A general theory of linear time-invariant adaptive feedforward systems with harmonic regressors

Establishes necessary and sufficient conditions for an adaptive system with a harmonic regressor (i.e., a regressor comprised exclusively of sinusoidal signals) to admit an exact linear time-invariant (LTI) representation. These conditions are important because a large number of adaptive systems used in practice have sinusoidal regressors, and the stability and performance of such systems having LTI representations can be completely analyzed by well-known methods. The theory is extended to applications where the LTI conditions do not hold, in which case the harmonic adaptive system can be written as the parallel connection of a purely LTI subsystem and a linear time-varying (LTV) subsystem. An explicit upper bound is established on the induced two-norm of the LTV block, which allows systematic treatment using emerging robust control methods applicable to LTI systems with norm-bounded LTV perturbations.

[1]  S. Conolly,et al.  State-space analysis of the adaptive notch filter , 1986, Proceedings of the IEEE.

[2]  Nurgun Erdol,et al.  Wavelet transform based adaptive filters: analysis and new results , 1996, IEEE Trans. Signal Process..

[3]  David S. Bayard,et al.  Exponential convergence of the tracking error in adaptive systems without persistent excitation , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[4]  Petros A. Ioannou,et al.  Adaptive Systems with Reduced Models , 1983 .

[5]  Marc Bodson,et al.  Design of adaptive feedforward algorithms using internal model equivalence , 1995 .

[6]  J. Glover Adaptive noise canceling applied to sinusoidal interferences , 1977 .

[7]  David S. Bayard An LTI/LTV decomposition of adaptive feedforward systems with sinusoidal regressors , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[8]  Bernard Widrow,et al.  Adaptive Signal Processing , 1985 .

[9]  David S. Bayard On the LTI properties of adaptive feedforward systems with tap delay-line regressors , 1999, IEEE Trans. Signal Process..

[10]  Andreas H. von Flotow,et al.  Comparison and extensions of control methods for narrow-band disturbance rejection , 1992, IEEE Trans. Signal Process..

[11]  Neil J. Bershad,et al.  Non-Wiener solutions for the LMS algorithm-a time domain approach , 1995, IEEE Trans. Signal Process..

[12]  Simon Andrew Collins Multi-axis analog adaptive feedforward cancellation of cryocooler vibration , 1994 .

[13]  A. Peterson,et al.  Transform domain LMS algorithm , 1983 .

[14]  Dennis R. Morgan,et al.  A control theory approach to the stability and transient analysis of the filtered-x LMS adaptive notch filter , 1992, IEEE Trans. Signal Process..

[15]  Stephen J. Elliott,et al.  A multiple error LMS algorithm and its application to the active control of sound and vibration , 1987, IEEE Trans. Acoust. Speech Signal Process..

[16]  David S. Bayard Mu robustness analysis of adaptive feedforward noise cancelling algorithms , 1999, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).

[17]  Alan J. Laub,et al.  The LMI control toolbox , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[18]  G. Zames On the input-output stability of time-varying nonlinear feedback systems Part one: Conditions derived using concepts of loop gain, conicity, and positivity , 1966 .

[19]  Anuradha M. Annaswamy,et al.  Stable Adaptive Systems , 1989 .

[20]  David S. Bayard Necessary and sufficient conditions for LTI representations of adaptive systems with sinusoidal regressors , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[21]  Alexander Weinmann Uncertain Models and Robust Control , 2002 .

[22]  William H. Beyer,et al.  CRC standard mathematical tables , 1976 .

[23]  K. Poolla,et al.  Robust performance against time-varying structured perturbations , 1995, IEEE Trans. Autom. Control..

[24]  David S. Bayard A modified augmented error algorithm for adaptive noise cancellation in the presence of plant resonances , 1998, Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207).

[25]  C. Desoer,et al.  Feedback Systems: Input-Output Properties , 1975 .

[26]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[27]  P. Khosla,et al.  Harmonic generation in adaptive feedforward cancellation schemes , 1994, IEEE Trans. Autom. Control..

[28]  Dennis R. Morgan,et al.  An analysis of multiple correlation cancellation loops with a filter in the auxiliary path , 1980, ICASSP.

[29]  R. Monopoli Model reference adaptive control with an augmented error signal , 1974 .

[30]  Zahidul H. Rahman,et al.  Narrow-band control experiments in active vibration isolation , 1994, Optics & Photonics.

[31]  J. Shamma Robust stability with time-varying structured uncertainty , 1994, IEEE Trans. Autom. Control..

[32]  Brian D. O. Anderson,et al.  Stability of adaptive systems: passivity and averaging analysis , 1986 .

[33]  J. Doyle,et al.  Robust and optimal control , 1995, Proceedings of 35th IEEE Conference on Decision and Control.