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66 CHAPTER 2. OVERVIEW OF THE UNDERLYING THEORY
The controllability grammian, Y is defined as,
Y =
∞
0
e
At
BB
T
e
A
T
t
dt,
and the observability grammian, X, is defined as
X =
∞
0
e
A
T
t
C
T
Ce
At
dt.
The grammians, X and Y , satisfy the Lyapunov equations,
AY + YA
T
+BB
T
=0
A
T
X+XA+C
T
C =0,
and this is typically how they are calculated. We can also see from the definitions that
X ≥ 0andY≥0. Actually Y>0 if and only if (A,B) is controllable and X>0ifand
only if (C, A) is observable.
Now consider the effect of a state transformation on these grammians. Define a new
state, ˆx,byˆx=Tx,whereT is invertible, to give
P(s)=
ˆ
A
ˆ
B
ˆ
C
D
=
TAT
−1
TB
CT
−1
D
.
The new grammians are
ˆ
Y = TYT
T
and
ˆ
X = T
−T
XT
−1
. The product of the
grammians,
ˆ
Y
ˆ
X is therefore given by,
ˆ
Y
ˆ
X = TYXT
−1
.
This illustrates that the eigenvalues of the product of the grammians is invariant under
state similarity transformation.