Nonlinear Control Sy.. - Free

1.6. PHASE-PLANE ANALYSIS OF LINEAR TIME-INVARIANT SYSTEMS 13
Figure 1.6: System trajectories of Example 1.8 (a) uncoupled system, (b) original system
directions, as expected given that both eigenvalues are positive. The equilibrium point in this
case is said to be an unstable node.
Example 1.9 Finally, consider the system.
[ x2 ] - [ 0 2 ] [ x2 J .
The eigenvalues in this case are Al = -1, 1\2 = 2. Applying the linear coordinate transformation
x = Ty, we obtain
yl = -yi
y2 = 2112
Figure 1.7 shows the trajectories of both the original and the uncoupled systems after the
coordinate transformation. Given the different sign of the eigenvalues, the equilibrium point
is attractive in one direction but repelling in the other. The equilibrium point in this case
is said to be a saddle.
CASE 2: Nondiagonalizable Systems
Assume that the eigenvalues of the matrix A are real and identical (i.e., Al = A2 = A). In
this case it may or may not be possible to associate two linearly independent eigenvalues
vl and v2 with the sole eigenvalue A. If this is possible the matrix A is diagonalizable
and the trajectories can be analyzed by the previous method. If, on the other hand, only
one linearly independent eigenvector v can be associated with A, then the matrix A is not