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56 Introduction to Control Theory

Figure 2.6.6: Damped pendulum: (a)time history; (b) phase plane

There are 2 equilibrium points located at (0, 0) and (1, 0), both of which
are unstable (check!). On the other hand, (1, 0) is convergent since all
trajectories will eventually converge to it after some time T. Before T however,
there is no guarantee that a trajectory will stay within some of (1, 0) no
matter how close the initial state is to (1, 0). In fact, suppose the system
starts at x(0), then the state-vector is given at any t 0 by

Copyright © 2004 by Marcel Dekker, Inc.