PROPT

<h2 id="description">Description</h2>
PROPT is a combined <a href="/facts/Mathematical_model/CGhyuxgz">modeling</a>, <a href="/facts/Code_generation_(compiler)/6Ce9gjPs">compilation</a> and solver engine, built upon the <a href="/facts/TomSym/88qFQLKD">TomSym</a> modeling class, for generation of highly complex optimal control problems. PROPT uses a <a href="/facts/Gauss_pseudospectral_method/ygL5PdTc">pseudospectral</a> <a href="/facts/Collocation_method/YMVkNS7m">Collocation method</a> (with Gauss or Chebyshev points) for solving optimal control problems. This means that the solution takes the form of a <a href="/facts/Polynomial/Lzak8VVx">Polynomial</a>, and this polynomial satisfies the DAE and the path <a href="/facts/Constraint_(mathematics)/QLFIFlFX">constraints</a> at the collocation points.
In general PROPT has the following main functions:

<ul><li>Computation of the constant <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a> used for the <a href="/facts/Derivative/7Ito70Dh">differentiation</a> and <a href="/facts/Integral/V7lyV997">integration</a> of the polynomials used to approximate the solution to the <a href="/facts/Trajectory_optimization/cjQAMobd">Trajectory optimization</a> problem.</li>
<li>Source transformation to turn user-supplied <a href="/facts/Expression_(mathematics)/MPrMlYbE">expressions</a> into MATLAB code for the cost function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 and constraint function 
 
 
 
 c
 
 
 {\displaystyle c}
 
 that are passed to a <a href="/facts/Nonlinear_programming/cSXGuX2m">Nonlinear programming</a> solver in <a href="/facts/TOMLAB/YtuEVte9">TOMLAB</a>. The source transformation package TomSym automatically generates first and second order derivatives.</li>
<li>Functionality for plotting and computing a variety of information for the solution to the problem.</li>
<li>Automatic detection of the following:
<ul><li>Linear and quadratic objective.</li>
<li>Simple bounds, linear and nonlinear constraints.</li>
<li>Non-optimized expressions.</li></ul></li>
<li>Integrated support for <a href="/facts/Smooth_function/mffL4ch2">non-smooth</a><a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> (hybrid) optimal control problems.</li>
<li>Module for automatic scaling of difficult space related problem.</li>
<li>Support for binary and integer variables, controls or states.</li></ul>
<h2 id="modeling">Modeling</h2>
The PROPT system uses the TomSym symbolic source transformation engine to model optimal control problems. It is possible to define <a href="/facts/Dependent_and_independent_variables/dINfzIF2">independent</a> variables, dependent functions, scalars and constant parameters:

 toms tf
 toms t
 p = tomPhase('p', t, 0, tf, 30);
 x0 = {tf == 20};
 cbox = {10 <= tf <= 40};

toms z1
 cbox = {cbox; 0 <= z1 <= 500};
 x0 = {x0; z1 == 0};

ki0 = [1e3; 1e7; 10; 1e-3];

<h3>States and controls</h3>
States and controls only differ in the sense that states need be continuous between phases.

 tomStates x1
 x0 = {icollocate({x1 == 0})};

tomControls u1
 cbox = {-2 <= collocate(u1) <= 1};
 x0 = {x0; collocate(u1 == -0.01)};

<h3>Boundary, path, event and integral constraints</h3>
A variety of boundary, path, event and integral constraints are shown below:

 cbnd = initial(x1 == 1); % Starting point for x1
 cbnd = final(x1 == 1); % End point for x1
 cbnd = final(x2 == 2); % End point for x2
 pathc = collocate(x3 >= 0.5); % Path constraint for x3
 intc = {integrate(x2) == 1}; % Integral constraint for x2
 cbnd = final(x3 >= 0.5); % Final event constraint for x3
 cbnd = initial(x1 <= 2.0); % Initial event constraint x1

<h2 id="single-phase-optimal-control-example">Single-phase optimal control example</h2>
Van der Pol Oscillator <a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
Minimize:

 
 
 
 
 
 
 
 
 J
 
 x
 ,
 t
 
 
 
 
 =
 
 
 
 x
 
 3
 
 
 (
 
 t
 
 f
 
 
 )
 
 
 
 
 
 
 {\displaystyle {\begin{matrix}J_{x,t}&=&x_{3}(t_{f})\\\end{matrix}}}

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
 
 
 
 
 
 
 
 
 {\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):

toms t
p = tomPhase('p', t, 0, 5, 60);
setPhase(p);

tomStates x1 x2 x3
tomControls u

% Initial guess
x0 = {icollocate({x1 == 0; x2 == 1; x3 == 0})
    collocate(u == -0.01)};

% Box constraints
cbox = {-10 <= icollocate(x1) <= 10
 -10 <= icollocate(x2) <= 10
 -10 <= icollocate(x3) <= 10
 -0.3 <= collocate(u) <= 1};

% Boundary constraints
cbnd = initial({x1 == 0; x2 == 1; x3 == 0});

% ODEs and path constraints
ceq = collocate({dot(x1) == (1-x2.^2).*x1-x2+u
    dot(x2) == x1; dot(x3) == x1.^2+x2.^2+u.^2});

% Objective
objective = final(x3);

% Solve the problem
options = struct;
options.name = 'Van Der Pol';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);

<h2 id="multi-phase-optimal-control-example">Multi-phase optimal control example</h2>
One-dimensional rocket <a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> with free end time and undetermined phase shift
Minimize:

 
 
 
 
 
 
 
 
 J
 
 x
 ,
 t
 
 
 
 
 =
 
 
 t
 C
 u
 t
 
 
 
 
 
 
 {\displaystyle {\begin{matrix}J_{x,t}&=&tCut\\\end{matrix}}}

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
 
 
 
 
 
 
 
 
 {\displaystyle {\begin{cases}{\frac {dx_{1}}{dt}}=x_{2}\\{\frac {dx_{2}}{dt}}=a-g\ (0<t<=tCut)\\{\frac {dx_{2}}{dt}}=-g\ (tCut<t<t_{f})\\x(t_{0})=[0\ 0]\\g=1\\a=2\\x_{1}(t_{f})=100\\\end{cases}}}

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

toms t
toms tCut tp2
p1 = tomPhase('p1', t, 0, tCut, 20);
p2 = tomPhase('p2', t, tCut, tp2, 20);

tf = tCut+tp2;

x1p1 = tomState(p1,'x1p1');
x2p1 = tomState(p1,'x2p1');
x1p2 = tomState(p2,'x1p2');
x2p2 = tomState(p2,'x2p2');

% Initial guess
x0 = {tCut==10
    tf==15
    icollocate(p1,{x1p1 == 50*tCut/10;x2p1 == 0;})
    icollocate(p2,{x1p2 == 50+50*t/100;x2p2 == 0;})};

% Box constraints
cbox = {
 1 <= tCut <= tf-0.00001
 tf <= 100
 0 <= icollocate(p1,x1p1)
 0 <= icollocate(p1,x2p1)
 0 <= icollocate(p2,x1p2)
 0 <= icollocate(p2,x2p2)};

% Boundary constraints
cbnd = {initial(p1,{x1p1 == 0;x2p1 == 0;})
    final(p2,x1p2 == 100)};

% ODEs and path constraints
a = 2; g = 1;
ceq = {collocate(p1,{
    dot(p1,x1p1) == x2p1
    dot(p1,x2p1) == a-g})
    collocate(p2,{
    dot(p2,x1p2) == x2p2
    dot(p2,x2p2) == -g})};

% Objective
objective = tCut;

% Link phase
link = {final(p1,x1p1) == initial(p2,x1p2)
    final(p1,x2p1) == initial(p2,x2p2)};

%% Solve the problem
options = struct;
options.name = 'One Dim Rocket';
constr = {cbox, cbnd, ceq, link};
solution = ezsolve(objective, constr, x0, options);

<h2 id="parameter-estimation-example">Parameter estimation example</h2>
Parameter estimation problem <a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
Minimize:

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

 
 
 
 
 
 {
 
 
 
 
 
 
 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
 
 
 
 
 
 
 
 
 {\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:

toms t p1 p2
x1meas = [0.264;0.594;0.801;0.959];
tmeas = [1;2;3;5];

% Box constraints
cbox = {-1.5 <= p1 <= 1.5
 -1.5 <= p2 <= 1.5};

%% Solve the problem, using a successively larger number collocation points
for n=[10 40]
    p = tomPhase('p', t, 0, 6, n);
    setPhase(p);
    tomStates x1 x2

% Initial guess
    if n == 10
        x0 = {p1 == 0; p2 == 0};
    else
        x0 = {p1 == p1opt; p2 == p2opt
            icollocate({x1 == x1opt; x2 == x2opt})};
    end

% Boundary constraints
    cbnd = initial({x1 == p1; x2 == p2});

% ODEs and path constraints
    x1err = sum((atPoints(tmeas,x1) - x1meas).^2);
    ceq = collocate({dot(x1) == x2; dot(x2) == 1-2*x2-x1});

% Objective
    objective = x1err;

%% Solve the problem
    options = struct;
    options.name   = 'Parameter Estimation';
    options.solver = 'snopt';
    solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);

% Optimal x, p for starting point
    x1opt = subs(x1, solution);
    x2opt = subs(x2, solution);
    p1opt = subs(p1, solution);
    p2opt = subs(p2, solution);
end

<h2 id="optimal-control-problems-supported">Optimal control problems supported</h2>
<ul><li>Aerodynamic trajectory control<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a></li>
<li><a href="/facts/Bang-bang_control/YpKToXYI">Bang-bang control</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a></li>
<li><a href="/facts/Chemical_engineering/Fb64amQf">Chemical engineering</a><a class="footnote-ref" id="fnref:8" href="#fn:8">8</a></li>
<li><a href="/facts/Dynamical_system/782gREl5">Dynamic systems</a><a class="footnote-ref" id="fnref:9" href="#fn:9">9</a></li>
<li>General optimal control</li>
<li>Large-scale linear control<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a></li>
<li>Multi-phase system control<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a></li>
<li><a href="/facts/Mechanical_engineering/cbId4fye">Mechanical engineering</a> design<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a></li>
<li>Nondifferentiable control<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a></li>
<li>Parameters estimation for dynamic systems<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a></li>
<li><a href="/facts/Singular_control/2U6uoFxM">Singular control</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="http://tomopt.com/">TOMLAB</a> - Developer and distributor of the software.</li>
<li><a href="http://tomsym.com/">TomSym</a> - Source transformation engine used in software.</li>
<li><a href="http://tomdyn.com/">PROPT</a> - Home page for PROPT.</li>
<li><a href="http://tomopt.com/tomlab/products/snopt/solvers/SNOPT.php">SNOPT</a> - Default solver used in PROPT.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Rutquist, Per; M. M. Edvall (June 2008). PROPT - Matlab Optimal Control Software (PDF). Pullman, WA: Tomlab Optimization Inc. <a href="http://tomopt.com/docs/TOMLAB_PROPT.pdf" target="_blank">http://tomopt.com/docs/TOMLAB_PROPT.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Banga, J. R.; Balsa-Canto, E.; Moles, C. G.; Alonso, A. A. (2003). "Dynamic optimization of bioprocesses: efficient and robust numerical strategies". Journal of Biotechnology. {{cite journal}}: Cite journal requires |journal= (help) <a href="/wiki/Template:Cite_journal" target="_blank">/wiki/Template:Cite_journal</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">"Van Der Pol Oscillator - Matlab Solution", PROPT Home Page June, 2008. <a href="http://tomdyn.com/examples/vanDerPol.html" target="_blank">http://tomdyn.com/examples/vanDerPol.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">"One Dimensional Rocket Launch (2 Free Time)", PROPT Home Page June, 2008. <a href="http://tomdyn.com/examples/onedimRocket.html" target="_blank">http://tomdyn.com/examples/onedimRocket.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">"Matlab Dynamic Parameter Estimation with PROPT", PROPT Home Page June, 2008. <a href="http://tomdyn.com/examples/parameterEstimation.html" target="_blank">http://tomdyn.com/examples/parameterEstimation.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Betts, J. (2007). "SOCS Release 6.5.0". THE BOEING COMPANY. {{cite journal}}: Cite journal requires |journal= (help) <a href="/wiki/Template:Cite_journal" target="_blank">/wiki/Template:Cite_journal</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Liang, J.; Meng, M.; Chen, Y.; Fullmer, R. (2003). "Solving Tough Optimal Control Problems by Network Enabled Optimization Server (NEOS)". School of Engineering, Utah State University USA, Chinene University of Hong Kong China. {{cite journal}}: Cite journal requires |journal= (help) <a href="/wiki/Template:Cite_journal" target="_blank">/wiki/Template:Cite_journal</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Carrasco, E. F.; Banga, J. R. (September 1998). "A HYBRID METHOD FOR THE OPTIMAL CONTROL OF CHEMICAL PROCESSES". University of Wales, Swansea, UK: UKACC International Conference on CONTROL 98. {{cite journal}}: Cite journal requires |journal= (help) <a href="/wiki/Template:Cite_journal" target="_blank">/wiki/Template:Cite_journal</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Vassiliadis, V. S.; Banga, J. R.; Balsa-Canto, E. (1999). "Second-order sensitivities of general dynamic systems with application to optimal control problems". Chemical Engineering Science. 54 (17): 3851–3860. Bibcode:1999ChEnS..54.3851V. doi:10.1016/S0009-2509(98)00432-1. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Luus, R. (2002). Iterative dynamic programming. Chapman and Hall/CRC. <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Fabien, B. C. (1998). "A Java Application for the Solution of Optimal Control Problems". Stevens Way, Box 352600 Seattle, WA 98195, USA: Mechanical Engineering, University of Washington. {{cite journal}}: Cite journal requires |journal= (help)CS1 maint: location (link) <a href="/wiki/Template:Cite_journal" target="_blank">/wiki/Template:Cite_journal</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Jennings, L. S.; Fisher, M. E. (2002). "MISER3: Optimal Control Toolbox User Manual, Matlab Beta Version 2.0". Nedlands, WA 6907, Australia: Department of Mathematics, The University of Western Australia. {{cite journal}}: Cite journal requires |journal= (help)CS1 maint: location (link) <a href="/wiki/Template:Cite_journal" target="_blank">/wiki/Template:Cite_journal</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Banga, J. R.; Seider, W. D. (1996). Floudas, C. A.; Pardalos, P. M. (eds.). Global Optimization of Chemical Processes using Stochastic Algorithms - State of the Art in Global Optimization: Computational Methods and Applications. Dordrecht, The Netherlands: Kluwer Academic Publishers. pp. 563–583. ISBN 0-7923-3838-3. <a href="0-7923-3838-3" target="_blank">0-7923-3838-3</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Dolan, E. D.; More, J. J. (January 2001). "Benchmarking Optimization Software with COPS". 9700 South Cass Avenue, Argonne, Illinois 60439: ARGONNE NATIONAL LABORATORY. {{cite journal}}: Cite journal requires |journal= (help)CS1 maint: location (link) <a href="/wiki/Template:Cite_journal" target="_blank">/wiki/Template:Cite_journal</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
</ol>

PROPT open-in-new

PROPT