Robust tube-based MPC of constrained piecewise affine systems with bounded additive disturbances

In this article, we propose a robust tube-based MPC formulation for a class of hybrid systems, namely autonomously switched PWA systems, with bounded additive disturbances. The term tube-based refers to those control techniques whose objective is to maintain all possible trajectories of the uncertain system inside a tube which is a set around the nominal (or reference) system trajectory, that is free from disturbances. Common methods in tube-based control systems consider an error dynamical system as the difference between the state of the nominal system and the state of the perturbed system. However, this definition of the error dynamical system leads to a complicated switched affine system for PWA systems. Therefore, we use a new notion of the reference system similar to the nominal system except that the switching between the various modes of the PWA system is driven by the state of the real system. Using this reference system instead of the nominal system leads us to an error dynamical system that can be modeled as a switched linear system. We employ a switched linear controller to stabilize this error system under arbitrary switching. This auxiliary controller forces the states of the uncertain system to remain in a tube confined to the invariant set around the state of the reference system. We add new constraints and tighten some other constraints of the nominal hybrid MPC for the reference system, in order to ensure convergence of the uncertain system and to guarantee robust exponential stability of the closed-loop system.     

Optimal burn-in for maximizing reliability of repairable non-series systems

Burn-in is a manufacturing process applied to products to eliminate early failures in the factory before the products reach the customers. Various methods have been proposed for determining an optimal burn-in time of a non-repairable system or a repairable series system, assuming that system burn-in improves all components in the system. In this paper, we establish the trade-off between the component reliabilities during system burn-in and develop an optimal burn-in time for repairable non-series systems to maximize reliability. One impediment to expressing the reliability of a non-series system is in that successive failures during system burn-in cannot be described precisely because a failed component is not detected until the whole system fails. For approximating the successive failures of a non-series system during system burn-in, we considered two types of repair: minimal repair at the time of system failure, and repair at the time of component or connection failure. The two types of repair provide bounds on the optimal system burn-in time of non-series systems.     

模块化可靠性系统的Signature计算方法研究

系统Signature是体现结构设计优良性的一组向量,描述系统设计对系统故障率的影响,在诸如系统可靠性指标分析、系统设计、系统寿命比较、寿命极限行为以及系统设计优化等方面展现出了强大的功能,成为可靠性研究领域越来越强有力的研究工具。而如何求解一个系统的Signature往往成为分析的关键一步,当系统庞大而复杂时,Signature计算难度将随着元件数目的增加呈指数增加,出现维数爆炸问题,这无疑对后续的分析造成巨大的障碍. 本文为了解决此问题,建立了基于模块化思想的系统Signature求解方法,并给出了基于模块化思想的模块化串、并联系统与模块化备份系统的求解方法,对比于传统算法,运用模块化思想大大减少了计算Signature的复杂度,能够有效减小计算量,缩减计算时间,并拓展了可求解Signature的系统范围。     

Randomized system trajectories

The notion of system trajectory of a time-varying input-output, dynamical system is reviewed. By introducing a probability measure on a class of such systems a stochastic system, the randomized system, is defined. The randomized system has a trajectory induced by the trajectories of the original systems. A theorem is proved giving fairly general conditions under which the randomized system trajectory is generated by a strongly continuous semigroup of bounded linear operators in a Banach space. An example is presented for a system represented by a quadratic integral operator.Research supported in part by National Science Foundation under Grant No. ECS-8005960.     

Finite-dimensional representation of delay systems

It is shown that given a state delayed system, one can construct a finite-dimensional system whose partial state replicates the response of the delay system up to a small solution of the homogeneous delay system. Thus, in the absence of small solutions, the response of the delay system can be exactly replicated by a finite-dimensional system. reserved.     

Model Dirac-Maxwell equations with pseudounitary symmetry

We introduce a new system of equations called a model system of Dirac-Maxwell equations, reproducing the main properties of the standard system. At the same time, the model system of equations differs from the standard system in several ways; in particular, it is a tensor system and has a new symmetry with respect to the pseudounitary group. We also propose a version of the model system of Dirac-Maxwell equations with local (gauge) pseudounitary symmetry. We show that any spinor solution of the standard system of Dirac-Maxwell equations can be obtained from the corresponding tensor solution of the model system. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 3, pp. 425–435, December, 2008.     

Poisson冲击下的$k/n(G)$系统的可靠性分析

本文研究了一类Poisson冲击下的$k/n(G)$系统(即$k$-out-of-$n$: $G$系统). 假定冲击的到达数形成一个参数为$\lambda$的Poisson过程, 且冲击的量服从某一分布. 当每次冲击到达时, 对系统中工作的部件独立地产生影响. 进而假定每一部件以一定的概率故障, 概率值是冲击量的函数. 且各次冲击独立地对系统造成损失, 直到工作部件数少于$k$系统故障为止. 在这些假定下, 我们获得了系统的可靠度函数和系统的平均工作时间. 进一步, 假定系统是可修的, 系统中有一个维修工, 并根据``先坏先修’’的维修规则对故障部件进行维修. 在维修时间服从指数分布的假设下, 系统状态转移服从Markov过程. 对该系统我们建立了状态转移方程, 并求得了系统可用度、稳态下的平均工作时间、平均停工时间和系统失效频率等可靠性指标. 最后, 我们还给出了一个简单例子来演示讨论的模型.     

Singularity Structure of Third-Order Dynamical Systems. II

The singularity structure of the solutions of a general third-order system, with polynomial right-hand sides of degree less than or equal to two, is studied about a movable singular point. An algorithm for transforming the given third-order system to a third-order Briot–Bouquet system is presented. The dominant behavior of a solution of the given system near a movable singularity is used to construct a transformation that changes the given system directly to a third-order Briot–Bouquet system. The results of Horn for the third-order Briot–Bouquet system are exploited to give the complete form of the series solutions of the given third-order system; convergence of these series in a deleted neighborhood of the singularity is ensured. This algorithm is used to study the singularity structure of the solutions of the Lorenz system, the Rikitake system, the three-wave interaction problem, the Rabinovich system, the Lotka–Volterra system, and the May–Leonard system for different sets of parameter values. The proposed approach goes far beyond the ARS algorithm.     

交通运输系统协调发展理论与模型研究

提出了交通运输系统协调度的评价分析模型.从系统论的观点出发,提出了交通运输系统协调理论的概念,探讨了交通运输系统随时间而不断演化变迁的规律,给出了交通运输系统协调发展基本步骤;并根据协调学原理,讨论了交通运输系统的协调性问题,提出了系统协调发展模型,对交通运输子系统内部及子系统之间及系统整体的协调发展问题进行了研究,探讨了交通运输可持续发展的系统协调管理过程,为进一步研究交通运输系统的可持续发展奠定了基础.     

Singularity Structure of Third-Order Dynamical Systems. I

A general third-order dynamical system with polynomial right-hand sides of finite degrees in the dependent variables is analyzed to unravel the singularity structure of its solutions about a movable singular point. To that end, the system is first transformed to a second-order Briot–Bouquet system and a third auxiliary equation via a transformation, similar to one used earlier by R. A. Smith in 1973–1974 for a general second-order dynamical system. This transformation imposes some constraints on the coefficients appearing in the general third-order system. The known results for the second-order Briot–Bouquet system are used to explicitly write out Laurent or psi-series solutions of the general third-order system about a movable singularity. The convergence of the relevant series solutions in a deleted neighborhood of the singularity is ensured. The theory developed here is illustrated with the help of the May–Leonard system.     

On a Stochastic Survival Model for a System Under Randomly Variable Environment

In many cases, the survival probability of a system depends not only on the intrinsic characteristic of the system itself but also on the randomly variable external environment under which the system is being operated. In this paper we study a stochastic survival model for a system under random shock process which affects the survival of the system in a complicated way. The lifetime distribution of the system is derived, and the effect of environmental factors on the failure process of the system is also investigated.     

Reduced systems in Markov chains and their applications in queueing theory

A reduced system is a smaller system derived in the process of analyzing a larger system. While solving for steady-state probabilities of a Markov chain, generally the solution can be found by first solving a reduced system of equations which is obtained by appropriately partitioning the transition probability matrix. In this paper, we catagorize reduced systems as standard and nonstandard and explore the existence of reduced systems and their properties relative to the original system. We also discuss first passage probabilities and means for the standard reduced system relative to the original system. These properties are illustrated while determining the steady-state probabilities and first passage time characteristics of a queueing system.     

Optimal Controller for Dithered Systems with Backlash or Hysteresis

Dither is a high-frequency signal introduced into a nonlinear system to improve the system performance. A nonlinearity with memory (backlash, hysteresis) is considered in this paper; a dither of a given amplitude is being injected at the input of the nonlinearity. The dither can effectively eliminate memory of the nonlinearity. The significance of the dither frequency lies in its effect on the deviation of the dithered system from its corresponding model, the smoothed system, in which the nonlinearity is a smooth function. The deviation amount can be improved as the dither frequency increases. If the dither has a sufficiently high frequency, the outputs of the smoothed system and the dithered system can be made as close as desired. This enables us to predict the stability of the dithered system by establishing the stability of the smoothed system.     

Hopf bifurcation analysis of integro-differential equation with unbounded delay

The complexity of a nonlinear dynamical system is controllable via a selection of system parameters. One representative behavior of such a complex system can be illustrated by Hopf bifurcation. This paper presents a Hopf bifurcation analysis of a kind of integro-differential equations with unbounded delay. Based on the Hopf bifurcation principle, a set of relationships among system parameters are obtained when a periodic orbit exists in the system. A numerical analysis is applied to solve the integro-differential delay equation. This paper proves the existence of Hopf bifurcation in the corresponding difference equations under the same system parameters as that in the integro-differential delay equations.     

Perturbation analysis of a condition number for convex inequality systems and global error bounds for analytic systems

In this paper several types of perturbations on a convex inequality system are considered, and conditions are obtained for the system to be well-conditioned under these types of perturbations, where the well-conditionedness of a convex inequality system is defined in terms of the uniform boundedness of condition numbers under a set of perturbations. It is shown that certain types of perturbations can be used to characterize the well-conditionedness of a convex inequality system, in which either the system has a bounded solution set and satisfies the Slater condition or an associated convex inequality system, which defines the recession cone of the solution set for the system, satisfies the Slater condition. Finally, sufficient conditions are given for the existence of a global error bound for an analytic system. It is shown that such a global error bound always holds for any inequality system defined by finitely many convex analytic functions when the zero vector is in the relative interior of the domain of an associated convex conjugate function.     

输入率可变且有差错服务的M/M/1排队系统的研究与应用

在到达系统的顾客数不变的情况下,顾客到达系统但是否进入系统接受服务对销售行业影响是巨大的.从排队长度对顾客输入率的影响着手,研究了顾客以泊松流到达系统,而到达系统的顾客进入系统接受服务的概率与队长有关的M/M/1排队模型,且系统服务会出差错.得出了进入系统的顾客流是泊松过程,且系统中的顾客数是生灭过程,并获得了该模型的平稳分布、顾客的平均输入率、系统的平均服务强度等多项指标,为销售行业调整自己的服务速度以影响排队长度及顾客输入率,进而提高自己的销售业绩提供了很有价值的参考.     

Effects of sourcing structure on performance in a multiple-product assemble-to-order supply chain

This paper addresses the strategic pricing decisions of a decentralized assemble-to-order system, and the effects of sourcing structure on system performance. We consider an assemble-to-order system consisting of two substitute products and three components manufactured by different suppliers. Demand for the products is price-sensitive. The selling prices for the products are jointly determined by firms in the system. We study the system under both centralized and decentralized decision-making situations. For the centralized system, we demonstrate that there exists a unique optimal pricing solution. For the decentralized system, we show that there exists a unique Nash equilibrium in the players’ pricing game. We further analyze the effects of different sourcing structures on the system performance and the assembler’s performance. We find that reduction of suppliers in the system does not guarantee improvement of system performance. Different sourcing structures have different impacts on performance. When the two dedicated components are sourced from a single supplier and the common component is sourced from another independent supplier, the system and the assembler’s performance become worse. We show that the decentralized system can be effectively coordinated and the system performance can be improved with a simple profit-sharing contract. Finally, we provide comprehensive numerical examples to demonstrate some important managerial insights.     

Fuzzy match and floating threshold strategy for expert system in traditional chinese medicine

An expert system is proposed in a computerized diagnosis and treatment system in traditional Chinese medicine. Fuzzy match and floating threshold is adopted in the expert system and the data base technique produces a conception of the large-scale expert system.     

Bifurcation of limit cycles at the equator

This paper studies center conditions and bifurcation of limit cycles from the equator for a class of polynomial differential system of order seven. By converting real planar system into complex system, we established the relation of focal values of a real system with singular point quantities of its concomitant system, and the recursion formula for the computation of singular point quantities of a complex system at the infinity. Therefore, the first 14 singular point quantities of a complex system at the infinity are deduced by using computer algebra system Mathematica. What’s more, the conditions for the infinity of the real system to be a center or 14 degree fine focus are derived, respectively. A system of order seven that bifurcates 12 limit cycles from the infinity is constructed for the first time.     

Stability analysis of a new kind n-unit series repairable system

In this paper, we deal with a new model of an n-unit series repairable system, in which a concept of a repairman with multiple-delayed vacation is introduced and the impact on the system reliability due to a replaceable facility is also considered. This paper is devoted to studying the unique existence and stability of the system solution. C₀-semigroup theory is used to prove the existence of a unique nonnegative time-dependent solution of the system. Then by analyzing the spectra distribution of the system operator, we prove that the dynamic solution of the system asymptotically converges to the nonnegative steady-state solution which is the eigenfunction corresponding to eigenvalue 0 of the system operator. Furthermore, we discuss the exponential stability of the system in a special case. Some reliability indices of the system are also studied and the optimal vacation time is analyzed at the end of the paper.