Efficient Sparse Approach for Solving Receding-Horizon Control Problems
Efficient Sparse Approach for Solving Receding-Horizon Control Problems
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DOI：
10.2514/1.60090
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发表时间：
2013-10
影响因子：
2.6
通讯作者：
Haijun Peng;Q. Gao;Zhigang Wu;W. Zhong
中科院分区：
文献类型：
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作者：
Haijun Peng;Q. Gao;Zhigang Wu;W. Zhong
REECEDING-HORIZONcontrol已成功应用于化工[1]、机械系统[2]、制导系统[3]等领域。滚动时域控制具有一个吸引人的特点，它具有一个移动的初始时间和移动的终端时间的性能指标，性能指标的时间间隔是有限的。由于性能指标的时间间隔是有限的，即使对于开环不稳定系统，也可以确定最优反馈律[4]。在航空航天工程中，滚动时域控制因其在X-33的精确进入制导[3]、卫星姿态稳定[5]以及绳系卫星的回收和部署[6]中的应用而引起了相当大的关注。与此同时，人们提出了许多求解滚动时域控制问题的数值方法。基于对积分的离散梯形近似和对导数的欧拉型近似，Lu [7]将滚动时域控制问题转化为二次规划问题，然后推导出解析控制律。Yan等人[8]和威廉姆斯[6]基于间接Legendre和Jacobi伪谱方法，使用一组线性方程组解决了滚动时域控制问题。用各种差分方法直接离散状态变量和控制变量，一般都属于直接法。用这种方法得到的二次规划问题也可基本上用线性方程组求解。间接Legendre [8]和Jacobi [6]伪谱方法将状态变量和协态变量展开为多项式，以不同离散点的状态变量和协态变量的值为展开系数，从而将Hamilton正则方程化为代数方程组。因此，求解线性方程组的效率和精度是滚动时域控制问题在线实现的关键。本说明中出现的研究方法的灵感来自于需要高性能的数值方法来解决recedhorizon控制问题。使用这种方法的动机是以下观察。对于大的状态空间模型和大的未知变量离散化，由上述方法得到的线性方程大多是具有非对称系数矩阵的大而密集的线性方程，因此，计算机内存存储和在线执行效率必然受到显著影响。本文提出了一种在线求解滚动时域控制问题的有效稀疏数值方法。利用变分原理和生成函数[9，10]，将后退控制问题转化为一组稀疏的对称正定线性方程组。最后，将该方法应用于航天器交会问题，与其他方法进行比较，验证了该方法的计算效率和准确性。
R ECEDING-HORIZONcontrol has been applied successfully in such fields as the chemical industry [1],mechanical systems [2], and guidance systems [3]. Receding-horizon control has an attractive feature in that it has a performance index of a moving initial time and a moving terminal time, and the time interval of the performance index is finite. Because the time interval of the performance index is finite, the optimal feedback law can be determined even for a system that is an open-loop unstable system [4]. In aerospace engineering, receding-horizon control has attracted considerable attention for its application in precision entry guidance for the X-33 [3], satellite attitude stabilization [5], and retrieval and deployment of tethered satellites [6]. Meanwhile, many numerical methods for solving receding-horizon control problems have been proposed. Based on Simpson-trapezoid approximations for the integral and Euler-type approximations for the derivatives, Lu [7] transformed the receding-horizon control problem into a quadratic programming problem and later derived the analytical control laws. Based on the indirect Legendre and Jacobi pseudospectral methods, Yan et al. [8] and Williams [6] solved the receding-horizon control problem using a set of linear equations. In general, the direct discretization of state and control variables by many kinds of difference methods belongs to direct method. The quadratic programming problem obtained from this method can also be solved essentially by linear equations. The indirect Legendre [8] and Jacobi [6] pseudospectral methods expanded the state and costate variables into polynomials with the values of the states and costates at the different discretization points as the expansion coefficients, and then the Hamiltonian canonical equation are reduced into a system of algebraic equations. Therefore, the efficiency and accuracy of solving linear equations are the key points in the online implementation of the receding-horizon control problem. The research method appearing in this Note is inspired by the need for high-performance numerical method for solving the recedinghorizon control problem. The use of this method is motivated by the following observations. For a large state-space model and a large discretization of unknown variables, the linear equation obtained from the aforementioned methods is mostly a large and dense linear equation with an asymmetrical coefficient matrix; thus, the computer memory storage and online implementation efficiency must be significantly influenced. In this Note, an efficient sparse numerical approach for solving the receding-horizon control problem online is proposed. With the variational principle and the generating function [9,10], the recedinghorizon control problem is transformed into a set of sparse symmetric positive definite linear equations. Finally, the proposed method is applied to a spacecraft rendezvous problem to demonstrate the computational efficiency and accuracy in comparison with other methods.