Efficient Sparse Approach for Solving Receding-Horizon Control Problems

Abstract

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.