Nonlinear Control Sy.. - Free

1.6. PHASE-PLANE ANALYSIS OF LINEAR TIME-INVARIANT SYSTEMS 11
Given that T is nonsingular, its inverse, denoted T-1 exists, and we can write
y = T-1ATy = Dy (1.17)
Transformation of the form D = T-'AT are very well known in linear algebra, and the
matrices A and D share several interesting properties:
Property 1: The matrices A and D share the same eigenvalues Al and A2. For this reason
the matrices A and D are said to be similar, and transformations of the form T-1AT are
called similarity transformations.
Property 2: Assume that the eigenvectors v1i v2 associated with the real eigenvalues A1, A2
are linearly independent. In this case the matrix T defined in (1.17) can be formed by placing
the eigenvectors vl and v2 as its columns. In this case we have that
DAl
0 A2
that is, in this case the matrix A is similar to the diagonal matrix D. The importance of
this transformation is that in the new coordinates y = [y1 y2]T the system is uncoupled,
i.e.,
[y20 '\2J [Y2
or
0]
yl = Alyl (1.18)
y2 = A2y2 (1.19)
Both equations can be solved independently, and the general solution is given by (1.10).
This means that the trajectories along each of the coordinate axes yl and Y2 are independent
of one another.
Several interesting cases can be distinguished, depending the sign of the eigenvalues
Al and A2. The following examples clarify this point.
Example 1.7 Consider the system
[± 2]-[ 0
1 -2] [x2 I .
The eigenvalues in this case are Al = -1,A2 = -2. The equilibrium point of a system where
both eigenvalues have the same sign is called a node. A is diagonalizable and D = T-1AT
with
T=[0 1 -3
0
1], D-[ 0 -2].
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