Nonlinear Filters. Simon Haykin

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СКАЧАТЬ 1 Baseline left-parenthesis nabla bold g Subscript i Baseline right-parenthesis EndMatrix comma i equals 1 comma ellipsis comma n Subscript y Baseline semicolon j equals 1 comma ellipsis comma k Subscript i Baseline period"/>

If is full‐rank, then the nonlinear system in (2.61) and (2.62) is locally weakly observable. It is worth noting that the observability matrix for continuous‐time linear systems (2.7) is a special case of the observability matrix for continuous‐time nonlinear systems (2.73). In other words, if and are linear functions, then (2.73) will be reduced to (2.7) [9, 24].

      The nonlinear system in (2.61) and (2.62) can be linearized about . Using Taylor series expansion and ignoring higher‐order terms, we will have the following linearized system:

      (2.74)

      (2.75)

Then, the observability test for linear systems can be applied to the following linearized system matrices:

(2.76)

(2.77)

In this way, the nonlinear observability matrix in (2.73) can be approximated by the observability matrix, which is constructed using and in (2.76) and (2.77). Although this approach may seem simpler, observability of the linearized system may not imply the observability of the original nonlinear system [9].

      2.6.2 Discrete‐Time Nonlinear Systems

      The state‐space model of a discrete‐time nonlinear system is represented by the following system of nonlinear equations:

(2.78)

(2.79)

      where is the system function, and is the measurement function. Similar to the discrete‐time linear case, starting from the initial cycle, system's output vectors at successive cycles till cycle can be written based on the initial state and input vectors as follows:

(2.80)СКАЧАТЬ