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The previous chapter presents methods for representing a class of dynamic systems with relatively small numbers of components, such as a harmonic resonator with one mass and spring. The results are models for deterministic mechanics, in which the state of every component of the system is represented and propagated explicitly. Another approach has been developed for extremely large dynamic systems, such as the ensemble of gas molecules in a reaction chamber. | Kalman Filtering TBicory and Practice Using FLlTKiB Second Edition Mohinder S. Grewal Angus P. Andrews Copyright 2001 John Wiley Sons Inc. ISBNs 0-471-39254-5 Hardback 0-471-26638-8 Electronic _3 Random Processes and Stochastic Systems A completely satisfactory definition of random sequence is yet to be discovered. G. James and R. C. James Mathematics Dictionary D. Van Nostrand Co. Princeton New Jersey 1959 3.1 CHAPTER FOCUS The previous chapter presents methods for representing a class of dynamic systems with relatively small numbers of components such as a harmonic resonator with one mass and spring. The results are models for deterministic mechanics in which the state of every component of the system is represented and propagated explicitly. Another approach has been developed for extremely large dynamic systems such as the ensemble of gas molecules in a reaction chamber. The state-space approach for such large systems would be impractical. Consequently this other approach focuses on the ensemble statistical properties of the system and treats the underlying dynamics as a random process. The results are models for statistical mechanics in which only the ensemble statistical properties of the system are represented and propagated explicitly. In this chapter some of the basic notions and mathematical models of statistical and deterministic mechanics are combined into a stochastic system model which represents the state of knowledge about a dynamic system. These models represent what we know about a dynamic system including a quantitative model for our uncertainty about what we know. In the next chapter methods will be derived for modifying the state of knowledge based on observations related to the state of the dynamic system. 56 3.1 CHAPTER FOCUS 57 3.1.1 Discovery and Modeling of Random Processes Brownian Motion and Stochastic Differential Equations. The British botanist Robert Brown 1773-1858 reported in 1827 a phenomenon he had observed while
