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# Lect_26 - Nonlinear Systems and Control Lecture 26...

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Unformatted text preview: Nonlinear Systems and Control Lecture # 26 Stabilization Feedback Lineaization – p.1/12 Consider the nonlinear system ˙ x = f ( x ) + G ( x ) u f (0) = 0 , x ∈ R n , u ∈ R m Suppose there is a change of variables z = T ( x ) , defined for all x ∈ D ⊂ R n , that transforms the system into the controller form ˙ z = Az + Bγ ( x )[ u − α ( x )] where ( A,B ) is controllable and γ ( x ) is nonsingular for all x ∈ D u = α ( x ) + γ − 1 ( x ) v ⇒ ˙ z = Az + Bv – p.2/12 v = − Kz Design K such that ( A − BK ) is Hurwitz The origin z = 0 of the closed-loop system ˙ z = ( A − BK ) z is globally exponentially stable u = α ( x ) − γ − 1 ( x ) KT ( x ) Closed-loop system in the x-coordinates: ˙ x = f ( x ) + G ( x ) bracketleftbig α ( x ) − γ − 1 ( x ) KT ( x ) bracketrightbig – p.3/12 What can we say about the stability of x = 0 as an equilibrium point of ˙ x = f ( x ) + G ( x ) bracketleftbig α ( x ) − γ − 1 ( x ) KT ( x ) bracketrightbig x = 0 is asymptotically stable because...
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