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# Lect_39 - Nonlinear Systems and Control Lecture 38...

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Nonlinear Systems and Control Lecture # 38 Observers High-Gain Observers Motivating Example – p. 1/1

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˙ x 1 = x 2 , ˙ x 2 = φ ( x, u ) , y = x 1 Let u = γ ( x ) stabilize the origin of ˙ x 1 = x 2 , ˙ x 2 = φ ( x, γ ( x )) Observer: ˙ ˆ x 1 = ˆ x 2 + h 1 ( y ˆ x 1 ) , ˙ ˆ x 2 = φ 0 x, u ) + h 2 ( y ˆ x 1 ) φ 0 ( x, u ) is a nominal model φ ( x, u ) ˜ x 1 = x 1 ˆ x 1 , ˜ x 2 = x 2 ˆ x 2 ˙ ˜ x 1 = h 1 ˜ x 1 + ˜ x 2 , ˙ ˜ x 2 = h 2 ˜ x 1 + δ ( x, ˜ x ) δ ( x, ˜ x ) = φ ( x, γ x )) φ 0 x, γ x )) – p. 2/1
Design H = bracketleftBigg h 1 h 2 bracketrightBigg such that A o = bracketleftBigg h 1 1 h 2 0 bracketrightBigg is Hurwitz Transfer function from δ to ˜ x : G o ( s ) = 1 s 2 + h 1 s + h 2 bracketleftBigg 1 s + h 1 bracketrightBigg Design H to make sup ω R bardbl G o ( ) bardbl as small as possible h 1 = α 1 ε , h 2 = α 2 ε 2 , ε > 0 G o ( s ) = ε ( εs ) 2 + α 1 εs + α 2 bracketleftBigg ε εs + α 1 bracketrightBigg – p. 3/1

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G o ( s ) = ε ( εs ) 2 + α 1 εs + α 2 bracketleftBigg ε εs + α 1 bracketrightBigg Observer eigenvalues are ( λ 1 ) and ( λ 2 )
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