High order optimal control of space trajectories with uncertain boundary conditions

A high order optimal control strategy is proposed in this work, based on the use of differential algebraic techniques. In the frame of orbital mechanics, differential algebra allows to represent, by high order Taylor polynomials, the dependency of the spacecraft state on initial conditions and environmental parameters. The resulting polynomials can be manipulated to obtain the high order expansion of the solution of two-point boundary value problems. Since the optimal control problem can be reduced to a two-point boundary value problem, differential algebra is used to compute the high order expansion of the solution of the optimal control problem about a reference trajectory. Whenever perturbations in the nominal conditions occur, new optimal control laws for perturbed initial and final states are obtained by the mere evaluation of polynomials. The performances of the method are assessed on lunar landing, rendezvous maneuvers, and a low-thrust Earth–Mars transfer.     

有限推力交会的最省燃料轨迹

给出了航天器有限推力交会的最省燃料轨迹。首先应用三角变换技术将推力约束 转化为没有任何约束的虚拟控制,进而利用直接优化方法,应用参数化控制方法以及强化技 术将控制向量表示为分段常值函数,将上述最优控制问题转化为非线性规划问题。应用经典 的参数优化方法即可求得最优控制律的一个近似解,通过增加参数个数,重复优化得到逼近 连续最优解的参数化解。仿真结果表明提出的控制方案是行之有效的。
     

Solution of nonlinear optimal control problems by the interpolating scaling functions

This paper presents a numerical method for solving nonlinear optimal control problems including state and control inequality constraints. The method is based upon interpolating scaling functions. The differential and integral expressions which arise in the system dynamics, the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

多饱和约束拦截弹参数时变系统控制器设计

针对拦截弹参数时变和执行机构饱和非线性影响控制系统性能,提出基于齐次多项式参数相关 Lyapunov 理论和非线性矩阵微分方程凸多面体转换算法相结合的鲁棒最优控制系统设计方案。 通过引入标准饱和约束算子,建立控制系统数学模型并给出控制系统结构;引入饱和归一化因子,把非线性参数时变微分方程转化成线性多胞型微分方程;基于齐次多项式参数相关Lyapunov理论和拓展Pólya’s定理,把参数时变多胞型微分方程转换成齐次多项式参数相关微分方程,并给出齐次多项式参数相关控制器结构;利用迭代算法把非线性矩阵不等式转换成线性矩阵不等式(LMIs),进而基于LMI凸优化理论获得控制器解,设计出的控制器是关于齐次多项式阶次g和Pólya’s松弛度d的函数;通过定点和制导控制系统仿真校验了该算法,仿真结果表明随着齐次多项式阶次g和Pólya’s松弛度d的增加,设计出的控制器在确保系统稳定的前提下改善控制系统性能。     

Optimization of Multi-Orbit Transfers between Noncoplanar Elliptic Orbits

Using the maximum principle formalism, the problem of optimizing interorbital transfer between two noncoplanar elliptic orbits is reduced to solution of a boundary value problem for a system of ordinary differential equations. In order to solve the resulting boundary value problem numerically, the numerical homotopic method or modified Newton's method is used. When solving the boundary value problem, the right-hand sides of differential equations of motion are averaged numerically. Efficient software is developed, and a large number of optimal trajectories are calculated using it. As a result of analysis of these numerical data, new high-quality results are obtained. Specifically, a bifurcation of optimal solutions is found, the existence of critical inclination is demonstrated, and a partial classification of the structure of optimal control is performed.     

基于Riccati方程解的再入飞行器制导律设计

提出一种再入飞行器纵向制导律设计方法。首先对纵向运动方程沿着实际轨道线性化,然后利用线性最优调节器原理设计制导律。在每个制导周期内求解代数R iccati方程,利用其正定解构造反馈控制律,与标准轨道的控制量叠加后形成全量控制,用于实际再入轨道的制导。仿真结果表明,所设计的制导律对再入初始偏差具有较强的鲁棒性,同时它也能较好地补偿由于气动参数和大气密度摄动造成的航程误差,从而保证落点精度。     

Some issues of time-optimal control over a spacecraft programmed turn

The problem of optimal turn of a spacecraft from an arbitrary initial position to a final specified angular position in a minimum time is considered and solved. A case is investigated, when the constraint on spacecraft’s angular momentum during the turn is essential. Based on the quaternion method a solution to the posed problem has been found, and an optimal control program is constructed taking the constraints on controlling moment into account. The optimal control is found in the class of regular motions. A condition (calculation expression) is presented for determining the moment to begin braking with the use of measurements of current motion parameters, which considerably improves the accuracy of putting the spacecraft into a preset position. For a dynamically symmetrical spacecraft the solution to the problem of optimal control by the spacecraft spatial turn is presented in analytical form (expressions in elementary functions). An example of mathematical modeling of the spacecraft motion dynamics under optimal control over reorientation is given.     

Optimal control of reorientation of a spacecraft using free trajectory method

The problem of optimal control over spatial reorientation of a spacecraft is considered. The functional having a sense of propellant consumption is minimized. The analytical solution to the formulated problem is presented. It is shown that the optimal solution can be found in the class of two-impulse control at which the spacecraft’s turn is performed along a free motion trajectory. In order to improve the accuracy of spacecraft guidance into a specified angular position, methods of control are suggested that realize the method of free trajectories. The synthesized controls are invariant with respect to both external perturbations and parametric errors. The results of mathematical modeling are presented that demonstrate high efficiency of developed control algorithms. Propellant consumption for realizing a programmed turn is numerically estimated taking into account considerable gravitational and aerodynamic moments acting upon the spacecraft.     

再入轨迹多约束模型预测静态凸规划方法

针对大升阻比可重复使用飞行器再入滑翔轨迹规划问题,提出了一种基于自适应分段Chebychev伪谱离散的多约束模型预测静态凸规划算法(P-CMPCP),实现了全量终端状态约束及多过程约束作用下再入轨迹的高精度迭代解算。考虑倾侧角翻转导致控制量不连续,传统规划方法难以同时严格满足终端位置、角度约束,且过程约束易超限,采用自适应分段伪谱离散策略,构建全程飞行状态及过程性能指标约束对各配点处控制调整量的敏感度关系,从而将非线性最优控制问题转化为静态凸规划问题,并利用内点法对各分段控制量进行了协调优化。本文所提方法无需对动力学模型进行简化或近似处理,即能以少量积分运算使轨迹满足再入全量模型约束,且控制量平滑,数值精度优于传统规划方法。     

空间飞行器动力学模型的动态反馈线性化

本文研究了空间飞行器非线性动力学模型的线性化问题，利用微分几何方法和反馈线性化理论，本文证明了推力方向相对体系不变，即受有“常方向推力”作用限制的空间飞行器的轨道－姿态耦合非线性动力学模型可用二阶动态补偿器／增广状态反馈／增广状态微分同胚变换实现精确线性化，提出了飞行器动态反馈线性化的分步算法，并给出了实现线性化的状态反馈／微分同胚的具体形式，从而将飞行器控制系统设计问题转化为线性系统问题。     

Optimal satellite formation reconfiguration actuated by inter-satellite electromagnetic forces

The inter-satellite electromagnetic forces generated by the magnetic dipoles on neighboring satellites provide an attractive control actuation alternative for satellite formation flight due to the prominent advantages of no propellant consumption or plume contamination. However, the internal force nature as well as the inherent high nonlinearity and coupling of electromagnetic forces bring unique dynamic characteristics and challenges. This paper investigates the nonlinear translational dynamics, trajectory planning and control of formation reconfiguration actuated by inter-satellite electromagnetic forces. The nonlinear translational dynamic model is derived by utilizing analytical mechanics theory; and analysis on the dynamic characteristics is put forward. Optimal reconfiguration trajectories of electromagnetic force actuated formation are studied by applying optimal control theory and the Gauss pseudospectral method. Considering the high nonlinearity and uncertainty in the dynamic model, an inner-and-outer loop combined control strategy based on feedback linearization theory and adaptive terminal sliding mode control is proposed with finite-time convergence capability and good robust performance. Theoretical analysis and numerical simulation results are presented to validate the feasibility of the proposed translational model, reconfiguration trajectory optimization approach and control strategy.     

稀疏拟谱最优控制法求解Goddard火箭问题

提出了一种新的基于直接转化法的求解基于常微分方程(ODE)和微分代数方程(DAE)的最优控制问题的数值方法.该方法通过Legendre-Gauss拟谱法同时离散化状态变量和控制变量,把最优控制问题转化为一个非线性规划问题,并利用改进的多相处理方法避免优化无控段,同时基于稀疏矩阵探索其一阶导数信息.数值结果表明,与传统的直接转换法相比,该方法是一种通用高效的精度较高的ODE/DAE最优控制直接数值求解法.最后,从工程观点出发,应用该方法成功求解了终端自由有路径约束的奇异最优控制问题Goddard火箭问题.     

失效航天器姿态接管的SDRE微分博弈控制

针对燃料耗尽的失效航天器姿态接管控制问题,提出多颗微卫星协同实现姿态稳定的状态相关黎卡提方程(SDRE)微分博弈控制方法。首先,将姿态接管问题转化为多颗微卫星的微分博弈问题,基于组合航天器的姿态模型和微卫星的性能指标函数建立多颗微卫星的非线性微分博弈模型,微卫星通过独立优化各自的性能指标函数得到控制策略。其次,引入状态相关系数矩阵,将非线性博弈转化为状态相关线性二次型博弈,采用SDRE方法更方便地逼近微卫星的博弈均衡策略。最终通过李雅普诺夫迭代法求解耦合状态相关黎卡提方程组得到微卫星的状态反馈控制器,实现微卫星的自主决策。数值仿真验证了多颗微卫星采用微分博弈控制方法实现姿态接管的有效性和容错性。     

基于伪谱法的固定采样实时最优制导方法研究

针对具有过程约束和终端状态约束的高超声速飞行器再入制导问题,给出了一种固定采样非线性实时最优制导算法,该算法通过连续在线计算开环最优控制的方式提供闭环反馈,避免了内环跟踪控制器的设计过程。利用通用伪谱优化软件包实现多约束非线性系统最优控制问题的在线求解。在考虑计算误差、预报误差、模型参数不确定性和干扰的情况下,对采用该算法构成的闭环控制系统的有界稳定性进行了理论分析与证明。仿真结果表明,该实时最优制导算法能有效地抑制飞行过程中不确定性和扰动的影响。     

Rotational and translational integrated control for Inner-formation Gravity Measurement Satellite System

Inner-Formation Gravity Measurement Satellite System (IFGMSS) is used to explore the Earth gravity using two satellites in an inner-formation flying mode. To fulfill the mission, relative position of the two satellites is required to be zeroed and attitude of the outer satellite should be stabilized in real-time. This paper proposed an integrated control scheme for the IFGMSS, and the main idea is to use only thrusters to control the relative position and attitude. The integrated control loop contains a control law's module and a control allocation law's module. The control law based on the feedback linearization method makes nonlinear dynamics counteracted and uses PD control law to reformulate the dynamics into a linear form. The integrated control allocation law is designed to assign the commanded control force and moment to each thruster dynamically. We transfer the control allocation problem into a linear programming (LP) problem and use the Optimal Theory to calculate the corresponding thrust of each thruster. Finally, an IFGMSS mission is simulated, where the two satellites fly in a circle orbit with a 300 km's altitude. Results using the integrated control scheme and the traditional separated control scheme are compared and analyzed. It has been found that the integrated control scheme is superior to the separated control scheme in output ability, level of redundancy and fuel cost.     

非凸二次约束下航天器姿态机动路径迭代规划方法

针对复杂约束下航天器姿态机动路径规划问题,首先描述和分析了航天器姿态机动过程中面临的动力学和运动学约束、有界约束、姿态指向约束,把姿态指向约束利用非凸二次型进行表述;其次从能量最优角度出发,将该约束机动问题归纳为非凸二次约束二次规划问题;然后引入线性松弛技术,将该问题转化成双线性规划问题,求出其中一个变量的凸包络和凹包络,降低求解复杂度,从而求出原问题的一个线性松弛。同时为了提高求解精度,提出一种基于评价函数的迭代规划算法,利用线性松弛求出的解作为初值,通过评价函数进行迭代规划,最终求出原问题的最优解。仿真结果表明该方法不仅可以满足复杂的姿态约束,得到全局姿态优化路径,而且能够降低能量消耗。     

基于序列二次规划算法的再入轨迹优化研究

介绍了序列二次规划算法在飞行器再入轨迹优化问题中的应用.首先引入了能量替代变量对无量纲运动方程进行推导,使得运动方程和优化问题易于处理,考虑严格的过程约束和终端约束,以攻角和倾侧角为控制变量,总加热量最小为性能指标;然后通过直接配点法将最优控制问题转化为非线性规划问题,选取各节点的状态量和控制量作为优化参数;最后应用序列二次规划算法对非线性规划问题进行求解.针对多约束的再入飞行器的轨迹优化时对初值敏感的问题,提出一种参考轨迹快速规划算法,提高了优化速度.仿真结果表明提出的方法能够较快地搜索到最优轨迹,满足所有约束且落点精度高.     

基于序列凸优化的高超声速滑翔式再入轨迹快速优化

针对具有热流、动压、过载以及多个禁飞区约束的再入轨迹优化问题,提出采用序列凸优化方法快速求解。利用归一化时间作为自变量解决终端时间自由问题,并引入辅助控制变量以减少序列优化结果中的高频振荡,在此基础上,通过线性化、离散化和非凸约束的凸化处理,将非凸非线性优化问题转化为二阶锥规划(Second Order Conic Programming,SOCP)问题,然后采用凸优化求解算法快速求解。数值优化结果与对比验证表明该方法能快速高效求解多约束条件下的再入轨迹优化问题,且计算效率和性能均优于传统的非线性规划方法。     

基于Reddy简化高阶剪切理论的复合材料对称角铺设矩形板横向弯曲一般解析解

采用复数级数法求解基于Reddy简化高阶剪切理论的复合材料对称角铺设矩形板横向弯曲问题。将待定位移函数展开为复数级数,代入该弯曲问题控制偏微分方程组,确定特征根和挠度待定常数与其他位移函数待定常数之间关系式。首次给出了该弯曲问题实数形式的一般解析解。将该一般解析解代入矩形板弯曲边界条件和角点条件,根据正弦级数的正交性建立关于挠度函数待定常数的线性代数方程组,求解此线性代数方程组可确定挠度函数待定常数。建立了该问题解析求解模式。将Reddy高阶剪切理论解析解与经典理论、一阶剪切理论解析解进行对比计算,验证了一般解析解,并给出数值算例。