Rapprochement between bounded‐error and stochastic estimation theory

There has been a recent surge of interest in estimation theory based on very simple noise descriptions; for example, the absolute value of a noise sample is simply bounded. to date, this line of work has not been critically compared with pre-existing work on stochastic estimation theory which uses more complicated noise descriptions. the present paper attempts to redress this by examining the rapprochement between the two schools of work. For example, we show that for many problems a bounded-error estimation approach is precisely equivalent in terms of the final result to the stochastic approach of Bayesian estimation. We also show that in spite of having the advantages of being simple and intuitive, bounded-error estimation theory is demanding on the quantitative accuracy of prior information. In contrast, we discuss how the assumptions underlying stochastic estimation theory are more complex but have the key feature that qualitative assumptions on the nature of a typical disturbance sequence can be made to reduce the importance of quantitative assumptions being correct. We also discuss how stochastic theory can be extended to deal with a problem at present tackled only by bounded-error estimation methods: the quantification of estimation errors arising from the presence of undermodelling.

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

[2]  P. R. Nelson.  The algebra of random variables , 1979 .

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

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

[5]  L. Ljung,et al.  Design variables for bias distribution in transfer function estimation , 1986, The 23rd IEEE Conference on Decision and Control.

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

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

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

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

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

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

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

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

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

[15]  R. Younce,et al.  Identification with non-parametric uncertainty , 1990, IEEE International Symposium on Circuits and Systems.

[16]  Pertti M. Mäkilä,et al.  Approximation of stable systems by laguerre filters , 1990, Autom..

[17]  J. P. Norton,et al.  Fast and robust algorithm to compute exact polytope parameter bounds , 1990 .

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

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

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

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

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

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

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