Skip to main content
Log in

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

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

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.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anderson, B.D.O., Moore, J.B.: Optimal Control Linear Quadratic Methods. Prentice-Hall, Englewood Cliffs (1990)

    MATH  Google Scholar 

  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. 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

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  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)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. Brauer, F., Nohel, J.A.: The Qualitative Theory of Ordinary Differential Equations. Dover, Mineola (1989)

    Google Scholar 

  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. 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. 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. 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. 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. 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. Friedland, B.: Advanced Control System Design. Prentice-Hall, Englewood Cliffs (1996)

    MATH  Google Scholar 

  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. 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

  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)

    Article  Google Scholar 

  19. Hu, X.: On state observers for nonlinear systems. Systems Control Lett. 17, 465–473 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. Isidori, A.: Nonlinear Control Systems. Springer, New York (1995)

    MATH  Google Scholar 

  23. Ito, K., Schroeter, J.D.: Reduced order feedback synthesis for viscous incompressible flows. Math. Comput. Model. 33, 173–192 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  24. Kirschner, D.: Using mathematics to understand HIV immune dynamics. Not. Am. Math. Soc., 191–202 (February 1996)

  25. Krstic, M., Kanellakopoulos, I., Kokotovic, P.: Nonlinear and Adaptive Control Design. Wiley, New York (1995)

    Google Scholar 

  26. Lewis, F.L., Syrmos, V.L.: Optimal Control. Wiley, New York (1995)

    Google Scholar 

  27. Markman, J., Katz, I.N.: An iterative algorithm for solving Hamilton–Jacobi type equations. SIAM J. Sci. Comput. 22(1), 312–329 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. 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. 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

  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. Shamma, J.S., Athens, M.: Analysis of gain scheduled control for nonlinear plants. IEEE Trans. Autom. Control 35(8), 898–907 (1990)

    Article  MATH  Google Scholar 

  37. Shamma, J.S., Cloutier, J.R.: Existence of SDRE stabilizing feedback. In: Proceedings of the American Control Conference, Arlington, VA, 2001

  38. Slotine, J.-J.E.: Applied Nonlinear Control. Prentice-Hall, Englewood Cliffs (1991)

    MATH  Google Scholar 

  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

  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. Thau, F.E.: Observing the state of non-linear dynamic systems. Int. J. Control 17, 471–479 (1973)

    MATH  Google Scholar 

  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)

    Article  Google Scholar 

  43. To, L.C., Tade, M.O., Kraetzl, M.: Robust Nonlinear Control of Industrial Evaporation Systems. World Scientific, River Edge (1999)

    MATH  Google Scholar 

  44. Wernli, A., Cook, G.: Suboptimal control for the nonlinear quadratic regulator problem. Automatica 11, 75–84 (1975)

    Article  MATH  Google Scholar 

  45. Zhou, K., Doyle, J., Glover, K.: Robust and Optimal Control. Prentice-Hall, Englewood Cliffs (1996)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to H. T. Banks.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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). https://doi.org/10.1007/s10589-007-9015-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-007-9015-2

Keywords

Navigation