nonlinear system Signals & systems Output signals Input signals Dynamic system

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1 nonlinear ssem Signals & ssems Inpu signals Dnamic ssem Oupu signals nonlinear ssems b meiling chen 009

2 nonlinear ssem Signal Classiicaion Coninuous signal Discree signal nonlinear ssems b meiling chen 009

3 Ssem classiicaion nonlinear ssem Finie-dimensional ssem lumped-parameers ssem described b dierenial equaions Linear ssems and nonlinear ssems Coninuous ime and discree ime ssems Time-invarian and ime varing ssems Ininie-dimensional ssem disribued parameers ssem described b parial dierenial equaions Power ransmission line Anennas Hea conducion Opical iber ec. nonlinear ssems b meiling chen 009

4 Coninuous-ime ssem ---characerized b dierenial equaions Deiniion: inpu and oupu o he ssem are coninuous uncions o he coninuous variable ime. nonlinear ssem Discree-ime ssem ---characerized b dieren equaions Deiniion: inpu and oupu o he ssem change a onl discree insans o ime. nonlinear ssems b meiling chen 009 4

5 nonlinear ssem Linear ime invarian LTI--coninuous Described b a linear dierenial equaion in ime domain can be ranserred o linear algebra orm b using Laplace ransorm. nonlinear ssems b meiling chen 009 5

6 nonlinear ssems b meiling chen LTI ssem Ordinar Linear dierenial equaion Z k k a k a m k a k b k b n k b m n 0 0 L L Ordinar Linear dierence equaion Coninuous ssem discree ssem nonlinear ssem

7 nonlinear ssems b meiling chen Linear ime-varing Nonlinear ime-invarian Linear ime-invarian Nonlinear ime-varing eamples nonlinear ssem

8 Linear ssem nonlinear ssem Deiniion: A ssem is linear i superposiion principle is saisied Lineari : a homogeneous principle muliplicaion b addiion principle c superposiion principle > a b nonlinear ssems b meiling chen 009 8

9 nonlinear ssems b meiling chen ] ] ] ] ] 5 ] ] ] ] ] ] ] 5 ] ] 5 5 ] 5 ] ] ] ] k k k k k k k k k Muliplicaion law saisied Addiion law saisied

10 Nonlinear ssem nonlinear ssems b meiling chen 009 0

11 Common nonlinear phenomena Ideal rela sgn u, 0,, u > 0 u 0 u < 0 Sauraion Eample: ampliier Ideal sauraion: sa u u sgn u u u > nonlinear ssems b meiling chen 009

12 Dead zone Eample: ampliier wih low inpu signals nonlinear ssems b meiling chen 009

13 Common nonlinear phenomena Back lash Eample:. Gear gap. hseresis loop 4 Dead zone Eample: ampliier wih low inpu signals nonlinear ssems b meiling chen 009

14 Common nonlinear phenomena 5 hseresis Eample: window comparaor/schmi rigger 6 parabola nonlinear ssems b meiling chen 009 4

15 Nonlinear ssem eamples nonlinear ssem Pendulum equaion ml && θ kl & θ mg sin θ 0 l θ m ml & θ kl & θ mgθ 0 For small θ nonlinear ssems b meiling chen 009 5

16 Nonlinear spring equaion m && B& k k 0 k m Nonlinear spring linear nonlinear B a hard spring k > 0 b so spring k < 0 nonlinear ssems b meiling chen 009 6

17 c linear spring k 0 k 0 ied k < 0 k > 0 Jump resonance : nonlinear ssem wih orce k > 0 m&& B& k k p cos nonlinear ssems b meiling chen 009 7

18 k < 0 m&& B& k k p cos 4 Harmonic oscillaion u A0 sin Linear ssem Ao G j sin G j] u A0 sin Nonlinear ssem A A sin θ ] sin θ ] L A sin θ ] nonlinear ssems b meiling chen 009 8

19 nonlinear ssems b meiling chen Eample: A u sin 0 u A A A u sin 4 sin 4 sin Subharmonic oscillaion Nonlinear ssem A u sin 0 L ] sin ] sin ] sin θ θ θ A A A

20 6 Limi ccle Eample: Van der pol s equaion & ε & & 0 ε > 0 << ε < 0 >> ε > 0 Sable limi ccle nonlinear ssems b meiling chen 009 0

21 & ε & 0 & << ε > 0 ε > 0 >> ε < 0 Unsable limi ccle nonlinear ssems b meiling chen 009