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# Lect_16 - Nonlinear Systems and Control Lecture 16 Feedback...

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Nonlinear Systems and Control Lecture # 16 Feedback Systems: Passivity Theorems – p.1/22

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a aA ±² a a A aA ±² A A ± ² u 1 u 2 e 1 e 2 y 1 y 2 H 1 H 2 + + + ˙ x i = f i ( x i ,e i ) , y i = h i ( x i ,e i ) y i = h i ( t,e i ) – p.2/22
Passivity Theorems Theorem 6.1: The feedback connection of two passive systems is passive Prove using V = V 1 + V 2 as a Lyapunov func. candidate Proof: Let V 1 ( x 1 ) and V 2 ( x 2 ) be the storage functions for H 1 and H 2 , respectively. If H i is memoryless, then take V i = 0 . Since H i is passive, we have e T i y i ˙ V i From the feedback connection, e T 1 y 1 + e T 2 y 2 = ( u 1 y 2 ) T y 1 +( u 2 + y 1 ) T y 2 = u T 1 y 1 + u T 2 y 2 u T y = u T 1 y 1 + u T 2 y 2 ˙ V 1 + ˙ V 2 = ˙ V – p.3/22

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Theorem 6.3: Consider the feedback connection of two dynamical systems. When u = 0 , the origin of the closed-loop system is asymptotically stable if each feedback component is either strictly passive , or output strictly passive and zero-state observable Furthermore, if the storage function for each component is radially unbounded, the origin is globally asymptotically stable Proof: H 1 is SP; H 2 e T 1 y 1 ˙ V 1 + ψ 1 ( x 1 ) , ψ 1 ( x 1 ) > 0 , x 1 n = 0 e T 2 y 2 ˙ V 2 + y T 2 ρ 2 ( y 2 ) , y T 2 ρ ( y 2 ) > 0 , y 2 n = 0 – p.4/22
e T 1 y 1 + e T 2 y 2 = ( u 1 y 2 ) T y 1 +( u 2 + y 1 ) T y 2 = u T 1 y 1 + u T 2 y 2 V ( x ) = V 1 ( x 1 ) + V 2 ( x 2 ) ˙ V u T y ψ 1 ( x 1 ) y T 2 ρ 2 ( y 2 ) u = 0 ˙ V ≤ − ψ 1 ( x 1 ) y T 2 ρ 2 ( y 2 ) ˙ V = 0 x 1 = 0 and y 2 = 0 y 2 ( t ) 0 e 1 ( t ) 0 x 1 ( t ) 0 )

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Lect_16 - Nonlinear Systems and Control Lecture 16 Feedback...

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