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# Lect_29 - Nonlinear Systems and Control Lecture 29...

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Nonlinear Systems and Control Lecture # 29 Stabilization Passivity-Based Control – p. 1/ ?

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˙ x = f ( x, u ) , y = h ( x ) f (0 , 0) = 0 u T y ˙ V = ∂V ∂x f ( x, u ) Theorem 14.4: If the system is (1) passive with a radially unbounded positive definite storage function and (2) zero-state observable, then the origin can be globally stabilized by u = φ ( y ) , φ (0) = 0 , y T φ ( y ) > 0 y negationslash = 0 – p. 2/ ?
Proof: ˙ V = ∂V ∂x f ( x, φ ( y )) ≤ − y T φ ( y ) 0 ˙ V ( x ( t )) 0 y ( t ) 0 u ( t ) 0 x ( t ) 0 Apply the invariance principle A given system may be made passive by (1) Choice of output, (2) Feedback, or both – p. 3/ ?

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Choice of Output ˙ x = f ( x ) + G ( x ) u, ∂V ∂x f ( x ) 0 , x No output is defined. Choose the output as y = h ( x ) def = bracketleftbigg ∂V ∂x G ( x ) bracketrightbigg T ˙ V = ∂V ∂x f ( x ) + ∂V ∂x G ( x ) u y T u Check zero-state observability – p. 4/ ?
Example ˙ x 1 = x 2 , ˙ x 2 = x 3 1 + u V ( x ) = 1 4 x 4 1 + 1 2 x 2 2 With u = 0 ˙ V = x 3 1 x 2 x 2 x 3 1 = 0 Take y = ∂V ∂x G = ∂V ∂x 2 = x 2 Is it zero-state observable?

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