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 · Ù Ý 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 = Σ 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 ϕ ( u ) for some function ϕ where u T ϕ ( u ) > , ∀ u negationslash = 0 output strictly passive if u T y ≥ y T ρ ( y ) for some function ρ where y T ρ ( y ) > , ∀ y negationslash = 0 – p.4/17 Sector Nonlinearity: h belongs to the sector [ α,β ] ( h ∈ [ α,β ] ) if αu 2 ≤ uh ( t,u ) ≤ βu 2 y= α u y= u β u (a) y α0 y= α u y= β u y (b) u α<0 Also, h ∈ ( α,β ] , h ∈ [ α,β ) , h ∈ ( α,β ) – p.5/17 αu 2 ≤ uh ( t,u ) ≤ βu 2 ⇔ [ h ( t,u ) − αu ][ h ( t,u ) − βu ] ≤ 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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This note was uploaded on 07/25/2008 for the course ME 859 taught by Professor Choi during the Spring '08 term at Michigan State University.

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Lect_14 - Nonlinear Systems and Control Lecture 14...

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