Optimal and Robust State Estimation

2Probability and Stochastic Processes

In stochastic processes the future is not uniquely determined, but we have at least probability relations enabling us to make predictions.

William Feller [47], p. 420.

Signals and messages containing information about electrical, mechanical, chemical, biological, and other processes are usually affected by various types of noise and disturbances, the values of which often cannot be ignored. In such cases, the deterministic approximation becomes too rough, and probabilistic methods are used to achieve the best results. Under the influence of noise, any process becomes random, and accurate information extraction about its features requires mathematical methods describing random variables, stochastic processes, and SDEs. This chapter provides a brief introduction to the concepts and foundations of the theory of probability and stochastic processes, which will be used later in the discussion of methods of state estimation.

2.1 Random Variables

In engineering practice we often deal with some kind of experiment and elements of its random outcomes that cannot be used directly. For example, tracking distance can be measured via time of arrival. Thus, to each we can assign a real number , call it random variable [191], and describe or simply ...