求解含复杂约束非线性最优控制问题的改进Gauss伪谱法

针对含有复杂约束条件的非线性最优控制问题,提出了一种改进的Gauss伪谱法 (Improved Gauss pseudospectral method, IGPM). 这类问题难以得到解析解,特别是有些问题不存在解析的模型, 一些参数只能通过查表得到,使得传统方法难以求解. 在传统的Gauss伪谱法的基础上,将非线性的终端状态积分约束等价地转化为线性形式,提出了IGPM, 通过协态映射定理可以计算出协态变量,检验最优性,使得IGPM具有间接法一样的精度. 并且给出了初始时刻协态变量和端点时刻控制变量的计算方法. 为了提高解的精度,基于IGPM提出了迭代算法, 最后将该算法应用于求解高超声速飞行器上升段轨迹优化问题,结果表明最优轨迹基本满足路径约束条件和最优性条件.     

求解最优控制问题的Chebyshev-Gauss伪谱法

提出了一种求解最优控制问题的Chebyshev-Gauss伪谱法, 配点选择为Chebyshev-Gauss点. 通过比较非线性规划问题的Kaursh-Kuhn-Tucker条件和伪谱离散化的最优性条件, 导出了协态和Lagrange乘子的估计公式. 在状态逼近中, 采用了重心Lagrange插值公式, 并提出了一种简单有效的计算状态伪谱微分矩阵的方法. 该法的独特优势是具有良好的数值稳定性和计算效率. 仿真结果表明, 该法能够高精度地求解带有约束的复杂最优控制问题.     

Direct Trajectory Optimization and Costate Estimation of Infinite-horizon Optimal Control Problems Using Collocation at the Flipped Legendre-Gauss-Radau Points

A pseudospectral method is presented for direct trajectory optimization and costate estimation of infinite-horizon optimal control problems using global collocation at flipped Legendre-Gauss-Radau points which include the end point +1. A distinctive feature of the method is that it uses a new smooth, strictly monotonically decreasing transformation to map the scaled left half-open interval τ∈(-1, +1] to the descending time interval t ∈ (+∞, 0]. As a result, the singularity of collocation at point +1 associated with the commonly used transformation, which maps the scaled right half-open interval τ∈[-1, +1) to the increasing time interval [0,+∞), is avoided. The costate and constraint multiplier estimates for the proposed method are rigorously derived by comparing the discretized necessary optimality conditions of a finite-horizon optimal control problem with the Karush-Kuhn-Tucker conditions of the resulting nonlinear programming problem from collocation. Another key feature of the proposed method is that it provides highly accurate approximation to the state and costate on the entire horizon, including approximation at t = +∞, with good numerical stability. Numerical results show that the method presented in this paper leads to the ability to determine highly accurate solutions to infinite-horizon optimal control problems.      

A Hermite‐Lobatto Pseudospectral Method for Optimal Control

A pseudospectral (PS) method based on Hermite interpolation and collocation at the Legendre‐Gauss‐Lobatto (LGL) points is presented for direct trajectory optimization and costate estimation of optimal control problems. A major characteristic of this method is that the state is approximated by the Hermite interpolation instead of the commonly used Lagrange interpolation. The derivatives of the state and its approximation at the terminal time are set to match up by using a Hermite interpolation. Since the terminal state derivative is determined from the dynamic, the state approximation can automatically satisfy the dynamic at the terminal time. When collocating the dynamic at the LGL points, the collocation equation for the terminal point can be omitted because it is constantly satisfied. By this approach, the proposed method avoids the issue of the Legendre PS method where the discrete state variables are over‐constrained by the collocation equations, hence achieving the same level of solution accuracy as the Gauss PS method and the Radau PS method, while retaining the ability to explicitly generate the control solution at the endpoints. A mapping relationship between the Karush‐Kuhn‐Tucker multipliers of the nonlinear programming problem and the costate of the optimal control problem is developed for this method. The numerical example illustrates that the use of the Hermite interpolation as described leads to the ability to produce both highly accurate primal and dual solutions for optimal control problems.     

求解最优控制问题的混合变量变分方法及其航天控制应用

针对非线性最优控制导出的Hamiltonian系统两点边值问题,提出一种以离散区段右端状态和左端协态为混合独立变量的数值求解方法, 将非线性Hamiltonian系统两点边值问题的求解通过混合独立变量变分原理转化为非线性方程组求解.所提出的算法综合了求解最优控制 的"直接法"和"间接法"的特征,既满足最优控制理论的一阶必要条件,又不需要对协态初值的准确猜测,避免了求解大规模非线性规划问题. 通过两个航天控制算例讨论了本文算法的精度和效率等问题.与近年来在航空航天控制中备受关注的高斯伪谱方法相比较,本文算法无论是在 精度还是效率上都具有明显的优势.     

A unified framework for the numerical solution of optimal control problems using pseudospectral methods

A unified framework is presented for the numerical solution of optimal control problems using collocation at Legendre-Gauss (LG), Legendre-Gauss-Radau (LGR), and Legendre-Gauss-Lobatto (LGL) points. It is shown that the LG and LGR differentiation matrices are rectangular and full rank whereas the LGL differentiation matrix is square and singular. Consequently, the LG and LGR schemes can be expressed equivalently in either differential or integral form, while the LGL differential and integral forms are not equivalent. Transformations are developed that relate the Lagrange multipliers of the discrete nonlinear programming problem to the costates of the continuous optimal control problem. The LG and LGR discrete costate systems are full rank while the LGL discrete costate system is rank-deficient. The LGL costate approximation is found to have an error that oscillates about the true solution and this error is shown by example to be due to the null space in the LGL discrete costate system. An example is considered to assess the accuracy and features of each collocation scheme.     

Solving a class of nonlinear optimal control problems via he’s variational iteration method

This paper presents an analytical approximate solution for a class of nonlinear quadratic optimal control problems. The proposed method consists of a Variational Iteration Method (VIM) together with a shooting method like procedure, for solving the extreme conditions obtained from the Pontryagin’s Maximum Principle (PMP). This method is applicable for a large class of nonlinear quadratic optimal control problems. In order to use the proposed method, a control design algorithm with low computational complexity is presented. Through the finite iterations of algorithm, a suboptimal control law is obtained for the nonlinear optimal control problem. Two illustrative examples are given to demonstrate the simplicity and efficiency of the proposed method.     

一种高效的快速近似控制向量参数化方法

控制向量参数化(Control vector parameterization, CVP) 方法是目前求解流程工业中最优操作问题的主流数值方法,然而,该方法的主要缺点之一是 计算效率较低,这是因为在求解生成的非线性规划(Nonlinear programming, NLP) 问题时,需要随着控制参数的调整,反复不断地求解相关的微分方程组,这也是CVP 方法中最耗时的部分.为了提高CVP 方法的计算效率,本文提出一种新颖的快速近似方法,能够有效减少微分方程组、函数值以及 梯度的计算量.最后,两个经典的最优控制问题上的测试结果及与国外成熟的最优控制 软件的比较研究表明:本文提出的快速近似CVP 方法在精度和效率上兼有良好的表现.     

Successive method for general multiple linear-quadratic controlproblem in discrete time

A successive method is proposed for the general multiple linear-quadratic control problem in discrete time. A family of auxiliary parametric linear-quadratic control problems is constructed such that its solution sequence converges to the optimal solution of the original problem. Theoretical analysis for the computational procedure and global convergence of the successive method is provided. The successive method utilizes the special structure of the problem, thus being computationally efficient and being able to furnish a closed-loop optimal control law     

A new solution to optimization-satisfaction problems by a penalty method

This paper is concerned with an optimization-satisfaction problem to determine an optimal solution such that a certain objective function is minimized, subject to satisfaction conditions against uncertainties of any disturbances or opponents' decisions. Such satisfaction conditions require that plural performance criteria are always less than specified values against any disturbances or opponents' decisions. Therefore, this problem is formulated as a minimization problem with the constraints which include max operations with respect to the disturbances or the opponents' decision variables. A new computational method is proposed in which a series of approximate problems transformed by applying a penalty function method to the max operations within the satisfaction conditions are solved by usual nonlinear programming. It is proved that a sequence of approximated solutions converges to a true optimal solution. The proposed algorithm may be useful for systems design under unknown parameters, process control under uncertainties, general approximation theory, and strategic weapons allocation problems.     

Initial condition of costate in linear optimal control using convex analysis

The Pontryagin Maximum Principle is one of the most important results in optimal control, and provides necessary conditions for optimality in the form of a mixed initial/terminal boundary condition on a pair of differential equations for the system state and its conjugate costate. Unfortunately, this mixed boundary value problem is usually difficult to solve, since the Pontryagin Maximum Principle does not give any information on the initial value of the costate. In this paper, we explore an optimal control problem with linear and convex structure and derive the associated dual optimization problem using convex duality, which is often much easier to solve than the original optimal control problem. We present that the solution to the dual optimization problem supplements the necessary conditions of the Pontryagin Maximum Principle, and elaborate the procedure of constructing the optimal control and its corresponding state trajectory in terms of the solution to the dual problem.     

Optimal control of switched systems based on parameterization of the switching instants

This paper presents a new approach for solving optimal control problems for switched systems. We focus on problems in which a prespecified sequence of active subsystems is given. For such problems, we need to seek both the optimal switching instants and the optimal continuous inputs. In order to search for the optimal switching instants, the derivatives of the optimal cost with respect to the switching instants need to be known. The most important contribution of the paper is a method which first transcribes an optimal control problem into an equivalent problem parameterized by the switching instants and then obtains the values of the derivatives based on the solution of a two point boundary value differential algebraic equation formed by the state, costate, stationarity equations, the boundary and continuity conditions, along with their differentiations. This method is applied to general switched linear quadratic problems and an efficient method based on the solution of an initial value ordinary differential equation is developed. An extension of the method is also applied to problems with internally forced switching. Examples are shown to illustrate the results in the paper.     

Adaptive Multilevel Correction Method for Finite Element Approximations of Elliptic Optimal Control Problems

In this paper we propose an adaptive multilevel correction scheme to solve optimal control problems discretized with finite element method. Different from the classical adaptive finite element method (AFEM for short) applied to optimal control which requires the solution of the optimization problem on new finite element space after each mesh refinement, with our approach we only need to solve two linear boundary value problems on current refined mesh and an optimization problem on a very low dimensional space. The linear boundary value problems can be solved with well-established multigrid method designed for elliptic equation and the optimization problems are of small scale corresponding to the space built with the coarsest space plus two enriched bases. Our approach can achieve the similar accuracy with standard AFEM but greatly reduces the computational cost. Numerical experiments demonstrate the efficiency of our proposed algorithm.     

A comparison of point and ball iterations in the contractive mapping case

In applications, one of the basic problems is to solve the fixed point equationx=Tx withT a contractive mapping. Two theorems which can be implemented computationally to verify the existence of a solutionx ^* to the equation and to obtain a convergent approximate solution sequence {x _n } are the classical Banach contraction mapping theorem and the newly established global convergence theorem of the ball algorithms in You, Xu and Liu [16]. These two theorems are compared on the basis of sensitivity, precision, computational complexity and efficiency. The comparison shows that except for computational complexity, the latter theorem is of far greater sensivity, precision and computational efficiency. This conclusion is supported by a number of numerical examples.     

基于浮动网格的路段检索方法

地图匹配是将车辆原始的GPS轨迹数据映射到实际道路网络上的过程,其中为GPS轨迹点检索候选路段是地图匹配的首要环节,然而不同的候选路段检索方式会直接影响地图匹配的准确性和效率.本文针对城市路网环境下的低频采样GPS轨迹数据,提出了一种基于浮动网格的路段检索方法.该方法利用GeoHash网格编码,采用浮动GeoHash网格的方式,为轨迹点检索候选路段.其次为了验证方法的可行性,本文通过隐马尔可夫模型,结合道路网络的拓扑结构以及轨迹的时空约束条件,采用增量的方式,利用维特比算法计算得到局部最优解.最后使用贪心策略,从已经得到的局部最优解中依次延伸得到全局最佳匹配路径.     

First-order corrections in optimal feedback control of singularly perturbed nonlinear systems

In this paper first-order correction terms, developed by using the method of matched asymptotic expansions, are incorporated in the feedback solution of a class of singularly perturbed nonlinear optimal control problems frequently encountered in aerospace applications. This improvement is based on an explicit solution of the integrals arising from the first-order matching conditions and leads to correct the initial values of the slow costate variables in the boundary layer. Consequently, a uniformly valid feedback control law, corrected to the first-order, can be synthesized. The new method is applied to an example of a constant speed minimum-time interception problem. Comparison of the zeroth- and first-order feedback control laws to the exact optimal solution demonstrates that first-order corrections greatly extend the domain of validity of the approximation obtained by singular perturbation methods.     

An indirect numerical method for the solution of a class of optimal control problems with singular arcs

An indirect numerical method is presented that solves a class of optimal control problems that have a singular arc occurring after an initial nonsingular arc. This method iterates on the subset of initial costate variables that enforce the junction conditions for switching to a singular arc, and the time of switching off of the singular arc to a final nonsingular arc, to reduce a terminal error function of the final conditions to zero. This results in the solution to the two-point boundary-value problem obtained using the minimum principle and some necessary conditions for singular arcs. The main advantage of this method is that the exact solution to the two-point boundary-value problem is obtained. The main disadvantage is that the sequence of controls for the problem must be known to apply this method. Two illustrative examples are presented.     

Necessary conditions for min-max problems and algorithms by a relaxation procedure

For decision making under uncertainty, a rational optimality criterion is min-max. Min-max problems such that the minimizer makes an optimal decision against the worst case that might be chosen by the maximizer are studied. This paper presents necessary conditions and computational methods for a min-max solution (not a saddle point solution). Those conditions are stated in a form like Kuhn-Tucker theorem. The computational methods are based on the relaxation procedure. A min-max problem such that the minimizer and the maximizer are subject to separate constraints is primarily studied. But it is shown that the obtained results can be applied for the unseparate constraint case by use of duality theory.     

Optimal Trajectory Planning for Flexible Link Manipulators with Large Deflection Using a New Displacements Approach

The main objective of the present paper is to determine the optimal trajectory of very flexible link manipulators in point-to-point motion using a new displacement approach. A new nonlinear finite element model for the dynamic analysis is employed to describe nonlinear modeling for three-dimensional flexible link manipulators, in which both the geometric elastic nonlinearity and the foreshortening effects are considered. In comparison to other large deformation formulations, the motion equations contain constant stiffness matrix because the terms arising from geometric elastic nonlinearity are moved from elastic forces to inertial, reactive and external forces, which are originally nonlinear. This makes the formulation particularly efficient in computational terms and numerically more stable than alternative geometrically nonlinear formulations based on lower-order terms. In this investigation, the computational method to solve the trajectory planning problem is based on the indirect solution of open-loop optimal control problem. The Pontryagin’s minimum principle is used to obtain the optimality conditions, which is lead to a standard form of a two-point boundary value problem. The proposed approach has been implemented and tested on a single-link very flexible arm and optimal paths with minimum effort and minimum vibration are obtained. The results illustrate the power and efficiency of the method to overcome the high nonlinearity nature of the problem.     

Chinese satellite photogrammetry without ground control points based on a public DEM using an efficient and robust DEM matching method

Photogrammetry based on high-resolution satellite image can acquire geospatial information within a large area rapidly and timely, but its geopositioning accuracy is highly dependent on ground control points. Under the background of global mapping, a public digital elevation model (DEM) assisted Chinese satellite image geopositioning scheme was proposed to realize satellite photogrammetry without ground control points. To make full use of public DEM advantages of consistent and high accuracy, public DEM was regarded as reference data and matched with the DEM extracted from image, and then the determined transformation parameters were applied to correct direct georeferencing results in object space. A fast least Z-difference method combined with least trimmed squares estimator was proposed to achieve DEM matching, which can not only automatically select corresponding point determination model, but also self-adaptively detect and eliminate difference between DEMs. Multi-groups of comparative experiments using Mapping Satellite-1 and ZiYuan-3 surveying satellite images were designed. Experimental results show that the DEM assisted geopositioning scheme exploits advantages of reference DEM, which can greatly improve the accuracy of photogrammetry without ground control points to a relatively high level. The geopositioning accuracy of image is largely determined by, but not confined to, the accuracy of reference DEM, but it is slightly affected by the resolution of reference DEM. If target DEM resolution is relatively high, the geopositioning accuracy of a single image assisted by Shuttle Radar Topography Mission (SRTM) DEM can satisfy the accuracy requirements of 1:50,000 scale mapping perfectly. This scheme also has good robustness and high computational efficiency.