Numerical Solution of the Discrete-Time Coupled Algebraic Riccati Equations

  • Ivan Ganchev Ivanov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5434)


We consider the numerical solution of a set of discrete-time coupled algebraic Riccati equations that arise in quadratic optimal control. Several iterations for computing a symmetric solution of this system are investigated and compared. New iterations are based on the properties of a Stein equation. It is necessary to solve a Stein equation at each step of considered algorithms. We will compare the corresponding solvers in regard of accuracy, number of iterations and time of executing. Several sets of test examples are used to demonstrate the performance.


Riccati Equation Linear Matrix Equation Jump Linear System Stein Equation Hermitian Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Costa, O.L.V., Aya, J.C.C.: Temporal Difference Methods for the Maximal Solution of Discrete-Time Coupled Algebraic Ricacti Equations. Journal of Optimization Theory and Application 109, 289–309 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Costa, O.L.V., Fragoso, M.D.: Stability Results for Discrete-Time Linear Systems with markovian jumping parameters. J. Math. Analysis and Applic. 179, 154–178 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Costa, O.L.V., Marques, R.P.: Maximal and Stabilizing Hermitian Solutions for Discrete-Time Coupled Algebraic Ricacti Equations. Mathematics of Control, Signals and Systems 12, 167–195 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gajic, Z., Borno, I.: Lyapunov iterations for optimal control of jump linear systems at steady state. IEEE Transaction on Authomatic Control 40, 1971–1975 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gajic, Z., Losada, R.: Monotonicity of algebraic Lyapunov iterations for optimal control of jump parameter linear systems. Systems & Control Letters 41, 175–181 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ivanov, I.: Stein Iterations for the Coupled Discrete-Time Riccati Equations (submitted)Google Scholar
  7. 7.
    Ivanov, I.: A method to solve the discrete-time coupled algebraic Riccati equations. Applied Mathematics and Computation (accepted)Google Scholar
  8. 8.
    Rami, M., Ghaoui, L.: LMI Optimization for Nonstandard Riccati Equations Arising in Stochastic Control. IEEE Transactions on Automatic Control 41, 1666–1671 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    do Val, J.B., Geromel, J.C., Costa, O.L.V.: Solutions for the linear quadratic control problem of Markov jump linear systems. J. Optimization Theory and Applications 103(2), 283–311 (1999)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Ivan Ganchev Ivanov
    • 1
  1. 1.Faculty of Economics and Business AdministrationSofia University ”St. Kliment Ohridski”SofiaBulgaria

Personalised recommendations