Difference between revisions of "Egerstedt standard problem"

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<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
 
<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
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  Image:MmlotkaRelaxed_12000_30_1.png| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation)  and <math>n_t=12000, \, n_u=400</math>.
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  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>.
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  Image:MmlotkaCIA 12000 30 1.png| Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=400</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
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  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.
 
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Model descriptions are available in
 
Model descriptions are available in
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* [[:Category:ACADO | ACADO code]] at [[Lotka Volterra fishing problem (ACADO)]]
 
 
* [[:Category:AMPL | AMPL code]] at [[Lotka Volterra fishing problem (AMPL)]]
 
* [[:Category:AMPL | AMPL code]] at [[Lotka Volterra fishing problem (AMPL)]]
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* [[:Category:APMonitor | APMonitor code]] at [[Lotka Volterra fishing problem (APMonitor)]]
 
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* [[:Category:Bocop | Bocop code]] at [[Lotka Volterra fishing problem (Bocop)]]
 
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* [[:Category:Casadi | Casadi code]] at [[Lotka Volterra fishing problem (Casadi)]]
 
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* [[:Category:JModelica | JModelica code]] at [[Lotka Volterra fishing problem (JModelica)]]
 
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* [[:Category:Julia/JuMP | JuMP code]] at [[Lotka Volterra fishing problem (JuMP)]]
 
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* [[:Category:Muscod | Muscod code]] at [[Lotka Volterra fishing problem (Muscod)]]
 
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* [[:Category:switch | switch code]] at [[Lotka Volterra fishing problem (switch)]]
 
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* [[:Category:TomDyn/PROPT | PROPT code]] at [[Lotka Volterra fishing problem (TomDyn/PROPT)]]
 
  
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== Variants ==
 
  
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There are several alternative formulations and variants of the above problem, in particular
 
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* a prescribed time grid for the control function <bib id="Sager2006" />, see also [[Lotka Volterra fishing problem (AMPL)]],
 
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* a time-optimal formulation to get into a steady-state <bib id="Sager2005" />,
 
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* the usage of a different target steady-state, as the one corresponding to <math> w(t) = 1</math> which is <math>(1 + c_1, 1 - c_0)</math>, see [[Lotka Volterra multi-arcs problem]]
 
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* different fishing control functions for the two species, see [[Lotka Volterra Multimode fishing problem]]
 
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* different parameters and start values.
 
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== Miscellaneous and Further Reading ==
 
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The Lotka Volterra fishing problem was introduced by Sebastian Sager in a proceedings paper <bib id="Sager2006" /> and revisited in his PhD thesis <bib id="Sager2005" />. These are also the references to look for more details.
 
  
 
== References ==
 
== References ==
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[[Category:ODE model]]
 
[[Category:ODE model]]
 
[[Category:Tracking objective]]
 
[[Category:Tracking objective]]
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[[Category:Chattering]]
 
 
[[Category:Sensitivity-seeking arcs]]
 
[[Category:Sensitivity-seeking arcs]]
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[[Category:Population dynamics]]
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Revision as of 14:29, 10 January 2018

Egerstedt standard problem
State dimension: 1
Differential states: 3
Discrete control functions: 3
Path constraints: 1
Interior point equalities: 3

The Egerstedt standard problemm 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
Link to Google Scholar. 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
Link to Google Scholar) 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}

for t \in [t_0, t_f]=[0,1] .

Reference Solutions

If the problem is relaxed, i.e., we demand that w(t) be in the continuous interval [0, 1] instead of the binary choice \{0,1\}, 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 n_t=6000, \, n_u=40 is x_3(t_f)=1.0.995906234. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is x_3(t_f) =3.20831942. 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)\} .

  • Reference solution plots

  • Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and n_t=6000, \, n_u=40.

  • Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and n_t=6000, \, n_u=40. The relaxed controls were approximated by Combinatorial Integral Approximation.


Source Code

Model descriptions are 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--115Link to Google Scholar
[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\"otschelLink to Google Scholar
[Sager2005]S. Sager (2005): Numerical methods for mixed--integer optimal control problems. (%edition%). Der andere Verlag, Tönning, Lübeck, Marburg, %pages%Link to Google Scholar
[Sager2006]S. Sager; H.G. Bock; M. Diehl; G. Reinelt; J.P. Schl\"oder (2009): Numerical methods for optimal control with binary control functions applied to a Lotka-Volterra type fishing problem. Springer, Recent Advances in OptimizationLink to Google Scholar




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

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