midsol06

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

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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 .

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

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