Nonlinear Filters. Simon Haykin

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СКАЧАТЬ reason that is called the state‐transition matrix is that it describes the dynamic behavior of the following autonomous system (a system with no input):

      (2.53)

      with being obtained from

      (2.54)

      Following a discussion on energy of the system output similar to the continuous‐time case, we reach the following definition for the discrete‐time observability Gramian matrix:

      (2.55)

As before, the system (2.49) and (2.50) is observable, if and only if the observability Gramian matrix is full‐rank (nonsingular) [9].

      2.5.3 Discretization of LTV Systems

This section generalizes the method presented before for discretization of continuous‐time LTI systems and describes how the continuous‐time LTV system (2.36) and (2.37) can be discretized. Solving the differential equation in (2.36), we obtain:

(2.56)

where is the state‐transition matrix as described in (2.40) and (2.41). Using zero‐order‐hold sampling results in a piecewise‐constant input , which remains constant at over the time interval . Setting and , from (2.56), we will have:

      (2.57)

      Therefore, dynamics of the discrete‐time equivalent of the continuous‐time system in (2.36) and (2.37) will be governed by the following state‐space model [19]:

      (2.58)

      (2.59)

      where

      (2.60)

2.6 Observability of Nonlinear Systems

      As mentioned before, observability is a global property for linear systems. However, for nonlinear systems, a weaker form of observability is defined, in which an initial state must be distinguishable only from its neighboring points. Two states and are indistinguishable, if their corresponding outputs are equal: for , where СКАЧАТЬ