Modified Legendre–Gauss–Radau Collocation Method for Optimal Control Problems with Nonsmooth Solutions

Eide, Joseph D.; Hager, William W.; Rao, Anil V.

doi:10.1007/s10957-021-01810-5

Cited by 6 publications

(3 citation statements)

References 38 publications

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“…For the initial value estimation problem of the indirect method, it can be obtained through random guessing or analytical estimation methods [2]. In order to improve the computational efficiency and approximation accuracy of the direct collocation method, scholars have investigated different grid discretization methods, numerical differentiation methods, and variable construction forms, such as Jacobi, Chebyshev, and Legendre grids [3]. Gauss Pseudospectral grids have higher numerical differentiation accuracy and can efficiently obtain the optimal solution.…”

Section: Introductionmentioning

confidence: 99%

Minimum-fuel low-thrust trajectory optimization using adaptive wavelet collocation and sequential convex programming

Qian,

Cui,

Zhao

et al. 2024

J. Phys.: Conf. Ser.

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The minimum-fuel transfer problem is a common challenge encountered in low-thrust orbit trajectory optimization, typically solved with Bang-Bang control solution. This paper proposes a novel approach to solve the optimization problem by employing an adaptive wavelet collocation method for grid refinement and a synergistic integration of sequential convex programming, which results in an effective and precise solution methodology. The method introduces a spline wavelet to convert the infinite-dimensional state and control variable into finite dimensions. The dynamic nonlinear programming (NLP) problem is then convexified and linearized into a second-order cone programming (SOCP) problem using a small perturbation method, which can be rapidly solved by convex optimization tools. By iteratively solving the sequential SOCP problems, state and control variables converge quickly and accurately to the optimal control solution. A numerical simulation of a low-thrust transfer trajectory from Earth to Mars is performed to demonstrate the effectiveness of the proposed method. The results show that the adaptive method achieves higher accuracy in determining the switching time compared to conventional methods.

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Section: Introductionmentioning

confidence: 99%

Minimum-fuel low-thrust trajectory optimization using adaptive wavelet collocation and sequential convex programming

Qian,

Cui,

Zhao

et al. 2024

J. Phys.: Conf. Ser.

View full text Add to dashboard Cite

show abstract

“…As a quick description, shooting methods mostly employ explicit techniques (such as Euler and Runge‐Kutta) to discretize the system dynamics normally with piecewise constant control variables (see e.g., References 17 and 18). In collocation methods (see e.g., References 19‐21), the system dynamics and control are considered as piecewise functions, approximated in different ways such as orthogonal polynomials. It should be pointed out that implicit discretization techniques (such as Hermite–Simpson) can also be viewed as collocation methods.…”

Section: Introductionmentioning

confidence: 99%

Perturbed‐analytic direct transcription for optimal control (PADOC)

Jafarimoghaddam

Soler

2023

Optim Control Appl Methods

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The herein coined PADOC (perturbed-analytic direct transcription for optimal control) stands for a new transcription method for direct trajectory optimization. We construct PADOC on a novel segmented decomposition method that provides series solutions for nonlinear problems. To transcribe the infinite dimensional problem into a finite one, PADOC comes along with a new solution approach, namely, the herein coined average nonlinear programming (aNLP). The aNLP theorem suggests generating a staircase optimal solution (low resolution) and turning it into a distributed optimal solution (high resolution) by exploiting the Hamiltonian of the problem. The analytic architecture provides PADOC with an analytic connection of a truncated order between the discrete nodes of the solution. This renders stability, accuracy within an analytic resolution, robustness, and a cut-down in the number of the decision variables in the frame of NLP. We also prove that the multipliers associated with the Karush-Kuhn-Tucker optimality conditions in the frame of NLP (as transcribed by PADOC) correspond to a backward analytic solution for the costate equations. We finally show distinct features of PADOC through some examples.

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“…In previous studies, several multi-phase pseudo-spectral methods have been proposed to solve optimal control problems with non-smooth states and discontinuous controls. For example, nonsmooth solutions of optimal control problems were achieved by modified LGR collocation method in reference [31]. Junsub et al [32] proposed integrated optimal guidance for the reentry and landing of a rocket using a multi-phase pseudo-spectral method.…”

Section: Introductionmentioning

confidence: 99%

Constraints Maintaining Mesh Refinement Method for Solving Whole Trajectory of Discontinuous Aerodynamics

Huang,

Wang,

Qiao

et al. 2023

IEEE Access

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Solving trajectories of multi-phase dynamic advanced vehicles is difficult due to the coexistence of discontinuous aerodynamics, long-range and multi-peak features. Despite traditional phase-by-phase solving method for trajectory is expressive and easy to optimize, it loses global optimality to some extent and the state estimation between phases is cumbersome. In contrast, an all-in-one solution method for the whole trajectory can carry out global search and avoid the inter-phase state estimation, but the expression of the problem formulation is more complicated, and it is difficult to converge to a feasible solution. In this paper, we formulate the whole trajectory solving problem based on a representative vehicle and propose a constraints maintaining mesh refinement method based on the pseudo-spectral method to solve the whole trajectory allin-one. First, the proposed method presets a knot on the trajectory to construct an optimal control problem with a two-phase dynamic constraint, which maintains continuity of variables between the discontinuous aerodynamics. Second, ph-refinement is used to deal with long-range and multi-peak features throughout the trajectory. Third, to make mesh refinement and iterative solution feasible under discontinuous aerodynamics, an interval extending operator is proposed to maintain the consistency of constraints during the process of mesh refinement. Finally, the effectiveness of the proposed method for solving the whole trajectory is verified by 7 comparison experiments. The results show that the proposed method can effectively handle the whole trajectory problem of discontinuous aerodynamics, as evidenced by feasibility, convergence, optimality, and superiority.INDEX TERMS hypersonic vehicles; optimal trajectory; mesh refinement; discontinuous aerodynamics

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Modified Legendre–Gauss–Radau Collocation Method for Optimal Control Problems with Nonsmooth Solutions

Cited by 6 publications

References 38 publications

Minimum-fuel low-thrust trajectory optimization using adaptive wavelet collocation and sequential convex programming

Minimum-fuel low-thrust trajectory optimization using adaptive wavelet collocation and sequential convex programming

Perturbed‐analytic direct transcription for optimal control (PADOC)

Constraints Maintaining Mesh Refinement Method for Solving Whole Trajectory of Discontinuous Aerodynamics

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