Ten years of genetic fuzzy systems: current framework and new trends

Fuzzy systems have demonstrated their ability to solve different kinds of problems in various application domains. Currently, there is an increasing interest to augment fuzzy systems with learning and adaptation capabilities. Two of the most successful approaches to hybridise fuzzy systems with learning and adaptation methods have been made in the realm of soft computing. Neural fuzzy systems and genetic fuzzy systems hybridise the approximate reasoning method of fuzzy systems with the learning capabilities of neural networks and evolutionary algorithms.The objective of this paper is to provide an account of genetic fuzzy systems, with special attention to genetic fuzzy rule-based systems. After a brief introduction to models and applications of genetic fuzzy systems, the field is overviewed, new trends are identified, a critical evaluation of genetic fuzzy systems for fuzzy knowledge extraction is elaborated, and open questions that remain to be addressed in the future are raised. The paper also includes some of the key references required to quickly access implementation details of genetic fuzzy systems.     

Preservation of reliability classes under the formation of coherent systems

The preservation of reliability aging classes under the formation of coherent systems is a relevant topic in reliability theory. Thus, it is well known that the new better than used class is preserved under the formation of coherent systems with independent components. However, surprisingly, the increasing failure rate class is not preserved in the independent and identically distributed case, that is, the components may have the (negative) aging increasing failure rate property, but the system does not have this property. In this paper, we study conditions for the preservation of the main reliability classes under the formation of general coherent systems. These results can be applied both for systems with independent or dependent components. We consider both the case of systems with identically distributed components and the case of systems with components having different distributions. Copyright © 2013 John Wiley & Sons, Ltd.     

A theory for synchronization of dynamical systems

In this paper, a theory for synchronization of multiple dynamical systems under specific constraints is developed from a theory of discontinuous dynamical systems. The concepts on synchronization of two or more dynamical systems to specific constraints are presented. The synchronization, desynchronization and penetration of multiple dynamical systems to multiple specified constraints are discussed, and the necessary and sufficient conditions for such synchronicity are developed. The synchronicity of two dynamical systems to a single specific constraint and to multiple specific constraints is investigated. Finally, the synchronization and the corresponding complexity for multiple slave systems with multiple master systems are discussed briefly. The meaning of synchronization for dynamical systems with constraints is extended as a generalized, universal concept. The theory presented in this paper may be as a universal theory for dynamical systems. The paper provides a theoretic frame work in order to control slave systems which can be synchronized with master systems through specific constraints in a general sense.     

Dynamical behavior of Lotka–Volterra competition systems: non-autonomous bistable case and the effect of telegraph noise

This article is concerned with the study of trajectory behavior of Lotka–Volterra competition bistable systems and systems with telegraph noises. We proved that for bistable systems, there exists a unique solution, bounded above and below by positive constants. The oscillatory situation of systems with telegraph noises is pointed out.     

Sensitivity of solutions to systems of optimality conditions under the violation of constraint qualifications

A survey is given of old and new results on the sensitivity of solutions to systems of optimality conditions with respect to parametric perturbations. Results of this kind play a key role in subtle convergence analysis of various constrained optimization algorithms. General systems of optimality conditions for problems with abstract constraints, Karush-Kuhn-Tucker systems for mathematical programs, and Lagrange systems for problems with equality constraints are examined. Special attention is given to the cases where the traditional constraint qualifications are violated.     

Output feedback stabilization for a class of nonlinear time-evolution systems

A variety of problems in nonlinear time-evolution systems such as communication networks, computer networks, manufacturing, traffic management, etc., can be modelled as min–max-plus systems in which operations of min, max and addition appear simultaneously. Systems with only maximum (or minimum) constraints can be modelled as max-plus system and handled by max-plus algebra which changes the original nonlinear system in the traditional sense into linear system in this framework. Min-max-plus systems are extensions of max-plus systems and nonlinear even in the max-plus algebra view. Output feedback stabilization for min–max-plus systems with min–max-plus inputs and max-plus outputs is considered in this paper. Max-plus projection representation for the closed-loop system with min–max-plus output feedback is introduced and the formula to calculate the cycle time is presented. Stabilization of reachable systems with at least one observable state and a further result for reachable and observable systems are worked out, during which max-plus output feedbacks are used to stabilize the systems. The method based on the max-plus algebra is constructive in nature.     

Existence of periodic solutions of sublinear Hamiltonian systems

The variational method is used to obtain some existence theorems of periodic solutions of sublinear systems with or not with impacts under suitable growth conditions. Compared with normal systems, impact systems need additional conditions to ensure the existence of periodic bouncing solutions.     

On the Non-Validity of the Order Reduction Method for Singularly Perturbed Control Systems

The order reduction method for singularly perturbed optimal control systems consists of setting the small parameter equal to zero and employing the differential system thus obtained. Although in many situations this provides the correct variational limit problem, it is established in this paper that when considering systems with non-scalar fast variables, the set of systems for which the order reduction method is invalid is dense in the class of systems under consideration. This extends previous results, where only systems with linear fast variables were considered. The present result complements a result established in a joint work with Artstein, where it was established that the order reduction method is valid for singularly perturbed optimal control systems with scalar fast variable.     

Generalizations of the concept of marginal synchronization of chaos

Generalizations of the concept of marginal synchronization between chaotic systems, i.e. synchronization with zero largest conditional Lyapunov exponent, are considered. Generalized marginal synchronization in drive–response systems is defined, for which the function between points of attractors of different systems is given up to a constant. Auxiliary system approach is shown to be able to detect this synchronization. Marginal synchronization in mutually coupled systems which can be viewed as drive–response systems with the response system influencing the drive system dynamics is also considered, and an example from solid-state physics is analyzed. Stability of these kinds of synchronization against changes of system parameters and noise is investigated. In drive–response systems generalized marginal synchronization is shown to be rather sensitive to the changes of parameters and may disappear either due to the loss of stability of the response system, or as a result of the blowout bifurcation. Nonlinear coupling of the drive system to the response system can stabilize marginal synchronization.     

Dynamics of controlled hybrid systems of aerial cable-ways

The dynamics of the hybrid systems of aerial cable-ways is investigated. The eigenvalue problems are considered for such hybrid systems with different assumptions. An overview of different methods for eigenvalue problems is given. In the research, the method of normal fundamental systems is applied, which turns out to be very effective for the considered problems. Changes in the dynamical characteristics of the systems depending on the controlled parameter are studied.     

一类非线性系统的局部镇定

研究一类非线性系统的局部状态反馈镇定问题．基于中心流形理论,给出一类非线性系统渐近镇定的充分条件,并设计出镇定系统的反馈控制律．文中利用具有齐次导数的Lyapunov函数方法,特别研究了一类平面非线性系统及具有二重零特征值的一类非线性系统的渐近镇定问题,给出了系统镇定的若干充分条件,并构造出控制律．文中的例表明了所得结果的有效性．     

A model of general systems

In this paper a mathematical model of general systems with not only one, but also more than one relation between the objects of the systems, was introduced. Among others, the concepts of partial systems, similarity of systems, embeddability of systems and free sum of systems were introduced. The following theorem was proved: there is no system whose object set consists of all systems.     

李三系分解的唯一性

李三系是从黎曼对称空间产生的三元运算的代数体系，近来得到许多数学家的重视．但是至今为止，李三系的研究集中在半单与单李三系上．本文主要讨论中心为零的李三系的一些基本问题，特别是分解唯一性问题．我们的结果包含了半单李三系的分解唯一性．     

A geometric approach for reachability and observability of linear switched impulsive systems

This paper is concerned with the reachability and observability of linear switched impulsive systems with singular impulse matrices. First some new concepts with respect to the reachability and unobservability are introduced. Especially, span reachability is proposed because the reachable sets of switched impulsive systems do not always constitute subspaces. Then the geometric characterization of the span reachable and unobservable sets is presented. Moreover, the relations between the span reachable set, unobservable set and the invariant subspaces of such systems are discussed. Finally, corresponding criteria applied to linear impulsive systems and linear switched systems are also discussed.     

一类共振条件下两个自由度系统的同宿轨道

研究一类在参数和强迫激励下发生共振时的两个自由度系统,利用多重尺度法证明存在锁频于Ω的周期解.在一定条件下可变换成Wiggins的系统,给出了判断这类系统同宿轨道存在的计算公式.     

Analysis of mixed problems of matching in hyperbolic systems of different orders

A mixed problem of matching in hyperbolic systems of different orders with irregular boundary conditions is considered. In particular, these systems include hyperbolic systems with coefficients having discontinuities of the first kind. Under certain conditions, an analytical representation of the solution of the problem is obtained by means of a finite integral transform.     

The dynamics of systems with rolling and gyroscopic systems with small generalized velocities and the realization of constraints

Approaches to the construction of mathematical models of systems with rolling and gyroscopic systems with dynamics characterized by the smallness of some of the generalized velocities are discussed. As a rule, a quasistatic approach is used in the modelling of such systems, within the limits of which the generalized accelerations corresponding to small generalized velocities are assumed to be equal to zero. Cases are indicated when the possibility, established by Kozlov, of obtaining the quasistatic equations of gyroscopic systems by the imposition of holonomic constraints is extended to systems with rolling. Additional conditions are formulated that enable one to estimate the error in the quasistatic equations of systems with rolling and gyroscopic systems. It is shown that they can be refined with respect to a small parameter, that is, the ratio of the characteristic values of the “small” and “finite” generalized velocities, using the Dirac formalism, based on an analysis of the constraints between the generalized coordinates and momenta of the system that arise on account of the degeneracy of its Lagrangian on changing to the quasistatic equations.     

一类时滞格动力系统的整体吸引子

本文研究了一类含时滞的格动力系统,证明了其整体吸引子的存在性.     

Autonomous Planar Systems of Riccati Type

The role of Riccati type systems in the plane along with the related linear, second order differential equation is examined. If $x$ and $y$ are the variables of the Riccati differential equation, then any integrable Riccati system has two independent invariant curves dependent upon these variables whose nature is easily determined from the solution of the linear equation. Each of these curves has the same cofactor. Other invariant curves depend upon $x$ alone and are shown to be less important. The systems have both Liouvillian and non--Liouvillian solutions and are easily transformable to symmetric systems. However, systems derived from them may not be symmetric in their transformed variables. Several systems from the literature are discussed with regard to the forms of the invariant curves presented in the paper. The relation of certain Riccati type systems is considered with respect to Abel differential equations.     

Maximum principle of stochastic controlled systems of functional type

This paper studies the optimal controls of stochastic systems of functional type with end constraints. The systems considered may be degenerate and the control region may be nonconvex. A stochastic maximum principle is derived. The method is based on the idea that stochastic systems are essentially infinite dimensional systems. The Project Supported by National Natural Science Fundation of China.