A new upper bound for the skewed structured singular value

The structured singular value (s.s.v)μ enables the study of robust stability and performance of a controller in the presence of real parametric uncertainties and complex uncertainties corresponding to neglected dynamics. In spite of the NP-hard characteristic of the problem, it is now possible to compute an interval for the s.s.v. μ using polynomial-time algorithms. The skewed s.s.v. ν was introduced by Fan and Tits in the context of robust performance analysis. The primary aim of this paper is to propose a new mixed ν upper bound, which is applicable to problems with a special, but practically important, structure. We then illustrate through a realistic missile example that certain problems naturally require the ν tool rather than the μ tool. Copyright © 1999 John Wiley & Sons, Ltd.     

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)     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Absolute stabilization of singular systems with ferromagnetic hysteresis nonlinearity

This paper is concerned with the absolute stabilization problem of a class of singular systems with feedback connected ferromagnetic hysteresis nonlinearities. Firstly, a novel differential-integral loop transformation framework is developed to achieve an augmented singular system model. Secondly, by constructing a new passive output derivative operator of hysteresis nonlinearity and establishing the bound condition of the solution of ferromagnetic hysteresis model, the equivalent absolute stability criterion of singular systems with hysteresis feedback is derived based on KYP method and LMIs technique. Furthermore, the strict LMIs conditions for absolute stabilization are obtained, which can easily be checked by the LMI toolbox in MATLAB. Finally, two examples are given to illustrate the effectiveness of the proposed method.     

由回差阵奇异值求稳定裕度的退化算法

本文研宄使用回差阵奇异值求多输入多输出(MIMO)线性定常系统稳定裕度的方法.首先对此方法进行退化分析,得到求解单输入单输出(SISO)线性定常系统稳定裕度的算法步骤.在此基础上,讨论退化所得算法与传统稳定裕度的关系;进一步地,详细分析此退化算法相比传统稳定裕度的优势,进而指出当系统的增益和相位同时变化时,系统Nyquist曲线g(jω)到临界点(-1,j0)的最短距离min|1+g(jω)|可作为一种更加合理的稳定裕度指标.最后,通过对实例进行数值仿真,说明本文所提退化算法可以克服传统稳定裕度局限性,同时与传统稳定裕度结合得到比较完整的SISO线性系统稳定裕度衡量体系.     

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…     

基于奇异值分解的Contourlet域稳健性数字水印算法*

提出了一种基于奇异值分解的Contourlet域数字水印算法。对置乱后的水印图像进行奇异值分解,在Contourlet域中选取合适的方向子带,利用得到的奇异值来调制系数矩阵,然后通过逆变换获取嵌入水印后的图像。在水印提取中只需要保存原始图像的奇异值,实现了水印的近似盲提取。最后,进行了一系列的攻击实验,证明了与传统小波域水印算法相比,该算法的不可见性和鲁棒性都有了较大的提高。     

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.     

Lower bounds for the smallest singular value

Let A = (a_ij) be an n × n complex matrix. Suppose that G(A), the undirected graph of A, has no isolated vertex. Let E be the set of edges of G(A). We prove that the smallest singular value of A, σ_n, satisfies: σ_n ≥ min σ_ij | (i, j) ∈ E, where g_ij ≡ a_i + a_j − [(a_i − a_j)² + (r_i + c_i)(r_j + c_j)]^1/2/2 with a_i ≡ |a_ii| and r_i,c_i are the i^th deleted absolute row sum and column sum of A, respectively. The result simplifies and improves that of Johnson and Szulc: σ_n ≥ min_i≠j σ_ij. (See [1].)     

An efficient method for solving nonlinear singular initial value problems

In [M.M. Hosseini, Modified Adomain decomposition method for specific second order ordinary differential equations, Appl. Math. Comput. 186 (2007), pp. 117–123] an efficient modification of Adomian decomposition method has been proposed for solving some cases of ordinary differential equations. In this paper, this method is generalized to more cases. The proposed method can be applied to linear, nonlinear, singular and nonsingular problems. Here, it is focused on nonlinear singular initial value problems of ordinary differential equations. The scheme is tested for some examples and the obtained results demonstrate reliability and efficiency of the proposed method.     

双正交提升小波和奇异值分解的彩色水印算法研究

在现有彩色图像水印提取技术基础上, 提出一种结合9/7双正交提升小波和图像矩阵奇异值分解的彩色图像水印加密算法。首先将彩色原始图像和彩色水印图像R、G、B分离, 将图像相应分量进行提升小波三级分解得到低频部分, 并将该低频信息进行奇异值分解, 把水印信息视做扰动矩阵按一定的量加在低频信息的对角阵上, 其中彩色水印图像预处理结合了骑士巡游变换和Arnold置乱算法。实验结果表明算法在确保水印不可见性的基础上有效地增强了水印的强度和鲁棒性, 实现了水印图像的盲检验。