Optimal motion planning of non-linear dynamic systems in the presence of obstacles and moving boundaries using SDRE: application on cable-suspended robot

In this paper a closed-loop non-linear optimal controller is designed via State Dependent Riccati Equation (SDRE) and employed for a spatial six-cable robot. SDRE provides a systematic and effective design of controller for non-linear systems. The power series approximation method is extended and used to solve SDRE. Trajectory tracking along with point to point movement is carried out. Moreover, two common constraints of optimal path planning, i.e., obstacles and moving boundaries are studied, and proper strategies are proposed to deal with these constraints. Obstacle avoidance technique used in this paper is based on the concept of Artificial Potential Field, while calculus of variations is applied to choose the optimal initial or end points from the moving boundaries. Capabilities of SDRE method provides an outstanding opportunity to be combined with the considered strategies. Simulations for the spatial six-cable robot are done, and Dynamic Load Carrying Capacity is computed to illustrate the efficiency of the employed procedure. Finally the results are validated by experimental tests conducted on ICaSbot which is a spatial six-cable robot manufactured in robotics research laboratory of Iran University of Science and Technology.     

哈密顿系统正则变换在时变最优控制中的应用

利用哈密顿系统正则变换和生成函数理论求解线性时变最优控制问题,构造了新的最优控制律形式并提出了控制增益计算的保结构算法. 利用生成函数求解最优控制导出的哈密顿系统两端边值问题,并构造线性时变系统的最优控制律,由第2类生成函数所构造的最优控制律避免了末端时刻出现无穷大反馈增益. 控制系统设计中需求解生成函数满足的时变矩阵微分方程组. 根据生成函数与哈密顿系统状态转移矩阵之间的关系,从正则变换的辛矩阵描述出发,导出了求解这组微分方程组的保结构递推算法.为了保持递推计算中的辛矩阵结构,哈密顿系统状态转移矩阵的计算中利用了Magnus级数.      

Lag synchronization of a class of chaotic systems with unknown parameters

Research on chaos synchronization of dynamical systems has been largely reported in literature. However, synchronization of different structure—uncertain dynamical systems—has received less attention. This paper addresses synchronization of a class of time-delay chaotic systems containing uncertain parameters. A unified scheme is established for synchronization between two strictly different time-delay uncertain chaotic systems. The synchronization is successfully achieved by designing an adaptive controller with the estimates of the unknown parameters and the nonlinear feedback gain. The result is rigorously proved by the Lyapunov stability theorem. Moreover, we illustrate the application of the proposed scheme by numerical simulation, which demonstrates the effectiveness and feasibility of the proposed synchronization method.     

Adaptive intelligence learning for nonlinear chaotic systems

In this paper, a reinforcement learning algorithm is proposed for a class of nonlinear differential chaotic systems. The nonlinear function of the chaotic systems is assumed to be bounded but the bounds are unknown. The unknown bounds need to be on-line adjusted. An adaptive optimal (or near optimal) control input with the reinforcement signal can be obtained compared with the current adaptive control for chaotic systems. The reinforcement signal is approximated by the neural networks. Based on Lyapunov analysis theory and by using Young’s inequalities, the closed-loop system is guaranteed to be stable. Finally, the simulation results are given to illustrate the effectiveness of the approach.     

Magnetic torque attitude control of a satellite using the state-dependent Riccati equation technique

A non-linear attitude control method for a satellite with magnetic torque rods using the state-dependent Riccati equation (SDRE) technique has been developed. The magnetic torque caused by the interaction with the Earth's magnetic field and the magnetic moment of torque rods plays a role of the control torque. The detailed equations of motion for this system are presented using angular velocity and quaternions. The SDRE controller is developed for the non-linear systems which can be formed in pseudo-linear representations using the state-dependent coefficient (SDC) method without linearization procedure. The aim of this control system is to achieve a stable attitude within 5°, and minimize the control effort. The stability regions for the SDRE controlled satellite system are estimated through the investigation of the stability conditions developed for pseudo-linear systems and the application of Lyapunov's theorem. For comparisons, the Linear Quadratic Regulator (LQR) method using the solution of the algebraic Riccati equation (ARE) is also applied to this non-linear system. The performance of the SDRE non-linear control system demonstrates more robustness and stability than the LQR control system when subjected to a wide range of initial conditions.     

Adaptive robust fuzzy control for a class of uncertain chaotic systems

In this paper, the output feedback control of uncertain chaotic systems is addressed via an adaptive robust fuzzy approach. Fuzzy logic systems are employed to approximate uncertain nonlinear functions in the chaotic systems. Because only partial information of the system’s states is needed to be known, an observer is given to estimate the unmeasured states. Compared with the existing results in the observer design, the prior knowledge on dynamic uncertainties is relaxed and a class of more general chaotic systems is considered as well as robustness to the approximation error is improved. It can be proven that the closed-loop system is stable in the sense that all the variables are bounded. Simulation example for the unified chaotic systems is given to verify the effectiveness of the proposed method. This work was supported in part by the National Natural Science Foundation of China (60874056) and the Foundation of Educational Department of Liaoning Province (2008312).     

Robust chaos synchronizations using an SDRE-based sub-optimal control approach

This paper presents a robust control via a sub-optimal approach, which is achieved by the SDRE control. The proposed approach is to transform a robust control problem into an optimal control problem, where the uncertainties are reflected in the performance index. Thus, the optimal control problem is solved by the SDRE control. Four types of robust control problems are presented, and each type is illustrated by a robust chaos synchronization example. The results show that the proposed approach can effectively regulate the error dynamics of the chaos synchronization.     

Disturbance-observer-based robust synchronization control of uncertain chaotic systems

In this paper, a robust synchronization control scheme is proposed for chaotic systems in the presence of system uncertainties and unknown external disturbances. For the synchronization error system, the compound disturbance which is estimated using the disturbance observer cannot be directly measured. If the gain matrix is properly chosen, the disturbance observer can approximate the unknown compound disturbance well. And then, the constrained robust synchronization control scheme is presented for uncertain chaotic systems based on the output of disturbance observer. In the design of a robust synchronization control scheme, the effect of unknown control input constraint has been explicitly considered to guarantee the synchronization performance. Numerical simulation results are presented to illustrate the effectiveness of the proposed constrained synchronization control scheme for uncertain chaotic systems.     

A projective synchronization scheme based on fuzzy adaptive control for unknown multivariable chaotic systems

In this paper, a projective synchronization problem of master–slave chaotic systems is investigated. More specifically, a fuzzy adaptive controller is investigated for a projective synchronization of uncertain multivariable chaotic systems. The adaptive fuzzy-logic systems are used to approximate the unknown functions. A decomposition property of the control gain matrix is used in the controller design and the stability analysis. A Lyapunov approach is employed to derive the parameter adaptation laws and prove the boundedness of all signals of the closed-loop system as well as the exponential convergence of the synchronization errors to an adjustable region. Numerical simulations are performed to verify the effectiveness of the proposed synchronization scheme.     

Combination–combination synchronization among four identical or different chaotic systems

Based on one drive system and one response system synchronization model, a new type of combination–combination synchronization is proposed for four identical or different chaotic systems. According to the Lyapunov stability theorem and adaptive control, numerical simulations for four identical or different chaotic systems with different initial conditions are discussed to show the effectiveness of the proposed method. Synchronization about combination of two drive systems and combination of two response systems is the main contribution of this paper, which can be extended to three or more chaotic systems. A universal combination of drive systems and response systems model and a universal adaptive controller may be designed to our intelligent application by our synchronization design.     

Using fractional-order integrator to control chaos in single-input chaotic systems

This paper deals with a fractional calculus based control strategy for chaos suppression in the 3D chaotic systems. It is assumed that the structure of the controlled chaotic system has only one control input. In the proposed strategy, the controller has three tuneable parameters and the control input is constructed as fractional-order integration of a linear combination of linearized model states. The tuning procedure is based on the stability theorems in the incommensurate fractional-order systems. To evaluate the performance of the proposed controller, the design method is applied to suppress chaotic oscillations in a 3D chaotic oscillator and in the Chen chaotic system.     

Finite-time chaos control of unified chaotic systems with uncertain parameters

This paper is concerned with finite-time chaos control of unified chaotic systems with uncertain parameters. Based on the finite-time stability theory in the cascade-connected system, a nonlinear control law is presented to achieve finite-time chaos control. The controller is simple and easy to be constructed. Simulation results for Lorenz, Lü, and Chen chaotic systems are provided to illustrate the effectiveness of the proposed scheme. Supported by the National Natural Science Foundation of China (Grant No. 60674024).     

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.

     

Directly adaptive fuzzy control of discrete-time chaotic systems by least squares algorithm with dead-zone

A new design scheme of directly adaptive fuzzy control for a class of discrete-time chaotic systems is proposed in this paper. The T-S fuzzy model is employed to represent the discrete-time chaotic systems. Then a fuzzy controller is designed and the unknown coefficients of the controller are identified by least squares algorithm with dead-zone. By Lyapunov method, all the signals involved in the closed-loop systems are shown to be bounded and the error between the system output and the reference output is proved to converge to a small neighborhood of zero. Simulation results demonstrate the effectiveness of the theoretical results.     

Adaptive control of a chaotic permanent magnet synchronous motor

This paper proposes a simple adaptive controller design method for a chaotic permanent magnet synchronous motor (PMSM) based on the sliding mode control theory which has given an effective means to design robust controllers for nonlinear systems with bounded uncertainties. The proposed sliding mode adaptive controller does not require any information on the PMSM parameter and load torque values, thus it is insensitive to model parameter and load torque variations. Simulation results are given to verify that the proposed method can be successfully used to control a chaotic PMSM under model parameter and load torque variations.     

Decentralized coordinated attitude control for satellite formation flying via the state-dependent Riccati equation technique

The goal of the present study is to develop a decentralized coordinated attitude control algorithm for satellite formation flying. To handle the non-linearity of the dynamic system, the problems of absolute and relative attitude dynamics are formulated for the state-dependent Riccati equation (SDRE) technique. The SDRE technique is for the first time utilized as a non-linear controller of the relative attitude control problem for satellite formation flying, and then the results are compared to those from linear control methods, mainly the PD and LQR controllers. The stability region for the SDRE-controlled system was obtained using a numerical method. This estimated stability region demonstrates that the SDRE controller developed in the present paper guarantees the globally asymptotic stability for both the absolute and relative attitude controls. Moreover, in order to complement a non-selective control strategy for relative attitude error in formation flying, a selective control strategy is suggested. This strategy guarantees not only a reduction in unnecessary calculation, but also the mission-failure safety of the attitude control algorithm for satellite formation. The attitude control algorithm of the formation flying was tested in the orbital-reference coordinate system for the sake of applying the control algorithms to Earth-observing missions. The simulation results illustrate that the attitude control algorithm based on the SDRE technique can robustly drive the attitude errors to converge to zero.     

Adaptive synchronization for uncertain complex dynamical network using fuzzy disturbance observer

This paper proposes a robust adaptive controller design method for synchronization of a complex dynamical network with uncertainty and disturbance. A fuzzy disturbance observer is used to estimate the overall disturbances without any prior knowledge about them. The proposed control method globally asymptotically synchronizes the network using adaptation laws obtained by using Lyapunov stability theory. The proposed method is applied to two chaotic systems and the results show the effectiveness of the approach.     

Achievement of chaotic synchronization trajectories of master–slave manipulators with feedback control strategy

This paper addresses a master-slave synchro- nization strategy for complex dynamic systems based on feedback control. This strategy is applied to 3-DOF pla- nar manipulators in order to obtain synchronization in such complicated as chaotic motions of end-effectors. A chaotic curve is selected from Duffing equation as the trajectory of master end-effector and a piecewise approximation method is proposed to accurately represent this chaotic trajectory of end-effectors. The dynamical equations of master-slave manipulators with synchronization controller are derived, and the Lyapunov stability theory is used to determine the stability of this controlled synchronization system. In numer- ical experiments, the synchronous motions of end-effectors as well as three joint angles and torques of master-slave manipulators are studied under the control of the proposed synchronization strategy. It is found that the positive gain matrix affects the implementation of synchronization con- trol strategy. This synchronization control strategy proves the synchronization＇s feasibility and controllability for com- plicated motions generated by master-slave manipulators.     

Chaos detection and parameter identification in fractional-order chaotic systems with delay

The paper first applies the 0–1 test for chaos to detecting chaos exhibited by fractional-order delayed systems. The results of the test reveal that there exists chaos in some fractional-order delayed systems with specific parameter values, which coincides with previous reports based on the phase portrait. In addition, it is very important to identify exactly the unknown specific parameters of fractional-order chaotic delayed systems in chaos control and synchronization. Thus, a method for parameter identification of fractional-order chaotic delayed systems based on particle swarm optimization (PSO) is presented. By treating the orders as parameters, the parameters and orders are identified through minimizing an objective function. PSO can efficiently find the optimal feasible solution of the objective function. Finally, numerical simulations on fractional-order chaotic logistic delayed system and fractional-order chaotic Chen delayed system show that the proposed method has effective performance of parameter identification.     

Optimal nonlinear observers for chaotic synchronization with message embedded

This paper presents an optimal nonlinear observer for synchronizing the transmitter-receiver pair with guaranteed optimal performance. In the proposed scheme, a generalized nonlinear state-space observer via uniform matrix transformations is constructed to estimate the transmitter state and the information signal, simultaneously. A nonlinear optimal design approach is used to synchronize chaotic systems. Solving the Hamilton–Jacobi–Bellman (H–J–B) equations we can obtain a linear optimal feedback scheme for piecewise-linear chaotic systems. Moreover, a robust scheme derived from the H _∞ optimization theory improves the synchronization performance of general nonlinear chaotic systems by suppressing the influence of their high order residual terms. Finally, two numerical simulation examples are illustrated by the chaotic Chua’s circuit system and the Lorenz chaotic system to demonstrate the effectiveness of our scheme.