Quadrotor (binary variant)

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

$\begin{array}{llclr} \min_{x, u, w} & 5 (x_{1} (t_{f}) - 6)^{2} & + & 5 (x_{3} (t_{f}) - 1)^{2} + (\sin (x_{5} (t_{f}) 0.5))^{2} + \int_{t_{0}}^{t_{f}} 5 ((w_{2} (τ) + w_{4} (τ) + w_{6} (τ))^{2} d τ \\ s.t. & {\dot{x}}_{1} & = & x_{2} (t), \\ {\dot{x}}_{2} & = & g \sin (x_{5} (t)) + \sum_{i \in [4]} c_{1, i} w_{i} (t) \frac{\sin (x_{5} (t))}{M}, \\ {\dot{x}}_{3} & = & x_{4} (t), \\ {\dot{x}}_{4} & = & g \cos (x_{5} (t)) - g + \sum_{i \in [4]} c_{1, i} w_{i} (t) \frac{\cos (x_{5} (t))}{M}, \\ {\dot{x}}_{5} & = & x_{6} (t), \\ {\dot{x}}_{6} & = & \sum_{i \in [4]} c_{2, i} w_{i} (t) L \frac{1}{I} \\ x (0) & = & (0, 0, 1, 0, 0, 0)^{T}, \\ w_{i} (t) & \in & {0, 1}, i = 1, \dots, 4 \\ \sum_{i = 1}^{4} w_{i} (t) & = & 1, \\ x_{3} (t) & \geq & 0, t \in [t_{0}, t_{f}] . \end{array}$

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 $15.5787932$ .

Reference solution plots
Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and $n_{t} = 12000, n_{u} = 25$ .
Differential states and binary cotnrol determined by an direct approach (Radau collocation) with ampl_mintoc and $n_{t} = 12000, n_{u} = 25$ . The relaxed controls were approximated by Combinatorial Integral Approximation.