The PROPT[1] MATLAB Optimal Control Software is a new generation platform for solving applied optimal control (with ODE or DAE formulation) and parameters estimation problems.
The platform was developed by MATLAB Programming Contest Winner, Per Rutquist in 2008.
The most recent version has support for binary and integer variables as well as an automated scaling module.
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: 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 only differ in the sense that states need be continuous between phases.
A variety of boundary, path, event and integral constraints are shown below: Van der Pol Oscillator [3] Minimize:
(
t
f
{\displaystyle {\begin{matrix}J_{x,t}&=&x_{3}(t_{f})\\\end{matrix}}}
Subject to:
{
d t
x
x (
{\displaystyle {\begin{cases}{\frac {dx_{1}}{dt}}=(1-x_{2}^{2})*x_{1}-x_{2}+u\\{\frac {dx_{2}}{dt}}=x_{1}\\{\frac {dx_{3}}{dt}}=x_{1}^{2}+x_{2}^{2}+u^{2}\\x(t_{0})=[0\ 1\ 0]\\t_{f}=5\\-0.3\leq u\leq 1.0\\\end{cases}}}
To solve the problem with PROPT the following code can be used (with 60 collocation points): One-dimensional rocket [4] with free end time and undetermined phase shift Minimize:
x , t
{\displaystyle {\begin{matrix}J_{x,t}&=&tCut\\\end{matrix}}}
Subject to:
{\displaystyle {\begin{cases}{\frac {dx_{1}}{dt}}=x_{2}\\{\frac {dx_{2}}{dt}}=a-g\ (0 The problem is solved with PROPT by creating two phases and connecting them: Parameter estimation problem [5] Minimize: {\displaystyle {\begin{matrix}J_{p}&=&\sum _{i=1,2,3,5}{(x_{1}(t_{i})-x_{1}^{m}(t_{i}))^{2}}\\\end{matrix}}} Subject to: {\displaystyle {\begin{cases}{\frac {dx_{1}}{dt}}=x_{2}\\{\frac {dx_{2}}{dt}}=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\\\end{cases}}} 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: