Modeling of normalized coprime factors with ν-metric uncertainty

Modeling of uncertain systems with normalized coprime factor description is investigated, where the experimental data is given by a finite set of frequency response measurement samples of the open loop plant that is linear, time-invariant, and possibly infinite-dimensional. The objective is not only to identify the nominal model but also to quantify the modeling error with sup-norm bounds in frequency domain. The uncertainty to be identified and quantified is chosen as the /spl nu/-metric, proposed by Vinnicombe (1993), because of its compatibility with H/sub /spl infin//-based robust control. An algorithm is developed to model the normalized coprime factors of the given plant using techniques of discrete Fourier analysis and balanced stochastic truncation and is shown to be robust in the presence of the worst case noise. Upper bounds are derived for the associated modeling error based on the minimum a priori information of the underlying model set and of the noise level in the measurement data. A simulation example is used to illustrate the effectiveness of the proposed algorithm.

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

[2]  Jie Chen,et al.  Worst Case Identification of Continuous time Systems via Interpolation , 1993 .

[3]  Carl N. Nett,et al.  Control oriented system identification: a worst-case/deterministic approach in H/sub infinity / , 1991 .

[4]  Guoxiang Gu,et al.  A class of algorithms for identification in H∞ , 1992, Autom..

[5]  Jie Chen,et al.  Worst-case system identification in H∞: validation of apriori information, essentially optimal algorithms, and error bounds , 1992, 1992 American Control Conference.

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

[7]  Mathukumalli Vidyasagar,et al.  Robust controllers for uncertain linear multivariable systems , 1984, Autom..

[8]  U. Desai,et al.  A transformation approach to stochastic model reduction , 1984 .

[9]  Tryphon T. Georgiou,et al.  Identification of linear systems: A graph point of view , 1992, 1992 American Control Conference.

[10]  P. Khargonekar,et al.  State-space solutions to standard H2 and H∞ control problems , 1988, 1988 American Control Conference.

[11]  Håkan Hjalmarsson,et al.  Some Reflections on Control Design Based on Experimental Data , 1994 .

[12]  Brian D. O. Anderson,et al.  Hilbert transforms from interpolation data , 1987, 26th IEEE Conference on Decision and Control.

[13]  R. Kálmán.  Realization of Covariance Sequences , 1982 .

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

[15]  P. Khargonekar,et al.  STATESPACE SOLUTIONS TO STANDARD 2 H AND H? CONTROL PROBLEMS , 1989 .

[16]  Michael Green,et al.  Balanced stochastic realizations , 1988 .

[17]  Ferenc Schipp,et al.  L∞ system approximation algorithms generated by ϑ summations , 1997, Autom..

[18]  T. Georgiou,et al.  On the computation of the gap metric , 1988, Proceedings of the 27th IEEE Conference on Decision and Control.

[19]  J. Tsitsiklis,et al.  Optimal asymptotic identification under bounded disturbances , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

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

[21]  Guoxiang Gu,et al.  Identification in ℋ∞ with nonuniformly spaced frequency response measurements , 1994 .

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

[23]  Roy S. Smith,et al.  Towards a Methodology for Robust Parameter Identification , 1990, 1990 American Control Conference.

[24]  George Zames,et al.  Optimal H/sup /spl infin// approximation by systems of prescribed order using frequency response data , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[25]  P. Khargonekar,et al.  Robust convergence of two-stage nonlinear algorithms for identification in H ∞ , 1992 .

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

[27]  Keith Glover,et al.  Robust control design using normal-ized coprime factor plant descriptions , 1989 .

[28]  Tryphon T. Georgiou,et al.  Realization of power spectra from partial covariance sequences , 1987, IEEE Trans. Acoust. Speech Signal Process..

[29]  M. Safonov,et al.  Relative-error H/sub infinity / identification from autocorrelation data-a stochastic realization method , 1992 .

[30]  T. Başar.  Feedback and Optimal Sensitivity: Model Reference Transformations, Multiplicative Seminorms, and Approximate Inverses , 2001 .

[31]  K. Glover,et al.  Robust stabilization of normalized coprime factor plant descriptions , 1990 .

[32]  M. Green,et al.  A relative error bound for balanced stochastic truncation , 1988 .

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

[34]  Pramod Khargonekar,et al.  A Time-Domain Approach to Model Validation , 1992, 1992 American Control Conference.

[35]  J. Partington.  Robust identification in H , 1992 .

[36]  Robert L. Kosut.  Uncertainty model unfalsification: a system identification paradigm compatible with robust control design , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[37]  Jie Chen,et al.  Worst case system identification in H∞: validation of a priori information, essentially optimal algorithms, and error bounds , 1995, IEEE Trans. Autom. Control..

[38]  Gene F. Franklin,et al.  An error bound for a discrete reduced order model of a linear multivariable system , 1987 .

[39]  P. Khargonekar,et al.  State-space solutions to standard H/sub 2/ and H/sub infinity / control problems , 1989 .

[40]  Jonathan R. Partington,et al.  Robust identification of strongly stabilizable systems , 1992 .

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

[42]  Raimund J. Ober,et al.  On the gap metric and coprime factor perturbations , 1993, Autom..

[43]  A. Pinkus.  n-Widths in Approximation Theory , 1985 .

[44]  C. Sanathanan,et al.  Transfer function synthesis as a ratio of two complex polynomials , 1963 .

[45]  D. Hinrichsen,et al.  An improved error estimate for reduced-order models of discrete-time systems , 1990 .

[46]  K. Glover.  All optimal Hankel-norm approximations of linear multivariable systems and their L, ∞ -error bounds† , 1984 .

[47]  D. S. Tracy,et al.  Generalized Inverse Matrices: With Applications to Statistics , 1971 .

[48]  Guoxiang Gu,et al.  Identification in H8 using Pick's interpolation , 1993 .

[49]  G. Vinnicombe.  Frequency domain uncertainty and the graph topology , 1993, IEEE Trans. Autom. Control..