Difference between revisions of "F-8 aircraft"

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The F-8 aircraft control problem is based on a very simple aircraft model. The control problem was introduced by \cite{Kaya2003} and aims at controlling an aircraft in a time--optimal way from an initial state to a terminal state.
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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 [[:Category:ODE Model|ODE Model]]. The interior point equality conditions fix both initial and terminal values of the differential states.
 
The mathematical equations form a small-scale [[:Category:ODE Model|ODE Model]]. The interior point equality conditions fix both initial and terminal values of the differential states.
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</math>
 
</math>
  
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<math>x_0</math> is the angle of attack in radians, <math>x_1</math> is the pitch angle, <math>x_2</math> is the pitch rate in rad/s, and the control function <math>w = w(t)</math> is the tail deflection angle in radians. This model goes back to \cite{Garrard1977}. See \cite{Kaya2003} for further references and details.
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<math>x_0</math> is the angle of attack in radians, <math>x_1</math> is the pitch angle, <math>x_2</math> is the pitch rate in rad/s, and the control function <math>w = w(t)</math> is the tail deflection angle in radians. This model goes back to Garrard<bibref>Garrard1977</bibref>.
  
 
== Initial and terminal values ==
 
== Initial and terminal values ==
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== Reference solutions ==
 
== Reference solutions ==
  
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If the problem is relaxed, i.e., we demand that <math>w(t)</math> be in the continuous interval <math>[0, 1]</math> instead of the binary choice <math>\{0,1\}</math>,
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The optimal objective value of this  problem given in Sager 2005<bibref>Sager2005</bibref> is <math>T = 5.73406</math>.
  
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The optimal objective value of this relaxed problem is <math>x_2(t_f) = 1.34408</math>. As follows from MIOC theory<bibref>Sager2008</bibref> this is the best lower bound on the optimal value of the original problem with the integer restriction on the control function. In other words, this objective value can be approximated arbitrarily close, if the control only switches often enough between 0 and 1. As no optimal solution exists, two suboptimal ones are shown, one with only two switches and an objective function value of <math>x_2(t_f) = 1.38276</math>, and one with 56 switches and <math>x_2(t_f) = 1.34416</math>.
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<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="3">
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  Image:lotkaRelaxedControls.png| Optimal relaxed control on a coarse control discretization grid and corresponding differential states.
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<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="4">
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  Image:lotkaindirektStates.png| Optimal relaxed control on a fine control discretization grid and corresponding differential states.
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  Image:lotkaRelaxedControls.png| Optimal relaxed control determined by an indirect approach and by a direct approach on different control discretization grids.
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  Image:lotka2Switches.png| Optimal integer control and corresponding differential states.
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  Image:lotkaindirektStates.png| Differential states and corresponding adjoint variables in the indirect approach.
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Image:lotka2Switches.png| Control and differential states with only two switches.
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  Image:lotka56Switches.png| Control and differential states with 56 switches.
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</gallery>
 
</gallery>
  
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The differential equations in C code:
 
The differential equations in C code:
 
<source lang="cpp">
 
<source lang="cpp">
 +
 +
  double x1 = xd[0];
 +
  double x2 = xd[1];
 +
  double x3 = xd[2];
 +
 +
  double u0 = -0.05236;
 +
  double u1 = 0.05236;
 +
  double f00, f10, f20;
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  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;
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  f20 = -4.208*x1 - 0.396*x3 - 0.47*x1*x1 - 3.564*x1*x1*x1
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          - 20.967*u0 + 6.265*x1*x1*u0 + 46*x1*u0*u0 + 61.4*u0*u0*u0;
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 +
  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;
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  f21 = -4.208*x1 - 0.396*x3 - 0.47*x1*x1 - 3.564*x1*x1*x1
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          - 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;
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  rhs[1] = u[0]*f11 + (1-u[0])*f10;
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  rhs[2] = u[0]*f21 + (1-u[0])*f20;
 
</source>
 
</source>
  

Revision as of 01:17, 1 November 2008

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} \displaystyle \min_{x, w, T} & T \\[1.5ex] \mbox{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

<bibreferences/>

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