Skip to main content
SpringerLink
Log in
Book cover

The Riccati Equation pp 243–262Cite as

The Infinite Horizon and the Receding Horizon LQ-Problems with Partial Stabilization Constraints

  • Chapter

Part of the book series: Communications and Control Engineering Series ((CCE))

Abstract

One of the main reasons why the Riccati equation and its generalizations has become very important in the theory of control, systems, and signals, is that it shows up in a very straightforward way in the analysis of two benchmark problems in control system design and signal filtering.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   89.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anderson, B.D.O.; Moore, J.B.: Linear Optimal Control. Prentice Hall, Englewood Cliffs, N.J. 1971

    MATH  Google Scholar 

  2. Anderson, B.D.O.; Moore, J.B.:Optimal Filtering. Prentice-Hall, Englewood Cliffs, N.J. 1979

    MATH  Google Scholar 

  3. Brunovsky, P.; Komornik, J.: The Riccati equation solution of the linear-quadratic problem with constrained terminal state. IEEE Transactions on Automatic Control AC-26 (1981) 398–402

    Google Scholar 

  4. Bucy, R.S.; Joseph, P.D.: Filtering for Stochastic Processes with Applications to Guidance. Wiley, New-York 1968

    MATH  Google Scholar 

  5. Callier, F.M.; Willems, J.L.:Criterion for the convergence of the solution of the Riccati differential equation. IEEE Transactions on Automatic Control AC-26 (1981) 1232–1242

    Google Scholar 

  6. Chan, W.; Goodwin, G.C.; Sin, K.S.: Convergence properties of the Riccati difference equation in optimal filtering of nonstabilizable systems. IEEE Transactions on Automatic Control AC-29(1984) 110–118

    Google Scholar 

  7. Coppel, W.A.: Matrix quadratic equations. Bulletin of the Australian Mathematical Society 10 (1974) 377–401

    Google Scholar 

  8. Davis, M.H.A.: Linear Estimation and Stochastic Control. Chapman and Hall, London 1977

    MATH  Google Scholar 

  9. De Souza, C.E.; Gevers, M.R.; Goodwin, G.C.: Riccati equations in optimal filtering of non-stabilizable systems having singular state transition matrices. IEEE Transactions on Automatic Control AC-31 (1986) 831–838

    Google Scholar 

  10. Faune, P.; Clerget, M.; Germain, F.: Opérateurs Rationnels Positifs. Dunod, Paris 1979

    Google Scholar 

  11. Geerts, A.H.W.: A necessary and sufficient condition for solvability of the linear-quadratic control problem without stability. Systems and Control Letters 11 (1988) 47–52

    Google Scholar 

  12. Kaiman, R.E.; Falb, P.L.; Arbib, M.L.: Topics in Mathematical System Theory. McGraw-Hill, New York 1969

    Google Scholar 

  13. Kalman, R.E.; Bucy, R.S.: New results in linear filtering and prediction theory. Transactions of the ASME, Ser.D. Journal of Basic Engineering 83D (1961)95–108

    Article  MathSciNet  Google Scholar 

  14. Kaiman, R.E.: Contributions to the theory of optimal control. Bol. Soc. Mat. Mexicana 5 (1960) 102–119

    Google Scholar 

  15. Knobloch, H.W.; Kwakernaak, H.; Lineare Kontrolltheorie. Springer-Verlag, Berlin 1985

    Book  MATH  Google Scholar 

  16. Kucera, V.: On nonnegative definite solutions to matrix quadratic equations. Automatica 8 (1972) 413–423

    Google Scholar 

  17. Kucera, V.: A contribution to matrix quadratic equations. IEEE Transactions on Automatic Control AC-17 (1972) 344–347

    Google Scholar 

  18. Kwakernaak, H.; Sivan, R.: Linear Optimal Control Systems. Wiley-Interscience, New York 1972

    MATH  Google Scholar 

  19. Molinari, B.P.: The time-invariant linear-quadratic optimal control problem. Automatica 13 (1977) 347–357

    Google Scholar 

  20. Poubelle, M.A.; Petersen, I.R.; Gevers, M.R.; Bitmead, R.R.: A miscellany of results on an equation of count J.F. Riccati. IEEE Transactions on Automatic Control AC-31 (1986) 651–653

    Google Scholar 

  21. Sasagawa, T.: A necessary and sufficient condition for the solution of the Riccati equation to be periodic. IEEE Transactions on Automatic Control AC-25 (1980) 564–566

    Google Scholar 

  22. Schweppe, F.C.: Uncertain Dynamic Systems. Prentice-Hall, Englewood Cliffs 1973

    Google Scholar 

  23. Shayman, MA.: Geometry of the algebraic Riccati equation, Part I and II. SLAM Journal on Control and Optimization 21 (1983) 375–409

    Google Scholar 

  24. Shayman, M.A.: On the periodic solutions of the matrix Riccati equation. Mathematical Systems Theory 16 (1983) 267–287

    Google Scholar 

  25. Willems, J.C.: Least-squares stationary optimal control and the algebraic Riccati equation. IEEE Transactions on Automatic Control AC-16(1971) 621–634

    Google Scholar 

  26. Willems, J.L.; Callier, F.M.: Large finite horizon and infinite horizon LQ-optimal control problems. Optimal Control Applications and Methods 4 (1983) pp 31–45

    Google Scholar 

  27. Wimmer, H.K.: The algebraic Riccati equation: conditions for the existence and uniqueness of solutions. Linear Algebra and Its Applications 58 (1984) 441–460

    Google Scholar 

  28. Lehtomaki, N.A.; Sandell, N.R. Jr.; Athans, M.: Robustness results in linear-quadratic Gaussian based multivariable control design. IEEE Transactions on Automatic Control AC-26 (1981) pp 75–93

    Google Scholar 

Download references

Authors
  1. Jacques L. Willems
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Frank M. Callier
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dipartimento di Elettronica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133, Milano, Italy

    Sergio Bittanti

  2. Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA, 93106, USA

    Alan J. Laub

  3. Department of Mathematics, University of Groningen, P.O. Box 800, NL-9700 AV, Groningen, The Netherlands

    Jan C. Willems

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Willems, J.L., Callier, F.M. (1991). The Infinite Horizon and the Receding Horizon LQ-Problems with Partial Stabilization Constraints. In: Bittanti, S., Laub, A.J., Willems, J.C. (eds) The Riccati Equation. Communications and Control Engineering Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58223-3_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-58223-3_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-63508-3

  • Online ISBN: 978-3-642-58223-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation