A modified Legendre-Gauss-Radau collocation method is developed for solving optimal control problems whose solutions contain a nonsmooth optimal control. The method includes an additional variable that defines the location of nonsmoothness. In addition, collocation constraints are added at the end of a mesh interval that defines the location of nonsmoothness in the solution on each differential equation that is a function of control along with a control constraint at the endpoint of this same mesh interval. The transformed adjoint system for the modified Legendre-Gauss-Radau collocation method along with a relationship between the Lagrange multipliers of the nonlinear programming problem and a discrete approximation of the costate of the optimal control problem is then derived. Finally, it is shown via example that the new method provides an accurate approximation of the costate.
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A pseudospectral method for optimal control based on collocation at the Gauss points
A Gauss collocation method is developed for solving
optimal control problems with convex control constraints.
The method has a local exponential convergence rate when
the solution of the continuous problem is smooth and the
Hamiltonian possesses a convexity property.
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- NSF-PAR ID:
- 10093310
- Date Published:
- Journal Name:
- 2018 IEEE Conference on Decision and Control (CDC)
- Page Range / eLocation ID:
- 2490 to 2495
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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A continuation method for solving singular optimal control problems is presented. Assuming that the structure of the optimal solution is known a priori, the time horizon of the optimal control problem is divided into multiple domains and is discretized using a multiple-domain Radau collocation formulation. The resulting nonlinear programming problem is then solved by implementing a continuation method over singular domains. The continuation method is then demonstrated on a minimum-time rigid body reorientation problem. The results obtained demonstrate that a continuation method can be used to obtain an accurate approximation to the optimal control on both a finite-order and an infinite-order singular arc.more » « less
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null (Ed.)A new method is developed for solving optimal control problems whose solutions are nonsmooth. The method developed in this paper employs a modified form of the Legendre–Gauss–Radau orthogonal direct collocation method. This modified Legendre–Gauss–Radau method adds two variables and two constraints at the end of a mesh interval when compared with a previously developed standard Legendre– Gauss–Radau collocation method. The two additional variables are the time at the interface between two mesh intervals and the control at the end of each mesh inter- val. The two additional constraints are a collocation condition for those differential equations that depend upon the control and an inequality constraint on the control at the endpoint of each mesh interval. The additional constraints modify the search space of the nonlinear programming problem such that an accurate approximation to the location of the nonsmoothness is obtained. The transformed adjoint system of the modified Legendre–Gauss–Radau method is then developed. Using this transformed adjoint system, a method is developed to transform the Lagrange multipliers of the nonlinear programming problem to the costate of the optimal control problem. Fur- thermore, it is shown that the costate estimate satisfies one of the Weierstrass–Erdmann optimality conditions. Finally, the method developed in this paper is demonstrated on an example whose solution is nonsmooth.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/CDC.2018.8618929
