Difference between revisions of "Van der Pol Oscillator (binary variant)"
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<math> | <math> | ||
\begin{array}{lll} | \begin{array}{lll} | ||
− | \min\limits_{x,y,w} & \int\limits_{t_0}^{t_f} & (x(t)^2+y(t)^2 | + | \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. | ||
− | The optimal objective value of the relaxed problem with <math> n_t= | + | 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"> | ||
− | Image: | + | Image:VanderpolRelaxed 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>. |
− | Image: | + | Image:VanderpolCIA_6000_100_1.pdf| 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> | ||
+ | == Source Code == | ||
+ | Model description is available in | ||
+ | * [[:Category:AMPL | AMPL code]] at [[Van der Pol Oscillator binary variant(AMPL)]] | ||
Latest revision as of 18:05, 10 January 2018
Van der Pol Oscillator (binary variant) | |
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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
Parameters
These fixed values are used within the model:
Reference Solutions
If the problem is relaxed, i.e., we demand that be in the continuous interval instead of the binary choice , the optimal solution can be determined by means of direct optimal control.
The optimal objective value of the relaxed problem with is . The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is .
Source Code
Model description is available in