Modern Control Engineering

Example Problems and Solutions 719

Consequently, for a certain x(0),

which implies that, for a certain x(0),x(0)cannot be determined from y(t). Therefore, the rank
of matrix Pmust be equal to n.
Next we shall obtain the sufficient condition. Suppose that rank P=n. Since

by premultiplying both sides of this last equation by eAtC, we get

If we integrate this last equation from 0 to t, we obtain

(9–124)

Notice that the left-hand side of this equation is a known quantity. Define

(9–125)

Then, from Equations (9–124) and (9–125), we have

(9–126)

where

It can be established that W(t)is a nonsingular matrix as follows: If @W(t)@were equal to 0, then

which means that

which implies that rank P<n. Therefore,@W(t)@Z0, or W(t)is nonsingular. Then, from Equa-
tion (9–126), we obtain

(9–127)

andx(0)can be determined from Equation (9–127).
Hence, we have proved that x(0)can be determined from y(t)if and only if rank P=n. Note
thatx(0)andy(t)are related by

y(t)=CeAt x(0)=a 0 (t) Cx(0)+a 1 (t) CAx(0)+p+an- 1 (t) CAn-^1 x(0)

x(0)=CW(t)D-^1 Q(t)

CeAt x= 0 , for 0 tt 1

x* WAt 1 B x=

3

t 1

0

7 CeAt x 72 dt= 0

W(t)=

3

t

0

eA*t C*CeAtdt

Q(t)=W(t) x(0)

Q(t)=

3

t

0

eA*t C* y(t)dt=known quantity

3

t

0

eA*t C* y(t)dt=

3

t

0

eA*t C* CeAt x( 0 )dt

eA*t C* y(t)=eA*t C* CeAt x( 0 )

y(t)=CeAt x(0)

y(t)=Cx(t)=CeAt x(0)= 0