Quadrotor (binary variant)

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Quadrotor (binary variant)
State dimension: 1
Differential states: 6
Discrete control functions: 4
Interior point equalities: 6

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


\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}

Parameters

These fixed values are used within the model.


\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}


Reference Solutions

If the problem is relaxed, i.e., we demand that w(t) is in the continuous interval [0, 1] 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  n_t=12000, \, n_u=25  is 13.0907346. The objective value of the solution with binary controls obtained by Combinatorial Integral Approximation (CIA) is 15.5787932.