PROPT

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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 platfrom was developed by MATLAB Programming Contest Winner, Per Rutquist in 2008.

[edit] Description

PROPT is a combined modeling, compilation and solver engine for generation of highly complex optimal control problems. PROPT uses a pseudospectral collocation method 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 five main functions:

• Computation of the constant matrices used for the differentiation and integration of the polynomials used to approximate the solution to the trajectory optimization problem.

• Text manipulation to turn user-supplied expressions into MATLAB code for the cost function f and constraint function c that are passed to a nonlinear programming solver in TOMLAB, ensuring that the code is compatible with MAD (TOMLAB package for automatic differentiation to achieve floating point precision for derivatives).

• Functionality for plotting and computing a variety of information for the solution to the problem.

• Partial automatic linearization for the following scenarios:
  • Minimal (or maximal) time problem
  • Problems with a linear objective (pure feasibility problems are also handled)
  • Linear systems with a fixed end time

• Integrated support for non-smooth^[2] (hybrid) optimal control problems.

[edit] Modeling

The PROPT system uses equations and expressions to model optimal control problems. It is possible to define independent variables, dependent functions, scalars and constant parameters:

t = proptIndependent('t', [0 0 0], [10 20 40]);
problem.variables.z1 = proptScalar(0, 500 , 10);
problem.parameters.ki0 = [1e3; 1e7; 10; 1e-3];

[edit] States and Controls

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

x.x1 = proptState(-inf , inf , [0; 1]);
x.u1 = proptControl(-2 , 1 , 0);

[edit] Boundary, path, event and integral constraints

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

expr.x1_i = '1';                    % Starting point for x1
expr.x1_f = '1';                    % End point for x1
expr.x2_f = '2';                    % End point for x2
eq.x3min    = 'x3 > 0.5 ';          % Path constraint for x3
eq.integral = 'quad_t *x2 - 1 = 0'; % Integral constraint for x2
eq.final3   = 'x3_f > 0.5 ';        % Final event constraint for x3
eq.init1    = 'x1_i < 2.0 ';        % Initial event constraint x1

[edit] Example

Van der Pol Oscillator

Minimize:

Subject to:

To solve the problem with PROPT the following code can be used:

tF = 5;
t  = proptIndependent('t', 0, tF);
 
clear x eq expr
x.x1 = proptState(-10, 10, 0);
x.x2 = proptState(-10, 10, 1);
x.x3 = proptState(-10, 10, 0);
x.u  = proptControl(-0.3, 1, -0.01);
 
expr.x1_i = '0';
expr.x2_i = '1';
expr.x3_i = '0';
 
eq.eq1    = 'x1_t = ((1-x2.^2).*x1-x2+u)';
eq.eq2    = 'x2_t = (x1)';
eq.eq3    = 'x3_t = (x1.^2+x2.^2+u.^2)';
 
problem = proptPhase(t, x, [], eq, expr, 60);
problem.cost = 'x3_f';
problem.name = 'Van Der Pol';
problem.language = 'Matlab, vectorized';
 
solution = proptSolve(problem);

[edit] Optimal Control Problems Supported

• Aerodynamic trajectory control^[3]
• Bang bang control^[4]
• Chemical engineering^[5]
• Dynamic systems^[6]
• General optimal control
• Large-scale linear control^[7]
• Multi-phase system control^[8]
• Mechanical engineering design^[9]
• Nondifferentiable control^[10]
• Parameters estimation for dynamic systems^[11]
• Singular control

[edit] References

[edit] External links

• TOMLAB - Developers and distributers of the software.
• MAD - Automatic differentiation package used in software.
• PROPT - Home page for PROPT.
• SNOPT - Default solver used in PROPT.