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.     

非线性最优控制系统的保辛摄动近似求解

非线性两端边值问题是在非线性最优控制计算中遇到的主要困难, 通常将其转化为线性两端边值问题的迭代求解．因此, 很有必要发展求解线性时变非齐次方程的两端边值问题的精确、高效算法. 本文通过引入区段混合能的概念, 将问题转化为区段的混合能矩阵及向量的求解, 进一步给出了它们的保辛摄动算法. 该算法具有很强的并行性, 高效而精确. 本文还指出经典的 Riccati 变换方法是该方法的一个特例. 数值算例验证了本文方法的有效性.     

非线性广义系统最优控制的最大值原理--有限维情形

本文利用Ekeland变分原理和Fattorini引理处理非线性广义系统最优控制问题,给出该问题解适合最大值原理的证明.     

基于对偶变量变分原理和两端动量独立变量的保辛方法

将广义位移和动量同时用拉格朗日多项式近似,并选择积分区间两端动量为独立变量,然后基于对偶变量变分原理导出了哈密顿系统的离散正则变换和对应的数值积分保辛算法．当位移和动量的拉格朗日多项式近似阶数满足一定条件时,可以自然导出保辛算法的不动点格式．通过数值算例讨论了位移和动量采用不同阶次插值所需最少Gauss积分点个数,并讨论了位移插值阶数、动量插值阶数以及Gauss积分点个数对保辛算法精度的影响,说明了上述不动点格式恰好是一种最优格式．     

化学反应器的非线性最优控制设计方法

本文针对具有非线性动态特性的连续搅拌釜式反应器，提出了一种基于分步变换的控制器设计方法，通过引入一种简单的中间非线性模型和相应的非线性状态变换，综合得到了一种新的非线性系统的非线性控制器设计方法，最后的控制仿真结果表明这种设计方法是有十分有效的。     

约束非线性系统多变量最优控制研究

近年来,非线性规划算法在最优控制领域中正受到越来越多的关注。该文深人研究并实现了一种新的非线性规划算法——FSQP算法,该算法具有所有迭代点均处于可行域之内、收敛速度较快的特点。提出了一种基于FSQP算法的约束非线性系统最优控制方法。然后,运用该方法解决了带有约束的复杂非线性系统的多变量时间最优控制问题,并通过计算机仿真表明了该控制算法的可行性和良好的控制效果。     

非线性系统的鲁棒采样最优控制

采样系统控制作为一种数字控制的直接设计方法,近年来引起了广泛的重视,另一方面系统的时域约束在工业控制中是不可避免的。利用实用稳定性理论,研究了具有输出约束的一类非线性系统的鲁棒采样最优控制问题,结果表示为一些矩阵不等式,最后给了出了一个迭代算法。     

离散时间非线性时滞系统最优控制的DISOPE算法

对于非线性时滞系统的最优控制,提出一种基于线性时滞模型和二次型性能指标问题的迭代处蒙混过关针时滞系统化为满足可尔可夫性质的增广状态系统,在模型和实际存在差异的情况下,该算法通过迭代求解时滞线性最优控制问题和参数估计问题,获得原问题的最优解,仿真实例表明该算法的有效性和实用性。     

A symplectic local pseudospectral method for solving nonlinear state‐delayed optimal control problems with inequality constraints

In this paper, a symplectic local pseudospectral (PS) method for solving nonlinear state‐delayed optimal control problems with inequality constraints is proposed. We first convert the original nonlinear problem into a sequence of linear quadratic optimal control problems using quasi‐linearization techniques. Then, based on local Legendre‐Gauss‐Lobatto PS methods and the dual variational principle, a PS method to solve these converted linear quadratic constrained optimal control problems is developed. The developed method transforms the converted problems into a coupling of a system of linear algebraic equations and a linear complementarity problem. The coefficient matrix involved is sparse and symmetric due to the benefit of the dual variational principle. Converged solutions can be obtained with few iterations because of the local PS method and quasi‐linearization techniques are used. The proposed method can be applied to problems with fixed terminal states or free terminal states, and the boundary conditions and constraints are strictly satisfied. Numerical simulations show that the developed method is highly efficient and accurate.     

Symplectic algorithms with mesh refinement for a hypersensitive optimal control problem

A symplectic algorithm with nonuniform grids is proposed for solving the hypersensitive optimal control problem using the density function. The proposed method satisfies the first-order necessary conditions for the optimal control problem that can preserve the structure of the original Hamiltonian systems. Furthermore, the explicit Jacobi matrix with sparse symmetric character is derived to speed up the convergence rate of the resulting nonlinear equations. Numerical simulations highlight the features of the proposed method and show that the symplectic algorithm with nonuniform grids is more computationally efficient and accuracy compared with uniform grid implementations. Besides, the symplectic algorithm has obvious advantages on optimality and convergence accuracy compared with the direct collocation methods using the same density function for mesh refinement.     

最优控制问题的Legendre 伪谱法求解及其应用

伪谱法通过全局插值多项式参数化状态和控制变量,将最优控制问题(OCP)转化为非线性规划问题(NLP)进行求解,是一类具有更高求解效率的直接法。总结Legendre伪谱法转化Bolza型最优控制问题的基本框架,推导OCP伴随变量与NLP问题KKT乘子的映射关系,建立基于拟牛顿法的LGL配点数值计算方法,并针对非光滑系统,进一步研究分段伪谱逼近策略。基于上述理论开发通用OCP求解器,并对3个典型最优控制问题进行求解,结果表明了所提出方法和求解器的有效性。     

线性–非二次最优控制问题的一种解法

讨论了求解状态终端无约束线性–非二次最优控制问题的拟Riccati方程方法, 并据此提出了计算无约束线性–非二次问题之数值解的方法; 然后将这个方法与一种能近似地化有约束问题为无约束问题的惩罚方法结合起来, 给出了一种算法, 可以计算状态终端有约束的线性–非二次最优控制问题之近似解.     

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．     

旋转曲面变换PSO 算法解非线性最优控制问题

针对利用粒子群优化算法进行多极值点函数优化时,存在陷入局部极小点和搜寻效率低的问题.提出旋转曲面变换方法,将被优化函数映射到一个同胚曲面上.它将当前局部极小点变换为全局最大点,并保持被优化函数值在当前局部极小点以下部分的形状不变,从而克服陷入局部极小点的问题.最后将其用于解一个非线性系统的最优控制问题,实验结果证明了该方法的可行性和有效性.     

非线性系统近似最优PD动态补偿控制

本文研究了一类基于动态补偿的非线性系统的近似最优PD控制的问题.用微分方程的逐次逼近理论将非线性系统的最优控制问题转化为求解线性非齐次两点边值序列问题,并提供了从时域最优状态反馈到频域最优PD控制器参数的优化方法,从而获取系统最优的动态补偿网络,设计出最优PD整定参数,给出其实现算法.最后仿真示例将所提出的方法与传统的线性二次型调节器（LQR）逐次逼近方法相比较,表明该方法具有良好的动态性能和鲁棒性.     

Robust nonlinear variable selective control for networked systems

This paper is concerned with the networked control of a class of uncertain nonlinear systems. In this way, Takagi–Sugeno (T-S) fuzzy modelling is used to extend the previously proposed variable selective control (VSC) methodology to nonlinear systems. This extension is based upon the decomposition of the nonlinear system to a set of fuzzy-blended locally linearised subsystems and further application of the VSC methodology to each subsystem. To increase the applicability of the T-S approach for uncertain nonlinear networked control systems, this study considers the asynchronous premise variables in the plant and the controller, and then introduces a robust stability analysis and control synthesis. The resulting optimal switching-fuzzy controller provides a minimum guaranteed cost on an H₂ performance index. Simulation studies on three nonlinear benchmark problems demonstrate the effectiveness of the proposed method.     

Event‐triggered optimal tracking control of nonlinear systems

We propose a novel event‐triggered optimal tracking control algorithm for nonlinear systems with an infinite horizon discounted cost. The problem is formulated by appropriately augmenting the system and the reference dynamics and then using ideas from reinforcement learning to provide a solution. Namely, a critic network is used to estimate the optimal cost while an actor network is used to approximate the optimal event‐triggered controller. Because the actor network updates only when an event occurs, we shall use a zero‐order hold along with appropriate tuning laws to encounter for this behavior. Because we have dynamics that evolve in continuous and discrete time, we write the closed‐loop system as an impulsive model and prove asymptotic stability of the equilibrium point and Zeno behavior exclusion. Simulation results of a helicopter, a one‐link rigid robot under gravitation field, and a controlled Van‐der‐Pol oscillator are presented to show the efficacy of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.     

A numerical method for solving a nonlinear 2-D optimal control problem with the classical diffusion equation

This paper presents a numerical solution for solving a nonlinear 2-D optimal control problem (2DOP). The performance index of a nonlinear 2DOP is described with a state and a control function. Furthermore, dynamic constraint of the system is given by a classical diffusion equation. It is preferred to use the Ritz method for finding the numerical solution of the problem. The method is based upon the Legendre polynomial basis. By using this method, the given optimisation nonlinear 2DOP reduces to the problem of solving a system of algebraic equations. The benefit of the method is that it provides greater flexibility in which the given initial and boundary conditions of the problem are imposed. Moreover, compared with the eigenfunction method, the satisfactory results are obtained only in a small number of polynomials order. This numerical approach is applicable and effective for such a kind of nonlinear 2DOP. The convergence of the method is extensively discussed and finally two illustrative examples are included to observe the validity and applicability of the new technique developed in the current work.