Linear systems with multiple singular values

Properties of linear dynamical systems are studied in the case of Hankel singular values of high multiplicity for monosingular and bisingular systems. Singular values of the Hankel operator take only one or two values for such systems. Properties of transfer functions are analysed. Algorithms of transfer function analysis and design are developed. Relations between frequency characteristics and Hankel singular values are established.     

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     

Hankel Singular Values and Vectors of a Class of Infinite-Dimensional Systems: Exact Hamiltonian Formulas for Control and Approximation Problems

This paper derives exact formulas for singular values and vectors of Hankel operators whose symbol is a product of a single-input single-output inner function and a multi-input multi-output rational function. This class of Hankel operators arises from the sensitivity minimization H ^∞ control problem with a rational weight function and the approximation problem of transfer functions having rational outer parts. It is shown that there is a Hamiltonian transcendental equation characterizing singular values which leads to a matrix function formula for singular vectors. Date received: May 12, 1998. Date revised: May 14, 1999.     

The all-pass property of optimal open-loop tracking systems

The structure of the optimal open-loop linear model-following system is investigated. It is shown that if the given plant is asymptotically stable but has zeros in the right half-plane, the stable optimal system contains an all-pass network whose transference possesses unity, singular values on the imaginary axis. In the special case of optimal tracking, it is shown that the resulting optimal transfer function matrix of the system is equal to the all-pass transfer function matrix which is normalized to be the identity matrix at the zero frequency.     

A linearization algorithm for optimizing control systems subject to singular value inequalities

The design of feedback linear multivariable systems subject to inequality constraints frequently leads to nonlinear programming problems involving singular values of matrix transfer functions. We present a readily implementable and globally convergent algorithm for solving such problems. It exploits the structure inherent in singular values differently than do other existing methods.     

On Hankel Singular Values and Reflected Zeros of Linear Dynamical Systems

This note discusses a relationship between the Hankel singular values and reflected zeros of linear systems. Our main result proves that the Hankel singular values of a linear continuous-time system increase (decrease) pointwise when one or more zeros of the transfer function are reflected with respect to the imaginary axis, that is, move from the left-(right-)half to the right-(left-)half of the complex plane. We also derive a similar result for linear discrete-time systems.      

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。     

Direct controllability and observability time domain conditions ofsingular systems

Controllability and observability criteria for singular systems using direct formulas in the time domain are introduced. The method used is based on a direct expansion of the transfer function matrix via a Leverrier-type algorithm. Hence, no decomposition of the state-space model is needed and the criteria are implementable on a digital computer. The compact form of the matrices might be helpful in other analysis as well as synthesis problems in singular systems     

函数S-粗集的随机特征及图像信息加密

在函数S-粗集的基础上,结合元素迁移的随机特征提出了函数S-随机粗集的概念,给出了图像生成的概念及图像加密定理和加密图像还原定理,并举例说明函数S-随机粗集在图像信息安全传递中的应用.     

Nondifferentiable optimization algorithm for designing control systems having singular value inequalities

It has been known for some time that many control system design requirements can be expressed as differentiable inequalities. More recently, it has been shown that important structural properties such as robustness and low noise sensitivity can be expressed as nondifferentiable inequalities involving the singular values of a system or return difference transfer function matrix. This paper presents an optimization algorithm which permits all these constraints to be considered.     

On the decay rate of Hankel singular values and related issues

This paper investigates the decay rate of the Hankel singular values of linear dynamical systems. This issue is of considerable interest in model reduction by means of balanced truncation, for instance, since the sum of the neglected singular values provides an upper bound for an appropriate norm of the approximation error. The decay rate involves a new set of invariants associated with a linear system, which are obtained by evaluating a modified transfer function at the poles of the system. These considerations are equivalent to studying the decay rate of the eigenvalues of the product of the solutions of two Lyapunov equations. The related problem of determining the decay rate of the eigenvalues of the solution to one Lyapunov equation will also be addressed. Very often these eigenvalues, like the Hankel singular values, are rapidly decaying. This fact has motivated the development of several algorithms for computing low-rank approximate solutions to Lyapunov equations. However, until now, conditions assuring rapid decay have not been well understood. Such conditions are derived here by relating the solution to a numerically low-rank Cauchy matrix determined by the poles of the system. Bounds explaining rapid decay rates are obtained under some mild conditions.     

Order estimation for linear time-invariant systems using frequencydomain identification methods

A two-step coarse-fine order estimation technique is proposed to determine the order of the numerator and the denominator polynomials of rational transfer function models for single-input/single-output (SISO) linear time-invariant systems. The coarse order estimation is based on rank detection by verification of the stochastic significance of the singular values of a linearized problem. The fine order estimation is based on a statistical analysis of the maximum likelihood cost function. The method is tested on measurements of low-(4 zeros, 6 poles) and high- (58 poles, 58 zeros) order systems     

A bisection method for computing the H∞ norm of a transfer matrix and related problems

We establish a correspondence between the singular values of a transfer matrix evaluated along the imaginary axis and the imaginary eigenvalues of a related Hamiltonian matrix. We give a simple linear algebraic proof, and also a more intuitive explanation based on a certain indefinite quadratic optimal control problem. This result yields a simple bisection algorithm to compute the H_∞ norm of a transfer matrix. The bisection method is far more efficient than algorithms which involve a search over frequencies, and the usual problems associated with such methods (such as determining how fine the search should be) do not arise. The method is readily extended to compute other quantities of system-theoretic interest, for instance, the minimum dissipation of a transfer matrix. A variation of the method can be used to solve the H_∞ Armijo line-search problem with no more computation than is required to compute a single H_∞ norm. Research supported in part by NSF under Grant ECS-85-52465, ONR under Grant N00014-86-K-0112, an IBM faculty development award, and Bell Communications Research.     

Design of robust <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript>-controller for energetic boiler plant

A problem of robust controller design for control of a boiler plant is solved using an H _∞ loop-shaping method with constraints imposed on regional pole placement of transfer functions of the closed-loop system, and linear matrix inequalities tools. Requirements to the closed-loop system are formulated both in the form of frequency constraints on singular values of transfer function of the open-loop system, and in the form of pole placement constraints for the transfer function of the closed-loop system as a given region on the complex plane. A reduction of the full-order H _∞-controller obtained in the design procedure is performed.     

Finite-time behavior of inner systems

In this paper, we investigate how nonminimum phase characteristics of a dynamical system affect its controllability and tracking properties. For the class of linear time-invariant dynamical systems, these characteristics are determined by transmission zeros of the inner factor of the system transfer function. The relation between nonminimum phase zeros and Hankel singular values of inner systems is studied and it is shown how the singular value structure of a suitably defined operator provides relevant insight about system invertibility and achievable tracking performance. The results are used to solve various tracking problems both on finite as well as on infinite time horizons. A typical receding horizon control scheme is considered and new conditions are derived to guarantee stabilizability of a receding horizon controller.     

面向矩阵秩函数准确估计的自表示子空间聚类方法

传统子空间聚类方法通常使用矩阵核范数代替矩阵秩函数进行低秩矩阵恢复,然而在目标优化过程中主要关注低秩矩阵大奇异值的影响,容易导致矩阵秩估计不准确的问题。为此,在分析矩阵奇异值长尾分布特点基础上,提出使用基于截断Schatten-p范数的低秩子空间聚类模型。该模型充分考虑小奇异值对低秩矩阵恢复过程的贡献,利用小奇异值信息拟合矩阵奇异值的长尾分布,通过对矩阵秩函数进行准确估计以提升子空间聚类性能。实验结果表明,与现有加权核范数子空间聚类WNNM-LRR和近邻约束子空间聚类BDR算法相比,在Extended Yale B数据集上的聚类准确性分别提升了11%和8%,所提方法能够更好地拟合数据奇异值分布以及生成准确的相似度矩阵。     

Optimal Hankel-norm approximation of stable systems with first-order stable weighting functions

Given a stable rational transfer function G(s) and weighting function W(s), the problem of finding of MacMillan degree k so as to minimise is considered. This problem is solved for W(s) =(s-β)/(s-α) with no assumptions on the signs of α and β. This gives rise to approximations where can be accurately bounded from above and below in terms of the Hankel singular values of WG (when α, β > 0).     

Properties of Zero-Free Spectral Matrices

In factor analysis, which is used for example in econometrics, by definition the number of latent variables has to exceed the number of factor variables. The associated transfer function matrix has more rows than columns, and when the factor variables are independent zero mean white noise sequences and the transfer function matrix is stable, then the output spectrum is singular. While a related paper focusses on the properties of such a nonsquare transfer function matrix, in this paper, we explore a number of properties of the spectral matrix and associated covariance sequence. In particular, a zero free minimum degree spectral factor can be computed with a finite number of rational calculations from the spectrum (in contrast to typical spectral factor calculations), assuming the spectrum fulfills a generic condition. Application of the result to Kalman filtering is indicated, and presentation of the results is also achieved using finite block Toeplitz matrices with entries obtained from the covariance of the latent variable vector.     

Generalized LQR control and Kalman filtering with relations to computations of inner-outer and spectral factorizations

We investigate the generalized linear quadratic regulator (LQR) control where the dimension of the control input is strictly greater than the dimension of the controlled output, and the weighting matrix on the control signal is singular. The dual problem is the generalized Kalman filtering where the dimension of the input noise process is strictly smaller than the dimension of the output measurement, and the covariance of the observation noise is singular. These two problems are intimately related to inner-outer factorizations for nonsquare stable transfer matrices with square inners of the smaller size. Such inner-outer factorizations are in turn related to spectral factorizations for power spectral density (PSD) matrices whose normal ranks are not full. We propose iterative algorithms and establish their convergence for inner-outer and spectral factorizations, which in turn solve the generalized LQR control and Kalman filtering.