Nonlinear Filters. Simon Haykin

Скачать книгу

x Superscript minus 2 Baseline left-parenthesis bold x Subscript k minus n Sub Subscript x Subscript plus 2 Baseline comma bold u Subscript k minus n Sub Subscript x Subscript plus 2 colon k Baseline right-parenthesis right-parenthesis 3rd Row vertical-ellipsis 4th Row bold g left-parenthesis bold upper F squared left-parenthesis bold x Subscript k minus 2 Baseline comma bold u Subscript k minus 2 colon k Baseline right-parenthesis right-parenthesis 5th Row bold g left-parenthesis bold upper F left-parenthesis bold x Subscript k minus 1 Baseline comma bold u Subscript k minus 1 colon k Baseline right-parenthesis right-parenthesis 6th Row bold g left-parenthesis bold x Subscript k Baseline comma bold u Subscript k Baseline right-parenthesis EndMatrix period"/>

      Defining

      (3.13)

the filtered state estimate is computed from the following discrete‐time system [9]:

      (3.14)

      (3.15)

      where is the observer gain, , and is the discrete nonlinear observability matrix, which is computed at sample time as:

      (3.16)

      The observer gain is determined in a way to guarantee local stability of the perturbed linear system for the reconstruction error in :

      (3.17)

      where and

      This approach for designing nonlinear observers is applicable to systems that are observable for every bounded input, , with and being uniformly Lipschitz continuous functions of the state:

      (3.18)

      (3.19)

      where and denote the corresponding Lipschitz constants. However, convergence is guaranteed only for a neighborhood around the true state [36].

3.4 Sliding‐Mode Observer

      The equivalent control approach allows for designing the discrete‐time sliding‐mode realization of a reduced‐order asymptotic observer [37]. Let us consider the following discrete‐time state‐space model:

(3.20)

Скачать книгу