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Optimization-Based Control Design Techniques and Tools

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Abstract

Structured output feedback controller synthesis is an exciting recent concept in modern control design, which bridges between theory and practice in so far as it allows for the first time to apply sophisticated mathematical design paradigms like H- or H2-control within control architectures preferred by practitioners. The new approach to structured H-control, developed by the authors during the past decade, is rooted in a change of paradigm in the synthesis algorithms. Structured design is no longer based on solving algebraic Riccati equations or matrix inequalities. Instead, optimization-based design techniques are required. In this essay we indicate why structured controller synthesis is central in modern control engineering. We explain why non-smooth optimization techniques are needed to compute structured control laws, and we point to software tools which enable practitioners to use these new tools in high-technology applications.

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Bibliography

  • Apkarian P (2013) Tuning controllers against multiple design requirements. In: American control conference (ACC), pp 3888–3893

    Google Scholar 

  • Apkarian P, Noll D (2006a) Controller design via nonsmooth multi-directional search. SIAM J Control Optim 44(6):1923–1949

    Article  MathSciNet  Google Scholar 

  • Apkarian P, Noll D (2006b) Nonsmooth H synthesis. IEEE Trans Aut Control 51(1):71–86

    Article  MathSciNet  Google Scholar 

  • Apkarian P, Noll D (2006c) Nonsmooth optimization for multidisk H synthesis. Eur J Control 12(3):229–244

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Apkarian P, Noll D (2018) Structured H-control of infinite dimensional systems. Int J Robust Nonlinear Control 28(9):3212–3238

    Article  MathSciNet  Google Scholar 

  • Apkarian P, Noll D (2019) Boundary control of partial differential equations using frequency domain optimization techniques. arXiv:1905.06786v1, pp 1–22

    Google Scholar 

  • Apkarian P, Noll D, Thevenet JB, Tuan HD (2003) A spectral quadratic-SDP method with applications to fixed-order H2 and H synthesis. Eur J Control 10(6):527–538

    Article  Google Scholar 

  • Apkarian P, Noll D, Rondepierre A (2007) Mixed H2H control via nonsmooth optimization. In: Proceedings of the 46th IEEE conference on decision and control, New Orleans, pp 4110–4115

    Google Scholar 

  • 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  Google Scholar 

  • Apkarian P, Dao MN, Noll D (2015a) Parametric robust structured control design. IEEE Trans Auto Control 60(7):1857–1869

    Article  MathSciNet  Google Scholar 

  • Apkarian P, Noll D, Ravanbod L (2015b) Computing the structured distance to instability. In: SIAM conference on control and its applications, pp 423–430. https://doi.org/10.1137/1.9781611974072.58

    Google Scholar 

  • Apkarian P, Noll D, Ravanbod L (2016) Nonsmooth bundle trust-region algorithm with applications to robust stability. Set-Valued Var Anal 24(1):115–148

    Article  MathSciNet  Google Scholar 

  • Apkarian P, Noll D, Ravanbod L (2018) Non-smooth optimization for robust control of infinite-dimensional systems. Set-Valued Var Anal 26(2):405–429

    Article  MathSciNet  Google Scholar 

  • Benner P, Sima V, Voigt M (2012) L-norm computation for continuous-time descriptor systems using structured matrix pencils. IEEE Trans Aut Control 57(1):233–238

    Article  MathSciNet  Google Scholar 

  • Boyd S, Balakrishnan V, Kabamba P (1989) A bisection method for computing the H norm of a transfer matrix and related problems. Math Control Signals Syst 2(3):207–219

    Article  MathSciNet  Google Scholar 

  • Bresch-Pietri D, Krstic M (2014) Output-feedback adaptive control of a wave PDE with boundary anti-damping. Automatica 50(5):1407–1415

    Article  MathSciNet  Google Scholar 

  • Burke J, Lewis A, Overton M (2005) A robust gradient sampling algorithm for nonsmooth, nonconvex optimization. SIAM J Optim 15:751–779

    Article  MathSciNet  Google Scholar 

  • DeQueiroz M, Rahn CD (2002) Boundary control of vibrations and noise in distributed parameter systems: an overview. Mech Syst Signal Process 16(1):19–38

    Article  Google Scholar 

  • Falcoz A, Pittet C, Bennani S, Guignard A, Bayart C, Frapard B (2015) Systematic design methods of robust and structured controllers for satellites. Application to the refinement of Rosetta’s orbit controller. CEAS Space J 7:319–334. https://doi.org/ 10.1007/ s12567- 015- 0099-8

    Article  Google Scholar 

  • Fares B, Apkarian P, Noll D (2001) An augmented Lagrangian method for a class of LMI-constrained problems in robust control theory. Int J Control 74(4):348–360

    Article  MathSciNet  Google Scholar 

  • Fares B, Noll D, Apkarian P (2002) Robust control via sequential semidefinite programming. SIAM J Control Optim 40(6):1791–1820

    Article  MathSciNet  Google Scholar 

  • Fridman E (2014) Introduction to time-delay systems. Systems and control, foundations and applications. Birkhuser, Basel

    Google Scholar 

  • Gahinet P, Apkarian P (2011) Structured H synthesis in MATLAB. In: Proceedings of IFAC world congress, Milan, pp 1435–1440

    Google Scholar 

  • Ganet-Schoeller M, Desmariaux J, Combier C (2017) Structured control for future european launchers. AeroSpaceLab 13:2–10

    Google Scholar 

  • Kocvara M, Stingl M (2003) A code for convex nonlinear and semidefinite programming. Optim Methods Softw 18(3):317–333

    Article  MathSciNet  Google Scholar 

  • Kolda TG, Lewis RM, Torczon V (2003) Optimization by direct search: new perspectives on some classical and modern methods. SIAM Rev 45(3):385–482

    Article  MathSciNet  Google Scholar 

  • Lemaréchal C (1975) An extension of Davidson’s methods to nondifferentiable problems. In: Balinski ML, Wolfe P (eds) Nondifferentiable optimization. Mathematical programming study, Springer, Berlin, Heidelberg, vol 3, pp 95–109

    Chapter  Google Scholar 

  • Lemaréchal C, Oustry F (2000) Nonsmooth algorithms to solve semidefinite programs. In: El Ghaoui L, Niculescu S-I (ed) SIAM advances in linear matrix inequality methods in control Series, SIAM, pp 57–77

    Google Scholar 

  • Lieslehto J (2001) PID controller tuning using evolutionary programming. In: American control conference, vol 4, pp 2828–2833

    Google Scholar 

  • Moelja AA, Meinsma G (2003) Parametrization of stabilizing controllers for systems with multiple i/o delays. IFAC Proc Vol 36(19):351–356. 4th IFAC workshop on time delay systems (TDS 2003), Rocquencourt, 8–10 Sept 2003

    Article  Google Scholar 

  • Noll D (2007) Local convergence of an augmented Lagrangian method for matrix inequality constrained programming. Optim Methods Softw 22(5):777–802

    Article  MathSciNet  Google Scholar 

  • Noll D, Apkarian P (2005) Spectral bundle methods for nonconvex maximum eigenvalue functions: first-order methods. Math Prog Ser B 104(2):701–727

    Article  Google Scholar 

  • Noll D, Torki M, Apkarian P (2004) Partially augmented Lagrangian method for matrix inequality constraints. SIAM J Optim 15(1):161–184

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Noll D, Prot O, Apkarian P (2009) A proximity control algorithm to minimize nonsmooth and nonconvex semi-infinite maximum eigenvalue functions. J Convex Anal 16(3 & 4):641–666

    MathSciNet  MATH  Google Scholar 

  • Oi A, Nakazawa C, Matsui T, Fujiwara H, Matsumoto K, Nishida H, Ando J, Kawaura M (2008) Development of PSO-based PID tuning method. In: International conference on control, automation and systems, pp 1917–1920

    Google Scholar 

  • Ravanbod L, Noll D, Apkarian P (2017) Branch and bound algorithm with applications to robust stability. J Glob Optim 67(3):553–579

    Article  MathSciNet  Google Scholar 

  • Robust Control Toolbox 42 (2012) The MathWorks Inc. Natick, MA

    Google Scholar 

  • Saldivar B, Mondié S, Loiseau J, Rasvan V (2013) Suppressing axial-torsional vibrations in drillstrings. J Control Eng Appl Inf, SRAIT 14:3–10

    Google Scholar 

  • Shamma JF, Athans M (1990) Analysis of gain scheduled control for nonlinear plants. IEEE Trans Aut Control 35(8):898–907

    Article  MathSciNet  Google Scholar 

  • Smyshlyaev A, Krstic M (2009) Boundary control of an anti-stable wave equation with anti-damping on the uncontrolled boundary. Syst Control Lett 58: 617–623

    Article  MathSciNet  Google Scholar 

  • Stein G, Doyle J (1991) Beyond singular values and loopshapes. AIAA J Guid Control 14:5–16

    Article  Google Scholar 

  • Zhou K, Doyle JC, Glover K (1996) Robust and optimal control. Prentice Hall, Upper Saddle River

    MATH  Google Scholar 

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Correspondence to Pierre Apkarian .

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Apkarian, P., Noll, D. (2019). Optimization-Based Control Design Techniques and Tools. In: Baillieul, J., Samad, T. (eds) Encyclopedia of Systems and Control. Springer, London. https://doi.org/10.1007/978-1-4471-5102-9_144-2

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  • DOI: https://doi.org/10.1007/978-1-4471-5102-9_144-2

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  • Publisher Name: Springer, London

  • Print ISBN: 978-1-4471-5102-9

  • Online ISBN: 978-1-4471-5102-9

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Chapter history

  1. Latest

    Optimization-Based Control Design Techniques and Tools
    Published:
    09 December 2019

    DOI: https://doi.org/10.1007/978-1-4471-5102-9_144-2

  2. Original

    Optimization-Based Control Design Techniques and Tools
    Published:
    22 March 2014

    DOI: https://doi.org/10.1007/978-1-4471-5102-9_144-1