Advertisement

Applied Mathematics and Optimization

, Volume 61, Issue 2, pp 145–166 | Cite as

On Optimal Feedback Control for Stationary Linear Systems

  • David L. Russell
Article

Abstract

We study linear-quadratic optimal control problems for finite dimensional stationary linear systems A X+B U=Z with output Y=C X+D U from the viewpoint of linear feedback solution. We interpret solutions in relation to system robustness with respect to disturbances Z and relate them to nonlinear matrix equations of Riccati type and eigenvalue-eigenvector problems for the corresponding Hamiltonian system. Examples are included along with an indication of extensions to continuous, i.e., infinite dimensional, systems, primarily of elliptic type.

Keywords

Stationary system Riccati equations Feedback invertibility 

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.: Linear Optimal Control. Prentice Hall, New York (1971) zbMATHGoogle Scholar
  2. 2.
    Halanay, A., Ionescu, V.: Time-Varying Discrete Linear Systems: Input-Output Operators, Riccati Equations, Disturbance Attenuation. Birkhauser, Basel (1994) Google Scholar
  3. 3.
    Kailath, T.: Linear Systems. Prentice-Hall, New York (1980) zbMATHGoogle Scholar
  4. 4.
    Kalman, R.E.: Contributions to the theory of optimal control. Bol. Soc. Mat. Mex. 2, 102–119 (1960) MathSciNetGoogle Scholar
  5. 5.
    Kalman, R.E., Bucy, R.S.: New results in linear filtering and prediction theory. J. Basic Eng. 83, 95–108 (1961) MathSciNetGoogle Scholar
  6. 6.
    Lasiecka, I., Triggiani, R.: Control Theory for Partial Differential Equations: Continuous and Approximation Theories, vols. I & II. Encyclopedia of Mathematics and Its Applications, vol. 74. Cambridge University Press, Cambridge (2000) Google Scholar
  7. 7.
    Lions, J.-P.: Control of Systems Governed by Partial Differential Equations. Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, vol. 170. Springer, Berlin (1971). Translated from the original French by S.K. Mitter zbMATHGoogle Scholar
  8. 8.
    Potter, J.E.: Matrix quadratic solutions. SIAM J. Appl. Math. 14, 496–501 (1966) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Russell, D.L.: Mathematics of Finite Dimensional Control Systems; Theory and Design. Dekker, New York (1979) zbMATHGoogle Scholar
  10. 10.
    Sennett, R.E.: Matrix Analysis of Structures. Waveland Press, Long Grove (1994) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of MathematicsVirginia TechBlacksburgUSA

Personalised recommendations