18 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Ackermann's formula | 3/3 | https://en.wikipedia.org/wiki/Ackermann's_formula | reference | science, encyclopedia | 2026-05-05T13:34:54.532302+00:00 | kb-cron |
x
˙
=
[
1
1
1
2
]
x
+
[
1
0
]
u
{\displaystyle \mathbf {\dot {x}} ={\begin{bmatrix}1&1\\1&2\end{bmatrix}}\mathbf {x} +{\begin{bmatrix}1\\0\end{bmatrix}}\mathbf {u} }
We know from the characteristic polynomial of A that the system is unstable since
det
(
s
I
−
A
)
=
(
s
−
1
)
(
s
−
2
)
−
1
=
s
2
−
3
s
+
2
,
{\displaystyle {\begin{aligned}\det(s\mathbf {I} -\mathbf {A} )&=(s-1)(s-2)-1\\&=s^{2}-3s+2,\end{aligned}}}
the matrix A will only have positive eigenvalues. Thus, to stabilize the system we shall put a feedback gain
k
=
[
k
1
k
2
]
.
{\displaystyle \mathbf {k} ={\begin{bmatrix}k_{1}&k_{2}\end{bmatrix}}.}
From Ackermann's formula, we can find a matrix k that will change the system so that its characteristic equation will be equal to a desired polynomial. Suppose we want
Δ
desired
(
s
)
=
s
2
+
11
s
+
30.
{\displaystyle \Delta _{\text{desired}}(s)=s^{2}+11s+30.}
Thus,
Δ
desired
(
A
)
=
A
2
+
11
A
+
30
I
{\displaystyle \Delta _{\text{desired}}(\mathbf {A} )=\mathbf {A} ^{2}+11\mathbf {A} +30\mathbf {I} }
and computing the controllability matrix yields
C
=
[
B
A
B
]
=
[
1
1
0
1
]
⟹
C
−
1
=
[
1
−
1
0
1
]
{\displaystyle {\begin{aligned}{\mathcal {C}}&={\begin{bmatrix}\mathbf {B} &\mathbf {AB} \end{bmatrix}}={\begin{bmatrix}1&1\\0&1\end{bmatrix}}\\[4pt]\implies {\mathcal {C}}^{-1}&={\begin{bmatrix}1&-1\\0&1\end{bmatrix}}\end{aligned}}}
Also, we have that
A
2
=
[
2
3
3
5
]
.
{\displaystyle \mathbf {A} ^{2}=\left[{\begin{smallmatrix}2&3\\3&5\end{smallmatrix}}\right].}
Finally, from Ackermann's formula
k
T
=
[
0
1
]
[
1
−
1
0
1
]
(
[
2
3
3
5
]
+
11
[
1
1
1
2
]
+
30
I
)
=
[
0
1
]
[
1
−
1
0
1
]
[
43
14
14
57
]
=
[
0
1
]
[
29
−
43
14
57
]
=
[
14
57
]
{\displaystyle {\begin{aligned}\mathbf {k} ^{\rm {T}}&={\begin{bmatrix}0&1\end{bmatrix}}{\begin{bmatrix}1&-1\\0&1\end{bmatrix}}\left({\begin{bmatrix}2&3\\3&5\end{bmatrix}}+11{\begin{bmatrix}1&1\\1&2\end{bmatrix}}+30\mathbf {I} \right)\\[2pt]&={\begin{bmatrix}0&1\end{bmatrix}}{\begin{bmatrix}1&-1\\0&1\end{bmatrix}}{\begin{bmatrix}43&14\\14&57\end{bmatrix}}\\[2pt]&={\begin{bmatrix}0&1\end{bmatrix}}{\begin{bmatrix}29&-43\\14&57\end{bmatrix}}\\[6pt]&={\begin{bmatrix}14&57\end{bmatrix}}\end{aligned}}}
== State observer design == Ackermann's formula can also be used for the design of state observers. Consider the linear discrete-time observed system
x
^
(
n
+
1
)
=
A
x
^
(
n
)
+
B
u
(
n
)
+
L
[
y
(
n
)
−
y
^
(
n
)
]
y
^
(
n
)
=
C
x
^
(
n
)
{\displaystyle {\begin{aligned}\mathbf {\hat {x}} (n+1)&=\mathbf {A{\hat {x}}} (n)+\mathbf {Bu} (n)+\mathbf {L} [\mathbf {y} (n)-\mathbf {\hat {y}} (n)]\\\mathbf {\hat {y}} (n)&=\mathbf {C{\hat {x}}} (n)\end{aligned}}}
with observer gain L. Then Ackermann's formula for the design of state observers is noted as
L
T
=
[
0
0
⋯
1
]
(
O
T
)
−
1
Δ
new
(
A
T
)
{\displaystyle \mathbf {L} ^{\rm {T}}={\begin{bmatrix}0&0&\cdots &1\end{bmatrix}}({\mathcal {O}}^{\rm {T}})^{-1}\Delta _{\text{new}}(\mathbf {A} ^{\rm {T}})}
with observability matrix
O
{\displaystyle {\mathcal {O}}}
. Here it is important to note, that the observability matrix and the system matrix are transposed:
O
T
{\displaystyle {\mathcal {O}}^{\rm {T}}}
and AT. Ackermann's formula can also be applied on continuous-time observed systems.
== See also == Full state feedback
== References ==
== External links == Chapter about Ackermann's Formula on Wikibook of Control Systems and Control Engineering