9 research outputs found
Convergence rate for a Gauss collocation method applied to unconstrained optimal control
A local convergence rate is established for an orthogonal collocation method
based on Gauss quadrature applied to an unconstrained optimal control problem.
If the continuous problem has a sufficiently smooth solution and the
Hamiltonian satisfies a strong convexity condition, then the discrete problem
possesses a local minimizer in a neighborhood of the continuous solution, and
as the number of collocation points increases, the discrete solution
convergences exponentially fast in the sup-norm to the continuous solution.
This is the first convergence rate result for an orthogonal collocation method
based on global polynomials applied to an optimal control problem
Convergence rate for a Radau collocation method applied to unconstrained optimal control
A local convergence rate is established for an orthogonal collocation method
based on Radau quadrature applied to an unconstrained optimal control problem.
If the continuous problem has a sufficiently smooth solution and the
Hamiltonian satisfies a strong convexity condition, then the discrete problem
possesses a local minimizer in a neighborhood of the continuous solution, and
as the number of collocation points increases, the discrete solution
convergences exponentially fast in the sup-norm to the continuous solution. An
earlier paper analyzes an orthogonal collocation method based on Gauss
quadrature, where neither end point of the problem domain is a collocation
point. For the Radau quadrature scheme, one end point is a collocation point.Comment: arXiv admin note: substantial text overlap with arXiv:1507.0826
Mesh Refinement Method for Solving Bang-Bang Optimal Control Problems Using Direct Collocation
A mesh refinement method is developed for solving bang-bang optimal control
problems using direct collocation. The method starts by finding a solution on a
coarse mesh. Using this initial solution, the method then determines
automatically if the Hamiltonian is linear with respect to the control, and, if
so, estimates the locations of the discontinuities in the control. The switch
times are estimated by determining the roots of the switching functions, where
the switching functions are determined using estimates of the state and costate
obtained from the collocation method. The accuracy of the switch times is then
improved on subsequent meshes by dividing the original optimal control problem
into multiple domains and including variables that define the locations of the
switch times. While in principle any collocation method can be used, in this
research the previously developed Legendre-Gauss-Radau collocation method is
employed because it provides an accurate approximation of the costate which in
turn improves the approximation of the switching functions. The method of this
paper is designed to be used with a previously developed mesh refinement method
in order to accurately approximate the solution in segments where the solution
is smooth. The method is demonstrated on three examples where it is shown to
accurately determine the switching structure of a bang-bang optimal control
problem. When compared with previously developed mesh refinement methods, the
results demonstrate that the method developed in this paper improves
computational efficiency when solving bang-bang optimal control problems.Comment: 22 pages, 9 figures, 3 tables
Convergence rate for a Gauss collocation method applied to constrained optimal control
A local convergence rate is established for a Gauss orthogonal collocation
method applied to optimal control problems with control constraints. If the
Hamiltonian possesses a strong convexity property, then the theory yields
convergence for problems whose optimal state and costate possess two square
integrable derivatives. The convergence theory is based on a stability result
for the sup-norm change in the solution of a variational inequality relative to
a 2-norm perturbation, and on a Sobolev space bound for the error in
interpolation at the Gauss quadrature points and the additional point -1. The
tightness of the convergence theory is examined using a numerical example.Comment: arXiv admin note: text overlap with arXiv:1605.0212
Comparison of Derivative Estimation Methods in Solving Optimal Control Problems Using Direct Collocation
A study is conducted to evaluate four derivative estimation methods when
solving a large sparse nonlinear programming problem that arises from the
approximation of an optimal control problem using a direct collocation method.
In particular, the Taylor series-based finite-difference, bicomplex-step, and
hyper-dual derivative estimation methods are evaluated and compared alongside a
well known automatic differentiation method. The performance of each derivative
estimation method is assessed based on the number of iterations, the
computation time per iteration, and the total computation time required to
solve the nonlinear programming problem. The efficiency of each of the four
derivative estimation methods is compared by solving three benchmark optimal
control problems. It is found that while central finite-differencing is
typically more efficient per iteration than either the hyper-dual or
bicomplex-step, the latter two methods have significantly lower overall
computation times due to the fact that fewer iterations are required by the
nonlinear programming problem when compared with central finite-differencing.
Furthermore, while the bicomplex-step and hyper-dual methods are similar in
performance, the hyper-dual method is significantly easier to implement.
Moreover, the automatic differentiation method is found to be substantially
less computationally efficient than any of the three Taylor series-based
methods. The results of this study show that the hyper-dual method offers
several benefits over the other three methods both in terms of computational
efficiency and ease of implementation.Comment: 26 pages, 2 figures, 3 table
A Warm Start Method for Solving Chance Constrained Optimal Control Problems
A warm start method is developed for efficiently solving complex chance
constrained optimal control problems. The warm start method addresses the
computational challenges of solving chance constrained optimal control problems
using biased kernel density estimators and Legendre-Gauss-Radau collocation
with an adaptive mesh refinement method. To address the computational
challenges, the warm start method improves both the starting point for the
chance constrained optimal control problem, as well as the efficiency of
cycling through mesh refinement iterations. The improvement is accomplished by
tuning a parameter of the kernel density estimator, as well as implementing a
kernel switch as part of the solution process. Additionally, the number of
samples for the biased kernel density estimator is set to incrementally
increase through a series of mesh refinement iterations. Thus, the warm start
method is a combination of tuning a parameter, a kernel switch, and an
incremental increase in sample size. This warm start method is successfully
applied to solve two challenging chance constrained optimal control problems in
a computationally efficient manner using biased kernel density estimators and
Legendre-Gauss-Radau collocation.Comment: 34 pages, 6 Figures, 8 Table
Convergence Rate for a Radau hp Collocation Method Applied to Constrained Optimal Control
For unconstrained control problems, a local convergence rate is established
for an -method based on collocation at the Radau quadrature points in each
mesh interval of the discretization. If the continuous problem has a
sufficiently smooth solution and the Hamiltonian satisfies a strong convexity
condition, then the discrete problem possesses a local minimizer in a
neighborhood of the continuous solution, and as either the number of
collocation points or the number of mesh intervals increase, the discrete
solution convergences to the continuous solution in the sup-norm. The
convergence is exponentially fast with respect to the degree of the polynomials
on each mesh interval, while the error is bounded by a polynomial in the mesh
spacing. An advantage of the -scheme over global polynomials is that there
is a convergence guarantee when the mesh is sufficiently small, while the
convergence result for global polynomials requires that a norm of the
linearized dynamics is sufficiently small. Numerical examples explore the
convergence theory.Comment: This paper subsumes arXiv: 1508.0378
Mesh Refinement Method for Solving Optimal Control Problems with Nonsmooth Solutions Using Jump Function Approximations
A mesh refinement method is described for solving optimal control problems
using Legendre-Gauss-Radau collocation. The method detects discontinuities in
the control solution by employing an edge detection scheme based on jump
function approximations. When discontinuities are identified, the mesh is
refined with a targeted -refinement approach whereby the discontinuity
locations are bracketed with mesh points. The remaining smooth portions of the
mesh are refined using previously developed techniques. The method is
demonstrated on two examples, and results indicate that the method solves
optimal control problems with discontinuous control solutions using fewer mesh
refinement iterations and less computation time when compared with previously
developed methods.Comment: 22 Pages, 8 Figures, 0 Table
Modified Legendre-Gauss-Radau Collocation Method for Solving Optimal Control Problems with Nonsmooth Solutions
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 interval. 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. Furthermore, 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.Comment: 36 pages, 8 figure