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2014-02-20T11:07:09Z

MichaelJung: /* Problems (add) */

__NOTOC__
This wiki contains a '''benchmark library''' of '''mixed-integer optimal control problems'''. The main intention is to provide algorithm developers with a set of challenging problems to evaluate their numerical optimization methods. An important focus is given on '''reproducibility''' of optimal solutions.
As, in contrast to say linear programming, there are no standard formats for the formulation of such problems, and they often show completely different characteristics, these pages dedicate some space for a thorough description of problem and solutions.

A more detailed description of the underlying concepts of this library can be found in the article <bibref>Sager2011b</bibref>

Sager, S., 'A benchmark library of mixed-integer optimal control problems',
Mixed Integer Nonlinear Programming, Eds. J. Lee and S. Leyffer, 2012, pp. 631-670.

A [http://mathopt.de/PUBLICATIONS/Sager2012b.pdf preprint] pdf is available.


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==[[:Category:Problem characterization|Problem characterization]] <span style="font-variant:small-caps" style="font-size:10px">[[Help:Adding a problem characterization|(add)]]</span>==

via ''[[:Category:Model characterization|mathematical model]]'' - via ''[[:Category:Solution characterization|optimal solution]]'' - via ''[[:Category:Application|application area]]''

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==[[:Category:MIOCP|Problems]] <span style="font-variant:small-caps" style="font-size:10px">[[Help:Adding a Problem|(add)]]</span>==

''[[Lotka Volterra fishing problem]] - [[Lotka Experimental Design]] - [[F-8 aircraft]] - [[Annihilation of calcium oscillations | Calcium]] - [[Annihilation of calcium oscillations with PLC activation inhibition| Calcium 2]]''

''[[Supermarket refrigeration system]] - [[Car testdrive]] - [[Fuller's problem]] - [[Bang-bang approximation of a traveling wave | 1D wave]] - [[Subway ride | Subway]] - [[Truck cruise control]]''
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==[[:Category:Help|Help]] <span style="font-variant:small-caps" style="font-size:10px">[[User:SebastianSager|(contact)]]</span>==

''[[Help:How to contribute|How to contribute]]'' - ''[[Help:How to cite|How to cite]]'' - ''[[Help:Using LaTeX|LaTeX]]'' - ''[[Help:Description guidelines | Guidelines]]
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Truck cruise control

2014-02-20T11:06:53Z

MichaelJung: Created page with "{{Dimensions |nd = 1 |nx = 2 |nw = 1 |nre = 2 }} Category:Model characterization The truck cruise control problem uses a quite simple truck mo..."

{{Dimensions
|nd = 1
|nx = 2
|nw = 1
|nre = 2
}}

[[Category:Model characterization]]

The truck cruise control problem uses a quite simple truck model. It realizes several objective criteria as fuel consumption, traveling time and driver comfort are competing against each other. The problem is formulated in terms of the traveled distance.

== Mathematical formulation (under construction) ==

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

<math>
\begin{alignat}{2}
& \displaystyle \min_{x, u, w} && \lambda_\text{dev} \Phi_\text{dev} + \lambda_\text{fuel} \Phi_\text{fuel} + \lambda_\text{comf} \Phi_\text{comf} \\[1.5ex]
& \mbox{s.t.}
& \dot{x}_0 & = \frac{1}{m x_0} \left( M_\text{acc} - \frac{i_A}{r_\text{stat}}M_\text{brk} - M_\text{air} - M_\text{road} \right) \\
&& \dot{x}_1 & = \frac{1}{x_0} Q_\text{fuel} (neng, u_0) \\
&& \dot{x}_2 & = -4.208 \; x_0 - 0.396 \; x_2 - 0.47 \; x_0^2 - 3.564 \; x_0^3 \\
&& & \quad - 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{alignat}
</math>

<math>\textstyle x_0</math> is the velocity of the truck in m/s, <math>\textstyle x_1</math> is the fuel consumption in liters, the control function <math>\textstyle u_0</math> is the induced engine torque in Nm, <math>\textstyle u_1</math> is the combined engine brake torque. This model goes back to Garrard<bibref>Garrard1977</bibref>.

The equality constraints <math>r^{\text{eq}}(\cdot)</math> will often fix the initial values, i.e., <math>x(0) = x_0</math>, or impose a [[:Category:Periodic | periodicity]] constraint.

== Extensions (under construction) ==
* For some problems the functions may as well depend explicitely on the time <math>t</math>.
* The differential equations might depend on [[:Category:State dependent switches | state-dependent switches]].
* The variables may include [[:Category:Boolean variables | boolean variables]].
* The underlying process might be a [[:Category:Multistage process | multistage process]].
* The dynamics might be [[:Category:Unstable | unstable]].
* There might be an underlying [[:Category:Network topology | network topology]].
* The integer control functions might have been (re)formulated by means of an [[:Category:Outer convexification|outer convexification]].

Note that a Lagrange term <math>\int_{t_0}^{t_f} L( x(t), u(t), v(t), q, \rho) \; \mathrm{d} t</math> can be transformed into a Mayer-type objective functional.

Main Page

2014-02-20T10:20:52Z

MichaelJung: /* Problems (add) */

__NOTOC__
This wiki contains a '''benchmark library''' of '''mixed-integer optimal control problems'''. The main intention is to provide algorithm developers with a set of challenging problems to evaluate their numerical optimization methods. An important focus is given on '''reproducibility''' of optimal solutions.
As, in contrast to say linear programming, there are no standard formats for the formulation of such problems, and they often show completely different characteristics, these pages dedicate some space for a thorough description of problem and solutions.

A more detailed description of the underlying concepts of this library can be found in the article <bibref>Sager2011b</bibref>

Sager, S., 'A benchmark library of mixed-integer optimal control problems',
Mixed Integer Nonlinear Programming, Eds. J. Lee and S. Leyffer, 2012, pp. 631-670.

A [http://mathopt.de/PUBLICATIONS/Sager2012b.pdf preprint] pdf is available.


<table bgcolor="#EEEEFF" width="100%" cellspacing="10px">
<tr valign="top">
<td width="80%" bgcolor="#EEEEFF">
==[[:Category:News|News]] <span style="font-variant:small-caps" style="font-size:10px">[[Current News|(add)]]</span>==

{{:Current News}}

</td>
</tr>
</table>

<table width="100%">


<tr valign="top">
<td width="50%">
==[[:Category:Problem characterization|Problem characterization]] <span style="font-variant:small-caps" style="font-size:10px">[[Help:Adding a problem characterization|(add)]]</span>==

via ''[[:Category:Model characterization|mathematical model]]'' - via ''[[:Category:Solution characterization|optimal solution]]'' - via ''[[:Category:Application|application area]]''

''[[:Category:ODE model|ODE model]]'' - ''[[:Category:Bang bang|Bang bang]]'' - ''[[:Category:Chattering|Chattering]]'' - ''[[:Category:AMPL|AMPL]]'' -
''[[:Category:AMPL/TACO|AMPL using TACO]]'' - ''[[:Category:C|C code]]'' - ''[[:Category:Optimica|optimica]]'' - ''[[:Category:Periodic|Periodicity]]''
</td>


<td width="50%" >

==[[:Category:MIOCP|Problems]] <span style="font-variant:small-caps" style="font-size:10px">[[Help:Adding a Problem|(add)]]</span>==

''[[Lotka Volterra fishing problem]] - [[Lotka Experimental Design]] - [[F-8 aircraft]] - [[Annihilation of calcium oscillations | Calcium]] - [[Annihilation of calcium oscillations with PLC activation inhibition| Calcium 2]]''

''[[Supermarket refrigeration system]] - [[Car testdrive]] - [[Fuller's problem]] - [[Bang-bang approximation of a traveling wave | 1D wave]] - [[Subway ride | Subway]] - [Truck cruise control]''
</td>
</tr>

<tr valign="top">


<td width="50%" >

==[[:Category:Help|Help]] <span style="font-variant:small-caps" style="font-size:10px">[[User:SebastianSager|(contact)]]</span>==

''[[Help:How to contribute|How to contribute]]'' - ''[[Help:How to cite|How to cite]]'' - ''[[Help:Using LaTeX|LaTeX]]'' - ''[[Help:Description guidelines | Guidelines]]
</td>


<td width="50%" >

==[[:Category:Community|Community]] <span style="font-variant:small-caps" style="font-size:10px">[[Help:Adding a User|(add)]]</span>==

[[:Category:Community|Contributors]] - ''[[External Links]]''

''Feel encouraged to participate!''

</td>
</tr>

</table>

[[Category:Main]]