Nonlinear Filters. Simon Haykin

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СКАЧАТЬ 2nd Column equals vertical-ellipsis 7th Row 1st Column bold y Superscript left-parenthesis n minus 1 right-parenthesis 2nd Column equals StartFraction partial-differential Over partial-differential bold x EndFraction left-bracket upper L Subscript bold f Superscript n minus 2 Baseline bold g 1 midline-horizontal-ellipsis upper L Subscript bold f Superscript n minus 2 Baseline bold g Subscript i Baseline midline-horizontal-ellipsis upper L Subscript bold f Superscript n minus 2 Baseline bold g Subscript n Sub Subscript y Subscript Baseline right-bracket Superscript upper T Baseline ModifyingAbove bold x With dot 8th Row 1st Column Blank 2nd Column equals left-bracket upper L Subscript bold f Superscript n minus 1 Baseline bold g 1 midline-horizontal-ellipsis upper L Subscript bold f Superscript n minus 1 Baseline bold g Subscript i Baseline midline-horizontal-ellipsis upper L Subscript bold f Superscript n minus 1 Baseline bold g Subscript n Sub Subscript y Subscript Baseline right-bracket Superscript upper T Baseline period EndLayout"/>

The aforementioned differential equations can be rewritten in the following compact form:

(2.67)

      where

      (2.68)

      and

      (2.69)

The system of nonlinear differential equations in (2.67) can be linearized about an initial state to develop a test for local observability of the nonlinear system (2.61) and (2.62) at this specific initial state, where the linearized test would be similar to the observability test for linear systems. Writing the Taylor series expansion of the function about and ignoring higher‐order terms, we will have:

      (2.70)

      Using Cartan's formula:

      (2.71)

      we obtain:

      (2.72)

Now, we can proceed with deriving the local observability test for nonlinear systems based on the aforementioned linearized system of equations. The nonlinear system in (2.61) and (2.62) is observable at , if there exists a neighborhood of and an ‐tuple of integers called observability indices such that [9, 24]:

      1  for .

      2 The row vectors of are linearly independent.

      From the row vectors , an observability matrix can be constructed for the continuous‐time nonlinear system in (2.61) and (2.62) as follows:

(2.73)СКАЧАТЬ