Digital Control Systems [MEE 4003] - Kckong.info

2.1. STATE SPACE REALIZATION OF DYNAMIC SYSTEMS 23
acteristic equation ofF is
[ ] −λ 1
det = λ 2 +3λ+2 = (λ+1)(λ+2) = 0
−2 −3−λ
Thus the eigenvalues ofF are λ 1 = −1 and λ 2 = −2. The eigenvalue equation is
[ ] −λ 1
v = 0
−2 −3−λ
and the associated eigenvectors (among many) are
[ ] 1
v 1 =
and v
−1
2 =
[ 1
−2
From the eigenvectors obtained, the transformation matrixV is set to
[ ] 1 1
V =
−1 −2
Finally, a new (diagonalized) state space model is set as
˙¯x = Λ¯x+Ḡu
y = ¯H¯x
¯x(0) = 0 ∈ R 2
]
where
[ ] −1 0
Λ = V −1 FV =
0 −2
[ ][ ] [ ]
2 1 0 1
Ḡ = V −1 G = =
−1 −1 1 −1
¯H = HV = [ 1 0 ][ ]
1 1
= [ 1 1 ]
−1 −2
The state of the diagonalized state space matrix is
∫ t
[ ][ ]
e
−(t−τ)
0 1
¯x(t) =
0
0 e −2(t−τ) dτ fort ≥ 0
−1
[ ]
1−e
=
−t
0.5e −2t fort ≥ 0
−0.5
and the output isy(t) = ¯H¯x(t) = 0.5+0.5e −2t −e −t fort ≥ 0.
Digital Control Systems, Sogang University
Kyoungchul Kong