Symplectic Approaches for Solving Two-Point Boundary-Value Problems

Peng, Haijun; Gao, Qiang; Wu, Zhigang; Zhong, Wanxie

doi:10.2514/1.55795

Cited by 59 publications

(35 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The detailed expressions of the Lagrange interpolation base function can be found in Peng et al [22,23].…”

Section: Symplectic Methods For Optimal Control Problemsmentioning

confidence: 99%

“…(32), the action is the function of only the state variables at the two ends of time intervals; thus, [22,23] ∂S j ∂x j = 0, j = 1, 2, . .…”

Section: Symplectic Methods For Optimal Control Problemsmentioning

confidence: 99%

“…3.1. In this section, a symplectic method for the optimal control problem without path constraints proposed by Peng et al [22][23][24] will be applied firstly. Then, several remarks about the application of the symplectic method to this study will be presented.…”

Section: Symplectic Methods For Optimal Control Problemsmentioning

confidence: 99%

“…The differential quotient for the action can be obtained as follows if Hamiltonian canonical equation is satisfied in the time interval [22,23] dS =λ where I denotes the identity matrix, ⊗ denotes the Kronecker product of two matrices, vectorsx j−1 andx j is the state variables at the left and the right end of j−th subinterval, vectorx j is defined by the combination of all the state variables at the interpolation points within the subinterval, i.e.,…”

Section: Symplectic Methods For Optimal Control Problemsmentioning

confidence: 99%

“…In this study, the new reconfiguration process mentioned above is called by reconfiguration with keeping in order to distinguish from traditional reconfigurations. The contributions of this paper mainly include following three points: (1) the optimal control problem statement with minimization of energy cost for a loose leaderfollowing spacecraft formation is established; (2) the collision avoidance between follower spacecrafts is taken into consideration; (3) an efficiently and easily implemented numerical algorithm which combines penalty techniques and structure persevering, symmetric, sparse and high efficient symplectic methods proposed by Peng et al [22][23][24], to solve nonlinear optimal control problems with path equality constraints, is developed. The key idea of the proposed algorithm is to employ penalty techniques, which convert constrained optimal control problems into optimal control problems without constraints, and then solve a sequence of resulting sparse symmetrical linear equations obtained by symplectic methods, and thus, the online implementation efficiency must be significantly increased.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Optimal control of loose spacecraft formations near libration points with collision avoidance

Peng

Zhong

2015

Nonlinear Dyn

Self Cite

View full text Add to dashboard Cite

This paper investigates the optimal control of loose leader-following spacecraft formations near libration points. The formation reconfigurations and keepings can be applied at the same time for the loose formations. A loose sphere leader-following formation, i.e., the leader spacecraft moves along the reference Halo orbits while the motions of all follower spacecrafts are restricted to a sphere formation with its center located at the leader, is explored in this study carefully. Collision avoidances among follower spacecrafts are taken into consideration since there are no constraints between followers and then the formation for all followers is loose. A symplectic penalty iteration algorithm is proposed to obtain the optimal solutions with minimization of energy cost. An effective penalty function to avoid collision is developed, and a high efficiency and structure-preserving symplectic method is introduced in the iteration algorithm. Several optimal control problems including formation reconfigurations and formation keepings are solved to show the effectiveness of the proposed algorithm. Numerical results show that errors of formations can be reduced to millimeters or smaller only by few control input and the collision can be avoided successfully by the developed penalty function.

show abstract

“…The detailed expressions of the Lagrange interpolation base function can be found in Peng et al [22,23].…”

Section: Symplectic Methods For Optimal Control Problemsmentioning

confidence: 99%

“…(32), the action is the function of only the state variables at the two ends of time intervals; thus, [22,23] ∂S j ∂x j = 0, j = 1, 2, . .…”

Section: Symplectic Methods For Optimal Control Problemsmentioning

confidence: 99%

Section: Symplectic Methods For Optimal Control Problemsmentioning

confidence: 99%

Section: Symplectic Methods For Optimal Control Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Optimal control of loose spacecraft formations near libration points with collision avoidance

Peng

Zhong

2015

Nonlinear Dyn

Self Cite

View full text Add to dashboard Cite

show abstract

Time-optimal reorientation of the rigid spacecraft using a pseudospectral method integrated homotopic approach

2014

Optim. Control Appl. Meth.

View full text Add to dashboard Cite

Summary This paper proposes a robust algorithm for time‐optimal rigid spacecraft reorientation trajectory generation. Based on the Pontryagin's maximum principle, the first‐order necessary optimality conditions are derived. These optimality conditions are numerically satisfied by adopting a pseudospectral method integrated homotopic approach to solve the associated shooting functions. First, the energy‐optimal reorientation solution is obtained using the Radau pseudospectral method, which has a spectral convergence speed and can give a precise estimation of the initial costates used to start the homotopic approach. Then, a modified homotopy scheme is given to deform the associated energy‐optimal solution to the desired time‐optimal solution continuously. Finally, for the inertially symmetric spacecraft reorientation problem, the newly found time‐optimal solutions are presented. The performance of the algorithm is illustrated by simulating a general asymmetric rigid spacecraft time‐optimal reorientation problem. Copyright © 2014 John Wiley & Sons, Ltd.

show abstract

A symplectic indirect approach for a class of nonlinear optimal control problems of differential‐algebraic systems

Shi

Peng

Wang

et al. 2021

Intl J Robust & Nonlinear

Self Cite

View full text Add to dashboard Cite

Differential-algebraic equations (DAEs) can model constrained dynamical systems and processes from practical engineering. Therefore, research on nonlinear optimal control problems of DAEs is of theoretical significance for optimal control of constrained systems, which can generate reference trajectories and control inputs for online control strategies. In terms of the numerical solution of this type of problem, research on indirect numerical methods is still insufficient and less research focuses on symplectic-preserving methods. In this article, a symplectic indirect approach is proposed for optimal control problems subject to index-1 DAEs. Necessary conditions of the optimal control problem constitute a Hamiltonian boundary value problem (HBVP) and there exists a symplectic structure in the Hamiltonian system. In the proposed approach, based on specified properties of generating functions, discrete equations can preserve the symplectic structure of the Hamiltonian system. In the iterative solution, the Jacobian matrices of the discrete equations are sparse and symmetric, which are very significant to save memory and improve efficiency in practical computation. In numerical examples, the proposed approach can provide highly accurate state variables and control inputs with fewer iterations. More accurate cost functional can be obtained. Problems from the chemistry process also can be solved effectively, it verifies the problem-solving ability of the proposed approach. K E Y W O R D Sconstrained systems, differential-algebraic equations, nonlinear optimal control, symplectic methods INTRODUCTIONOriginating from practical engineering applications such as motion planning of robots and optimal strategies of processes, optimal control problems 1-4 have been widely researched in the past decades. Generally, there are a defined cost functional and dynamical constraints in the optimal control problem. The aim is to find optimal state variables and control inputs that fulfill dynamical constraints and minimize the cost functional. The cost functional is a scalar function with respect to state variables and control inputs, which is defined according to practical requirements. For example, the task is to 2712

show abstract

Symplectic Approaches for Solving Two-Point Boundary-Value Problems

Cited by 59 publications

References 9 publications

Optimal control of loose spacecraft formations near libration points with collision avoidance

Optimal control of loose spacecraft formations near libration points with collision avoidance

Time-optimal reorientation of the rigid spacecraft using a pseudospectral method integrated homotopic approach

A symplectic indirect approach for a class of nonlinear optimal control problems of differential‐algebraic systems

Contact Info

Product

Resources

About