Modelling of systems in the presence of unknown-but-bounded errors attracts the permanent attention of researchers in the fields of parameter estimation, identification, control and optimization. It is a natural counterpart to the stochastic approach (with such traditional tools as the least-squares method, Kalman filtering and maximum-likelihood estimates), where all perturbations are assumed to be random with known distributions. However, often the statistics of these perturbations are not available and it is more natural to consider them as belonging to some bounded sets with no other hypothesis on their distributions. This approach originates in the pioneering works by Schweppe [Citation1], Withenhausen [Citation2] and Bertsekas and Rhodes [Citation3] at the end of the 1960s. Later this technique has been developed by many researchers including those providing papers for the present issue (see the monographs [Citation4 – Citation Citation Citation Citation Citation9], the survey papers [Citation10, Citation11] and special issues of journals [Citation12 – Citation Citation14]).
The unknown-but-bounded model of perturbations becomes very useful for experts in modelling dynamical systems. There were special sessions on this subject at the 3rd and 4th IMACS Symposia on Mathematical Modelling held in Vienna in 2000 and 2003 respectively. The talks at the sessions were the basis for the present special issue. We are proud to report that top experts in the field have contributed to it. The contents of the issue provide an extensive presentation of modern developments and techniques in the set-membership approach to modelling. Numerous applications to industry, space and economic models are also provided.
References
References
- Schweppe , FC . 1968 . Recursive state estimation: unknown but bounded errors and system inputs . IEEE Transactions on Automatic Control , 13 : 22 – 28 .
- Witsenhausen , HS . 1968 . Sets of possible states of linear systems given perturbed observations . IEEE Transctions on Automatic Control , 13 : 556 – 558 .
- Bertsekas , DP and Rhodes , IB . 1971 . Recursive state estimation for a set-membership description of uncertainty . IEEE Transactions on Automatic Control , 16 : 117 – 128 .
- Schweppe FC 1973 Uncertain Dynamic Systems Englewood Cliffs New Jersey Prentice Hall
- Kurzhanski AB 1977 Control and Observation under Uncertainty Moscow Nauka (in Russian)
- Chernousko FL 1988 Estimation of Phase State for Dynamical Systems Moscow Nauka (in Russian)
- Chernousko FL 1994 State Estimation for Dynamic Systems Boca Raton Florida CRC Press
- Kurzhanski AB Valyi I 1996 Ellipsoidal Calculus for Estimation and Control Boston Massachusetts Birkhäuser
- Milanese M Norton J Pet-Lahanier H Walter E (Eds) 1996 Bounding Approaches to System Identification New York Plenum Press
- Combettes , PL . 1993 . The foundations of set-theoretic estimation . Proceedings of the IEEE , 81 : 182 – 208 .
- Durieu , C , Walter , E and Polyak , B . 2001 . Multi-input multi-output ellipsoidal state bounding . Journal of Optimization Theory and Applications , 111 : 273 – 303 .
- Walter E (Ed.) 1990 Special issue on parameter identifications with error bound Mathematics and Computers in Simulation 32 447 607
- Norton J (Ed.) 1994 Special issue on bounded-error estimation, 1 International Journal of Adaptive Control and Signal Processing 8 1 118
- Norton J (Ed.) 1995 Special issue on bounded-error estimation, 2 International Journal of Adaptive Control and Signal Processing 9 1 132