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Unformatted text preview: 1 Outline Motivation Linearization s tems stems Linearization Periodic Solutions & Limit Cycles Existence of Periodic Solutions ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys Motivation Phase portrait of a nonlinear system near its singular points resembles stemsstems near its singular points resembles linear behavior The local phase portrait is approximated by linearizing the nonlinear system Important conclusions can be drawn ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys about the global behavior of a nonlinear system by examining its local behavior near the singular points 2 Linearization Nonlinear Systems x x = & Consider the autonomous 2nd order nonlinear system s tems stems ) x ( f x = = = f f A x A x 2 1 1 1 ~ ~ x x & nonlinear system Linearization of the system about a singular point x*: ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys = = = x x f f A x A x 2 2 1 2 , x x = x x x ~ Types of Singular Points The type of singular point x * of a 2 nd order nonlinear system is determined from the stemsstems eigenvalues ( 1 and 2 ) of its lineariation about x * : Type Condition Node 1 and 2 are real, and 1 2 >0 Saddle and are real and <0 ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys Saddle 1 and 2 are real, and 1 2 <0 Focus 1 and 2 are complex conjugates with nonzero real parts Center 1 and 2 are pure imaginary 3 Revolving Pendulum Phase Portrait 5.5 6 3 s tems stems 2.5 3 3.5 4 4.5 51 1 2 x2 ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys 0.5 1 1.5 221 1 2 3 4 532 x1 Van der Pol Example ( ) 1 2 = + + x x x x & & & Differential Equation : stemsstems ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys 4 Van der Pole System Phase Portrait 2 3 s tems stems21 1 x2 ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys2.521.510.5 0.5 1 1.5 2 2.53 x1 The closed trajectory in the phase plane corresponds to a periodic solution Periodic Solutions Many systems like pendulum and Van der Pol systems have periodic stemsstems der Pol systems have periodic trajectories Periodic trajectories correspond to closed curves in the phase plane A isolated closed curve (e g Van der ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys A isolated closed curve (e.g., Van der Pol System) is called a limit cycle 5 Existence of Periodic Solutions In practice it is often important to predict existence of periodic trajectories s tems stems In the absence of an analytical solution, the...
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This note was uploaded on 09/22/2011 for the course ME 6402 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
 Staff

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