Math Editor

Nonlinear Control Report 3

Prepared by: Ibrahim Mohamed El-bastawisi

Odd function non linearity has an output of y=b1​x+b3​x3+b5​x5+⋯ where x is a sinusoidal input x=Xsinωt

show that N(X)=b1​+43​b3​X2+85​X4+⋯

Solution

y=b1​Xsinωt+b3​X3sin3ωt+b5​X5sin5ωt+⋯

use some trigonometric spells 🪄

sin3x=43sinx−sin3x​

sin5x=161​(sin(5x)−5sin(3x)+10sin(x))

discard higher order harmonics

sin3x=43​sinx

sin5x=85​sinx

or in general the first harmonic of odd powers of sinx is

(using Euler's formula and Binomial theorem) :

sinnx=2n/2(nk​)​sinx where k=2n±1​

substitute

y=b1​Xsinωt+43​b3​X3sinωt+85​b5​X5sinωt+⋯

N(X)=x(ωt)y(ωt)​=Xsinωtb1​Xsinωt+43​b3​X3sinωt+85​b5​X5sinωt+⋯​

N(X)=b1​+43​b3​X2+85​b5​X4+⋯#