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

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Unformatted text preview: Nonlinear Systems and Control Lecture # 5 Limit Cycles p. 1/ ? ? Oscillation: A system oscillates when it has a nontrivial periodic solution x ( t + T ) = x ( t ) , t Linear (Harmonic) Oscillator: z = bracketleftBigg bracketrightBigg z z 1 ( t ) = r cos( t + ) , z 2 ( t ) = r sin( t + ) r = radicalBig z 2 1 (0) + z 2 2 (0) , = tan 1 bracketleftbigg z 2 (0) z 1 (0) bracketrightbigg p. 2/ ? ? 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/ ? ? Limit Cycles: Example: Negative Resistance Oscillator C i C i e i ie E e e i a C C C C XX v (b) i = h(v) p. 4/ ? ?...
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Lect_5 - Nonlinear Systems and Control Lecture # 5 Limit...

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