Skip to main content
SpringerLink
Log in
Book cover

Control of Linear Parameter Varying Systems with Applications pp 61–89Cite as

Parametric Gain-scheduling Control via LPV-stable Realization

  • Chapter
  • First Online:

Abstract

It has been recently shown that, given a plant described by a parametric transfer function, any compensator that internally stabilizes the plant for each constant value of the parameter can be realized in such a way that the closed-loop stability is guaranteed under arbitrary variations of the parameter (LPV stability), provided that certain necessary and sufficient stabilizability conditions are satisfied. The realization of such an LPV stabilizing compensator is based on the Youla–Kucera parametrization of all stabilizing compensators; precisely, closed-loop LPV-stability can be ensured by taking an LPV-stable realization of the Youla–Kucera parameter. In this chapter, the technique is further explored, and several issues concerning the practical implementation of the control are considered. The applications include pointwise optimality with guaranteed LPV stability, online parameter tuning, and parametric pole assignment. Some design examples are worked out to show the features of the proposed approach.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    A parametric transfer function of the form (3.1) is called proper if the degree of d(s, w) as a polynomial function of s is strictly greater than the degree of every entry of N(s, w) as a polynomial function of s.

  2. 2.

    The reason for introducing the dummy signal v(t) = 0 will become clear later.

  3. 3.

    With ℒ we denote the Laplace operator.

References

  1. Apkarian P, Gahinet P (1995) A convex characterization of gain-scheduled ℋ ∞ controllers. IEEE Trans Automat Contr 40(5):853–863

    Article  MathSciNet  MATH  Google Scholar 

  2. Barabanov NE (1988) Lyapunov indicator of discrete inclusion. I. Autom Remote Contr 49(2):152–157

    MathSciNet  MATH  Google Scholar 

  3. Barabanov NE (1988) Lyapunov indicator of discrete inclusion. II. Autom Remote Contr 49(4):283–287

    MathSciNet  MATH  Google Scholar 

  4. Barabanov NE (1988) Lyapunov indicator of discrete inclusion. III. Autom Remote Contr 49(5):558–566

    MathSciNet  MATH  Google Scholar 

  5. Becker G, Packard A (1994) Robust performance of linear parametrically varying systems using parametrically-dependent linear feedback. Syst Contr Lett 23(3):205–215

    Article  MathSciNet  MATH  Google Scholar 

  6. Bliman PA (2003) Stabilization of LPV systems. In: Proceedings of 42nd IEEE Conf decision and control, Maui, HW, pp 6103–6108

    Google Scholar 

  7. Bliman PA (2004) A convex approach to robust stability for linear systems with uncertain scalar parameters. SIAM J Contr Optim 42(6):2016–2042

    Article  MathSciNet  MATH  Google Scholar 

  8. Blanchini F (1995) Nonquadratic Lyapunov functions for robust control. Automatica 31(3):451–461

    Article  MathSciNet  MATH  Google Scholar 

  9. Blanchini F, Casagrande D, Miani S, Viaro U (2010) Stable LPV realization of parametric transfer functions and its application to gain-scheduling control design. IEEE Trans Automat Contr 55(10):2271–2281

    Article  MathSciNet  Google Scholar 

  10. Blanchini F, Miani S (2003) Stabilization of LPV systems: state feedback, state estimation and duality. SIAM J Contr Optim 42(1):76–97

    Article  MathSciNet  MATH  Google Scholar 

  11. Blanchini F, Miani S (2008) Set-theoretic methods in control, Birkhäuser, Boston, MA

    Google Scholar 

  12. Blanchini F, Miani S, Mesquine F (2009) A separation principle for linear switching systems and parametrization of all stabilizing controllers. IEEE Trans Automat Contr 54(2):279–292

    Article  MathSciNet  Google Scholar 

  13. Blanchini F, Miani S, Savorgnan C (2007) Stability results for linear parameter varying and switching systems. Automatica 43(10):1817–1823

    Article  MathSciNet  MATH  Google Scholar 

  14. Boyd S, El Ghaoui L, Feron E, Balakrishnan V (2004) Linear matrix inequalities in system and control theory, SIAM, Philadelphia, PA

    Google Scholar 

  15. Brayton RK, Tong CH (1979) Stability of dynamical systems: a constructive approach. IEEE Trans Circ Syst 26(4):224–234

    Article  MathSciNet  MATH  Google Scholar 

  16. Brayton RK, Tong CH (1980) Constructive stability and asymptotic stability of dynamical systems. IEEE Trans Circ Syst 27(11):1121–1130

    Article  MathSciNet  MATH  Google Scholar 

  17. Chesi G (2003) Robust analysis of linear systems affected by time-invariant hypercubic parameter uncertainty. In: Proceedings of 42nd IEEE Conf decision and control, Maui, HW, pp 5019–5024

    Google Scholar 

  18. Chesi G, Garulli A, Tesi A, Vicino A, (2007) Robust stability of polytopic systems via polynomially parameter-dependent Lyapunov functions. Automatica 43(2):309–316

    Article  MathSciNet  MATH  Google Scholar 

  19. Chesi G, Garulli A, Tesi A, Vicino A, (2005) Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: an LMI approach. IEEE Trans Automat Contr 50(3):365–370

    Article  MathSciNet  Google Scholar 

  20. Chesi G, Garulli A, Tesi A, Vicino A, (2009) Homogeneous polynomial forms for robustness analysis of uncertain systems. Springer, Berlin Heidelberg

    Book  MATH  Google Scholar 

  21. Dinh M, Scorletti G, Fromion V, Magarotto E (2005) Parameter dependent \({\mathcal{H}}_{\infty }\) control by finite dimensional LMI optimization: application to trade-off dependent control. Int J Robust Nonlin Contr 15(9):383–406

    Article  MathSciNet  MATH  Google Scholar 

  22. Feron E, Apkarian P, Gahinet P (1996) Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functions. IEEE Trans Automat Contr 41(7):1041–1046

    Article  MathSciNet  MATH  Google Scholar 

  23. Helmerson A (1998) μ synthesis and LFT gain scheduling with real uncertainties. Int J Robust Nonlin Contr 8(7):631–642

    Article  Google Scholar 

  24. Hespanha J, Morse AS (2002) Switching between stabilizing controllers. Automatica 38(11):1905–1917

    Article  MathSciNet  MATH  Google Scholar 

  25. Hu T, Huang B, Lin Z (2004) Absolute stability with a generalized sector condition. IEEE Trans Automat Contr 49(4):535–548

    Article  MathSciNet  Google Scholar 

  26. Iwasaki T, Tsiotras P, Zhang X (2005) State-feedback controller synthesis for parameter-dependent LTI systems. In: Proceedings of american control conference, Portland, OR, pp 593–597

    Google Scholar 

  27. Kucera V (1991) Analysis and design of discrete linear control systems. Prentice Hall, Englewood Cliffs, NJ

    Google Scholar 

  28. Lawrence DA, Rugh WJ (1995) Gain scheduling dynamic controllers for a nonlinear plant.Automatica 31(3):381–390

    Google Scholar 

  29. Liberzon D (2003) Switching in systems and control, systems and control: foundations & applications, Birkhäuser, Boston, MA

    Google Scholar 

  30. Molchanov AP, Pyatnitskii ES (1986) Lyapunov functions that define necessary and sufficient conditions for absolute stability of nonlinear nonstationary control systems. I. Autom Remote Contr 47:344–354

    MATH  Google Scholar 

  31. Molchanov AP, Pyatnitskii ES (1986) Lyapunov functions that define necessary and sufficient conditions for absolute stability of nonlinear nonstationary control systems. II. Autom Remote Contr 47:443–451

    MATH  Google Scholar 

  32. Molchanov AP, Pyatnitskii ES (1986) Lyapunov functions that define necessary and sufficient conditions for absolute stability of nonlinear nonstationary control systems. III. Autom Remote Contr 47:620–630

    MATH  Google Scholar 

  33. Rugh WJ, Shamma JS (2000) Research on gain scheduling. Automatica 36(10):1401–1425

    Article  MathSciNet  MATH  Google Scholar 

  34. Sanchez-Pena R, Sznaier M (1998) Robust systems, theory and applications. Wiley, New York

    Google Scholar 

  35. Scorletti G, El Ghaoui L (1992) Improved LMI conditions for gain scheduling and related control problems. Int J Robust Nonlin Contr 8(10):845–877

    Article  MathSciNet  Google Scholar 

  36. Shahruz SM, Behtash S (1992) Designing controllers for linear parameter-varying systems by the gain scheduling technique. J Math Anal Appl 168(1):195–217

    Article  MathSciNet  MATH  Google Scholar 

  37. Shamma JS (1996) Linearization and gain scheduling. In: Levine W (ed) The control handbook, CRC Press, pp 388–396

    Google Scholar 

  38. Shamma JS, Athans M (1990) Analysis of gain scheduled control for nonlinear plants.IEEE Trans Automat Contr  35(8):898–907

    Google Scholar 

  39. Shamma JS, Athans M (1991) Guaranteed properties of gain scheduled control for linear parameter-varying plants. Automatica 27(3):559–564

    Article  MathSciNet  MATH  Google Scholar 

  40. Youla D, Jabr H, Bongiorno J (1976) Modern Wiener–Hopf design of optimal controllers. ii. The multivariable case. IEEE Trans Automat Contr 21(3):319–338

    Google Scholar 

  41. Zelentsowsky AL (1994) Nonquadratic Lyapunov functions for robust stability analysis of linear uncertain system. IEEE Trans Automat Contr 39(1):135–138

    Article  Google Scholar 

  42. Zhou K, Doyle J, Glower K (1996) Robust and optimal control, Prentice Hall, Englewood, Cliffs, NJ

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Udine, 33100, Udine, Italy

    Franco Blanchini

  2. Department of Electrical, Managerial and Mechanical Engineering, University of Udine, 33100, Udine, Italy

    Daniele Casagrande, Stefano Miani & Umberto Viaro

Authors
  1. Franco Blanchini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniele Casagrande
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stefano Miani
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Umberto Viaro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Franco Blanchini .

Editor information

Editors and Affiliations

  1. Dept. Mechanical Engineering, University of Houston, Calhoun Road 4800, Houston, 77204-4006, Texas, USA

    Javad Mohammadpour

  2. , Department of Mathematics, University of Stuttgart, Pfaffenwaldring, Stuttgart, 70569, Germany

    Carsten W. Scherer

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Blanchini, F., Casagrande, D., Miani, S., Viaro, U. (2012). Parametric Gain-scheduling Control via LPV-stable Realization. In: Mohammadpour, J., Scherer, C. (eds) Control of Linear Parameter Varying Systems with Applications. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-1833-7_3

Download citation

  • DOI: https://doi.org/10.1007/978-1-4614-1833-7_3

  • Published:

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4614-1832-0

  • Online ISBN: 978-1-4614-1833-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation