# F-8 aircraft

F-8 aircraft | |
---|---|

State dimension: | 1 |

Differential states: | 3 |

Discrete control functions: | 1 |

Path constraints: | 2 |

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 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, relaxed control function shows bang bang behavior. The problem is furthermore interesting as it should be reformulated equivalently.

## Contents

## Mathematical formulation

For almost everywhere the mixed-integer optimal control problem is given by

is the angle of attack in radians, is the pitch angle, is the pitch rate in rad/s, and the control function is the tail deflection angle in radians. This model goes back to Garrard[Garrard1977]**Author: ** *Garrard, W.L.; Jordan, J.M.***Journal: ** *Automatica***Pages: ** *497--505***Title: ** *Design of Nonlinear Automatic Control Systems***Volume: ** *13***Year: ** *1977*

.

In the control problem, both initial and terminal values of the differential states are fixed.

## Reformulation

The control w(t) is restricted to take values from a finite set only. Hence, the control problem can be reformulated equivalently to

with . Note that there is a bijection between optimal solutions of the two problems.

## Reference solutions

We provide here a comparison of different solutions reported in the literature. The numbers show the respective lengths of the switching arcs with the value of on the upper or lower bound (given in the second column). *Claim* denotes what is stated in the respective publication, *Simulation* shows values obtained by a simulation with a Runge-Kutta-Fehlberg method of 4th/5th order and an integration tolerance of .

Arc | w(t) | Lee et al.[Lee1997a]Author: Lee, H.W.J.; Teo, K.L.; Rehbock, V.; Jennings, L.S.Journal: Dynamic Systems and ApplicationsPages: 243--262Title: Control Parametrization Enhancing Technique for Time-Optimal Control ProblemsVolume: 6Year: 1997 |
Kaya[Kaya2003]Author: C.Y. Kaya; J.L. NoakesJournal: Journal of Optimization Theory and ApplicationsPages: 69--92Title: A Computational Method for Time-Optimal ControlVolume: 117Year: 2003 |
Sager[Sager2005]Address: Tönning, Lübeck, MarburgAuthor: S. SagerEditor: ISBN 3-89959-416-9Publisher: Der andere VerlagTitle: Numerical methods for mixed--integer optimal control problemsUrl: http://mathopt.de/PUBLICATIONS/Sager2005.pdfYear: 2005 |
Schlueter/ Gerdts | Sager |
---|---|---|---|---|---|---|

1 | 1 | 0.00000 | 0.10292 | 0.10235 | 0.0 | 1.13492 |

2 | 0 | 2.18800 | 1.92793 | 1.92812 | 0.608750 | 0.34703 |

3 | 1 | 0.16400 | 0.16687 | 0.16645 | 3.136514 | 1.60721 |

4 | 0 | 2.88100 | 2.74338 | 2.73071 | 0.654550 | 0.69169 |

5 | 1 | 0.33000 | 0.32992 | 0.32994 | 0.0 | 0.0 |

6 | 0 | 0.47200 | 0.47116 | 0.47107 | 0.0 | 0.0 |

Claim: Infeasibility | - | 1.00E-10 | 7.30E-06 | 5.90E-06 | 3.29169e-06 | 2.21723e-07 |

Claim: Objective | - | 6.03500 | 5.74217 | 5.72864 | 4.39981 | 3.78086 |

Simulation: Infeasibility | - | 1.75E-03 | 1.64E-03 | 5.90E-06 | 3.29169e-06 | 2.21723e-07 |

Simulation: Objective | - | 6.03500 | 5.74218 | 5.72864 | 4.39981 | 3.78086 |

The best known optimal objective value of this problem given is given by . The corresponding solution is shown in the rightmost plot. The solution of bang-bang type switches three times, starting with .

Integer control and corresponding differential states from Sager[Sager2005]

**Address:***Tönning, Lübeck, Marburg***Author:***S. Sager***Editor:***ISBN 3-89959-416-9***Publisher:***Der andere Verlag***Title:***Numerical methods for mixed--integer optimal control problems***Url:***http://mathopt.de/PUBLICATIONS/Sager2005.pdf***Year:***2005*

.

### jModelica

Objective : 5.12799232 infeasibility : 6.2235588037251599e-10

## Source Code

Model descriptions are available in

- AMPL code at F-8 aircraft (AMPL)
- Muscod code at F-8 aircraft (Muscod)
- JModelica code at F-8 aircraft (JModelica)

## Variants

- a prescribed time grid for the control function, see F-8 aircraft (AMPL),

## Miscellaneous and Further Reading

See [Kaya2003]**Author: ** *C.Y. Kaya; J.L. Noakes***Journal: ** *Journal of Optimization Theory and Applications***Pages: ** *69--92***Title: ** *A Computational Method for Time-Optimal Control***Volume: ** *117***Year: ** *2003*

and [Sager2005]**Address: ** *Tönning, Lübeck, Marburg***Author: ** *S. Sager***Editor: ** *ISBN 3-89959-416-9***Publisher: ** *Der andere Verlag***Title: ** *Numerical methods for mixed--integer optimal control problems***Url: ** *http://mathopt.de/PUBLICATIONS/Sager2005.pdf***Year: ** *2005*

for details.

## References

[Garrard1977] | Garrard, W.L.; Jordan, J.M. (1977): Design of Nonlinear Automatic Control Systems. Automatica, 13, 497--505 | |

[Kaya2003] | C.Y. Kaya; J.L. Noakes (2003): A Computational Method for Time-Optimal Control. Journal of Optimization Theory and Applications, 117, 69--92 | |

[Lee1997a] | Lee, H.W.J.; Teo, K.L.; Rehbock, V.; Jennings, L.S. (1997): Control Parametrization Enhancing Technique for Time-Optimal Control Problems. Dynamic Systems and Applications, 6, 243--262 | |

[Sager2005] | S. Sager (2005): Numerical methods for mixed--integer optimal control problems. (%edition%). Der andere Verlag, Tönning, Lübeck, Marburg, %pages% |