PROPT is a software package intended to solve dynamic optimization problems. Such problems are usually described by
The goal of PROPT is to enable the formulation of such problem descriptions as seamlessly as possible, without having to worry about the mathematics of the actual solver. Once a problem has been properly defined, PROPT will take care of all the necessary steps in order to return a solution.^{2}
PROPT uses pseudospectral collocation methods (and other options) 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 (Note that both the DAE and the path constraints can be violated between collocation points). The default choice is to use Gauss points as collocation points, although the user can specify any set of points to use.
It should be noted that the code is written in a general way, allowing for a DAE rather than just an ODE formulation with path constraints.
Parameter estimation for dynamic systems is intrinsically supported by the framework as scalar decision variables can be introduced in the formulation.
A PROPT problem is defined with tomSym objects and standard Matlab expressions (usually in cell arrays), which contain information about different aspects of the problem.
In general an optimal control consists of the following:
PROPT has many built-in features:
Note: The tomSym and PROPT softwares are the first Matlab modules to enable complete source transformation for optimal control problem.
In PROPT, each state is represented by a scalar x_{0} and a vector x = [
x_{1} |
⋮ |
x_{N} |
], such that x_{i} corresponds to the value of the state at the i:th collocation point.
State equations are written on vector form, and hence apply at all collocation points. The initial point is treated separately, so the state equations do not apply at the initial point.
The final state is not a free variable. Instead, it is computed via the interpolating polynomial.^{3}
PROPT does not use Pontryagin’s maximum principle, rather it uses a pseudospectral method, however the results are mathematically equivalent. This means that if the solver prints "an optimal solution was found", the solution satisfies necessary, but not sufficient, conditions of optimality. It is guaranteed that the returned solution cannot be improved by an infinitesimal change in the trajectory, but there may exist completely different trajectories that are better.
A recommended procedure for finding other solutions is to set conservative bounds for all variables (states, controls and parameters) and change the solver to "multiMin". This will run a large number of optimizations from random starting points. If they all converge to the same minimum, this is an indication, but no guarantee, that the solution is indeed the global optimum.
In order guarantee that the global optimum is obtained, one must either solve the Hamilton-Jacobi-Bellman equation (which PROPT does not) or show that the problem is convex and therefore only has one optimum (which may not be the case). If ezsolve claims that the problem type is "LP" or "QP", the problem is convex, and the solution is a global optimum.
It is also worth mentioning that a solution computed by PROPT only satisfies the ODE and constraints in the specified collocation points. There is no guarantee that the solution holds between those points. A common way of testing the integrity of a solution is to re-run the computation using twice as many collocation points. If nothing changes, then there was probably enough points in the first computation.