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Polytopic Observers for LPV Discrete-Time Systems

  • Chapter
Robust Control and Linear Parameter Varying Approaches

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 437))

Abstract

The main goal of this work is to give a general treatment on observer synthesis for LPV systems in the framework of Linear Matrix Inequalities.A special Parameter Dependent Lyapunov Function, called poly-quadratic Lyapunov function, is considered. It incorporates the parameter variations for LPV systems with polytopic parameter dependence and allows to guarantee a so-called poly-quadratic stability which is sufficient to ensure Global Asymptotic Stability. The concept of polytopic observers is introduced. A LMI-based method for the synthesis of this type of observers is proposed. The case when LPV systems are subjected to disturbances or when the parameter is known with a bounded level of uncertainty is further addressed. Conditions to guarantee performances like Input-to-State Stability (ISS), bounded peak-to-peak gain and L2 gain are given. The design of polytopic Unknown Input Observers both in the deterministic and in the noisy or uncertain cases is also presented. Finally, two illustrative examples dealing with polytopic observers for chaos synchronization and air path management of a turbocharged Spark Ignition engine are detailed.

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References

  1. Allison, D.C.S., Noga, M.T.: Computing the three-dimensional convex hull. Computer Physics Communications 103(1), 74–82 (1997)

    Article  Google Scholar 

  2. Apkarian, P., Gahinet, P., Becker, G.: A convex characterization of gain-scheduled H  ∞  Controllers. IEEE Transactions on Automatic Control 40(5), 853–864 (1995)

    Article  MATH  Google Scholar 

  3. Balas, G., Bokor, J., Szabo, Z.: Invariant subspaces for lpv systems and their applications. IEEE Trans. on Automatic Control 48(11), 2065–2069 (2003)

    Article  MathSciNet  Google Scholar 

  4. Bara, G.I., Daafouz, J., Kratz, F., Ragot, J.: Parameter-dependent state observer design for affine LPV systems. International Journal of Control 74(16), 1601–1611 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Becker, G., Packard, A., Philbrick, D., Balas, G.: Control of parametrically-dependent linear systems: a single quadratic lyapunov approach. In: Proc. of American Control Conference, San Fransisco (June 1993)

    Google Scholar 

  6. Bliman, P.-A.: Nonconservative lmi approach to robust stability for systems with uncertain scalar parameters. In: Proceeding of 41st IEEE Control and Decision Conference, Las Vegas, USA (December 2002)

    Google Scholar 

  7. Bruzelius, F.: Linear Parameter-Varying Systems: an approach to gain scheduling. PhD thesis, Department of Signals and Systems, Chalmers University of Technology, Goteborg, Sweden (2004)

    Google Scholar 

  8. Casavola, A., Famularo, D., Franze, G.: A feedback min-max mpc algorithm for lpv systems subject to bounded rates of change of parameters. IEEE Transactions on Automatic Control 47(7) (2002)

    Google Scholar 

  9. Chatterjee, S., Chatterjee, S.: A note of finding extreme points in multivariable space. Computational Statistics and Data Analysis 10(1), 87–92 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  10. Colin, G., Bloch, G., Chamaillard, Y., Anstett, F.: Two air path observers for turbocharged SI engines with VCT. Control Engineering Practice 17(5), 571–578 (2009)

    Article  Google Scholar 

  11. Daafouz, J., Bernussou, J.: Parameter dependent lyapunov functions for discrete time systems with time varying parametric uncertainties. Systems and Control Letters 43, 355–359 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Daafouz, J., Millérioux, G., Iung, C.: A poly-quadratic stability based approach for switched systems. International Journal of Control 75, 1302–1310 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Daafouz, J., Millérioux, G., Rosier, L.: Observer design with guaranteed bound for lpv systems. In: Proc. of the 16th IFAC World Congress, Prague, Czech Republic, July 4-8 (2005)

    Google Scholar 

  14. Darouach, M., Zazadinski, M., Xu, S.J.: Full-order observers for linear systems with unknown inputs. IEEE Trans. on Automatic Control 39(3), 606–609 (1994)

    Article  MATH  Google Scholar 

  15. Eddy, W.F.: A new convex hull algorithm for planar sets. ACM Transactions on Mathematical Software 3, 398–403 (1977)

    Article  MATH  Google Scholar 

  16. Farhood, M., Dullerud, G.E.: Control of nonstationary LPV systems. Automatica 44, 2108–2119 (2010)

    Article  MathSciNet  Google Scholar 

  17. Feron, E., Apkarian, P., Gahinet, P.: Analysis and synthesis of robust control systems via parameter-dependent lyapunov functions. IEEE Trans. on Automatic Control 41, 1041–1046 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gahinet, P., Apkarian, P., Chilali, M.: Parameter-dependent lyapunov functions for real parametric uncertainty. IEEE Trans. on Automatic Control 41(3), 436–442 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Graham, R.L.: An efficient algorithm for determining the convex hull of a finite planar set. Information Processing Letters 2(1), 132–133 (1973)

    Google Scholar 

  20. Heemels, M., Daafouz, J., Millérioux, G.: Observer-based control of discrete-time lpv systems with uncertain parameters. IEEE Trans. on Automatic Control 55(9), 2130–2135 (2010)

    Article  Google Scholar 

  21. Jungers, M., Oliveira, R., Peres, P.: Mpc for lpv systems with bounded parameter variations. International Journal of Control 84(1) (2011)

    Google Scholar 

  22. El Ghaoui, L., Niculescu, S.-I. (eds.): Advances in Linear Matrix Inequality Methods in Control. SIAM’s Advances in Design and Control (2000)

    Google Scholar 

  23. Lee, S.M., Park, J.H.: Output feedback model predictive control for LPV systems using parameter-dependent Lyapunov-function. Applied Mathematics and Computation 190, 671–676 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Leith, D.J., Leithead, W.E.: Survey of gain scheduling analysis and design. International Journal of Control 73, 1001–1025 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Millerioux, G., Anstett, F., Bloch, G.: Considering the attractor structure of chaotic maps for observer-based synchronization problems. Mathematics and Computers in Simulation 68(1), 67–85 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Millérioux, G., Daafouz, J.: Global chaos synchronization and robust filtering in noisy context. IEEE Trans. on Circuits and Systems I: Fundamental Theory and Applications 48(10), 1170–1176 (2001)

    Article  Google Scholar 

  27. Millérioux, G., Daafouz, J.: Performances of unknown input observers for chaotic lpv maps in a stochastic context. In: Proc. of 12th International IEEE Workshop on Nonlinear Dynamics of Electronic Systems, NDES 2004, Evora, Portugal (May 2004)

    Google Scholar 

  28. Millérioux, G., Daafouz, J.: Performances of unknown input observers for chaotic lpv maps in a stochastic context. Nonlinear Dynamics 44, 205–212 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Millérioux, G., Daafouz, J.: Flatness of switched linear discrete-time systems. IEEE Trans. on Automatic Control 54(3), 615–619 (2009)

    Article  Google Scholar 

  30. Millérioux, G., Rosier, L., Bloch, G., Daafouz, J.: Bounded state reconstruction error for lpv systems with estimated parameters. IEEE Trans. on Automatic Control 49(8), 1385–1389 (2004)

    Article  Google Scholar 

  31. Montagner, V.F., Oliveira, R., Leite, V., Peres, P.: Gain scheduled state feedback control of discrete-time systems with time-varying uncertainties: an lmi approach. In: Procc. of the 44th IEEE Conf. on Deciion and Control (CDC 2005), Seville, Spain (December 2005)

    Google Scholar 

  32. Packard, A., Balas, G.: Theory and application of linear parameter-varying control techniques. In: Proceedings of the American Control Conference, Tutorial Workshop (1997)

    Google Scholar 

  33. Pardalos, P.M., Li, Y., Hager, W.W.: Linear programming approaches to the convex hull problem in ℝm. Computers and Mathematics with Applications 29(7), 23–29 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  34. Park, B.P., Verriest, E.I.: Canonical forms for linear time-varying multivariable discrete systems. In: Proceedings. 28th Annual Allerton Conf. on Communication, Control and Computers, Urbana, USA (1990)

    Google Scholar 

  35. Preparata, F.P., Shamos, M.I.: Computational Geometry. Springer (1985)

    Google Scholar 

  36. Rugh, W., Shamma, J.: Research on gain scheduling. Automatica 36, 1401–1425 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  37. Sato, M.: Filter design for lpv systems using quadratically parameter-dependent lyapunov functions. Automatica 42, 2017–2023 (2006)

    Article  MATH  Google Scholar 

  38. Scherer, C.: LPV control and full block multipliers. Automatica 37, 361–375 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  39. Silverman, L.M.: Inversion of multivariable linear systems. IEEE Trans. Automatic Control 14(3), 270–276 (1969)

    Article  MathSciNet  Google Scholar 

  40. Toth, R., Felici, F., Heuberger, P.S.C., Van den Hof, P.M.J.: Discrete time lpv i/o and state-space representations, differences of behavior and pitfalls of interpolation. In: Proc. of the European Control Conference (ECC 2007), Kos, Greece (2007)

    Google Scholar 

  41. Wu: Control of Linear Parameter Varying Systems. PhD thesis, University of Carlifonia at Berkeley, Mechanical Engineering (1995)

    Google Scholar 

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Authors and Affiliations

  1. CRAN Centre de Recherche en Automatique de Nancy (UMR CNRS 7039), Université de Lorraine, Campus Sciences, BP 70239, 54506, VANDOEUVRE Cedex, France

    Meriem Halimi, Gilles Millerioux & Jamal Daafouz

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  1. Meriem Halimi
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  2. Gilles Millerioux
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  3. Jamal Daafouz
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Editors and Affiliations

  1. Systems (Dpt Automatique), UMR CNRS 5216, GIPSA-lab, Department of Control, Rue de la Houille Blanche 11, Saint Martin d'Heres Cedex, 38 402, France

    Olivier Sename

  2. , Computer & Automation Research Institute, Hungarian Academy of Sciences, Kende u. 13-17, Budapest, 1111, Hungary

    Peter Gaspar

  3. , Computer & Automation Research Institute, Hungarian Academy of Sciences, Kende u. 13-17, Budapest, 1111, Hungary

    József Bokor

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Halimi, M., Millerioux, G., Daafouz, J. (2013). Polytopic Observers for LPV Discrete-Time Systems. In: Sename, O., Gaspar, P., Bokor, J. (eds) Robust Control and Linear Parameter Varying Approaches. Lecture Notes in Control and Information Sciences, vol 437. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36110-4_5

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  • DOI: https://doi.org/10.1007/978-3-642-36110-4_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-36109-8

  • Online ISBN: 978-3-642-36110-4

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