes a stochastically balanced realization by means of the singular value decomposition of a weighted block Hankel matrix derived by a “block LQ decomposition”. Extension to a stochastic subspace identification method explains how the proposed abstract algorithm is implemented in system identification.     

On the matrix feedback shift register synthesis for matrix sequences矩阵序列的矩阵反馈移位寄存器综合问题

In this paper, a generalization of the linear feedback shift register synthesis problem is presented for synthesizing minimum-length matrix feedback shift registers (MFSRs for short) to generate prescribed matrix sequences and so a new complexity measure, that is, matrix complexity, is introduced. This problem is closely related to the minimal partial realization in linear systems and so can be solved through any minimal partial realization algorithm. All minimum-length MFSRs capable of generating a given matrix sequence with finite length are characterized and a necessary and sufficient condition for the uniqueness issue is obtained. Furthermore, the asymptotic behavior of the matrix complexity profile of random vector sequences is determined.     

Minimal realization of transfer function matrices via oneorthogonal transformation

The minimal realization of a given arbitrary transfer function matrix G(s) is obtained by applying one orthogonal similarity transformation to the controllable realization of G( s). The similarity transformation is derived by computing the QR or the singular value decomposition of a matrix constructed from the coefficients of G(s). It is emphasized that the procedure has not been proved to be numerically stable. Moreover, the matrix to be decomposed is larger than the matrices factorized during the step-by-step procedures given     

New results in realization theory for linear time-varying analytic systems

The realization of linear time-varying systems specified by an analytic weighting pattern is approached in a novel manner using an algebraic framework defined over the ring of analytic functions. Realizations are given by a state representation consisting of a first-order vector differential equation and an output equation, both with analytic coefficients. Various new criteria for realizability are derived, including conditions given in terms of the finiteness of modules over the ring of analytic functions generated by the elementary rows or columns of a (generalized) Hankel matrix. These results are related to local criteria for realizability specified in terms of the rank of matrix functions, as developed in the work of Silverman and Meadows [5], [8], [9] and Kalman [7]. It is shown that the construction of minimal realizations reduces to the problem of computing a basis for a finite free module defined over the ring of analytic functions. A minimal realization algorithm is then derived using a constructive procedure for computing bases for finite free modules over a Bezout domain. The Silverman-Meadows realization algorithm [5] is a special case of the procedure given here. In the last part of the paper, the realization algorithm is applied to the problem of system reduction.     

Robust admissibility and admissibilisation of uncertain discrete singular time-delay systems

This paper is concerned with the characterisation of robust admissibility and admissibilisation for uncertain discrete-time singular system with interval time-varying delay. Considering the norm-bounded uncertainty and the interval time-varying delay, a new comparison model is introduced to transform the original singular system into two connected subsystems. After this transformation, a singular system without uncertainty and delay can be handled by the Lyapunov–Krasovskii functional method. By virtue of the scaled small gain theorem, an admissibility condition of the original singular system is proposed in terms of linear matrix inequalities. Moreover, the problem of robust admissibilisation of uncertain discrete singular time-varying system is also studied by iterative linear matrix inequality algorithm with initial condition optimisation. Several numerical examples are used to illustrate that the results are less conservative than existing ones.