# Module 7 - ECE 514 Nonlinear Adaptive Control Module 7...

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1 ECE 514 Nonlinear & Adaptive Control Module 7 PASSIVITY & RELATED CONCEPTS

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2 Topics 1. Passivity Concepts 2. PR & SPR 3. KYP lemma 4. L2 & Lyapunov Stability 5. Passivity Theorems Material covered in Chapter 6 of Khalil.
3 Goal of Module 1. To understand concepts of passivity, and those of positive realness 2. To be able to analyze feedback systems using such concepts. 3. To apply concepts to robustness analysis.

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4 Memoryless Functions
5 Passive Elements u y + u y

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6 Excess and Shortage of Passivity u y (.) ϕ u y ~ y +
7 Excess and Shortage of Passivity u y (.) ρ ~ u + +

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8 Sector Nonlinearities u y y u α = y u β = 0 α> y u = yu = 0 α<
9 Sector Nonlinearities Definition: 11 22 2 2 12 2 1 A memoryless function :[0, ) is said to belong to the sector: [0, ] if ( , ) 0 [, ] i f [ ( , ) ]0 [0, ], 0 if ( , )[ ( , ) ] 0 ] , 0 i f [ Pp T T TT T h uhtu Ku h t u K u KK K h t u h t u K u K K K K h ∞× •∞ −≥ •= > = > \\ (, ) ][ (, ) ] 0 In all cases, the inequalities should hold for all ( , ). T tu Ku htu Ku tu −−

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10 Transformation of General Sector y 1 K + + 1 K (, ) y htu = +
11 Passivity for Static Nonlinearities Definition: The system ( , ) is Passive if 0 Lossless if 0 Input-feedforward passive if ( ), for some function Input-strictly passive if ( ), ( ) 0, 0 Output-feedback T T TT T yh t u uy uy u u uy u u u u u ϕ ϕϕ = •≥ •= > passive if ( ), for some function Output-strictly passive if ( ) 0 All inequalities hold for all ( , ). T uy y y u y yy y tu ρ ρρ > u y (, .) ht

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12 State Models Given the square nonlinear system: (,) where : is locally Lipschitz : is continuous (0,0) 0 and 0 np n p xf x u yh x u f h fh = = ×→ == i \\ \ \ u y hxu x f xu = i x
13 Passivity Concepts for State-Space The system ( , ), ( , ) is said to be passive if a continuously differentiable positive semindefinite (storage) function ( ) : (,) , (,) It is said to be Lossless if Tn p T xf x u y h x u Vx V u y V f xu xu x uy V == ≥= × •= i i i \\ Input-feedforward passive if ( ), for some function Input-strictly passive if ( ), ( ) 0, 0 Output-feedback passive if ( ), for some function Output-s TT T uy V u u uy V u u u u u uy V y y ϕϕ ρρ •≥ + + > + i i i trictly passive if ( ) 0 Strictly passive if ( ) for some positive-definite All inequalities hold for all ( , ).

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## This note was uploaded on 05/06/2010 for the course ECE 514 taught by Professor Chaoukit.abdallah during the Spring '09 term at University of New Brunswick.

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Module 7 - ECE 514 Nonlinear Adaptive Control Module 7...

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