F-8 aircraft

F-8 aircraft
State dimension:	1
Differential states:	3
Discrete control functions:	1
Interior point equalities:	6

The F-8 aircraft control problem is based on a very simple aircraft model. The control problem was introduced by Kaya and Noakes in 2003<bibref>Kaya2003</bibref> and aims at controlling an aircraft in a time-optimal way from an initial state to a terminal state.

The mathematical equations form a small-scale ODE Model. The interior point equality conditions fix both initial and terminal values of the differential states.

The optimal integer control functions shows bang bang behavior. The problem is furthermore interesting as it should be reformulated equivalently.

Mathematical formulation

For $t \in [0, T]$ almost everywhere the mixed-integer optimal control problem is given by

$\begin{array}{llcl} \min_{x, w, T} & T \\ s.t. & {\dot{x}}_{0} & = & - 0.877 x_{0} + x_{2} - 0.088 x_{0} x_{2} + 0.47 x_{0}^{2} - 0.019 x_{1}^{2} - x_{0}^{2} x_{2} \\ + 3.846 x_{0}^{3} - 0.215 w + 0.28 x_{0}^{2} w + 0.47 x_{0} w^{2} + 0.63 w^{3} \\ {\dot{x}}_{1} & = & x_{2} \\ {\dot{x}}_{2} & = & - 4.208 x_{0} - 0.396 x_{2} - 0.47 x_{0}^{2} - 3.564 x_{0}^{3} \\ x (0) & = & x_{0}, \\ x (T) & = & x_{T}, \\ w (t) & \in & {- 0.05236, 0.05236} . \end{array}$

$x_{0}$ is the angle of attack in radians, $x_{1}$ is the pitch angle, $x_{2}$ is the pitch rate in rad/s, and the control function $w = w (t)$ is the tail deflection angle in radians. This model goes back to Garrard<bibref>Garrard1977</bibref>.

Initial and terminal values

Both initial as terminal values of the differential states are fixed.

$\begin{array}{rcl} x_{0} & = & (0.4655, 0, 0)^{T}, \\ x_{T} & = & (0, 0, 0)^{T} . \end{array}$

Reference solutions

The optimal objective value of this problem given in Sager 2005<bibref>Sager2005</bibref> is $T = 5.73406$ .

Reference solution plots
Optimal relaxed control on a coarse control discretization grid and corresponding differential states.
Optimal relaxed control on a fine control discretization grid and corresponding differential states.
Optimal integer control and corresponding differential states.

Source Code

C

The differential equations in C code:

  double x1 = xd[0];
  double x2 = xd[1];
  double x3 = xd[2];

  double u0 = -0.05236;
  double u1 = 0.05236;
  double f00, f10, f20;
  double f01, f11, f21;
  
  f00 = -0.877*x1 + x3 - 0.088*x1*x3 + 0.47*x1*x1 - 0.019*x2*x2
           -x1*x1*x3 + 3.846*x1*x1*x1 - 0.215*u0 + 0.28*x1*x1*u0 + 0.47*x1*u0*u0 + 0.63*u0*u0*u0;
  f10 = x3;
  f20 = -4.208*x1 - 0.396*x3 - 0.47*x1*x1 - 3.564*x1*x1*x1
           - 20.967*u0 + 6.265*x1*x1*u0 + 46*x1*u0*u0 + 61.4*u0*u0*u0;

  f01 = -0.877*x1 + x3 - 0.088*x1*x3 + 0.47*x1*x1 - 0.019*x2*x2
           -x1*x1*x3 + 3.846*x1*x1*x1 - 0.215*u1 + 0.28*x1*x1*u1 + 0.47*x1*u1*u1 + 0.63*u1*u1*u1;
  f11 = x3;
  f21 = -4.208*x1 - 0.396*x3 - 0.47*x1*x1 - 3.564*x1*x1*x1
           - 20.967*u1 + 6.265*x1*x1*u1 + 46*x1*u1*u1 + 61.4*u1*u1*u1;

  rhs[0] = u[0]*f01 + (1-u[0])*f00;
  rhs[1] = u[0]*f11 + (1-u[0])*f10;
  rhs[2] = u[0]*f21 + (1-u[0])*f20;

Miscellaneous and Further Reading

See <bibref>Kaya2003</bibref> and <bibref>Sager2005</bibref> for details.

References