Page 129 - Modern Control of DC-Based Power Systems
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Background 93
Then, the transformation matrix can be calculated using the following
equation:
T 5 W c H (3.13)
where W c is the controllability matrix and H is calculated from the coef-
ficients of the original characteristic polynomial:
?
2 3
a n21 a n22 a 1 1
?
a n22 a n23
6 1 0 7
6 7
H 5 6 ^ ^ 1 ^ ^ 7 (3.14)
6 7
1 ? 0 0
a 1
4 5
1 0 ? 0 0
knowing the desired closed-loop poles μ .. . μ , the desired characteristic
n
1
polynomial is calculated to determine the coefficients α 1 .. . α n :
n n21
s 2 μ .. . s 2 μ 5 s 1 α 1 s (3.15)
1 n 1 .. . 1 α n21 s 1 α n
The required state feedback gain matrix K can be calculated as
follows:
K 5 ½α n 2 a n j .. . α 1 2 a 1 T 21 (3.16)
j
Another option to determine the state feedback gain matrix is
Ackermann’s formula:
21
K 5 0 0 .. . 1 W c Φ AðÞ (3.17)
where W c is the controllability matrix. Φ AðÞ is the characteristic equa-
tion of A and can be obtained from the following equation:
n
Φ AðÞ 5 A 1 α 1 A n21 1 .. . 1 α n21 A 1 α n I (3.18)
As before, α i are the coefficients of the desired characteristic polyno-
mial after applying the state feedback K and moving the poles to the
desired places.
3.5 OBSERVER
As mentioned in the previous section, the underlying preassump-
tion for using the pole placement technique is the access to all state
