System identification with generalized orthonormal basis functions

A least-squares identification method is studied that estimates a finite number of expansion coefficients in the series expansion of a transfer function, where the expansion is in terms of recently introduced generalized basis functions. The basis functions are orthogonal in H2, and generalize the pulse, Laguerre and Kautz bases. One of their important properties is that, when chosen properly, they can substantially increase the speed of convergence of the series expansion. This leads to accurate approximate models with only a few coefficients to be estimated. Explicit bounds are derived for the bias and variance errors that occur in parameter estimates as well as in the resulting transfer function estimates.

[1]  O. Bosgra,et al.  Approximate system identification using system based orthonormal functions , 1990, 29th IEEE Conference on Decision and Control.

[2]  W. Kautz.  Transient synthesis in the time domain , 1954 .

[3]  Morris J. Gottlieb.  Concerning Some Polynomials Orthogonal on a Finite or Enumerable Set of Points , 1938 .

[4]  Munther A. Dahleh,et al.  On slowly time-varying systems , 1991, Autom..

[5]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[6]  Le Yi Wang.  Lipschitz continuity of inner-outer factorization , 1991 .

[7]  Bo Wahlberg.  On the Use of Orthogonalized Exponentials in System Identification , 1990 .

[8]  G. Clowes,et al.  Choice of the time-scaling factor for linear system approximations using orthonormal Laguerre functions , 1965 .

[9]  T. Georgiou,et al.  Optimal robustness in the gap metric , 1990 .

[10]  Le Yi Wang,et al.  Local–global double algebras for slow H∞ adaptation; the case of l2 disturbances , 1989 .

[11]  G. Dumont,et al.  An optimum time scale for discrete Laguerre network , 1993, IEEE Trans. Autom. Control..

[12]  K. Glover,et al.  Robust stabilization of normalized coprime factor plant descriptions with H/sub infinity /-bounded uncertainty , 1989 .

[13]  P. V. D. Hof,et al.  System identification with generalized orthonormal basis functions , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[14]  V. E. Benes,et al.  Statistical Theory of Communication , 1960 .

[15]  B. Wahlberg.  System identification using Laguerre models , 1991 .

[16]  L. Ljung,et al.  Asymptotic properties of black-box identification of transfer functions , 1985 .

[17]  B. Wahlberg.  System identification using Kautz models , 1994, IEEE Trans. Autom. Control..

[18]  Malcolm C. Smith.  Well-posedness of H ∞ optimal control problems , 1990 .

[19]  George Zames,et al.  Lipschitz continuity of H interpolation , 1990 .

[20]  D. K. De Vries,et al.  Identification of model uncertainty for control design , 1994 .

[21]  U. Grenander,et al.  Toeplitz Forms And Their Applications , 1958 .

[22]  Bo Wahlberg,et al.  Convergence rates for inverse Toeplitz matrix forms , 1989 .

[23]  N. Nikol’skiĭ,et al.  Treatise on the Shift Operator , 1986 .

[24]  C. Desoer.  Slowly varying discrete system xi+1=Aixi , 1970 .

[25]  Petros G. Voulgaris,et al.  On slowly varying systems: L∞ to L∞ performance and implications to robust adaptive control , 1992, 1992 American Control Conference.

[26]  George Zames,et al.  Logarithmic Variation Criteria for the Stability of Systems with Time-Varying Gains , 1968 .

[27]  P. V. D. Hof,et al.  A generalized orthonormal basis for linear dynamical systems , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[28]  Lennart Ljung,et al.  System Identification: Theory for the User , 1987 .

[29]  E. Kamen,et al.  Control of slowly-varying linear systems , 1989 .

[30]  M. Dahleh,et al.  Optimal rejection of persistent and bounded disturbances: continuity properties and adaptation , 1990 .

[31]  P.M.J. Van den Hof,et al.  An instrumental variable procedure for the identification of probabilistic frequency response uncertainty regions , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[32]  B. Wahlberg.  Laguerre and Kautz Models , 1994 .

[33]  P. N. Paraskevopoulos,et al.  Parametric identification of discrete-time SISO systems , 1979 .

[34]  M. Steinbuch,et al.  Control relevant identification of a compact disc pick-up mechanism , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[35]  Norbert Wiener,et al.  Extrapolation, Interpolation, and Smoothing of Stationary Time Series , 1964 .