By Jean-Jacques Slotine, Weiping Li
Covers in a innovative type a few research instruments and layout ideas without delay acceptable to nonlinear keep watch over difficulties in excessive functionality structures (in aerospace, robotics and automobile areas).
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Additional info for Applied Nonlinear Control
F). The name comes from the fact that all trajectories are ellipses and the singu1arl:y point is the center of these ellipses. The phase portrait of the undamped mass spring system belongs to this category. :ly determined by the nature of their singularity points. This, however, is not true for nonlinear systems. ;hould be kept in mind. Phase plane analysis of nonlinear systems is related to that of linear systems, because the local behavior of a nonlinear system can be approximated b y the behavior of a linear system.
In this section, we state th:ee simple classical theorems to that effect. These theorems are easy to understand and apply. The first theorem to be presented reveals a simple relationship bc,tween the existence of a limit cycle and the number of singular points it encloses. ers, and foci enclosed by a limit cycle, and S to represent the number of enclosed saddle points. 1 (Poincare) If a limit cycle exists in the second-order ailtonomous system ( 2. I ) , then N = S + 1 . This theorem is sometimes called the index theorem.
Let us consider some additional examples of stable, unstable. and semi-stable limit cycles. 7: stable, unstable, and semi-stable limit cycles Consider the following nonlinear systems (a) iI= x 2 - x , ( x 1 2 + . I,? r2? 13) (c) ~ 1 = ~ 2 - x l ( . r 1 2 -+ . 14) 36 Phase Plane Analysis Chap. 2 Let us study system (a) first. 12) are transformed as When the state starts on the unit circle, the above equation shows that ;(t) = 0. When r < 1, then i > 0. This imp11l:s [hat the state tends to the circle from inside.
Applied Nonlinear Control by Jean-Jacques Slotine, Weiping Li