Pseudospectral methods for solving infinite-horizon optimal control problems

Garg, Divya; Hager, William W.; Rao, Anil V.

doi:10.1016/j.automatica.2011.01.085

Cited by 288 publications

(207 citation statements)

References 11 publications

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“…The integral cost function can also be discretized by GaussLobatto quadrature rules, which provide highly accurate results for approximating integrals. 24,25) The optimization problem is thereby converted into a nonlinear programming problem (NLP), 26,27) which can be solved by a well-developed parameter optimization algorithm. Thus, the integral and differential equations in eq.…”

Section: Solving Methods Investigationmentioning

confidence: 99%

Solving the Transient Cost-Related Optimization Problem for Copper Flash Smelting Process with Legendre Pseudospectral Method

et al. 2013

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In the present contribution, a scheme to solve transient cost-related optimization problem of dynamic transient procedure for copper flash smelting process is investigated. Taking the actual copper flash smelting process at a Smelter in China as the research object, the transient costrelated optimization problem is presented by integrating transient time, transient resources consumption and the state fluctuation constraint. Then, it was considered as the operational parameters trajectories optimization problem that generates minimum-time and minimum-resources consumption during dynamic transient procedure. The dynamic relationship and the working state fluctuation are constructed in the form of constraints in the resulting optimization problem formulation. The Legendre Pseudospectral method is used to solve the constrained, nonlinear optimization problem. The results of numerical experiments of dynamic transient procedure for copper flash smelting process are given to illustrate the proposed scheme.

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Section: Solving Methods Investigationmentioning

confidence: 99%

Solving the Transient Cost-Related Optimization Problem for Copper Flash Smelting Process with Legendre Pseudospectral Method

et al. 2013

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“…[18][19][20] In the Radau pseudospectral method, the state and its time derivative are approximated, respectively, in each mesh interval S k , (k = 1, . .…”

Section: Radau Pseudospectral Methods For Problem Mmentioning

confidence: 99%

“…In even more recent years, a great deal of research has been done on the class of direct collocation pseudospectral methods. 3,12,16,[18][19][20][21] In a pseudospectral method, the state is approximated using a basis of either Lagrange of Chebyshev polynomials and the dynamics are collocated at points associated with a Gaussian quadrature. The most common collocation points, which are the roots of a linear combination of Legendre polynomials or derivatives of Legendre polynomials, are Legendre-Gauss (LG), Legendre-Gauss-Radau (LGR), and Legendre-Gauss-Lobatto (LGL) points.…”

Section: Introductionmentioning

confidence: 99%

“…The method developed in this paper utilizes a modified version of the Radau pseudospectral method. [18][19][20] Specifically, the optimal control problem is divided into a mesh such that the times of the mesh points are included as variables in the optimal control problem. This leads to an optimal control problem where it is desired to determine not only the optimal state and control in each mesh interval, but it is also desired to determine the optimal values of mesh point times.…”

Section: Introductionmentioning

confidence: 99%

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Direct Trajectory Optimization and Costate Estimation of State Inequality Path-Constrained Optimal Control Problems Using a Radau Pseudospectral Method

Francolin¹,

Rao²

2012

AIAA Guidance, Navigation, and Control Conference

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A Radau pseudospectral method is derived for solving state-inequality path constrained optimal control problems. The continuous-time state-inequality path constrained optimal control problem is modified by applying a set of tangency conditions at the entrance of the activity of the path constraint. It is shown that the first-order optimality condition of the nonlinear programming problem associated with the Radau pseudospectral method is equivalent to the Radau pseudospectral discretized first-order optimality conditions of the modified continuous-time optimal control problem. The method is applied to a classical state-inequality path constrained optimal control problem and it is found that the solution accuracy is improved significantly when compared with the accuracy of the solution obtained using the original Radau pseudospectral discretization.

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“…Convergence in an h method is then achieved by increasing the number of mesh intervals [6][7][8]. In a p method, the state is approximated using few mesh intervals (often a single mesh interval is used), and convergence is achieved by increasing the degree of the polynomial [9][10][11][12][13][14][15][16]. In an hp method, both the number of mesh intervals and the degree of the polynomial within each mesh interval is varied, and convergence is achieved through the appropriate combination of the number of mesh intervals and the polynomial degrees within each interval [17,18].…”

mentioning

confidence: 99%

Exploiting Sparsity in Direct Collocation Pseudospectral Methods for Solving Optimal Control Problems

Patterson¹,

Rao²

2012

Journal of Spacecraft and Rockets

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In a direct collocation pseudospectral method, a continuous-time optimal control problem is transcribed to a finitedimensional nonlinear programming problem. Solving this nonlinear programming problem as efficiently as possible requires that sparsity at both the first-and second-derivative levels be exploited. In this paper, a computationally efficient method is developed for computing the first and second derivatives of the nonlinear programming problem functions arising from a pseudospectral discretization of a continuous-time optimal control problem. Specifically, in this paper, expressions are derived for the objective function gradient, constraint Jacobian, and Lagrangian Hessian arising from the previously developed Radau pseudospectral method. It is shown that the computation of these derivative functions can be reduced to computing the first and second derivatives of the functions in the continuous-time optimal control problem. As a result, the method derived in this paper reduces significantly the amount of computation required to obtain the first and second derivatives required by a nonlinear programming problem solver. The approach derived in this paper is demonstrated on an example where it is found that significant computational benefits are obtained when compared against direct differentiation of the nonlinear programming problem functions. The approach developed in this paper improves the computational efficiency of solving nonlinear programming problems arising from pseudospectral discretizations of continuous-time optimal control problems.

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Pseudospectral methods for solving infinite-horizon optimal control problems

Cited by 288 publications

References 11 publications

Solving the Transient Cost-Related Optimization Problem for Copper Flash Smelting Process with Legendre Pseudospectral Method

Solving the Transient Cost-Related Optimization Problem for Copper Flash Smelting Process with Legendre Pseudospectral Method

Direct Trajectory Optimization and Costate Estimation of State Inequality Path-Constrained Optimal Control Problems Using a Radau Pseudospectral Method

Exploiting Sparsity in Direct Collocation Pseudospectral Methods for Solving Optimal Control Problems

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