Managements of scalar and vector rogue waves in a partially nonlocal nonlinear medium with linear and harmonic potentials

Nonlinear Dynamics - This paper proposes a series of fractional-order control methods (FOCMs) based on fractional calculus (FC) for a class of general nonlinear systems. In order to deal with the...     

A General Formulation and Solution Scheme for Fractional Optimal Control Problems

Accurate modeling of many dynamic systems leads to a set of Fractional Differential Equations (FDEs). This paper presents a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The fractional derivative is described in the Riemann–Liouville sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of FDEs. The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler–Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic system. The formulation is used to derive the control equations for a quadratic linear fractional control problem. An approach similar to a variational virtual work coupled with the Lagrange multiplier technique is presented to find the approximate numerical solution of the resulting equations. Numerical solutions for two fractional systems, a time-invariant and a time-varying, are presented to demonstrate the feasibility of the method. It is shown that (1) the solutions converge as the number of approximating terms increase, and (2) the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs. It is hoped that the simplicity of this formulation will initiate a new interest in the area of optimal control of fractional systems.     

A General Formulation and Solution Scheme for Fractional Optimal Control Problems

Accurate modeling of many dynamic systems leads to a set of Fractional Differential Equations (FDEs). This paper presents a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The fractional derivative is described in the Riemann–Liouville sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of FDEs. The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler–Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic system. The formulation is used to derive the control equations for a quadratic linear fractional control problem. An approach similar to a variational virtual work coupled with the Lagrange multiplier technique is presented to find the approximate numerical solution of the resulting equations. Numerical solutions for two fractional systems, a time-invariant and a time-varying, are presented to demonstrate the feasibility of the method. It is shown that (1) the solutions converge as the number of approximating terms increase, and (2) the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs. It is hoped that the simplicity of this formulation will initiate a new interest in the area of optimal control of fractional systems.     

Nonlinear problems of shell theory and their solution methods (review)

Conclusions In the present review we considered several problems of nonlinear shell theory, their deformation and stability. Speciailsts in many areas have devoted much attention recently to problems of the nonlinear behavior of evolutionary systems. New scientific trends related to the study of stability problems have been generated. They can be called the Hopf bifurcation theory; or, the theory of attractors, referring to the study of unstable trajectory behavior in phase spaces; or, catastrophe theory, applied in solving multiparametric stability problems; or, the theory of phase transitions and critical phenomena, developed in special areas of physics; or, a theory considering nonequilibrium phase transitions and self-organization complex spatial and temporal ordered structures (synergetics).The use of methods developed by these scientific trends in shell theory enables deep understanding of effects of stability loss, in both equilibrium and motion, and helps to create investigation methods.Institute of Mechanics, Ukrainian Academy of Sciences, Kiev. Kiev Institute of Structural Engineering. Translated from Prikladnaya Mekhanika, Vol. 27, No. 10, pp. 3–23, October, 1991.     

Dynamics and optimal control of multibody systems using fractional generalized divide-and-conquer algorithm

In this paper, a new framework is presented for the dynamic modeling and control of fully actuated multibody systems with open and/or closed chains as well as disturbance in the position, velocity, acceleration, and control input of each joint. This approach benefits from the computed torque control method and embedded fractional algorithms to control the nonlinear behavior of a multibody system. The fractional Brunovsky canonical form of the tracking error is proposed for a generalized divide-and-conquer algorithm (GDCA) customized for having a shortened memory buffer and faster computational time. The suite of a GDCA is highly efficient. It lends itself easily to the parallel computing framework, that is used for the inverse and forward dynamic formulations. This technique can effectively address the issues corresponding to the inverse dynamics of fully actuated closed-chain systems. Eventually, a new stability criterion is proposed to obtain the optimal torque control using the new fractional Brunovsky canonical form. It is shown that fractional controllers can robustly stabilize the system dynamics with a smaller control effort and a better control performance compared to the traditional integer-order control laws.

     

No-chatter variable structure control for fractional nonlinear complex systems

This paper concerns the problem of robust control of uncertain fractional-order nonlinear complex systems. After establishing a simple linear sliding surface, the sliding mode theory is used to derive a novel robust fractional control law for ensuring the existence of the sliding motion in finite time. We use a nonsmooth positive definitive function to prove the stability of the controlled system based on the fractional version of the Lyapunov stability theorem. In order to avoid the chattering, which is inherent in conventional sliding mode controllers, we transfer the sign function of the control input into the first derivative of the control signal. The proposed sliding mode approach is also applied for control of a class of nonlinear fractional-order systems via a single control input. Simulation results indicate that the proposed fractional variable structure controller works well for stabilization of hyperchaotic and chaotic complex fractional-order nonlinear systems. Moreover, it is revealed that the control inputs are free of chattering and practical.     

Dynamic behavior of fractional order Duffing chaotic system and its synchronization via singly active control

With the increasingly deep studies in physics and technology,the dynamics of fractional order nonlinear systems and the synchronization of fractional order chaotic systems have become the focus in scientific research.In this paper,the dynamic behavior including the chaotic properties of fractional order Duffing systems is extensively investigated.With the stability criterion of linear fractional systems,the synchronization of a fractional non-autonomous system is obtained.Specifically,an effective singly active control is proposed and used to synchronize a fractional order Duffing system.The numerical results demonstrate the effectiveness of the proposed methods.     

Forced sliding mode control for chaotic systems synchronization

Synchronization of chaotic systems is considered to be a common engineering problem. However, the proposed laws of synchronization control do not always provide robustness toward the parametric perturbations. The purpose of this article is to show the use of synergy-cybernetic approach for the construction of robust law for Arneodo chaotic systems synchronization. As the main method of design of robust control, the method of design of control with forced sliding mode of the synergetic control theory is considered. To illustrate the effectiveness of the proposed law, in this article it is compared with the classical sliding mode control and adaptive backstepping. The distinctive features of suggested robust control law are the more good compensation of parametric perturbations (better performance indexes—the root-mean-square error (RMSE), average absolute value (AVG) of error) without designing perturbation observers, the ability to exclude the chattering effect, less energy consuming and a simpler analysis of the stability of a closed-loop system. The study of the proposed control law and the change of its parameters and the place of parametric perturbation’s application is carried out. It is possible to significantly reduce the synchronization error and RMSE, as well as AVG of error by reducing some parameters, but that leads to an increase in control signal amplitude. The place of application of parametric disturbances (slave or master system) has no effect on the RMSE and AVG of error. Offered approach will allow a new consideration for the design of robust control laws for chaotic systems, taking into account the ideas of directed self-organization and robust control. It can be used for synchronization other chaotic systems.

     

A Lyapunov-based control scheme for robust stabilization of fractional chaotic systems

This paper concerns the problem of robust stabilization of autonomous and non-autonomous fractional-order chaotic systems with uncertain parameters and external noises. We propose a simple efficient fractional integral-type sliding surface with some desired stability properties. We use the fractional version of the Lyapunov theory to derive a robust sliding mode control law. The obtained control law is single input and guarantees the occurrence of the sliding motion in a given finite time. Furthermore, the proposed nonlinear control strategy is able to deal with a large class of uncertain autonomous and non-autonomous fractional-order complex systems. Also, Rigorous mathematical and analytical analyses are provided to prove the correctness and robustness of the introduced approach. At last, two illustrative examples are given to show the applicability and usefulness of the proposed fractional-order variable structure controller.     

Observer-based fractional-order adaptive type-2 fuzzy backstepping control of uncertain nonlinear MIMO systems with unknown dead-zone

Nonlinear Dynamics - A new problem of observer-based fractional adaptive type-2 fuzzy backstepping control for a class of fractional-order MIMO nonlinear dynamic systems with dead-zone input...     

拟哈密顿系统非线性随机最优控制

主要介绍近十几年来拟哈密顿系统非线性随机最优控制理论方法及其应用的研究成果, 包括基于拟哈密顿系统随机平均法与随机动态规划原理的非线性随机最优控制基本策略, 即响应极小化控制、随机稳定化、首次穿越损坏最小化控制、以概率密度为目标的控制, 为将它们应用于工程实际而作的部分可观测系统最优控制、有界控制、时滞控制、半主动控制、极小极大控制的进一步研究, 以及综合考虑这些实际问题的非线性随机最优控制的综合策略, 非线性随机最优控制在滞迟系统、分数维系统等中的若干应用, 介绍与这些研究有关的背景, 并指出今后有待进一步研究的问题.     

Nonlinear dynamics and neo-piagetian theories in problem solving: perspectives on a new epistemology and theory development

In this study, an attempt is made to integrate Nonlinear Dynamical Systems theory and neo-Piagetian theories applied to creative mental processes, such as problem solving. A catastrophe theory model is proposed, which implements three neo-Piagetian constructs as controls: the functional M-capacity as asymmetry and logical thinking and the degree of field dependence independence as bifurcation. Data from achievement scores of students in tenth grade physics were analyzed using dynamic difference equations and statistical regression techniques. The cusp catastrophe model proved superior comparing to the pre-post linear counterpart and demonstrated nonlinearity at the behavioral level. The nonlinear phenomenology, such as hysteresis effects and bifurcation, is explained by an analysis, which provides a causal interpretation via the mathematical theory of self-organization and thus building bridges between NDS-theory concepts and neo-Piagetian theories. The contribution to theory building is made, by also addressing the emerging philosophical, - ontological and epistemological- questions about the processes of problem solving and creativity.     

Conserved quantities and adiabatic invariants for fractional generalized Birkhoffian systems

Based on Riemann-Liouville fractional derivatives, conserved quantities and adiabatic invariants for fractional generalized Birkhoffian systems are investigated. Firstly, fractional generalized Birkhoff equations are obtained by studying fractional generalized Pfaff-Birkhoff principle. Secondly, the definition of fractional generalized quasi-symmetry is given, the criteria of fractional generalized quasi-symmetry and the corresponding conserved quantity are achieved for fractional generalized Birkhoffian systems. Thirdly, perturbation to symmetry and adiabatic invariants for disturbed fractional generalized Birkhoffian systems are presented. Finally, an example is given to illustrate the results.     

Synchronization and stabilization of fractional order nonlinear systems with adaptive fuzzy controller and compensation signal

In this paper, a direct adaptive fuzzy controller with compensation signal is presented to control and stabilize a class of fractional order systems with unknown nonlinearities. Based on a Lyapunov function candidate the global Mittag–Leffler stability is proved and a new fractional order adaptation law is derived. The adaptation law adjusts free parameters of the fuzzy controller and bounds them by utilizing a novel fractional order projection algorithm. Furthermore, due to the use of compensation term, the proposed approach does not demand suitable membership functions in the fuzzy system. In addition, the stability of the closed-loop system is guaranteed by utilizing a supervisory controller. Numerical simulations show the validity and effectiveness of the introduced scheme for various fractional order nonlinear models that perturbed by disturbance and uncertainty.     

Homogeneous flow field effect on the control of Maxwell materials

The controllability of viscoelastic fields is a fundamental concept that defines some essential capabilities and limitations of the resulting materials. In this paper, we study the controllability of different homogeneous flow fields of viscoelastic fluids governed by the upper convected Maxwell model. The approach is largely based on the nonlinear geometric control theory. Through the analysis of the control Lie algebra, we find the submanifolds in the state space on which the homogeneous flow fields are weakly controllable. Our approach can be generalized to more complicated systems.     

分数阶对称金融模型复杂度分析及有限时间同步

分析了一类分数阶对称金融非线性系统的复杂度特性,利用有限时间同步理论设计控制器,实现了有限时间同步。根据分数阶系统定义和Adomain分解法对该系统的非线性项进行Adomain分解,结合分解系数定义系统的表达式,将其离散化。基于谱熵复杂度及C0复杂度的基本算法,利用Matlab仿真其复杂度曲线及复杂度图谱。为进一步探究对称金融非线性系统的动力学特性,利用有限时间同步理论设计误差控制器,实现有限时间同步,仿真结果表明该控制器可使系统在极短的时间内实现同步且鲁棒性好。     

Finite-time chaos control and synchronization of fractional-order nonautonomous chaotic (hyperchaotic) systems using fractional nonsingular terminal sliding mode technique

In this paper, a novel fractional-order terminal sliding mode control approach is introduced to control/synchronize chaos of fractional-order nonautonomous chaotic/hyperchaotic systems in a given finite time. The effects of model uncertainties and external disturbances are fully taken into account. First, a novel fractional nonsingular terminal sliding surface is proposed and its finite-time convergence to zero is analytically proved. Then an appropriate robust fractional sliding mode control law is proposed to ensure the occurrence of the sliding motion in a given finite time. The fractional version of the Lyapunov stability is used to prove the finite-time existence of the sliding motion. The proposed control scheme is applied to control/synchronize chaos of autonomous/nonautonomous fractional-order chaotic/hyperchaotic systems in the presence of both model uncertainties and external disturbances. Two illustrative examples are presented to show the efficiency and applicability of the proposed finite-time control strategy. It is worth to notice that the proposed fractional nonsingular terminal sliding mode control approach can be applied to control a broad range of nonlinear autonomous/nonautonomous fractional-order dynamical systems in finite time.     

Introduction to Nonlinear Dynamics of Electronic Systems: Tutorial

The study of chaos has generated enormous interest in exploring the complexity of the behavior in nature and in technology. Many of the important features of chaotic dynamical systems can be seen using experimental and computational methods in simple nonlinear mechanical systems or electronic circuits. Starting with the study of a chaotic nonlinear mechanical system (driven damped pendulum) or a nonlinear electronic system (circuit Chua) we introduce the reader into the concepts of chaos order in Sharkovsky's sense, and topological invariants (topological entropy and topological frequencies). The Kirchhoff's circuit laws are a pair of laws that deal with the conservation of charge and energy in electric circuits, and the algebraic theory of graphs characterizes these linear systems in terms of cycles and cocycles (or cuts). Here we discuss methods (topological semiconjugacy to piecewise linear maps and Markov graphs) to find a similar situation for the nonlinear dynamics, to understanding chaotic dynamics. Thus to chaotic dynamics we associate a Markov graph, where the dynamical and topological invariants will be seen as graph theoretical quantities.     

Chaos suppression in a fractional order financial system using intelligent regrouping PSO based fractional fuzzy control policy in the presence of fractional Gaussian noise

Financial systems are known to have irregular and erratic fluctuations due to diverse influences and often result in economic crisis and huge financial losses. Recent models of financial systems show that they behave chaotically and have long range memory dependence. Mitigating these undesirable chaotic natures of financial systems by appropriate control policies is important in order to reduce investment risks and improve economic performance. In this paper, a fractional order fuzzy control policy is employed to suppress the chaotic dynamics of a representative chaotic fractional order financial system. An intelligent Regrouping Particle Swarm Optimization (Reg-PSO) is used to design the numeric weights of the control policy and the methodology is demonstrated by credible simulations. The designed fractional fuzzy control policies are shown to work well with respect to conventional fuzzy control policies in the presence of persistent and anti-persistent noise, which can be due to additional extraneous influences on the system.     

Fractal sets in control systems

In this paper the Hausdorff measure of sets of integral and fractional dimensions is introduced and applied to control systems.A new concept,namely,pseudo-self-similar set is also introduced.The existence and uniqueness of such sets are then proved,and the formula for calculating the dimension of self-similar sets is extended to the psuedo-self-similar case.Using the previous theorem,we show that the reachable set of a control system may have fractional dimensions.We hope that as a new approach the geometry of fractal sets will be a proper tool to analyze the controllability and observability of nonlinear systems.