A novel optimal tracking control scheme for a class of discrete-time nonlinear systems using generalised policy iteration adaptive dynamic programming algorithm

In this paper, a novel iterative adaptive dynamic programming (ADP) algorithm, called generalised policy iteration ADP algorithm, is developed to solve optimal tracking control problems for discrete-time nonlinear systems. The idea is to use two iteration procedures, including an i-iteration and a j-iteration, to obtain the iterative tracking control laws and the iterative value functions. By system transformation, we first convert the optimal tracking control problem into an optimal regulation problem. Then the generalised policy iteration ADP algorithm, which is a general idea of interacting policy and value iteration algorithms, is introduced to deal with the optimal regulation problem. The convergence and optimality properties of the generalised policy iteration algorithm are analysed. Three neural networks are used to implement the developed algorithm. Finally, simulation examples are given to illustrate the performance of the present algorithm.     

基于ADP算法的带时滞及饱和的非线性系统最优控制

针对控制时滞及带饱和的一类离散时间非线性系统的最优控制问题,通过重构性能指标函数和对应的系统变换,处理了性能指标函数中的控制耦合项;继而引入一个合适的泛函,解决了控制带饱和问题.给出了一个新的性能指标函数,利用迭代自适应动态规划(ADP)算法获得最优控制.为实现该算法,采用神经网络逼近函数来求解最优控制问题.仿真结果验证了方法的有效性.     

Optimal tracking control of a class of nonlinear discrete-time switched systems using adaptive dynamic programming

In this paper, an infinite-horizon optimal tracking control scheme is proposed for a class of nonlinear discrete-time switched systems. First, via system transformation, the optimal tracking problem is converted into designing an optimal regulator for the tracking error dynamics. And then, with convergence analysis in terms of value function and control policy, the iterative adaptive dynamic programming (ADP) algorithm is introduced to obtain the infinite-horizon optimal tracking controller which makes the value function close to its optimal value function. Next, two neural networks are used as parametric structures to implement the ADP algorithm, which aim at approximating the value function and the control policy, respectively. Finally, a simulation example is included to complement the theoretical discussions.     

Finite horizon optimal control of discrete-time nonlinear systems with unfixed initial state using adaptive dynamic programming

In this paper, we aim to solve the finite horizon optimal control problem for a class of discrete-time nonlinear systems with unfixed initial state using adaptive dynamic programming (ADP) approach. A new ε-optimal control algorithm based on the iterative ADP approach is proposed which makes the performance index function converge iteratively to the greatest lower bound of all performance indices within an error according to ε within finite time. The optimal number of control steps can also be obtained by the proposed ε-optimal control algorithm for the situation where the initial state of the system is unfixed. Neural networks are used to approximate the performance index function and compute the optimal control policy, respectively, for facilitating the implementation of the ε-optimal control algorithm. Finally, a simulation example is given to show the results of the proposed method.     

Neuro-optimal tracking control for a class of discrete-time nonlinear systems via generalized value iteration adaptive dynamic programming approach

In this paper, a novel value iteration adaptive dynamic programming (ADP) algorithm, called “generalized value iteration ADP” algorithm, is developed to solve infinite horizon optimal tracking control problems for a class of discrete-time nonlinear systems. The developed generalized value iteration ADP algorithm permits an arbitrary positive semi-definite function to initialize it, which overcomes the disadvantage of traditional value iteration algorithms. Convergence property is developed to guarantee that the iterative performance index function will converge to the optimum. Neural networks are used to approximate the iterative performance index function and compute the iterative control policy, respectively, to implement the iterative ADP algorithm. Finally, a simulation example is given to illustrate the performance of the developed algorithm.     

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 control of unknown nonaffine nonlinear discrete-time systems based on adaptive dynamic programming

An intelligent-optimal control scheme for unknown nonaffine nonlinear discrete-time systems with discount factor in the cost function is developed in this paper. The iterative adaptive dynamic programming algorithm is introduced to solve the optimal control problem with convergence analysis. Then, the implementation of the iterative algorithm via globalized dual heuristic programming technique is presented by using three neural networks, which will approximate at each iteration the cost function, the control law, and the unknown nonlinear system, respectively. In addition, two simulation examples are provided to verify the effectiveness of the developed optimal control approach.     

An integrated optimal control algorithm for discrete-time nonlinear stochastic system

Consider a discrete-time nonlinear system with random disturbances appearing in the real plant and the output channel where the randomly perturbed output is measurable. An iterative procedure based on the linear quadratic Gaussian optimal control model is developed for solving the optimal control of this stochastic system. The optimal state estimate provided by Kalman filtering theory and the optimal control law obtained from the linear quadratic regulator problem are then integrated into the dynamic integrated system optimisation and parameter estimation algorithm. The iterative solutions of the optimal control problem for the model obtained converge to the solution of the original optimal control problem of the discrete-time nonlinear system, despite model-reality differences, when the convergence is achieved. An illustrative example is solved using the method proposed. The results obtained show the effectiveness of the algorithm proposed.     

Stable iterative adaptive dynamic programming algorithm with approximation errors for discrete-time nonlinear systems

In this paper, a novel iterative adaptive dynamic programming (ADP) algorithm is developed to solve infinite horizon optimal control problems for discrete-time nonlinear systems. When the iterative control law and iterative performance index function in each iteration cannot be accurately obtained, it is shown that the iterative controls can make the performance index function converge to within a finite error bound of the optimal performance index function. Stability properties are presented to show that the system can be stabilized under the iterative control law which makes the present iterative ADP algorithm feasible for implementation both on-line and off-line. Neural networks are used to approximate the iterative performance index function and compute the iterative control policy, respectively, to implement the iterative ADP algorithm. Finally, two simulation examples are given to illustrate the performance of the present method.     

Finite horizon optimal control of non-linear discrete-time switched systems using adaptive dynamic programming with ε-error bound

In this paper, we aim to solve the finite-horizon optimal control problem for a class of non-linear discrete-time switched systems using adaptive dynamic programming(ADP) algorithm. A new ε-optimal control scheme based on the iterative ADP algorithm is presented which makes the value function converge iteratively to the greatest lower bound of all value function indices within an error according to ε within finite time. Two neural networks are used as parametric structures to implement the iterative ADP algorithm with ε-error bound, which aim at approximating the value function and the control policy, respectively. And then, the optimal control policy is obtained. Finally, a simulation example is included to illustrate the applicability of the proposed method.     

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.     

Approximate optimal solution of the DTHJB equation for a class of nonlinear affine systems with unknown dead-zone constraints

In this paper, an optimal control scheme of a class of unknown discrete-time nonlinear systems with dead-zone control constraints is developed using adaptive dynamic programming (ADP). First, the discrete-time Hamilton–Jacobi–Bellman (DTHJB) equation is derived. Then, an improved iterative ADP algorithm is constructed which can solve the DTHJB equation approximately. Combining with Riemann integral, detailed proofs of existence and uniqueness of the solution are also presented. It is emphasized that this algorithm allows the implementation of optimal control without knowing internal system dynamics. Moreover, the approach removes the requirements of precise parameters of the dead-zone. Finally, simulation studies are given to demonstrate the performance of the present approach using neural networks.     

离散时间最优控制—–评论动态规划

阐述离散时间最优控制的特点.对比3种求解离散时间最优控制的解法,即:1)用非线性规划求解离散时间最优控制;2)用无约束优化求解离散时间最优控制;3)动态规划及其数值解.1)和2)都适用于多维静态优化,计算效率较高,是高级方法.在名义上,3)为动态优化.实际上,3)为一维分段无约束静态优化,计算效率较低,是初级方法.本文并用数字实例进一步阐明动态规划及其数值解在求解方面较差,故动态规划及其数值解已失去实用价值.在求解离散时间最优控制问题方面,无法与非线性规划求解相匹敌.     

Optimal Fixed-Point Tracking Control for Discrete-Time Nonlinear Systems via ADP

Based on adaptive dynamic programming (ADP), the fixed-point tracking control problem is solved by a value iteration (Ⅵ) algorithm. First, a class of discrete-time (DT) nonlinear system with disturbance is considered. Second, the convergence of a Ⅵ algorithm is given. It is proven that the iterative cost function precisely converges to the optimal value, and the control input and disturbance input also converges to the optimal values. Third, a novel analysis pertaining to the range of the discount factor is presented, where the cost function serves as a Lyapunov function. Finally, neural networks (NNs) are employed to approximate the cost function, the control law, and the disturbance law. Simulation examples are given to illustrate the effective performance of the proposed method.      

Policy Iteration for Optimal Control of Discrete-Time Time-Varying Nonlinear Systems

Aimed at infinite horizon optimal control problems of discrete time-varying nonlinear systems, in this paper, a new iterative adaptive dynamic programming algorithm, which is the discrete-time time-varying policy iteration (DTTV) algorithm, is developed. The iterative control law is designed to update the iterative value function which approximates the index function of optimal performance. The admissibility of the iterative control law is analyzed. The results show that the iterative value function is non-increasingly convergent to the Bellman-equation optimal solution. To implement the algorithm, neural networks are employed and a new implementation structure is established, which avoids solving the generalized Bellman equation in each iteration. Finally, the optimal control laws for torsional pendulum and inverted pendulum systems are obtained by using the DTTV policy iteration algorithm, where the mass and pendulum bar length are permitted to be time-varying parameters. The effectiveness of the developed method is illustrated by numerical results and comparisons.      

Neural-network-based approach to finite-time optimal control for a class of unknown nonlinear systems

This paper proposes a novel finite-time optimal control method based on input–output data for unknown nonlinear systems using adaptive dynamic programming (ADP) algorithm. In this method, the single-hidden layer feed-forward network (SLFN) with extreme learning machine (ELM) is used to construct the data-based identifier of the unknown system dynamics. Based on the data-based identifier, the finite-time optimal control method is established by ADP algorithm. Two other SLFNs with ELM are used in ADP method to facilitate the implementation of the iterative algorithm, which aim to approximate the performance index function and the optimal control law at each iteration, respectively. A simulation example is provided to demonstrate the effectiveness of the proposed control scheme.     

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．     

基于迭代神经动态规划的数据驱动非线性近似最优调节

利用数据驱动控制思想,建立一种设计离散时间非线性系统近似最优调节器的迭代神经动态规划方法.提出针对离散时间一般非线性系统的迭代自适应动态规划算法并且证明其收敛性与最优性.通过构建三种神经网络,给出全局二次启发式动态规划技术及其详细的实现过程,其中执行网络是在神经动态规划的框架下进行训练.这种新颖的结构可以近似代价函数及其导函数,同时在不依赖系统动态的情况下自适应地学习近似最优控制律.值得注意的是,这在降低对于控制矩阵或者其神经网络表示的要求方面,明显地改进了迭代自适应动态规划算法的现有结果,能够促进复杂非线性系统基于数据的优化与控制设计的发展.通过两个仿真实验,验证本文提出的数据驱动最优调节方法的有效性.     

锑浮选过程加药量自适应迭代学习控制

针对现有的加药量控制方法需要浮选过程动态模型或是鲁棒性不足的问题, 提出一种基于自适应动态规划 (ADP) 的浮选过程加药量自适应迭代学习控制方法. 首先, 将药剂量控制问题转化为两级优化问题 (问题 1 和问题 2). 其中, 基于前馈控制原理求解问题 1 得出前馈补偿分量以抑制外界扰动. 然后, 采用基于值迭代的 ADP 算法, 求解问题 2 以得到最优反馈增益, 从而设计一个数据驱动的最优加药量控制策略使最终的生产指标 (精矿品位和尾矿品位) 跟踪给定值, 且药剂量消耗最少. 最后, 通过工业生产数据进行仿真验证, 证明所提方法的收敛性和稳定性.     

Dual iterative adaptive dynamic programming for a class of discrete-time nonlinear systems with time-delays

In this paper, a new dual iterative adaptive dynamic programming (ADP) algorithm is developed to solve optimal control problems for a class of nonlinear systems with time-delays in state and control variables. The idea is to use the dynamic programming theory to solve the expressions of the optimal performance index function and control. Then, the dual iterative ADP algorithm is introduced to obtain the optimal solutions iteratively, where in each iteration, the performance index function and the system states are both updated. Convergence analysis is presented to prove the performance index function to reach the optimum by the proposed method. Neural networks are used to approximate the performance index function and compute the optimal control policy, respectively, for facilitating the implementation of the dual iterative ADP algorithm. Simulation examples are given to demonstrate the validity of the proposed optimal control scheme.