Lect_3 - Nonlinear Systems and Control Lecture 3...

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Nonlinear Systems and Control Lecture # 3 Second-Order Systems – p.1/16
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˙ x 1 = f 1 ( x 1 , x 2 ) = f 1 ( x ) ˙ x 2 = f 2 ( x 1 , x 2 ) = f 2 ( x ) Let x ( t ) = ( x 1 ( t ) , x 2 ( t )) be a solution that starts at initial state x 0 = ( x 10 ,x 20 ) . The locus in the x 1 x 2 plane of the solution x ( t ) for all t 0 is a curve that passes through the point x 0 . This curve is called a trajectory or orbit The x 1 x 2 plane is called the state plane or phase plane The family of all trajectories is called the phase portrait The vector field f ( x ) = ( f 1 ( x ) , f 2 ( x )) is tangent to the trajectory at point x because dx 2 dx 1 = f 2 ( x ) f 1 ( x ) – p.2/16
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Vector Field diagram Represent f ( x ) as a vector based at x ; that is, assign to x the directed line segment from x to x + f ( x ) a a a a a a a A x 1 x 2 f ( x ) x = (1 , 1) x + f ( x ) = (3 , 2) Repeat at every point in a grid covering the plane – p.3/16
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-5 0 5 -6 -4 -2 0 2 4 6 x 1 x 2 ˙ x 1 = x 2 , ˙ x 2 = 10 sin x 1 – p.4/16
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This note was uploaded on 07/25/2008 for the course ME 859 taught by Professor Choi during the Spring '08 term at Michigan State University.

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Lect_3 - Nonlinear Systems and Control Lecture 3...

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