Introduction to LTV Systems Computation of the State Transition Matrix Discretization of Continuous Time Systems
Changing Coordinates
Changing of coordinates of an LTI system: basically means we’re
scaling the coordinates in a different way
Assume that T ∈ R
n×n
is a nonsingular transformation matrix
Define ˜x = T
−1
x. Recall that ˙x = Ax + Bu, then:
˙
˜x = (T
−1
AT )˜x + T
−1
Bu =
˜
A˜x +
˜
Bu
with initial conditions ˜x (0) = T
−1
x(0)
Remember the diagonal canonical form? We can get to it if the
transformation T is the matrix containing the eigenvectors of A
What if the matrix is not diagonlizable? Well, we can still write
A = TJT
−1
, which means that
˜
A = J is the new state-space matrix
via the eigenvector transformation
In fact, you can show that if A = TJT
−1
with j Jordan blocks (i.e.,
J = diag(J
1
, J
2
, . . . , J
j
), then after the transformation ˜x = T
−1
x,
the LTI system becomes decoupled:
˙
˜x
1
= J
1
˜x
1
,
˙
˜x
2
= J
2
˜x
2
, . . . ,
˙
˜x
j
= J
j
˜x
j
.
©Ahmad F. Taha Module 04 — Linear Time-Varying Systems 6 / 26