Estimation of model quality

This paper provides an introduction to recent work on the problem of quantifying errors in the estimation of models for dynamic systems. This is a very large field. We therefore concentrate on approaches that have been motivated by the need for reliable models for control system design. This will involve a discussion of efforts that go under the titles of ‘estimation in tH∞’, ‘worst-case estimation’, ‘estimation in l1’ and ‘stochastic embedding of undermodelling’. A central theme of this survey is to examine these new methods with reference to the classic bias/variance tradeoff in model structure selection.

[1]  P. Erdos,et al.  An extremal problem in the theory of interpolation , 1964 .

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

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

[4]  C. A. Jacobson,et al.  Identification in H∞: the continuous-time case , 1990, 1990 American Control Conference.

[5]  Arthur J. Helmicki,et al.  Identification in H ∞ : linear algorithms , 1990 .

[6]  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.

[7]  G. Belforte,et al.  Optimal input design for worst-case system identification in l 1 /l 2 /l ∞ , 1993 .

[8]  Pramod P. Khargonekar,et al.  The least squares algorithm, parametric system identification and bounded noise , 1993, Autom..

[9]  Guoxiang Gu,et al.  Worst Case Identification of Continuous time Systems via Interpolation , 1993, 1993 American Control Conference.

[10]  G. Goodwin,et al.  A stochastic embedding approach for quantifying uncertainty in the estimation of restricted complexity models , 1989 .

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

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

[13]  Jonathan R. Partington,et al.  Worst-case identification in e 2 : linear and nonlinear algorithms , 1994 .

[14]  Guoxiang Gu,et al.  Identification in H∞ with nonuniformly spaced frequency response measurements , 1992, 1992 American Control Conference.

[15]  P.M. Mäkilä,et al.  Worst-case control-relevant identification , 1995, Autom..

[16]  Robert Bitmead,et al.  Adaptive frequency response identification , 1987, 26th IEEE Conference on Decision and Control.

[17]  Lennart Ljung,et al.  Model quality: the roles of prior knowledge and data information , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[18]  M. B. Zarrop,et al.  Book Review: Adaptive Optimal Control: the thinking man's GPC , 1991 .

[19]  R. Tempo.  Robust estimation and filtering in the presence of bounded noise , 1987, 26th IEEE Conference on Decision and Control.

[20]  Y. F. Huang,et al.  On the value of information in system identification - Bounded noise case , 1982, Autom..

[21]  Brian D. O. Anderson,et al.  On Adaptive Robust Control and Control-Relevant System Identification , 1992, 1992 American Control Conference.

[22]  D.J.N. Limebeer,et al.  System identification for H/sup infinity / control , 1992, [1992] Proceedings of the 31st IEEE Conference on Decision and Control.

[23]  Mario Milanese,et al.  H∞ Identification with Mixed Parametric and Nonparametric Models , 1994 .

[24]  Anders Ahlén,et al.  A probabilistic approach to multivariable robust filtering and open-loop control , 1995, IEEE Trans. Autom. Control..

[25]  G. Zames.  On the metric complexity of casual linear systems: ε -Entropy and ε -Dimension for continuous time , 1979 .

[26]  G. Goodwin,et al.  Robust Frequency Response Estimation Accounting for Noise and Undermodelling , 1992, 1992 American Control Conference.

[27]  R. Tempo,et al.  Optimal algorithms theory for robust estimation and prediction , 1985 .

[28]  E. C. Titchmarsh,et al.  The theory of functions , 1933 .

[29]  R. Tempo,et al.  Maximum likelihood estimators and worst case optimal algorithms for system identification , 1988 .

[30]  Brian D. O. Anderson,et al.  A new approach to adaptive robust control , 1993 .

[31]  Palle Andersen,et al.  Estimation of Frequency Domain Model Uncertainties with Application to Robust Controller Design , 1994 .

[32]  T. Zhou.  Input-output extrapolation-minimization theorem and its application to model validation and robust identification.The modelling of uncertainty in control systems , 1994 .

[33]  H. Piet-Lahanier,et al.  Bounded-error tracking of time-varying parameters , 1992, [1992] Proceedings of the 31st IEEE Conference on Decision and Control.

[34]  Allan Pinkus,et al.  Proof of the conjectures of Bernstein and Erdös concerning the optimal nodes for polynomial interpolation , 1978 .

[35]  E. Fogel.  System identification via membership set constraints with energy constrained noise , 1979 .

[36]  C. E. Rohrs,et al.  Identification with nonparametric uncertainty , 1992 .

[37]  K. Poolla,et al.  On the time complexity of worst-case system identification , 1994, IEEE Trans. Autom. Control..

[38]  P. Khargonekar,et al.  Linear and nonlinear algorithms for identification in H∞ with error bounds , 1991, 1991 American Control Conference.

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

[40]  G. Goodwin,et al.  Quantification of Uncertainty in Estimation using an Embedding Principle , 1989 .

[41]  Stephen P. Boyd,et al.  Robust control design for ellipsoidal plant set , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[42]  N. Akhiezer,et al.  The Classical Moment Problem. , 1968 .

[43]  Roberto Tempo.  Robust and Optimal Algorithms for Worst-Case Parametric System Identification , 1994 .

[44]  Antonio Vicino,et al.  Information-Based Complexity and Nonparametric Worst-Case System Identification , 1993, J. Complex..

[45]  J. Partington,et al.  Robust approximation and identification in H∞ , 1991, 1991 American Control Conference.

[46]  L. Ljung.  Asymptotic variance expressions for identified black-box transfer function models , 1984, The 23rd IEEE Conference on Decision and Control.

[47]  L. Ljung,et al.  Hard frequency-domain model error bounds from least-squares like identification techniques , 1992 .

[48]  M. Nimchek.  Banach spaces of analytic functions , 1996 .

[49]  J. Traub,et al.  Perspectives on information-based complexity , 1992, math/9201269.

[50]  J. Norton.  Identification of parameter bounds for ARMAX models from records with bounded noise , 1987 .

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

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

[53]  K. POOLLAUniv,et al.  Statistical Testability of Uncertainty Models , 1994 .

[54]  Pramod P. Khargonekar,et al.  A comparison of classical stochastic estimation and deterministic robust estimation , 1992 .

[55]  V. Wertz,et al.  Adaptive Optimal Control: The Thinking Man's G.P.C. , 1991 .

[56]  Jie Chen,et al.  The Caratheodory-Fejer problem and H∞/l1 identification: a time domain approach , 1995, IEEE Trans. Autom. Control..

[57]  József Bokor,et al.  Identification with generalized orthonormal basis functions - statistical analysis and error bounds , 1994 .

[58]  F. Schweppe.  Recursive state estimation: Unknown but bounded errors and system inputs , 1967 .

[59]  Sheldon M. Ross,et al.  Stochastic Processes , 2018, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

[60]  Pramod P. Khargonekar,et al.  Parameter identification in the presence of non-parametric dynamic uncertainty , 1990, Autom..

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

[62]  John Doyle,et al.  Model validation: a connection between robust control and identification , 1992 .

[63]  Antonio Vicino,et al.  Optimal estimation theory for dynamic systems with set membership uncertainty: An overview , 1991, Autom..

[64]  Graham C. Goodwin,et al.  Quantification of Uncertainty in Estimation using an Embedding Principle , 1989, 1989 American Control Conference.

[65]  Fred C. Schweppe,et al.  Uncertain dynamic systems , 1973 .

[66]  Jie Chen,et al.  Optimal Non-Parametric System Identification From Arbitrary Corrupt Finite Time Series: A Control-Oriented Approach , 1992, 1992 American Control Conference.

[67]  J. Deller.  Set membership identification in digital signal processing , 1989, IEEE ASSP Magazine.

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

[69]  Sandor M. Veres,et al.  Structure selection for bounded-parameter models: consistency conditions and selection criterion , 1991 .

[70]  C. Jacobson,et al.  Worst Case System Identification in l1: Optimal Algorithms and Error Bounds , 1991, 1991 American Control Conference.

[71]  Ruud J.P. Schrama,et al.  An iterative scheme for identification and control design based on coprime factorizations , 1992, 1992 American Control Conference.

[72]  Er-Wei Bai,et al.  Adaptive quantification of model uncertainties by rational approximation , 1991 .

[73]  P.M.J. Van den Hof,et al.  Quantification of Uncertainty in Transfer function Estimation: A Mixed Deterministic-Probabilistic Approach , 1993 .

[74]  Graham C. Goodwin,et al.  Model Error Quantification for Robust Control based on Quasi-Bayesian Estimation in Closed Loop , 1991, 1991 American Control Conference.

[75]  G. Goodwin,et al.  Quantification of Uncertainty in Estimation , 1990 .

[76]  T. H. Gronwall.  A sequence of polynomials connected with the $n$th roots of unity , 1921 .

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

[78]  Richard G. Hakvoort,et al.  Worst-Case System Identification in H∞: Error Bounds and Optimal Models , 1993 .

[79]  Beresford N. Parlett.  Some basic information on information-based complexity theory , 1992 .

[80]  S. Gunnarsson,et al.  A convergent iterative restricted complexity control design scheme , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[81]  Roberto Tempo,et al.  IBC: A Working Tool for Robust Parametric Identification , 1992, 1992 American Control Conference.

[82]  Håkan Hjalmarson,et al.  A Unifying View of Disturbances in Identification , 1994 .

[83]  Gilead Tadmor,et al.  A note on H 8 system identification with probabilistic a priori information , 1993 .

[84]  Petre Stoica,et al.  Decentralized Control , 2018, The Control Systems Handbook.

[85]  P. L. Combettes.  The foundations of set theoretic estimation , 1993 .

[86]  P. Mäkilä.  Robust identification and Galois sequences , 1991 .

[87]  Paul M.J. Van den Hof,et al.  Frequency domain curve fitting with maximum amplitude criterion and guaranteed stability , 1994 .

[88]  G. Zames,et al.  Uncertainty Principles and Identification n-Widths for LTI and Slowly Varying Systems , 1992, 1992 American Control Conference.

[89]  P. Andersen,et al.  Estimated frequency domain model uncertainties used in robust controller design-a /spl mu/-approach , 1994, 1994 Proceedings of IEEE International Conference on Control and Applications.

[90]  J. Partington.  Robust identification and interpolation in H , 1991 .

[91]  Guoxiang Gu,et al.  Identification in hH∞ via convex programming , 1993, 1993 American Control Conference.

[92]  R. Kosut.  Adaptive Robust Control via Transfer Function Uncertainty Estimation , 1988, 1988 American Control Conference.

[93]  E. Hannan.  The asymptotic theory of linear time-series models , 1973, Journal of Applied Probability.

[94]  Jonathan R. Partington,et al.  Algorithms for identification in H∞ with unequally spaced function measurements , 1993 .

[95]  L. Brutman.  On the Polynomial and Rational Projections in the Complex Plane , 1980 .

[96]  Harry L. Trentelman,et al.  Essays on control : perspectives in the theory and its applications , 1993 .

[97]  Graham C. Goodwin,et al.  Estimated Transfer Functions with Application to Model Order Selection , 1992 .

[98]  J. P. Norton,et al.  Identification and application of bounded-parameter models , 1985, Autom..

[99]  M. Milanese,et al.  Estimation theory and uncertainty intervals evaluation in presence of unknown but bounded errors: Linear families of models and estimators , 1982 .

[100]  P. V. D. Hof,et al.  Quantification of model uncertainty from data: input design, interpolation, and connection with robust control design specifications , 1992, 1992 American Control Conference.

[101]  C. N. Nett,et al.  Fundamentals of control-oriented system identification and their application for identification in H∞ , 1991, 1991 American Control Conference.

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

[103]  M. Zwaan.  An introduction to hilbert space , 1990 .

[104]  Jonathan R. Partington,et al.  Worst-case identification in Banach spaces , 1992 .

[105]  E. Walter,et al.  Estimation of parameter bounds from bounded-error data: a survey , 1990 .

[106]  Allan Pinkus,et al.  On the Erdös Conjecture Concerning Minimal Norm Interpolation on the Unit Circle , 1980 .

[107]  A. Helmicki,et al.  H∞ identification of stable LSI systems: a scheme with direct application to controller design , 1989, 1989 American Control Conference.

[108]  Mario Milanese,et al.  Properties of least squares estimates in set membership identification, , 1995, Autom..

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

[110]  Munther A. Dahleh,et al.  Controller design for plants with structured uncertainty , 1993, Autom..

[111]  P. Mäkilä.  Worst-case input-output identification , 1992 .

[112]  László Gerencsér,et al.  Topics in Stochastic Systems: Modelling, Estimation, and Adaptive Control , 1991 .

[113]  J. Tsitsiklis,et al.  The sample complexity of worst-case identification of FIR linear systems , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[114]  Stephen P. Boyd,et al.  Set-membership identification of systems with parametric and nonparametric uncertainty , 1992 .

[115]  Paul Erdös,et al.  Problems and results on the theory of interpolation. II , 1958 .

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

[117]  E. Walter,et al.  Exact recursive polyhedral description of the feasible parameter set for bounded-error models , 1989 .

[118]  Sandor M. Veres.  Limited Complexity and Parallel Implementation of Polytope Updating , 1992, 1992 American Control Conference.

[119]  Håkan Hjalmarsson.  Aspects on Incomplete Modeling in System Identification , 1993 .

[120]  G. Stein,et al.  Robust performance of adaptive controllers with general uncertainty structure , 1990, 29th IEEE Conference on Decision and Control.

[121]  R. Kosut.  Adaptive control via parameter set estimation , 1988 .

[122]  Hidenori Kimura,et al.  Time domain identification for robust control , 1993 .

[123]  Pramod P. Khargonekar,et al.  A Comparison of Classical Stochastic Estimation and Deterministic Robust Estimation , 1991, 1991 American Control Conference.

[124]  P Erdős.  PROBLEMS AND RESULTS ON T ' HE CONVERGENCE AND DIVERGENCE PROPERTIES OF THE LAGRANGE INTERPOLATION POLYNOMIALS AND SOME EXTREMAL PROBLEMS , .

[125]  Brian D. O. Anderson,et al.  On robust performance improvement through the windsurfer approach to adaptive robust control , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[126]  Robert R. Bitmead,et al.  Iterative Control Design Approaches 1 , 1993 .

[127]  L. Ljung,et al.  The role of model validation for assessing the size of the unmodelled dynamics , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[128]  M. Milanese,et al.  Optimal inner bounds of feasible parameter set in linear estimation with bounded noise , 1991 .

[129]  A. Helmicki,et al.  Identification in H∞: a robustly convergent, nonlinear algorithm , 1990, 1990 American Control Conference.

[130]  Roberto Tempo.  Robust estimation and filtering in the presence of bounded noise , 1987 .

[131]  Philip Parker,et al.  Frequency domain descriptions of linear systems , 1988 .

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

[133]  P. Erdős.  Problems and results on the theory of interpolation. II , 1964 .

[134]  C. Jacobson,et al.  Worst case system identification in l 1 ne-equation> 1 : optimal algorithms and error bounds , 1992 .

[135]  P. Khargonekar,et al.  Linear and nonlinear algorithms for identification in H/sub infinity / with error bounds , 1992 .

[136]  Michael Athans,et al.  A Frequency-Domain Estimator for Use in Adaptive Control Systems , 1987, 1987 American Control Conference.

[137]  Antonio Vicino,et al.  Estimation theory for nonlinear models and set membership uncertainty , 1991, Autom..

[138]  Zhou,et al.  Identification in H¿ via convex programming , 1993 .

[139]  Ruud J. P. Schrama.  Accurate identification for control: the necessity of an iterative scheme , 1992 .

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

[141]  H. Woxniakowski.  Information-Based Complexity , 1988 .