66 CHAPTER 2. OVERVIEWOF THE UNDERLYING THEORY
The controllability grammian, Yis defined as,
Y=Z∞
0
eAtBBTeATtdt,
and the observability grammian, X, is defined as
X=Z∞
0
eATtCTCeAtdt.
The grammians, Xand Y, satisfy the Lyapunov equations,
AY +YA
T+BBT=0
A
T
X+XA+CTC=0,
and this is typically how they are calculated. We can also see from the definitions that
X≥0andY≥0. Actually Y>0if and only if (A,B) is controllableand X>0ifand
only if (C,A) is observable.
Now consider the effectof a state transformation on these grammians. Define a new
state, ˆx,by ˆx=Tx,whereTisinvertible, to give
P(s)=ˆ
Aˆ
B
ˆ
C
D
=
TAT−1TB
CT−1D.
The new grammians are ˆ
Y=TYTTand ˆ
X=T−TXT−1. The product of the
grammians, ˆ
Yˆ
Xis therefore given by,
ˆ
Yˆ
X=TYXT−1.
This illustrates that the eigenvaluesof the product of the grammians is invariant under
state similaritytransformation.