Difference between revisions of "D'Onofrio model (binary variant)"
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& \dot{x}_3 & = & \sum\limits_{i=1}^{4} w_i\;c_{1,i}, \\ [1.5ex] | & \dot{x}_3 & = & \sum\limits_{i=1}^{4} w_i\;c_{1,i}, \\ [1.5ex] | ||
& x_2 & \leq & x_2^{max}, \\ | & x_2 & \leq & x_2^{max}, \\ | ||
| − | & x_3 & \leq & x_3^{max}, | + | & x_3 & \leq & x_3^{max},\\ |
& 1 &=& \sum\limits_{i=1}^{3}w_i(t), \\ | & 1 &=& \sum\limits_{i=1}^{3}w_i(t), \\ | ||
& w_i(t) &\in& \{0, 1\}, \quad i=1\ldots 4. | & w_i(t) &\in& \{0, 1\}, \quad i=1\ldots 4. | ||
Revision as of 16:19, 11 January 2018
| D'Onofrio model (binary variant) | |
|---|---|
| State dimension: | 1 |
| Differential states: | 4 |
| Discrete control functions: | 4 |
| Path constraints: | 2 |
This site describes a D'Onofrio model variant with four binary controls instead which of only two continuous controls. The continuous controls are replaced via the outer convexifacation method.
Mathematical formulation
For ![t \in [t_0, t_f]](https://mintoc.de/images/math/5/5/8/55823791d9100bcb5461801aff4f6edd.png)
the optimal control problem is given by
![\begin{array}{llcl}
\displaystyle \min_{x, u} & x_0(t_f) &+& \alpha \int_{t_0}^{t_f} u_0(t)^2 \text{d}t \\[1.5ex]
\mbox{s.t.} & \dot{x}_0 & = & - \zeta x_0 \text{ln} \left( \frac{x_0}{x_1} \right) - \sum\limits_{i=1}^{4} w_i\;c_{1,i}\; F \; x_0 , \\
& \dot{x}_1 & = & b x_0 - \mu x_1 - d x_0^{\frac{2}{3}}x_1 -\sum\limits_{i=1}^{4} w_i c_{0,i} G x_1 - \sum\limits_{i=1}^{4} w_i\;c_{1,i} \eta x_1, \\
& \dot{x}_2 & = & \sum\limits_{i=1}^{4} w_i\;c_{0,i}, \\
& \dot{x}_3 & = & \sum\limits_{i=1}^{4} w_i\;c_{1,i}, \\ [1.5ex]
& x_2 & \leq & x_2^{max}, \\
& x_3 & \leq & x_3^{max},\\
& 1 &=& \sum\limits_{i=1}^{3}w_i(t), \\
& w_i(t) &\in& \{0, 1\}, \quad i=1\ldots 4.
\end{array}](https://mintoc.de/images/math/e/9/f/e9f859e26cc8f2421afdd069e3a9c8f9.png)
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.](https://mintoc.de/images/math/6/5/5/6556b72c5eff2a3dcb8b0c40da03d1fd.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 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 
.
- 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
