Estimation Theory for Dynamic Systems with Unknown but Bounded Uncertainty: An Overview

Abstract In many problems such as linear and nonlinear regressions, parameter and state estimation of dynamic systems, state space and time series prediction, interpolation, smoothing, functions approximation, one has to evaluate some unknown variable using available data. The data are always associated with some uncertainty and it is necessary to evaluate how this uncertainty affects the estimated variables. Typically the problem is approached assuming a probabilistic description of uncertainty and applying statistical estimation theory. An interesting alternative approach, referred to as set membership or Unknown But Bounded (UBB) error description, has been investigated since the late 60's. In this approach, uncertainty is described by an additive noise which is known only to have given integral (typically l 2 or l 1 ) or componentwise ( l ∞ ) bounds. In this paper we review the main results of this theory, with special attention to the most recent advances obtained in the case of componentwise bounds.

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

[2]  Antonio Vicino,et al.  Robust Parameter Estimation for Linear Models with Set-membership Uncertainty , 1988 .

[3]  R. Tempo,et al.  Optimal error predictors for economic models , 1988 .

[4]  Antonio Vicino,et al.  Guaranteed specification control design of inertial platforms for a high accuracy calibration device , 1990, 29th IEEE Conference on Decision and Control.

[5]  J. Norton,et al.  Problems in identifying the dynamics of biological systems from very short records , 1986, 1986 25th IEEE Conference on Decision and Control.

[6]  M. Milanese,et al.  Computation of nonconservative stability perturbation bounds for systems with nonlinearly correlated uncertainties , 1990 .

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

[8]  E. Walter,et al.  Mathematical equivalence of two ellipsoid algorithms for bounded-error estimation , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[9]  C. Micchelli.  Optimal Sampling Design for Parameter Estimation and P-Widths under Stochastic and Deterministic Noise , 1989 .

[10]  H. Witsenhausen.  Sets of possible states of linear systems given perturbed observations , 1968 .

[11]  Antonio Vicino,et al.  Conditionally optimal algorithms and estimation of reduced order models , 1988, J. Complex..

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

[13]  J. Norton.  Recursive computation of inner bounds for the parameters of linear models , 1989 .

[14]  M. Milanese,et al.  Advanced modeling and identification techniques for metabolic processes. , 1984, Critical reviews in biomedical engineering.

[15]  J. Ecker.  Geometric Programming: Methods, Computations and Applications , 1980 .

[16]  Yih-Fang Huang,et al.  Asymptotically convergent modified recursive least-squares with data-dependent updating and forgetting factor , 1985, 1985 24th IEEE Conference on Decision and Control.

[17]  M. Milanese,et al.  Optimal Approximation of Feasible Parameter Set in Estimation with Unknown but Bounded Noise , 1991 .

[18]  A. Vicino,et al.  Optimal inner bounds of feasible parameter set in linear estimation with bounded noise , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[19]  Enrico Canuto,et al.  Coordinate Measuring Machines and Machine Tools Selfcalibration and Error Correction , 1987 .

[20]  R. Tempo,et al.  Robust Estimation and Prediction Using Arma Models: A Nonstatistical Approach , 1988 .

[21]  M. Smit.  A novel approach to the solution of indirect measurement problems with minimal error propagation , 1983 .

[22]  Antonio Vicino,et al.  Strongly optimal algorithms and optimal information in estimation problems , 1986, J. Complex..

[23]  M. Milanese,et al.  A note on the derivation and use of reduced-order models , 1976 .

[24]  J. Norton,et al.  Structure identification of parameter-bounding models by use of noise-structure bounds , 1989 .

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

[26]  Adriaan van den Bos,et al.  Nonlinear least-absolute-values and minimax model fitting , 1985, Autom..

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

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

[29]  G. Belforte,et al.  An Improved Parameter Identification Algorithm for Signals with Unknown-But-Bounded Errors , 1985 .

[30]  V. Broman,et al.  A compact algorithm for the intersection and approximation of N -dimensional polytopes , 1990 .

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

[32]  Eric Walter,et al.  Experiment design in a bounded-error context: Comparison with D-optimality , 1989, Autom..

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

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

[35]  Yakov Z. Tsypkin,et al.  Robust identification , 1980, Autom..

[36]  Charles A. Micchelli,et al.  A Survey of Optimal Recovery , 1977 .

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

[38]  V. Broman,et al.  Polytopes, a novel approach to tracking , 1986, 1986 25th IEEE Conference on Decision and Control.

[39]  J. R. Deller,et al.  An alternative adaptive sequential regression algorithm and its application to the recognition of cerebral palsy speech , 1987 .

[40]  J. R. Deller,et al.  Set-membership theory applied to linear prediction analysis of speech , 1987, ICASSP '87. IEEE International Conference on Acoustics, Speech, and Signal Processing.

[41]  G. V. Straten.  Analytical methods for parameter-space delimination and application to shallow-lake phytoplankton-dynamics modeling , 1985 .

[42]  E. Walter,et al.  Further results on recursive polyhedral description of parameter uncertainty in the bounded-error context , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[43]  J. Norton.  Bounding Techniques for Model-Structure Selection , 1989 .

[44]  Antonio Vicino,et al.  Optimal error and GMDH predictors: A comparison with some statistical techniques , 1987 .

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

[46]  E. Walter,et al.  Identifiability and distinguishability of fundamental parameters in catalytic methanation , 1986 .

[47]  G. Belforte,et al.  Optimal sampling schedule for parameter estimation of linear models with unknown but bounded measurement errors , 1987 .

[48]  Eric Walter,et al.  A distribution-free criterion for robust identification, with applications in system modelling and image processing , 1985, Autom..

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

[50]  Patrick L. Combettes,et al.  Stability of the linear prediction filter: a set theoretic approach , 1988, ICASSP-88., International Conference on Acoustics, Speech, and Signal Processing.

[51]  Gustavo Belforte,et al.  A bounded error approach to the tuning of a digital voltmeter , 1988 .

[52]  R. Tempo,et al.  Optimality of central and projection algorithms for bounded uncertainty , 1986 .