Lect_4 - Nonlinear Systems and Control Lecture 4...

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Unformatted text preview: Nonlinear Systems and Control Lecture # 4 Qualitative Behavior Near Equilibrium Points & Multiple Equilibria – p. 1/ ? ? The qualitative behavior of a nonlinear system near an equilibrium point can take one of the patterns we have seen with linear systems. Correspondingly the equilibrium points are classified as stable node, unstable node, saddle, stable focus, unstable focus, or center Can we determine the type of the equilibrium point of a nonlinear system by linearization? – p. 2/ ? ? Let p = ( p 1 ,p 2 ) be an equilibrium point of the system ˙ x 1 = f 1 ( x 1 ,x 2 ) , ˙ x 2 = f 2 ( x 1 ,x 2 ) where f 1 and f 2 are continuously differentiable Expand f 1 and f 2 in Taylor series about ( p 1 ,p 2 ) ˙ x 1 = f 1 ( p 1 ,p 2 ) + a 11 ( x 1 − p 1 ) + a 12 ( x 2 − p 2 ) + H . O . T . ˙ x 2 = f 2 ( p 1 ,p 2 ) + a 21 ( x 1 − p 1 ) + a 22 ( x 2 − p 2 ) + H . O . T . a 11 = ∂f 1 ( x 1 ,x 2 ) ∂x 1 vextendsingle vextendsingle vextendsingle vextendsingle x = p , a 12 = ∂f 1 (...
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