Lect_14 - Nonlinear Systems and Control Lecture # 14...

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Unformatted text preview: Nonlinear Systems and Control Lecture # 14 Passivity Memoryless Functions & State Models p.1/17 Memoryless Functions a y u (b) power inflow = uy Resistor is passive if uy p.2/17 u y (a) u y (b) u y (c) Passive Passive Not passive y = h ( t,u ) , h [0 , ] Vector case: y = h ( t,u ) , h T = bracketleftBig h 1 , h 2 , , h p bracketrightBig power inflow = p i =1 u i y i = u T y p.3/17 Definition: y = h ( t,u ) is passive if u T y lossless if u T y = 0 input strictly passive if u T y u T ( u ) for some function where u T ( u ) > , u negationslash = 0 output strictly passive if u T y y T ( y ) for some function where y T ( y ) > , y negationslash = 0 p.4/17 Sector Nonlinearity: h belongs to the sector [ , ] ( h [ , ] ) if u 2 uh ( t,u ) u 2 y= u y= u u (a) y 0 y= u y= u y (b) u <0 Also, h ( , ] , h [ , ) , h ( , ) p.5/17 u 2 uh ( t,u ) u 2 [ h ( t,u ) u ][ h ( t,u ) u ] Definition: A memoryless function h ( t,u ) is said to belong to the sector [0 , ] if u T h ( t,u ) [ K 1 , ] if u T [ h ( t,u ) K 1 u ] [0 ,K 2 ] with K 2 = K T 2 > if h T ( t,u )[ h ( t,u ) K 2 u ] [ K 1 ,K 2 ] with K = K 2 K 1 = K T > if [...
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Lect_14 - Nonlinear Systems and Control Lecture # 14...

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