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.     

Body-Fitted Adjoint Formulation of the Euler Equations

The shape optimization problem governed by the Euler equations is posed in a fixed reference plane. The boundary control is exerted by a parametric mapping from the physical plane to the reference fixed plane. The adjoint equations are derived in such fixed plane. By using this approach remeshing is unnecessary; furthermore, as in many practical applications the parametric mapping can be easily differentiated, the computation of mesh sensitivities is avoided.     

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。     

Shallow Water Flow Analysis with Moving Boundary Technique Using Least-squares Bubble Function

This paper presents a computational simulation method for a river problem. For the actual flow problem, it is necessary to compute flow velocity, water elevation and water region at the same time. For the basic formulation, the unsteady shallow water equations are used. As the numerical approach, implicit FEM is proposed by bubble function. To control numerical stability and accuracy, LSBF (Least-Squares Bubble Function) is used to solve the finite element equations. Also, the fixed boundary technique is combined to deal with wet and dry areas in the moving finite element mesh. Some numerical tests are shown to check this method.     

LARGE RANGE ANALYSIS FOR NONLINEAR DYNAMIC SYSTEMS——ELEMENT MAPPING METHOD

This paper presents a new method for global analysis of nonlinear system. By means of transforming the nonlinear dynamic problems into point mapping forms which are single-valued and continuous, the state space can be regularly divided into a certain number of finitely small triangle elements on which the non-linear mapping can be approximately substituted by the linear mapping given by definition. Hence, the large range distributed problem of the mapping fixed points will be simplified as a process for solving a set of linear equations. Still further, the exact position of the fixed points can be found by the iterative technique. It is convenient to judge the stability of fixed points and the shrinkage zone in the state space by using the deformation matrix of linear mapping. In this paper, the attractive kernel for the stationary fixed points is defined, which makes great advantage for describing the attractive domains of the fixed points. The new method is more convenient and effective than the cell mapping methodl^[1]. And an example for two-dimensional mapping is given.     

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.     

LARGE RANGE ANALYSIS FOR NONLINEAR DYNAMIC SYSTEMS——ELEMENT MAPPING METHOD

This paper presents a new method for global analysis of nonlinear system. By means of transforming the nonlinear dynamic problems into point mapping forms which are single-valued and continuous, the state space can be regularly divided into a certain number of finitely small triangle elements on which the non-linear mapping can be approximately substituted by the linear mapping given by definition. Hence, the large range distributed problem of the mapping fixed points will be simplified as a process for solving a set of linear equations. Still further, the exact position of the fixed points can be found by the iterative technique. It is convenient to judge the stability of fixed points and the shrinkage zone in the state space by using the deformation matrix of linear mapping. In this paper, the attractive kernel for the stationary fixed points is defined, which makes great advantage for describing the attractive domains of the fixed points. The new method is more convenient and effective than the cell     

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。     

Optimal control problems with incomplete and different integral time domains in the objective and constraints

Optimal control problem with incomplete and different integral time domains is a class of very common practical engineering problems. In traditional way, the integral items are transformed to the transient items and treated as artificial states to reduce the complexity of programming. However, its main disadvantage is time wasting for the considered problems. In this paper, an efficient computational method is therefore proposed for this type of problem, where the integral time domains can be either fixed or variable. By employing the control vector parameterization and a timescaling transformation, the original problem is converted to an approximate optimal parameter selection problem. Moreover, new gradient formulae for the cost and constraint functions are derived. With these gradient formulae, standard gradient-based optimization methods can be easily applied to solve the generated approximate problem. For illustration, three classical numerical examples are tested. The research results, which save 10–22 % of time, show the effectivity of the proposed approach.     

Non-smooth predictive control for mechanical transmission systems with backlash-like hysteresis

This paper proposes a non-smooth predictive control approach for mechanical transmission systems described by dynamic models with preceded backlash-like hysteresis. In this type of system, the work platform is driven by a DC motor through a gearbox. The work platform is represented by a linear dynamic sub-model connected in series with a backlash-like hysteresis inherent in gearbox. Here, backlash-like hysteresis is modeled as a non-smooth function with multi-valued mapping. In this case, the conventional model predictive control for such system cannot be implemented directly since the gradients of the control objective function with respect to control variables do not exist at non-smooth points. In order to solve this problem, a non-smooth receding horizon strategy is proposed. Moreover, the stability of predictive control of such non-smooth dynamic systems is analyzed. Finally, a numerical example and a simulation study on a mechanical transmission system are presented for validating the proposed method.     

Pseudospectral method for optimal propellantless rendezvous using geomagnetic Lorentz force

A charged spacecraft is subject to the Lorentz force when it orbits a central body with a magnetic field. The induced Lorentz force provides a new mean of propellantless electromagnetic propulsion for orbital control. Modeling the Earth magnetic field as a tilted dipole that co-rotates with the Earth, this paper develops a nonlinear dynamical model that describes the relative motion of the Lorentz spacecraft about an arbitrary reference orbit. Based on the proposed dynamical model, feasibility of Lorentz-propelled rendezvous with no restrictions on the initial states is investigated. The rendezvous problem is then formulated as an optimal control problem, and solved with the Gauss pseudospectral method (GPM). Numerical simulations substantiate the validity of proposed model and method, and results show that the propellantless rendezvous is achieved at both fixed and free final time.     

An Edge Crack Problem in a Semi-Infinite Plane Subjected to Concentrated Forces

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The adjoining cell mapping and its recursive unraveling,part I: Description of adaptive and recursive algorithms

A new type of cell mapping, referred to as an adjoining cell mapping, is developed in this paper for autonomous dynamical systems employing the cellular state space. It is based on an adaptive time integration employed to compute an associated cell mapping for the system. This technique overcomes the problem of determining an appropriate duration of integration time for the simple cell mapping method. Employing the adjoining mapping principle, the first type of algorithm developed here is an adaptive mapping unraveling algorithm to determine equilibria and limit cycles of the dynamical system in a way similar to that of the simple cell mapping. In addition, it is capable of providing useful information regarding the behavior of dynamical systems possessing pathological dynamics and of systems with rapidly changing vector field. The adjoining property inherent in the adjoining cell mapping method, in general, permits development of new recursive algorithms for unraveling dynamics. The required computer memory for a practical implementation of such algorithms is considerably less than that required by the simple cell mapping algorithm since they allow for a recursive partitioning of state space for trajectory analysis. The second type of algorithm developed in this paper is a recursive unraveling algorithm based on adaptive integration and recursive partitioning of state space into blocks of cells with a view toward its practical implementation. It can find equilibria of the system in the same manner as the simple cell mapping method but is more efficient in locating periodic solutions.     

含J2摄动的有限时间航天器追逃博弈问题研究

航天器追逃博弈是航天器在轨捕获任务的一个重要问题,具有极高的军民两用双重价值．针对有限时间且考虑J2摄动的航天器追逃博弈问题,本文提出了一种精确的求解方法．该方法的核心思想是将有限时间的航天器追逃博弈问题建模为有限时间二人零和对策,则博弈中两航天器的最优控制策略可以转化为有限时间二人零和对策的鞍点解．在鞍点解的求解过程中,本文首先基于考虑J2摄动的非线性动力学方程,将两航天器动力学方程和始末边值条件与鞍点解必要条件结合得到两点边值问题,然后提出一种结合遗传算法和配点法的混合算法求解该两点边值问题以得到精确的鞍点解．本文利用数值仿真对所提方法的有效性进行了验证．结果表明：(i) 在航天器追逃博弈过程中,J2摄动对两航天器的最优控制策略具有较大影响;(ii) 所提方法能够精确求解出两航天器在有限时间的追逃博弈过程中的最优控制策略．     

基于模糊神经网络的惯性导航系统预热控制

针对惯性导航系统预热中存在的非线性问题,采用了基于模糊神经网络的控制系统。讨论了模糊系统与神经网络的优缺点,给出了适于非线性时滞、基于BP网络的控制方案,并通过仿真说明方案的可行性。     

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...     

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.     

惯导系统在动基座条件下对准的新途径

提出一种解决惯导系统在动基座上对准问题的新途径。试验表明：用最小二乘法作参数估计能缩短对准时间,并得到高精度的参数估计。这种方案可应用于舰船上。     

Well-Posedness of Surface Wave Equations Above a Viscoelastic Fluid

In this article, we prove the local well-posedness, for arbitrary initial data with certain regularity assumptions, of the equations of a Viscoelastic Fluid of Johnson–Segalman type in a domain with a free surface. Managing more general constitutive laws is also briefly depicted. The 2D geometry is defined by a solid fixed bottom and an upper free boundary submitted to surface tension. The proof relies on a Lagrangian formulation. First we solve two intermediate problems through a fixed point using mainly (Allain in Appl Math Optim 16:37–50, 1987) for the Navier–Stokes part. Then we solve the whole Lagrangian problem on [0, T ₀] for T ₀ small enough through a contraction mapping. Since the Lagrangian solution is regular enough and the change of coordinates invertible, we can come back to an Eulerian one.     

Bending of an Anisotropic Elliptic Plate on an Elastic Foundation

An approach is proposed to solve a stress–strain problem for anisotropic rigidly fixed plates on an elastic foundation. The problem is solved by the method of successive approximations. At each approximation, the deflection is represented as polynomials whose coefficients are determined from a system of linear algebraic equations. Study is made of the influence of the reinforcement angle and the modulus of subgrade reaction on the deflections and the bending moments in an orthotropic plate.