Which methods can one use to estimate the initial values of the co-state equation when solving nonlinear optimal control problem using indirect numerical methods.
In such a formulation, you must be more certain. If a posteriori data are available, you may apply smoothing to estimate the initial state.
You could solve the optimal control problem by a direct shooting method. Then "some adjoint information" of the discretized problem can provide very accurate costate estimations. In a recent article I have discussed the relation of the discretized direct shooting problem and the continuous problem. For the convenience a add a preprint that also contains further references.
Dear Ralf,
I would suggest to use firstly a direct method (transcrition to a nonlinear programming problem) using the combination of AMPL (which yield exact first and second order gradients via AD) and, e.g., the interior point solver IPOPT (Waechter, Biegler, open source). IPOPT provides you with estimates of the adjoint variables, may be with the oposite sign, depending if you use a minimum or maximum principle. In addition, these estimates must be scaled due to the discretization you have used for discretizing the ODEs.
However, due to my experience the approximation of simply the initial conditions
of the adjoints is not precise enough to integrate the coupled system of state and co-state equations by just solving an initial value problem over the entire interval.
This is usually caused by the nonlinearity.
If the nonlinearity is not too strong you might have luck. Otherwise use multiple shooting.
Last, but most important question: Do you really need to use an indirect method?
Why not to use the above combination followed by an a posteriori check of the main necessary condition, the control law. This works very well.
Kind regards,
Hans Josef
Sorry,
please replace "Dear Ralf" by Dear Belinda"; see also the answer of Ralf!
I have meet the same question of choosing the proper initial co-state approximation for PMP optimal control of hybrid vehicle. Conventionally, shooting method are used to catch the co-state.,which is not convenient for real-time control.I am trying to look for another way to estimate the initial value .Who can give me some advise?
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