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.     

Computation of the real structured singular value via pole migration

The paper introduces a new computationally efficient algorithm to determine a lower bound on the real structured singular value μ. The algorithm is based on a pole migration approach where an optimization solver is used to compute a lower bound on real μ independent of a frequency sweep. A distinguishing feature of this algorithm from other frequency independent one‐shot tests is that multiple localized optima (if they exist) are identified and returned from the search. This is achieved by using a number of alternative methods to generate different initial conditions from which the optimization solver can initiate its search from. The pole migration algorithm presented has also been extended to determine lower bounds for complex parametric uncertainties as well as full complex blocks. However, the results presented are for strictly real and repeated parametric uncertainty problems as this class of problem is the focus of this paper and are in general the most difficult to solve. Copyright © 2014 John Wiley & Sons, Ltd.     

A numerical method for computing the structured singular value

This article suggests a new approach to computing Doyle's structured singular value (SSV) of a matrix. The SSV is a notion important in robust control and several iteration schemes exist for approximation a solution [1,2,4,-8,10,11].Our idea is to pick a special case of the general problem, which we believe to be natural and give an algorithm for studying it based on ‘off the shelf’ packages. Once we are committed to this ‘special case’ we discuss a very general plant uncertainty problem; it embraces real as well as complex plant perturbations of many kinds. The idea is simple and we believe very natural to the problem.     

Off-line robust fault diagnosis using the generalized structured singular value

This paper presents a new approach to the problem of off-line model-based fault diagnosis for multivariable uncertain systems. The proposed method uses the generalized structured singular value and is based on frequency-domain model invalidation tools. A novel step-by-step methodology for off-line fault identification and symptom-aided diagnostic is developed. Fault detectability and isolability issues are discussed within the new framework. Experimental results obtained by using real data from a three-tank system are reported, showing the potential of the proposed techniques.     

An upper bound of structured singular value

An improved upper bound of structured singular value for mixed uncertainties with purely real uncertainty blocks is proposed.     

Real structured singular value conditions for the strong D-stability

Recently, Chen et al. (Systems Control Lett. 24 (1995) 19) proposed conditions for D-stability and strong D-stability in terms of structured singular values. In this paper, simpler conditions for the strong D-stability are derived.     

Bounds on generalized structured singular values via the Perron root of matrix majorants

The purpose of this paper is to present several bounds upon the structured singular value. We first adopt a generalized notion of the structured singular value which is useful for problems where uncertainties are assumed to be bounded in an l_p-induced matrix norm. Two different type of bounds, in terms of Perron root and interaction parameters respectively, are given for the new structured singular value and their relations are discussed. These bounds are useful in that they are easy to compute and may be further analyzed to provide insights useful in design.     

Structured singular value of a repeated complex full-block uncertainty

The structured singular value (SSV), or $\mu$, is used to assess the robust stability and performance of an uncertain linear time-invariant system. Existing algorithms compute upper and lower bounds on the SSV for structured uncertainties that contain repeated (real or complex) scalars and/or nonrepeated complex full-blocks. This paper presents algorithms to compute bounds on the SSV for the case of repeated complex full-blocks. This specific class of uncertainty is relevant for the input-output analysis of many convective systems, such as fluid flows. Specifically, we present a power iteration to compute the SSV lower bound for the case of repeated complex full-blocks. This generalizes existing power iterations for repeated complex scalars and nonrepeated complex full-blocks. The upper bound can be formulated as a semi-definite program (SDP), which we solve using a standard interior-point method to compute optimal scaling matrices associated with the repeated full-blocks. Our implementation of the method only requires gradient information, which improves the computational efficiency of the method. Finally, we test our proposed algorithms on an example model of incompressible fluid flow. The proposed methods provide less conservative bounds as compared to prior results, which ignore the repeated full-block structure.     

基于截断奇异值低秩矩阵恢复的Canny边缘检测算法

针对Canny算法在处理噪声图像时存在的不足,为提高其准确性和鲁棒性,提出一种基于截断奇异值的低秩矩阵恢复方法,以及一种更加准确的双噪声凸优化模型和求解方法。使用经典Canny边缘检测算法作用于分解后去除冗余信息的主成分上,将图像的边缘检测转化为对主成分的边缘检测,可以在有效地去除脉冲噪声和高斯噪声干扰的同时,更好地保留边缘信息。为验证其有效性,在不同噪声浓度以及混合噪声情况下进行实验,结果分析表明,基于低秩矩阵恢复的边缘检测算法可以更好地保留完整的边缘信息,提高边缘检测的准确性及鲁棒性。     

关于结构奇异值的一个注记

１引言Ｄｏｙｌｅ在１９８２年提出的结构奇异值（μ）方法是分析和综合结构式不确定系统的有力工具［１,２］．基于结构奇异值分析的小μ定理［２］给出了具有多个摄动块的线性动态系统鲁棒稳定的充要条件．而鲁棒性能定理［２］则进一步地将鲁棒稳定性问题和鲁棒性能问题统一成μ分析问题．然而．我们注意到,在所有研究结构奇异值的文献中,均要求块对角摄动矩阵中每个子摄动块是方的．这一要求无疑大大限制了μ方法的应用,因为非方摄动块在系统中是经常存在的．此时对Ｄｏｙｌｅ给出的结构奇异值的上界函数［１］必须进行修正．２非方…     

Polynomial methods for the structured singular value with real parameters

We consider the structured singular value problem with real parametric uncertainty only. Using techniques from algebraic geometry, we propose two algorithms that in principle can yield the precise value of the structured singular value at a fixed frequency. Their ability to do so depends upon their ability to find all common roots to a system of polynomial equations. The first algorithm is applicable to problems with two real parameters each of multiplicity two. The second algorithm is applicable to problems with n distinct real parameters. These algorithms have proved useful in applications to aerospace control law analysis.     

A measure of worst-case H∞ performance and of largest acceptable uncertainty

The structured singular value (SSV or μ) is known to be an effective tool for assessing robust performance of linear time-invariant models subject to structured uncertainty. Yet all single μ analysis provides is a bound β on the uncertainty under which stability as well as H_∞ performance level of κ/β are guaranteed, where κ is preselectable. In this paper, we introduce a related quantity ν which provides answers for the following questions: (i) given β, determine the smallest with the property that, for any uncertainty bounded by β, an H∞ performance level of is guaranteed; (ii) conversely, given , determined the largest β with the property that, again, for any uncertainty bounded by β, an H_∞ performance level of is guaranteed. Properties of this quantity are established and approaches to its computation are investigated. Both unstructured uncertainty and structured uncertainty are considered.     

A study of the gap between the structured singular value and itsconvex upper bound for low-rank matrices

The size of the smallest structured destabilizing perturbation for a linear time-invariant system can be calculated via the structured singular value (μ). The function μ can be bounded above by the solution of a convex optimization problem, and in general there is a gap between μ and the convex bound. This paper gives an alternative characterization of μ which is used to study this gap for low-rank matrices. The low-rank characterization provides an easily computed bound which can potentially be significantly better than the standard convex bound. This is used to find new examples with larger gaps than previously known     

An implicit small gain condition and an upper bound for the real structured singular value

In this paper we develop an upper bound for the real structured singular value that has the form of an implicit small gain theorem. The implicit small gain condition involves a shifted plant whose dynamics depend upon the uncertainty set bound and, unlike prior bounds, does not require frequency-dependent scales or multipliers. Numerical results show that the implicit small gain bound compares favorably with real-μ bounds that involve frequency-dependent scales and multipliers.     

最大奇异值移位的鲁棒图像信息隐藏

为了提高图像信息隐藏的鲁棒性和不可感知性,提出了一种基于图像块最大奇异值移位的鲁棒信息隐藏算法.算法首先对图像分块并对每个图像块进行奇异值分解,根据图像块最大奇异值的大小选择合适的阈值,并对图像块的最大奇异值进行区间划分,通过将最大奇异值移位到与秘密信息比特相对应的区间实现秘密信息的嵌入.实验结果表明,该算法具有较大的嵌入容量,且在嵌入相同大小的秘密信息时与其他同类算法相比,具有更好的图像视觉质量和鲁棒性.该算法更能适应噪声环境下的信息隐藏.     

Improved upper bounds for the mixed structured singular value

In this paper, we take a new look at the mixed structured singular value problem, a problem of finding important applications in robust stability analysis. Several new upper bounds are proposed using a very simple approach which we call the multiplier approach. These new bounds are convex and computable by using linear matrix inequality (LMI) techniques. We show, most importantly, that these upper bounds are actually lower bounds of a well-known upper bound which involves the so-called D-scaling (for complex perturbations) and G-scaling (for real perturbations)     

A detailed comparative analysis of all practical algorithms to compute lower bounds on the structured singular value

μ analysis is one of the most efficient techniques to evaluate the stability margins and the performance levels of linear time-invariant systems in the presence of structured time-invariant uncertainties. The exact computation of the structured singular value μ is known to be NP hard in the general case, but several methods have been developed in the last 30 years to compute accurate and reliable bounds. In this paper, all existing μ lower bound algorithms are reviewed and the most relevant ones are evaluated on a wide set of real-world benchmarks, corresponding to various fields of application, system dimensions and structures of the uncertainties. The results are thoroughly analyzed and simple improvements to the existing algorithms are proposed to approach the exact value of μ with a reasonable computation cost. Conclusions show that non-conservative values can be obtained in almost all cases. A brief extension to skew-μ analysis confirms the good results obtained in the classical μ case.     

Robustness analysis of flexible structures: practical algorithms

When analysing the robustness properties of a flexible system, the classical solution, which consists of computing lower and upper bounds of the structured singular value (s.s.v.) at each point of a frequency gridding, appears unreliable. This paper describes two algorithms, based on the same technical result: the first one directly computes an upper bound of the maximal s.s.v. over a frequency interval, while the second one eliminates frequency intervals, inside which the s.s.v. is guaranteed to be below a given value. Various strategies are then proposed, which combine these two techniques, and also integrate methods for computing a lower bound of the s.s.v. The computational efficiency of the scheme is illustrated on a real‐world application, namely a telescope mock‐up which is significant of a high order flexible system. Copyright © 2003 John Wiley & Sons, Ltd.