Lect_40 - Nonlinear Systems and Control Lecture 40...

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Nonlinear Systems and Control Lecture # 40 Observers High-Gain Observers Stabilization – p. 1/1

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˙ x = Ax + ( x,z,u ) ˙ z = ψ ( x,z,u ) y = Cx ζ = q ( x,z ) u R p , y R m , ζ R s , x R ρ , z R A,B,C are block diagonal matrices A i = 0 1 · · · · · · 0 0 0 1 · · · 0 . . . . . . 0 · · · · · · 0 1 0 · · · · · · · · · 0 ρ i × ρ i , B i = 0 0 . . . 0 1 ρ i × 1 – p. 2/1
C i = bracketleftBig 1 0 · · · · · · 0 bracketrightBig 1 × ρ i , ρ = m summationdisplay i =1 ρ i Normal form Mechanical and electromechanical systems Example: Magnetic Suspension ˙ x 1 = x 2 ˙ x 2 = g k m x 2 L 0 ax 2 3 2 m ( a + x 1 ) 2 ˙ x 3 = 1 L ( x 1 ) bracketleftbigg Rx 3 + L 0 ax 2 x 3 ( a + x 1 ) 2 + u bracketrightbigg – p. 3/1

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Stabilizing (partial) state feedback controller: u = γ ( x,ζ ) ˙ ϑ = Γ( ϑ,x,ζ ) , u = γ ( ϑ,x,ζ ) Closed-loop system under state feedback: ˙ X = f ( X ) , X = ( x,z,ϑ ) The origin of ˙ X = f ( X ) is asymptotically stable
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