A radial basis collocation method for Hamilton–Jacobi–Bellman equations

In this paper we propose a semi-meshless discretization method for the approximation of viscosity solutions to a first order Hamilton–Jacobi–Bellman (HJB) equation governing a class of nonlinear optimal feedback control problems. In this method, the spatial discretization is based on a collocation scheme using the global radial basis functions (RBFs) and the time variable is discretized by a standard two-level time-stepping scheme with a splitting parameter θ. A stability analysis is performed, showing that even for the explicit scheme that θ=0, the method is stable in time. Since the time discretization is consistent, the method is also convergent in time. Numerical results, performed to verify the usefulness of the method, demonstrate that the method gives accurate approximations to both of the control and state variables.     

Hamilton–Jacobi–Bellman Equations and Approximate Dynamic Programming on Time Scales

The time scales calculus is a key emerging area of mathematics due to its potential use in a wide variety of multidisciplinary applications. We extend this calculus to approximate dynamic programming (ADP). The core backward induction algorithm of dynamic programming is extended from its traditional discrete case to all isolated time scales. Hamilton–Jacobi–Bellman equations, the solution of which is the fundamental problem in the field of dynamic programming, are motivated and proven on time scales. By drawing together the calculus of time scales and the applied area of stochastic control via ADP, we have connected two major fields of research.      

Hybrid Hamilton–Jacobi–Poisson wall distance function model

Expensive to compute wall distances are used in key turbulence models and also for the modeling of peripheral physics. A potentially economical, robust, readily parallel processed, accuracy improving, differential equation based distance algorithm is described. It is hybrid, partly utilising an approximate Poisson equation. This also allows auxiliary front propagation direction/velocity information to be estimated, effectively giving wall normals. The Poisson normal can be used fully, in an approximate solution of the eikonal equation (the exact differential equation for wall distance). Alternatively, a weighted fraction of this Poisson front direction (effectively, front velocity, in terms of the eikonal equation input) information and that implied by the eikonal equation can be used. Either results in a hybrid Poisson–eikonal wall distance algorithm. To improve compatibility of wall distance functions with turbulence physics a Laplacian is added to the eikonal equation. This gives what is termed a Hamilton–Jacobi equation. This hybrid Poisson–Hamilton–Jacobi approach is found to be robust on poor quality grids. The robustness largely results from the elliptic background presence of the Poisson equation. This elliptic component prevents fronts propagated from solid surfaces, by the hyperbolic eikonal equation element, reflecting off zones of rapidly changing grid density. Where this reflection (due to poor grid quality) is extreme, the transition of front velocity information from the Poisson to Hamilton–Jacobi equation can be done more gradually. Consistent with turbulence modeling physics, under user control, the hybrid equation can overestimate the distance function strongly around convex surfaces and underestimate it around concave. If the former trait is not desired the current approach is amenable to zonalisation. With this, the Poisson element is automatically removed around convex geometry zones.     

Neural-Network-Based Near-Optimal Control for a Class of Discrete-Time Affine Nonlinear Systems With Control Constraints

In this paper, the near-optimal control problem for a class of nonlinear discrete-time systems with control constraints is solved by iterative adaptive dynamic programming algorithm. First, a novel nonquadratic performance functional is introduced to overcome the control constraints, and then an iterative adaptive dynamic programming algorithm is developed to solve the optimal feedback control problem of the original constrained system with convergence analysis. In the present control scheme, there are three neural networks used as parametric structures for facilitating the implementation of the iterative algorithm. Two examples are given to demonstrate the convergence and feasibility of the proposed optimal control scheme.     

不确定离散系统的神经网络自适应控制方法研究

本文研究了不确定离散系统的神经网络自适应控制器的设计。因为它不需要假设系统状态是可测的,一个观测器用来估计不可测状态。与现有离散系统的结果相比,该控制器具有较少的直接自适应参数。因此,可以很方便地实现工程算法。利用Lyapunov分析方法,所有的闭环系统的信号是保证最终有界（UUB）,并且能够实现系统输出跟踪参考信号到有界紧集。一个仿真例子,验证了该方法的有效性。     

Feedback-Linearization-Based Neural Adaptive Control for Unknown Nonaffine Nonlinear Discrete-Time Systems

A new feedback-linearization-based neural network (NN) adaptive control is proposed for unknown nonaffine nonlinear discrete-time systems. An equivalent model in affine-like form is first derived for the original nonaffine discrete-time systems as feedback linearization methods cannot be implemented for such systems. Then, feedback linearization adaptive control is implemented based on the affine-like equivalent model identified with neural networks. Pretraining is not required and the weights of the neural networks used in adaptive control are directly updated online based on the input–output measurement. The dead-zone technique is used to remove the requirement of persistence excitation during the adaptation. With the proposed neural network adaptive control, stability and performance of the closed-loop system are rigorously established. Illustrated examples are provided to validate the theoretical findings.      

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.     

非线性系统的神经网络自适应逆控制

提出了非线性系统的神经网络自适应逆控制方法。设计中使用了2个神经网络,经离线训练的NN1实现非线性系统的逆,在线网络NN2用于补偿逆误差和系统的动态特性变化,对一非线性系统的仿真结果表明,神经网络自适应逆控制能够提高系统的动态性能,并且具有较好的鲁棒性。     

Control of Nonlinear Systems Using a Self-Organising Neural Network

Two applications of Self Organising Map (SOM) networks in the context of nonlinear control are introduced, one in approximate feedback linearisation and the second in optimal control. It is shown that a modified SOM can be used to approximately Input/Output (I/O) linearise and to control nonlinear systems using a combination of the SOM learning algorithm, and a biologically inspired optimisation algorithm known as chemotaxis. A proof to guarantee the stability of the closed loop during the training of the network and the operation of the whole system is included. The results are illustrated with simulations of a single link manipulator.     

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.     

一类非线性系统的神经网络预测控制

针对一类具有特殊模型的非线性系统本文提出了一种新型神经网络预测控制算法。该算法利用线性系统预测控制技术和神经网络的非线性映射及并行处理能力来求实际控制量,避免了解非线性方程和非线性预测控制所需的在线数值寻优计算,减少了计算量和计算时间。仿真结果表明了该算法的何效性。     

An Iterative Learning Control of Nonlinear Systems Using Neural Network Design

In this paper, a feedforward neural network with sigmoid hidden units is used to design a neural network based iterative learning controller for nonlinear systems with state dependent input gains. No prior offline training phase is necessary, and only a single neural network is employed. All the weights of the neurons are tuned during the iteration process in order to achieve the desired learning performance. The adaptive laws for the weights of neurons and the analysis of learning performance are determined via Lyapunov‐like analysis. A projection learning algorithm is used to prevent drifting of weights. It is shown that the tracking error vector will asymptotically converges to zero as the iteration goes to infinity, and the all adjustable parameters as well as internal signals remain bounded.     

Fixed-Final-Time-Constrained Optimal Control of Nonlinear Systems Using Neural Network HJB Approach

In this paper, fixed-final time-constrained optimal control laws using neural networks (NNS) to solve Hamilton-Jacobi-Bellman (HJB) equations for general affine in the constrained nonlinear systems are proposed. An NN is used to approximate the time-varying cost function using the method of least squares on a predefined region. The result is an NN nearly -constrained feedback controller that has time-varying coefficients found by a priori offline tuning. Convergence results are shown. The results of this paper are demonstrated in two examples, including a nonholonomic system.     

Analysis of Nonlinear Discrete-Time Systems with Higher-Order Iterative Learning Control

In this paper, an iterative learning control method is proposed for a class of nonlinear discrete-time systems with well-defined relative degree, which uses the output data from several previous operation cycles to enhance tracking performance. A new analysis approach is developed, by which the iterative learning control is shown to guarantee the convergence of the output trajectory to the desired one within bound and the bound is proportional to the bound on resetting errors. It is further proved effective to overcome initial shifts and the resultant output trajectory can be assessed as iteration increases. Numerical simulation is carried out to verify the theoretical results and exhibits that the proposed updating law possesses good transient behavior of learning process so that the convergence speed is improved.     

Polynomial-Time Probabilistic Controllability Analysis of Discrete-Time Piecewise Affine Systems

This paper proposes a probabilistic approach to the controllability analysis for discrete-time piecewise affine (PWA) systems. Three kinds of randomized algorithms, which are based on random sampling of the mode sequence and/or the initial state, for determining with a probabilistic accuracy if the system is controllable are presented: a positive one-sided error algorithm, a negative one-sided error algorithm, and a two-sided error algorithm. It is proven that these are polynomial-time algorithms with respect to several variables of the problem. It is also shown with some examples, for which it is hopeless to check the controllability in a deterministic way, that these algorithms are efficient.     

Observer-based Adaptive Neural Network Output-feedback Control for Nonlinear Strict-feedback Discrete-time Systems

This paper focuses on an observer-based output-feedback controller design for a nonlinear discrete-time system. The major characteristics of this system is that all of the subsystems are in strict-feedback form and all the states of the system are not measurable. An output tracking control problem is firstly considered in this paper. NNs are utilized to approximate unknown functions, while a state observer is designed to approximatethe unvailable states. An adaptive controller is designed on the basis of the backstepping technique. On the basis of the Lyapunov analysis approach, the boundedness of all the signals is provided. The feasibility of the proposed scheme is verified through a simulation example.

     

一类非线性系统神经网络控制的稳定性分析

从理论上研究将神经网络用于非线性系统控制，通过对神经网络的训练，实现一类非线性系统的定点跟踪。证明了神经网络学习算法的收敛性不仅与系统的动态特性有关，而且与网络的初始条件有关。仿真结果表明，适当选取网络的初值和加权的调节速率，可以实现非线性系统的定点跟踪。     

未知仿射非线性系统的执行器容错控制设计

针对一类多输入单输出未知仿射非线性系统研究了在执行器发生故障情况下的容错控制问题,基于反步设计和自适应模糊逼近理论给出了控制方案使闭环系统能够同时容忍执行器的卡死和失效故障。在每一步设计中采用一个模糊逼近器来逼近未知函数,其中包括系统函数和由故障引起的未知动态。然后基于Lyapunov稳定性理论设计自适应律在线调节逼近器的参数,使其可以自动补偿故障对系统的影响。最终得到的控制器能够在执行器发生卡死和失效故障的情况下利用未卡死执行器的有效部分保证闭环系统是稳定的,并且输出能够以指定的精度跟踪给定参考信号。仿真算例证实了该方法的可行性和有效性。     

SISO仿射非线性系统自适应变结构BPNN控制

为了更好地消除抖振,提高复杂非线性系统的控制效果,针对一类典型SISO仿射非线性系统,提出了一种新的变结构BP神经网络(back propagation neural network,BPNN)自适应控制策略(VSYNC);其中,对于系统未知非线性函数,将神经网络用作估计器;对控制输入加入连续函数项,其可以根据状态点和滑动切换面之间的距离自适应地调节不连续的控制变量,从而使变结构控制策略得到了显著提高;所提出的控制方法能够有效地抑制周围切换面的抖振,保证了系统地动态性能,同时还能够消除系统的稳态误差;仿真结果表明,该方法具有较高的控制精度和鲁棒性。     

Robust Output Feedback Tracking Control for Time-Delay Nonlinear Systems Using Neural Network

In this paper, the problem of robust output tracking control for a class of time-delay nonlinear systems is considered. The systems are in the form of triangular structure with unmodeled dynamics. First, we construct an observer whose gain matrix is scheduled via linear matrix inequality approach. For the case that the information of uncertainties bounds is not completely available, we design an observer-based neural network (NN) controller by employing the backstepping method. The resulting closed-loop system is ensured to be stable in the sense of semiglobal boundedness with the help of changing supplying function idea. The observer and the controller designed are both independent of the time delays. Finally, numerical simulations are conducted to verify the effectiveness of the main theoretic results obtained