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.     

胞映射搜索最优控制路径的新策略

在胞映射求解最优控制问题的现有方法的基础上，提出了一种利用胞映射搜索最优控制路径的新策略。该策略基于负步长逆向数值积分技巧，采用了一种新的搜索过程，使得求解问题的效率大为提高。同现有的方法相比，新策略节省了几倍，甚至十几倍的时间。而且计算结果可靠，有很强的工程应用价值，为求解高维的最优控制问题可能会提供一定的基础。     

Stochastic Optimal Control of Nonlinear Systems via Short-Time Gaussian Approximation and Cell Mapping

A novel strategy to obtain global solutions of stochasticoptimal control problems with fixed state terminal conditions and controlbounds is proposed in this paper. The solution is global in the sense that theoptimal control solutions for all the initial conditions in a region of thestate space are obtained. The method makes use of Bellman's principle ofoptimality, the cumulant neglect closure method and the short-time Gaussianapproximation. A Markov chain with a control dependent transition probabilitymatrix is built using the generalized cell mapping method. This allows toevaluate the transient and steady state response of the controlled system. Themethod is applied to several linear and nonlinear systems leading to excellentcontrol performances.     

Optimal Control of Populations of Competing Species

The optimal control problem of populations of two competing species isstudied by using the simple cell mapping method. Two control strategiesfor pursuing the extinction of one species are considered. The controlsinclude the inclusion of population of the other species from anexternal habitat and the parametric manipulation of the intrinsicgrowing rates of the targeted species. A control to maintain bothspecies alive in the long term is also studied by manipulating thecoupling coefficients.     

Optimal Control of the Equilibrium State of a Prey-Predator Model

The control problem of the equilibrium state of prey-predator model has been studied. The equilibrium states of prey-predator model are found. The optimal control law is derived from the conditions that ensure the asymptotic stability of the equilibrium state of this model using the Lyapunov function. The general solution of the equations of the perturbed state as a function of time is obtained. Graphical and numerical simulation studies of the obtained results are presented.     

The Geometry of the Solution Set of Nonlinear Optimal Control Problems

In an optimal control problem one seeks a time-varying input to a dynamical systems in order to stabilize a given target trajectory, such that a particular cost function is minimized. That is, for any initial condition, one tries to find a control that drives the point to this target trajectory in the cheapest way. We consider the inverted pendulum on a moving cart as an ideal example to investigate the solution structure of a nonlinear optimal control problem. Since the dimension of the pendulum system is small, it is possible to use illustrations that enhance the understanding of the geometry of the solution set. We are interested in the value function, that is, the optimal cost associated with each initial condition, as well as the control input that achieves this optimum. We consider different representations of the value function by including both globally and locally optimal solutions. Via Pontryagin’s maximum principle, we can relate the optimal control inputs to trajectories on the smooth stable manifold of a Hamiltonian system. By combining the results we can make some firm statements regarding the existence and smoothness of the solution set.     

Control of an Adaptive Refinement Technique of Generalized Cell Mapping by System Dynamics

Generalized cell mapping is an efficient and powerful numerical tool for the prediction of the long-term behavior and global analysis of nonlinear dynamic systems. The only drawback of this method is the enormous computational effort it requires for high-dimensional systems. We overcome this problem by adaptively refining a very rough starting cell grid, where the adaptation is controlled by the long-term dynamics of the system. We illustrate the efficiency of our approach by examples.     

Optimal Control of a Vortex Trapped by an Airfoil with a Cavity

An airfoil with a cavity traps a vortex; the lift increases but the vortex shows great receptivity to upstream disturbances. A simple potential flow model confirms that the vortex stability basin is of a reduced extent. In this paper we present a control technique stabilizing the vortex position based on a potential flow model. The actuators are sources/sinks at the wall and the suction/blowing law is obtained by the adjoint optimization method. This revised version was published online in July 2006 with corrections to the Cover Date.     

Dynamic Stability Problems of Anisotropic Cylindrical Shells via a Simplified Analysis

An analytical–numerical method involving a small number of generalized coordinates is presented for the analysis of the nonlinear vibration and dynamic stability behaviour of imperfect anisotropic cylindrical shells. Donnell-type governing equations are used and classical lamination theory is employed. The assumed deflection modes approximately satisfy simply supported boundary conditions. The axisymmetric mode satisfying a relevant coupling condition with the linear, asymmetric mode is included in the assumed deflection function. The shell is statically loaded by axial compression, radial pressure and torsion. A two-mode imperfection model, consisting of an axisymmetric and an asymmetric mode, is used. The static-state response is assumed to be affine to the given imperfection. In order to find approximate solutions for the dynamic-state equations, Hamiltons principle is applied to derive a set of modal amplitude equations. The dynamic response is obtained via numerical time-integration of the set of nonlinear ordinary differential equations. The nonlinear behaviour under axial parametric excitation and the dynamic buckling under axial step loading of specific imperfect isotropic and anisotropic shells are simulated using this approach. Characteristic results are discussed. The softening behaviour of shells under parametric excitation and the decrease of the buckling load under step loading, as compared with the static case, are illustrated.     

Cell mapping methods-beyond global analysis of nonlinear dynamic systems

在20世纪80年代由徐皆苏教授创建的胞映射方法一直受非线性科学界同仁的欢迎.近几年胞映射方法有了许多新的应用和算法.本文介绍了一些控制应用和算法的文献.另外,还介绍和讨论胞映射方法应用与多目标优化问题的研究和方法,多目标优化控制设计和非线性代数方程找零解.文中指出胞映射方法在并行计算的帮助下,现在可以解决中等高维空间中的各类问题,新的应用还会不断出现.     

Application of Cell Mapping methods to a discontinuous dynamic system

The Cell Mapping method is a robust tool for investigating nonlinear dynamic systems. It is capable of finding the attractors and corresponding basins of attraction of a system under investigation. To investigate the applicability of the Cell Mapping method to discontinuous systems, a forced zero-stiffness impact oscillator is chosen as an application. The numerical integration algorithm, the basic element in the Cell Mapping method, is adjusted to overcome the discontinuity. Four types of Cell Mapping techniques are applied: Simple Cell Mapping, Generalized Cell Mapping, Interpolated Cell Mapping, and Mixed Cell Mapping. The last type is a new modification to existing types. Each type of Cell Mapping is briefly explained. The results are compared to the exact solutions. The Interpolated Cell Mapping and Mixed Cell Mapping methods are found to produce the most accurate results for this case.     

An Optimal Nonlinear Feedback Control Strategy for Randomly Excited Structural Systems

A strategy for optimal nonlinear feedback control of randomlyexcited structural systems is proposed based on the stochastic averagingmethod for quasi-Hamiltonian systems and the stochastic dynamicprogramming principle. A randomly excited structural system isformulated as a quasi-Hamiltonian system and the control forces aredivided into conservative and dissipative parts. The conservative partsare designed to change the integrability and resonance of the associatedHamiltonian system and the energy distribution among the controlledsystem. After the conservative parts are determined, the system responseis reduced to a controlled diffusion process by using the stochasticaveraging method. The dissipative parts of control forces are thenobtained from solving the stochastic dynamic programming equation. Boththe responses of uncontrolled and controlled structural systems can bepredicted analytically. Numerical results for a controlled andstochastically excited Duffing oscillator and a two-degree-of-freedomsystem with linear springs and linear and nonlinear dampings, show thatthe proposed control strategy is very effective and efficient.     

最优控制群桩扩张位移的安全度

本文主要通过最优控制理论来研究群桩沉击时挤压扩张位移的最优控制方案。在对综合性隔阻系数和安全度的规划后,可以有效地制订最佳的减弱和控制群桩扩张位移对相邻结构的破坏量实施方案,从而提高相邻结构的安全度。从文中工程实例的分析可以明显反映出所提供的最优控制群桩扩张位移方程的实用性、简便性,井具有一定的经济价值。     

连体结构基于时域显式法的瞬时最优控制算法

论文针对受到非平稳地震激励作用下的连体结构提出一种基于时域显式法的瞬时最优控制算法.该算法可以利用有限元模型施加脉冲荷载快速建立受控模型,可以避免系统状态部分可观测的问题,而且可以根据需要单独调控子结构受控效果的优点.一连体高层建筑结构算例用来说明该算法,并将该算法与传统瞬时最优控制算法比较,结果表明论文算法具有良好的控制效果.     

Optimal control of hyperbolic H-hemivariational inequalities with state constraints

The optimal control problems of hyperbolic H-hemivariational inequalities with the state constraints and nonnornotone multivalued mapping term are considered. The optimal solutions are obtained. In addition, their approximating problems are also studied.     

回滞非线性结构的瞬态最优控制

本文对一种国滞非线性基础隔振的主从结构模型用瞬态最优控制法进行振动控制研究。利用四阶Ｒｕｎｇｅ－Ｋｕｔｔａ积分格式统一处理最优控制方程，可直接逐步积分求出系统在瞬态最优控制下的最优控制力与系统响应。分别对主从结构无主动控制及有主动控制时的两种情况（包含或不包含ｖｂ反馈）进行计算。结果表明瞬态最优控制可有效地抑制振动。     

基于Gauss伪谱法的3D刚体摆姿态最优控制

研究Gauss伪谱法求解3D刚体摆姿态最优控制问题．针对其最优姿态控制问题,既要满足由任意位置运动到平衡位置姿态运动规划问题,又要满足系统含有动力学约束的力学模型问题,提出基于四元数来描述3D刚体摆的数学模型,建立3D刚体摆姿态的动力学和运动学方程,为了解决3D刚体摆在平衡位置处的姿态最优控制问题,设计基于Gauss伪谱算法的最优姿态开环控制器,得到了3D刚体摆的姿态最优控制轨迹,得到满足的可行解,通过仿真实验验证了其开环解在平衡位置的控制姿态最优性．     

空间机械臂非完整运动规划的最优控制

讨论空间机械臂系统的运动规划问题，利用系统的非完整特性，将空间机械臂动力学方程转化为非线性控制系统的状态方程，给出一种最优控制的数值自满。通过仿真计算，表明该算法的有效性。     

A multi-objective optimal PID control for a nonlinear system with time delay

It is generally difficult to design feedback controls of nonlinear systems with time delay to meet time domain specifications such as rise time, overshoot, and tracking error. Furthermore, these time domain specifications tend to be conflicting to each other to make the control design even more challenging. This paper presents a cell mapping method for multi-objective optimal feedback control design in time domain for a nonlinear Duffing system with time delay. We first review the multi-objective optimization problem and its formulation for control design. We then introduce the cell mapping method and a hybrid algorithm for global optimal solutions. Numerical simulations of the PID control are presented to show the features of the multi-objective optimal design.     

Optimal control of parabolic variational inequalities with state constraint

1　IntroductionandProblemWeshallstudytheoptimalcontrolproblemsgovernedbynonlinearparabolicvariationalinequalitiesoftheformy′+Ay +β(y) ∈Bu+f(a.e .(x,t)∈Q =Ω× [0 ,t]) ,y(0 ) =y0 , ( )withthestateconstraintF(y) S ,andthecostfunctionalI(y,u) .Whereβisadiscontinuous,nonlinearandnonmonotonemulti_valuedmapping .Theoptimalcontrolproblemsofthedifferentialsystemshavebeenstudiedforalongtime.Manyscholars,suchasJ.L .Lions ,V .Barbu ,D .Tiba,andF .Mignotetal.,haveresearchedtheoptimalcontrolpr…