Nonlinear Filters. Simon Haykin

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Название Nonlinear Filters
Автор произведения Simon Haykin
Жанр Программы
Серия
Издательство Программы
Год выпуска 0
isbn 9781119078159
Nonlinear Filters - Simon Haykin
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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"/>

      (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)StartLayout 1st Row 1st Column 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 2nd Column equals Start 6 By 1 Matrix 1st Row ModifyingAbove bold y With Ì‚ Subscript k vertical-bar k minus n Sub Subscript x Subscript Baseline 2nd Row ModifyingAbove bold y With Ì‚ Subscript k vertical-bar k minus n Sub Subscript x Subscript plus 1 Baseline 3rd Row vertical-ellipsis 4th Row ModifyingAbove bold y With Ì‚ Subscript k vertical-bar k minus 3 Baseline 5th Row ModifyingAbove bold y With Ì‚ Subscript k vertical-bar k minus 2 Baseline 6th Row ModifyingAbove bold y With Ì‚ Subscript k vertical-bar k minus 1 Baseline EndMatrix 2nd Row 1st Column Blank 2nd Column equals Start 6 By 1 Matrix 1st Row bold g left-parenthesis bold 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 minus 1 Baseline right-parenthesis comma bold u Subscript k Baseline right-parenthesis 2nd Row bold g left-parenthesis bold 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 minus 1 Baseline right-parenthesis comma bold u Subscript k Baseline right-parenthesis 3rd Row vertical-ellipsis 4th Row bold g left-parenthesis bold f squared left-parenthesis bold x Subscript k minus 2 Baseline comma bold u Subscript k minus 2 colon k minus 1 Baseline right-parenthesis comma bold u Subscript k Baseline right-parenthesis 5th Row bold g left-parenthesis bold f left-parenthesis bold x Subscript k minus 1 Baseline comma bold u Subscript k minus 1 Baseline right-parenthesis comma bold u Subscript k Baseline right-parenthesis 6th Row bold g left-parenthesis bold x Subscript k Baseline comma bold u Subscript k Baseline right-parenthesis EndMatrix period EndLayout

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

      (3.12)bold g left-parenthesis bold upper F Superscript i Baseline left-parenthesis bold x Subscript k minus i Baseline comma bold u Subscript k minus i colon k Baseline right-parenthesis right-parenthesis equals bold g left-parenthesis bold f Superscript i Baseline left-parenthesis bold x Subscript k minus i Baseline comma bold u Subscript k minus i colon k minus 1 Baseline right-parenthesis comma bold u Subscript k Baseline right-parenthesis comma

      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

      (3.14)StartLayout 1st Row 1st Column bold z Subscript k minus n Sub Subscript x Subscript plus 2 vertical-bar k plus 1 Baseline equals 2nd Column bold z Subscript k minus n Sub Subscript x Subscript plus 2 vertical-bar k Baseline plus script upper O Superscript negative 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 1 colon k minus 1 Baseline right-parenthesis 2nd Row 1st Column Blank 2nd Column times left-parenthesis bold upper B left-parenthesis bold y Subscript k plus 1 Baseline minus bold g left-parenthesis bold z Subscript k plus 1 vertical-bar k Baseline right-parenthesis right-parenthesis plus bold upper L left-parenthesis bold y Subscript k minus n Sub Subscript x Subscript plus 1 Baseline minus bold g left-parenthesis bold z Subscript k minus n Sub Subscript x Subscript plus 1 vertical-bar k Baseline right-parenthesis right-parenthesis right-parenthesis comma EndLayout

      (3.15)StartLayout 1st Row 1st Column ModifyingAbove x With caret Subscript k vertical-bar k Baseline equals 2nd Column bold upper F Superscript n Super Subscript x Superscript 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 1 colon k Baseline right-parenthesis comma EndLayout

      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 k 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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