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Nonlinear Dynamics

, Volume 70, Issue 1, pp 693–707 | Cite as

Observer-based controller for discrete-time systems: a state dependent Riccati equation approach

  • Mohamed F. Hassan
Original Paper

Abstract

In this paper, an observer-based controller for discrete-time nonlinear dynamical systems is proposed. After transforming the nonlinear system to a linear structure having state-dependent coefficient matrices (SDC), a recursive regularized least-square (RLS) state estimator is developed. The observed states are then used to generate either a constrained or unconstrained state feedback controller using the state dependent Riccati equation (SDRE) approach. The stability of the observer-based control system is rigorously analyzed in a theoretical frame work. Applications to different numerical examples as well as to a practical case study demonstrate the effectiveness of the proposed procedure.

Keywords

State-Dependent Riccati Equation (SDRE) Nonlinear estimation Stability Discrete-time systems Power systems 

References

  1. 1.
    Banks, H.T., Bortz, D.M., Holte, S.E.: Incorporation of variability into the modeling of viral delays in HIV infection dynamics. Math. Biosci. 183, 63–91 (2003) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Banks, H.T., Beeler, S.C., Kepler, G.M., Tran, H.T.: Reduced order modeling and control of thin film growth in an HPCVD reactor. SIAM J. Appl. Math. 62(4), 1251–1280 (2002) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Parrish, D.K., Ridgely, D.B.: Attitude control of a satellite using the SDRE method. In: Proc. of the American Control Conference 1997, Albuquerque, NM, pp. 942–946 (1997) Google Scholar
  4. 4.
    Zhou, K., Doyle, J., Glover, K.: Robust and Optimal Control. Prentice-Hall, Englewood Cliffs (1996) MATHGoogle Scholar
  5. 5.
    Isidori, A.: Nonlinear Control Systems. Springer, New York (1995) MATHGoogle Scholar
  6. 6.
    Shamma, J.S., Athens, M.: Analysis of gain scheduled control for nonlinear plants. IEEE Trans. Autom. Control 35(8), 898–907 (1990) MATHCrossRefGoogle Scholar
  7. 7.
    Krstic, M., Kanellakopoulos, I., Kokotovic, P.: Nonlinear and Adaptive Control Design. Wiley, New York (1995) Google Scholar
  8. 8.
    Peng, C.-C., Hsue, A.W.-J., Chen, C.-L.: Variable structure based robust backstepping controller design for nonlinear systems. Nonlinear Dyn. 63, 253–262 (2011) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Bowong, S.: Adaptive synchronization of chaotic systems with unknown bounded uncertainties vis backstepping approach. Nonlinear Dyn. 49, 59–70 (2007) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Slotine, J.-J.: Applied Nonlinear Control. Prentice-Hall, Englewood Cliffs (1991) MATHGoogle Scholar
  11. 11.
    Huang, Y.J., Wang, Y.J.: Steady-state analysis for a class of sliding mode controlled systems using describing function method. Nonlinear Dyn. 30, 223–241 (2002) MATHCrossRefGoogle Scholar
  12. 12.
    Koofigar, H.R., Hosseinnia, S., Sheikoleslam: Robust adaptive nonlinear control for uncertain control-affine systems and its applications. Nonlinear Dyn. 56, 13–22 (2009) MATHCrossRefGoogle Scholar
  13. 13.
    Li, T., Wang, D., Chen, N.: Adaptive fuzzy control of uncertain MIMO nonlinear systems in block-triangular forms. Nonlinear Dyn. 63, 105–123 (2011) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Liu, Y.J., Wang, Z.F.: Adaptive fuzzy controller design of nonlinear systems with unknown gain sign. Nonlinear Dyn. 58(4), 687–695 (2009) MATHCrossRefGoogle Scholar
  15. 15.
    Beeler, S.C., Tran, H.T., Banks, H.T.: Feedback control methodologies for nonlinear systems. J. Optim. Theory Appl. 107(1), 1–33 (2000) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Wernli, A., Cook, G.: Suboptimal control for the nonlinear quadratic regulator problem. Automatica 11, 75–84 (1975) MATHCrossRefGoogle Scholar
  17. 17.
    Marcek, C.P., Cloutier, J.R.: Control design for the nonlinear benchmark problem via the state-dependent Riccati equation method. Int. J. Robust Nonlinear Control 8, 401–433 (1998) CrossRefGoogle Scholar
  18. 18.
    Friedland, B.: Advanced Control System Design. Prentice-Hall, Englewood Cliffs (1996) MATHGoogle Scholar
  19. 19.
    Banks, H.T., Lewis, B.M., Tran, H.T.: Nonlinear feedback controllers and compensators: a state dependent Riccati equation approach. Comput. Optim. Appl. 37, 177–218 (2007) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Cloutier, J.R., D’Souza, C.N., Marcek, C.P.: Nonlinear regulation and nonlinear H control via the state-dependent Riccati equation technique: Part I. Theory, Part II. Examples. In: Proceedings of the First International Conference on Nonlinear Problems in Aviation and Aerospace 1996, Daytona Beach, FL, pp. 117–141 (1996) Google Scholar
  21. 21.
    Marcek, C.P., Cloutier, J.R.: Missile longitudinal autopilot design using the state-dependent Riccati equation method. In: Proceeding of the International Conference on Nonlinear Problems in Aviation and Aerospace 1996, pp. 387–396. Daytona, Beach, FL (1996) Google Scholar
  22. 22.
    Cimen, T.: State-dependent Riccati equation (SDRE): a survey. In: Proceeding of the 17th World Congress, The International Federation of Automatic Control, Seoul, Korea 2008, 6–11 July, pp. 3761–3774 (2008) Google Scholar
  23. 23.
    Cloutier, J.R., Zipfel, P.H.: Hypersonic guidance via the state-dependent Riccati equation control method. In: Proceeding of the IEEE Conference on Control Applications, Hawaii, HI, 1999, pp. 219–224 (1999) Google Scholar
  24. 24.
    Marcek, C.P., Cloutier, J.R.: Full envelope missile longitudinal autopilot design using the state-dependent Riccati equation method. In: Proceedings of the AIAA Guidance, Navigation and Control Conference, New Orleans, LA, 1997, pp. 1697–1705 (1997) Google Scholar
  25. 25.
    Sun, P., Liu, K.: Missile autopilot design based on state-dependent Riccati equation. In: International Asia Conference on Information in Control, Automation and Robotics 2009, pp. 134–138 (2009) CrossRefGoogle Scholar
  26. 26.
    Hammett, K.D., Hall, C.D., Ridgely, D.B.: Controllability issues in state-dependent Riccati equation control. J. Guid. Control Dyn. 21, 767–773 (1998) CrossRefGoogle Scholar
  27. 27.
    Stansbery, D.T., Cloutier, J.R.: Position and attitude control of a spacecraft using the state-dependent Riccati equation technique. In: Proceedings of the American Control Conference 2000, Chicago, IL, pp. 1867–1871 (2000) Google Scholar
  28. 28.
    Singh, S.N., Yimm, W.: State feedback control of an aeroelastic system with structure nonlinearity. Aerosp. Sci. Technol. 7, 23–31 (2003) MATHCrossRefGoogle Scholar
  29. 29.
    Tadi, M.: State-dependent Riccati equation for control of aeroelastic flutter. J. Guid. Control Dyn. 26, 914–917 (2003) CrossRefGoogle Scholar
  30. 30.
    Banks, H.T., Beeler, S.C., Kepler, G.M., Tran, H.T.: Reduced order modeling and control of thin film growth in an HPCVD reactor. SIAM J. Appl. Math. 62, 1251–1280 (2002) MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Erdem, E.B., Alleyne, A.G.: Experimental real-time SDRE control of an under actuated robot. In: Proceedings of the 40th IEEE Conference on Decision and Control 2001, Piscataway, NJ, pp. 219–224 (2001) Google Scholar
  32. 32.
    Erdem, E.B., Alleyne, A.G.: Design of a class of nonlinear controllers via state-dependent Riccati equation. IEEE Trans. Control Syst. Technol. 12, 2986–2991 (2004) CrossRefGoogle Scholar
  33. 33.
    Naik, M.S., Singh, S.N.: State-dependent Riccati equation-based robust dive plane control of AUV with control constraints. Ocean Eng. 34, 1711–1723 (2007) CrossRefGoogle Scholar
  34. 34.
    Steinfeldt, B.A., Tsiotras, P.: A state-dependent Riccati equation approach to atmospheric entry guidance. In: Guidance, Navigation, and Control Conference 2020, Toronto, Ontario, Canada, pp. 1–20 (2010) Google Scholar
  35. 35.
    Pergher, R., Bottega, V., Molter, A.: Musculetendinidiun postural stabilization control based on state-dependent. Riccati equation. In: 2nd International Conference on Engineering Optimization 2010, Lispon, Portugal, pp. 1–8 (2010) Google Scholar
  36. 36.
    Sayed, A.H.: A framework for state-space estimation with uncertain models. IEEE Trans. Autom. Control 46(7), 998–1013 (2001) MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Kirk, D.E.: Optimal Control Theory: An Introduction. Prentice Hall, Englewood Cliffs (1970) Google Scholar
  38. 38.
    Holet, J.M.: Discrete Gronwall lemma and applications. http://www.math.purdue.edu/
  39. 39.
    Struble, R.A.: Nonlinear Differential Equations. McGraw-Hill, New York (1962) MATHGoogle Scholar
  40. 40.
    Kundur, P.: Power System Stability and Control. McGraw-Hill, New York (1994) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department of Electrical EngineeringKuwait UniversitySafatKuwait

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