Nonlinear Filters. Simon Haykin

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СКАЧАТЬ right-parenthesis right-parenthesis"/>. In such cases, a difference between the actual and the nominal values of the state (a perturbation in the state) leads to a difference between the actual and the nominal values of the input (a perturbation in the input), and in effect therefore, . Otherwise, can be zero. Using the Taylor series expansion and neglecting the higher‐order terms, we obtain the following perturbation state‐space model:

(2.88)

(2.89)

      where and , respectively, denote the Jacobian matrices obtained by taking the derivatives of with respect to and . Similarly, and are Jacobians of with respect to and .

Now, the continuous‐time state‐space model in (2.88) and (2.89) can be treated as an LTV system. The discrete‐time equivalent of (2.88) and (2.89) is obtained as:

      (2.90)

      (2.91)

      where is the solution to

      (2.92)

      with the initial condition:

      (2.93)

      when we set and . As before, is given by:

      (2.94)

      So far, this chapter has been focused on studying the observability of deterministic systems. Section 2.7 discusses the observability of stochastic systems.

2.7 Observability of Stochastic Systems

      Before proceeding with defining observability for stochastic systems, we need to recall a few concepts from information theory [26]:

      Definition 2.3 Entropy is a measure of our uncertainty about an event in Shannon's information theory. Specifically, СКАЧАТЬ