Difference between revisions of "Van der Pol Oscillator (binary variant)"

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(Created page with "{{Dimensions |nd = 1 |nx = 2 |nw = 3 |nre = 2 }}<!-- Do not insert line break here or Dimensions Box moves up in the layout... -->This site describ...")
 
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<math>
 
<math>
 
\begin{array}{lll}
 
\begin{array}{lll}
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\min\limits_{x,y,w}  & \int\limits_{t_0}^{t_f} & (x(t)^2+y(t)^2+u(t)^2) dt\\
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\min\limits_{x,y,w}  & \int\limits_{t_0}^{t_f} & (x(t)^2+y(t)^2 dt\\
 
s.t. & \dot x & = y,\\
 
s.t. & \dot x & = y,\\
 
& \dot y & =  \sum\limits_{i=1}^{3} c_{i}\;  w_i \;(1-x^2) y-x,\\
 
& \dot y & =  \sum\limits_{i=1}^{3} c_{i}\;  w_i \;(1-x^2) y-x,\\
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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 means of direct optimal control.  
 
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 means of direct optimal control.  
  
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The optimal objective value of the relaxed problem with  <math> n_t=12000, \, n_u=400 </math> is <math>x_2(t_f) =1.82875272</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>x_2(t_f) =1.82878681</math>.   
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The optimal objective value of the relaxed problem with  <math> n_t=6000, \, n_u=60 </math> is <math>1.30167235</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>1.30273681</math>.   
  
 
<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:VanderpolCIA_6000_100_1.pdf| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation)  and <math>n_t=6000, \, n_u=60</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:MmlotkaCIA 12000 30 1.png| Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=6000, \, n_u=60</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
 
</gallery>
 
</gallery>
  

Revision as of 18:58, 10 January 2018

Van der Pol Oscillator (binary variant)
State dimension: 1
Differential states: 2
Discrete control functions: 3
Interior point equalities: 2

This site describes a Van der Pol Oscillator variant with three binary controls instead of only one continuous control.

Mathematical formulation

The mixed-integer optimal control problem is given by

\begin{array}{lll} \min\limits_{x,y,w} & \int\limits_{t_0}^{t_f} & (x(t)^2+y(t)^2 dt\\ s.t. & \dot x & = y,\\ & \dot y & = \sum\limits_{i=1}^{3} c_{i}\; w_i \;(1-x^2) y-x,\\ & x(0) & =1,\\ & y(0) & =0,\\ & 1 &= \sum\limits_{i=1}^{3}w_i(t), \\ & w_i(t) &\in \{0, 1\}, \quad i=1\ldots 3. \end{array}

Parameters

These fixed values are used within the model:

[t_0,t_f]=[0,20], c_1=-1, c_2=0.75, c_3=-2.

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 means of direct optimal control.

The optimal objective value of the relaxed problem with n_t=6000, \, n_u=60 is 1.30167235. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is 1.30273681.

  • Reference solution plots

  • Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and n_t=6000, \, n_u=60.

  • Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and n_t=6000, \, n_u=60. The relaxed controls were approximated by Combinatorial Integral Approximation.

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