Module 2 - Nonlinear & Adaptive Control Module 2 ECE...

Info iconThis preview shows pages 1–12. Sign up to view the full content.

View Full Document Right Arrow Icon
Nonlinear & Adaptive Control odule Module 2 ECE 514 Electrical & Computer Engineering The University of New Mexico 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
In This Part 1. Linearization of nonlinear systems 2. 2-D Behavior 3. Examples of nonlinear phenomena and stems systems 4. Matlab & Simulink This part contains section 2.1 of Chapter 2 in Khalil’s book. 2
Background image of page 2
Linear Approximation Consider a sufficiently small ball around n r B R i s x {: } ni is Bx R x x ε =∈ < he linearization of at is given by ) f x & i x The linearization of at is given by () xf s xx where i i i x zA z f A = = & s x = Example: 12 n a x b x = −− & & 0 , 0 ss π ⎤⎡ ⇒= = ⎥⎢ ⎣ ⎦ 21 2 sin xa = 00 ⎣⎦ 1 01 cos f ax b x ⎡⎤ = ⎢⎥ 1 , A ab = 2 A = 3
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Examples of Linearizations hen the two linearizations are 12 21 2 zz za z b z = =− & & 212 z b z = & & Then the two linearizations are 1 in B 2 in B n xx a x b x = & & 2 sin xa 2 B 2 B 1 B π 4
Background image of page 4
Fixed Points for 2 D Linear System x Consider where is in 22 Introduce where with det M 0 xM z M R × =∈ Ax dt dx = x 2 R 1 Then M zA M z zM A M z == && 1 1 0 or or r k MA M J λ αβ ⎡⎤ ⎤⎡ 2 0 0 λβ α ⎢⎥ ⎢⎥ ⎣⎦ (a) 12 0 & real ≠≠ 1 11 1 1 1 0 ( ) t zz z t z e =⇒ = & 2 2 1 2 2 2 0 21 ( ) t z t z e zc z = = & Thus 5 2 1 20 10 () z c z = stable node 0 < < where
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Fixed Points in 2 D 12 ,0 λλ < > stable node unstable node 6 21 0 : saddle point λ < <
Background image of page 6
Fixed Points in 2 D (b) 2 0 & real λ =≠ 12 11 2 0 2 0 2 2 0 22 () ( ) , tt zz k z zt e z k zt zt ze =+ = = & & when 0, kz c z == 0 < 0 > + = 20 2 20 10 2 1 ln , 0 z z k z z z z k 7 one or both eigenvalues are zero equilibrium subspace 0 < 0 >
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Fixed Points in 2 D ) ± (c) 1,2 j λα β 112 21 2 zzz zz z α βα =− =+ & & 22 12 1 2 tan rz z z θ = rr = = & & 0 () t rt re tt θβ = = + Let 1 z 0 0 < 0 > 0 = 8 stable focus unstable focus center
Background image of page 8
Linearization in 2D 11 1 2 22 1 2 (, ) xf x x x x = = & & ( ) 0 linearization ( ) ss f xz A x z = ⇒= & ss me that the nat re of this sing lar point in the linear s s is Assume that the nature of this singular point in the linear sys. is [node, focus, center, saddle] α What is the nature of the singular point in the nonlinear system ? Answer: Same as its linearized counterpart, except for center! Center for the linear system doesn’t mean center in the nonlinear system. An equilibrium of a nonlinear system such that the linearization has no eigenvalues on the imaginary axis is called hyperbolic. Thus, around hyperbolic uilibria the nonlinear system and its linearized counterpart behave similarly s x 9 equilibria, the nonlinear system and its linearized counterpart behave similarly.
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Example of Lineazrization x & 12 21 2 sin xx xa x b x = =− & where 1, 0.5 ab = = 1 1 00 1 stable 0 s xA ⎡⎤ == ⎢⎥ −− ⎣⎦ 2 01 nstable A π 1 unstable 0 s x 10
Background image of page 10
When Linearization Does Not Work A scalar nonlinear system 2 dx when linearized around the equilibrium point 0, leaves x dt = us with 0 = dx Which gives no information about the nonlinear system dt 11
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 12
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 63

Module 2 - Nonlinear &amp; Adaptive Control Module 2 ECE...

This preview shows document pages 1 - 12. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online