Optimization of Stability Robustness Bounds for Linear Discrete-Time Systems

In this paper, existing stability robustness measures for the perturbation of both continuous-time and discrete-time systems are reviewed. Optimized robustness bounds for discrete-time systems are derived. These optimized bounds are obtained reducing the conservatism of existing bounds by (a) using the structural information on the perturbation and (b) changing the system coordinates via a properly chosen similarity transformation matrix. Numerical examples are used to illustrate the proposed reduced conservatism bounds.     

Equivalent Optimization Formulations and Error Bounds for Variational Inequality Problems

We investigate whether some merit functions for variational inequality problems (VIP) provide error bounds for the underlying VIP. Under the condition that the involved mapping F is strongly monotone, but not necessarily Lipschitz continuous, we prove that the so-called regularized gap function provides an error bound for the underlying VIP. We give also an example showing that the so-called D-gap function might not provide error bounds for a strongly monotone VIP.This research was supported by United College and by a direct grant of the Chinese University of Hong Kong. The authors thank the referees for helpful comments and suggestions.     

Optimization Approach to the Robustness of Linear Delay Systems

By using the Lyapunov equation approach and an improved Razumikhin-type theorem, this paper presents a new robust stability criterion for a linear system subject to delayed time-varying nonlinear perturbations. Then, by using a parameter optimization technique, an efficient algorithm is derived for determining a desirable matrix for the Lyapunov equation. As a consequence, less conservative robust stability bounds for the perturbed system are achieved. Numerical examples are included to demonstrate the effectiveness of the proposed approach.     

Robustness bounds of Hurwitz and Schur polynomials

In this paper, the problem of robustness bounds of Hurwitz and Schur polynomials is addressed. For weightedL ₂-norm perturbations of a Hurwitz polynomialp(s) or a Schur polynomialp(z), a new method is developed for calculating the maximal perturbation bound under which stability is preserved. We show that such a robustness bound is related to the minimum of a rational function. The new method is superior to the previous one developed by Soh, Berger, and Dabke in Ref. 1. Our approach also provides solutions for the perturbation polynomial p(s) or p(z) with minimal coefficient norm which causep(s)+p(s) orp(z)+p(z) to be unstable.     

Family of Projected Descent Methods for Optimization Problems with Simple Bounds

This paper presents a family of projected descent direction algorithms with inexact line search for solving large-scale minimization problems subject to simple bounds on the decision variables. The global convergence of algorithms in this family is ensured by conditions on the descent directions and line search. Whenever a sequence constructed by an algorithm in this family enters a sufficiently small neighborhood of a local minimizer satisfying standard second-order sufficiency conditions, it gets trapped and converges to this local minimizer. Furthermore, in this case, the active constraint set at is identified in a finite number of iterations. This fact is used to ensure that the rate of convergence to a local minimizer, satisfying standard second-order sufficiency conditions, depends only on the behavior of the algorithm in the unconstrained subspace. As a particular example, we present projected versions of the modified Polak–Ribière conjugate gradient method and the limited-memory BFGS quasi-Newton method that retain the convergence properties associated with those algorithms applied to unconstrained problems.     

Robustness of perturbation analysis estimators for queueing systems with unknown distributions

Sample-path-based stochastic gradient estimators for performance measures of queueing systems rely on the assumption that a probability distribution of the random vector of interest (e.g., a service or interarrival time sequence) is given. In this paper, we address the issue of dealing with unknown probability distributions and investigate the robustness of such estimators with respect to possibly erroneous distribution choices. We show that infinitesimal perturbation analysis (IPA) can be robust in this sense and, in some cases, provides distribution-independent estimates. Comparisons with other gradient estimators are provided, including experimental results. We also show that finite perturbation analysis (FPA), though only providing gradient approximations, possesses some attractive robustness properties with respect to unknown distribution parameters. An application of FPA estimation is included for a queueing system performance optimization problem involving customers with real-time constraints.This work was supported in part by the National Science Foundation Grant ECS-88-01912 and by the Office of Naval Research Contract N00014-87-K-0304.The authors wish to thank Dr. Jack Holtzman for several useful comments and suggestions.     

奇异系统的某些性质及其与奇异脉冲系统的稳定性等价

本文研究了与特征值相关的奇异系统解的构造及其性态，并讨论了奇异脉冲系统与奇异系统的稳定性等价。     

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.     

多参数多状态变量离散型有势非线性稳定问题的活化方法*

本文针对多参数变量和多状态变量的离散型有势系统的非线性稳定问题,提出了活化方法,导出了活化势函数和活化平衡方程.活化方法是弹性稳定理论中Liapunov-Schmidt方法的改进和提高,它比通常的摄动方法更加一般化、规范化.活化势函数可变换成标准突变势函数,活化平衡方程可作为分岔方程.本文的研究将促进弹性稳定理论与突变理论和分岔理论的结合.     

A Note on Error Bounds for Convex and Nonconvex Programs

Given a single feasible solution and a single infeasible solution of a mathematical program, we provide an upper bound to the optimal dual value. We assume that satisfies a weakened form of the Slater condition. We apply the bound to convex programs and we discuss its relation to Hoffman-like bounds. As a special case, we recover a bound due to Mangasarian [11] on the distance of a point to a convex set specified by inequalities.     

Exponential Stability for Uncertain Dynamic Systems with Time-Varying Delays: LMI Optimization Approach

In this paper, we consider dynamic systems with uncertainties and time-varying delays. Based on the Lyapunov method and convex optimization approach, a delay-dependent criterion for exponential stability of the system is derived in terms of LMI (linear matrix inequality). In order to solve effectively the LMI convex optimization problem, an interior-point algorithm is utilized in this work. Numerical examples are illustrated to show the effectiveness of our results.     

Stability of Linear Inequality Systems in a Parametric Setting

In this paper, we propose a parametric approach to the stability theory for the solution set of a semi-infinite linear inequality system in the n-dimensional Euclidean space . The main feature of this approach is that the coefficient perturbations are modeled through the so-called mapping of parametrized systems, which assigns to each parameter, ranging in a metric space, a subset of . Each vector of this image set provides the coefficients of an inequality in and the whole image set defines the inequality system associated with the parameter. Thus, systems associated with different parameters are not required to have the same number (cardinality) of inequalities. The paper is focused mainly on the structural stability of the feasible set mapping, providing a characterization of the Berge lower-semicontinuity property. The role played by the strong Slater qualification is analyzed in detail.This research has been partially supported by Grant PB98-0975 from DGES (Spain), Grant BFM2002-04114-C02 (01–02) from MCYT (Spain), FEDER (European Union), and Bancaja-UMH (Spain).     

The Robustness of Strong Stability of Positive Homogeneous Difference Systems Under Parameter Perturbations

In this article, we study the robustness of strong stability of the homogeneous difference systems via the concept of strong stability radii: complex, real and positive radii under parameter perturbations. We also show that in the case of positive systems, these radii coincide and can be computed by a simple formula. The results generalize those obtained in [5 B.T. Anh and D.D.X. Thanh ( 2008 ). The robustness of strong stability of positive homogeneous difference equations . J. Appl. Math. Article ID 124269; doi : 10 . 1155/2008/124269  [Google Scholar]].     

Normal Vectors on Manifolds of Critical Points for Parametric Robustness of Equilibrium Solutions of ODE Systems

Summary. {Equilibrium solutions of systems of parameterized ordinary differential equations \dot x = f(x, α) , x ∈ R ⁿ , α∈ R ^m can be characterized by their parametric distance to manifolds of critical solutions at which the behavior of the system changes qualitatively. Critical points of interest are bifurcation points and points at which state variable constraints or output constraints are violated. We use normal vectors on manifolds of critical points to measure the distance between these manifolds and equilibrium solutions as suggested in I. Dobson [J. Nonlinear Sci., 3:307-327, 1993], where systems of equations to calculate normal vectors on codimension-1 bifurcations were presented. We present a scheme to derive systems of equations to calculate normal vectors on manifolds of critical points which (i) generalizes to bifurcations of arbitrary codimension, (ii) can be applied to state variable constraints and output constraints, (iii) implies that the normal vector defining system of equations is of size c ₁ n+ c ₂ m+ c ₃ , c _i ∈ R , i.e., no bilinear terms nm or higher-order terms occur, (iv) reduces the number of equations for normal vectors on Hopf bifurcation manifolds compared to previous work, and (v) simplifies the proof of regularity of the normal vector system. As an application of this scheme, we present systems of equations for normal vectors to manifolds of output/state variable constraints, to manifolds of saddle-node, Hopf, cusp, and isola bifurcations, and we give illustrative examples of their use in engineering applications.} Received September 27, 2000; accepted December 10, 2001 Online publication March 11, 2002 Communicated by Y. G. Kevrekidis Communicated by Y. G. Kevrekidis rid="     

Bounds for moments of cubic and quartic Dirichlet L-functions

We study the $2k$-th moment of central values of the family of primitive cubic and quartic Dirichlet $L$-functions. We establish sharp lower bounds for all real $k\ge 1/2$ unconditionally for the cubic case and under the Lindelöf hypothesis for the quartic case. We also establish sharp lower bounds for all real $0\le k<1/2$ and sharp upper bounds for all real $k\ge 0$ for both the cubic and quartic cases under the generalized Riemann hypothesis (GRH). As an application of our results, we establish quantitative non-vanishing results for the corresponding $L$-values.     

一类平面系统稳定区域的最佳估计及其在电力系统分析中的应用

本文利用现代动力系统几何理论研究一类平面系统的平衡点的吸引区域估计问题 ,并将其应用于一类具体的电力系统 .     

线性离散大系统的稳定性

本文直接运用线性范数型Liapunov函数处理线性定常和时变离散大系统的稳定性,得到了渐近稳定若干判别准则.     

Stability Analysis of Controlled, Partially-Connected, Distributed-Parameter Systems Through Perturbation Methods

A stability result for locally-controlled, interconnected, distributed-parameter systems (DPS) is developed. Using a special perturbation operator, exponential stability is shown to be a function of both the value of the perturbation operator and the characteristics of the interconnected DPS. Proof of the bound is shown using the expansion of the matrix operator and the solution vector through a set of gauge functions. Each expansion term is power matched and individually bound using stationary phase methods. Special consideration is given to interconnected systems of a structural dynamic nature.     

时变线性系统特征根判稳的推广

本文研究了二维时变线性系统零解的稳定性，给出了一些允许系数矩阵主对角元变号，系数矩阵特征根变号的判定稳定性的充分条件。