Haijun Peng

Symplectic Approaches for Solving Two-Point Boundary-Value Problems

T HE two-point boundary-value problem (TPBVP) plays a fundamental role in optimal control problems of aerospace engineering, including the problem of spacecraft orbit transfer [1], the optimal reconfiguration of spacecraft formations [2], and continuous thrust rendezvous problems [3]. Therefore, many techniques and methods for solving TPBVP have been proposed and developed [4–7]. The optimal control problem is as follows. The dynamic system is

H 2-norm computation of linear time-varying periodic systems via the periodic Lyapunov differential equation

A new reliable algorithm for computing -norm of linear time-varying periodic (LTP) systems in continuous-time domain is proposed in this article. The solving of -norm is firstly transformed into the solving of the periodic Lyapunov differential equation (PLDE). Then, the key point in this article is that a new method based on the interval mixed energy matrix and the interval transition matrix computed by the structure-preserving Magnus series method is proposed for solving the PLDE. With the solutions of PLDE, the -norm of LTP systems is evaluated by a simple first-order ordinary differential equation. Finally, the effectiveness and the high accuracy of the proposed algorithms are demonstrated by two numerical examples.

H∞ norm computation of linear continuous-time periodic systems by a structure-preserving algorithm

A new reliable structure-preserving algorithm for computing H∞ norm of linear continuous-time periodic systems is proposed in this paper. In the computation of the H∞ norm, no Riccati differential equations are needed to solve and only eigenvalues of a monodromy matrix of the associated periodic Hamiltonian system will be evaluated. First, the monodromy matrix is expressed as the product of state transition matrices of the Hamiltonian system. Second, these state transition matrices, which have been proved to be symplectic matrices, are evaluated by a structure-preserving Magnus series method. Then, in order to preserve the standard symplectic form of the monodromy matrix, the structure-preserving matrices obtained by state transition matrices are employed to compute the monodromy matrix. At last, the effectiveness and the high accuracy of the proposed structure-preserving algorithm are demonstrated by numerical examples.

Efficient Sparse Approach for Solving Receding-Horizon Control Problems

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.

Fourier expansion based recursive algorithms for periodic Riccati and Lyapunov matrix differential equations

Combining Fourier series expansion with recursive matrix formulas, new reliable algorithms to compute the periodic, non-negative, definite stabilizing solutions of the periodic Riccati and Lyapunov matrix differential equations are proposed in this paper. First, periodic coefficients are expanded in terms of Fourier series to solve the time-varying periodic Riccati differential equation, and the state transition matrix of the associated Hamiltonian system is evaluated precisely with sine and cosine series. By introducing the Riccati transformation method, recursive matrix formulas are derived to solve the periodic Riccati differential equation, which is composed of four blocks of the state transition matrix. Second, two numerical sub-methods for solving Lyapunov differential equations with time-varying periodic coefficients are proposed, both based on Fourier series expansion and the recursive matrix formulas. The former algorithm is a dimension expanding method, and the latter one uses the solutions of the homogeneous periodic Riccati differential equations. Finally, the efficiency and reliability of the proposed algorithms are demonstrated by four numerical examples.

Symplectic algorithms with mesh refinement for a hypersensitive optimal control problem

A symplectic algorithm with nonuniform grids is proposed for solving the hypersensitive optimal control problem using the density function. The proposed method satisfies the first-order necessary conditions for the optimal control problem that can preserve the structure of the original Hamiltonian systems. Furthermore, the explicit Jacobi matrix with sparse symmetric character is derived to speed up the convergence rate of the resulting nonlinear equations. Numerical simulations highlight the features of the proposed method and show that the symplectic algorithm with nonuniform grids is more computationally efficient and accuracy compared with uniform grid implementations. Besides, the symplectic algorithm has obvious advantages on optimality and convergence accuracy compared with the direct collocation methods using the same density function for mesh refinement.

Parametric variational solution of linear-quadratic optimal control problems with control inequality constraints

A parametric variational principle and the corresponding numerical algorithm are proposed to solve a linear-quadratic (LQ) optimal control problem with control inequality constraints. Based on the parametric variational principle, this control problem is transformed into a set of Hamiltonian canonical equations coupled with the linear complementarity equations, which are solved by a linear complementarity solver in the discrete-time domain. The costate variable information is also evaluated by the proposed method. The parametric variational algorithm proposed in this paper is suitable for both time-invariant and time-varying systems. Two numerical examples are used to test the validity of the proposed method. The proposed algorithm is used to astrodynamics to solve a practical optimal control problem for rendezvousing spacecrafts with a finite low thrust. The numerical simulations show that the parametric variational algorithm is effective for LQ optimal control problems with control inequality constraints.

A novel extended precise integration method based on Fourier series expansion for the H2-norm of linear time-varying periodic systems

ABSTRACT A new reliable algorithm for computing the H2-norm of linear time-varying periodic (LTP) systems via the periodic Lyapunov differential equation (PLDE) is proposed. By taking full advantage of the periodicity, the transition matrix of the underlying LTP system associated with the PLDE is effectively computed by developing a novel extended precise integration method based on Fourier series expansion, where the time-consuming work for the computation of the matrix exponential and its related integrals in every sub-interval is avoided. Then, a highly accurate and efficient algorithm for the PLDE is derived using the block form of the transition matrix. Thus, the H2-norm is evaluated by solving a simple first-order ordinary differential equation. Finally, two numerical examples are presented and compared with other algorithms to verify the numerical accuracy and efficiency of the proposed algorithm.

Fast Model Predictive Control Method for Large-Scale Structural Dynamic Systems: Computational Aspects

A fast and accurate model predictive control method is presented for dynamic systems representing large-scale structures. The fast model predictive control formulation is based on highly efficient computations of the state transition matrix, that is, the matrix exponential, using an improved precise integration method. The enhanced efficiency for model predictive control is achieved by exploiting the sparse structure of the matrix exponential at each discrete time step. Accuracy is maintained using the precise integration method. Compared with the general model predictive control method, the reduced central processing unit (CPU) time required by the fast model predictive control scheme can result in a shorter control update interval and a lower online computational burden. Therefore, the proposed method is more efficient for large-scale structural dynamic systems.

Symplectic method based on generating function for receding horizon control of linear time-varying systems

Abstract A novel method for solving the linear receding-horizon control (RHC) problem with time-varying coefficients is proposed based on a generating function and the standard symplectic form of Hamiltonian systems. In contrast to other methods used to solve the linear RHC problem, the generating function is utilized to avoid directly online integrating the differential Riccati equation (DRE). Solutions to the DRE at discrete time points have been obtained by applying the generating function at each computation step. The derivation of the coefficient includes calculating the state transition matrices of the linear Hamiltonian system using the Magnus method, which preserves the symplectic structure of the Hamiltonian system. Numerical simulations of spacecraft rendezvous demonstrate that the proposed symplectic method obtains highly precise results for relatively long discretization sizes, and then yields computational efficiency improvements of one to two orders of magnitude compared with conventional backward sweep methods and the Legendre pseudospectral methods.