Tham khảo tài liệu 'new approaches in automation and robotics part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Bilinear Time Series in Signal Analysis 113 2. Bilinear time series models Large amount of dynamical systems may be described with set of conservation equations in the following form Ax t Bu t 2 Nkuk t x t 7 flt k 1 where the last term creates the bilinear part of the equation. Bilinear equations are the natural way of description of a number of chemical technological processes like decantation distillation and extraction as well as biomedical systems . Mohler 1999 Nise 2000 . Though the nature of many processes is bilinear identification of the model 7 can be difficult at least because some of the state or input variables may be immeasurable. This is the case of many biological or biomedical processes. Often the discrete set of the output observation yi for i 1 . n is the only information on the considered process. In such cases bilinear time series model 8 which explains relation between the set of the output data only may be considered. K A z-1 y c z-1 e yE Pk -kVi-i 8 k 1 1 1 Bilinear time series models have been mentioned in control engineering since early seventieth. Schetzen Theorem Schetzen 1980 states that any stable time variant process may be modeled as time invariant bilinear time series. General structure of bilinear time series model 8 is complex enough to make its analysis very difficult. Therefore in practise the particular model structures are being analysed. Stochastic processes are completely characterized by their probabilistic structure . probability or probability density p y . Therrien 1992 . However in practice probabilistic structure of a considered system is unknown and therefore the system analysis is performed on the ground of its statistical moments. The moments for any stochastic process with any probabilistic density p y are expressed as My E y 9 where E is an operator of expected value E y E yp y p. 10 x Central moments are Myr E yi - p r 11 When the structure of particular bilinear model is simple the moments and the .
