The sensitive of state identifiers and \(\varepsilon\)-observability (Q1899660)

Linear finite-dimensional control systems are considered. Using the methods taken from the theory of generalized functions both necessary and sufficient conditions for approximate observability in the presence of various types of noise are formulated and proved. Moreover, state identifiers sensitivity with respect to the parameters is also investigated. The existence of an optimum choice for the parameters of the observer is proved and a procedure for calculating the spectrum of the observer is presented. This procedure guarantees the observability of the system with a specific degree of accuracy. The theoretical considerations are illustrated by two simple numerical examples. Similar observability problems have been recently considered in the paper [\textit{A. W. Olbrot} and \textit{A. Sosnowski}, Int. J. Control 47, 1455-1465 (1988; Zbl 0646.93012)].