PROPT

Description

PROPT is a combined modeling, compilation and solver engine, built upon the TomSym modeling class, for generation of highly complex optimal control problems. PROPT uses a pseudospectral Collocation method (with Gauss or Chebyshev points) for solving optimal control problems. This means that the solution takes the form of a Polynomial, and this polynomial satisfies the DAE and the path constraints at the collocation points.

In general PROPT has the following main functions:

Modeling

The PROPT system uses the TomSym symbolic source transformation engine to model optimal control problems. It is possible to define independent variables, dependent functions, scalars and constant parameters:

States and controls

States and controls only differ in the sense that states need be continuous between phases.

Boundary, path, event and integral constraints

A variety of boundary, path, event and integral constraints are shown below:

Single-phase optimal control example

Van der Pol Oscillator

Minimize:

J x , t = x 3 ( t f )

Subject to:

{ d x 1 d t = ( 1 − x 2 2 ) ∗ x 1 − x 2 + u d x 2 d t = x 1 d x 3 d t = x 1 2 + x 2 2 + u 2 x ( t 0 ) = [ 0   1   0 ] t f = 5 − 0.3 ≤ u ≤ 1.0

To solve the problem with PROPT the following code can be used (with 60 collocation points):

Multi-phase optimal control example

One-dimensional rocket with free end time and undetermined phase shift

Minimize:

J x , t = t C u t

Subject to:

{ d x 1 d t = x 2 d x 2 d t = a − g   ( 0 < t <= t C u t ) d x 2 d t = − g   ( t C u t < t < t f ) x ( t 0 ) = [ 0   0 ] g = 1 a = 2 x 1 ( t f ) = 100

The problem is solved with PROPT by creating two phases and connecting them:

Parameter estimation example

Parameter estimation problem

Minimize:

J p = ∑ i = 1 , 2 , 3 , 5 ( x 1 ( t i ) − x 1 m ( t i ) ) 2

Subject to:

{ d x 1 d t = x 2 d x 2 d t = 1 − 2 ∗ x 2 − x 1 x 0 = [ p 1   p 2 ] t i = [ 1   2   3   5 ] x 1 m ( t i ) = [ 0.264   0.594   0.801   0.959 ] | p 1 : 2 | <= 1.5

In the code below the problem is solved with a fine grid (10 collocation points). This solution is subsequently fine-tuned using 40 collocation points:

Optimal control problems supported