Discrete-time neural network output feedback control of nonlinear discrete-time systems in non-strict form

An adaptive neural network (NN)-based output feedback controller is proposed to deliver a desired tracking performance for a class of discrete-time nonlinear systems, which are represented in non-strict feedback form. The NN backstepping approach is utilized to design the adaptive output feedback controller consisting of: (1) an NN observer to estimate the system states and (2) two NNs to generate the virtual and actual control inputs, respectively. The non-causal problem encountered during the control design is overcome by using a dynamic NN which is constructed through a feedforward NN with a novel weight tuning law. The separation principle is relaxed, persistency of excitation condition (PE) is not needed and certainty equivalence principle is not used. The uniformly ultimate boundedness (UUB) of the closed-loop tracking error, the state estimation errors and the NN weight estimates is demonstrated. Though the proposed work is applicable for second order nonlinear discrete-time systems expressed in non-strict feedback form, the proposed controller design can be easily extendable to an nth order nonlinear discrete-time system.     

Online Adaptive Approximate Optimal Tracking Control with Simplified Dual Approximation Structure for Continuous-time Unknown Nonlinear Systems

This paper proposes an online adaptive approximate solution for the infinite-horizon optimal tracking control problem of continuous-time nonlinear systems with unknown dynamics. The requirement of the complete knowledge of system dynamics is avoided by employing an adaptive identifier in conjunction with a novel adaptive law, such that the estimated identifier weights converge to a small neighborhood of their ideal values. An adaptive steady-state controller is developed to maintain the desired tracking performance at the steady-state, and an adaptive optimal controller is designed to stabilize the tracking error dynamics in an optimal manner. For this purpose, a critic neural network (NN) is utilized to approximate the optimal value function of the Hamilton-Jacobi-Bellman (HJB) equation, which is used in the construction of the optimal controller. The learning of two NNs, i.e., the identifier NN and the critic NN, is continuous and simultaneous by means of a novel adaptive law design methodology based on the parameter estimation error. Stability of the whole system consisting of the identifier NN, the critic NN and the optimal tracking control is guaranteed using Lyapunov theory; convergence to a near-optimal control law is proved. Simulation results exemplify the effectiveness of the proposed method.      

Control of Nonaffine Nonlinear Discrete-Time Systems Using Reinforcement-Learning-Based Linearly Parameterized Neural Networks

A nonaffine discrete-time system represented by the nonlinear autoregressive moving average with eXogenous input (NARMAX) representation with unknown nonlinear system dynamics is considered. An equivalent affinelike representation in terms of the tracking error dynamics is first obtained from the original nonaffine nonlinear discrete-time system so that reinforcement-learning-based near-optimal neural network (NN) controller can be developed. The control scheme consists of two linearly parameterized NNs. One NN is designated as the critic NN, which approximates a predefined long-term cost function, and an action NN is employed to derive a near-optimal control signal for the system to track a desired trajectory while minimizing the cost function simultaneously. The NN weights are tuned online. By using the standard Lyapunov approach, the stability of the closed-loop system is shown. The net result is a supervised actor-critic NN controller scheme which can be applied to a general nonaffine nonlinear discrete-time system without needing the affinelike representation. Simulation results demonstrate satisfactory performance of the controller.     

Neural-Network-Based State Feedback Control of a Nonlinear Discrete-Time System in Nonstrict Feedback Form

In this paper, a suite of adaptive neural network (NN) controllers is designed to deliver a desired tracking performance for the control of an unknown, second-order, nonlinear discrete-time system expressed in nonstrict feedback form. In the first approach, two feedforward NNs are employed in the controller with tracking error as the feedback variable whereas in the adaptive critic NN architecture, three feedforward NNs are used. In the adaptive critic architecture, two action NNs produce virtual and actual control inputs, respectively, whereas the third critic NN approximates certain strategic utility function and its output is employed for tuning action NN weights in order to attain the near-optimal control action. Both the NN control methods present a well-defined controller design and the noncausal problem in discrete-time backstepping design is avoided via NN approximation. A comparison between the controller methodologies is highlighted. The stability analysis of the closed-loop control schemes is demonstrated. The NN controller schemes do not require an offline learning phase and the NN weights can be initialized at zero or random. Results show that the performance of the proposed controller schemes is highly satisfactory while meeting the closed-loop stability.      

Discrete-Time Adaptive Backstepping Nonlinear Control via High-Order Neural Networks

This paper deals with adaptive tracking for discrete-time multiple-input-multiple-output (MIMO) nonlinear systems in presence of bounded disturbances. In this paper, a high-order neural network (HONN) structure is used to approximate a control law designed by the backstepping technique, applied to a block strict feedback form (BSFF). This paper also includes the respective stability analysis, on the basis of the Lyapunov approach, for the whole controlled system, including the extended Kalman filter (EKF)-based NN learning algorithm. Applicability of the scheme is illustrated via simulation for a discrete-time nonlinear model of an electric induction motor.     

Reinforcement learning-based output feedback control of nonlinear systems with input constraints.

A novel neural network (NN)-based output feedback controller with magnitude constraints is designed to deliver a desired tracking performance for a class of multi-input and multi-output (MIMO) strict feedback nonlinear discrete-time systems. Reinforcement learning is proposed for the output feedback controller, which uses three NNs: 1) an NN observer to estimate the system states with the input-output data, 2) a critic NN to approximate certain strategic utility function, and 3) an action NN to minimize both the strategic utility function and the unknown dynamics estimation errors. Using the Lyapunov approach, the uniformly ultimate boundedness (UUB) of the state estimation errors, the tracking errors and weight estimates is shown.     

An adaptive dynamic programming method for observer-based sliding mode control of connected vehicles subject to deception attacks

This article investigates the problem of optimal observer-based sliding mode control (SMC) of connected vehicles subject to deception attacks and disturbances with adaptive dynamic programming (ADP) method. For a group of vehicles with unknown nonlinear dynamics term and disturbance, this article aims to give a control methodology to achieve secure tracking of the desired spacing, velocity and acceleration. A neural network (NN) and an observer are constructed to estimate the unknown nonlinear term and the states, respectively. Then, a SMC scheme incorporating NN approximation is developed and an off-policy ADP method is used to implement the optimal control of sliding mode dynamics. The proposed method can ensure individual stability and string stability of the set of vehicles. Numerical simulations are conducted to demonstrate the validity of the proposed controller.     

Control of a class of nonlinear discrete-time systems usingmultilayer neural networks

A multilayer neural-network (NN) controller is designed to deliver a desired tracking performance for the control of a class of unknown nonlinear systems in discrete time where the system nonlinearities do not satisfy a matching condition. Using the Lyapunov approach, the uniform ultimate boundedness of the tracking error and the NN weight estimates are shown by using a novel weight updates. Further, a rigorous procedure is provided from this analysis to select the NN controller parameters. The resulting structure consists of several NN function approximation inner loops and an outer proportional derivative tracking loop. Simulation results are then carried out to justify the theoretical conclusions. The net result is the design and development of an NN controller for strict-feedback class of nonlinear discrete-time systems.     

Fuzzy adaptive single neuron NN control of brushless DC motor

Inherently, the brushless DC motor (BLDCM) is a nonlinear plant. So, it is hard to get a good performance by using the conventional PI controller for the speed control of BLDCM. In this paper, a fuzzy adaptive single neuron neural networks (NN) controller for BLDCM is developed. The fuzzy logic system (FLS) is adopted to adjust the parameter K of single neuron NN controller online. By this way, performance of the system can be improved. Performances of the proposed fuzzy adaptive single neuron NN controller are compared with the performances of conventional PI controller and normal single neuron NN controller. The experimental results demonstrate that a good control performance is achieved. The using of fuzzy adaptive single neuron NN makes the drive system robust, accurate, and insensitive to parameter variations.     

Adaptive fuzzy APSO based inverse tracking-controller with an application to DC motors

This paper introduces the use of the adaptive particle swarm optimization (APSO) for adapting the weights of fuzzy neural networks (FNN) on line. The fuzzy neural network is used for identification of the dynamics of a DC motor with nonlinear load torque. Then the motor speed is controlled using an inverse controller to follow a required speed trajectory. The parameters of the DC motor are assumed unknown as well as the nonlinear load torque characteristics. In the first stage a nonlinear fuzzy neural network (FNN) is used to approximate the motor control voltage as a function of the motor speed samples. In the second stage, the above mentioned approximator is used to calculate the control signal (the motor voltage) as a function of the speed samples and the required reference trajectory. Unlike the conventional back-propagation technique, the adaptation of the weights of the FNN approximator is done on-line using adaptive particle swarm optimization (APSO). The APSO is based on the least squares error minimization with random initial condition and without any off-line pre-training. Simulation results are presented to prove the effectiveness of the proposed control technique in achieving the tracking performance.     

Synchronization and Tracking Control for Dual‐motor Driving Servo Systems with Friction Compensation

A nonsingular fast terminal sliding mode (NFTSM) controller is designed by incorporating the variable gain neural network (NN) observer, which is utilized to guarantee motor speed synchronization and load position tracking of dual‐motor driving servo systems. By designing the variable gain NN observer, the states and uncertain nonlinearities of servo systems are estimated with fast convergence rate and small steady‐state error, where the effects from external disturbance are suppressed as well. Based on the estimated states, the cross‐coupling synchronization strategy and NFTSM tracking scheme are designed to achieve the rapid speed synchronization and precise load tracking, where the NNs are introduced to approximate and compensate friction nonlinearities. In particular, a novel nonlinear synchronization factor characterizing the degree of speed synchronization is proposed to achieve switching between synchronization control and tracking control, which is proven to deal with the coupling problem of synchronization and tracking. Finally, the comparative simulations and experiments are included to verify the reliability and effectiveness.     

基于NARX网络的无刷直流电机自适应逆控制

针对无刷直流电机(bnmhless DC motor,BLDCM)非线性的特点,引入了一种基于神经网络的自适应逆控制方法.该方案中,用非线性自回归(NARX)动态网络做为模型辨识器和控制器.辨识器采用了BP(back propagation)算法在线调整参数,并获取被控时象精确的Jacobian信息,再由实时递归学习算法(RTRL)实现对控制器的在线整定.仿真结果表明,方法具有响应速度较快、无超调的优点,且具备较强的自适应性和鲁棒性.     

基于EKF和SMC的永磁同步电机无传感器矢量控制

采用扩展卡尔曼滤波(EKF)算法估计永磁同步电机(PMSM)转速和转子位置,构成转速、电流双闭环的无传感器矢量控制系统.针对扩展卡尔曼滤波为有偏估计、对模型误差鲁棒性差等问题,提出了基于指数趋近律的滑模转速控制器.为提高转速环抗负载转矩扰动能力,设计负载转矩观测器并将观测结果引入到电流控制器的输入端,作为速度控制器前馈补偿的控制输入.仿真实验结果表明,与传统采用PI(proportional-integral)转速控制器的系统相比,文中所提控制策略具有转速跟踪误差小、响应快、无超调、抗负载扰动能力强等优点.     

基于神经网络与粒子滤波的柔性臂控制方法研究

基于奇异摄动法将单连杆柔性臂系统分解为慢变、快变子系统,采用混合控制方法;设计了基于粒子滤波的神经网络控制器来线性化慢子系统,使其跟踪期望轨迹;采用粒子滤波训练神经网络克服了BP算法收敛速度慢、易陷入局部极小值的缺陷,及扩展卡尔曼滤波方法带来的模型线性化损失;对于快变系统采用最优控制方法;仿真结果表明:在神经网络训练误差收敛速度及精度方面,粒子滤波要比BP及卡尔曼滤波要好;组合控制方法能有效地抑制柔性臂弹性振动,轨迹跟踪迅速准确,精度方面也是前者最优。     

Multiple model-based event-triggered adaptive control of a class of discrete-time nonlinear systems

In this study, the problem of event-triggered-based adaptive control (ETAC) for a class of discrete-time nonlinear systems with unknown parameters and nonlinear uncertainties is considered. Both neural network (NN) based and linear identifiers are used to approximate the unknown system dynamics. The feedback output signals are transmitted, and the parameters and the NN weights of the identifiers are tuned in an aperiodic manner at the event sample instants. A switching mechanism is provided to evaluate the approximate performance of each identifier and decide which estimated output is utilised for the event-triggered controller design, during any two events. The linear identifier with an auxiliary output and an improved adaptive law is introduced so that the nonlinear uncertainties are no longer assumed to be Lipschitz. The number of transmission times are significantly reduced by incorporating multiple model schemes into ETAC. The boundedness of both the parameters of identifiers and the system outputs is demonstrated though the Lyapunov approach. Simulation results demonstrate the effectiveness of the proposed method.     

Control of a nonholonomic mobile robot using neural networks

A control structure that makes possible the integration of a kinematic controller and a neural network (NN) computed-torque controller for nonholonomic mobile robots is presented. A combined kinematic/torque control law is developed using backstepping and stability is guaranteed by Lyapunov theory. This control algorithm can be applied to the three basic nonholonomic navigation problems: tracking a reference trajectory, path following, and stabilization about a desired posture. Moreover, the NN controller proposed in this work can deal with unmodeled bounded disturbances and/or unstructured unmodeled dynamics in the vehicle. Online NN weight tuning algorithms do not require off-line learning yet guarantee small tracking errors and bounded control signals are utilized.     

Real‐Time Results for High Order Neural Identification and Block Control Transformation Form Using High Order Sliding Modes

In this paper, real‐time results for a novel continuous‐time adaptive tracking controller algorithm for nonlinear multiple input multiple output systems are presented. The control algorithm includes the combination of a recurrent high order neural network with block control transformation using a high order sliding modes technique as control law. A neural network is used to identify the dynamic plant behavior where a filtered error algorithm is used to train the neural identifier. A decentralized high order sliding mode, named the twisting algorithm, is used to design chattering‐reduced independent controllers to solve the trajectory tracking problem for a robot arm with three degrees of freedom. Stability analyses are given via a Lyapunov approach.     

Control of a nonholomic mobile robot: Backstepping kinematics into dynamics

A dynamical extension that makes possible the integration of a kinematic controller and a torque controller for nonholonomic mobile robots is presented. A combined kinematic/torque control law is developed using backstepping, and asymptotic stability is guaranteed by Lyapunov theory. Moreover, this control algorithm can be applied to the three basic nonholonomic navigation problems: tracking a reference trajectory, path following, and stabilization about a desired posture. The result is a general structure for controlling a mobile robot that can accommodate different control techniques, ranging from a conventional computed-torque controller, when all dynamics are known, to robust-adaptive controllers if this is not the case. A robust-adaptive controller based on neural networks (NNs) is proposed in this work. The NN controller can deal with unmodeled bounded disturbances and/or unstructured unmodeled dynamics in the vehicle. On-line NN weight tuning algorithms that do not require off-line learning yet guarantee small tracking errors and bounded control signals are utilized. © 1997 John Wiley & Sons, Inc.     

Extended observer-based hybrid tracking control strategy for networked system with FDI attacks

This paper studies the speed tracking control of networked control systems (NCSs) with external disturbance and false data injection (FDI) attacks. First, the system model with external disturbances and FDI attacks is built. Then, an extended observer based on discrete time sliding function and neural network (NN) is proposed to observe the extended states and suppress the effect of external disturbance and FDI attacks. Furthermore, a novel hybrid discrete-time sliding mode control (HDSMC) strategy combining discrete time sliding mode control with super-twisting control is designed to perform closed-loop control of the system, in which the exponential term and nonlinear term are constructed to restrain the jitters. The convergence and reachability of the sliding motion are proofed. Finally, the validity and feasibility of the proposed methods are proved by simulations and experiments.     

NN-adaptive output feedback tracking control for a class of discrete-time non-affine systems with a dynamic compensator

The problem of tracking control for a class of uncertain non-affine discrete-time nonlinear systems with internal dynamics is addressed. The fixed point theorem is first employed to ensure the control problem in question is solvable and well-defined. Based on it, an adaptive output feedback control scheme based on neural network (NN) is presented. The proposed control algorithm consists of two parts: a dynamic compensator is introduced to stabilise the linear portion of the tracking error system; a single-hidden-layer neural network (SHL NN) approximation mechanism is introduced to cancel the uncertainties resulting from the non-affine function, where the recursive weight update rules of NN estimation are derived from the discrete-time version of Lyapunov control theory. Ultimate boundedness of the error signals is shown through Lyapunov’s direct method and the discrete-time version of input-to-state stability (ISS) theory. Finally, a model of automatical underwater vehicle (AUV) is considered to show the effectiveness of the proposed control scheme.