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.     

Legacy recovery and robust augmentation structured design for the VEGA launcher

This paper presents a systematic robust control framework based on the structured approach to address the synthesis of the atmospheric ascent‐flight control system of a launch vehicle. To introduce this synthesis framework, the control problem is first formulated to recover the classically designed baseline rigid‐body controller of the actual VEGA launcher VV05 mission. This legacy recovery builds the necessary background for a good understanding of the problem and increases confidence for its transfer to the Space industry. Subsequently, it is shown how to systematically augment the robustness of the design from the synthesis stage against wind turbulence perturbations and parametric uncertainty. The resulting controller is verified via classical stability margins and robust structured singular value analyses and finally validated using nonlinear, time‐domain simulations in a Monte Carlo campaign. It is highlighted that this robust synthesis framework allows to obtain a controller with improved robust stability and global performance, and more importantly, it provides a more systematic methodology for design.     

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.     

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.     

A non‐smooth lower bound on ν

A non‐smooth optimization technique to directly compute a lower bound on the skew structured singular value ν is developed. As corroborated by several real‐world challenging applications, the proposed technique can provide tighter lower bounds when compared with currently available techniques. Moreover, in many cases, the determined lower bound equals the true value of ν. Thanks to the efficiency of the non‐smooth technique, the algorithm can be applied to problems involving even a significant number of uncertain parameters. Another appealing feature of the proposed non‐smooth approach is that the dimension of repeated scalar uncertainties in the overall structured uncertainty matrix has little impact on the computational time. The technique can be used to compute a lower bound on the structured singular value μ as well. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

State‐space µ analysis for an experimental drive‐by‐wire vehicle

This paper considers the application of the skewed structured singular value to the robust stability of systems subject to strictly real parametric uncertainty. Three state‐space formulations that counteract the discontinuous nature of this problem are detailed. It is shown that the calculation of the supremum of the structured singular value over a frequency range using these formulations transforms into a single skewed structured singular value calculation. Similar to the structured singular value, the exact calculation of the skewed structured singular value is an NP‐hard problem. In this work, two efficient algorithms that determine upper and lower bounds on the skewed structured singular value are presented. These algorithms are critically assessed using a series of robustness analysis tests on a safety‐critical experimental drive‐by‐wire vehicle. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

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.     

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.     

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.     

Development of a skew μ upper bound

Exploitation of the NP hard, mixed μ problem structure provides a polynomial time algorithm that approximates μ with usually reasonable answers. When the problem is extended to the skew μ problem, an extension of the existing method to the skew μ formulation is required. The focus of this paper is to extend the μ upper bound derivation to the skew μ upper bound and show its direct computation by way of an Osborne balancing algorithm and an linear matrix inequality (LMI) algorithm. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

On the synthesis of robust PID controllers for plants with structured and unstructured uncertainty

In this paper, we consider the problems of synthesizing PID controllers for robust stability and performance for a given linear time‐invariant plant subject to both parametric and H_∞‐norm‐bounded perturbations. Using results from the area of parametric robust control, synthesis problems are converted into simultaneous stabilization of a family of complex segment polynomials. The results on H_∞ PID synthesis are then used to devise a design procedure for determining the admissible PID gain values. One of the important features of the proposed method is that it constructively characterizes the approximated set of all admissible PID controllers. This characterization can facilitate the optimal design of any additional design requirements. Copyright © 2005 John Wiley & Sons, Ltd.     

正规和奇异H∞控制器的统一表达式

给出了一种新的H∞控制器设计方法,通过引入设计时可选的非奇异实数阵,取消了控制器设计时对D矩阵的秩限制.适用于正规的H∞控制问题和奇异的H∞控制问题.对状态反馈等四种典型问题和输出反馈控制问题,给出了控制器存在的充分必要条件.控制器通过Riccati方程的解,用参数化方法表示.输出反馈控制器,通过解两个Riccati方程得到.讨论了控制器的相关特性.     

Synthesis and analysis of robust control compensators for Space descent & landing

In this article, a complete modelling, synthesis, and analysis methodology of control compensators for descent and landing (D&L) on small planetary bodies is presented. These missions are scientifically very rewarding but technically extremely challenging due to the complex and poorly known environment around those bodies, which calls for the ability to manage competing robustness and performance requirements. While this issue is typically addressed via the redefinition of D&L guidance strategies, here, it is tackled through the augmentation with a simple yet robust control compensator. This compensator is designed using linear fractional transformation modelling to capture the interplay with uncertain gravity fields and the recently developed structured optimisation framework, which has been proved particularly suitable for industry‐oriented applications. The proposed approach is completely generic but uses the scenario of a landing on the Martian moon Phobos as an illustrative example. Different compensators are then verified and compared analytically via the structured singular value μ and through high‐fidelity Monte Carlo simulation.     

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。     

Stability radii of positive discrete-time systems under affine parameter perturbations

In this paper we study stability radii of positive linear discrete-time systems under affine parameter perturbations. It is shown that real and complex stability radii of positive systems coincide for arbitrary perturbation structures, in particular, for blockdiagonal disturbances as considered in μ-analysis. Estimates and computable formulae are derived for these stability radii. The results are derived for arbitrary perturbation norms induced by monotonic vector norms (e.g. p-norms, 1⩽p⩽∞). © 1998 John Wiley & Sons, Ltd.