midsol06 - Problem 1(15 points Consider the following...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Problem 1. (15 points) Consider the following system ˙ ζ - 2 ζ = θ +2 ζu, ¨ θ + θ 2 ± 1+ ˙ θ ² = u + θ. (This system does not arise from any physical application but its structure and its nonlinear terms mimic phenomena that appear in mechanical, electrical and bio-chemical systems). (a) (5 points) Treating ζ as the output and u as the input, derive a state space representation of the system. (b) (5 points) Let u = 0 and Fnd all the equilibrium points of the state space system. (c) (5 points) ±or each equilibrium found in (b), compute the linearization of the system (in state space form, i.e., Fnd F , G , H , J ). Solution: (a) Calling x 1 = θ , x 2 = ˙ θ ,and x 3 = ζ ,weget ˙ x 1 = x 2 , ˙ x 2 = x 1 - x 2 1 (1 + x 2 )+ u, ˙ x 3 = x 1 x 3 x 3 u, and since ζ is the output, then y = x 3 . This can be written as ˙ x = f ( x, u ) , y = h ( x, u ) , where f ( x, u )= x 2 x 1 - x 2 1 (1 + x 2 u x 1 x 3 x 3 u ,h ( x, u x 3 . (b) ±or Fnding the equilibria we set ˙ x =0and u = 0. Then, we need to solve the equation f ( x, 0) = 0, i.e., 0= x 2 , x 1 - x 2 1 (1 + x 2 ) , x 1 x 3 .
Background image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 3

midsol06 - Problem 1(15 points Consider the following...

This preview shows document pages 1 - 2. Sign up to view the full document.

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