Difference between revisions of "Egerstedt standard problem"
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| − | -->The '''Egerstedt standard | + | -->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. |
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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. | 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)= | + | 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"> | <gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2"> | ||
Latest revision as of 16:09, 19 September 2019
| Egerstedt standard problem | |
|---|---|
| State dimension: | 1 |
| Differential states: | 3 |
| Discrete control functions: | 3 |
| Path constraints: | 1 |
| Interior point equalities: | 3 |
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 [Egerstedt2006]Author: M. Egerstedt; Y. Wardi; H. Axelsson
Journal: IEEE Transactions on Automatic Control
Pages: 110--115
Title: Transition-time optimization for switched-mode dynamical systems
Volume: 51
Year: 2006
. It has been used since then in many MIOCP research studies (e.g. [Jung2013]Author: M. Jung; C. Kirches; S. Sager
Booktitle: Facets of Combinatorial Optimization -- Festschrift for Martin Gr\"otschel
Editor: M. J\"unger and G. Reinelt
Pages: 387--417
Publisher: Springer Berlin Heidelberg
Title: On Perspective Functions and Vanishing Constraints in Mixed-Integer Nonlinear Optimal Control
Url: http://www.mathopt.de/PUBLICATIONS/Jung2013.pdf
Year: 2013) for benchmarking of MIOCP algorithms.
Mathematical formulation
The mixed-integer optimal control problem after partial outer convexification is given by
![\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}](https://mintoc.de/images/math/2/0/8/208ecb2b5782a3106028f82d6458c1ba.png)
for ![t \in [t_0, t_f]=[0,1]](https://mintoc.de/images/math/3/4/e/34e97321e38e35c980a25b5da6ebb7b8.png)
.
Reference Solutions
If the problem is relaxed, i.e., we demand that 
be in the continuous interval ![[0, 1]](https://mintoc.de/images/math/c/c/f/ccfcd347d0bf65dc77afe01a3306a96b.png)
instead of the binary choice 
, 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 
is 
. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is 
. The binary control solution was evaluated in the MIOCP by using a Merit function with additional Lagrange term ![100 \max\limits_{t\in[0,1]}\{0,0.4-x_2(t)\}](https://mintoc.de/images/math/0/f/a/0fabc73d7f238f7afc86bde9b0bbf37f.png)
.
- Reference solution plots
Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and
.
Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and
. The relaxed controls were approximated by Combinatorial Integral Approximation.
Source Code
Model description is available in
References
| [Egerstedt2006] | M. Egerstedt; Y. Wardi; H. Axelsson (2006): Transition-time optimization for switched-mode dynamical systems. IEEE Transactions on Automatic Control, 51, 110--115 | |
| [Jung2013] | M. Jung; C. Kirches; S. Sager (2013): On Perspective Functions and Vanishing Constraints in Mixed-Integer Nonlinear Optimal Control. Facets of Combinatorial Optimization -- Festschrift for Martin Gr\"otschel | |
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
