Since the input is constant the convolution integral is easily evaluated and
z(t)=e
t
z(0) +
−1
[e
t
−I]V
−1
Bu
k
(5.97)
Thus the transform, equation (5.82), enables the state vector to be determined:
x(t)=Ve
t
[V
−1
x(0) +
−1
V
−1
Bu
k
]−A
−1
Bu
k
≡e
At
[x(0) +A
−1
Bu
k
]−A
−1
Bu
k
(5.98)
The derivation ofequation (5.98) makesuse of thefollowingproperty ofthe matrix
exponential:
−1
e
t
≡e
t
−1
(5.99)
and the similarity transform:
A
−1
=V
−1
V
−1
(5.100)
Again, the outputresponse is obtained by substituting the state vectorx(t),equation
(5.98), into the output equation to give
y(t)=CVe
t
[V
−1
x(0)
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