Skip to main content
SpringerLink
Log in

Optimal compensators for nonlinear analytic discrete time processes

  • Published:
Dynamics and Control

Abstract

Construction methodology is presented, by which optimal Controllers, Estimators and Compensators can be synthesized for nonlinear discrete time processes. The nonlinearities in both the process and the measurement equations are expanded into Taylor series in the state vector, holding an arbitrary, but consistent level of truncation. Then, both the state conditional probability evolution, and the associated Dynamic Programming Equation can be solved for off line, ahead of time, retaining the same truncation level as that of the process model. This yield non-anticipative control laws that are consistently optimal, i.e., are as close to the optimum as the state models are to the exact ones. In the perfect information case the present controllers are nonlinear feedback-feedforward laws. The feedback part consists of polynomials of the state. The feedforward part includes moments of the process dynamic noise and possible forcing functions. All the gains are known ahead of time. In the fully stochastic situation, algorithms are obtained for on line update of the state conditional probability distribution, as the compensator feeds back various moments of that distribution in real time with all the gains known ahead of time.

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

Access this article

Log in via an institution

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. Lukes, D. L., “Optimal regulation of nonlinear dynamical systems,”SIAM J. Control, Vol. 7, pp. 75–100. 1969.

    Article  Google Scholar 

  2. Willemstein, A. P., “Optimal regulation of nonlinear dynamical systems on a finite interval,”SIAM J. Control and Optimization, Vol. 15, pp. 1050–1069, 1977.

    Article  Google Scholar 

  3. Bryson, A. E. and Ho, U.-C.,Applied Optimal Control. New York: John Wiley & Sons, 1975.

    Google Scholar 

  4. Hermann, R. and Krener, A. J., “Nonlinear controllability and observability,”IEEE Trans. on Automatic Control, Vol. AC 22, pp. 728–740, 1977.

    Article  Google Scholar 

  5. Fliess, M., “Lie Brackets and optimal nonlinear feebback regulation,”Proc. of the 9th IFAC World Congress, Budapest, Vol. 5, pp. 78–82, 1984.

    Google Scholar 

  6. Fliess, M. and Hazewinkel, M. (eds.),Algebraic Methods in Nonlinear Control Theory, D. Reidel: Dordrecht, 1986.

    Google Scholar 

  7. Beaman, J. J., “Non-linear quadratic Gaussian control,”International Journal of Control, Vol. 39, pp. 343–361, 1984.

    Google Scholar 

  8. Cox, H., “On the estimation of state variables and parameters for noisy dynamic systems,”IEEE Trans. on Automatic Control, Vol. 9, pp. 5–12, 1964.

    Article  Google Scholar 

  9. Jazwinski, A. H.,Stochastic Processes and Filtering Theory, New York: Academic Press, 1970.

    Google Scholar 

  10. Blankenship, G. L., Lin, C.-H. and Marcus, S. I., “Asymptotic expansions and Lie algebras for some nonlinear filtering problems,”IEEE Trans. on Automatic Control, Vol. AC-28, 1983.

  11. Tse, E. and Bar Shalom, Y., “Adaptive dual-control for stochastic nonlinear systems with free end time,”IEEE Trans. on Automatic Control, Vol. AC 20, pp. 670–675, 1975.

    Article  Google Scholar 

  12. Casler, R. J., “Dual control guidance strategy for homing interceptors taking angle-only measurements,” it AIAA J. Guidance & Control, Vol. 1, pp. 63–70, 1978.

    Google Scholar 

  13. Speyer, J. L. and Hull, D. G., “Estimation enhancement by trajectory modulation for homing missiles,” {jtProc. of the American Control Conference}, Arlington, VA., pp. 978–984, 1982.

  14. Bellman, R. E.,Adaptive Control Processes: A Guided Tour. Princeton, NJ: Princeton University Press, 1961.

    Google Scholar 

  15. Shefer, M.,Estimation and Control with Cubic Nonlinearities, Ph.D. Thesis, Standford, CA: Stanford University, 1983.

    Google Scholar 

  16. Shefer, M. and Breakwell, J. V., “Estimation and control with cubic nonlinearities,”Journal of Optimization Theory and Applications, Vol. 53, pp. 1–7, 1987.

    Article  Google Scholar 

  17. Shefer, M. and Breakwell, J. V., “Perturbation guidance laws for perfect information interceptors with symmetrical nonlinearities,”AIAA Journal of Guidance, Control and Dynamics, Vol. 9, pp. 32–36, 1986.

    Google Scholar 

  18. Shefer, M. and Breakwell, J. V., “An opitmal nonlinear compensator,”Proc. of the 27th IEEE Conference on Decision and Control, Austin, Texas, pp. 466–468, 1988.

  19. Shefer, M., “Discrete optimal tracking with prescribed disturbances,”AIAA Journal of Guidance, Control and Dynamics, Vol. 11, pp. 371, 1988.

    Google Scholar 

  20. Shefer, M., “The intrinsic approach to space robotic manipulators,”Proc. of the AIAA Guidance, Navigation and Control Conference, Portland, Oregon, 1990.

Download references

Author information

Authors and Affiliations

  1. Missile Division, RAFAEL, P.O.B. 2250, Haifa, Israel

    Moti Shefer & John V. Breakwell

Authors
  1. Moti Shefer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. John V. Breakwell
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shefer, M., Breakwell, J.V. Optimal compensators for nonlinear analytic discrete time processes. Dynamics and Control 5, 37–54 (1995). https://doi.org/10.1007/BF01968534

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01968534

Keywords

Search

Navigation