Advertisement

Computational Optimization and Applications

, Volume 37, Issue 2, pp 177–218 | Cite as

Nonlinear feedback controllers and compensators: a state-dependent Riccati equation approach

  • H. T. BanksEmail author
  • B. M. Lewis
  • H. T. Tran
Article

Abstract

State-dependent Riccati equation (SDRE) techniques are rapidly emerging as general design and synthesis methods of nonlinear feedback controllers and estimators for a broad class of nonlinear regulator problems. In essence, the SDRE approach involves mimicking standard linear quadratic regulator (LQR) formulation for linear systems. In particular, the technique consists of using direct parameterization to bring the nonlinear system to a linear structure having state-dependent coefficient matrices. Theoretical advances have been made regarding the nonlinear regulator problem and the asymptotic stability properties of the system with full state feedback. However, there have not been any attempts at the theory regarding the asymptotic convergence of the estimator and the compensated system. This paper addresses these two issues as well as discussing numerical methods for approximating the solution to the SDRE. The Taylor series numerical methods works only for a certain class of systems, namely with constant control coefficient matrices, and only in small regions. The interpolation numerical method can be applied globally to a much larger class of systems. Examples will be provided to illustrate the effectiveness and potential of the SDRE technique for the design of nonlinear compensator-based feedback controllers.

Keywords

Nonlinear feedback control Nonlinear compensator State-dependent Riccati equations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anderson, B.D.O., Moore, J.B.: Optimal Control Linear Quadratic Methods. Prentice-Hall, Englewood Cliffs (1990) zbMATHGoogle Scholar
  2. 2.
    Banks, H.T., Beeler, S.C., Kepler, G.M., Tran, H.T.: Feedback control of thin film growth in an HPCVD reactor via reduced order models. In: Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL. IEEE, Los Alamitos (2001) Google Scholar
  3. 3.
    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). CRSC Technical report CRSC-TR00-33, NCSU zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    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). CRSC Technical report CRSC-TR01-25, NCSU zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Beeler, S.C.: Modeling and control of thin film growth in a chemical vapor deposition reactor. Ph.D. dissertation, North Carolina State University, Raleigh (2000) Google Scholar
  6. 6.
    Beeler, S.C., Tran, H.T., Banks, H.T.: Feedback control methodologies for nonlinear systems. J. Optim. Theory Appl. 107(1) 1–33 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Beeler, S.C., Tran, H.T., Banks, H.T.: State estimation and tracking control of nonlinear dynamical systems. In: Desch, W., Kappel, F., Kunisch, K. (eds.) Control and Estimation of Distributed Parameter Systems, International Series of Numerical Mathematics, vol. 143, pp. 1–24. Birkhäuser, Basel (2002). CRSC Technical report CRSC-TR00-19, NCSU Google Scholar
  8. 8.
    Brauer, F., Nohel, J.A.: The Qualitative Theory of Ordinary Differential Equations. Dover, Mineola (1989) Google Scholar
  9. 9.
    Cloutier, J.R., Mracek, C.P., Ridgely, D.B., Hammett, K.D.: State-dependent Riccati equation techniques: theory and applications. In: Notes from the SDRE Workshop Conducted at the American Control Conference, Philadelphia, PA. IEEE, Los Alamitos (1998) Google Scholar
  10. 10.
    Cloutier, J.R., D’Souza, C.N., Mracek, C.P.: Nonlinear regulation and nonlinear h control via the state-dependent Riccati equation technique: part 1. Theory. In: Proceedings of the First International Conference on Nonlinear Problems in Aviation and Aerospace, Daytona Beach, FL. European Conference Publishers, London (1996) Google Scholar
  11. 11.
    Cloutier, J.R., Stansbery, D.T.: Nonlinear, hybrid bank-to-turn/skid-to-turn autopilot design. In: Proceedings of the AIAA Guidance, Navigation, and Control Conference, Montreal, Canada. AIAA, Reston (2001) Google Scholar
  12. 12.
    Doyle, J., Huang, Y., Primbs, J., Freeman, R., Murray, R., Packard, A., Krstic, M.: Nonlinear control: Comparisons and case studies. In: Notes from the Nonlinear Control Workshop conducted at the American Control Conference, Albuquerque, NM. IEEE, Los Alamitos (1998) Google Scholar
  13. 13.
    Erdem, E.B., Alleyne, A.G.: Experimental real-time SDRE control of an underactuated robot. In: Proceedings of the American Control Conference, San Diego, CA. IEEE, Los Alamitos (1999) Google Scholar
  14. 14.
    Erdem, E.B., Alleyne, A.G.: Globally stabilizing second-order nonlinear systems by SDRE control. In: Proceedings of the American Control Conference, San Diego, CA. IEEE, Los Alamitos (1999) Google Scholar
  15. 15.
    Friedland, B.: Advanced Control System Design. Prentice-Hall, Englewood Cliffs (1996) zbMATHGoogle Scholar
  16. 16.
    Friedland, B.: Feedback control of systems with parasitic effects. In: Proceedings of the American Control Conference, Albuquerque, New Mexico. IEEE, Los Alamitos (1997) Google Scholar
  17. 17.
    Hammett, K.D.: Control of Nonlinear Systems via state feedback state-dependent Riccati equation techniques. Ph.D. dissertation, Air Force Institute of Technology, Wright-Patterson AFB, Ohio, 1997 Google Scholar
  18. 18.
    Hammett, K.D., Hall, C.D., Ridgely, D.B.: Controllability issues in nonlinear state-dependent Riccati equation control. AIAA J. Guid. Control Dyn. 21(5), 767–773 (1998) CrossRefGoogle Scholar
  19. 19.
    Hu, X.: On state observers for nonlinear systems. Systems Control Lett. 17, 465–473 (1991) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Huang, Y., Lu, W.M.: Nonlinear optimal control: alternatives to Hamilton–Jacobi equation. In: Proceedings of the IEEE Conference on Decision and Control, Kobe, Japan. IEEE, Los Alamitos (1996) Google Scholar
  21. 21.
    Hull, R.A., Cloutier, J.R., Mracek, C.P., Stansbery, D.T.: State-dependent Riccati equation solution of the toy nonlinear optimal control problem. In: Proceedings of the American Control Conference, Philadelphia, PA. IEEE, Los Alamitos (1998) Google Scholar
  22. 22.
    Isidori, A.: Nonlinear Control Systems. Springer, New York (1995) zbMATHGoogle Scholar
  23. 23.
    Ito, K., Schroeter, J.D.: Reduced order feedback synthesis for viscous incompressible flows. Math. Comput. Model. 33, 173–192 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Kirschner, D.: Using mathematics to understand HIV immune dynamics. Not. Am. Math. Soc., 191–202 (February 1996) Google Scholar
  25. 25.
    Krstic, M., Kanellakopoulos, I., Kokotovic, P.: Nonlinear and Adaptive Control Design. Wiley, New York (1995) Google Scholar
  26. 26.
    Lewis, F.L., Syrmos, V.L.: Optimal Control. Wiley, New York (1995) Google Scholar
  27. 27.
    Markman, J., Katz, I.N.: An iterative algorithm for solving Hamilton–Jacobi type equations. SIAM J. Sci. Comput. 22(1), 312–329 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Mracek, 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. AIAA, Reston (1997) Google Scholar
  29. 29.
    Mracek, C.P., Cloutier, J.R.: Control designs for the nonlinear benchmark problem via the state-dependent Riccati equation method. Int. J. Robust Nonlinear Control 8(4–5), 401–433 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Mracek, C.P., Cloutier, J.R.: Control designs for the nonlinear benchmark problem via the state-dependent Riccati equation method. Int. J. Robust Nonlinear Control 8, 401–433 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Palumbo, N.F., Jackson, T.: Development of a fully integrated missile guidance and control system: a state-dependent Riccati differential equation approach. In: Proceedings of the Conference on Control Applications, Hawaii. IEEE, Los Alamitos (1999) Google Scholar
  32. 32.
    Parrish, D.K., Ridgely, D.B.: Attitude control of a satellite using the SDRE method. In: Proceedings of the American Control Conference, Albuquerque, NM. IEEE, Los Alamitos (1997) Google Scholar
  33. 33.
    Parrish, D.K., Ridgely, D.B.: Control of an artificial human pancreas using the SDRE method. In: Proceedings of the American Control Conference, Albuquerque, NM. IEEE, Los Alamitos (1997) Google Scholar
  34. 34.
    Qu, Z., Cloutier, J.R., Mracek, C.P.: A new suboptimal nonlinear control design technique. In: Proceedings of the 13th IFAC World Congress, San Francisco, CA, 1996 Google Scholar
  35. 35.
    Rodman, L.: On extremal solutions of the algebraic Riccati equation. In: Byrnes, C.I., Martin, C.F. (eds.) Algebraic and Geometric Methods in Linear Systems Theory, Lectures in Applied Mathematics, vol. 18. American Mathematical Society, Providence (1980) Google Scholar
  36. 36.
    Shamma, J.S., Athens, M.: Analysis of gain scheduled control for nonlinear plants. IEEE Trans. Autom. Control 35(8), 898–907 (1990) zbMATHCrossRefGoogle Scholar
  37. 37.
    Shamma, J.S., Cloutier, J.R.: Existence of SDRE stabilizing feedback. In: Proceedings of the American Control Conference, Arlington, VA, 2001 Google Scholar
  38. 38.
    Slotine, J.-J.E.: Applied Nonlinear Control. Prentice-Hall, Englewood Cliffs (1991) zbMATHGoogle Scholar
  39. 39.
    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, Chicago, IL, 2000 Google Scholar
  40. 40.
    Sznaier, M., Cloutier, J.R., Hull, R.A., Jacques, D., Mracek, C.P.: Receding horizon control Lyapunov function approach to suboptimal regulation of nonlinear systems. AIAA J. Guid. Control Dyn. 23(3), 399–405 (2000) Google Scholar
  41. 41.
    Thau, F.E.: Observing the state of non-linear dynamic systems. Int. J. Control 17, 471–479 (1973) zbMATHGoogle Scholar
  42. 42.
    Theodoropoulou, A., Adomaitis, R.A., Zafiriou, E.: Model reduction for optimization of rapid thermal chemical vapor deposition systems. IEEE Trans. Semicond. Manuf. 11, 85–98 (1998) CrossRefGoogle Scholar
  43. 43.
    To, L.C., Tade, M.O., Kraetzl, M.: Robust Nonlinear Control of Industrial Evaporation Systems. World Scientific, River Edge (1999) zbMATHGoogle Scholar
  44. 44.
    Wernli, A., Cook, G.: Suboptimal control for the nonlinear quadratic regulator problem. Automatica 11, 75–84 (1975) zbMATHCrossRefGoogle Scholar
  45. 45.
    Zhou, K., Doyle, J., Glover, K.: Robust and Optimal Control. Prentice-Hall, Englewood Cliffs (1996) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Center for Research in Scientific Computation, Department of MathematicsNorth Carolina State UniversityRaleighUSA
  2. 2.MIT Lincoln LaboratoryMassachussetts Institute of TechnologyLexingtonUSA

Personalised recommendations