Newton's Method for Discrete Algebraic Riccati Equations when the Closed-Loop Matrix Has Eigenvalues on the Unit Circle

When Newton's method is applied to find the maximal symmetric solution of a discrete algebraic Riccati equation (DARE), convergence can be guaranteed under moderate conditions. In particular, the initial guess does not need to be close to the solution. The convergence is quadratic if the Frechet derivative is invertible at the solution. When the closed-loop matrix has eigenvalues on the unit circle, the derivative at the solution is not invertible. The convergence of Newton's method is shown to be either quadratic or linear with the common ratio $\frac{1}{2}$, provided that the eigenvalues on the unit circle are all semisimple. The linear convergence appears to be dominant, and the efficiency of the Newton iteration can be improved significantly by applying a double Newton step at the right time.

[1]  Harald K. Wimmer,et al.  Normal Forms of Symplectic Pencils and the Discrete-Time Algebraic Riccati Equation* , 1991 .

[2]  Richard H. Bartels,et al.  Algorithm 432 [C2]: Solution of the matrix equation AX + XB = C [F4] , 1972, Commun. ACM.

[3]  V. Mehrmann.  The Autonomous Linear Quadratic Control Problem , 1991 .

[4]  G. Hewer.  An iterative technique for the computation of the steady state gains for the discrete optimal regulator , 1971 .

[5]  P. Dooren.  A Generalized Eigenvalue Approach for Solving Riccati Equations , 1980 .

[6]  G. Reddien.  On Newton’s Method for Singular Problems , 1978 .

[7]  C. Kelley,et al.  Convergence Acceleration for Newton’s Method at Singular Points , 1982 .

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

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

[10]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[11]  H. Keller,et al.  CONVERGENCE RATES FOR NEWTON'S METHOD AT SINGULAR POINTS* , 1983 .

[12]  Alan J. Laub,et al.  A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case , 1998 .

[13]  C. Kelley,et al.  Newton’s Method at Singular Points. I , 1980 .

[14]  Alan J. Laub,et al.  A Collection of Benchmark Examples for the Numerical Solution of Algebraic Riccati Equations II: Dis , 1995 .

[15]  C. Kelley,et al.  A New Acceleration Method for Newton’s Method at Singular Points , 1983 .

[16]  A. Emami-Naeini,et al.  Comments on "On the numerical solution of the discrete-time algebraic Riccati equation" , 1980 .

[17]  P. Lancaster,et al.  Factorization of selfadjoint matrix polynomials with constant signature , 1982 .

[18]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[19]  C. Kelley.  A Shamanskii-Like acceleration scheme for nonlinear equations at singular roots , 1986 .

[20]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[21]  V. Hernandez,et al.  On the Kleinman iteration for periodic nonstabilizable systems , 1997, 1997 European Control Conference (ECC).

[22]  Vasile Sima,et al.  An efficient Schur method to solve the stabilizing problem , 1981 .

[23]  M. R. Osborne,et al.  Analysis of Newton’s Method at Irregular Singularities , 1983 .

[24]  L. Kantorovich,et al.  Functional analysis in normed spaces , 1952 .

[25]  G. Goodwin,et al.  Convergence properties of the Riccati difference equation in optimal filtering of nonstabilizable systems , 1984 .

[26]  A. Ran,et al.  Existence and comparison theorems for algebraic Riccati equations for continuous- and discrete-time systems , 1988 .

[27]  Charles R. Johnson.  The Theory of Matrices. Second Edition (with Applications) (Peter Lancaster and Miron Tismenetsky) , 1987 .