Linearization of Nonlinear Physical Effects - Mechatronics

Linearization of Nonlinear Physical Effects• Many real-world nonlinearities involve a “smooth”curvilinear relation between an independent variable x anda dependent variable y: y = f(x)• A linear approximation to the curve, accurate in theneighborhood of a selected operating point, is the tangentline to the curve at this point.• This approximation is given conveniently by the first twoterms of the Taylor series expansion of f(x):xxy f (x) x x dfy y x xdx2dfd f 2dxxxdx 2!xxxxLinearization K. Craig 12dfy y x xdxyˆ Kxˆxx

Linearizationfor aNonlinear Spring22dfd f xx00 0 2dxxxdx 2!xxy f (x ) x xdfy y x x0 0dxxx00 0dfy y x xyˆ Kxˆ0 0dxxx0Linearization K. Craig 2

• For another example, in liquid-level control systems, whenthe tank is not prismatic, a nonlinear volume/heightrelationship exists and causes a nonlinear systemdifferential equation. For a conical tank of height H andtop radius R we would have:VV2R h23H3H32 3 2 2R h R h h ˆ2 2• Often a dependent variable y is related nonlinearly toseveral independent variables x 1 , x 2 , x 3 , etc. according tothe relation: y=f(x 1 , x 2 , x 3 , …).HLinearization K. Craig 3

• We may linearize this relation using the multivariable form ofthe Taylor series: f fy f x 1,x 2,x 3, x1 x1 x 2 x2 x 1x x 21,x 2 ,x 3, x 1,x 2 ,x 3, f x3 x3x3 x 1,x 2 ,x 3, f f fy y xˆ ˆ ˆ1 x2 x3 x x x1 x 1,x 2 ,x 3, 2 x 31,x 2 ,x 3, x 1,x 2 ,x 3,yˆ K xˆ K ˆ1 12x2 K3x3ˆThe partial derivatives can be thought of as the sensitivity of thedependent variable to small changes in that independent variable.Linearization K. Craig 4

Example: Magnetic LevitationSystemApplicationsinclude magneticbearings forvacuum pumps,conveyor systems inclean rooms, highspeedlevitatedtrains, andelectromagneticautomotive valveactuators.ElectromagnetInfrared LEDLevitated BallPhototransistorLinearization K. Craig 6

iMagnetic Levitation SystemElectromagnetf x,iEquation of Motion: imx mg C x2 2 At Equilibrium:mg iC x iC x2 2 2 2 +xBall (mass m)mgLinearization: C C C x C i x x x x 2 2 2i i 2 i 2 i ˆ ˆ2 2 3 2 mxˆ mg C C x C i x x x i 2 2 i 2ˆ2 i ˆ2 3 2 mxˆC x C i x x 2 i 2ˆ 2 i ˆ3 2Linearization K. Craig 7

Use of Experimental Testing in Multivariable Linearizationf f (i, y)mfff f i , y y y i i m 0 0 0 0yii ,yi ,y0 00 0Linearization K. Craig 8