View source for Quadrotor (binary variant)

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-->This site describes a Quadrotor helicoptor problem variant where the continuous control is replaced via outer convexification with binary controls.


== Mathematical formulation ==

The mixed-integer optimal control problem is given by

<p>
<math>
\begin{array}{llclr}
 \displaystyle \min_{x,u, w} &  5(x_1(t_f)-6)^2&+&5(x_3(t_f)-1)^2+(\sin(x_5(t_f)0.5))^2 +\int\limits_{t_0}^{t_f} 5( (w_2(\tau)+w_4(\tau)+w_6(\tau))^2  \ d \tau   \\[1.5ex]
 \mbox{s.t.} 
 & \dot{x}_1 & = &  x_2(t), \\
 & \dot{x}_2 & = & g \sin( x_5(t)) + \sum\limits_{i\in [4]}c_{1,i}w_i(t)\frac{\sin(x_5(t))}{M},   \\
 & \dot{x}_3 & = & x_4(t),   \\
 & \dot{x}_4 & = & g \cos( x_5(t))-g+  \sum\limits_{i\in [4]}c_{1,i}w_i(t)\frac{\cos(x_5(t))}{M},   \\
 & \dot{x}_5 & = & x_6(t),   \\
 & \dot{x}_6 & = &  \sum\limits_{i\in [4]}c_{2,i}w_i(t)L \frac{1}{I}  \\[1.5ex]
 & x(0) &=& (0, 0, 1, 0 , 0, 0)^T, \\
 & w_i(t) &\in&  \{0, 1\}, i=1,\ldots,4 \\
 & \sum\limits_{i=1}^{4}w_i(t) &=& 1, \\
& x_3(t) & \geq & 0, \quad t\in[t_0,t_f].
\end{array} 
</math>
</p>

== Parameters ==

These fixed values are used within the model.

<math>
\begin{array}{rcl}
[t_0, t_f] &=& [0, 7.5],\\
(g, M, L, I) &=& (9.8, 1.3, 0.305, 0.0605),\\
c_1 &=& (0,0.001,0,0),\\
c_2 &=& (0,0,-0.001,0.001),
\end{array}
</math>


== Reference Solutions ==

If the problem is relaxed, i.e., we demand that <math>w(t)</math> is in the continuous interval <math>[0, 1]</math> rather than being binary, the optimal solution can be determined by means of direct optimal control. 

The optimal objective value of the relaxed problem with  <math> n_t=12000, \, n_u=25  </math> is <math>13.0907346</math>. The objective value of the solution with binary controls obtained by Combinatorial Integral Approximation (CIA) is <math>15.5787932</math>.  

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
 Image:Quadrotor_bin_Relaxed_12000_25.pdf| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation)  and <math>n_t=12000, \, n_u=25</math>.
 Image:Quadrotor_bin_CIA_12000_25.pdf| Differential states and binary cotnrol determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=25</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
</gallery>








<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->
[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Tracking objective]]
[[Category:Chattering]]
[[Category:Sensitivity-seeking arcs]]
[[Category:Population dynamics]]


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