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.     

A numerical technique for solving fractional optimal control problems

This paper presents a numerical method for solving a class of fractional optimal control problems (FOCPs). The fractional derivative in these problems is in the Caputo sense. The method is based upon the Legendre orthonormal polynomial basis. The operational matrices of fractional Riemann-Liouville integration and multiplication, along with the Lagrange multiplier method for the constrained extremum are considered. By this method, the given optimization problem reduces to the problem of solving a system of algebraic equations. By solving this system, we achieve the solution of the FOCP. Illustrative examples are included to demonstrate the validity and applicability of the new technique.     

Sinc-Galerkin method for solving nonlinear weakly singular two point boundary value problems

A study of Sinc-Galerkin method based on double exponential transformation for solving a class of nonlinear weakly singular two point boundary value problems with non-homogeneous boundary conditions is given. The properties of the Sinc-Galerkin approach are utilized to reduce the computation of nonlinear problem to nonlinear system of equations with unknown coefficients. This method tested on several test examples. We compare our numerical results with several numerical results of existing methods. The demonstrated results confirm that proposed method is considerably efficient, accurate nature and rapidly converge.     

Use of fast direct methods for mildly nonlinear elliptic difference equations

We consider in a rectangle the dirichelet mildly nonlinear elliptic boundary value problem. The numerical solution by finite-difference equations leads to the problem of solving special nonlinear systems. We propose a numerical technique founded on the application of a fast double-sweep method to the linear system induced by modified Picard iteration. Computer results are given and a comparison is made with other methods which illustrate the effectiveness of the method.     

A constrained optimal control of nonlinear systems

A new method for optimal control of nonlinear systems with input constraint is discussed in this paper. The system is optimized by minimizing a quadratic performance index. Considering this problem as a nonlinear programming problem, the necessary and sufficient conditions for optimal control are derived, which are later simplified to an integral equation. This integral equation becomes a necessary condition. The existence of a solution of this integral equation is studied, and a method of solving it is discussed. A numerical example is worked out at the end.     

A higher-order compact ADI method with monotone iterative procedure for systems of reaction-diffusion equations

This paper is concerned with an existing compact finite difference ADI method, published in the paper by Liao et al. (2002) [3], for solving systems of two-dimensional reaction-diffusion equations with nonlinear reaction terms. This method has an accuracy of fourth-order in space and second-order in time. The existence and uniqueness of its solution are investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear reaction terms. The convergence of the finite difference solution to the continuous solution is proved. An efficient monotone iterative algorithm is presented for solving the resulting discrete system, and some techniques for the construction of upper and lower solutions are discussed. An application using a model problem gives numerical results that demonstrate the high efficiency and advantages of the method.     

Acceleration-constrained time-optimal control inndimensions

The minimum-time acceleration of a generalized point mass from an initial position and velocity to the origin inndimensions is solved by transforming the problem to an equivalent problem in two dimensions and analytically integrating the system differential equations. Computation of the optimal control is thereby reduced to the solution of five simultaneous nonlinear equations. A numerical continuation method is presented for solving these equations by starting at the known solution of a related single-dimensional problem and progressing incrementally to the desired solution. The problem and solution method are illustrated by a numerical example.     

A polynomial approach for optimal control of switched nonlinear systems

Optimal control problems for switched nonlinear systems are investigated. We propose an alternative approach for solving the optimal control problem for a nonlinear switched system based on the theory of moments. The essence of this method is the transformation of a nonlinear, nonconvex optimal control problem, that is, the switched system, into an equivalent optimal control problem with linear and convex structure, which allows us to obtain an equivalent convex formulation more appropriate to be solved by high‐performance numerical computing. Consequently, we propose to convexify the control variables by means of the method of moments obtaining semidefinite programs. Copyright © 2013 John Wiley & Sons, Ltd.     

Least squares solutions of bilinear equations

The problem of finding a least squares solution for a system of bilinear equations is investigated. Sufficient conditions to have a unique minimum are given in the case of random inputs. Three methods, the normalized iterative method, the over-parameterization method and the numerical method are presented for solving the problem along with their convergence properties. Simulation examples are provided.     

一种求解非线性约束优化全局最优的新方法

本文提出了一种求解非线性约束优化的全局最优的新方法—它是基于利用非线性互补函数和不断增加新的约束来重复解库恩-塔克条件的非线性方程组的新方法。因为库恩-塔克条件是非线性约束优化的必要条件,得到的解未必是非线性约束优化的全局最优解,为此,本文首次给出了通过利用该优化问题的先验知识,不断地增加约束来限制全局最优解范围的方法,一些仿真例子表明提出的方法和理论有效的,并且可行的。     

基于双线性模型的动态系统优化和参数估计集成方法

针对双线性模型与实际系统之间的差异，提出一种基于双线性模型求解非线性动态系统最优控制的迭代算法。该算法通过重复求解修正的基于双线性模型的优化控制问题和参数估计问题，获得实际系统的最优解。同时提出求解修正的基于双线性模型的优化控制问题的一种新的分解方法，克服了非线性和双线性两点边值问题求解的困难。仿真例子表明该算法的有效性和实用性。     

Improvement of Discrete Control Processes in Problems with Mixed Constraints

The discrete problem of optimal control with mixed constraints was considered from the standpoint of numerical solution. Computational methods are usually based on solving a series of improvement problems. The nonlinear discrete problem and that of nonlinear functional improvement for the variational system were compared. A procedure for constructing an admissible control process improving the original problem from the solution of the reduced problem was described.     

Finite element procedures for nonlinear structures in moving coordinates. Part 1: Infinite bar under moving axial loads

This paper deals with analyzing nonlinear structures under high-speed moving loads by use of the finite element method. The stationary response of an infinite bar posed on a Winkler foundation under constant moving loads is investigated. Instead of the transient analysis, the stationary solution of this problem is obtained by solving a static system in a reference frame which moves with the load for reducing the computation cost. To overcome the difficulty due to numerical instabilities when considering very fast loads (supersonic loads), a new procedure to govern the finite element formulation in moving coordinates is proposed. Comparing numerical solutions with analytical ones shows that the proposed method is valid for all values of load speed. Last, an example of nonlinear elastic foundation is considered to outline the nonlinear effects.     

Application of shifted Jacobi pseudospectral method for solving (in)finite-horizon min–max optimal control problems with uncertainty

The difficulty of solving the min–max optimal control problems (M-MOCPs) with uncertainty using generalised Euler–Lagrange equations is caused by the combination of split boundary conditions, nonlinear differential equations and the manner in which the final time is treated. In this investigation, the shifted Jacobi pseudospectral method (SJPM) as a numerical technique for solving two-point boundary value problems (TPBVPs) in M-MOCPs for several boundary states is proposed. At first, a novel framework of approximate solutions which satisfied the split boundary conditions automatically for various boundary states is presented. Then, by applying the generalised Euler–Lagrange equations and expanding the required approximate solutions as elements of shifted Jacobi polynomials, finding a solution of TPBVPs in nonlinear M-MOCPs with uncertainty is reduced to the solution of a system of algebraic equations. Moreover, the Jacobi polynomials are particularly useful for boundary value problems in unbounded domain, which allow us to solve infinite- as well as finite and free final time problems by domain truncation method. Some numerical examples are given to demonstrate the accuracy and efficiency of the proposed method. A comparative study between the proposed method and other existing methods shows that the SJPM is simple and accurate.     

面向数据的句法分析消歧

面向数据的分析技术(Data-Oriented Parsing,DOP)是一种概率分析策略,其概率模型的主要目的在于为一个给定的句子找到最可能的分析,即分析消歧。实际上,有关算法计算复杂度的大量研究证明,该类消歧问题属于NP-完全问题。因此,为有效实现最可能的分析,国外学者提出许多近似分析算法。本文主要论述在 DOP 框架中,基于 Monte Carlo 方法找到最可能分析的近似分析算法,并说明该方法可在合理的算法时间代价范围内实现,而且在统计上受控,以确保所获得的近似解确实对应着分析消歧后的精确解。     

Application of Sinc-collocation method for solving a class of nonlinear Fredholm integral equations

In this paper, the numerical solution of nonlinear Fredholm integral equations of the second kind is considered by two methods. The methods are developed by means of the Sinc approximation with the single exponential (SE) and double exponential (DE) transformations. These numerical methods combine a Sinc collocation method with the Newton iterative process that involves solving a nonlinear system of equations. We provide an error analysis for the methods. So far approximate solutions with polynomial convergence have been reported for this equation. These methods improve conventional results and achieve exponential convergence. Some numerical examples are given to confirm the accuracy and ease of implementation of the methods.     

H infinity Control of nonlinear discrete-time systems based on dynamical fuzzy models

This paper presents a solution to H infinity control problem for a class of discrete-time nonlinear systems. This class of nonlinear systems can be represented by a discretetime dynamical fuzzy model. A suitable quadratic L yapunov function is used to establish asymptotic stability with an l₂-norm bound gamma of the closed-loop system. Furthermore, a constructive algorithm is developed to obtain the stabilizing feedback control law. The controller design algorithm involves solving a set of suitable algebraic Riccati equations. An example is given to illustrate the application of the method.     

Iterative method for fuzzy equations

In this paper, we propose numerical solution for solving a system of fuzzy nonlinear equations based on Fixed point method. The convergence theorem is proved in detail. In this method the algorithm is illustrated by solving several numerical examples.     

On one boundary problem for nonlinear stationary controlled system

The object of study is nonlinear stationary controlled system of ordinary differential equations with constant disturbance in the right part. The problem of constructing the synthesising control function providing the transfer of this system from the initial state to the origin is considered. The sufficiently simple for numerical implementation algorithm of solution of the above-mentioned problem is obtained. It is shown that for local null controllability of the considered system, it is sufficient that the conditions of the Kalman's type were satisfied. In addition, the estimates restricting the choice of initial conditions and external disturbances under which the transfer is guaranteed are obtained. The main idea of the method of construction of the desired control function consists in reducing the original problem to stabilisation of a special kind linear non-stationary system and solving the Cauchy problem for an auxiliary system of ordinary differential equations closed by stabilising control. The simplicity of the realisation of this algorithm is determined by the construction of the auxiliary system and its stabilisation that could be obtained by analytical methods. The effectiveness of the method is illustrated by solving the problem of crane control and its numerical simulation.     

A modified predictor-corrector scheme for the Klein–Gordon equation

A numerical method based on a three-time level finite-difference scheme has been proposed for the solution of the two forms of the Klein–Gordon equation. The method, which is analysed for local truncation error and stability, leads to the solution of a nonlinear system. To avoid solving it, a predictor-corrector scheme using as predictor a second-order explicit scheme is proposed. The procedure of the corrector is modified by considering, as known, the already evaluated corrected values instead of the predictor ones. This modified scheme is applied to problems possessing periodic, kinks and soliton waves. The accuracy as well as the long-time behaviour of the proposed scheme is discussed and comparisons with the relevant known in the bibliography schemes are given.