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非线性系统闭环反馈控制保辛算法研究及应用

Research on the Symplectic Conservative Approach for Solving Nonlinear Closed-Loop Feedback Control Problems and the Applications

【作者】 江新

【导师】 陈飙松; 彭海军;

【作者基本信息】 大连理工大学 , 计算力学, 2014, 硕士

【摘要】 最优控制的数值算法在最优控制领域已经处于越来越重要的地位。由于考虑的问题变得越来越复杂,很多问题无法通过存粹的理论分析解决,因必须研究高效的数值算法来进行近似计算。一般的最优控制数值算法都只关注解得精度,会忽视掉一些系统固有的属性。而辛数值算法在保证求解精度的同时,能够在对系统进行数值离散的时候尽可能的保持系统的固有属性。本文旨在针对最优控制问题的辛数值算法进行一些研究,并将其应用于一些当今热门的航天问题的求解。本文针对非线性最优控制系统提出了保辛的数值计算方法,并将其适用范围进一步扩展到了控制受限的非线性最优控制问题。另外为了能够准确的求解实际应用中一些受到外部环境干扰的非线性最优控制问题,本文设计了有效的闭环反馈控制器。而后,基于本文提出的创新算法,研究了绳系卫星系统的闭环反馈控制问题,以及Halo轨道上航天器交会的闭环反馈控制问题。本文的具体工作如下:1.提出了一种求解非线性系统闭环反馈控制问题的保辛算法。首先,通过拟线性化方法将非线性系统最优控制问题转化为线性非齐次Hamilton系统两点边值问题的迭代格式求解。然后,通过参变量变分原理与生成函数构造了保辛的数值算法,且该算法保持了原Hamilton系统的辛几何性质。最后,通过时间步的递进完成状态与控制变量的更新,进而达到闭环控制的目的。2.为了求解控制受限的非线性系统的最优控制问题,本文利用拟线性化方法与参变量变分原理将所研究问题最终转化为了与线性互补问题耦合的Hamilton两点边值问题的求解。为了研究复杂环境下的各类非线性最优控制问题,设计了一种有效的闭环反馈控制策略,通过在各个离散点对开环控制的快速计算实现实现了闭环反馈控制。3.应用非线性系统闭环反馈控制的保辛算法求解绳系卫星系统子星释放与回收过程的闭环反馈控制问题。首先通过第二类Lagrange方程推导出二体绳系卫星系统的动力学方程,然后利用求解非线性闭环反馈最优控制的保辛算法进行分析求解。数值仿真表明:相对Legendre伪谱方法,保辛算法求解绳系卫星系统的闭环反馈控制问题具有较高的计算速度与收敛速度。此外,通过数值仿真绳系卫星系统的开环控制与闭环反馈控制问题,结果表明:在绳系卫星的初始状态存在偏差的情况下,使用开环控制会导致系统在终端无法达到稳定状态,而使用闭环反馈控制则能在一段时间内抵消初始状态向量偏差对系统产生的影响,最终达到稳定状态。4.提出了一种在日-地系统的平动点轨道上连续小推力航天器交会的闭环反馈控制策略。在航天器交会的控制模型中考虑了初始状态误差、驱动器饱和极限、测量误差以及从工程角度考虑时引入的外部干扰。所提出的非线性闭环反馈控制器并不能显示解析,而是通过在每一个更新点对开环轨道的反复计算实现。为了保证计算的效率,使用了求解开环最优控制问题的保辛算法。借助于拟线性话方法,开环最优控制问题被转化为了与线性互补问题耦合的稀疏线性对称方程组的求解,计算效率得到了显著的提高。对航天器交会问题的数值模拟很好的验证了闭环反馈控制器的鲁棒性、高精度以及实时制导的特性。

【Abstract】 The numerical algorithms of optimal control has been increasingly important in the field of optimal control. As the problems are becoming more and more complex, sometimes we cannot study them by purely theoretical analysis. Thus it is necessary to design efficient numerical methods for approximate calculation. In most cases the numerical algorithms of optimal control cares only about the precision of computation, and thus ignore many intrinsic properties of the system. Symplectic conservative algorithms can keep the intrinsic properties of the system as well as meet the precision requirement. Thus this paper aims at the study of symplectic conservative numerical methods for optimal control and its applications in some hot aerospace problems. We presented a symplectic conservative numerical method for solving nonlinear optimal control problems, and extended this method into solving problems with control inequality constraints. In addition, we designed an efficient feedback control strategy, which can allow us to study some nonlinear optimal control problems when considering various disturbances from the outside environment. Then by using the novel numerical algorithms we presented, we studied the feedback control problems of the tethered satellite system and the spacecraft rendezvous problem between Halo orbits. The specific work of this paper are as follows:1. A symplectic approach was proposed to solve the nonlinear closed-loop feedback control problems in this paper. First, the optimal control problems of the nonlinear system were transformed into the iteration form of linear Hamilton system’s two-point boundary value problems. Second, a symplectic numerical approach was deduced based on dual variable principle and generating function. This method can keep the symplectic geometry structure of the Hamilton system. Last, update the state vector and control input by the forwarding of time steps and thus achieve the goal of closed-loop control.2. In order to solve the nonlinear optimal control problems with control inequality constraints, we transformed the original problems into the Hamiltonian two-point boundary value problems coupled with linear complementary problems by using the qusi-linearization method. In addition, we designed an efficient closed-loop feedback control strategy which can allow us to study the nonlinear optimal control problems under various disturbances under complex environment. The feedback control strategy was realized by the fast computation of open-loop control problems at each feedback point. 3. The symplectic conservative approach for solving nonlinear receding horizon control problems was applied on the closed-loop feedback control problems of the subsatellite’s deploy and retrieval process of tethered satellite system. First, the dynamic equations of two-body tethered satellite system were deduced based on Second Lagrange equations. Then we analysis and solve the problem using the symplectic conservative approach for solving nonlinear closed-loop feedback control problems. The numerical simulation showed that compared with the Legendre pseudospectral method, the symplectic approach has desirable computation and iteration speed when solving feedback control problems of tethered satellite system. Furthermore, the numerical simulations of the open-loop control and closed-loop feedback control problems of tethered satellite system showed that with the presence of initial errors, the open-loop control could not lead the system to a stable state, while the closed-loop feedback control can eliminate the initial errors within a certain period of time and the final state was still stable.4. A nonlinear closed-loop feedback control strategy for the spacecraft rendezvous problem with finite low-thrust between libration orbits in the Sun-Earth system was presented. The model of spacecraft rendezvous takes the perturbations in initial states, the actuator saturation limits, the measurement errors, and the external disturbance forces into consideration from an engineering point of view. The proposed nonlinear closed-loop feedback control strategy is not analytically explicit; rather, it is implemented by a rapid re-computation of the open-loop optimal control at each update instant. To guarantee the computational efficiency, a novel numerical algorithm for solving the open-loop optimal control is given. With the aid of the quasilinearization method, the open-loop optimal control problem is replaced successfully by a series of sparse symmetrical linear equations coupled with linear complementary problem, and the computational efficiency can be significantly increased. The numerical simulations of spacecraft rendezvous problems in the paper well demonstrate the robustness, high precision and dominant real-time merits of the proposed closed-loop feedback control strategy.

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