Stability of a Riccati equation arising in recursive parameter estimation under lack of excitation

Stability properties of the Riccati equation in a recently suggested antiwindup algorithm for recursive parameter estimation are analyzed. Convergence of the resulting dynamic system is implied by that of a linear time-varying difference matrix equation. By means of converging matrix products theory, the linear mapping associated with the system is shown to be a paracontraction with respect to a certain norm. Therefore, measured in that norm, the solution to the matrix equation will not diverge notwithstanding excitation properties of the data. Thus the purpose of anti-windup is achieved.

[1]  S. Gunnarsson.  Combining tracking and regularization in recursive least squares identification , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[2]  A. T. Neto,et al.  Stabilization via static output feedback , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[3]  Howard M. Schwartz,et al.  Analysis of the Kalman filter based estimation algorithm: an orthogonal decomposition approach , 2004, Autom..

[4]  Robert E. Skelton,et al.  Static output feedback controllers: stability and convexity , 1998, IEEE Trans. Autom. Control..

[5]  W. Beyn,et al.  Infinite products and paracontracting matrices , 1997 .

[6]  S. Bittanti,et al.  The Riccati equation , 1991 .

[7]  Torbjörn Wigren.  Fast converging and low complexity adaptive filtering using an averaged Kalman filter , 1998, IEEE Trans. Signal Process..

[8]  I. Daubechies,et al.  Sets of Matrices All Infinite Products of Which Converge , 1992 .

[9]  Yong-Yan Cao,et al.  A new necessary and sufficient condition for static output feedback stabilizability and comments on "Stabilization via static output feedback" , 1998, IEEE Trans. Autom. Control..

[10]  W. Rugh.  Linear System Theory , 1992 .

[11]  Lennart Ljung,et al.  Adaptation and tracking in system identification - A survey , 1990, Autom..

[12]  F. Gustafsson.  AVOIDING WINDUP IN RECURSIVE PARAMETER ESTIMATION , 2002 .

[13]  I. Daubechies,et al.  Corrigendum/addendum to: Sets of matrices all infinite products of which converge , 2001 .

[14]  Liyu Cao,et al.  A directional forgetting algorithm based on the decomposition of the information matrix , 2000, Autom..

[15]  D. Hartfiel.  Nonhomogeneous Matrix Products , 2002 .

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

[17]  Olga Holtz.  On convergence of infinite matrix products , 2005, math/0512590.

[18]  Chaouki T. Abdallah,et al.  Static output feedback: a survey , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[19]  J. Geromel,et al.  Numerical comparison of output feedback design methods , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).