Orthonormal Basis Functions for Continuous-Time Systems and Lp Convergence

Abstract. In this paper, model sets for linear-time-invariant continuous-time systems that are spanned by fixed pole orthonormal bases are investigated. These bases generalize the well-known Laguerre and two-parameter Kautz cases. It is shown that the obtained model sets are everywhere dense in the Hardy space H1(Π) under the same condition as previously derived by the authors for the denseness in the (Π is the open right half plane) Hardy spaces Hp(Π), 1<p<∞. As a further extension, the paper shows how orthonormal model sets, that are everywhere dense in Hp(Π), 1≤p<∞, and which have a prescribed asymptotic order, may be constructed. Finally, it is established that the Fourier series formed by orthonormal basis functions converge in all spaces Hp(Π) and (D is the open unit disk) Hp(D), 1<p<∞. The results in this paper have application in system identification, model reduction, and control system synthesis.

[1]  Graham C. Goodwin,et al.  Iterative algorithm for robust performance optimization , 1993 .

[2]  Jonathan R. Partington,et al.  Interpolation in Normed Spaces from the Values of Linear Functionals , 1994 .

[3]  P. Duren.  Theory of H[p] spaces , 1970 .

[4]  B. Ninness,et al.  A unifying construction of orthonormal bases for system identification , 1997, IEEE Trans. Autom. Control..

[5]  Jerry M. Mendel,et al.  A Unified Approach to the Synthesis of Orthonormal Exponential Functions Useful in Systems Analysis , 1966, IEEE Trans. Syst. Sci. Cybern..

[6]  O. Morg.  A Unifying Construction of Orthonormal Bases for System Identification , 1997 .

[7]  L. Wang,et al.  Frequency smoothing using Laguerre model , 1992 .

[8]  Brett Ninness,et al.  Orthonormal basis functions for modelling continuous-time systems , 1999, Signal Process..

[9]  Jozsef Bokor,et al.  System identification with generalized orthonormal basis functions , 1995, Autom..

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

[11]  D. C. Ross.  Orthonormal exponentials , 1964, IEEE Transactions on Communication and Electronics.

[12]  P. Duren.  Theory of Hp Spaces , 2000 .

[13]  Christopher J. BISHOPAbstra,et al.  Orthogonal Functions , 2022 .

[14]  Jonathan R. Partington,et al.  Robust identification in the disc algebra using rational wavelets and orthonormal basis functions , 1996 .

[15]  B. Epstein.  Orthogonal families of analytic functions , 1965 .

[16]  Bo Wahlberg,et al.  On approximation of stable linear dynamical systems using Laguerre and Kautz functions , 1996, Autom..

[17]  Jonathan R. Partington,et al.  Approximation of delay systems by fourier-laguerre series , 1991, Autom..

[18]  P. V. D. Hof,et al.  A generalized orthonormal basis for linear dynamical systems , 1995, IEEE Trans. Autom. Control..

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

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

[21]  Tomás Oliveira e Silva,et al.  Optimality conditions for truncated Laguerre networks , 1994, IEEE Trans. Signal Process..

[22]  Pertti M. Mäkilä,et al.  Laguerre series approximation of infinite dimensional systems , 1990, Autom..

[23]  J. Schoukens,et al.  Parametric identification of transfer functions in the frequency domain-a survey , 1994, IEEE Trans. Autom. Control..

[24]  H. Hjalmarsson,et al.  Generalized Fourier and Toeplitz Results for Rational Orthonormal Bases , 1999 .

[25]  L. Ljung,et al.  Subspace-based multivariable system identification from frequency response data , 1996, IEEE Trans. Autom. Control..

[26]  Z. Nehari.  Bounded analytic functions , 1950 .

[27]  Paul W. Broome,et al.  Discrete Orthonormal Sequences , 1965, JACM.

[28]  P. R. Clement.  Applications of generalized laguerre functions , 1985 .

[29]  P. M. Mäkilät.  Laguerre methods and H ∞ identification of continuous-time systems , 1991 .

[30]  P. Khargonekar,et al.  Approximation of infinite-dimensional systems , 1989 .

[31]  Brett Ninness,et al.  Rational Basis Functions for Robust Identification from Frequency and Time-Domain Measurements , 1998, Autom..

[32]  T. E. O. Silva.  Optimality conditions for truncated Kautz networks with two periodically repeating complex conjugate poles , 1995, IEEE Trans. Autom. Control..

[33]  I. Horowitz.  Synthesis of feedback systems , 1963 .

[34]  Brett Ninness,et al.  Orthonormal basis functions for continuous-time systems: Completeness and Lp-convergence , 1999, 1999 European Control Conference (ECC).

[35]  Brett Ninness,et al.  Identification of power transformer models from frequency response data: A case study , 1998, Signal Process..