mintOC - User contributions [en]

User:MartinSchlueter

2009-10-28T16:20:14Z

MartinSchlueter: /* Personal */

== Personal ==
{|
|-
| '''Name''': || Martin Schlueter
|-
| '''Affiliation''': || University of Birmingham
|-
| '''Background''': || Mathematics, Optimization
|}

== Contact ==
{|
|-
| '''Address''': || Watson Building, Birmingham B15 2TT, United Kingdom
|-
| '''Web''': || www.midaco-solver.com [http://www.midaco-solver.com]
|-
| '''Email''': || ''click here'' [http://www.midaco-solver.com/contact.html]
|}


[[Category:Community]]

User:MartinSchlueter

2009-10-28T16:19:32Z

MartinSchlueter: /* Personal */

== Personal ==
{|
|-
| '''Name''': || Martin Schlueter
|-
| '''Affiliation''': || University of Birmingham
|-
| '''Background''': || Global Optimization
|}

== Contact ==
{|
|-
| '''Address''': || Watson Building, Birmingham B15 2TT, United Kingdom
|-
| '''Web''': || www.midaco-solver.com [http://www.midaco-solver.com]
|-
| '''Email''': || ''click here'' [http://www.midaco-solver.com/contact.html]
|}


[[Category:Community]]

F-8 aircraft

2009-04-11T10:04:39Z

MartinSchlueter: /* Reference solutions */

{{Dimensions
|nd = 1
|nx = 3
|nw = 1
|nre = 6
}}

The F-8 aircraft control problem is based on a very simple aircraft model. The control problem was introduced by Kaya and Noakes and aims at controlling an aircraft in a time-optimal way from an initial state to a terminal state.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. The interior point equality conditions fix both initial and terminal values of the differential states.

The optimal, relaxed control function shows [[:Category:Bang bang|bang bang]] behavior. The problem is furthermore interesting as it should be reformulated equivalently.

== Mathematical formulation ==

For <math>t \in [0, T]</math> almost everywhere the mixed-integer optimal control problem is given by

<math>
\begin{array}{llcl}
\displaystyle \min_{x, w, T} & T \\[1.5ex]
\mbox{s.t.} & \dot{x}_0 &=& -0.877 \; x_0 + x_2 - 0.088 \; x_0 \; x_2 + 0.47 \; x_0^2 - 0.019 \; x_1^2 - x_0^2 \; x_2 + 3.846 \; x_0^3 \\
&&& - 0.215 \; w + 0.28 \; x_0^2 \; w + 0.47 \; x_0 \; w^2 + 0.63 \; w^3 \\
& \dot{x}_1 &=& x_2 \\
& \dot{x}_2 &=& -4.208 \; x_0 - 0.396 \; x_2 - 0.47 \; x_0^2 - 3.564 \; x_0^3 \\
&&& - 20.967 \; w + 6.265 \; x_0^2 \; w + 46 \; x_0 \; w^2 + 61.4 \; w^3 \\
& x(0) &=& (0.4655,0,0)^T, \\
& x(T) &=& (0,0,0)^T, \\
& w(t) &\in& \{-0.05236,0.05236\}.
\end{array}
</math>

<math>x_0</math> is the angle of attack in radians, <math>x_1</math> is the pitch angle, <math>x_2</math> is the pitch rate in rad/s, and the control function <math>w = w(t)</math> is the tail deflection angle in radians. This model goes back to Garrard<bibref>Garrard1977</bibref>.

In the control problem, both initial and terminal values of the differential states are fixed.

== Reformulation ==

The control w(t) is restricted to take values from a finite set only. Hence, the control problem can be reformulated equivalently to

<math>
\begin{array}{llcl}
\displaystyle \min_{x, w, T} & T \\[1.5ex]
\mbox{s.t.} & \dot{x}_0 &=& -0.877 \; x_0 + x_2 - 0.088 \; x_0 \; x_2 + 0.47 \; x_0^2 - 0.019 \; x_1^2 - x_0^2 \; x_2 + 3.846 \; x_0^3 \\
&&& - \left( 0.215 \; \xi - 0.28 \; x_0^2 \; \xi - 0.47 \; x_0 \; \xi^2 - 0.63 \; \xi^3 \right) \; w \\
&&& - \left( - 0.215 \; \xi + 0.28 \; x_0^2 \; \xi - 0.47 \; x_0 \; \xi^2 + 0.63 \; \xi^3 \right) \; (1 - w) \\
& \dot{x}_1 &=& x_2 \\
& \dot{x}_2 &=& -4.208 \; x_0 - 0.396 \; x_2 - 0.47 \; x_0^2 - 3.564 \; x_0^3 \\
&&& - \left( 20.967 \; \xi - 6.265 \; x_0^2 \; \xi -46 \; x_0 \; \xi^2 - 61.4 \; \xi^3 \right) \; w \\
&&& - \left( - 20.967 \; \xi + 6.265 \; x_0^2 \; \xi -46 \; x_0 \; \xi^2 + 61.4 \; \xi^3 \right) \; (1 - w) \\
& x(0) &=& (0.4655,0,0)^T, \\
& x(T) &=& (0,0,0)^T, \\
& w(t) &\in& \{0,1\},
\end{array}
</math>

with <math>\xi = 0.05236</math>. Note that there is a bijection between optimal solutions of the two problems.

== Reference solutions ==

We provide here a comparison of different solutions reported in the literature. The numbers show the respective lengths <math>t_i - t_{i-1}</math> of the switching arcs with the value of <math>w(t)</math> on the upper or lower bound (given in the second column). ''Claim'' denotes what is stated in the respective publication, ''Simulation'' shows values obtained by a simulation with a Runge-Kutta-Fehlberg method of 4th/5th order and an integration tolerance of <math>10^{-8}</math>.

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Arc !! w(t) !! Lee et al.<bibref>Lee1997a</bibref> !! Kaya<bibref>Kaya2003</bibref> !! Sager<bibref>Sager2005</bibref> !! [[User:MartinSchlueter | Schlueter]]/[[User:Matthias.gerdts | Gerdts]] !! Sager
|-
| align=center | 1 || align=center | 1 || 0.00000 || 0.10292 || 0.10235 || 0.0 || 1.13492
|-
| align=center | 2 || align=center | 0 || 2.18800 || 1.92793 || 1.92812 || 0.608750 || 0.34703
|-
| align=center | 3 || align=center | 1 || 0.16400 || 0.16687 || 0.16645 || 3.136514 || 1.60721
|-
| align=center | 4 || align=center | 0 || 2.88100 || 2.74338 || 2.73071 || 0.654550 || 0.69169
|-
| align=center | 5 || align=center | 1 || 0.33000 || 0.32992 || 0.32994 || 0.0 || 0.0
|-
| align=center | 6 || align=center | 0 || 0.47200 || 0.47116 || 0.47107 || 0.0 || 0.0
|-
| Claim: Infeasibility || align=center | - || 1.00E-10 || 7.30E-06 || 5.90E-06 || 3.29169e-06 || 2.21723e-07
|-
| Claim: Objective || align=center | - || 6.03500 || 5.74217 || 5.72864 || 4.39981 || 3.78086
|- style="font-style:italic;background-color:#ffffcc;"
! Simulation: Infeasibility || align=center | - || 1.75E-03 || 1.64E-03 || 5.90E-06 || 3.29169e-06 || 2.21723e-07
|- style="font-style:italic;background-color:#ffffcc;"
! Simulation: Objective || align=center | - || 6.03500 || 5.74218 || 5.72864 || 4.39981 || 3.78086
|}

The best known optimal objective value of this problem given is given by <math>T = 3.78086</math>. The corresponding solution is shown in the rightmost plot. The solution of bang-bang type switches three times, starting with <math>w(t) = 1</math>.

<gallery caption="Reference solution plots" widths="240px" heights="167px" perrow="3">
Image:f8aircraftRelaxedCoarse.png| Locally optimal relaxed control on a coarse control discretization grid and corresponding differential states.
Image:f8aircraftRelaxedFine.png| Locally optimal relaxed control on a fine, adaptively chosen control discretization grid and corresponding differential states.
Image:f8aircraftInteger.png| Integer control and corresponding differential states from Sager<bibref>Sager2005</bibref>.

Image:f8aircraftSchlueter.png| Integer control and corresponding differential states from [[User:MartinSchlueter | Schlueter]]/[[User:Matthias.gerdts | Gerdts]] solution.
Image:f8aircraftSager2009.png| Best known integer control and corresponding differential states from Sager solution.
</gallery>

== Source Code ==

=== C ===

The differential equations in C code:
<source lang="cpp">
double x1 = xd[0];
double x2 = xd[1];
double x3 = xd[2];

double u0 = -0.05236;
double u1 = 0.05236;
double f00, f10, f20;
double f01, f11, f21;

f00 = - 0.877*x1 + x3 - 0.088*x1*x3 + 0.47*x1*x1 - 0.019*x2*x2
- x1*x1*x3 + 3.846*x1*x1*x1 - 0.215*u0 + 0.28*x1*x1*u0 + 0.47*x1*u0*u0 + 0.63*u0*u0*u0;
f10 = x3;
f20 = - 4.208*x1 - 0.396*x3 - 0.47*x1*x1 - 3.564*x1*x1*x1
- 20.967*u0 + 6.265*x1*x1*u0 + 46*x1*u0*u0 + 61.4*u0*u0*u0;

f01 = - 0.877*x1 + x3 - 0.088*x1*x3 + 0.47*x1*x1 - 0.019*x2*x2
- x1*x1*x3 + 3.846*x1*x1*x1 - 0.215*u1 + 0.28*x1*x1*u1 + 0.47*x1*u1*u1 + 0.63*u1*u1*u1;
f11 = x3;
f21 = - 4.208*x1 - 0.396*x3 - 0.47*x1*x1 - 3.564*x1*x1*x1
-20.967*u1 + 6.265*x1*x1*u1 + 46*x1*u1*u1 + 61.4*u1*u1*u1;

rhs[0] = u[0]*f01 + (1-u[0])*f00;
rhs[1] = u[0]*f11 + (1-u[0])*f10;
rhs[2] = u[0]*f21 + (1-u[0])*f20;
</source>

== Miscellaneous and Further Reading ==
See <bibref>Kaya2003</bibref> and <bibref>Sager2005</bibref> for details.

== References ==
<bibreferences/>


[[Category:Bang bang]]
[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Aeronautics]]

User:MartinSchlueter

2009-04-11T10:00:49Z

MartinSchlueter: /* Contact */

== Personal ==
{|
|-
| '''Name''': || Martin Schlueter
|-
| '''Affiliation''': || University of Birmingham
|-
| '''Background''': || Optimization, Stochastic Metaheuristics
|}

== Contact ==
{|
|-
| '''Address''': || Watson Building, Birmingham B15 2TT, United Kingdom
|-
| '''Web''': || www.midaco-solver.com [http://www.midaco-solver.com]
|-
| '''Email''': || ''click here'' [http://www.midaco-solver.com/contact.html]
|}


[[Category:Community]]
[[Category:MIOCP]]

User:MartinSchlueter

2009-04-11T10:00:29Z

MartinSchlueter: /* Contact */

== Personal ==
{|
|-
| '''Name''': || Martin Schlueter
|-
| '''Affiliation''': || University of Birmingham
|-
| '''Background''': || Optimization, Stochastic Metaheuristics
|}

== Contact ==
{|
|-
| '''Address''': || Watson Building, Birmingham B15 2TT, United Kingdom
|-
| '''Web''': || www.midaco-solver.com [http://www.midaco-solver.com]
|-
| '''Email''': || ''see link'' [http://www.midaco-solver.com/contact.html]
|}


[[Category:Community]]
[[Category:MIOCP]]

User:MartinSchlueter

2009-04-11T09:59:28Z

MartinSchlueter: /* Contact */

== Personal ==
{|
|-
| '''Name''': || Martin Schlueter
|-
| '''Affiliation''': || University of Birmingham
|-
| '''Background''': || Optimization, Stochastic Metaheuristics
|}

== Contact ==
{|
|-
| '''Address''': || Watson Building, Birmingham B15 2TT, United Kingdom [http://www.midaco-solver.com/contact.html]
|-
| '''Web''': || www.midaco-solver.com [http://www.midaco-solver.com]
|-
| '''Email''': || [http://www.midaco-solver.com/contact.html]
|}


[[Category:Community]]
[[Category:MIOCP]]

User:MartinSchlueter

2009-04-11T09:58:09Z

MartinSchlueter: /* Personal */

== Personal ==
{|
|-
| '''Name''': || Martin Schlueter
|-
| '''Affiliation''': || University of Birmingham
|-
| '''Background''': || Optimization, Stochastic Metaheuristics
|}

== Contact ==
{|
|-
| '''Address''': || University of Birmingham, Birmingham B15 2TT, United Kingdom
|-
| '''Web''': || www.midaco-solver.com [http://www.midaco-solver.com]
|-
| '''Email''': || [http://www.midaco-solver.com/contact.html]
|}


[[Category:Community]]
[[Category:MIOCP]]

User:MartinSchlueter

2009-04-11T09:43:53Z

MartinSchlueter: /* Contact */

== Personal ==
{|
|-
| '''Name''': || Martin Schlueter
|-
| '''Affiliation''': || University of Birmingham, School of Mathematics
|-
| '''Background''': || Optimization, Stochastic Metaheuristics, Blackbox MINLP's
|}

== Contact ==
{|
|-
| '''Address''': || University of Birmingham, Birmingham B15 2TT, United Kingdom
|-
| '''Web''': || www.midaco-solver.com [http://www.midaco-solver.com]
|-
| '''Email''': || [http://www.midaco-solver.com/contact.html]
|}


[[Category:Community]]
[[Category:MIOCP]]

User:Matthias.gerdts

2009-04-11T09:36:03Z

MartinSchlueter: /* Contact */

__NOTOC__


== Personal ==
{|
|-
| '''Name''': || Prof. Dr. Matthias Gerdts
|-
| '''Affiliation''': || University of Wuerzburg
|-
| '''Background''': || mathematics, optimal control, optimization
|}

== Contact ==
{|
|-
| '''Address''': || Institute of Mathematics, University of Wuerzburg, Am Hubland, 97074 Wuerzburg, Germany
|-
| '''Web''': || [http://www.mathematik.uni-wuerzburg.de/~gerdts]
|-
| '''Email''': || gerdts@mathematik.uni-wuerzburg.de
|}


[[Category:Community]]
[[Category:MIOCP]]

User:MartinSchlueter

2009-04-11T09:34:03Z

MartinSchlueter: /* Personal */

User:MartinSchlueter

2009-04-11T09:32:23Z

MartinSchlueter: /* Contact */

== Personal ==
{|
|-
| '''Name''': || Martin Schlueter
|-
| '''Affiliation''': || University of Birmingham
|-
| '''Background''': || Optimization, Stochastic Metaheuristics, Blackbox MINLP's
|}

== Contact ==
{|
|-
| '''Address''': || University of Birmingham, Birmingham B15 2TT, United Kingdom
|-
| '''Web''': || www.midaco-solver.com [http://www.midaco-solver.com]
|-
| '''Email''': || [http://www.midaco-solver.com/contact.html]
|}

User:MartinSchlueter

2009-04-11T09:31:08Z

MartinSchlueter: /* Contact */

== Personal ==
{|
|-
| '''Name''': || Martin Schlueter
|-
| '''Affiliation''': || University of Birmingham
|-
| '''Background''': || Optimization, Stochastic Metaheuristics, Blackbox MINLP's
|}

== Contact ==
{|
|-
| '''Address''': || University of Birmingham, School of Mathematics, Birmingham B15 2TT, United Kingdom
|-
| '''Web''': || www.midaco-solver.com [http://www.midaco-solver.com]
|-
| '''Email''': || [http://www.midaco-solver.com/contact.html]
|}

User:MartinSchlueter

2009-04-11T09:30:07Z

MartinSchlueter: /* Personal */

== Personal ==
{|
|-
| '''Name''': || Martin Schlueter
|-
| '''Affiliation''': || University of Birmingham
|-
| '''Background''': || Optimization, Stochastic Metaheuristics, Blackbox MINLP's
|}

== Contact ==
{|
|-
| '''Address''': || University of Birmingham, School of Mathematics, Birmingham B15 2TT, United Kingdom
|-
| '''Web''': || www.midaco-solver.com [http://www.midaco-solver.com]
|-
| '''Email''': || [http://www.midaco-solver/contact.html]
|}

User:MartinSchlueter

2009-04-11T09:28:46Z

MartinSchlueter: /* Contact */

== Personal ==
{|
|-
| '''Name''': || Martin Schlueter
|-
| '''Affiliation''': || University of Birmingham
|-
| '''Background''': || optimization, stochastic metaheuristics, blackbox MINLP's
|}

== Contact ==
{|
|-
| '''Address''': || University of Birmingham, School of Mathematics, Birmingham B15 2TT, United Kingdom
|-
| '''Web''': || www.midaco-solver.com [http://www.midaco-solver.com]
|-
| '''Email''': || [http://www.midaco-solver/contact.html]
|}

User:MartinSchlueter

2009-04-11T09:26:16Z

MartinSchlueter: /* Contact */

== Personal ==
{|
|-
| '''Name''': || Martin Schlueter
|-
| '''Affiliation''': || University of Birmingham
|-
| '''Background''': || optimization, stochastic metaheuristics, blackbox MINLP's
|}

== Contact ==
{|
|-
| '''Address''': || University of Birmingham, Watson Building, Birmingham B15 2TT, United Kingdom
|-
| '''Web''': || www.midaco-solver.com [http://www.midaco-solver.com]
|-
| '''Email''': || info_AT_midaco-solver.com
|}

User:MartinSchlueter

2009-04-11T09:25:36Z

MartinSchlueter: /* Contact */

== Personal ==
{|
|-
| '''Name''': || Martin Schlueter
|-
| '''Affiliation''': || University of Birmingham
|-
| '''Background''': || optimization, stochastic metaheuristics, blackbox MINLP's
|}

== Contact ==
{|
|-
| '''Address''': || Watson Building, Birmingham B15 2TT, United Kingdom
|-
| '''Web''': || www.midaco-solver.com [http://www.midaco-solver.com]
|-
| '''Email''': || info_AT_midaco-solver.com
|}

User:MartinSchlueter

2009-04-11T09:24:42Z

MartinSchlueter: /* Contact */

== Personal ==
{|
|-
| '''Name''': || Martin Schlueter
|-
| '''Affiliation''': || University of Birmingham
|-
| '''Background''': || optimization, stochastic metaheuristics, blackbox MINLP's
|}

== Contact ==
{|
|-
| '''Address''': || Theoretical & Computational Optimization Group, School of Mathematics,
| University of Birmingham, Birmingham B15 2TT, United Kingdom
|-
| '''Web''': || www.midaco-solver.com [http://www.midaco-solver.com]
|-
| '''Email''': || info_AT_midaco-solver.com
|}

User:MartinSchlueter

2009-04-11T09:23:08Z

MartinSchlueter: /* Contact */

== Personal ==
{|
|-
| '''Name''': || Martin Schlueter
|-
| '''Affiliation''': || University of Birmingham
|-
| '''Background''': || optimization, stochastic metaheuristics, blackbox MINLP's
|}

== Contact ==
{|
|-
| '''Address''': || Theoretical & Computational Optimization Group, School of Mathematics,
| University of Birmingham, Birmingham B15 2TT, United Kingdom
|-
| '''Web''': || www.midaco-solver.com [http://www.midaco-solver.com]
|-
| '''Email''': || info_AT_midaco-solver.com
|}

User:MartinSchlueter

2009-04-11T09:22:47Z

MartinSchlueter: /* Contact */

== Personal ==
{|
|-
| '''Name''': || Martin Schlueter
|-
| '''Affiliation''': || University of Birmingham
|-
| '''Background''': || optimization, stochastic metaheuristics, blackbox MINLP's
|}

== Contact ==
{|
|-
| '''Address''': || Theoretical & Computational Optimization Group, School of Mathematics,
| University of Birmingham, Birmingham B15 2TT, United Kingdom
|-
| '''Web''': || www.midaco-solver.com [http://www.midaco-solver.com]
|-
| '''Email''': || info_AT_midaco-solver.com
|}

User:MartinSchlueter

2009-04-11T09:21:59Z

MartinSchlueter: /* Contact */

== Personal ==
{|
|-
| '''Name''': || Martin Schlueter
|-
| '''Affiliation''': || University of Birmingham
|-
| '''Background''': || optimization, stochastic metaheuristics, blackbox MINLP's
|}

== Contact ==
{|
|-
| '''Address''': || Theoretical & Computational Optimization Group, School of Mathematics,
|- University of Birmingham, Birmingham B15 2TT, United Kingdom
|-
| '''Web''': || www.midaco-solver.com [http://www.midaco-solver.com]
|-
| '''Email''': || info_AT_midaco-solver.com
|}

User:MartinSchlueter

2009-04-11T09:20:54Z

MartinSchlueter: /* Contact */

== Personal ==
{|
|-
| '''Name''': || Martin Schlueter
|-
| '''Affiliation''': || University of Birmingham
|-
| '''Background''': || optimization, stochastic metaheuristics, blackbox MINLP's
|}

== Contact ==
{|
|-
| '''Address''': || Theoretical & Computational Optimization Group, School of Mathematics,
University of Birmingham, Birmingham B15 2TT, United Kingdom
|-
| '''Web''': || www.midaco-solver.com [http://www.midaco-solver.com]
|-
| '''Email''': || info_AT_midaco-solver.com
|}

User:MartinSchlueter

2009-04-11T09:19:30Z

MartinSchlueter: Created page with ' == Personal == {| |- | '''Name''': || Martin Schlueter |- | '''Affiliation''': || University of Birmingham |- | '''Background''': || optimization, stochastic metaheuristics,…'

== Personal ==
{|
|-
| '''Name''': || Martin Schlueter
|-
| '''Affiliation''': || University of Birmingham
|-
| '''Background''': || optimization, stochastic metaheuristics, blackbox MINLP's
|}

== Contact ==
{|
|-
| '''Address''': || Theoretical & Computational Optimization Group, School of Mathematics,
University of Birmingham, Birmingham B15 2TT, United Kingdom
|-
| '''Web''': || [http://www.midaco-solver.com]
|-
| '''Email''': || martin.schlueter@midaco-solver.com
|}