Chaos control of uncertain time‐delay chaotic systems with input dead‐zone nonlinearity

This article presents an adaptive sliding mode control (SMC) scheme for the stabilization problem of uncertain time‐delay chaotic systems with input dead‐zone nonlinearity. The algorithm is based on SMC, adaptive control, and linear matrix inequality technique. Using Lyapunov stability theorem, the proposed control scheme guarantees the stability of overall closed‐loop uncertain time‐delay chaotic system with input dead‐zone nonlinearity. It is shown that the state trajectories converge to zero asymptotically in the presence of input dead‐zone nonlinearity, time‐delays, nonlinear real‐valued functions, parameter uncertainties, and external disturbances simultaneously. The selection of sliding surface and the design of control law are two important issues, which have been addressed. Moreover, the knowledge of upper bound of uncertainties is not required. The reaching phase and chattering phenomenon are eliminated. Simulation results demonstrate the effectiveness and robustness of the proposed scheme. © 2014 Wiley Periodicals, Inc. Complexity 21: 13–20, 2016     

Sector-condition-based results for adaptive control and synchronization of chaotic systems under input saturation

This paper addresses the design of adaptive feedback controllers for two problems (namely, stabilization and synchronization) of chaotic systems with unknown parameters by considering input saturation constraints. A novel generalized sector condition is developed to deal with the saturation nonlinearities for synthesizing the nonlinear and the adaptive controllers for the stabilization and synchronization control objectives. By application of the proposed sector condition and rigorous regional stability analysis, control and adaptation laws are formulated to guarantee local stabilization of a nonlinear system under actuator saturation. Further, simple control and adaptation laws are developed to synchronize two chaotic systems under uncertain parameters and input saturation nonlinearity. Numerical simulation results for Rössler and FitzHugh–Nagumo models are provided to demonstrate the effectiveness of the proposed adaptive stabilization and synchronization control methodologies.     

Anti-synchronization of uncertain unified chaotic systems with dead-zone nonlinearity

This paper addresses chaos anti-synchronization of uncertain unified chaotic systems with dead-zone input nonlinearity. Using the sliding mode control technique and Lyapunov stability theory, a proportional–integral (PI) switching surface is proposed to ensure the stability of the closed-loop error system in sliding mode. Then a sliding mode controller (SMC) is proposed to guarantee the hitting of the switching surface even with uncertainties and the control input containing dead-zone nonlinearity. Some simulation results are included to demonstrate the effectiveness and feasibility of the proposed synchronization scheme.     

Synchronization of different dimensional chaotic systems with time varying parameters, disturbances and input nonlinearities

In this paper, the problems of robust exponential generalized and robust exponential Q-S chaos synchronization are investigated between different dimensional chaotic systems. We consider the more practical and realistic cases when unknown time varying parameters with uncertainties, environmental disturbances, and nonlinearity of input control signals are present. The adaptive technique is employed to design the appropriate controllers and the validity of the proposed controllers are proved using Lyapunov stability theorem. Furthermore, numerical simulations are performed to show the efficiency of the presented scheme.     

Switching adaptive controllers to control fractional‐order complex systems with unknown structure and input nonlinearities

This article investigates the chaos control problem for the fractional‐order chaotic systems containing unknown structure and input nonlinearities. Two types of nonlinearity in the control input are considered. In the first case, a general continuous nonlinearity input is supposed in the controller, and in the second case, the unknown dead‐zone input is included. In each case, a proper switching adaptive controller is introduced to stabilize the fractional‐order chaotic system in the presence of unknown parameters and uncertainties. The control methods are designed based on the boundedness property of the chaotic system's states, where, in the proposed methods the nonlinear/linear dynamic terms of the fractional‐order chaotic systems are assumed to be fully unknown. The analytical results of the mentioned techniques are proved by the stability analysis theorem of fractional‐order systems and the adaptive control method. In addition, as an application of the proposed methods, single input adaptive controllers are adopted for control of a class of three‐dimensional nonlinear fractional‐order chaotic systems. And finally, some numerical examples illustrate the correctness of the analytical results. © 2014 Wiley Periodicals, Inc. Complexity 21: 211–223, 2015     

Adaptive terminal sliding mode control subject to input nonlinearity for synchronization of chaotic gyros

Under the existence of system uncertainties, external disturbances, and input nonlinearity, complete synchronization and anti-synchronization between two chaotic gyros are achieved by introducing a novel adaptive terminal sliding mode (ATSM) controller. In the literature, by taking account of input nonlinearity, the magnitudes of bounded nonlinear dynamics of synchronous error system were required in the designed sliding mode controller. In this study, the proposed ATSM controller associated with time-varying feedback gains can tackle nonlinear dynamics according to the novel adaptive rules. These feedback gains are not necessary to be determined in advance but updated by the adaptive rules without known the magnitudes of bounded nonlinear dynamics, system uncertainties, and external disturbances. Sufficient conditions to guarantee stable synchronization are given in the sense of the Lyapunov stability theorem, and the numerical simulations are performed to verify the effectiveness of presented schemes.     

Unknown nonlinear chaotic gyros synchronization using adaptive fuzzy sliding mode control with unknown dead-zone input

In this paper, the problem of synchronizing two chaotic gyros in the presence of uncertainties, external disturbances and dead-zone nonlinearity in the control input is studied while the structure of the gyros, parameters of the dead-zone and the bounds of uncertainties and external disturbances are unknown. The dead-zone nonlinearity in the control input might cause the perturbed chaotic system to show unpredictable behavior. This is due to the high sensitivity of these systems to small changes in their parameters. Thereby, the effect of these issues should not be ignored in the control design for these systems. In order to eliminate the effects from the dead-zone nonlinearity, in this paper, a robust adaptive fuzzy sliding mode control scheme is proposed to overcome the synchronization problem for a class of unknown nonlinear chaotic gyros. The main contribution of our paper in comparison with other works that attempt to solve the problem of dead-zone in the synchronization of chaotic gyros is that we assume that the structure of the system, uncertainties, external disturbances, and dead-zone are fully unknown. Simulation results are provided to illustrate the effectiveness of the proposed method.     

Robust tracking and model following for uncertain time‐delay systems with input nonlinearity

This article proposes a novel adaptive sliding mode control (SMC) scheme to realize the problem of robust tracking and model following for a class of uncertain time‐delay systems with input nonlinearity. It is shown that the proposed robust tracking controller guarantees the stability of overall closed‐loop system and achieves zero‐tracking error in the presence of input nonlinearity, time‐delays, time‐varying parameter uncertainties and external disturbances. The selection of sliding surface and the existence of sliding mode are two important issues, which have been addressed. This scheme assures robustness against input nonlinearity, time‐delays, parameter uncertainties, and external disturbances. Moreover, the knowledge of the upper bound of uncertainties is not required and chattering phenomenon is eliminated. Both theoretical analysis and illustrative examples demonstrate the validity of the proposed scheme. © 2014 Wiley Periodicals, Inc. Complexity 21: 66–73, 2015     

Adaptive type-2 fuzzy backstepping control of uncertain fractional-order nonlinear systems with unknown dead-zone

A new problem of adaptive type-2 fuzzy fractional control with pseudo-state observer for commensurate fractional order dynamic systems with dead-zone input nonlinearity is considered in presence of unmatched disturbances and model uncertainties; the control scheme is constructed by using the backstepping and adaptive technique. To avoid the complexity of backstepping design process, the dynamic surface control is used. Also, Interval type-2 Fuzzy logic systems (IT2FLS) are used to approximate the unknown nonlinear functions. By using the fractional adaptive backstepping, fractional control laws are constructed; this method is applied to a class of uncertain fractional-order nonlinear systems. In order to better control performance in reducing tracking error, the PSO algorithm is utilized for tuning the controller parameters. Stability of the system is proven by the Mittag–Leffler method. It is shown that the proposed controller guarantees the boundedness property for the system and also the tracking error can converge to a small neighborhood of the origin. The efficiency of the proposed method is illustrated with simulation examples.     

电液位置伺服系统的鲁棒自适应控制

针对由于参数不确定性、非线性等因素导致的电液位置伺服系统跟踪控制问题,基于Lyapunov(李雅普诺夫)稳定性理论,提出了一种具有参数自适应能力的鲁棒自适应反步方法.通过设计的自适应律来抑制由于参数不确定性对系统跟踪控制性能的影响,设计的鲁棒控制律使得系统具有全局一致渐近稳定性能.此外,还对伺服阀换向引起的不连续性进行了近似处理.以伺服阀控对称缸系统为控制对象,仿真结果表明,和传统的PD控制方法相比,在参数不确定性的情况下,该控制方法使得电液伺服系统的位置跟踪误差波动较小,且能以较快速度渐近收敛到0,同时所需要的伺服阀输入电压信号值也更小,相关不确定参数在经过较短时间后均可以收敛到其稳定值,从而验证了所提出算法的有效性.     

不确定非自治混沌陀螺仪在非线性输入下的有限时间稳定性

陀螺仪是一个非常有趣,又是永恒的非线性非自治动力系统课题,它可以显示出非常复杂的动力学行为,如混沌现象.在一个给定的有限时间内,研究非线性非自治陀螺仪鲁棒稳定性问题.假设陀螺仪系统受到模型不确定的外部扰动而摄动,系统参数并不知道,同时考虑了非线性输入的影响.为未知参数提出了适当的自适应律.以自适应律和有限时间控制理论为基础,提出非连续有限时间控制理论,来研究系统的有限时间稳定性.解析证明了闭循环系统的有限时间稳定性及其收敛性.若干数值仿真结果表明,该文的有限时间控制法是有效的,同时验证了该文的理论结果.     

Synchronization of nonlinear master–slave systems under input delay and slope‐restricted input nonlinearity

This article addresses the synchronization of nonlinear master–slave systems under input time‐delay and slope‐restricted input nonlinearity. The input nonlinearity is transformed into linear time‐varying parameters belonging to a known range. Using the linear parameter varying (LPV) approach, applying the information of delay range, using the triple‐integral‐based Lyapunov–Krasovskii functional and utilizing the bounds on nonlinear dynamics of the nonlinear systems, nonlinear matrix inequalities for designing a simple delay‐range‐dependent state feedback control for synchronization of the drive and response systems is derived. The proposed controller synthesis condition is transformed into an equivalent but relatively simple criterion that can be solved through a recursive linear matrix inequality based approach by application of cone complementary linearization algorithm. In contrast to the conventional adaptive approaches, the proposed approach is simple in design and implementation and is capable to synchronize nonlinear oscillators under input delays in addition to the slope‐restricted nonlinearity. Further, time‐delays are treated using an advanced delay‐range‐dependent approach, which is adequate to synchronize nonlinear systems with either higher or lower delays. Furthermore, the resultant approach is applicable to the input nonlinearity, without using any adaptation law, owing to the utilization of LPV approach. A numerical example is worked out, demonstrating effectiveness of the proposed methodology in synchronization of two chaotic gyro systems. © 2015 Wiley Periodicals, Inc. Complexity 21: 220–233, 2016     

Robust controlling hyperchaos of the Rössler system subject to input nonlinearities by using sliding mode control

A sliding mode control is designed to stabilize the well-known hyperchaos of Rössler system to equilibrium points subject to sector nonlinear input. The proposed control law is robust against both the input nonlinearity and external disturbance. The error bound can be arbitrarily set by assigning the corresponding dynamics to the sliding surfaces when the desired state is not an equilibrium point. Simulation results show that the system state can be regulated to an equilibrium point in the state space. It is also seen that the system still possesses advantage of fast response and good transient performance even though the control input is nonlinear.     

RBF‐based discrete sliding mode control for robust tracking of uncertain time‐delay systems with input nonlinearity

In this article, a control scheme combining radial basis function neural network and discrete sliding mode control method is proposed for robust tracking and model following of uncertain time‐delay systems with input nonlinearity. The proposed robust tracking controller guarantees the stability of overall closed‐loop system and achieves zero‐tracking error in the presence of input nonlinearity, time‐delays, time‐varying parameter uncertainties, and external disturbances. The salient features of the proposed controller include no requirement of a priori knowledge of the upper bound of uncertainties and the elimination of chattering phenomenon and reaching phase. Simulation results are presented to demonstrate the effectiveness of the proposed scheme. © 2015 Wiley Periodicals, Inc. Complexity 21: 194–201, 2016     

Chaos suppression of a class of unknown uncertain chaotic systems via single input

This paper deals with the design of a robust adaptive control scheme for chaos suppression of a class of chaotic systems. We assume that model uncertainties and external disturbances disturb the system’s dynamics. The bounds of both model uncertainties and external disturbances are assumed to be unknown in advance. Moreover, it is assumed that the nonlinear terms of the chaotic system dynamics are unknown bounded. Based on the global boundedness feature of the chaotic systems’ trajectories, a simple one input adaptive sliding mode control approach is proposed to suppress the chaos of the uncertain chaotic system. Furthermore, using a dynamical sliding manifold the discontinuous sign function in the control input is diverted to the first derivative of the control input to eliminate the chattering. Finally, the robustness of the proposed approach is mathematically proved and numerically illustrated.     

Adaptive synchronization of two coupled chaotic Hindmarsh–Rose neurons by controlling the membrane potential of a slave neuron

This paper addresses the problem of an adaptive synchronization of two chaotic Hindmarsh–Rose (HR) neurons coupled with a gap junction via a single control input. Two adaptive control approaches (i.e., a nonlinear control when the entire state variables are available and a linear feedback control when only the membrane potential is available) are proposed to guarantee the asymptotic synchronization of the state trajectories of two coupled HR neurons having unknown parameters. Numerical simulations demonstrating and verifying the effectiveness of the proposed control methods are provided.     

计及位置不确定的喷雾机喷杆系统仿形姿态渐近跟踪控制北大核心CSCD

针对喷雾机喷杆仿形系统中同时存在负载变化、未建模不确定项、物理参数摄动以及外部干扰等问题,提出了一种基于小波网络逼近的具有自适应性和鲁棒性的反步控制方法.首先,将含有不确定、未知和非线性项的喷杆仿形系统建立为完整的数学模型,将其等价转化为具有严格反馈的状态空间形式;其次,采用设计的小波基元去构造神经网络,在满足最优误差有界条件下逼近反步法中虚拟等效控制部分,选取自适应更新律估计系统中存在的未知参数,引入鲁棒补偿项减小复合干扰对系统的不利影响,降低了输入指令信号的阶次要求;最后,通过构造合适的Lyapunov函数,应用稳定性理论证明了闭环系统位置跟踪误差渐近收敛到原点.仿真结果表明,所提控制方法可实现喷雾机喷杆位置姿态快速升降机动调整,有效地增强了喷杆系统的鲁棒稳定性和控制精度.     

Chaos synchronization between two different chaotic systems with uncertainties, external disturbances, unknown parameters and input nonlinearities

In this paper, the problem of chaos synchronization between two different uncertain chaotic systems with input nonlinearities is investigated. Both master and slave systems are perturbed by model uncertainties, external disturbances and unknown parameters. The bounds of the model uncertainties and external disturbances are assumed to be unknown in advance. First, a simple linear sliding surface is selected. Then, appropriate adaptive laws are derived to tackle the model uncertainties, external disturbances and unknown parameters. Subsequently, based on the adaptive laws and Lyapunov stability theory, a robust adaptive sliding mode control law is designed to guarantee the existence of the sliding motion. Two illustrative examples are presented to verify the usefulness and applicability of the proposed technique.     

Robust decentralized adaptive control for uncertain large-scale delayed systems with input nonlinearities

The paper is concerned with the problem of robust stabilization for uncertain large-scale time-varying delayed systems with input nonlinearities. Based on the sliding mode control, a memoryless decentralized adaptive sliding mode controller (DASMC) is developed. The proposed controller ensures the occurrence of the sliding manifold of the composite system even subjected to input nonlinearity. It shows that the uncertain nonlinear large-scale system also possesses the property of insensitivity to uncertainties and disturbances as a linear system does. A numerical example is given to verify the validity of the developed memoryless DASMC.     

A practical projective synchronization approach for uncertain chaotic systems with dead-zone input

In this paper, a practical projective synchronization problem of master–slave chaotic systems is investigated. More specifically, a fuzzy adaptive slave chaotic system subject to dead-zone nonlinearity in the input channel is proposed using only the measurable output of the master system thanks to a suitable observer. A practical projective synchronization between the master and slave systems is achieved by an adequate fuzzy adaptive control system. The underlying parameter adaptation design as well as stability analysis are carried out using a Lyapunov based approach. Unlike the previous works, in the design of the proposed synchronization scheme, we do not require to know the uncertainties function and that the dynamics of the original synchronization error are strictly positive real (SPR). In fact, herein, the uncertainties function is estimated by a fuzzy adaptive system and the dynamics of the original synchronization error are augmented by a low pass filter designed to satisfy the SPR condition. Simulation results are given to show the effectiveness of the proposed practical projective synchronization scheme.