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# Lect_2 - Nonlinear Systems and Control Lecture 2 Examples...

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Unformatted text preview: Nonlinear Systems and Control Lecture # 2 Examples of Nonlinear Systems – p.1/17 Pendulum Equation θ mg l • ml ¨ θ = − mg sin θ − kl ˙ θ x 1 = θ, x 2 = ˙ θ – p.2/17 ˙ x 1 = x 2 ˙ x 2 = − g l sin x 1 − k m x 2 Equilibrium Points: 0 = x 2 0 = − g l sin x 1 − k m x 2 ( nπ, 0) for n = 0 , ± 1 , ± 2 , .. . Nontrivial equilibrium points at (0 , 0) and ( π, 0) – p.3/17 Pendulum without friction: ˙ x 1 = x 2 ˙ x 2 = − g l sin x 1 Pendulum with torque input: ˙ x 1 = x 2 ˙ x 2 = − g l sin x 1 − k m x 2 + 1 ml 2 T – p.4/17 Tunnel-Diode Circuit « ¨ « ¨ « ¨ « ¨ C Ú C ª ª ª Ú E ×- i C i C C £ £ C C £ £ i Ú XX a 0.5 1-0.5 0.5 1 i=h(v) v,V i,mA (b) i C = C dv C dt , v L = L di L dt x 1 = v C , x 2 = i L , u = E – p.5/17 i C + i R − i L = 0 ⇒ i C = − h ( x 1 ) + x 2 v C − E + Ri L + v L = 0 ⇒ v L = − x 1 − Rx 2 + u ˙ x 1 = 1 C [ − h ( x 1 ) + x 2 ] ˙ x 2 = 1 L [ − x 1 − Rx 2 + u ] Equilibrium Points: 0 = − h ( x 1 ) + x 2 0 = − x 1 − Rx 2 + u – p.6/17 h ( x 1 ) = E R − 1 R x 1 0.5 1 0.2 0.4 0.6 0.8 1 1.2 Q Q Q 1 2 3 v R i R – p.7/17 Mass–Spring System...
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Lect_2 - Nonlinear Systems and Control Lecture 2 Examples...

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