外文文献及翻译--自适应动态规划综述.doc

外文文献：Adaptive Dynamic Programming: An IntroductionAbstract: In this article, we introduce some recent research trends within the field of adaptive/approximate dynamic programming (ADP), including the variations on the structure of ADP schemes, the development of ADP algorithms and applications of ADP schemes. For ADP algorithms, the point of focus is that iterative algorithms of ADP can be sorted into two classes: one class is the iterative algorithm with initial stable policy; the other is the one without the requirement of initial stable policy. It is generally believed that the latter one has less computation at the cost of missing the guarantee of system stability during iteration process. In addition, many recent papers have provided convergence analysis associated with the algorithms developed. Furthermore, we point out some topics for future studies.IntroductionAs is well known, there are many methods for designing stable control for nonlinear systems. However, stability is only a bare minimum requirement in a system design. Ensuring optimality guarantees the stability of the nonlinear system. Dynamic programming is a very useful tool in solving optimization and optimal control problems by employing the principle of optimality. In 16, the principle of optimality is expressed as: “An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision.” There are several spectrums about the dynamic programming. One can consider discrete-time systems or continuous-time systems, linear systems or nonlinear systems, time-invariant systems or time-varying systems, deterministic systems or stochastic systems, etc.We first take a look at nonlinear discrete-time (timevarying) dynamical (deterministic) systems. Time-varying nonlinear systems cover most of the application areas and discrete-time is the basic consideration for digital computation. Suppose that one is given a discrete-time nonlinear (timevarying) dynamical systemwhere represents the state vector of the system and denotes the control action and F is the system function. Suppose that one associates with this system the performance index (or cost)where U is called the utility function and g is the discount factor with 0 , g # 1. Note that the function J is dependent on the initial time i and the initial state x( i ), and it is referred to as the cost-to-go of state x( i ). The objective of dynamic programming problem is to choose a control sequence u(k), k5i, i11,c, so that the function J (i.e., the cost) in (2) is minimized. According to Bellman, the optimal cost from time k is equal toThe optimal control u* 1k2 at time k is the u1k2 which achieves this minimum, i.e.,Equation (3) is the principle of optimality for discrete-time systems. Its importance lies in the fact that it allows one to optimize over only one control vector at a time by working backward in time.In nonlinear continuous-time case, the system can be described byThe cost in this case is defined asFor continuous-time systems, Bellmans principle of optimality can be applied, too. The optimal cost J*(x0)5min J(x0, u(t) will satisfy the Hamilton-Jacobi-Bellman EquationEquations (3) and (7) are called the optimality equations of dynamic programming which are the basis for implementation of dynamic programming. In the above, if the function F in (1) or (5) and the cost function J in (2) or (6) are known, the solution of u(k ) becomes a simple optimization problem. If the system is modeled by linear dynamics and the cost function to be minimized is quadratic in the state and control, then the optimal control is a linear feedback of the states, where the gains are obtained by solving a standard Riccati equation 47. On the other hand, if the system is modeled by nonlinear dynamics or the cost function is nonquadratic, the optimal state feedback control will depend upon solutions to the Hamilton-Jacobi-Bellman (HJB) equation 48 which is generally a nonlinear partial differential equation or difference equation. However, it is often computationally untenable to run true dynamic programming due to the backward numerical process required for its solutions, i.e., as a result of the well-known “curse of dimensionality” 16, 28. In 69, three curses are displayed in resource management and control problems to show the cost function J , which is the theoretical solution of the Hamilton-Jacobi- Bellman equation, is very difficult to obtain, except for systems satisfying some very good conditions. Over the years, progress has been made to circumvent the “curse of dimensionality” by building a system, called “critic”, to approximate the cost function in dynamic programming (cf. 10, 60, 61, 63, 70, 78, 92, 94, 95). The idea is to approximate dynamic programming solutions by using a function approximation structure such as neural networks to approximate the cost function.The Basic Structures of ADPIn recent years, adaptive/approximate dynamic programming (ADP) has gained much attention from many researchers in order to obtain approximate solutions of the HJB equation, cf. 2, 3, 5, 8, 1113, 21, 22, 25, 30, 31, 34, 35, 40, 46, 49, 52, 54, 55, 63, 70, 76, 80, 83, 95, 96, 99, 100. In 1977, Werbos 91 introduced an approach for ADP that was later called adaptive critic designs (ACDs). ACDs were proposed in 91, 94, 97 as a way for solving dynamic programming problems forward-in-time. In the literature, there are several synonyms used for “Adaptive Critic Designs” 10, 24, 39, 43, 54, 70, 71, 87, including “Approximate Dynamic Programming” 69, 82, 95, “Asymptotic Dynamic Programming” 75, “Adaptive Dynamic Programming” 63, 64, “Heuristic Dynamic Programming” 46, 93, “Neuro-Dynamic Programming” 17, “Neural Dynamic Programming” 82, 101, and “Reinforcement Learning” 84.Bertsekas and Tsitsiklis gave an overview of the neurodynamic programming in their book 17. They provided the background, gave a detailed introduction to dynamic programming, discussed the neural network architectures and methods for training them, and developed general convergence theorems for stochastic approximation methods as the foundation for analysis of various neuro-dynamic programming algorithms. They provided the core neuro-dynamic programming methodology, including many mathematical results and methodological insights. They suggested many useful methodologies for applications to neurodynamic programming, like Monte Carlo simulation, on-line and off-line temporal difference methods, Q-learning algorithm, optimistic policy iteration methods, Bellman error methods, approximate linear programming, approximate dynamic programming with cost-to-go function, etc. A particularly impressive success that greatly motivated subsequent research, was the development of a backgammon playing program by Tesauro 85. Here a neural network was trained to approximate the optimal cost-to-go function of the game of backgammon by using simulation, that is, by letting the program play against itself. Unlike chess programs, this program did not use lookahead of many steps, so its success can be attributed primarily to the use of a properly trained approximation of the optimal cost-to-go function.To implement the ADP algorithm, Werbos 95 proposed a means to get around this numerical complexity by using “approximate dynamic programming” formulations. His methods approximate the original problem with a discrete formulation. Solution to the ADP formulation is obtained through neural network based adaptive critic approach. The main idea of ADP is shown in Fig. 1.He proposed two basic versions which are heuristic dynamic programming (HDP) and dual heuristic programming (DHP).HDP is the most basic and widely applied structure of ADP 13, 38, 72, 79, 90, 93, 104, 106. The structure of HDP is shown in Fig. 2. HDP is a method for estimating the cost function. Estimating the cost function for a given policy only requires samples from the instantaneous utility function U, while models of the environment and the instantaneous reward are needed to find the cost function corresponding to the optimal policy.In HDP, the output of the critic network is J, which is the estimate of J in equation (2). This is done by minimizing the following error measure over timewhere J(k)5J 3x(k), u(k), k, WC4 and WC represents the parameters of the critic network. When Eh50 for all k, (8) implies thatDual heuristic programming is a method for estimating the gradient of the cost function, rather than J itself. To do this, a function is needed to describe the gradient of the instantaneous cost function with respect to the state of the system. In the DHP structure, the action network remains the same as the one for HDP, but for the second network, which is called the critic network, with the costate as its output and the state variables as its inputs.The critic networks training is more complicated than that in HDP since we need to take into account all relevant pathways of backpropagation.This is done by minimizing the following error measure over timewhere J 1k2 /x1k2 5J 3x1k2, u1k2, k, WC4/x1k2 and WC represents theparameters of the critic network. When Eh50 for all k, (10) implies that2. Theoretical DevelopmentsIn 82, Si et al summarizes the cross-disciplinary theoretical developments of ADP and overviews DP and ADP; and discusses their relations to artificial intelligence, approximation theory, control theory, operations research, and statistics.In 69, Powell shows how ADP, when coupled with mathematical programming, can solve (approximately) deterministic or stochastic optimization problems that are far larger than anything that could be solved using existing techniques and shows the improvement directions of ADP.In 95, Werbos further gave two other versions called “actiondependent critics,” namely, ADHDP (also known as Q-learning 89) and ADDHP. In the two ADP structures, the control is also the input of the critic networks. In 1997, Prokhorov and Wunsch 70 presented more algorithms according to ACDs.They discussed the design families of HDP, DHP, and globalized dual heuristic programming (GDHP). They suggested some new improvements to the original GDHP design. They promised to be useful for many engineering applications in the areas of optimization and optimal control. Based on one of these modifications, they present a unified approach to all ACDs. This leads to a generalized training procedure for ACDs. In 26, a realization of ADHDP was suggested: a least squares support vector machine (SVM) regressor has been used for generating the control actions, while an SVM-based tree-type neural network (NN) is used as the critic. The GDHP or ADGDHP structure minimizes the error with respect to both the cost and its derivatives. While it is more complex to do this simultaneously, the resulting behavior is expected to be superior. So in 102, GDHP serves as a reconfigurable controller to deal with both abrupt and incipient changes in the plant dynamics due to faults. A novel fault tolerant control (FTC) supervisor is combined with GDHP for the purpose of improving the performance of GDHP for fault tolerant control. When the plant is affected by a known abrupt fault, the new initial conditions of GDHP are loaded from dynamic model bank (DMB). On the other hand, if the fault is incipient, the reconfigurable controller maintains performance by continuously modifying itself without supervisor intervention. It is noted that the training of three networks used to implement the GDHP is in an online fashion by utilizing two distinct networks to implement the critic. The first critic network is trained at every iterations while the second one is updated with a copy of the first one at a given period of iterations.All the ADP structures can realize the same function that is to obtain the optimal control policy while the computation precision and running time are different from each other. Generally speaking, the computation burden of HDP is low but the computation precision is also low; while GDHP has better precision but the computation process will take longer time and the detailed comparison can be seen in 70. In 30, 33 and 83, the schematic of direct heuristic dynamic programming is developed. Using the approach of 83, the model network in Fig. 1 is not needed anymore. Reference 101 makes significant contributions to model-free adaptive critic designs. Several practical examples are included in 101 for demonstration which include single inverted pendulum and triple inverted pendulum. A reinforcement learning-based controller design for nonlinear discrete-time systems with input constraints is presented by 36, where the nonlinear tracking control is implemented with filtered tracking error using direct HDP designs. Similar works also see 37. Reference 54 is also about model-free adaptive critic designs. Two approaches for the training of critic network are provided in 54: A forward-in-time approach and a backward-in-time approach. Fig. 4 shows the diagram of forward-intimeapproach. In this approach, we view J(k) in (8) as the output of the critic network to be trained and choose U(k)1gJ(k11) as the training target. Note that J(k) and J(k11) are obtained using state variables at different time instances. Fig. 5 shows the diagram of backward-in-time approach. In this approach, we view J(k11) in (8) as the output of the critic network to be trained and choose ( J(k)2U(k)/g as the training target. The training ap proach of 101 can be considered as a backward- in-time ap proach. In Fig. 4 and Fig. 5, x(k11) is the output of the model network.An improvement and modification to the two network architecture, which is called the “single network adaptive critic (SNAC)” was presented in 65, 66. This approach eliminates the action network. As a consequence, the SNAC architecture offers three potential advantages: a simpler architecture, lesser computational load (about half of the dual network algorithms), and no approximate error due to the fact that the action network is eliminated. The SNAC approach is applicable to a wide class of nonlinear systems where the optimal control (stationary) equation can be explicitly expressed in terms of the state and the costate variables. Most of the problems in aerospace, automobile, robotics, and other engineering disciplines can be characterized by the nonlinear control-affine equations that yield such a relation. SNAC-based controllers yield excellent tracking performances in applications to microelectronic mechanical systems, chemical reactor, and high-speed reentry problems. Padhi et al. 65 have proved that for linear systems (where the mapping between the costate at stage k11 and the state at stage k is linear), the solution obtained by the algorithm based on the SNAC structure converges to the solution of discrete Riccati equation.译文:自适应动态规划综述摘要：自适应动态规划(Adaptive dynamic programming, ADP) 是最优控制领域新兴起的一种近似最优方法, 是当前国际最优化领域的研究热点. ADP 方法利用函数近似结构来近似哈密顿 雅可比 贝尔曼(Hamilton-Jacobi-Bellman, HJB)方程的解, 采用离线迭代或者在线更新的方法, 来获得系统的近似最优控制策略, 从而能够有效地解决非线性系统的优化控制问题. 本文按照ADP 的结构变化、算法的发展和应用三个方面介绍ADP 方法. 对目前ADP 方法的研究成果加以总结, 并对这一研究领域仍需解决的问题和未来的发展方向作了进一步的展望。关键词：自适应动态规划, 神经网络, 非线性系统, 稳定性引言动态系统在自然界中是普遍存在的, 对于动态系统的稳定性分析长期以来一直是研究热点, 且已经提出了一系列方法. 然而控制科技工作者往往在保证控制系统稳定性的基础上还要求其最优性. 本世纪5060 年代, 在空间技术发展和数字计算机实用化的推动下, 动态系统的优化理论得到了迅速的发展, 形成了一个重要的学科分支: 最优控制. 它在空间技术、系统工程、经济管理与决策、人口控制、多级工艺设备的优化等许多领域都有越来越广泛的应用. 1957 年Bellman 提出了一种求解最优控制问题的有效工具: 动态规划(Dynamic programing,DP) 方法1. 该方法的核心是贝尔曼最优性原理,即: 多级决策过程的最优策略具有这种性质, 不论初始状态和初始决策如何, 其余的决策对于由初始决策所形成的状态来说, 必定也是一个最优策略. 这个原理可以归结为一个基本的递推公式, 求解多级决策问题时, 要从末端开始, 到始端为止, 逆向递推.该原理适用的范围十分广泛, 例如离散系统、连续系统、线性系统、非线性系统、确定系统以及随机系统等。下面分别就离散和连续两种情况对DP方法的基本原理进行说明. 首先考虑离散非线性系统。假设一个系统的动态方程为其中, 为系统的状态向量, 为控制输入向量。系统相应的代价函数(或性能指标函数)的形式为其中, 初始状态x(k) = xk 给定, l(x(k),u(k),k) 是效用函数, r为折扣因子且满足0 r =1。控制目标就是求解容许决策(或控制) 序列u(k), k = i, i+ 1, , 使得代价函数(2) 最小。根据贝尔曼最优性原理, 始自第k 时刻任意状态的最小代价包括两部分, 其中一部分是第k 时刻内所需最小代价, 另一部分是从第k +1 时刻开始到无穷的最小代价累加和, 即相应的k 时刻的控制策略u(k) 也达到最优, 表示为接下来, 考虑连续非线性(时变) 动态(确定) 系统的最优控制问题. 考察如下的连续时间系统:其中, F(x，u，t) 为任意连续函数。 求一容许控制策略u(t) 使得代价函数(或性能指标函数)最小. 我们可以通过离散化的方法将连续问题转换为离散问题, 然后通过离散动态规划方法求出最优控制, 当离散化时间间隔趋于零时, 两者必趋于一致。 通过应用贝尔曼最优性原理, 可以得到DP 的连续形式为可以看出, 上式是J* (x(t)，t) 以x(t)、t 为自变量的一阶非线性偏微分方程, 在数学上称其为哈密顿雅可比贝尔曼(Hamilton-Jacobi-Bellman, HJB)方程。如果系统是线性的且代价函数是状态和控制输入的二次型形式, 那么其最优控制策略是状态反馈的形式, 可以通过求解标准的黎卡提方程得到. 如果系统是非线性系统或者代价函数不是状态和控制输入的二次型形式, 那么就需要通过求解HJB 方程进而获得最优控制策略. 然而, HJB 方程这种偏微分方程的求解是一件非常困难的事情. 此外, DP方法还有一个明显的弱点: 随着x 和u 维数的增加, 计算量和存储量有着惊人的增长, 也就是我们平常所说的维数灾 问题1-2. 为了克服这些弱点, Werbos 首先提出了自适应动态规划(Adaptive dynamic programming, ADP) 方法的框架3, 其主要思想是利用一个函数近似结构(例如神经网络、模糊模型、多项式等) 来估计代价函数, 用于按时间正向求解DP 问题。近些年来, ADP 方法获得了广泛的关注, 也产生了一系列的同义词, 例如: 自适应评价设计4-7、启发式动态规划8-9、神经元动态规划10-11、自适应动态规划12 和增强学习13 等. 2006 年美国科学基金会组织的2006 NSF Workshop and Out-reach Tutorials on Approximate Dynamic Pro-gramming 研讨会上, 建议将该方法统称为Adap-tive/Approximate dynamic programming. Bert-sekas 等在文献1011 中对神经元动态规划进行了总结, 详细地介绍了动态规划、神经网络的结构和训练算法, 提出了许多应用神经元动态规划的有效方法. Si 等总结了ADP 方法在交叉学科的发展,讨论了DP 和ADP 方法与人工智能、近似理论、控制理论、运筹学和统计学的联系14. 在文献15中, Powell 展示了如何利用ADP 方法求解确定或者随机最优化问题, 并指出了ADP 方法的发展方向. Balakrishnan 等在文献16 中从有模型和无模型两种情况出发, 对之前利用ADP 方法设计动态系统反馈控制器的方法进行了总结. 文献17 从要求初始稳定和不要求初始稳定的角度对ADP 方法做了介绍. 本文将基于我们的研究成果, 在之前研究的基础上, 概述ADP 方法的最新进展。ADP 的结构发展为了执行ADP 方法, Werbos 提出了两种基本结构: 启发式动态规划(Heuristic dynamic pro-gramming, HDP) 和二次启发式规划(Dual heuris-tic programming, DHP), 其结构如图1 和图2 所示4.HDP