Stability Analysis of Fast Recursive Least Squares Algorithm: Application to Adaptive Filtering

1. Stability Analysis of Fast Recursive Least Squares Algorithm: Application to Adaptive Filtering LATSI laboratory, Department of Electronic, Faculty of Engineering Sciences, University of Blida, Algeria f_ykhlef@yahoo.fr By Farid Ykhlef

2. Presentation Outline (Overview) • Adaptive Identification • Fast Recursive Least SquaresAlgorithm • New numerically stable version of the FRLS Algorithm • Stability Analysis of the FRLS Algorithm • Simulation Results • Conclusion

3. + Unknown System + _ Adaptive Filter Adaptive Identification • System identification structure

4. Adaptive Identification • The coefficients of the adaptive filter are adjusted by a FRLS algorithm, according to the following equations: • with

5. FRLS algorithm Prediction Part Filtering Part Fast Recursive Least SquaresAlgorithm and Stability problem • The FRLS algorithm: Fast Transversal Filters (FTF) version

6. Fast Recursive Least SquaresAlgorithm • : input signal • : desired signal • : a priori error • : adaptation gain (or Kalman gain) • / : forward and backward predictor • : adaptive filter

7. Fast Recursive Least SquaresAlgorithm • FRLS algorithm can produce a good trade-off between convergence speed and computational complexity • FRLS algorithm suffers from numerical instability when operating under the effects of finite precision arithmetic • The instability of FRLS algorithm is due to thepropagation of thenumerical errors • Several numerical solutions of stabilization, with stationary signals, are proposed in the literature

8. New numerically stable version of the FRLS Algorithm (NS- FRLS) • It is possible to maintain stability by using a few equations. • The stabilizationmethod is based on a first order model of the propagation of the numerical errors. • The general principle is to modify the numerical properties of the algorithm without modifying its theoretical behavior

9. New numerically stable version of the FRLS Algorithm (NS- FRLS) • Using some known relationships between the different backward a priori prediction errors, we define a ”control variable” , theoretically null, given by : backward a priori prediction errors, theoretically equal, calculated differently.

10. New numerically stable version of the FRLS Algorithm (NS- FRLS) • controls the propagation of the numerical errors in the algorithm • To stabilize the algorithm, we use the variable to calculate a backward a priori prediction error: • To simplify and to ensure the numerical stability of this version in the stationary case, it is necessary to choose a forgetting factor: stability condition : a real number higher than 2

11. Stability Analysis of the NS-FRLS Algorithm • The equations of numerical errors propagation for the recursive variables of the NS-FRLS algorithm according to the state linear model are • with

12. Stability Analysis of the NS-FRLS Algorithm • The approximate transition matrix for errors propagationis given by: Stability The algorithm is stable when the eigen-values of this matrix F(t) are close to 1 in magnitude.

13. Stability Analysis of the NS-FRLS Algorithm • In this work, the instability study of FRLS algorithm is conducted entirely around the sub-matrix . • contains a companion matrix, given by :

14. Stability Analysis of the NS-FRLS Algorithm The roots of the backward predictor vector = The eigen-values of the matrix

15. Simulation Results • The goal of these simulations is to analyze the position of the zeros of the forward/backward predictors in the z-plane (unit circle) in order to verify the numerical stability. • To test the numerical behavior of the prediction part of the numerically stable FRLS algorithm, the input signals used for this simulation are AR signals. • So, we use fifth-order AR processes characterized by the following poles: • AR5f (poles far from the unit circle): • AR5c (some poles close to the unit circle):

16. Simulation Results State of stability for various orders

17. Simulation Results • The table shows : • NS-FRLS algorithm is stable for the AR5f signal λ. • NS-FRLS algorithm unstable for the AR5c signal for a forgetting factor λ very close to the stability condition. • NS-FRLS algorithm is stable for the AR5c signal when we increase the value of the forgetting factor λ.

18. Simulation Results • Position of the estimated zeros of the forward /backward predictors in steady-state for the AR5f signal: poles far from the unit circle (λ=0.9333(p=3), N=5). No divergence detected ○ Predicted pole × Real pole

19. Simulation Results • Position of the estimated zeros of the forward/backward predictors just before the divergence for the AR5c signal: poles close to the unit circle (λ=0.9333(p=3), N=5). a divergence was detected ○ Predicted pole × Real pole

20. Conclusion • The results reveal that • for the signals of the same kind as the AR5f signal, the choice of a forgetting factor very close to the stability condition allows to maintain the stability of the algorithm NS-FRLS. • for the signals of the AR5c kind, the stability is only assured when we choose a forgetting factor extremely higher than the stability condition. • The NS-FRLS algorithms is stable for a suitable choice of the forgetting factor; its minimal limit depends in fact on the nature of the input signal.