Analysis and Optimization of Stability Robustness Bounds for Discretized Systems

In this paper, robustness bounds for the perturbations of continuous-time systems to ensure the stability of their discretized counterparts are developed. Both zero-order hold and P-step matrix integrators are considered. The effect of the sampling time on the robustness bounds is studied via examples. To determine how well a simulated system will retain the robustness properties of the continuous-time system being simulated, a new criterion for the selection of the simulation method and time step is introduced. Both implicit and explicit robustness measures for sampled-data systems are obtained.     

Logarithmic sample bounds for Sample Average Approximation with capacity- or budget-constraints

Sample Average Approximation (SAA) is used to approximately solve stochastic optimization problems. In practice, SAA requires much fewer samples than predicted by existing theoretical bounds that ensure the SAA solution is close to optimal. Here, we derive new sample-size bounds for SAA that, for certain problems, are logarithmic (existing bounds are polynomial) in problem dimension. Notably, our new bounds provide a theoretical explanation for the success of SAA for many capacity- or budget-constrained optimization problems.     

Perturbation Theory for Simultaneous Bases of Singular Subspaces

New perturbation theorems are proved for simultaneous bases of singular subspaces of real matrices. These results improve the absolute bounds previously obtained in [6] for general (complex) matrices. Unlike previous results, which are valid only for the Frobenius norm, the new bounds, as well as those in [6] for complex matrices, are extended to any unitarily invariant matrix norm. The bounds are complemented with numerical experiments which show their relevance for the algorithms computing the singular value decomposition. Additionally, the differential calculus approach employed allows to easily prove new sin perturbation theorems for singular subspaces which deal independently with left and right singular subspaces.     

Riesz界与条件数

在多分辨分析与小波分析中,人们经常要估计Riesz基的上下界.在有限维空间中,这等价于计算Riesz基所对应Gramian矩阵的条件数.本文给出Riesz基与条件数的关系并且讨论了Riesz基加入元素后对Riesz界产生的影响.     

Comparison of Two Sets of First-Order Conditions as Bases of Interior-Point Newton Methods for Optimization with Simple Bounds

In this paper, we compare the behavior of two Newton interior-point methods derived from two different first-order necessary conditions for the same nonlinear optimization problem with simple bounds. One set of conditions was proposed by Coleman and Li; the other is the standard KKT set of conditions. We discuss a perturbation of the CL conditions for problems with one-sided bounds and the difficulties involved in extending this to problems with general bounds. We study the numerical behavior of the Newton method applied to the systems of equations associated with the unperturbed and perturbed necessary conditions. Preliminary numerical results for convex quadratic objective functions indicate that, for this class of problems, the Newton method based on the perturbed KKT formulation appears to be the more robust.     

矩阵Hoffman-Wielandt型乘法相对扰动界

<正>1引言设A是一个方阵,(?)是它的扰动矩阵.特征值的加法扰动和乘法扰动是矩阵特征值的两种不同类型的扰动.当(?)=A+E时,称(?)是A的加法扰动矩阵;当(?)=D_1~*AD_2时,其中     

关于正规矩阵对广义特征值新的扰动界限

本文考虑了正规矩阵对的任意扰动时广义特征值的变化情况,给出了正规矩阵对任意扰动的Hoffman-Wielandt型扰动界,推广了正规矩阵对的相应的扰动结果.     

ARNOLDI TYPE ALGORITHMS FOR LARGE UNSYMMETRIC MULTIPLE EIGENVALUE PROBLEMS

1.IntroductionTheLanczosalgorithm[Zo]isaverypowerfultoolforextractingafewextremeeigenvaluesandassociatedeigenvectorsoflargesymmetricmatrices[4'5'22].Sincethe1980's,considerableattentionhasbeenpaidtogeneralizingittolargeunsymmetricproblems.Oneofitsgen...     

Bounds for the reliability functions of coherent systems with heterogeneous components

The computation of the reliability function of a (complex) coherent system is a difficult task. Hence, sometimes, we should simply work with some bounds (approximations). The computation of these bounds has been widely studied in the case of coherent systems with independent and identically distributed (IID) components. However, few results have been obtained in the case of heterogeneous (non ID) components. In this paper, we derive explicit bounds for systems with heterogeneous (independent or dependent) components. Also some stochastic comparisons are obtained. Some illustrative examples are included where we compare the different bounds proposed in the paper.     

受控马尔可夫跳线性系统的随机鲁棒稳定性

本文讨论了受控连续和离散时间马尔可夫跳线性系统的随机鲁棒稳定性,并且给出了此时该系统发生马尔可夫跳的转移速率的一个界.     

Interval techniques for enclosures of regions of reachability and controllability and for guaranteed state and parameter estimation of dynamical systems

Interval techniques are a powerful means for calculation of enclosures of the regions of reachability and controllability of dynamical systems with uncertainties during analysis and design of controllers. In this contribution, both discrete-time and continuous-time dynamical systems are considered. Using suitable algorithms, guaranteed state enclosures can be determined for systems with uncertain parameters, uncertain initial conditions, nonlinearities, and time-varying characteristics. Although both uncertain system parameters and bounded control variables are assumed to be represented by interval boxes in the following, they have to be distinguished in reachability and controllability analysis. Typically, robustness specifications for controllers of dynamical systems are given in terms of bounds on the system's time response which must not be violated for any possible operating condition. Hence, reachability as well as controllability of states have to be proven for all possible parameter values but for at least one admissible control sequence. Robust control strategies for nonlinear systems usually rely on knowledge of all current states. However, the complete state vector is not always directly accessible for measurement. In this case, observers are applicable to reconstruct non-measurable state variables. Furthermore, they can reduce the uncertainties of the measured quantities by model-based recursive computation of estimates and fusion of information gathered by different measurement devices. If guaranteed bounds of all uncertain parameters of a dynamical system (including the sensor characteristics) and conservative bounds of all disturbances can be specified, the presented interval observer provides guaranteed enclosures of all reachable states. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Truncated VSV Solutions to Symmetric Rank-Deficient Problems

Symmetric VSV decompositions are new rank-revealing decompositions that exploit and preserve symmetry. Truncated VSV solutions are stabilized solutions computed by neglecting blocks in the VSV decomposition with small norm. We compare the truncated VSV solutions with truncated SVD solutions and give perturbation bounds for the VSV solutions. Numerical examples illustrate our results.     

关于TLS和LS解的扰动分析

1．引言本文采用卜］的记号．最小二乘（LS）和总体最小二乘（TLS）是科学计算中的两种重要方法．尤是TLS，近来已有多篇论文讨论［1－6，8－16］．奇异值分解（SVD）和CS分解是研究TLS和LS的重要工具．令ACm，BCm,C=(A,B)，A和C的SVD分别为（1．1）（1．2）其中P51为某个正整数，U，U，V，V均为西矩阵，UI，UI，VI，VI为上述矩阵的前P列，z1一山。g（。1，…，内），】2＝di。g（内十l，…，。小】1＝dl。g（61；…，站，】2二diag（4＋1；…，dk），。l三··2。120和dl三…三d。20分别为C和A的奇异值，Z＝mhfm．n十以…     

A dynamic bounding algorithm for approximating multi-state two-terminal reliability

This paper modifies Jane and Laih’s (2008) exact and direct algorithm to provide sequences of upper bounds and lower bounds that converge to the NP-hard multi-state two-terminal reliability. Advantages of the modified algorithm include (1) it does not require a priori the lower and/or upper boundary points of the network, (2) it derives a series of increasing lower bounds and a series of decreasing upper bounds simultaneously, guaranteed to enclose the exact reliability value, and (3) trade-off between accuracy and execution time can be made to ensure an exact difference between the upper and lower bounds within an acceptable time. Examples are analyzed to illustrate the bounding algorithm, and to compare the bounding algorithm with existing algorithms. Computational experiments on a large network are conducted to realize the performance of the bounding algorithm.     

Exact Bounds for the Expectations of Order Statistics from Non-Negative Populations

Some new exact bounds for the expected values of order statistics, under the assumption that the parent population is non-negative, are obtained in terms of the population mean. Similar bounds for the differences of any two order statistics are also given. It is shown that the existing bounds for the general case can be improved considerably under the above assumption.     

Exponential stabilizability and robustness analysis for discrete-time nonlinear systems

In this paper, we derive some results on the exponential stabilizability and robustness analysis for discrete-time nonlinear control systems. Using the discrete Gronwall’s inequality, we also derive an important absolute estimate for the robustness index of the controlled discrete-time nonlinear system.     

Hybrid Impulsive Control of Stochastic Systems with Multiplicative Noise Under Markovian Switching

A problem of state output feedback stabilization of discrete-time stochastic systems with multiplicative noise under Markovian switching is considered. Under some appropriate assumptions, the stability of this system under pure impulsive control is given. Further under hybrid impulsive control, the output feedback stabilization problem is investigated. The hybrid control action is formulated as a combination of the regular control along with an impulsive control action. The jump Markovian switching is modeled by a discrete-time Markov chain. The control input is simultaneously applied to both the stochastic and the deterministic terms. Sufficient conditions based on stochastic semi-definite programming and linear matrix inequalities (LMIs) for both stochastic stability and stabilization are obtained. Such a nonconvex problem is solved using the existing optimization algorithms and the nonconvex CVX package. The robustness of the stability and stabilization concepts against all admissible uncertainties are also investigated. The parameter uncertainties we consider here are norm bounded. Two examples are given to demonstrate the obtained results.     

An optimal perturbation bound

Cai and Zhang establish separate perturbation bounds for distances with spectral and Frobenius norms (Cai T, Zhang A. Rate‐optimal perturbation bounds for singular subspaces with applications to high‐dimensional statistics. The Annals of Statistics. 2018; Vol. 46, No. 1: 60?89). We extend their theorem to each unitarily invariant norm. It turns out that our estimation is optimal as well.     

Non-Exponential Stabilization of Nonlinear Discrete-Time Systems

In this paper, we investigate the stabilization problem for a class of nonlinear discrete-time control systems. We consider semilinear systems with a nonlinear perturbation satisfying a Hölder-type condition. Based on a new discrete inequality of Gronwall type, we establish sufficient conditions for the global and local weak stabilizability of semilinear and nonlinear discrete-time systems, respectively.     

A Posteriori Error Bounds for Gaussian Elimination

Explicit bounds are constructed for the error in the solutionof a system of linear algebraic equations obtained by Gaussianelimination using floating-point arithmetic. The bounds takeaccount of inherent errors in the data and all abbreviations(choppings or roundings) introduced during the process of solution.The bounds are strict and agree with the estimate for the maximumerror obtained by linearized perturbation theory. The formulationof the bounds avoids the need for specially directed roundingprocedures in the hardware or software; in consequence the boundscan be evaluated on most existing computers. The cost of computingthe bounds is comparable with the cost of computing the originalsolution.