Module3 p1 - Nonlinear & Adaptive Control Module 3 P M d l...

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onlinear & Adaptive Control Nonlinear & Adaptive Control Module 3-Part 1 Mathematical Preliminaries 1
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Topics In This Module 1. Existence of Periodic Orbits in 2-D 2. Vectors Spaces 3. Linear Independence asis Vectors 4. Basis Vectors 2
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Second-Order Systems Consider the autonomous 2 nd -order system dx here is continuously differentiable We ) ( x f dt = where is continuously differentiable. We have already seen what the behavior around xed points looks like for nonlinear systems ) ( x f fixed points looks like for nonlinear systems. We shall now look at the behavior about osed orbits 3 closed orbits.
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Existence of Periodic orbits The existence of ISOLATED periodic orbits is peculiar to nonlinear systems. he Poincare- endixon theorem gives The Poincare Bendixon theorem gives conditions for existence of periodic orbits. endixon’s criterion is used to rule out the Bendixon s criterion is used to rule out the existence of periodic orbits. 4
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olated closed trajectories Limit Cycles-Periodic Orbits ± Isolated closed trajectories ± Analysis is in general difficult. For 2 nd -order nonlinear systems, two simple results exist: Bendixon criterion and oincare endixon theorem 5 Poincare-Bendixon theorem.
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Bendixson’s Theorem B Connected but not mply connected Ω Ω B A , : set connected a is lying curve a by connected be can , B A AB Ω A simply connected Ω set ithin the entirely w : if set connected simply a is D and connected is D c (annular region) set connected a is D d me and ass erentiab uously di is conti , R x), x onsider f( ed set. C ly connect be a simp R Let D THEOREM: 2 2 d does not on of D an ny subregi ero over a ntically z is not ide x f x f f(x) ume le and ass fferentiab nuously di f is conti + = 2 2 1 1 6 f(x) x of c orbits no periodi hen D has n in D. T change sig = &
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Example 1 0 ) ( 2 x x x x ε + = & Let the following 2-D system be given: ) ( 2 2 1 1 2 2 1 2 1 x x x x + = & 2 2 1 2 2 2 1 2 2 1 1 s rajector osed o ) 0 , 0 ( ) , ( 0 ) ( ) ( x x x x x f x f x f > + = + = in ies trajector closed No R 7
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Example 2 (Example 2.10 Khalil) Consider the following 2-D nonlinear system: 1 dx 3 2 2 2 x x x bx ax dx x dt + = = and let D be the whole plane. Then 1 2 1 2 1 dt 2 1 ) ( x b x f = and no periodic orbits can exist if 0 < b 8
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Limit Sets in ) ( sequence a if ) ( of point limit (positive) a is point A . ), ( of y) (trajector solution a be ) ( Let 1 2 2 = + = R t t x R z R x x f x t x n n & . as and ) ( y with analogousl defined is set limit Negative ). ( of , set, limite (positive) the called is ) ( of points limit all of set The . as ) ( and as such that 1
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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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Module3 p1 - Nonlinear & Adaptive Control Module 3 P M d l...

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