Fuzzy stability and synchronization of hyperchaos systems

This paper studies stability and synchronization of hyperchaos systems via a fuzzy-model-based control design methodology. First, we utilize a Takagi–Sugeno fuzzy model to represent a hyperchaos system. Second, we design fuzzy-model-based controllers for stability and synchronization of the system, based on so-called “parallel distributed compensation (PDC)”. Third, we reduce a question of stabilizing and synchronizing hyperchaos systems to linear matrix inequalities (LMI) so that convex programming techniques can solve these LMIs efficiently. Finally, the generalized Lorenz hyperchaos system is employed to illustrate the effectiveness of our designing controller.     

A neural fuzzy control system with structure and parameter learning

A general connectionist model, called neural fuzzy control network (NFCN), is proposed for the realization of a fuzzy logic control system. The proposed NFCN is a feedforward multilayered network which integrates the basic elements and functions of a traditional fuzzy logic controller into a connectionist structure which has distributed learning abilities. The NFCN can be constructed from supervised training examples by machine learning techniques, and the connectionist structure can be trained to develop fuzzy logic rules and find membership functions. Associated with the NFCN is a two-phase hybrid learning algorithm which utilizes unsupervised learning schemes for structure learning and the backpropagation learning scheme for parameter learning. By combining both unsupervised and supervised learning schemes, the learning speed converges much faster than the original backpropagation algorithm. The two-phase hybrid learning algorithm requires exact supervised training data for learning. In some real-time applications, exact training data may be expensive or even impossible to obtain. To solve this problem, a reinforcement neural fuzzy control network (RNFCN) is further proposed. The RNFCN is constructed by integrating two NFCNs, one functioning as a fuzzy predictor and the other as a fuzzy controller. By combining a proposed on-line supervised structure-parameter learning technique, the temporal difference prediction method, and the stochastic exploratory algorithm, a reinforcement learning algorithm is proposed, which can construct a RNFCN automatically and dynamically through a reward-penalty signal (i.e., “good” or “bad” signal). Two examples are presented to illustrate the performance and applicability of the proposed models and learning algorithms.     

Polynomial growth rates

We consider linear equations v^′=A(t)v with a polynomial asymptotic behavior, that can be stable, unstable and central. We show that this behavior is exhibited by a large class of differential equations, by giving necessary and sufficient conditions in terms of generalized “polynomial” Lyapunov exponents for the existence of polynomial behavior. In particular, any linear equation in block form in a finite-dimensional space, with three blocks having “polynomial” Lyapunov exponents respectively negative, positive, and zero, has a nonuniform version of polynomial trichotomy, which corresponds to the usual notion of trichotomy but now with polynomial growth rates. We also obtain sharp bounds for the constants in the notion of polynomial trichotomy. In addition, we establish the persistence under sufficiently small nonlinear perturbations of the stability of a nonuniform polynomial contraction.     

The rate of convergence for the cyclic projections algorithm III: Regularity of convex sets

The cyclic projections algorithm is an important method for determining a point in the intersection of a finite number of closed convex sets in a Hilbert space. That is, for determining a solution to the “convex feasibility” problem. This is the third paper in a series on a study of the rate of convergence for the cyclic projections algorithm. In the first of these papers, we showed that the rate could be described in terms of the “angles” between the convex sets involved. In the second, we showed that these angles often had a more tractable formulation in terms of the “norm” of the product of the (nonlinear) metric projections onto related convex sets.In this paper, we show that the rate of convergence of the cyclic projections algorithm is also intimately related to the “linear regularity property” of Bauschke and Borwein, the “normal property” of Jameson (as well as Bakan, Deutsch, and Li’s generalization of Jameson’s normal property), the “strong conical hull intersection property” of Deutsch, Li, and Ward, and the rate of convergence of iterated parallel projections. Such properties have already been shown to be important in various other contexts as well.     

L-fuzzy quantifiers of type determined by fuzzy measures

The aim of this paper is, first, to introduce two new types of fuzzy integrals, namely, -fuzzy integral and →-fuzzy integral. The first integral is based on a fuzzy measure of L-fuzzy sets and the second one on a complementary fuzzy measure of L-fuzzy sets, where L is a complete residuated lattice. Some of their properties and a relation to the fuzzy (Sugeno) integral are investigated. Second, using these integrals, two classes of monadic L-fuzzy quantifiers of type 1 are defined. These L-fuzzy quantifiers can be used for modeling the semantics of natural language quantifiers like “all”, “some”, “many”, “none”, “at most half”, etc. Several semantic properties of these L-fuzzy quantifiers are studied.     

Hopf bifurcation in doubly fed induction generator under vector control

This paper first presents the Hopf bifurcation phenomena of a vector-controlled doubly fed induction generator (DFIG) which is a competitive choice in wind power industry. Using three-phase back-to-back pulse-width-modulated (PWM) converters, DFIG can keep stator frequency constant under variable rotor speed and provide independent control of active and reactive power output. Main results are illustrated by “exact” cycle-by-cycle simulations. The detailed mathematical model of the closed-loop system is derived and used to analyze the observed bifurcation phenomena. The loci of the Jacobian’s eigenvalues are computed and the analysis shows that the system loses stability via a Hopf bifurcation. Moreover, the boundaries of Hopf bifurcation are also given to facilitate the selection of practical parameters for guaranteeing stable operation.     

采用模糊滑模变结构方案控制Chua混沌系统

基于模糊动态模型 ,研究了 Chua混沌系统的稳定控制问题 .将非线性混沌系统模糊化为局部线性模型 .用 Lyapunov稳定性理论设计出 ,确保模糊动态模型全局渐近稳定的变结构控制器 .仿真验证了方案的有效性 .模糊控制器简单 ,规则少 .     

Fuzzy logic in measurements

Our main interest in this paper is to translate from “natural language” into “system theoretical language”. This is of course important since a statement in system theory can be analyzed mathematically or computationally. We assume that, in order to obtain a good translation, “system theoretical language” should have great power of expression. Thus we first propose a new frame of system theory, which includes the concepts of “measurement” as well as “state equation”. And we show that a certain statement in usual conversation, i.e., fuzzy modus ponens with the word “very”, can be translated into a statement in the new frame of system theory. Though our result is merely one example of the translation from “natural language” into “system theoretical language”, we believe that our method is fairly general.     

On deformation of associative algebras and graph homology

Deformation theory of associative algebras and in particular of Poisson algebras is reviewed. The role of an “almost contraction” leading to a canonical solution of the corresponding Maurer–Cartan equation is noted. This role is reminiscent of the Homotopical Perturbation Lemma, with the infinitesimal deformation cocycle as “initiator.”Applied to star-products, we show how Moyal's formula can be obtained using such an almost contraction and conjecture that the “merger operation” provides a canonical solution at least in the case of linear Poisson structures.     

Willis环状脑动脉瘤系统的自适应模糊控制

针对不确定非线性生物系统—W illis环状脑动脉瘤系统,利用高斯型模糊逻辑系统的逼近能力及新构造的Lyapunov函数,基于模糊建模提出了一种自适应模糊控制器设计的新方案.该方案把逼近误差引入到控制器设计条件中用以改善系统的动态性能.不但设计简单还保证了控制方法的鲁棒性与稳定性.通过反向传播算法调整模糊基函数参数及递归最小二乘法调整参数向量,θ更新控制律,实现了理想跟踪.从理论上研究了脑动脉瘤内血流速度的非线性行为及控制,具有实际意义.仿真结果表明该控制方法的有效性.     

Occurrence and underlying mechanism of multi-stripe chaotic attractors

This paper is concerned with the generation of multi-stripe chaotic attractors. Simple periodic nonlinear functions are employed to transform the original chaotic attractors to a pattern with multiple “parallel” or “rectangular” stripes. The relationship between the system parameters related to some periodic functions and the shape of the generated attractor is analyzed. Theoretic analysis about the underlying mechanism of generating the parallel stripes in the attractors is given. A general creation mechanism of multi-stripe attractors of the Lorenz system and other well-known chaotic systems is derived from the proposed unified approach.     

A reliable technique for solving the weakly singular second-kind Volterra-type integral equations

In this paper, the Adomian decomposition method and the phenomenon of the self-canceling “noise” terms are used for solving the weakly singular Volterra-type, linear and nonlinear, integral equations. The solution obtained is in the form of a convergent power series with elegantly computable terms. Comparing this scheme with many collocation-type methods that use the nonpolynomial basis shows that the present approach is effective and powerful. Many test modeling problems from mathematical physics, linear and nonlinear, are discussed to illustrate the effectiveness and the performance of the decomposition method.     

A general fuzzy control scheme for nonlinear processes with stability analysis

In this paper we develop a general fuzzy control scheme for nonlinear processes. Assuming little knowledge about the dynamics of the controlled process, the proposed scheme starts by probing the process at different points in its operating region to generate a fuzzy quantisation. A simple local controller is then designed at each fuzzy locality. A fuzzy inference mechanism then links up tje local controllers to form a global controller which can be further refined by the learning algorithm. By employing a newly developed structure-adaptive fuzzy modelling scheme, the appropriate fuzzy rule-base for the inference mechanism can be extracted stably and efficiently. The conditions for the stability of the global controller are rigourously established. Simulation results are presented to illustrate the effectiveness of the scheme.     

Fuzzy Relation Equations for Compression/Decompression Processes of Colour Images in the RGB and YUV Colour Spaces

We use particular fuzzy relation equations for compression/decompression of colour images in the RGB and YUV spaces, by comparing the results of the reconstructed images obtained in both cases. Our tests are made over well known images of 256×256 pixels (8 bits per pixel in each band) extracted from Corel Gallery. After the decomposition of each image in the three bands of the RGB and YUV colour spaces, the compression is performed using fuzzy relation equations of “min - →_t” type, where “t” is the Lukasiewicz t-norm and “→_t” is its residuum. Any image is subdivided in blocks and each block is compressed by optimizing a parameter inserted in the Gaussian membership functions of the fuzzy sets, used as coders in the fuzzy equations. The decompression process is realized via a fuzzy relation equation of max-t type. In both RGB and YUV spaces we evaluate and compare the root means square error (RMSE) and the consequentpeak signal to noise ratio (PSNR) on the decompressed images with respect to the original image under several compression rates.     

Stability analysis of sampled-data fuzzy controller for nonlinear systems based on switching T–S fuzzy model

This paper investigates the system stability of a sampled-data fuzzy-model-based control system, formed by a nonlinear plant and a sampled-data fuzzy controller connected in a closed loop. The sampled-data fuzzy controller has an advantage that it can be implemented using a microcontroller or a digital computer to lower the implementation cost and time. However, discontinuity introduced by the sampling activity complicates the system dynamics and makes the stability analysis difficult compared with the pure continuous-time fuzzy control systems. Moreover, the favourable property of the continuous-time fuzzy control systems which is able to relax the stability analysis result vanishes in the sampled-data fuzzy control systems. A Lyapunov-based approach is employed to derive the LMI-based stability conditions to guarantee the system stability. To facilitate the stability analysis, a switching fuzzy model consisting of some local fuzzy models is employed to represent the nonlinear plant to be controlled. The comparatively less strong nonlinearity of each local fuzzy model eases the satisfaction of the stability conditions. Furthermore, membership functions of both fuzzy model and sampled-data fuzzy controller are considered to alleviate the conservativeness of the stability analysis result. A simulation example is given to illustrate the merits of the proposed approach.     

Indirect adaptive fuzzy observer and controller design based on interval type-2 T-S fuzzy model

This paper presents the design scheme of the indirect adaptive fuzzy observer and controller based on the interval type-2 (IT2) T-S fuzzy model. The nonlinear systems can be well approximated by IT2 T-S fuzzy model, in which the fuzzy rules’ antecedents are interval type-2 fuzzy sets and consequents are linear state equations. The proposed IT2 T-S fuzzy model is a combination of IT2 fuzzy system and T-S fuzzy model, and also inherits the benefits of type-2 fuzzy logic systems, which is able to directly handle uncertainties and can minimize the effects of uncertainties in rule-based fuzzy system. These characteristics can improve the accuracy of the system modeling and reduce the number of system rules. The proposed method using feedback control, adaptive laws, and on-line object parameters are adjusted to ensure observation error bounded. In addition, using Lyapunov synthesis approach and Lipschitz condition, the stability analysis is conducted. The simulation results show that the proposed method can handle unpredicted disturbance and data uncertainties very well in advantage of the effectiveness of observation and control.     

Passage to the limit in Proposition I, Book I of Newton's Principia

The purpose of this paper is to analyze the way in which Newton uses his polygon model and passes to the limit in Proposition I, Book I of his Principia. It will be evident from his method that the limit of the polygon is indeed the orbital arc of the body and that his approximation of the actual continuous force situation by a series of impulses passes correctly in the limit into the continuous centripetal force situation. The analysis of the polygon model is done in two ways: (1) using the modern concepts of force, linear momentum, linear impulse, and velocity, and (2) using Newton's concepts of motive force and quantity of motion. It should be clearly understood that the term “force” without the adjective “motive,” is used in the modern sense, which is that force is a vector which is the time rate of change of the linear momentum. Newton did not use the word “force” in this modern sense. The symbol F denotes modern force. For Newton “force” was “motive force,” which is measured by the change in the quantity of motion of a body. Newton's “quantity of motion” is proportional to the magnitude of the modern vector momentum. Motive force is a scalar and the symbol F_m is used for motive force.     

Analysis of stochastic numerical schemes for the evolution equations of geophysics

We present and study the stability and convergence, and order of convergence of a numerical scheme used in geophysics, namely, the stochastic version of a deterministic “implicit leapfrog” scheme which has been developed for the approximation of the so-called barotropic vorticity model. Two other schemes which might be useful in the context of geophysical applications are also introduced and discussed.     

In search of Robbins stability

We speculate on whether a certain p-adic stability phenomenon, observed by David Robbins empirically for Dodgson condensation, appears in other nonlinear recurrence relations that “unexpectedly” produce integer or nearly-integer sequences. We exhibit an example (number friezes) where this phenomenon provably occurs.     

Fuzzy guaranteed cost controller for trajectory tracking in nonlinear systems

In this paper, we propose a fuzzy logic based guaranteed cost controller for trajectory tracking in nonlinear systems. Takagi–Sugeno (T–S) fuzzy model is used to represent the dynamics of a nonlinear system and the controller design is carried out using this fuzzy model. State feedback law is used for building the fuzzy controller whose performance is evaluated using a quadratic cost function. For designing the fuzzy logic based controller which satisfies guaranteed performance, linear matrix inequality (LMI) approach is used. Sufficient conditions are derived in terms of matrix inequalities for minimizing the performance function of the controller. The performance function minimization problem with polynomial matrix inequalities is then transformed into a problem of minimizing a convex performance function involving standard LMIs. This minimization problem can be solved easily and efficiently using the LMI optimization techniques. Our controller design method also ensures that the closed-loop system is asymptotically stable. Simulation study is carried out on a two-link robotic manipulator tracking a reference trajectory. From the results of the simulation study, it is observed that our proposed controller tracks the reference trajectory closely while maintaining a guaranteed minimum cost.