A symplectic sequence iteration approach for nonlinear optimal control problems with state-control constraints

Li, Mingwu; Peng, Haijun; Zhong, Wanxie

doi:10.1016/j.jfranklin.2015.03.019

Cited by 21 publications

(5 citation statements)

References 32 publications

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“…Moreover, Maleki and Hashim 22 introduced an iterative‐adaptive PS method using the QL scheme to solve nonlinear time‐delay OCPs with inequality constraints. More recently, the QL scheme is also named as successive convexification which is based on symplectic discretization 34‐36 …”

Section: Formulation Of the Iterative Radau Ps Methodsmentioning

confidence: 99%

On the modification and convergence of unconstrained optimal control using pseudospectral methods

Ghassemi

Maleki

Allame

2020

Optim Control Appl Methods

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A modified pseudospectral (PS) method is presented based on direct Legendre interpolation for unconstrained optimal control problems (OCP). The conditions for the convergence of the proposed PS method are provided and it is proved that the method possesses the spectral accuracy for solutions in appropriate Sobolev spaces. Different from existing convergence results in the literature, in this new analysis neither special cases of the problem nor relaxing the discretized dynamics is considered. Moreover, the proof does not use necessary conditions of optimal control. An iterative procedure is then proposed for nonlinear OCPs. The nonlinear problem is first replaced with a sequence of linear‐quadratic OCPs by utilizing the quasilinearization technique. Then, this sequence of problems are successively solved using the modified PS method. In some nonlinear OCPs where the resulting nonlinear programming problem is dense, the PS method becomes computationally intractable. In such a this problems, the iterative PS method reduces the computational complexity and saves work. In addition, a new efficient costate estimation procedure is derived by employing the necessary optimality conditions. This new costate estimation method is flexible in that it allows us to define three different schemes based on the structure of the problem. Several examples of varying complexity are included to demonstrate the efficiency and accuracy of the proposed methods.

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Section: Formulation Of the Iterative Radau Ps Methodsmentioning

confidence: 99%

On the modification and convergence of unconstrained optimal control using pseudospectral methods

Ghassemi

Maleki

Allame

2020

Optim Control Appl Methods

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“…Symplectic methods for nonlinear problems have been investigated in recent years. For example, Peng et al 17 proposed the hp symplectic pseudospectral method, and Li et al 18 studied the symplectic sequence iteration method.…”

Section: Introductionmentioning

confidence: 99%

Numeric‐analytical solution of an LC circuit with nonlinear capacitor based on the symplectic conservative perturbation method

Yang

Wang

et al. 2021

Int J Numerical Modelling

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The LC circuit with a nonlinear capacitor is analyzed using the symplectic conservative perturbation method based on the symplectic matrix. By treating the electric charge and the flux linkage as dual variables, the LC circuit system can be transformed into a Hamiltonian system. Then we solve the nonlinear Hamiltonian matrix equation by the perturbation method. The Hamiltonian matrix can be cast into linear and nonlinear parts. Based on the exact solution of the linear part, the nonlinear part is solved perturbatively by a canonical transformation. Since the coefficient matrix of the obtained equation is still a Hamiltonian matrix, symplectic conservation is guaranteed. The validity of the method is demonstrated, and the stability, precision, and the efficiency of the method are discussed by comparing it with the fourth‐order Runge–Kutta method (RK4), the ordinary perturbation method, the small‐parameter perturbation method based on displacement (SPPD), and the harmonic balance (HB) method. Numerical results show that the proposed method has better stability, precision, and efficiency compared with the other methods and thus has great advantages in long‐term simulations.

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“…In [5,6], the authors combined homotopy perturbation methods and iterative RKM to solve problem (1). In [7][8][9], Peng et al proposed an alternative iterative method called symplectic method to solve nonlinear BVPs. Furthermore, methods such as sinccollocation methods and Jacobi matrix methods were also considered in [10,11].…”

Section: Introductionmentioning

confidence: 99%

A High-Order Numerical Method for a Nonlinear System of Second-Order Boundary Value Problems

Jing

Guo

2020

Mathematical Problems in Engineering

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This paper is concerned with a high-order numerical scheme for nonlinear systems of second-order boundary value problems (BVPs). First, by utilizing quasi-Newton’s method (QNM), the nonlinear system can be transformed into linear ones. Based on the standard Lobatto orthogonal polynomials, we introduce a high-order Lobatto reproducing kernel method (LRKM) to solve these linear equations. Numerical experiments are performed to investigate the reliability and efficiency of the presented method.

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A symplectic sequence iteration approach for nonlinear optimal control problems with state-control constraints

Cited by 21 publications

References 32 publications

On the modification and convergence of unconstrained optimal control using pseudospectral methods

On the modification and convergence of unconstrained optimal control using pseudospectral methods

Numeric‐analytical solution of an LC circuit with nonlinear capacitor based on the symplectic conservative perturbation method

A High-Order Numerical Method for a Nonlinear System of Second-Order Boundary Value Problems

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