Nonrecursive solution for the discrete algebraic Riccati equation and X + A*X-1A=L

Abstract In this paper, we present two new algebraic algorithms for the solution of the discrete algebraic Riccati equation. The first algorithm requires the nonsingularity of the transition matrix and is based on the solution of a standard eigenvalue problem for a new symplectic matrix; the proposed algorithm computes the extreme solutions of the discrete algebraic Riccati equation. The second algorithm solves the Riccati equation without the assumption of the nonsingularity of the transition matrix; the proposed algorithm is based on the solution of the matrix equation X + A*X-1A=L, where A is a singular matrix and L a positive definite matrix.

[1]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[2]  T. K. Nguyen.  Numerical solution of discrete-time algebraic Riccati equation , 1999 .

[3]  Mohamed A. Ramadan,et al.  On the existence of a positive definite solution of the matrix equation , 2001, Int. J. Comput. Math..

[4]  A. Laub.  A schur method for solving algebraic Riccati equations , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[5]  G. Gaalman.  Comments on "A nonrecursive algebraic solution for the discrete Riccati equation" , 1980 .

[6]  J. Engwerda.  On the existence of a positive definite solution of the matrix equation X + A , 1993 .

[7]  A. Ran,et al.  Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X + A*X-1A = Q , 1993 .

[8]  W.-W. Lin,et al.  A generalized structure-preserving doubling algorithm for generalized discrete-time algebraic Riccati equations , 2005 .

[9]  Leiba Rodman,et al.  Stable Hermitian solutions of discrete algebraic Riccati equations , 1992, Math. Control. Signals Syst..

[10]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

[11]  R. E. Kalman,et al.  A New Approach to Linear Filtering and Prediction Problems , 2002 .

[12]  Wen-Wei Lin,et al.  Convergence Analysis of Structure-Preserving Doubling Algorithms for Riccati-Type Matrix Equations , 2006, SIAM J. Matrix Anal. Appl..

[13]  Improved perturbation estimates for the matrix equations X±A∗X−1A=Q , 2004 .

[14]  D. Vaughan.  A nonrecursive algebraic solution for the discrete Riccati equation , 1970 .

[15]  Petros G. Voulgaris,et al.  On optimal ℓ∞ to ℓ∞ filtering , 1995, Autom..

[16]  Iterative methods for nonlinear matrix equations X + A⋆X−αA = I , 2005 .

[17]  Leiba Rodman,et al.  Algebraic Riccati equations , 1995 .

[18]  E. K. Gatcombe.  Discussion: “The Measurement of Oil-Film Thickness in Gear Teeth” (MacConochie, I. O., and Cameron, A., 1960, ASME J. Basic Eng., 82, pp. 29–34) , 1960 .

[19]  Vlad Ionescu,et al.  On computing the stabilizing solution of the discrete-time Riccati equation , 1992 .

[20]  V. Hasanov,et al.  On two perturbation estimates of the extreme solutions to the equations X ± A*X−1A = Q☆ , 2006 .

[21]  Sokratis K. Katsikas,et al.  A survey of recursive algorithms for the solution of the discrete time Riccati Equation , 1997 .

[22]  M. Adam,et al.  Riccati Equation Solution Method for the Computation of the Extreme Solutions of X+A*X-1A=Q and X-A*X-1A=Q , 2009, 2009 16th International Conference on Systems, Signals and Image Processing.