View source for Egerstedt standard problem

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-->The '''Egerstedt standard problem''' is the problem is of an academic nature and was proposed by Egerestedt to highlight the features of an Hybrid System algorithm in 2006 <bib id="Egerstedt2006" />. It has been used since then in many MIOCP research studies (e.g. <bib id="Jung2013" />) for benchmarking of MIOCP algorithms.


== Mathematical formulation ==

The mixed-integer optimal control problem after partial outer convexification is given by

<p>
<math>
\begin{array}{llclr}
 \displaystyle \min_{x, \omega} & x_3(t_f)   \\[1.5ex]
 \mbox{s.t.} 
 & \dot{x}_1 & = & -x_1\omega_1 + (x_1+x_2)\omega_2+(x_1-x_2)\omega_3, \\
 & \dot{x}_2 & = & (x_1+2x_2)\omega_1+(x_1-2x_2)\omega_2+(x_1+x_2)\omega_3, \\
 & \dot{x}_3 & = & x_1^2+x_2^2,  \\[1.5ex]
 & x(0) &=& (0.5, 0.5, 0)^T, \\
 & x_2(t) & \geq & 0.4, \\
 & 1 &=& \sum\limits_{i=1}^3\omega_i(t), \\
 & \omega(t) &\in&  \{0, 1\}, 
\end{array} 
</math>
</p>
for <math>t \in [t_0, t_f]=[0,1] </math>.

== Reference Solutions ==

If the problem is relaxed, i.e., we demand that <math>w(t)</math> be in the continuous interval <math>[0, 1]</math> instead of the binary choice <math>\{0,1\}</math>, the optimal solution can be determined by  using a direct method such as collocation or Bock's direct multiple shooting method. 

The optimal objective value of the relaxed problem with  <math> n_t=6000, \, n_u=40  </math> is <math>x_3(t_f)=0.995906234</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>x_3(t_f) =3.20831942</math>.  The binary control solution was evaluated in the MIOCP by using a Merit function with additional Lagrange term <math> 100 \max\limits_{t\in[0,1]}\{0,0.4-x_2(t)\}  </math>.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
 Image:EgerstedtRelaxed 6000 150 1.png| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation)  and <math>n_t=6000, \, n_u=40</math>.
 Image:EgerstedtCIA 6000 150 1.png| Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=6000, \, n_u=40</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
</gallery>



== Source Code ==

Model description is available in
* [[:Category:AMPL | AMPL code]] at [[Egerstedt standard problem (AMPL)]]



== References ==
<biblist />

<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->
[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Tracking objective]]
[[Category:Sensitivity-seeking arcs]]



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We present numerical results for a benchmark MIOCP from a previous study [157] with the
addition of switching constraints. In its original form, the problem was:


After partial outer convexification with respect to the integer control v, the binary
convexified counterpart problem reads