Event-triggered adaptive fuzzy tracking control of nonlinear MIMO systems

This paper addresses the problem of an adaptive fuzzy event-triggered control (ETC) for uncertain multi-input and multi-output nonlinear systems. To reduce the communication burden of the network control systems, a novel state-dependent event-triggering condition is designed to decide when to update the controllers. By combining the backstepping and event-trigged techniques, the adaptive fuzzy ETC strategies are developed and the resulting closed-loop system is semi-global bounded. Finally, the analytical results are substantiated using simulation studies.     

Online adaptive optimal control for continuous-time nonlinear systems with completely unknown dynamics

An online adaptive optimal control is proposed for continuous-time nonlinear systems with completely unknown dynamics, which is achieved by developing a novel identifier-critic-based approximate dynamic programming algorithm with a dual neural network (NN) approximation structure. First, an adaptive NN identifier is designed to obviate the requirement of complete knowledge of system dynamics, and a critic NN is employed to approximate the optimal value function. Then, the optimal control law is computed based on the information from the identifier NN and the critic NN, so that the actor NN is not needed. In particular, a novel adaptive law design method with the parameter estimation error is proposed to online update the weights of both identifier NN and critic NN simultaneously, which converge to small neighbourhoods around their ideal values. The closed-loop system stability and the convergence to small vicinity around the optimal solution are all proved by means of the Lyapunov theory. The proposed adaptation algorithm is also improved to achieve finite-time convergence of the NN weights. Finally, simulation results are provided to exemplify the efficacy of the proposed methods.     

Observer-critic structure-based adaptive dynamic programming for decentralised tracking control of unknown large-scale nonlinear systems

In this paper, a decentralised tracking control (DTC) scheme is developed for unknown large-scale nonlinear systems by using observer-critic structure-based adaptive dynamic programming. The control consists of local desired control, local tracking error control and a compensator. By introducing the local neural network observer, the subsystem dynamics can be identified. The identified subsystems can be used for the local desired control and the control input matrix, which is used in local tracking error control. Meanwhile, Hamiltonian-Jacobi-Bellman equation can be solved by constructing a critic neural network. Thus, the local tracking error control can be derived directly. To compensate the overall error caused by substitution, observation and approximation of the local tracking error control, an adaptive robustifying term is employed. Simulation examples are provided to demonstrate the effectiveness of the proposed DTC scheme.     

离散非线性零和博弈的事件驱动最优控制方案

在求解离散非线性零和博弈问题时,为了在有效降低网络通讯和控制器执行次数的同时保证良好的控制效果,本文提出了一种基于事件驱动机制的最优控制方案.首先,设计了一个采用新型事件驱动阈值的事件驱动条件,并根据贝尔曼最优性原理获得了最优控制对的表达式.为了求解该表达式中的最优值函数,提出了一种单网络值迭代算法.利用一个神经网络构建评价网.设计了新的评价网权值更新规则.通过在评价网、控制策略及扰动策略之间不断迭代,最终获得零和博弈问题的最优值函数和最优控制对.然后,利用Lyapunov稳定性理论证明了闭环系统的稳定性.最后,将该事件驱动最优控制方案应用到了两个仿真例子中,验证了所提方法的有效性.     

Finite-horizon neuro-optimal tracking control for a class of discrete-time nonlinear systems using adaptive dynamic programming approach

In this paper, a finite-horizon neuro-optimal tracking control strategy for a class of discrete-time nonlinear systems is proposed. Through system transformation, the optimal tracking problem is converted into designing a finite-horizon optimal regulator for the tracking error dynamics. Then, with convergence analysis in terms of cost function and control law, the iterative adaptive dynamic programming (ADP) algorithm via heuristic dynamic programming (HDP) technique is introduced to obtain the finite-horizon optimal tracking controller which makes the cost function close to its optimal value within an ?-error bound. Three neural networks are used as parametric structures to implement the algorithm, which aims at approximating the cost function, the control law, and the error dynamics, respectively. Two simulation examples are included to complement the theoretical discussions.     

Optimal and robust control of a class of nonlinear systems using dynamically re-optimised single network adaptive critic design

Following the philosophy of adaptive optimal control, a neural network-based state feedback optimal control synthesis approach is presented in this paper. First, accounting for a nominal system model, a single network adaptive critic (SNAC) based multi-layered neural network (called as NN₁) is synthesised offline. However, another linear-in-weight neural network (called as NN₂) is trained online and augmented to NN₁ in such a manner that their combined output represent the desired optimal costate for the actual plant. To do this, the nominal model needs to be updated online to adapt to the actual plant, which is done by synthesising yet another linear-in-weight neural network (called as NN₃) online. Training of NN₃ is done by utilising the error information between the nominal and actual states and carrying out the necessary Lyapunov stability analysis using a Sobolev norm based Lyapunov function. This helps in training NN₂ successfully to capture the required optimal relationship. The overall architecture is named as ‘Dynamically Re-optimised single network adaptive critic (DR-SNAC)’. Numerical results for two motivating illustrative problems are presented, including comparison studies with closed form solution for one problem, which clearly demonstrate the effectiveness and benefit of the proposed approach.     

Event-triggered adaptive tracking control for nonlinear systems with input saturation and unknown control directions

This article is concerned with event-triggered adaptive tracking control design of strict-feedback nonlinear systems, which are subject to input saturation and unknown control directions. In the design procedure, a smooth nonlinear function is employed to approximate the saturation function so that the controller can be designed under the framework of backstepping. The Nussbaum gain technique is employed to address the issue of the unknown control directions. A predetermined time convergent performance function and a nonlinear mapping technique are introduced to guarantee that the tracking error can converge in the predetermined time with a fast convergence rate and a high accuracy. Then the event-triggered adaptive prescribed performance tracking control strategy is proposed, which not only ensures the boundedness of all the closed-loop signals and the convergence of tracking error but also reduces the communication burden from the controller to the actuator. At last, the simulation study further tests the availability of the proposed control strategy.     

Improved value iteration for nonlinear tracking control with accelerated learning

In this article, an adaptive critic scheme with a novel performance index function is developed to solve the tracking control problem, which eliminates the tracking error and possesses the adjustable convergence rate in the offline learning process. Under some conditions, the convergence and monotonicity of the accelerated value function sequence can be guaranteed. Combining the advantages of the adjustable and general value iteration schemes, an integrated algorithm is proposed with a fast guaranteed convergence, which involves two stages, namely the acceleration stage and the convergence stage. Moreover, an effective approach is given to adaptively determine the acceleration interval. With this operation, the fast convergence of the new value iteration scheme can be fully utilized. Finally, compared with the general value iteration, the numerical results are presented to verify the fast convergence and the tracking performance of the developed adaptive critic design.     

非线性零和微分对策的事件触发自适应动态规划算法

针对一类非线性零和微分对策问题,本文提出了一种事件触发自适应动态规划(event-triggered adaptive dynamic programming,ET--ADP)算法在线求解其鞍点.首先,提出一个新的自适应事件触发条件.然后,利用一个输入为采样数据的神经网络(评价网络)近似最优值函数,并设计了新型的神经网络权值更新律使得值函数、控制策略及扰动策略仅在事件触发时刻同步更新.进一步地,利用Lyapunov稳定性理论证明了所提出的算法能够在线获得非线性零和微分对策的鞍点且不会引起Zeno行为.所提出的ET--ADP算法仅在事件触发条件满足时才更新值函数、控制策略和扰动策略,因而可有效减少计算量和降低网络负荷.最后,两个仿真例子验证了所提出的ET--ADP算法的有效性.     

Optimal prescribed performance tracking control of nonlinear motor driven systems via adaptive dynamic programming

Although optimal regulation problem has been well studied, resolving optimal tracking control via adaptive dynamic programming (ADP) has not been completely resolved, particularly for nonlinear uncertain systems. In this paper, an online adaptive learning method is developed to realize the optimal tracking control design for nonlinear motor driven systems (NMDSs), which adopts the concept of ADP, unknown system dynamic estimator (USDE), and prescribed performance function (PPF). To this end, the USDE in a simple form is first proposed to address the NMDSs with bounded disturbances. Then, based on the estimated unknown dynamics, we define an optimal cost function and derive the optimal tracking control. The derived optimal tracking control is divided into two parts, that is, steady-state control and optimal feedback control. The steady-state control can be obtained with the tracking commands directly. The optimal feedback control can be obtained via the concept of ADP based on the PPF; this contributes to improving the convergence of critic neural network (CNN) weights and tracking accuracy of NMDSs. Simulations are provided to display the feasibility of the designed control method.     

Neural-network-observer-based optimal control for unknown nonlinear systems using adaptive dynamic programming

In this paper, an observer-based optimal control scheme is developed for unknown nonlinear systems using adaptive dynamic programming (ADP) algorithm. First, a neural-network (NN) observer is designed to estimate system states. Then, based on the observed states, a neuro-controller is constructed via ADP method to obtain the optimal control. In this design, two NN structures are used: a three-layer NN is used to construct the observer which can be applied to systems with higher degrees of nonlinearity and without a priori knowledge of system dynamics, and a critic NN is employed to approximate the value function. The optimal control law is computed using the critic NN and the observer NN. Uniform ultimate boundedness of the closed-loop system is guaranteed. The actor, critic, and observer structures are all implemented in real-time, continuously and simultaneously. Finally, simulation results are presented to demonstrate the effectiveness of the proposed control scheme.     

Event-triggered adaptive consensus control of nonlinear multi-agent systems with unknown control directions and actuator saturation

This article addresses the event-triggered adaptive consensus control of nonlinear multi-agent systems with unknown control direction and actuator saturation. A new robust adaptive control algorithm based on an event-triggered mechanism is designed. The smooth Lipschitz function approximates the saturated nonlinear function, while the Nussbaum function handles unknown control directions and residual terms. The event-triggered mechanism is designed to determine the time of communication, significantly reducing the communication burden. An additional estimator is utilized to deal with unknown parameters involved in neighbor dynamics and prevent information exchange to consistency errors between connected subsystems. The results show that all the signals of the closed-loop system are uniformly bounded, and the consensus tracking error converges to a bounded set. Meanwhile, Zeno's behavior is eliminated. Simulation results confirm the superiority of the proposed method.     

Online optimal tracking control of continuous-time linear systems with unknown dynamics by using adaptive dynamic programming

In this paper, a novel theoretic formulation based on adaptive dynamic programming (ADP) is developed to solve online the optimal tracking problem of the continuous-time linear system with unknown dynamics. First, the original system dynamics and the reference trajectory dynamics are transformed into an augmented system. Then, under the same performance index with the original system dynamics, an augmented algebraic Riccati equation is derived. Furthermore, the solutions for the optimal control problem of the augmented system are proven to be equal to the standard solutions for the optimal tracking problem of the original system dynamics. Moreover, a new online algorithm based on the ADP technique is presented to solve the optimal tracking problem of the linear system with unknown system dynamics. Finally, simulation results are given to verify the effectiveness of the theoretic results.     

Adaptive critic learning for event-triggered safe control of nonlinear safety-critical systems

In this paper, an event-triggered safe control method based on adaptive critic learning (ACL) is proposed for a class of nonlinear safety-critical systems. First, a safe cost function is constructed by adding a control barrier function (CBF) to the traditional quadratic cost function; the optimization problem with safety constraints that is difficult to deal with by classical ACL methods is solved. Subsequently, the event-triggered scheme is introduced to reduce the amount of computation. Further, combining the properties of CBF with the ACL-based event-triggering mechanism, the event-triggered safe Hamilton–Jacobi–Bellman (HJB) equation is derived, and a single critic neural network (NN) framework is constructed to approximate the solution of the event-triggered safe HJB equation. In addition, the concurrent learning method is applied to the NN learning process, so that the persistence of excitation (PE) condition is not required. The weight approximation error of the NN and the states of the system are proven to be uniformly ultimately bounded (UUB) in the safe set with the Lyapunov theory. Finally, the availability of the presented method can be validated through the simulation.     

Event-triggered fuzzy adaptive control of incommensurate fractional order nonlinear systems

This paper focuses on the event-triggered control of nonstrict-feedback incommensurate fractional order systems with external disturbances. To release more communication resources and reduce the triggered time instant, we adopt the relative-threshold-based event-triggered strategy and prove that the Zeno behavior does not exist. We utilize the fuzzy systems to approximate the nonlinear systems and design an adaptive update law to estimate the approximation errors and external disturbances. To conquer the dimension explosion phenomenon in the backstepping process, we design a fractional order filter and apply the dynamic surface control technique. The closed-loop system is semiglobally stable and the tracking error converges to a small neighborhood of equilibrium point under the proposed control approach. Finally, the validity of the developed controller is verified by the simulation examples.     

非线性离散系统的近似最优跟踪控制

研究非线性离散系统的最优跟踪控制问题. 通过在由最优控制问题所导致的非线性两点边值问题中引入灵敏度参数, 并对它进行Maclaurin级数展开, 将原最优跟踪控制问题转化为一族非齐次线性两点边值问题. 得到的最优跟踪控制由解析的前馈反馈项和级数形式的补偿项组成. 解析的前馈反馈项可以由求解一个Riccati差分方程和一个矩阵差分方程得到. 级数补偿项可以由一个求解伴随向量的迭代算法近似求得. 以连续槽式反应器为例进行仿真验证了该方法的有效性．     

不对称约束多人非零和博弈的自适应评判控制

本文针对连续时间非线性系统的不对称约束多人非零和博弈问题, 建立了一种基于神经网络的自适应评判控制方法. 首先, 本文提出了一种新颖的非二次型函数来处理不对称约束问题, 并且推导出最优控制律和耦合Hamilton-Jacobi方程. 值得注意的是, 当系统状态为零时, 最优控制策略是不为零的, 这与以往不同. 然后, 通过构建单一评判网络来近似每个玩家的最优代价函数, 从而获得相关的近似最优控制策略. 同时, 在评判学习期间发展了一种新的权值更新规则. 此外, 通过利用Lyapunov理论证明了评判网络权值近似误差和闭环系统状态的稳定性. 最后, 仿真结果验证了本文所提方法的有效性     

Event-triggered adaptive fuzzy control for nonstrict-feedback nonlinear systems with output hysteresis

This paper concentrates upon the issue of adaptive fuzzy tracing control for a class of nonstrict-feedback nonlinear systems output with hysteresis via an event-triggered strategy. To handle the difficulty caused by the nonstrict nonlinear systems, the variable separation technique is introduced. The design difficulty of output hysteresis is addressed by employing a hysteresis inverse function and Nussbaum function to compensate unmeasurable state signal. Meanwhile, the fuzzy logic system (FLS) is used to estimate the unknown function at each step of recursion. Moreover, by devising the relative threshold event-triggered mechanism (ETM), the frequency of actuators and controllers can be largely decreased. Thus, the adaptive fuzzy event-triggered tracing control strategy is proposed by combining the barrier Lyapunov function and backstepping technique. With the proposed scheme, it is theoretically demonstrated that all signals in the closed-loop system are bounded, and the tracing errors are driven to a small neighborhood of the origin under the output constraint. Eventually, two examples demonstrate the efficacy of the proposed control strategy.     

未知饱和控制系统有穷域最优控制

针对带有饱和执行器且局部未知的非线性连续系统的有穷域最优控制问题,设计了一种基于自适应动态规划(ADP)的在线积分增强学习算法,并给出算法的收敛性证明.首先,引入非二次型函数处理控制饱和问题.其次,设计一种由常量权重和时变激活函数构成的单一网络,来逼近未知连续的值函数,与传统双网络相比减少了计算量.同时,综合考虑神经网络产生的残差和终端误差,应用最小二乘法更新神经网络权重,并且给出基于神经网络的迭代值函数收敛到最优值的收敛性证明.最后,通过两个仿真例子验证了算法的有效性.     

Adaptive neural tracking control for a class of stochastic nonlinear systems

This paper investigates the problem of adaptive neural control design for a class of single‐input single‐output strict‐feedback stochastic nonlinear systems whose output is an known linear function. The radial basis function neural networks are used to approximate the nonlinearities, and adaptive backstepping technique is employed to construct controllers. It is shown that the proposed controller ensures that all signals of the closed‐loop system remain bounded in probability, and the tracking error converges to an arbitrarily small neighborhood around the origin in the sense of mean quartic value. The salient property of the proposed scheme is that only one adaptive parameter is needed to be tuned online. So, the computational burden is considerably alleviated. Finally, two numerical examples are used to demonstrate the effectiveness of the proposed approach. Copyright © 2012 John Wiley & Sons, Ltd.