# lec16 - Outline Motivation Integrability The Frobenius...

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1 Outline ± Motivation stems ± Integrability & The Frobenius Theorem ± Input-State Feedback Linearization ± Illustrative Examples ME6402, Nonlinear Control Sys Motivation ± Input-State feedback linearization can be used to transform a nonlinear system to one that can be exactly linearized. ± It has been successfully applied to many nonlinear systems like helicopters, aircrafts, and robots. ± There are powerful necessary and sufficient conditions for feedback linearizabilty ± These conditions are satisfied for a certain class of nonlinear systems

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2 More on Lie Bracket and Derivative Properties • Bilinearity: [a 1 f 1 +a 2 f 2 ,g]=a 1 [f 1 ,g]+a 2 [f 2 ,g] stems Skew symmetry: [f,g]=-[g,f] [f,f]=0 • Jacobi identity: L [f,g] h=L f L g h- L g L f h See lemma 6.1 of text for proof Other Lie Bracket Notation : ad f g:=[f,g], ad f 2 g: [f [f g and more generally ME6402, Nonlinear Control Sys g:=[f,[f,g] ]] , [ , , [ ad times g f f g i i f 3 2 1 L = State Transformation ± Feedback linearization, requires transformation to a nonlinear extension transformation to a nonlinear extension of the CCF for linear systems ± The state transformation must be a diffeomorphism ± The function φ : Ω ⊂ R n R n is a diffeomorphism if its smooth and its inverse φ -1 exists and is smooth. ± Example: φ (x)=Tx ⇒ φ -1 (z)=T -1 z so that φ -1 o φ (x)= φ -1 ( φ (x))=x
3 Example of Nonlinear State Transformation Example: + = φ ) sin( ) ( 2 2 1 1 x x x x x stems ME6402, Nonlinear Control Sys State Transformation Lemma Lemma : Let φ (x) be a smooth transformation from Ω ⊂ R n R n and suppose that ∂φ / x is nonsingular at some x 0 in Ω . Then φ (x) defines a local diffeomorphism in a subregion of Ω . Example: + = φ ) sin( ) ( 2 2 1 1 x x x x x

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4 Input-State Feedback Linearization Problem Given the nonlinear system u x g x f x ) ( ) ( + = & stems Find a diffiomorphism z= ϕ (x) to transform the system to the following nonlinear CCF: = = z z z z 3 2 2 1 M & & ME6402, Nonlinear Control Sys + = u x g x f z n n n ) ( ) ( & Where u can be used to accomplish feedback linearization and pole-placement.
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lec16 - Outline Motivation Integrability The Frobenius...

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