Partially observable linear systems under dependent noises
The author starts with some comments on the role of noise in the Arts and Sciences: The artists create ``deliberate noise'', ``the best masterpieces of literature, music, modern fine art etc. are those where a clear idea, traditionally related to such concept as love, is presented under a skilful veil of ``deliberate noise''. On the contrary, scientists fight against noise; ``A scientific discovery is a law of nature extracted from a noisy medium and refined''. But is love an idea or a concept? And is it as clear as the author means? If Art draws on ambiguous feelings in reality, there is certainly not necessarily a need for an activity of superimposed dissimulation as the author suggests. Further many respectable scientists do not believe that randomness is intrinsic to nature, so the mentioned activity of extraction would not make sense. Conversely other respectable scientists believe that noise is part of the world, so that getting rid of it would be an act of dissimulation. Actually engineers (are they seen then as artists?) reinject noise in their deterministic formulas as a modelling tool. Also ``fighting'' may not be a correct description of scientific activity in general, curiosity and love being a motivation for scientific enquiry in some cases. It would be interesting to see the reasons for this formal separation of the Arts and the Sciences. It is not isolated, going in line with the common pejorative expression in French ``flou artistique''. Let us mention that other scientists defended an exactly opposite standpoint, like \textit{W. Lowen} [Int. J. Gen. Syst. 30, 509-512 (2001; Zbl 0999.93007)], being unconvincing as well. This book deals with key issues in control theory, namely the interaction between optimal control and observation and/or estimation issues. The specificity is that here the observation noise and the system noise may be dependent and coloured. Moreover quantities evolve in possibly infinite-dimensional spaces. As a consequence, some elements of functional analysis and operator theory are recalled in the first chapters. Chapter four introduces stochastic differential equations, random processes in Hilbert spaces and estimation under various types of noises. The author considers shifted white noise and wide band noise, which generalize the case of colored noises arising from a white noise at the input of a linear system to a situation with point delays and distributed delays respectively. Most of the rest of the book is devoted to optimal control (linear quadratic case) and estimation. One obtains a generalized separation principle. This is used as well as duality in the sequel for obtaining controllers and estimators. The author studies first the wide band case and then the shifted white noise being approximated by the previous one. Chapter eleven is a study on duality in the more general cases covered by this book. The filtering problem, seen classically as dual to a control problem under the separation principle, becomes a smoothing problem under the extended separation principle of the author. The final chapter deals with problems of stochastic controllability, relating them to corresponding deterministic criteria. Issues of stability and infinite horizon as well as nonlinearity remain open. The bibliography contains 95 entries. There is an index of notations and a modest index. The book, which presents work of the author and colleagues in Azerbaijan and research institutes of the Soviet Union, will be of interest to researchers in optimal control and estimation.
scientific article; zbMATH DE number 1879198