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Design of a flight control architecture using a non-convex bundle method

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Abstract

We design a feedback control architecture for longitudinal flight of an aircraft. The multi-level architecture includes the flight control loop to govern the short-term dynamics of the aircraft, and the autopilot to control the long-term modes. Using \(H_\infty \) performance and robustness criteria, the problem is cast as a non-convex and non-smooth optimization program. We present a non-convex bundle method, prove its convergence, and show that it is apt to solve the longitudinal flight control problem.

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References

  1. Tischler MB (1996) Advances in aircraft flight control. Taylor & Francis, Bristol

    Google Scholar 

  2. Stevens BL, Lewis FL (1992) Aircraft control and simulation. Wiley, New York

    Google Scholar 

  3. Apkarian P, Noll D, Prot O (2008) A trust region spectral bundle method for nonconvex eigenvalue optimization. SIAM J Optim 19(1):281–306

    Article  MathSciNet  MATH  Google Scholar 

  4. Noll D, Prot O, Rondepierre A (2008) A proximity control algorithm to minimize nonsmooth and nonconvex functions. Pacific J Optim 4(3):569–602

    MathSciNet  Google Scholar 

  5. Noll D (2010) Cutting plane oracles to minimize nonsmooth and nonconvex functions. J Set-Valued Variat Anal 18(3–4):531–568

    Article  MathSciNet  MATH  Google Scholar 

  6. Apkarian P, Noll D (2006) Nonsmooth \({H}_\infty \) synthesis. IEEE Trans Autom Control 51(1):71–86

    Article  MathSciNet  Google Scholar 

  7. Polak E (1997) Optimization: algorithms and consistent approximations. Spinger, Berlin

    MATH  Google Scholar 

  8. Apkarian P, Noll D, Rondepierre A (2008) Mixed \({H}_2/{H}_\infty \) control via nonsmooth optimization. SIAM J Control Optim 47(3):1516–1546

    Article  MathSciNet  MATH  Google Scholar 

  9. Alazard D (2002) Robust \({H}_2\) design for lateral flight control of a highly flexible aircraft. J Guidance Control Dyn 25(6):502–509

    Article  Google Scholar 

  10. Apkarian P, Noll D (2006) Nonsmooth optimization for multidisk \({H}_\infty \) synthesis. Eur J Control 12(3):229–244

    Article  MathSciNet  Google Scholar 

  11. Apkarian P, Noll D (2007) Nonsmooth optimization for multiband frequency domain control design. Automatica 43(4):724–731

    Article  MathSciNet  MATH  Google Scholar 

  12. Spingarn JE (1981) Submonotone subdifferentials of Lipschitz functions. Trans Am Math Soc 264:77–89

    Article  MathSciNet  MATH  Google Scholar 

  13. Rockafellar RT, Wets RJ-B (1998) Variational analysis. Springer, Berlin

    Book  MATH  Google Scholar 

  14. Georgiev P, Daniilidis A (2004) Approximate convexity and submonotonicity. J Math Anal Appl 291:117–144

    MathSciNet  Google Scholar 

  15. Boyd S, Balakrishnan V (1990) A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its \({L}_\infty \)-norm. Syst Control Lett 15:1–7

    Article  MathSciNet  MATH  Google Scholar 

  16. Bompart V, Noll D, Apkarian P (2007) Second order non smooth optimization for feedback control. Numer Math 107(3):433–454

    Article  MathSciNet  MATH  Google Scholar 

  17. Burke JV, Lewis AS, Overton ML (2002) Two numerical methods for optimizing matrix stability. Linear Algebra Appl 351–352:117–145

    Article  MathSciNet  Google Scholar 

  18. Blondel V, Tsitsiklis J (1997) NP-hardness of some linear control design problems. SIAM J Control Optim 35(6):2118–2127

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

Financial support by Fondation de Recherche pour l’Aéronautique et l’Espace (FNRAE) under research grant Survol and by Fondation d’Entreprise EADS (FEADS) under research grant Technicom is gratefully acknowledged.

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Authors and Affiliations

  1. Université Paul Sabatier, Institut de Mathématiques, Toulouse, France

    Marion Gabarrou & Dominikus Noll

  2. Institut Supérieur de l’Aéronautique et de l’Espace, Toulouse, France

    Daniel Alazard

Authors
  1. Marion Gabarrou
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  2. Daniel Alazard
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  3. Dominikus Noll
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Corresponding author

Correspondence to Dominikus Noll.

Appendix

Appendix

The numerical data for the specific flight point Mach\(=0.7\), Altitude\(=5000\,ft\) used in (5) are

$$\begin{aligned}&A= \left[\!\! \begin{array}{lllll} -0.0120&-9.8040&-14.8800&0&0 \\ 0.0004&0&0.8524&0&-0.0000 \\ -0.0004&0&-0.8524&1.0000&0.0000 \\ 0&0&-2.6650&-0.2783&0\\ 0&234.1000&0&0&0\end{array}\!\!\right], \\&B= \left[\!\!\begin{array}{ll} 4.9580&0\\ 0&0.3113 \\ 0&-0.3113 \\ 0&-4.9360 \\ 0&0 \end{array}\!\!\right], \\&C=\left[ \!\!\begin{array}{lllll} 1.0000&0&0&0&0 \\ 0&1.0000&0&0&0 \\ 0.0085&0&13.5409&-0.7092&-0.0001 \\ 0&0&0&1.0000&0 \\ 0&0&0&\,&1.0000 \end{array}\!\! \right], \\&D = \left[\!\! \begin{array}{ll} 0&0 \\ 0&0 \\ 0&-5.1535 \\ 0&0 \\ 0&0 \end{array} \!\!\right]. \end{aligned}$$

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Gabarrou, M., Alazard, D. & Noll, D. Design of a flight control architecture using a non-convex bundle method. Math. Control Signals Syst. 25, 257–290 (2013). https://doi.org/10.1007/s00498-012-0093-z

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  • DOI: https://doi.org/10.1007/s00498-012-0093-z

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