Nonlinear Filters. Simon Haykin

Nonlinear Filters

k minus 1 Baseline right-parenthesis comma bold u Subscript k Baseline right-parenthesis 4th Row bold g left-parenthesis bold x Subscript k Baseline comma bold u Subscript k Baseline right-parenthesis EndMatrix period"/>

Let bold f Superscript i denote the th composite of the system function bold f as:

(3.10) bold f Superscript i Baseline equals ModifyingBelow bold f ring bold f ring midline-horizontal-ellipsis ring bold f With presentation form for vertical right-brace Underscript i times Endscripts period

Given an initial state and the sequence bold u Subscript k minus n Sub Subscript x Subscript plus 1 colon k , the corresponding sequence of predicted outputs is obtained as:

(3.11)

To simplify the notation, let us define bold upper F such that

(3.12)

then we have:

bold upper Phi left-parenthesis bold x Subscript k minus n Sub Subscript x Subscript plus 1 colon k Baseline comma bold u Subscript k minus n Sub Subscript x Subscript plus 1 colon k Baseline right-parenthesis equals Start 6 By 1 Matrix 1st Row bold g left-parenthesis bold upper F Superscript n Super Subscript x Superscript minus 1 Baseline left-parenthesis bold x Subscript k minus n Sub Subscript x Subscript plus 1 Baseline comma bold u Subscript k minus n Sub Subscript x Subscript plus 1 colon k Baseline right-parenthesis right-parenthesis 2nd Row bold g left-parenthesis bold upper F Superscript n Super Subscript 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) bold z Subscript k minus n Sub Subscript x Subscript plus i vertical-bar k Baseline equals bold upper F Superscript i minus 1 Baseline left-parenthesis bold z Subscript k minus n Sub Subscript x Subscript plus 1 vertical-bar k Baseline comma bold u Subscript k minus n Sub Subscript x Subscript plus i minus 1 colon i minus 1 Baseline right-parenthesis comma

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

(3.14)

(3.15)

where bold upper L is the observer gain, bold upper B equals left-bracket 1 0 midline-horizontal-ellipsis 0 right-bracket Superscript upper T , and script upper O is the discrete nonlinear observability matrix, which is computed at sample time as:

(3.16) script upper O left-parenthesis bold x comma bold u Subscript k minus n Sub Subscript x Subscript plus 2 colon k Baseline right-parenthesis equals StartFraction partial-differential bold upper Phi left-parenthesis bold x comma bold u Subscript k minus n Sub Subscript x Subscript plus 2 colon k Baseline right-parenthesis Over partial-differential bold x EndFraction period

The observer gain is determined in a way to guarantee local stability

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