Feedback control of chaotic systems

A suboptimal feedback controller implemented by a multilayer feed-forward neural network is presented to control the unpredictable behavior of chaotic systems. The controller has been tested on the Lorenz and the Rössler systems using numerical simulation. Results show that chaotic systems, subject to feedback control, can be tamed to behave like a system having point attractors with associated basins of attraction.     

Computability in planar dynamical systems

In this paper we explore the problem of computing attractors and their respective basins of attraction for continuous-time planar dynamical systems. We consider C ¹ systems and show that stability is in general necessary (but may not be sufficient) to attain computability. In particular, we show that (a) the problem of determining the number of attractors in a given compact set is in general undecidable, even for analytic systems and (b) the attractors are semi-computable for stable systems. We also show that the basins of attraction are semi-computable if and only if the system is stable.     

Multistability and New Attraction Basins of Almost-Periodic Solutions of Delayed Neural Networks

In this paper, we investigate multistability of almost-periodic solutions of recurrently connected neural networks with delays (simply called delayed neural networks). We will reveal that under some conditions, the space Rn can be divided into 2n subsets, and in each subset, the delayed n -neuron neural network has a locally stable almost-periodic solution. Furthermore, we also investigate the attraction basins of these almost-periodic solutions. We reveal that the attraction basin of almost-periodic trajectory is larger than the subset, where the corresponding almost-periodic trajectory is located. In addition, several numerical simulations are presented to corroborate the theoretical results.     

基于胞参考点映射法对转子/定子碰摩响应的全局分析

利用胞参考点映射法对两自由度(四维)分段光滑的转子/定子碰摩模型的全局响应特性进行了计算,着重对解析分析得到的碰摩系统典型的吸引子共存现象进行了验证,确定了相应吸引子的吸引域.此外,通过计算转子/定子碰摩系统全局特性随系统参数变化的演化特点,考察了吸引子与其吸引域边界对参数变化的敏感性,讨论了其对实际转子/定子碰摩系统...     

Optimal Newton–Secant like methods without memory for solving nonlinear equations with its dynamics

We construct two optimal Newton–Secant like iterative methods for solving nonlinear equations. The proposed classes have convergence order four and eight and cost only three and four function evaluations per iteration, respectively. These methods support the Kung and Traub conjecture and possess a high computational efficiency. The new methods are illustrated by numerical experiments and a comparison with some existing optimal methods. We conclude with an investigation of the basins of attraction of the solutions in the complex plane.     

The Ellipsoidal Invariant Set of Fractional Order Systems Subject to Actuator Saturation: The Convex Combination Form

The domain of attraction of a class of fractional order systems subject to saturating actuators is investigated in this paper. We show the domain of attraction is the convex hull of a set of ellipsoids. In this paper, the Lyapunov direct approach and fractional order inequality are applied to estimating the domain of attraction for fractional order systems subject to actuator saturation. We demonstrate that the convex hull of ellipsoids can be made invariant for saturating actuators if each ellipsoid with a bounded control of the saturating actuators is invariant. The estimation on the contractively invariant ellipsoid and construction of the continuous feedback law are derived in terms of linear matrix inequalities (LMIs). Two numerical examples illustrate the effectiveness of the developed method.      

Galerkin approximationwith Legendre polynomials for a continuous-time nonlinear optimal control problem

We investigate the use of an approximation method for obtaining near-optimal solutions to a kind of nonlinear continuous-time (CT) system. The approach derived from the Galerkin approximation is used to solve the generalized Hamilton-Jacobi-Bellman (GHJB) equations. The Galerkin approximation with Legendre polynomials (GALP) for GHJB equations has not been applied to nonlinear CT systems. The proposed GALP method solves the GHJB equations in CT systems on some well-defined region of attraction. The integrals that need to be computed are much fewer due to the orthogonal properties of Legendre polynomials, which is a significant advantage of this approach. The stabilization and convergence properties with regard to the iterative variable have been proved. Numerical examples show that the update control laws converge to the optimal control for nonlinear CT systems.     

Multistability of complex-valued neural networks with distributed delays

This paper addresses the multistability problem for the complex-valued neural networks with appropriate real–imaginary-type activation functions and distributed delays. Based on the geometrical properties of the activation functions and the fixed point theory, several sufficient criteria are obtained which not only guarantee the existence of \(9^n\) equilibrium points but also assure the local exponential stability for the \(4^n\) equilibrium points of them. Furthermore, the attraction basins of the \(4^n\) equilibrium points are also estimated, which infers that the attraction basins could be enlarged under some mild restrictions. Finally, one numerical example is provided to illustrate the effectiveness of the obtained results.

     

Estimating the domain of attraction of nonlinear feedback systems

An iterative procedure is developed for estimating the domain of attraction of a class of nonlinear systems. The procedure uses an approximation of the system by a Carleman linearization and is suitable for automatic computations. The algorithm is applied to the transient stability problem in electrical power systems and a favorable comparison with previous criteria is obtained.     

The convergence properties of a clipped Hopfield network and itsapplication in the design of keystream generator

We first present a modified Hopfield network, the clipped Hopfield network, with synaptic weights assigned to three values {-1,0,+1}. We give the necessary conditions under which a set of 2n binary vectors can be stored as stable points of the network. We show that in the parallel updating mode, for most of the state vectors, the network will always converge to these 2n stable points. We further demonstrate that these 2n stable points can be divided into two groups, the alpha group and the beta group, each with n stable points. It is shown that the basins of attraction of the stable points in the alpha group are evenly distributed, and the basins of attraction of the stable points in the beta group are also evenly distributed. By ways of application, we show that this class of Hopfield network can be used to build a cryptographically secure keystream generator.     

Fault-tolerant control of delta operator systems with actuator saturation and effectiveness loss

This paper studies the problem of robust fault-tolerant control against the actuator effectiveness loss for delta operator systems with actuator saturation. Ellipsoids are used to estimate the domain of attraction for the delta operator systems with actuator saturation and effectiveness loss. Some invariance set conditions used for enlarging the domain of attraction are expressed by linear matrix inequalities. Discussions on system performance optimisation are presented in this paper, including reduction on computational complexity, expansion of the domain of attraction and disturbance rejection. Two numerical examples are given to illustrate the effectiveness of the developed techniques.     

Stabilization of one class of nonlinear systems

Consideration was given to the problem of stabilization for one class of systems that are nonlinear and uncontrollable in the first approximation. The stabilization problem was solved by considering the nonlinear approximation. The stabilizing control was obtained by the method of Lyapunov function which was constructed as a quadratic form. To determine matrix of this quadratic form, a singular matrix Lyapunov equation was solved. For the system of nonlinear approximation, the stabilizing control was determined explicitly. It was proved that the resulting control solves the problem of stabilizing the original nonlinear system. An ellipsoidal estimate of the attraction domain of the zero stationary point was given.     

Attractor Spaces as Modules: A Semi-Eliminative Reduction of Symbolic AI to Dynamic Systems Theory

I propose a semi-eliminative reduction of Fodors concept of module to the concept of attractor basin which is used in Cognitive Dynamic Systems Theory (DST). I show how attractor basins perform the same explanatory function as modules in several DST based research program. Attractor basins in some organic dynamic systems have even been able to perform cognitive functions which are equivalent to the If/Then/Else loop in the computer language LISP. I suggest directions for future research programs which could find similar equivalencies between organic dynamic systems and other cognitive functions. This type of research could help us discover how (and/or if) it is possible to use Dynamic Systems Theory to more accurately model the cognitive functions that are now being modeled by subroutines in Symbolic AI computer models. If such a reduction of subroutines to basins of attraction is possible, it could free AI from the limitations that prompted Fodor to say that it was impossible to model certain higher level cognitive functions.     

Queueing systems with service interruptions: An approximation model

We present a single queueing model which can be used to analyse queueing systems with service interruptions. The model is based on a diffusion approximation using an instantaneous return process which reflects the particular queueing system under consideration. Applications to queueing systems with preemptive resume priority and breakdowns are given in this paper as well as numerical results showing the accuracy of the approximation.     

Extension of WAF Type Methods to Non-Homogeneous Shallow Water Equations with Pollutant

This paper deals with the extension of the WAF method to discretize Shallow Water Equations with pollutants. We consider two different versions of the WAF method, by approximating the intermediate waves using the flux of HLL or the direct approach of HLLC solver. It is seen that both versions can be written under the same form with different definitions for the approximation of the velocity waves. We also propose an extension of the method to non-homogeneous systems. In the case of homogeneous systems it is seen that we can rewrite the third component of the numerical flux in terms of an intermediate wave speed approximation. We conclude that—in order to have the same relation for non-homogeneous systems—the approximation of the intermediate wave speed must be modified. The proposed extension of the WAF method preserves all stationary solutions, up to second order accuracy, and water at rest in an exact way, even with arbitrary pollutant concentration. Finally, we perform several numerical tests, by comparing it with HLLC solver, reference solutions and analytical solutions.     

Numerical and computational aspects of some block-preconditioners for saddle point systems

Linear systems with two-by-two block matrices are usually preconditioned by block lower- or upper-triangular systems that require an approximation of the related Schur complement. In this work, in the finite element framework, we consider one special such approximation, namely, the element-wise Schur complement. It is sparse and its construction is perfectly parallelizable, making it an appropriate ingredient when building preconditioners for iterative solvers executed on both distributed and shared memory computer architectures. For saddle point matrices with symmetric positive (semi-)definite blocks we show that the Schur complement is spectrally equivalent to the so-constructed approximation and derive spectral equivalence bounds. We also illustrate the quality of the approximation for nonsymmetric problems, where we observe the same good numerical efficiency.Furthermore, we demonstrate the computational and numerical performance of the corresponding preconditioned iterative solution method on a large scale model benchmark problem originating from the elastic glacial isostatic adjustment model discretized using the finite element method.     

Constrained piecewise linear systems with disturbances: Controller design via convex invariant sets

This paper presents two control strategies under the time optimal control and model predictive control frameworks for constrained piecewise linear systems with bounded disturbances (PWLBD systems). Each of the proposed approaches uses an inner convex polytopal approximation of the non‐convex domains of attraction and results in simplified control laws that can be determined off‐line via multi‐parametric programming. These control strategies rely on invariant sets of PWLBD systems. Thereby, approaches for the computation of the disturbance invariant outer bounds of the minimal disturbance invariant set, F_∞, and convex polytopal disturbance invariant sets are presented. The effectiveness of the approaches is assessed through numerical examples. Copyright © 2007 John Wiley & Sons, Ltd.     

Uncertainty of asymptotic dynamics in bioresource management simulation

Specific features of simulating a system that experiences structural changes in the course of its development are studied by the example of a controlled biological population. The problem of whether nonattracting chaotic sets can arise in dynamic systems with more than one attractor is considered in this context. The formalism of hybrid automata as applied to simulation problems of biological processes is described. Specific features of the phase portrait of the developed dynamic model are characterized by locally disconnected boundaries of basins of attraction of two attractors. The conclusion on limited predictability of the dynamics of some controlled natural systems is made, which is a consequence of uncertainty with respect to the motion of the system towards one of possible stable states.     

Constrained stabilization of continuous-time linear systems

In this paper we consider the stabilization problem for linear continuous-time systems, under state and control constraints. We show that the largest domain of attraction to the origin can be arbitrarily closely approximated by a polyhedral domain of attraction associated to a certain (continuous) feedback stabilizing control and we show how to use existing numeric procedures for discrete-time systems to solve the continuous-time problem. We propose a new discontinuous stabilizing control law for scalar-input systems which has the advantage of being successfully applicable to systems with quantized control.     

On the computation of the robust viability kernel for switched systems

This paper presents an algorithm of computing the robust viability kernel for discrete-time switched systems with arbitrary switching rule and bounded disturbance based on the robust one-step set and Pontryagin difference. The algorithm can be implemented by performing the relevant set computations using polyhedral algebra and computational geometry software when switched systems are linear. In addition, we propose a method of computing inner approximation of the robust viability kernel for switched systems based on the set theory. The convergence of the iterative algorithm is proved using a null controllable set. Finally, two examples are provided to show the effectiveness of the proposed method.