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 $15.5787932$.