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Lect_5 - Nonlinear Systems and Control Lecture 5 Limit...

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Nonlinear Systems and Control Lecture # 5 Limit Cycles – p. 1/ ?
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Oscillation: A system oscillates when it has a nontrivial periodic solution x ( t + T ) = x ( t ) , t 0 Linear (Harmonic) Oscillator: ˙ z = bracketleftBigg 0 β β 0 bracketrightBigg z z 1 ( t ) = r 0 cos( βt + θ 0 ) , z 2 ( t ) = r 0 sin( βt + θ 0 ) r 0 = radicalBig z 2 1 (0) + z 2 2 (0) , θ 0 = tan 1 bracketleftbigg z 2 (0) z 1 (0) bracketrightbigg – p. 2/ ?
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The linear oscillation is not practical because It is not structurally stable. Infinitesimally small perturbations may change the type of the equilibrium point to a stable focus (decaying oscillation) or unstable focus (growing oscillation) The amplitude of oscillation depends on the initial conditions The same problems exist with oscillation of nonlinear systems due to a center equilibrium point (e.g., pendulum without friction) – p. 3/ ?
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Limit Cycles: Example: Negative Resistance Oscillator v (b) i = h(v) – p. 4/ ?
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˙ x 1 = x 2 ˙ x 2 = x 1 εh ( x 1 ) x 2 There is a unique equilibrium point at the origin A = ∂f ∂x vextendsingle vextendsingle vextendsingle vextendsingle x =0 = 0 1 1 εh (0) λ 2 + εh (0) λ + 1 = 0 h (0) < 0 Unstable Focus or Unstable Node
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