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.     

非线性动力学系统全局分析之外的胞映射方法新发展

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井行计算的帮助下,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.     

时滞耦合质量摆动力吸振器减振系统的等峰优化理论与实验

摆式调谐质量阻尼器因其便于安装、维修、更换,且经济实用,广泛应用于结构减振.它通过将摆的自振频率调谐到接近主系统的控制频率,使摆产生与主系统相反的振动,从而抑制或消除主系统的振动.本文通过对主系统无阻尼的被动减振系统和主系统有阻尼的时滞反馈主动减振系统进行多目标优化设计,实现了对主系统幅频响应曲线的等峰控制和共振峰与反共振峰差值的有效控制.首先,建立了时滞耦合质量摆动力吸振器减振系统的力学模型和振动微分方程,通过对主系统无阻尼的被动减振系统进行等峰优化,获得了减振系统的最优频率比和质量摆的最优阻尼比.对于主系统存在阻尼的被动减振系统,在该优化参数下主系统的幅频响应曲线等峰优化失效.其次,对于主系统存在阻尼的时滞反馈优化控制系统,采用CTCR方法得到了反馈增益系数和时滞的稳定区域.在保证系统稳定的前提下,通过调节反馈增益系数和时滞量两个控制参数能够实现对主系统幅频响应曲线的等峰控制.再次,对共振点处主系统振幅放大因子时滞敏感度和反馈增益系数敏感度进行分析,表明共振点幅值对反馈增益系数比对时滞更为敏感.最后,通过实验分别在频域和时域内对理论结果进行了验证.研究表明,通过采用时滞反馈对摆式调...     

线性时滞系统的离散最优控制

介绍了对线性时滞系统进行最优控制的设计，将具有时滞控制的线性系统离散后引入增广状态向量。获得不显含时滞的差分方程，根据时滞量的两种分类情况采用连续和离散形式的性能指标函数导出了最优控制律。控制律包含当前状态和此前若干步状态向量的叠加，最优控制律直接从时滞方程中得到，可保证系统的稳定性，此方法亦适用于大时滞的情况。数值算例验证了控制策略的有效性。     

Solution of Fixed Final State Optimal Control Problems via Simple Cell Mapping

A strategy is proposed to solve the fixed final state optimalcontrol problem using the simple cell mapping method. A non-uniform timestep simple cell mapping is developed to create a general database fromwhich solutions of various optimal control problems can be obtained. Atwo-stage backward search algorithm is proposed to eliminate degeneratedpaths often associated with the simple cell mapping. The proposed methodcan accurately delineate the switching curves and eliminate false limitcycles in the solution. The method is applied to two optimal controlproblems with bang-bang control. The well-known minimum time controlproblem of moving a point mass from any initial condition to the originof the phase plane is studied first. This example has exact solutionsavailable which provide a yardstick to examine the accuracy of themethod. The cell size dependence of the solution accuracy is studiednumerically. The second example is a variable stiffness feedback controlproblem with tuning range saturation. The strategy proposed is able toprovide the switching curves in the phase plane. This result has notbeen obtained before.     

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.     

Control strategy of optimal deployment for spacecraft solar array system with initial state uncertainty

A control strategy combining feedforward control and feedback control is presented for the optimal deployment of a spacecraft solar array system with the initial state uncertainty. A dynamic equation of the spacecraft solar array system is established under the assumption that the initial linear momentum and angular momentum of the system are zero. In the design of feedforward control, the dissipation energy of each revolute joint is selected as the performance index of the system. A Legendre pseudospectral method (LPM) is used to transform the optimal control problem into a nonlinear programming problem. Then, a sequential quadratic programming algorithm is used to solve the nonlinear programming problem and offline generate the optimal reference trajectory of the system. In the design of feedback control, the dynamic equation is linearized along the reference trajectory in the presence of initial state errors. A trajectory tracking problem is converted to a two-point boundary value problem based on Pontryagin’s minimum principle. The LPM is used to discretize the two-point boundary value problem and transform it into a set of linear algebraic equations which can be easily calculated. Then, the closed-loop state feedback control law is designed based on the resulting optimal feedback control and achieves good performance in real time. Numerical simulations demonstrate the feasibility and effectiveness of the proposed control strategy.     

柔性机械臂的时滞最优跟踪控制

采用最优跟踪控制方法对柔性机械臂线性化主动控制中的时滞问题进行研究。首先采用一种积分变换,将包含时滞项的动力学方程转换成形式上不包含时滞项的标准形式,然后根据最优跟踪控制理论设计控制律。在每一步控制律的计算中,不但包含有当前步的状态反馈,而且包含有前若干步控制的线性组合。文中还给出了一个模态坐标的提取方法。仿真结果显示,若对时滞不处理,控制系统会在很小时滞量时出现发散;当控制系统中的时滞量不大时,本文的线性化时滞控制设计能够取得较好的控制效果;当系统存在较大时滞量时,线性化时滞控制设计的有效范围是有限的。     

OPTIMAL CONTROL OF NONLINEAR VIBRATION RESONANCES OF SINGLE-WALLED NANOTUBE BEAMS

An optimal time-delay feedback control method is provided to mitigate the primary resonance of a single-walled carbon nanotube （SWCNT） subjected to a Lorentz force excited by a longitudinal magnetic field. The nonlinear governing equations of motion for the SWCNT under longitudinal magnetic field are derived and the modulation equations are obtained by using the method of multiple scales. The regions of the stable feedback gain are worked out by using the stability conditions of eigenvalue equation. Taking the attenuation ratio as the objective function and the stable vibration regions as constrained conditions, the optimal control parameters are worked out by using minimum optimal method. The optimal controllers are designed to control the dynamic behaviors of tile nonlinear vibration systems. It is found that the optimal feedback gain obtained by the optimal method can enhance the control performance of the primary resonance of SWCNT devices.     

Active control for a flexible beam with nonlinear hysteresis and time delay

A new active control method is presented to attenuate vibrations of a flexible beam with nonlinear hysteresis and time delay. The nonlinear and hysteretic behavior of the system is illustrated by the Bouc—Wen model. By specific transformation and augmentation of state parameters, we can convert the motion equation of the system with explicit time delay to the standard state space representation without any explicit time delay. Then the instantaneous optimal control method and Runge—Kutta method in fourth-order are applied to the controller design with time delay. Finally, in order to verify the effectivity of the time-delay controller proposed, numerical simulations are implemented. It is indicated by the simulation results that the control performance will deteriorate if neglect the time delay in process of the controller design and proposed time delay controller works well with both small and large time delay problems.     

考虑时滞的主动悬架系统控制策略对比研究全文替换

本文以二自由度四分之一汽车悬架系统为研究对象,采用不同控制策略对考虑时滞的悬架系统控制特性进行研究,并对控制效果进行对比分析．首先,采用第二类拉氏方程建立考虑时滞的二自由度悬架控制系统的动力学模型;然后分别基于状态变换法和H_∞控制理论,设计系统的时滞反馈控制律．其中状态变换法主要通过系统状态变量的转换,将系统时滞控制方程转换成不显含时滞的动力学方程,然后采用传统的二次型最优控制方法对系统进行控制．H_∞控制主要通过使用Lyapunov-Krasovskii泛函和自由权矩阵法,设计考虑时滞的H_∞控制律;最后在Matlab/Simulink平台上对控制系统进行仿真分析,并在相同时滞下对两种控制结果进行对比．研究表明,两种控制策略在考虑时滞的情况下均可保证系统的稳定性,且H_∞控制相比于状态变换法有更好的控制效果．     

EXACT LINEARIZATION BASED MULTIPLE-SUBSPACE ITERATIVE RESOLUTION TO AFFINE NONLINEAR CONTROL SYSTEM

To the optimal control problem of affine nonlinear system, based on differential geometry theory, feedback precise linearization was used. Then starting from the simulative relationship between computational structural mechanics and optimal control, multiple-substructure method was inducted to solve the optimal control problem which was linearized. And finally the solution to the original nonlinear system was found. Compared with the classical linearizational method of Taylor expansion, this one diminishes the abuse of error expansion with the enlargement of used region.     

滞变智能隔震结构的序列最优控制算法

将地震波简化为一系列脉冲,在每个时间步长上重新构造控制目标函数,建立了一种新的时域内的结构振动最优控制算法,并在滞变模型的智能隔震结构中用状态转移法加以实现.文中采用Bouc-Wen模型描述结构恢复力,并利用等效线性化方法处理非线性运动方程,分别导出了状态反馈和输出加权的两种表达式.文末选用作者承担设计过的实际隔震建筑中的一个单体工程作为算例,比较了输入三种不同地震波时各种算法的控制效果.结果表明,在相同控制能量下,本文算法对滞变结构能有效地削减响应峰值,综合性能优于现有的两种时域内的结构最优控制算法.     

Delayed full-state feedback control of airfoil flutter using sliding mode control method

This paper studies the delayed feedback control of flutter of a two-dimensional airfoil using a sliding mode control (SMC) method. The dynamic equation of airfoil flutter is firstly established using the Lagrange method, in which the cubic hardening spring nonlinearity of pitch stiffness is considered. Then, the state equation with time delay is transformed into a standard state equation with implicit time delay by a special integral transformation. Next a nonlinear time-delay controller is designed using the SMC method. Finally the effectiveness of the proposed controller is verified through numerical simulations. Simulation results indicate that time delay in the control system has significant influence on the control performance. Control failure may happen if time delay is not considered in control design. The time-delay controller proposed is effective in suppressing the airfoil flutter with either small or large control time delay.     

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。     

风浪联合发电系统水动力学研究进展

随着化石能源枯竭和全球变暖等环境问题的日益严重,海洋可再生能源(海上风能、波浪能和潮流能)成为研究热点. 为了有效开发海洋可再生能源,降低成本,多种能源综合开发成为现阶段的趋势. 海上风能与波浪能结合具有广阔的应用前景,联合发电系统不断创新. 水动力性能是联合发电系统与波浪相互作用的重要基础. 本文简要介绍多种应用在联合发电系统上的水动力学数值模拟方法,包括线性频域、线性时域、势流非线性方法标识码基于 Navier-Stokes 方程的黏性方法,对现有文献的水动力学数值模拟方法进行综述,从计算效率和精度标识码析其优缺点,且进一步阐述水动力控制优化的技术原理与实验技术主要科研难点,为联合发电系统的水动力设计提供依据. 得到以下主要结论:从计算效率上看,线性频域方法最优,其次为线性时域、势流非线性、黏性方法,从计算精标识码,与前者恰好相反;综合考虑计算效率和精度,采用考虑黏性修正的势流方法来研究是一个切实可行的方案;模标识码方法和优化控制技术目前还不够成熟,尚处于探索阶段.      

哈密顿系统正则变换在时变最优控制中的应用

利用哈密顿系统正则变换和生成函数理论求解线性时变最优控制问题,构造了新的最优控制律形式并提出了控制增益计算的保结构算法. 利用生成函数求解最优控制导出的哈密顿系统两端边值问题,并构造线性时变系统的最优控制律,由第2类生成函数所构造的最优控制律避免了末端时刻出现无穷大反馈增益. 控制系统设计中需求解生成函数满足的时变矩阵微分方程组. 根据生成函数与哈密顿系统状态转移矩阵之间的关系,从正则变换的辛矩阵描述出发,导出了求解这组微分方程组的保结构递推算法.为了保持递推计算中的辛矩阵结构,哈密顿系统状态转移矩阵的计算中利用了Magnus级数.      

ACTIVE CONTROL OF A FLEXIBLE CANTILEVER PLATE WITH MULTIPLE TIME DELAYS

Active control of a flexible cantilever plate with multiple time delays is investigated using the discrete optimal control method. A controller with multiple time delays is presented. In this controller, time delay effect is incorporated in the mathematical model of the dynamic system throughout the control design and no approximations and assumptions are made in the controller derivation, so the system stability is easily guaranteed. Furthermore, this controller is available for both small time delays and large time delays. The feasibility and efficiency of the proposed controller are verified through numerical simulations in the end of this paper.     

控制存在延时的建筑结构地震作用下的最优控制方法

直接从时滞微分方程进行控制律设计,对控制存在延时的建筑结构在地震作用下的最优控制方法进行了研究。在控制时滞量为采样周期的整数倍和非整数倍的两种情况下,通过采用零阶保持器,将包含时滞的连续系统转化为形式上不包含时滞的标准离散线性系统,然后进行控制律的设计。所得出的控制律表达式中,除了含有当前的状态反馈外,还包含有前若干步控制项的线形组合。最后对某三自由度结构模型进行了仿真计算,结果表明,延时对控制效果有较大的影响,延时并非愈短愈好。     

Time-delayed stochastic optimal control of strongly non-linear systems with actuator saturation by using stochastic maximum principle

A time-delayed stochastic optimal bounded control strategy for strongly non-linear systems under wide-band random excitations with actuator saturation is proposed based on the stochastic averaging method and the stochastic maximum principle. First, the partially averaged Itô equation for the system amplitude is derived by using the stochastic averaging method for strongly non-linear systems. The time-delayed feedback control force is approximated by a control force without time delay based on the periodically random behavior of the displacement and velocity of the system. The partially averaged Itô equation for the system energy is derived from that for the system amplitude by using Itô formula and the relation between system amplitude and system energy. Then, the adjoint equation and maximum condition of the partially averaged control problem are derived based on the stochastic maximum principle. The saturated optimal control force is determined from maximum condition and solving the forward–backward stochastic differential equations (FBSDEs). For infinite time-interval ergodic control, the adjoint variable is stationary process and the FBSDE is reduced to a ordinary differential equation. Finally, the stationary probability density of the Hamiltonian and other response statistics of optimally controlled system are obtained from solving the Fokker–Plank–Kolmogorov (FPK) equation associated with the fully averaged Itô equation of the controlled system. For comparison, the optimal control forces obtained from the time-delayed bang–bang control and the control without considering time delay are also presented. An example is worked out to illustrate the proposed procedure and its advantages.