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9/6/2014 1 Module1 NonLinear Systems ase ane na ys s T o genera te motio n trajec torie s corres pondi ng to variou s initial conditions in the phase plane. T o examine the qualitative features of the trajectories. In such a way, information concerning stability and other motion patterns of the system can be obtained. Basic Idea A gra phi cal met hod :  to vi sual iz e what goes on in a nonlinear system without solving the nonlinear equations analytically. Limi tation : limit ed  fo r seco nd-order (or fir st- order) dynamic system; howeve r, some practi cal contr ol systems can be approximated as second-order systems . Phase Portrait of  a Massspring System Massspring system , Solution: ; 0 ) ( ) ( .. = +  t  y t  y 0 . ) 0 ( ; 0 ) 0 (  y  y  y = = t t t  y t  y sin cos ) ( . 0 = = Equation of  the trajectories:  2 0 . 2 2  y  y  y  = + System response corresponding to various initial conditions is directly displayed on the phase plane. The system trajectories neither converge to the origin nor diverge to infinity . They simply circle around the origin, indicating the marginal nature of the system’s stability. enera esc r p on o econ - r er ys em Free motion of any secon d-o rde r non-li nea r syst em can always be described by an equation of the form: The state of the system for any gi ven inst ant, can be repres ent ed by the a poi nt of coor dinates in a system of rectangular coordinates. 0 ) , ( ) , ( . . . .. = + +  y  y  y h  y  y  y g  y ) , ( .  y  y

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