target="_blank" rel="nofollow" href="#fb3_img_img_b9de217b-8384-5790-a35e-d4ea28dc9fb1.png" alt="bold-script upper L left-parenthesis bold x left-parenthesis t right-parenthesis comma bold u left-parenthesis t right-parenthesis right-parenthesis equals Start 4 By 1 Matrix 1st Row Start 3 By 1 Matrix 1st Row upper L Subscript bold f Superscript 0 Baseline bold g 1 2nd Row vertical-ellipsis 3rd Row upper L Subscript bold f Superscript 0 Baseline bold g Subscript n Sub Subscript y Subscript Baseline EndMatrix 2nd Row Start 3 By 1 Matrix 1st Row upper L Subscript bold f Superscript 1 Baseline bold g 1 2nd Row vertical-ellipsis 3rd Row upper L Subscript bold f Superscript 1 Baseline bold g Subscript n Sub Subscript y Subscript Baseline EndMatrix 3rd Row vertical-ellipsis 4th Row Start 3 By 1 Matrix 1st Row upper L Subscript bold f Superscript n minus 1 Baseline bold g 1 2nd Row vertical-ellipsis 3rd Row upper L Subscript bold f Superscript n minus 1 Baseline bold g Subscript n Sub Subscript y Subscript Baseline EndMatrix EndMatrix period"/>
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:
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:
