Skip to main content
SpringerLink
Log in

Cumulant minimization and robust control

  • Conference paper
  • First Online:
Stochastic Theory and Adaptive Control

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 184))

Abstract

The use of higher order spectral information has recently attracted a great deal of attention in the estimation literature. Emphasis there has been placed upon the cumulants of the random variable in question. A related development in the stochastic control literature has to do with risk sensitive optimal control. Glover has pointed out the connections between the risk sensitive control problem and H ∞ methods. In this paper we show the relation between a natural approximation to the risk-sensitive problem, by means of series expansion, and the problem of minimizing linear combinations of cumulants of the performance function. Special attention is given to the linear combination of cumulants one and two, the mean and variance respectively. Using the theory for this minimal cost variance formulation, we present numerical results for a single degree-of-freedom building excited by an earthquake process.

This work has been supported in part by the National Science Foundation under Grant BCS 90-06781, and in part by the Frank M. Freimann Chair in Electrical Engineering at the University of Notre Dame. The ideas herein were first announced at the Stochastic Theory and Adaptive Control Workshop, University of Kansas, September 26–28, 1991.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Peter Whittle, Risk Sensitive Optimal Control. New York: John Wiley & Sons, 1990.

    MATH  Google Scholar 

  2. K. Glover and J. C. Doyle, “State-Space Formulae for All Stabilizing Controllers that Satisfy an H ∞-Norm Bound and Relations to Risk Sensitivity,” Systems and Control Letters, Volume 11, pp. 167–172, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  3. Tamer Basar and Pierre Bernhard, H ∞-Optimal Control and Related Minimax Design Problems. Boston: Birkhäuser, 1991.

    Google Scholar 

  4. D. H. Jacobson, “Optimal Stochastic Linear Systems with Exponential Performance Criteria and Their Relationship to Deterministic Differential Games,” IEEE Transactions on Automatic Control, AC-18, pp. 124–131, 1973.

    Article  Google Scholar 

  5. Michael K. Sain, “On Minimal-Variance Control of Linear Systems with Quadratic Loss,” Ph.D Thesis, Department of Electrical Engineering and Coordinated Science Laboratory, University of Illinois, Urbana, Illinois, January 1965.

    Google Scholar 

  6. M. K. Sain, “Control of Linear Systems According to the Minimal Variance Criterion—A New Approach to the Disturbance Problem,” IEEE Transactions on Automatic Control, AC-11, No. 1, pp. 118–122, January 1966.

    Article  Google Scholar 

  7. P. Whittle, “A Risk-Sensitive Maximum Principle: The Case of Imperfect State Observation,” IEEE Transactions on Automatic Control, AC-36, No. 7, pp. 793–801, July 1991.

    Article  MathSciNet  Google Scholar 

  8. Jason L. Speyer, John Deyst, and David H. Jacobson, “Optimization of Stochastic Linear Systems with Additive Measurement and Process Noise Using Exponential Performance Criteria,” IEEE Transactions on Automatic Control, AC-19, No. 4, pp. 358–366, August 1974.

    Article  MathSciNet  Google Scholar 

  9. J. L. Speyer, “An Adaptive Terminal Guidance Scheme Based on an Exponential Cost Criterion with Application to Homing Missile Guidance,” IEEE Transactions on Automatic Control, AC-21, pp. 371–375, 1976.

    Article  Google Scholar 

  10. P. R. Kumar and J. H. van Schuppen, “On the Optimal Control of Stochastic Systems with an Exponential-of-Integral Performance Index,” Journal of Mathematical Analysis and Applications, Volume 80, pp. 312–332, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  11. P. Whittle, “Risk-Sensitive Linear/Quadratic/Gaussian Control,” Advances in Applied Probability, Volume 13, pp. 764–777, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  12. A. Bensoussan and J. H. van Schuppen, “Optimal Control of Partially Observable Stochastic Systems with an Exponential-of-Integral Performance Index,” SIAM Journal on Control and Optimization, Volume 23, pp. 599–613, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  13. P. Whittle and J. Kuhn, “A Hamiltonian Formulation of Risk-Sensitive Linear/Quadratic/Guassian Control,” International Journal of Control, Volume 43, pp. 1–12, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  14. W. Fleming and W. McEneaney, ‘Risk Sensitive Optimal Control,” Workshop on Stochastic Theory and Adaptive Control, University of Kansas, September 26–28, 1991.

    Google Scholar 

  15. K. Glover, “Minimum Entropy and Risk-Sensitive Control: The Continuous Time Case,” Proceedings 28th IEEE Conference on Decision and Control, pp. 388–391, December 1989.

    Google Scholar 

  16. S. R. Liberty, “Characteristic Functions of LQG Control,” Ph.D. Dissertation, Department of Electrical Engineering, University of Notre Dame, August 1971.

    Google Scholar 

  17. S. R. Liberty and R. C. Hartwig, “On the Essential Quadratic Nature of LQG Control-Performance Measure Cumulants,” Information and Control, Volume 32, Number 3, pp. 276–305, 1976.

    Article  MathSciNet  MATH  Google Scholar 

  18. M. K. Sain and S. R. Liberty, “Performance Measure Densities for a Class of LQG Control Systems,” IEEE Transactions on Automatic Control, AC-16, Number 5, pp. 431–439, October 1971.

    Article  MathSciNet  Google Scholar 

  19. S. R. Liberty and R. C. Hartwig, “Design-Performance-Measure Statistics for Stochastic Linear Control Systems,” IEEE Transactions on Automatic Control, AC-23, Number 6, pp. 1085–1090, December 1978.

    Article  Google Scholar 

  20. M. K. Sain and C. R. Souza, “A Theory for Linear Estimators Minimizing the Variance of the Error Squared,” IEEE Transactions on Information Theory, IT-14, Number 5, pp. 768–770, September 1968.

    Article  Google Scholar 

  21. A. B. Baggeroer, “A State-Variable Approach to the Solution of Fredholm Integral Equations,” IEEE Transactions on Information Theory, IT-15, pp. 557–570, September 1969.

    Article  MathSciNet  Google Scholar 

  22. M.V. Maguire, “On a Nonsymmetric Riccati Equation of Stochastic Control,” M.S. Thesis, Department of Electrical Engineering, University of Notre Dame, Notre Dame, Indiana, 1971.

    Google Scholar 

  23. L. L. Chung, A. M. Reinhorn, and T. T. Soong, “Experiments on Active Control of Seismic Structures,” Journal of Engineering Mechanics, Volume 114, Number 2, pp. 241–256, February 1988.

    Article  Google Scholar 

  24. J. N. Yang and M. J. Lin, “Building Critical-Mode Control: Nonstationary Earthquake,” Journal of Engineering Mechanics, Volume 109, Number 6, pp. 1375–1389, December 1983.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, University of Notre Dame, 46556, Notre Dame, Indiana, USA

    Michael K. Sain

  2. Department of Electrical Engineering, University of Notre Dame, 46556, Notre Dame, Indiana, USA

    Chang-Hee Won

  3. Department of Civil Engineering and Geological Sciences, University of Notre Dame, 46556, Notre Dame, Indiana, USA

    B. F. Spencer

Authors
  1. Michael K. Sain
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chang-Hee Won
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. F. Spencer
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

T. E. Duncan B. Pasik-Duncan

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer-Verlag

About this paper

Cite this paper

Sain, M.K., Won, CH., Spencer, B.F. (1992). Cumulant minimization and robust control. In: Duncan, T.E., Pasik-Duncan, B. (eds) Stochastic Theory and Adaptive Control. Lecture Notes in Control and Information Sciences, vol 184. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0113257

Download citation

  • DOI: https://doi.org/10.1007/BFb0113257

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55962-7

  • Online ISBN: 978-3-540-47327-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation