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Minimizing memory effects of a system

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Abstract

Given a stable linear time-invariant system with tunable parameters, we present a method to tune these parameters in such a way that undesirable responses of the system to past excitations, known as system ringing, are avoided or reduced. This problem is addressed by minimizing the Hankel norm of the system, which quantifies the influence of past inputs on future outputs. We indicate by way of examples that minimizing the Hankel norm has a wide scope for possible applications. We show that the Hankel norm minimization program may be cast as an eigenvalue optimization problem, which we solve by a nonsmooth bundle algorithm with a local convergence certificate. Numerical experiments are used to demonstrate the efficiency of our approach.

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References

  1. Apkarian P, Noll D (2006) Nonsmooth \(H_\infty \) synthesis. IEEE Trans Automat Control 51(1):71–86

    Article  MathSciNet  Google Scholar 

  2. 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  MATH  MathSciNet  Google Scholar 

  3. Bellman R (1959) Kronecker products and the second method of Lyapunov. Math Nachr 20:17–19

    Article  MathSciNet  Google Scholar 

  4. Bompart V, Apkarian P, Noll D (2007) Non-smooth techniques for stabilizing linear systems. In: Proceedings of the American Control Conference, New York, pp 1245–1250

  5. Boyd S, Barratt C (1991) Linear controller design: limits of performance. Prentice Hall, New York

    MATH  Google Scholar 

  6. Bronstein MD (1979) Smoothness of roots of polynomials depending on parameters. Sibirsk Mat Zh 20(3): 493–501. English Transl. in (1980) Siberian Math J, vol 20, pp 347–352

  7. Burke JV, Overton ML (1994) Differential properties of the spectral abscissa and the spectral radius for analytic matrix-valued mappings. Nonlinear Anal 23(4):467–488

    Article  MATH  MathSciNet  Google Scholar 

  8. Clarke FH (1983) Pptimization and nonsmooth analysis. Wiley, New York

    Google Scholar 

  9. Dao MN, Noll D (2013) Minimizing the memory of a system. In: Proceedings of the Asian Control Conference, Istanbul

  10. Dennis JE Jr, Schnabel RB (1983) Numerical methods for unconstrained optimization and nonlinear equations. Prentice Hall, New Jersey

    MATH  Google Scholar 

  11. Gabarrou M, Alazard D, Noll D (2013) Design of a flight control architecture using a non-convex bundle method. Math Control Signals Syst 25(2):257–290

    Article  MATH  MathSciNet  Google Scholar 

  12. Glover K (1984) All optimal Hankel-norm approximations of linear multivariable systems and their \(L^\infty \)-error bounds. Int J Control 39(6):1115–1193

    Article  MATH  MathSciNet  Google Scholar 

  13. Mangasarian OL (1969) Nonlinear programming. McGraw-Hill Book Co., New York

    MATH  Google Scholar 

  14. Mifflin R (1982) A modification and extension of Lemaréchal’s algorithm for nonsmooth minimization. Nondifferential and variational techniques in optimization (Lexington, Ky., 1980). Math Program Stud 17:77–90

    Article  MATH  MathSciNet  Google Scholar 

  15. Nesterov Y (2007) Smoothing technique and its applications in semidefinite optimization. Math Program Ser A 110(2):245–259

    Article  MATH  MathSciNet  Google Scholar 

  16. Noll D (2010) Cutting plane oracles to minimize non-smooth non-convex functions. Set Value Var Anal 18(3–4):531–568

    Article  MATH  MathSciNet  Google Scholar 

  17. Noll D, Prot O, Rondepierre A (2008) A proximity control algorithm to minimize nonsmooth and nonconvex functions. Pac J Optim 4(3):571–604

    MATH  MathSciNet  Google Scholar 

  18. Overton ML (1992) Large-scale optimization of eigenvalues. SIAM J Optim 2(1):88–120

    Article  MATH  MathSciNet  Google Scholar 

  19. Parusiński A, Rainer A (2014) A new proof of Bronshtein’s theorem. arXiv:1309.2150v2.

  20. Polak E (1997) Optimization: algorithms and consistent approximations. Applied Mathematical Sciences 124. Springer, New York

    Book  Google Scholar 

  21. Rainer A (2011) Smooth roots of hyperbolic polynomials with definable coefficients. Israel J Math 184:157–182

    Article  MATH  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

  23. Skogestad S, Postlethwaite I (2005) Multivariable feedback dontrol: analysis and design. Wiley, Chichester

    Google Scholar 

  24. Spingarn JE (1981) Submonotone subdifferentials of Lipschitz functions. Trans Am Math Soc 264(1):77–89

    Article  MATH  MathSciNet  Google Scholar 

  25. van den Dries L (1998) Tame topology and o-minimal structures. London Math Soc Lecture Note Ser 248. Cambridge University Press, Cambridge

    Book  Google Scholar 

  26. Zhou K, Doyle JC, Glover K (1996) Robust and optimal control. Prentice Hall, New Jersey

    MATH  Google Scholar 

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Acknowledgments

The authors acknowledge helpful discussions with Dr. Armin Rainer (University of Vienna).

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Authors and Affiliations

  1. Department of Mathematics and Informatics, Hanoi National University of Education, Hanoi, Vietnam

    Minh Ngoc Dao

  2. Institut de Mathématiques, Université de Toulouse, Toulouse, France

    Minh Ngoc Dao & Dominikus Noll

Authors
  1. Minh Ngoc Dao
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  2. Dominikus Noll
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Correspondence to Minh Ngoc Dao.

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Dao, M.N., Noll, D. Minimizing memory effects of a system. Math. Control Signals Syst. 27, 77–110 (2015). https://doi.org/10.1007/s00498-014-0135-9

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  • DOI: https://doi.org/10.1007/s00498-014-0135-9

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