Use of the energy integral in optimal control of the spacecraft spatial attitude

In this paper, we study the problem of spacecraft optimal rotation into the position with given attitude. The rotation takes a fixed time. The control program is optimized by minimizing the energy integral. We present the analytic solution of this problem and obtain formalized equations and computational expressions for constructing the optimal rotation program. We also present an example and the results of mathematical simulation of the spacecraft motion dynamics under optimal control, which demonstrate the practical realizability of our method for the spacecraft spatial attitude control.     

Optimal spacecraft terminal attitude control synthesis by the quaternion method

We consider an optimal control problem of spacecraft retargeting from an arbitrary initial position to a given terminal angular position. The objective is to minimize a path functional. Using the quaternion method, we obtain an analytic solution of this problem and present formalized equations and computational expressions for synthesizing an optimal manoeuvre program. We also present an example and the results of mathematical simulation of the spacecraft motion dynamics under an optimal control, which demonstrate the practical implementability of the control method developed in the present paper.     

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.     

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．     

论基础数学工具的创新运用

在建立叶轮机三元流面理论时,吴仲华巧妙地、创造性地提出了全导数型的沿流面方向偏导数的概念和表达式,导出了两类流面气动力学数学模型,不但数学形式简洁,而且具有明晰的物理意义,有利于分析和求解。在吴氏流面理论的启示下,作者不断致力于基础数学工具和概念的推广和创新运用,挖掘并发挥其潜在功能。本文将通过流体力学、传热学以及最优控制论方面的一些实例来综合介绍个人的心得和体会。     

Optimal solution error quantification in variational data assimilation involving imperfect models

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition. If the model is ‘perfect,’ the optimal solution (analysis) error rises because of the presence of the input data errors (background and observation errors). Then, this error is quantified by the covariance matrix, which can be approximated by the inverse Hessian of an auxiliary control problem. If the model is not perfect, the optimal solution error includes an additional component because of the presence of the model error. In this paper, we study the influence of the model error on the optimal solution error covariance, considering strong and weak constraint data assimilation approaches. For the latter, an additional equation describing the model error dynamics is involved. Numerical experiments for the 1D Burgers equation illustrate the presented theory. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical approximation of optimal control of unsteady flows using SQP and time decomposition

In this paper, we present numerical approximations of optimal control of unsteady flow problems using sequential quadratic programming method (SQP) and time domain decomposition. The SQP method is considered superior due to its fast convergence and its ability to take advantage of existing numerical techniques for fluid flow problems. It iteratively solves a sequence of linear quadratic optimal control problems converging to the solution of the non‐linear optimal control problem. The solution to the linear quadratic problem is characterized by the Karush–Kuhn–Tucker (KKT) optimality system which in the present context is a formidable system to solve. As a remedy various time domain decompositions, inexact SQP implementations and block iterative methods to solve the KKT systems are examined. Numerical results are presented showing the efficiency and feasibility of the algorithms. Copyright © 2004 John Wiley & Sons, Ltd.     

Optimal control of quadratic functionals for affine nonlinear systems

In this paper we analyze the optimal control problem for a class of affine nonlinear systems under the assumption that the associated Lie algebra is nilpotent. The Lie brackets generated by the vector fields which define the nonlinear system represent a remarkable mathematical instrument for the control of affine systems. We determine the optimal control which corresponds to the nilpotent operator of the first order. In particular, we obtain the control that minimizes the energy of the given nonlinear system. Applications of this control to bilinear systems with first order nilpotent operator are considered.     

Hysteresis and Horizontal Redistribution in Porous Media

It is well known that multiphase flow in porous media exhibits hysteretic behaviour. This is caused by different fluid–fluid behaviour if the flux reverses. For example, for flow of water in unsaturated soils the process of imbibition and drainage behaves differently. In this paper we study a new model for hysteresis that extends the current playtype hysteresis model in which the scanning curves between drainage and imbibition are vertical. In our approach the scanning curves are non-vertical and can be constructed to approximate experimentally observed scanning curves. Furthermore our approach does not require any book-keeping when the flux reverses at some point in space. Specifically, we consider the problem of horizontal redistribution to illustrate the strength of the new model. We show that all cases of redistribution can be handled, including the unconventional flow cases. For an infinite column, our analysis involves a self-similar transformation of the equations. We also present a numerical approach (L-scheme) for the partial differential equations in a finite domain to recover all redistribution cases of the infinite column provided time is not too large.     

基于遗传算法的抗震钢框架多目标优化设计

考虑抗震钢框架优化问题具有多目标的特点,在遗传算法的基础上对抗震钢框架多目 标优化设计进行了探讨. 在无约束Pareto排序遗传算法的基础上,提出了一个简单、实用 而又可以避免采用罚函数的全新排序方法,在此基础上形成了一种求解有约束多目标优化 问题的Pareto遗传算法(CMOPGA), 并给出了具体的算法流程图. 以钢框架重量最轻和结构 总动应变能最小为目标,基于相关的设计规范,给出了抗震钢框架多目标优化问题的一种合 理提法. 采用CMOPGA对一个两跨六层抗震钢框架实例进行了多目标优化设计,并提出了一 个在Pareto最优解集的基础上选取妥协解的相对最小距离妥协原则. 算例结果表明,采用 CMOPGA求解抗震钢框架多目标优化问题是可行和有效的.     

Fuzzy logic adaptive Kalman filtering in the control of irrigation canals

A fuzzy logic adaptive Kalman filtering methodology was developed for the automatic control of an irrigation canal system under unknown disturbances (water withdrawals) acting in the canal. Using a linearized finite difference model of open channel flow, the canal operation problem was formulated as an optimal control problem and an algorithm for gate opening in the presence of arbitrary external disturbances (changes in flow rates) was derived. Based on the linear optimal control theory, the linear quadratic regulator (LQR), assuming all the state variables (flow depths and flow rates) were available, was designed to generate control input (optimal gate opening). As it was expensive to measure all the state variables (flow rates and flow depths) in a canal system, a fuzzy logic adaptive Kalman filter and traditional Kalman filter were designed to estimate the values for the state variables that were not measured but were needed in the feedback loop. The performances of the state estimators designed using the fuzzy logic adaptive Kalman filter methodology and the traditional Kalman filtering technique were compared with the results obtained using the LQR (target loop function). The results of the present study indicated that the performance of the fuzzy logic adaptive Kalman filter was far superior to the performance of the observer design based upon the traditional Kalman filter approach. The obvious advantages of the fuzzy logic adaptive Kalman filter were the prevention of filter divergence and ease of implementation. As the fuzzy logic adaptive Kalman filter requires smaller number of state variables for the acceptable accuracy therefore, it would need less computational effort in the control of irrigation canals. Copyright © 2009 John Wiley & Sons, Ltd.     

Cluster-based control of a separating flow over a smoothly contoured ramp

The ability to manipulate and control fluid flows is of great importance in many scientific and engineering applications. The proposed closed-loop control framework addresses a key issue of model-based control: The actuation effect often results from slow dynamics of strongly nonlinear interactions which the flow reveals at timescales much longer than the prediction horizon of any model. Hence, we employ a probabilistic approach based on a cluster-based discretization of the Liouville equation for the evolution of the probability distribution. The proposed methodology frames high-dimensional, nonlinear dynamics into low-dimensional, probabilistic, linear dynamics which considerably simplifies the optimal control problem while preserving nonlinear actuation mechanisms. The data-driven approach builds upon a state space discretization using a clustering algorithm which groups kinematically similar flow states into a low number of clusters. The temporal evolution of the probability distribution on this set of clusters is then described by a control-dependent Markov model. This Markov model can be used as predictor for the ergodic probability distribution for a particular control law. This probability distribution approximates the long-term behavior of the original system on which basis the optimal control law is determined. We examine how the approach can be used to improve the open-loop actuation in a separating flow dominated by Kelvin–Helmholtz shedding. For this purpose, the feature space, in which the model is learned, and the admissible control inputs are tailored to strongly oscillatory flows.     

Optimal synthesis in the problem of uniaxial stabilization for a satellite in the case of phase constraints

The optimal stabilization problem for the pitching oscillations of a satellite is considered. For the reduced system, an optimal control problem with phase constraints is formulated and a controllability domain is constructed. A necessary optimality condition (the maximum principle for the optimal control problem with phase constraints) is used to determine the optimal trajectories, the nature of interaction between a trajectory and the phase constraint boundary, and the number of boundary segments. An optimal synthesis is proposed in the constructed domain.     

Optimal limited-rate slewing of a flexible spacecraft

The paper presents an efficient method of finding the optimal program control for the reorientation of a spacecraft with flexible appendages at a limited slewing rate. The appendages are incorporated into the mathematical model based on the quasistatic approximation. The problem is solved analytically by parametrizing the functional of a multipoint boundary-value problem. The optimal solution is illustrated graphically for different parts of the attainability domains     

A Model of Discontinuous, Incompressible Two-Phase Flow

Lack of hyperbolicity is a recurring problem for models of two-phase flow assuming the form of systems of balance laws. In particular, smooth solutions occur only for very special initial data, and the standard results on the local structure of discontinuous weak solutions do not apply to such nonhyperbolic systems. A simple example is inviscid, incompressible two-fluid flow with a single pressure. We suggest that such an unattractive mathematical feature may result from the mathematical derivation of the model, rather than from the underlying physical assumptions. In particular, for the case described above we present an alternative treatment which leads to a consistent model for piecewise smooth, discontinuous solutions. We obtain admissibility conditions for the anticipated discontinuities by considering the limit of vanishing viscosity with a convenient dissipation term.      

Relieved control in cyclic systems

Summary It is desirable to design a controlled cyclic system in such a way that its uncontrolled motion differs as little as possible from the optimal controlled motion. In the present paper, the problem of the synthesis of a cyclic transportation modulus control is solved along these principles. For the resulting double-channel optimal control problem an exact solution is developed. Comments are made about technical realizations of the results.     

Low-rank solution of an optimal control problem constrained by random Navier-Stokes equations

We develop a low-rank tensor decomposition algorithm for the numerical solution of a distributed optimal control problem constrained by two-dimensional time-dependent Navier-Stokes equations with a stochastic inflow. The goal of optimization is to minimize the flow vorticity. The inflow boundary condition is assumed to be an infinite-dimensional random field, which is parametrized using a finite- (but high-) dimensional Fourier expansion and discretized using the stochastic Galerkin finite element method. This leads to a prohibitively large number of degrees of freedom in the discrete solution. Moreover, the optimality conditions in a time-dependent problem require solving a coupled saddle-point system of nonlinear equations on all time steps at once. For the resulting discrete problem, we approximate the solution by the tensor-train (TT) decomposition and propose a numerically efficient algorithm to solve the optimality equations directly in the TT representation. This algorithm is based on the alternating linear scheme (ALS), but in contrast to the basic ALS method, the new algorithm exploits and preserves the block structure of the optimality equations. We prove that this structure preservation renders the proposed block ALS method well posed, in the sense that each step requires the solution of a nonsingular reduced linear system, which might not be the case for the basic ALS. Finally, we present numerical experiments based on two benchmark problems of simulation of a flow around a von Kármán vortex and a backward step, each of which has uncertain inflow. The experiments demonstrate a significant complexity reduction achieved using the TT representation and the block ALS algorithm. Specifically, we observe that the high-dimensional stochastic time-dependent problem can be solved with the asymptotic complexity of the corresponding deterministic problem.     

A non-iterative transformation method for Newton's free boundary problem

In book II of Newton's Principia Mathematica of 1687 several applicative problems are introduced and solved. There, we can find the formulation of the first calculus of variations problem that leads to the first free boundary problem of history. The general calculus of variations problem is concerned with the optimal shape design for the motion of projectiles subject to air resistance. Here, for Newton's optimal nose cone free boundary problem, we define a non-iterative initial value method which is referred in the literature as a transformation method. To define this method we apply invariance properties of Newton's free boundary problem under a scaling group of point transformations. Finally, we compare our non-iterative numerical results with those available in the literature and obtained via an iterative shooting method. We emphasize that our non-iterative method is faster than shooting or collocation methods and does not need any preliminary computation to test the target function as the iterative method or even provide any initial iterate. Moreover, applying Buckingham Pi-Theorem we get the functional relation between the unknown free boundary and the nose cone radius and height.     

HYBRID OPTIMIZATION STRATEGY FOR MOTION PLANNING OF SPACE ROBOT SYSTEM WITH PRISMATIC JOINT

研究了自由漂浮带滑移铰空间机器人非完整运动规划的最优控制问题,提出一种由高斯伪谱法求解可行解与直接打靶法求解最优解相结合的混合优化策略.首先,根据多体系统动力学理论建立空间机器人的动力学模型,给定系统的初始和目标位形,将空间机器人运动规划问题描述成博尔察（Bolza）型最优控制问题;然后,利用高斯伪谱法将最优控制问题离散为非线性规划问题,求解在较少勒让德-高斯（Legendre-Gauss,LG）点时状态变量和控制变量对应的可行解;最后,在LG点处离散控制变量,作为直接打靶法的初值,利用序列二次规划算法求解空间机器人系统的优化运动轨迹和最优控制输入.通过数值仿真,系统优化运动轨迹光滑平稳,最优控制输入也能很好地满足各种约束条件,仿真结果验证了该混合优化策略的鲁棒性和有效性.     

Distributed optimal control of a nonstandard system of phase field equations

We investigate a distributed optimal control problem for a phase field model of Cahn–Hilliard type. The model describes two-species phase segregation on an atomic lattice under the presence of diffusion; it has been introduced recently in Colli et?al. (SIAM J Appl Math), on the basis of the theory developed in Podio-Guidugli (Ric. Mat. 55:105–118, 2006), and consists of a system of two highly nonlinearly coupled PDEs. For this reason, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality.