Applications of optimization methods to robust stability of linear systems

We study the robust stability problem for a family of polynomials. We allow for all the coefficients of the polynomials to be affinely perturbed, where the size of the perturbation is measured by an arbitrary convex function. We apply optimization techniques, and in particular convex duality methods, to derive simple formulas for the stability radius, to find a minimal perturbation which destroys stability, and to obtain necessary and sufficient conditions for robust stability. Our framework is general enough to cover many applications. As special cases, we obtain many results recently reported in the literature.The work of the first author was partially supported by AFOSR Grant 91-008 and NSF Grant DMS-92-01297.     

Index to Volume 1 (1992)

Abstract

Since its introduction, APL has frequently been touted as an ideal programming language for statistical applications. Among the attractive features of APL for statistics are its extensibility, the presence of primitives for operations such as sorting, matrix inversion, and arranging data, and powerful facilities for handling matrices and other arrays. Newer programming languages incorporating some of the features of APL—notably New S and XLisp-Stat—have been designed specifically for statistical applications. Is there still a niche for APL?

We believe that APL2 continues to offer some significant benefits for statistical computation, including user-defined operators, nested arrays, and convenient implementation of arrays of any dimension. We use these characteristics of the language in the course of designing an extensible computing environment for data analysis and programming based on APL2 that incorporates some of the features of modern statistical programming languages, such as data objects, symbolic model specification, missing-data handling, and automatic search of a path of files. The system, which includes interactive statistical graphics, general linear models, and robust estimation methods, has been implemented using IBM's APL2 for PC-compatibles—both the standard version of this system and the freeware TryAPL2. The latter provides students with a free environment for modern data analysis and one in which they can explore the design of statistical software.     

化学计量学中的稳健估计方法

稳健统计学是八十年未才基本定型的统计学分支，它是针对实际情况中假设模型常常只是对实际数据的一种近似而导致传统统计学推断失误而发展起来的。稳健统计学构造一些新的具有稳健性的方法，使得在假设模型满足时，稳健方法具有接近最优的性能；在实际数据与假设模型有差别时，其性能仍为次优的；而在实际数据与假设模型差别大时，统计方法的性能也不会变得过差。本文介绍了稳健估计的一般概念。综述了化学计量学中的稳健估计方法，     

Robust and easily retrievable Pd/Ti_3C_2T_x■graphene hydrogels for efficient catalytic hydrogenation of nitroaromatic compounds

Ti_3C_2T_x has been emerging as an attractive platform to prepare composite catalysts,and their assembly into integrated catalytic mate rials repre sents a key step forward toward practical applications.Howeve r,the swelling behavior of Ti_3C_2T_x leads to significant structure change,which challenges the stability of Ti_3C_2T_x-based integrated functional materials for catalytic applications.Here we report a facile synthesis of Pd/Ti_3C_2T_x■graphene hydrogels in which Pd/Ti_3C_2T_x are spatially encapsulated in the 3 D porous graphene framework.The porous interconnected structure not only affords efficient mass transfer and desirable functional accessibility to catalytic active sites,but also effectively buffers the swelling behavior of Ti_3C_2T_x.When applied for catalytic hydrogenation of nitroaromatic compounds,the mechanically robust Pd/Ti_3C_2T_x■graphene hydrogels exhibit efficient activities,easy separability,and good cyclability.This work is expected to promote the application of Ti_3C_2T_x-based functional materials for practical applications involving interactions with salt solutions,such as supercapacitors,catalysis,and water purification.     

ℓ1-Minimization with Magnitude Constraints in the Frequency Domain

In this paper, we study the -optimal control problem with additional constraints on the magnitude of the closed-loop frequency response. In particular, we study the case of magnitude constraints at fixed frequency points (a finite number of such constraints can be used to approximate an -norm constraint). In previous work, we have shown that the primal-dual formulation for this problem has no duality gap and both primal and dual problems are equivalent to convex, possibly infinite-dimensional, optimization problems with LMI constraints. Here, we study the effect of approximating the convex magnitude constraints with a finite number of linear constraints and provide a bound on the accuracy of the approximation. The resulting problems are linear programs. In the one-block case, both primal and dual programs are semi-infinite dimensional. The optimal cost can be approximated, arbitrarily well from above and within any predefined accuracy from below, by the solutions of finite-dimensional linear programs. In the multiblock case, the approximate LP problem (as well as the exact LMI problem) is infinite-dimensional in both the variables and the constraints. We show that the standard finite-dimensional approximation method, based on approximating the dual linear programming problem by sequences of finite-support problems, may fail to converge to the optimal cost of the infinite-dimensional problem.     

基于NOMA的无线携能D2D通信鲁棒能效优化算法

针对频谱短缺、基站负荷过高、通信系统功耗较大等问题,考虑不完美的信道状态信息,该文提出一种基于非正交多址接入的无线携能(SWIPT)D2D网络鲁棒能效(EE)最大化资源分配算法(SREA).考虑用户的服务质量约束以及最大发射功率约束,基于随机信道不确定性建立鲁棒能效最大化资源分配模型.利用Dinkelbach和变量替换方法,将原NP-hard问题转换为确定性的凸优化问题,通过拉格朗日对偶理论求得解析解.仿真结果表明,所提算法在保证蜂窝用户通信质量的同时,能够有效提高D2D用户的能效性和鲁棒性能.     

Robust supply chain network design with service level against disruptions and demand uncertainties: A real-life case

We have developed a stochastic mathematical formulation for designing a network of multi-product supply chains comprising several capacitated production facilities, distribution centres and retailers in markets under uncertainty. This model considers demand-side and supply-side uncertainties simultaneously, which makes it more realistic in comparison to models in the existing literature. In this model, we consider a discrete set as potential locations of distribution centres and retailing outlets and investigate the impact of strategic facility location decisions on the operational inventory and shipment decisions of the supply chain. We use a path-based formulation that helps us to consider supply-side uncertainties that are possible disruptions in manufacturers, distribution centres and their connecting links. The resultant model, which incorporates the cut-set concept in reliability theory and also the robust optimisation concept, is a mixed integer nonlinear problem. To solve the model to attain global optimality, we have created a transformation based on the piecewise linearisation method. Finally, we illustrate the model outputs and discuss the results through several numerical examples, including a real-life case study from the agri-food industry.     

Robust disturbance rejection method for uncertain system with disturbances of unknown frequencies

In this paper, we discuss a robust disturbance rejection method for dealing with disturbances of unknown frequencies. Unlike many other approaches, the method proposed here does not require the disturbance frequencies of the separate harmonics to be estimated. The current approach is based on disturbance reduction and disturbance suppression. This novel disturbance reduction controller consists of an inverse of the nominal model with an input deduction and a high gain integral term. The proposed controller can reduce both periodic and nonperiodic unknown disturbances with uncertainties in both stable and unstable systems. In addition, undesired responses caused by residual disturbances and residual modeling uncertainties are suppressed by combining the novel disturbance reduction controller with a sliding mode controller. The simulation results demonstrate that the proposed disturbance rejection method performs well under different disturbance inputs including random signals.     

Deleting Outliers in Robust Regression with Mixed Integer Programming

In robust regression we often have to decide how many are the unusual observations, which should be removed from the sample in order to obtain better fitting for the rest of the observations. Generally, we use the basic principle of LTS, which is to fit the majority of the data, identifying as outliers those points that cause the biggest damage to the robust fit. However, in the LTS regression method the choice of default values for high break down-point affects seriously the efficiency of the estimator. In the proposed approach we introduce penalty cost for discarding an outlier, consequently, the best fit for the majority of the data is obtained by discarding only catastrophic observations. This penalty cost is based on robust design weights and high break down-point residual scale taken from the LTS estimator. The robust estimation is obtained by solving a convex quadratic mixed integer programming problem, where in the objective function the sum of the squared residuals and penalties for discarding observations is minimized. The proposed mathematical programming formula is suitable for small-sample data. Moreover, we conduct a simulation study to compare other robust estimators with our approach in terms of their efficiency and robustness.