EEL5173 Fall 2009 Lecture _7

# EEL5173 Fall 2009 - v Solution to the State Equations r r x(k 1 = Ax(k Bu(k Z cfw_x1(k 1 zX 1 z zx1(0 Z cfw_x(k 1 zX z zx(0 r 2 2 = 2 Z cfw_x(k 1 =

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v. Solution to the State Equations {} {} {} {} ) ( ) ( ) 0 ( ) ( ) 0 ( ) ( ) 0 ( ) ( ) 0 ( ) ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) ( ) ( ) 1 ( 2 2 1 1 2 1 z BU z X A x z z X z zx z zX zx z zX zx z zX k x Z k x Z k x Z k x Z k Bu k x A k x n n n + = = = + + + = + + = + M M r r r

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(1) The State Transition Matrix: {} {} { } {} { } ) ( ) ( ) 0 ( ) ( ) ( ) ( ) 0 ( ) ( ) ( ) ( ) 0 ( ) ( ) ( ) ( ) ( ) 0 ( ) ( )] ( ) 0 ( [ ) ( ) ( ) ( ) 0 ( ) ( ) ( ) ( ) 0 ( ) ( ) ( 1 1 1 1 1 1 1 1 1 1 1 1 1 1 z BU A zI Z x A zI z Z z BU A zI Z x A zI z Z z BU A zI x A zI z Z k x z BU A zI x A zI z z BU x z A zI z X z BU x z z X A zI z BU x z z X A z X z + = + = + = + = + = + = + = r r r r r r r r r r r r { } {} ) 1 ( ) ( ) ( ) ( ) ( 1 1 1 1 1 1 Φ = = = Φ k z A zI z Z A zI Z A zI z Z k
Solution: ), ( ) 1 ( ) 0 ( ) ( ) ( 1 0 i Bu i k x k k x k i = Φ + Φ = r r for k 1 . Alternative Representation of the State Transition Matrix: () k k k k A k k Bu Bu A Bu A x A k x Bu ABu x A Bu Bu x A A Bu x A x Bu x A x

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## This note was uploaded on 07/15/2011 for the course EEL 5173 taught by Professor Tung during the Fall '09 term at FSU.

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EEL5173 Fall 2009 - v Solution to the State Equations r r x(k 1 = Ax(k Bu(k Z cfw_x1(k 1 zX 1 z zx1(0 Z cfw_x(k 1 zX z zx(0 r 2 2 = 2 Z cfw_x(k 1 =

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