Nonlinear Control Sy.. - Free

12 CHAPTER 1. INTRODUCTION
Figure 1.5: System trajectories of Example 1.7: (a) original system, (b) uncoupled system.
In the new coordinates y = Tx, the modified system is
yi = -Yi
y2 = -2Y2
or y = Dy, which is uncoupled. Figure 1.5 shows the trajectories of both the original system
[part (a)] and the uncoupled system after the coordinate transformation [part (b)]. It is
clear from part (b) that the origin is attractive in both directions, as expected given that both
eigenvalues are negative. The equilibrium point is thus said to be a stable node. Part (a)
retains this property, only with a distortion of the coordinate axis. It is in fact worth nothing
that Figure 1.5 (a) can be obtained from Figure 1.5 (b) by applying the linear transformation
of coordinates x = Ty.
Example 1.8 Consider the system
[ 22 j = [ 0 2 ] [ x2 ] .
The eigenvalues in this case are Al = 1, A2 = 2. Applying the linear coordinate transformation
x = Ty, we obtain
y1 = yi
y2 = 2Y2
Figure 1.6 shows the trajectories of both the original and the uncoupled systems after the
coordinate transformation. It is clear from the figures that the origin is repelling on both