Bulirsch
R.
10.1007/978-3-0348-7539-4_20
doi
theory
reliability
necessary condition
estimates
control problem
procedure
false
glider
direct method
solution
sensitive dependence
upwind
extension
https://doi.org/10.1007/978-3-0348-7539-4_20
initial guess
step
switching structure
adjoint
relationship
multipoint boundary value problem
direct collocation
1993-01-01
low accuracy
difficulties
control theory
1993
domain
numerical solution
structure
large domains
adjoint variables
competition
way
convergence
iterative solution
control
trajectory optimization problem
maximization
multipliers
accuracy
direct collocation method
iteration
shooting method
hang glider
possibility
Combining Direct and Indirect Methods in Optimal Control: Range Maximization of a Hang Glider
optimal control
optimal control problem
chapters
inequality constraints
optimization problem
indirect method
When solving optimal control problems, indirect methods such as multiple shooting suffer from difficulties in finding an appropriate initial guess for the adjoint variables. For, this initial estimate must be provided for the iterative solution of the multipoint boundary-value problems arising from the necessary conditions of optimal control theory. Direct methods such as direct collocation do not suffer from this problem, but they generally yield results of lower accuracy and their iteration may even terminate with a non-optimal solution. Therefore, both methods are combined in such a way that the direct collocation method is at first applied to a simplified optimal control problem where all inequality constraints are neglected as long as the resulting problem is still well-defined. Because of the larger domain of convergence of the direct method, an approximation of the optimal solution of this problem can be obtained easier. The fusion between direct and indirect methods is then based on a relationship between the Lagrange multipliers of the underlying nonlinear programming problem to be solved by the direct method and the adjoint variables appearing in the necessary conditions which form the boundary-value problem to be solved by the indirect method. Hence, the adjoint variables, too, can be estimated from the approximation obtained by the direct method. This first step then facilitates the subsequent extension and competition of the model by homotopy techniques and the solution of the arising boundary-value problems by the indirect multiple shooting method. Proceeding in this way, the high accuracy and reliability of the multiple shooting method, especially the precise computation of the switching structure and the possibility to verify many necessary conditions, is preserved while disadvantages caused by the sensitive dependence on an appropriate estimate of the solution are considerably cut down. This procedure is described in detail for the numerical solution of the maximum-range trajectory optimization problem of a hang glider in an upwind which provides an example for a control problem where appropriate initial estimates for the adjoint variables are hard to find.
computation
disadvantages
appropriate initial estimates
problem
model
variables
273-288
https://scigraph.springernature.com/explorer/license/
multiple shooting
nonlinear programming problem
multiple shooting method
conditions
detail
collocation method
first step
optimal control theory
Lagrange multipliers
collocation
indirect multiple shooting method
example
homotopy technique
method
high accuracy
chapter
approximation
guess
non-optimal solutions
appropriate estimates
optimal solution
boundary value problem
constraints
range maximization
subsequent extension
2022-08-04T17:19
shooting
results
programming problem
technique
fusion
initial estimates
dependence
appropriate initial guess
precise computation
978-3-0348-7541-7
978-3-0348-7539-4
Optimal Control
Miele
A.
von Stryk
Oskar
Edda
Nerz
Mathematical Sciences
Hans Josef
Pesch
Applied Mathematics
Roland
Bulirsch
Numerical and Computational Mathematics
J.
Stoer
Springer Nature
pub.1004585554
dimensions_id
Well
K.
Mathematisches Institut, Technische Universität München, Postfach 20 24 20, D-8000, München 2, Germany
Mathematisches Institut, Technische Universität München, Postfach 20 24 20, D-8000, München 2, Germany
Springer Nature - SN SciGraph project