9/6/2014 1 Module‐1 Non‐LinearSystemsaseanena 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 othermotion 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 . PhasePortraitofaMass‐springSystem Mass‐spring system , Solution: ; 0 ) ( ) ( .. = + ty ty 0 . ) 0 ( ; 0 ) 0 ( y y y = = ttty ty sin cos ) ( . 0 − = = Equationofthetrajectories: 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 nordiverge 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 ofrectangular coordinates. 0 ) , ( ) , ( . . . .. = + + y y y h y y y g y ) , ( . y y