一种最优控制的胞映射算法

由Hsu提出的利用胞映射进行最优控制的方法可以发展成实用的工程控制技术.本文以研究算法为出发点,利用胞映射的特性,结合实际的计算与可能,作了一系列旨在提高计算速度和节省内存的改进;并给出了简单有效的区分同耗最优控制对的统一的方法.使用平常的微型计算机也能有效地解决实际工程问题的最优控制.     

非线性动力系统全局分析的变胞胞映射法与转子／轴承系统的全局稳定性

在Ｐｏｉｎｃａｒｅ映射及胞映理论的基础上，提出了一种非线性动力系统全局分析的新方法－－变胞胞映射法，这种新方法改变了原胞映射法中胞在胞空间分布的不合理性及运算逻辑的不合理性，更适用于高维、大求解域非线性动力系统的求解。应用此方法，对具有非线性油膜力的Ｊｅｆｆｃｏｔ转子轴承系统进行了全局分析，绘制了系统分岔后的全局吸引域图，解释了一些工程中常见的非线性现象。     

非线性轨迹优化问题的保辛自适应求解方法

非线性轨迹优化问题一般是一个非线性最优控制问题。将非线性系统的最优控制问题导入到哈密顿体系的辛几何空间当中,基于对偶变量变分原理提出了求解非线性最优控制问题的一种保辛自适应方法。以时间区段两端协态作为独立变量,在时间区段内采用拉格朗日插值近似状态和协态变量,并利用对偶变量变分原理将非线性最优控制问题转化为非线性方程组的求解,保持了哈密顿系统的辛几何结构。并进一步,提出了基于多层次迭代的自适应算法,提高了非线性最优控制问题的求解效率。数值实验验证了该算法在求解非线性轨迹优化问题中的有效性。     

胞映射方法的研究和进展

介绍了胞映射方法的研究和进展. 归纳了目前胞映射方法的几种主要研究方法, 主要包括简单胞映射、广义胞映射、图胞映射、图胞映射的符号分析方法、图胞映射的面向集合方法、邻接胞映射、庞加莱型的简单胞映射、插值胞映射以及胞参照点映射方法, 分析了各类方法的基本特点和特色, 简述了这几种胞映射方法的最新国内外进展, 综述了胞映射方法在控制及相关领域的应用研究及进展, 给出了胞映射方法研究的一些展望, 提出了胞映射方法研究可能率先突破的几个研究方向.     

迭代图胞映射方法的快速生成实现

在迭代图胞映射方法的框架下,基于摄动微分多项式的思想讨论了常微分方程的快速求解,将所得结果与迭代图胞映射方法有机结合,有效地解决了迭代图胞映射动力系统的快速生成问题,克服了微分方程动力系统生成迭代图胞映射系统过程中耗时较多、效率低下的不足,大大提高了计算效率。通过对典型非线性系统——杜芬方程的应用分析,证实了该方法的有...     

流体混沌混合过程的可视化方法研究

为解决流体混沌混合的可视化问题，本文提出采用逆向庞加莱胞映射方法，来控制混沌系统数值模拟过程中的初始误差敏感，用插值胞映射的方法，来提高模拟效率。并用这一方法，成功地实现了扭转弯管中的混沌混合过程的计算机动画模拟，研究表明，本文所提出的逆向庞加莱胞映射和插值映射相结合，可以大幅度提高模拟精度和与速度，从而实现流体混沌混合的过程模拟。     

有限变形问题过程最优控制算法

在非增量算法的基础上,提出了用最优控制变分原理形成过程最优控制迭代求解的基本思路,并给出求解的基本控制方程。这一工作为有限变形力学问题的数值求解提供了一个新的处理方法。     

最优轨迹规划方法及其平动点航天器编队均衡耗能重构应用

由于均衡耗能航天器编队能够提高整体航天器编队服役时间,针对平动点航天器编队重构的均衡耗能最优轨迹规划问题,提出一种以状态、协态和控制三类变量插值为核心的求解非线性最优控制问题的新方法。基于连续时间表达的非线性最优控制问题通过变分原理转化为非线性方程组的求解,并进一步推导非线性方程组显示格式的Jacobi矩阵提高非线性方程组的计算效率。本文方法既满足最优控制理论的一阶必要条件又具有较大的收敛域;同时,不需要对协态初值准确猜测,避免了大规模非线性规划问题的求解。通过对中心航天器位置固定和无中心航天器两种情况的数值模拟,结果表明,本文方法对航天器编队重构轨迹规划问题能够达到均衡耗能的目标,具有一定的应用价值。     

最优轨迹规划方法及其平动点航天器编队均衡耗能重构应用

由于均衡耗能航天器编队能够提高整体航天器编队服役时间,针对平动点航天器编队重构的均衡耗能最优轨迹规划问题,提出一种以状态、协态和控制三类变量插值为核心的求解非线性最优控制问题的新方法。基于连续时间表达的非线性最优控制问题通过变分原理转化为非线性方程组的求解,并进一步推导非线性方程组显示格式的Jacobi矩阵提高非线性方程组的计算效率。本文方法既满足最优控制理论的一阶必要条件又具有较大的收敛域;同时,不需要对协态初值准确猜测,避免了大规模非线性规划问题的求解。通过对中心航天器位置固定和无中心航天器两种情况的数值模拟,结果表明,本文方法对航天器编队重构轨迹规划问题能够达到均衡耗能的目标,具有一定的应用价值。     

基于插值与点映射相结合的全局分析方法

利用插值的基本思想并结合点映射，提出了一种插值与点映射相结合的全局分析。通过吸引子之间的比较，判断相胞的顶点是否在同一吸引子的吸引域之内，从而识别出边界胞与非边界胞，并重点对边界胞进行处理，进而确定出相胞内各个相点的初值特性。通过比较与分析，本算法可以克服插值胞映射所存在的不足，算法简单且容易在计算机上实现。文中分析了算法产生误差的原因，给出了相应的处理方法。     

Optimal Bounded Control for Minimizing the Response of Quasi Non-Integrable Hamiltonian Systems

A strategy for designing optimal bounded control to minimize theresponse of quasi non-integrable Hamiltonian systems is proposed basedon the stochastic averaging method for quasi non-integrable Hamiltoniansystems and the stochastic dynamical programming principle. Theequations of motion of a controlled quasi non-integrable Hamiltoniansystem are first reduced to an one-dimensional averaged Itô stochasticdifferential equation for the Hamiltonian by using the stochasticaveraging method for quasi non-integrable Hamiltonian systems. Then, thedynamical programming equation for the control problem of minimizing theresponse of the averaged system is formulated based on the dynamicalprogramming principle. The optimal control law is derived from thedynamical programming equation and control constraints without solvingthe equation. The response of optimally controlled systems is predictedthrough solving the Fokker–Planck–Kolmogrov (FPK) equation associatedwith completely averaged Itô equation. Finally, two examples are workedout in detail to illustrate the application and effectiveness of theproposed control strategy.     

STOCHASTIC OPTIMAL CONTROL FOR THE RESPONSE OF QUASI NON-INTEGRABLE HAMILTONIAN SYSTEMS~

A strategy is proposed based on the stochastic averaging method for quasi nonintegrable Hamiltonian systems and the stochastic dynamical programming principle. The proposed strategy can be used to design nonlinear stochastic optimal control to minimize the response of quasi non-integrable Hamiltonian systems subject to Gaussian white noise excitation. By using the stochastic averaging method for quasi non-integrable Hamiltonian systems the equations of motion of a controlled quasi non-integrable Hamiltonian system is reduced to a one-dimensional averaged Ito stochastic differential equation. By using the stochastic dynamical programming principle the dynamical programming equation for minimizing the response of the system is formulated.The optimal control law is derived from the dynamical programming equation and the bounded control constraints. The response of optimally controlled systems is predicted through solving the FPK equation associated with It5 stochastic differential equation. An example is worked out in detail to illustrate the application of the control strategy proposed.     

非线性最小二乘跟踪的对偶变量变分方法

最小二乘跟踪方法是近几年提出的一种计算动力系统跟踪轨迹的方法.基于最小二乘跟踪的灵敏度分析算法可以有效避免传统的非线性系统灵敏度分析方法中的病态初值问题,因此其在混沌系统灵敏度分析方面有着重要的应用.针对非线性的最小二乘跟踪问题,首先将其重新描述为带有约束的非线性最优控制问题,引入协态变量并将系统的哈密顿函数表示为关于状态变量和协态变量的函数.然后将目标函数的积分时间离散化,根据对偶变量变分原理,以离散区间两端的状态变量作为独立变量,用Lagrange插值多项式近似离散区间内的状态变量和协态变量,进而将非线性最优控制问题转化为求解非线性方程组问题.这种算法无需对原问题做线性化处理,避免了复杂的线性化过程以及可能因此造成的误差,同时为求解非线性最小二乘跟踪问题提供了新的思路.根据最小二乘方法可以得到两条设计参数有微小变化的状态轨迹,基于这两条状态轨迹可进一步计算出系统关于设计参数的灵敏度,范德波振子作为数值算例验证了该方法在求解最小二乘跟踪问题以及计算非线性系统灵敏度时的有效性.      

Removing of vortices in incompressible flows by distributed controls

We consider an optimal control problem of fluids flow. The fluid motion is governed by the incompressible time-dependent Navier-Stokes equations. A new optimal control formulation for the reduction and possibly extinction of vortices is proposed. A cost functional based on a local dynamical systems characterization of vortices is investigated. The resulting functional is a non-convex function of the velocity gradient tensor. The optimality system describing first order necessary optimality conditions is derived. The gradient and the second derivative of the cost functional with respect to the distributed control are established.     

Fixed Final Time Optimal Control via Simple Cell Mapping

A cell mapping strategy based on Bellman'sPrinciple of optimality (BP) is proposed to solve the fixed final time optimalcontrol of strongly nonlinear systems with state and control bounds. Thefixed final time problem is transformed into a fixed final time-fixed finalstate optimal control problem by reversing the time. This transformationallows to solve the problem in the framework of the BP. Backward searchingalgorithms within the cell mapping context are used to obtain the solution ofthe new problem. This approach reduces considerably the computational effortrequired for the original problem when it is solved by a forward searchingapproach. The vibration control problem of damped and undamped variablestiffness oscillators with bounded stiffness tuning range is studied todemonstrate the effectiveness of the approach. A numerical study on theconvergence of the method is also presented.     

A Constrained Control Approach for the Automated Choice of an Optimal Progress Variable for Chemistry Tabulation

Flame structure look-up table generating strategies, e.g. the flamelet-progress variable approach, are based on the definition of a suitable progress variable. This variable is usually obtained from a linear combination of the species’ mass fractions contained in the system. The most important issue is the unique mapping between the progress variable and the independent variable, e.g. space or time. This is ensured by claiming the monotonicity of the progress variable. For simple fuel-oxidiser compositions, this can be performed by analysing the flame structure. However, in the case of complex chemical systems, finding such a progress variable is a non-trivial task, since a lack of monotonicity in the main species mass fractions often exists. In this article it is investigated how a valid progress variable can be found by an automated procedure. Some recent investigations use the fact that finding a monotonous progress variable is equivalent to solving a constrained optimisation problem. The new approach is to construct an algorithm to find a valid progress variable, which is also optimal in the sense that the gradients of the tabulated quantities are minimized. This leads to smoother data and reduces the interpolation effort when reading from the table. The method solves a nonlinear optimal control problem, where the monotonicity of the progress variable is ensured as an inequality constraint by using a genetic algorithm. Finally the performance of the new algorithm is evaluated for homogeneous reactor calculations and laminar diffusion flamelet look-up tables.     

Nonlinear robust and optimal control of robot manipulators

In this paper, we propose a new optimal control method for robust control of nonlinear robot manipulators. Many industrial robot systems are required to perform relatively large angular movement with sufficient accuracy. In real circumstances, highly nonlinear manipulator dynamics and uncertainties such as unknown load placed on the manipulator, external disturbance, and joint friction make the precise control of manipulators a very challenging task. The main contribution of this work is to develop a new robust control strategy to accomplish the precise control of robot manipulators under load uncertainty using a nonlinear optimal control formulation and solution. This methodology is based on the underlying relation between the robust stability and performance optimality. A class of robust control problems can be transformed to an equivalent optimal control problem by incorporating the uncertainty bounds into the cost functional. The θ-D optimal control approach is utilized to find an approximate closed-form feedback solution to the resultant nonlinear optimal control problem via a perturbation process. Numerical simulations show that the proposed robust controller is able to control the robot manipulator precisely under large load variations.     

Dynamical systems‐based optimal control of incompressible fluids

For optimal control problems related to fluid flow the choice of an adequate cost functional for suppression of vortices is of significant importance. In this research we propose a cost functional based on a local dynamical systems characterization of vortices. The resulting functional is a non‐convex function of the velocity gradient tensor. The resulting optimality system describing first order necessary optimality conditions is derived, a possible strategy for numerical realization of the optimal control problem is provided and a numerical feasibility study is conducted. Copyright © 2004 John Wiley & Sons, Ltd.     

Exponential synchronization of chaotic Lur’e systems with delayed feedback control

The exponential synchronization problem is studied in this paper for a class of chaotic Lur’e systems by using delayed feedback control. An augmented Lyapunov functional based approach is proposed to deal with this issue. A delay-dependent condition is established such that the controlled slave system can exponentially synchronize with the master system. It is shown that the delayed feedback gain matrix and the exponential decay rate can be obtained by solving a set of linear matrix inequalities. The decay coefficient can be also easily calculated. Finally, as an example, the Chua’s circuit is used to illustrate the effectiveness of the developed approach and the improvement over some existing results.     

Discrete optimal control for Birkhoffian systems

In this paper, we investigate a discrete variational optimal control for mechanical systems that admit a Birkhoffian representation. Instead of discretizing the original equations of motion, our research is based on a direct discretization of the Pfaff–Birkhoff–d’Alembert principle. The resulting discrete forced Birkhoffian equations then serve as constraints for the minimization of the objective functional. In this way, the optimal control problem is transformed into a finite-dimensional optimization problem, which can be solved by standard methods. This approach yields discrete dynamics, which is more faithful to the continuous equations of motion and consequently yields more accurate solutions to the optimal control problem which is to be approximated. We illustrate the method numerically by optimizing the control for the damped oscillator.