State feedback control
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Consider a linear continuous-time invariant system with a state-space representation
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{\displaystyle {\begin{aligned}\mathbf {\dot {x}} (t)&=\mathbf {Ax} (t)+\mathbf {Bu} (t)\\\mathbf {y} (t)&=\mathbf {Cx} (t)\end{aligned}}}
where x u A , B , C transfer function
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{\displaystyle {\begin{aligned}G(s)&=\mathbf {C} (s\mathbf {I} -\mathbf {A} )^{-1}\mathbf {B} \\[4pt]&=\mathbf {C} \ {\frac {\operatorname {adj} (s\mathbf {I} -\mathbf {A} )}{\det(s\mathbf {I} -\mathbf {A} )}}\ \mathbf {B} .\end{aligned}}}
where det is the determinant and adj is the adjugate .
Since the denominator of the right equation is given by the characteristic polynomial of A G are eigenvalues of A unstable , or has a slow response or any other characteristic that does not specify the design criteria, it could be advantageous to make changes to it. The matrices A , B , C feedback loop with a gain k x u
If the system is controllable , there is always an input u (t )x 0 x (t )u (t ) = r (t ) − kx (t )
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{\displaystyle {\begin{aligned}\mathbf {\dot {x}} (t)&=\mathbf {Ax} (t)+\mathbf {B} [\mathbf {r} (t)-\mathbf {kx} (t)]\\[2pt]&=[\mathbf {A} -\mathbf {Bk} ]\mathbf {x} (t)+\mathbf {Br} (t),\\[4pt]\mathbf {y} (t)&=\mathbf {Cx} (t).\end{aligned}}}
In this new realization, the poles will be dependent on the characteristic polynomial Δnew of A − Bk
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{\displaystyle \Delta _{\text{new}}(s)=\det {\bigl (}s\mathbf {I} -(\mathbf {A} -\mathbf {Bk} ){\bigr )}.}
Computing the characteristic polynomial and choosing a suitable feedback matrix can be a challenging task, especially in larger systems. One way to make computations easier is through Ackermann's formula. For simplicity's sake, consider a single input vector with no reference parameter r
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{\displaystyle {\begin{aligned}\mathbf {u} (t)&=-\mathbf {k} ^{\rm {T}}\mathbf {x} (t)\\[2pt]\mathbf {\dot {x}} (t)&=\mathbf {Ax} (t)-\mathbf {Bk} ^{\rm {T}}\mathbf {x} (t),\end{aligned}}}
where k T
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{\displaystyle \mathbf {k} ^{\rm {T}}={\begin{bmatrix}0&\cdots &0&1\end{bmatrix}}\,{\mathcal {C}}^{-1}\Delta _{\text{new}}(\mathbf {A} ),}
in which Δnew (A ) is the desired characteristic polynomial evaluated at matrix A
C
{\displaystyle {\mathcal {C}}}
controllability matrix of the system.
This proof is based on Encyclopedia of Life Support Systems entry on Pole Placement Control.[ 3] controllable . The characteristic polynomial of
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{\displaystyle \mathbf {A} _{\rm {CL}}:=(\mathbf {A} -\mathbf {Bk} ^{\rm {T}})}
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{\displaystyle \Delta (\mathbf {A} _{\rm {CL}})=(\mathbf {A} _{\rm {CL}})^{n}+\sum _{k=0}^{n-1}\alpha _{k}\mathbf {A} _{\rm {CL}}^{k}}
Calculating the powers of A CL
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{\displaystyle {\begin{aligned}(\mathbf {A} _{\rm {CL}})^{0}=&\ (\mathbf {A} -\mathbf {Bk} ^{\rm {T}})^{0}=\mathbf {I} \\[4pt](\mathbf {A} _{\rm {CL}})^{1}=&\ (\mathbf {A} -\mathbf {Bk} ^{\rm {T}})^{1}=\mathbf {A} -\mathbf {Bk} ^{\rm {T}}\\[4pt](\mathbf {A} _{\rm {CL}})^{2}=&\ (\mathbf {A} -\mathbf {Bk} ^{\rm {T}})^{2}\\[2pt]&=\mathbf {A} ^{2}-\mathbf {ABk} ^{\rm {T}}-\mathbf {Bk} ^{\rm {T}}\mathbf {A} +(\mathbf {Bk} ^{\rm {T}})^{2}\\[2pt]&=\mathbf {A} ^{2}-\mathbf {ABk} ^{\rm {T}}-(\mathbf {Bk} ^{\rm {T}})[\mathbf {A} -\mathbf {Bk} ^{\rm {T}}]\\[2pt]&=\mathbf {A} ^{2}-\mathbf {ABk} ^{\rm {T}}-\mathbf {Bk} ^{\rm {T}}\mathbf {A} _{\rm {CL}}\\[4pt]\vdots \ &\\[4pt](\mathbf {A} _{\rm {CL}})^{n}=&\ (\mathbf {A} -\mathbf {Bk} ^{\rm {T}})^{n}\\[2pt]&=\mathbf {A} ^{n}-\mathbf {A} ^{n-1}\mathbf {Bk} ^{\rm {T}}-\mathbf {A} ^{n-2}\mathbf {Bk} ^{\rm {T}}\mathbf {A} _{\rm {CL}}-\ldots -\mathbf {Bk} ^{\rm {T}}\mathbf {A} _{\rm {CL}}^{n-1}\end{aligned}}}
Replacing the previous equations into Δ(A CL ) yields
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{\displaystyle {\begin{aligned}\Delta (\mathbf {A} _{\rm {CL}})&=\overbrace {(\mathbf {A} ^{n}-\mathbf {A} ^{n-1}\mathbf {Bk} ^{\rm {T}}-\mathbf {A} ^{n-2}\mathbf {Bk} ^{\rm {T}}\mathbf {A} _{\rm {CL}}-\ldots -\mathbf {Bk} ^{\rm {T}}\mathbf {A} _{\rm {CL}}^{n-1})} ^{(\mathbf {A} _{\rm {CL}})^{n}}+\overbrace {\ldots +\alpha _{2}(\mathbf {A} ^{2}-\mathbf {ABk} ^{\rm {T}}-\mathbf {Bk} ^{\rm {T}}\mathbf {A} _{\rm {CL}})+\alpha _{1}(\mathbf {A} -\mathbf {Bk} ^{\rm {T}})+\alpha _{0}\mathbf {I} } ^{\sum _{k=0}^{n-1}\alpha _{k}\mathbf {A} _{\rm {CL}}^{k}}\\[4pt]&=(\mathbf {A} ^{n}+\alpha _{n-1}\mathbf {A} ^{n-1}+\ldots +\alpha _{2}\mathbf {A} ^{2}+\alpha _{1}\mathbf {A} +\alpha _{0}\mathbf {I} )-(\mathbf {A} ^{n-1}\mathbf {Bk} ^{\rm {T}}+\mathbf {A} ^{n-2}\mathbf {Bk} ^{\rm {T}}\mathbf {A} _{\rm {CL}}+\ldots +\mathbf {Bk} ^{\rm {T}}\mathbf {A} _{\rm {CL}}^{n-1})+\ldots -\alpha _{2}(\mathbf {ABk} ^{\rm {T}}+\mathbf {Bk} ^{\rm {T}}\mathbf {A} _{\rm {CL}})-\alpha _{1}(\mathbf {Bk} ^{\rm {T}})\\[4pt]&=\Delta (\mathbf {A} )-(\mathbf {A} ^{n-1}\mathbf {Bk} ^{\rm {T}}+\mathbf {A} ^{n-2}\mathbf {Bk} ^{\rm {T}}\mathbf {A} _{\rm {CL}}+\ldots +\mathbf {Bk} ^{\rm {T}}\mathbf {A} _{\rm {CL}}^{n-1})-\ldots -\alpha _{2}(\mathbf {ABk} ^{\rm {T}}+\mathbf {Bk} ^{\rm {T}}\mathbf {A} _{\rm {CL}})-\alpha _{1}(\mathbf {Bk} ^{\rm {T}})\end{aligned}}}
Rewriting the above equation as a matrix product and omitting terms that k T
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{\displaystyle \Delta (\mathbf {A} _{\rm {CL}})=\Delta (\mathbf {A} )-{\begin{bmatrix}\mathbf {B} &\mathbf {AB} &\cdots &\mathbf {A} ^{n-1}\mathbf {B} \end{bmatrix}}{\begin{bmatrix}\star \\\vdots \\\mathbf {k} ^{\rm {T}}\end{bmatrix}}}
From the Cayley–Hamilton theorem , Δ(A CL ) = 0 , thus
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{\displaystyle {\begin{bmatrix}\mathbf {B} &\mathbf {AB} &\cdots &\mathbf {A} ^{n-1}\mathbf {B} \end{bmatrix}}{\begin{bmatrix}\star \\\vdots \\\mathbf {k} ^{\rm {T}}\end{bmatrix}}=\Delta (\mathbf {A} )}
Note that
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{\displaystyle {\mathcal {C}}={\begin{bmatrix}\mathbf {B} &\mathbf {AB} &\cdots &\mathbf {A} ^{n-1}\mathbf {B} \end{bmatrix}}}
controllability matrix of the system. Since the system is controllable,
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{\displaystyle {\begin{bmatrix}\star \\\vdots \\\mathbf {k} ^{\rm {T}}\end{bmatrix}}={\mathcal {C}}^{-1}\Delta (\mathbf {A} )}
To find k T
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{\displaystyle {\begin{bmatrix}0&\cdots &0&1\end{bmatrix}}}
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{\displaystyle {\begin{bmatrix}0&\cdots &0&1\end{bmatrix}}{\begin{bmatrix}\star \\\vdots \\\mathbf {k} ^{\rm {T}}\end{bmatrix}}={\begin{bmatrix}0&\cdots &0&1\end{bmatrix}}\,{\mathcal {C}}^{-1}\Delta (\mathbf {A} )}
Thus,
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{\displaystyle \mathbf {k} ^{\rm {T}}={\begin{bmatrix}0&\cdots &0&1\end{bmatrix}}\,{\mathcal {C}}^{-1}\Delta (\mathbf {A} )}
Consider[ 4]
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{\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
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{\displaystyle {\begin{aligned}\det(s\mathbf {I} -\mathbf {A} )&=(s-1)(s-2)-1\\&=s^{2}-3s+2,\end{aligned}}}
the matrix A
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{\displaystyle \mathbf {k} ={\begin{bmatrix}k_{1}&k_{2}\end{bmatrix}}.}
From Ackermann's formula, we can find a matrix k
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{\displaystyle \Delta _{\text{desired}}(s)=s^{2}+11s+30.}
Thus,
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{\displaystyle \Delta _{\text{desired}}(\mathbf {A} )=\mathbf {A} ^{2}+11\mathbf {A} +30\mathbf {I} }
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{\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
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{\displaystyle \mathbf {A} ^{2}=\left[{\begin{smallmatrix}2&3\\3&5\end{smallmatrix}}\right].}
Finally, from Ackermann's formula
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{\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
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Ackermann's formula can also be used for the design of state observers . Consider the linear discrete-time observed system
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{\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
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{\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}}}
transposed :
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T
{\displaystyle {\mathcal {O}}^{\rm {T}}}
A T
Ackermann's formula can also be applied on continuous-time observed systems.
![{\displaystyle {\begin{aligned}G(s)&=\mathbf {C} (s\mathbf {I} -\mathbf {A} )^{-1}\mathbf {B} \\[4pt]&=\mathbf {C} \ {\frac {\operatorname {adj} (s\mathbf {I} -\mathbf {A} )}{\det(s\mathbf {I} -\mathbf {A} )}}\ \mathbf {B} .\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9972684db7cd56e41e5ef2574747c806a6731f44)
![{\displaystyle {\begin{aligned}\mathbf {\dot {x}} (t)&=\mathbf {Ax} (t)+\mathbf {B} [\mathbf {r} (t)-\mathbf {kx} (t)]\\[2pt]&=[\mathbf {A} -\mathbf {Bk} ]\mathbf {x} (t)+\mathbf {Br} (t),\\[4pt]\mathbf {y} (t)&=\mathbf {Cx} (t).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/443a00d6c1f4e521da8071fcd63f5efd1216afe0)
![{\displaystyle {\begin{aligned}\mathbf {u} (t)&=-\mathbf {k} ^{\rm {T}}\mathbf {x} (t)\\[2pt]\mathbf {\dot {x}} (t)&=\mathbf {Ax} (t)-\mathbf {Bk} ^{\rm {T}}\mathbf {x} (t),\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f1aa529ea5908956682d4dd75840921caa17d61)
^{1}=&\ (\mathbf {A} -\mathbf {Bk} ^{\rm {T}})^{1}=\mathbf {A} -\mathbf {Bk} ^{\rm {T}}\\[4pt](\mathbf {A} _{\rm {CL}})^{2}=&\ (\mathbf {A} -\mathbf {Bk} ^{\rm {T}})^{2}\\[2pt]&=\mathbf {A} ^{2}-\mathbf {ABk} ^{\rm {T}}-\mathbf {Bk} ^{\rm {T}}\mathbf {A} +(\mathbf {Bk} ^{\rm {T}})^{2}\\[2pt]&=\mathbf {A} ^{2}-\mathbf {ABk} ^{\rm {T}}-(\mathbf {Bk} ^{\rm {T}})[\mathbf {A} -\mathbf {Bk} ^{\rm {T}}]\\[2pt]&=\mathbf {A} ^{2}-\mathbf {ABk} ^{\rm {T}}-\mathbf {Bk} ^{\rm {T}}\mathbf {A} _{\rm {CL}}\\[4pt]\vdots \ &\\[4pt](\mathbf {A} _{\rm {CL}})^{n}=&\ (\mathbf {A} -\mathbf {Bk} ^{\rm {T}})^{n}\\[2pt]&=\mathbf {A} ^{n}-\mathbf {A} ^{n-1}\mathbf {Bk} ^{\rm {T}}-\mathbf {A} ^{n-2}\mathbf {Bk} ^{\rm {T}}\mathbf {A} _{\rm {CL}}-\ldots -\mathbf {Bk} ^{\rm {T}}\mathbf {A} _{\rm {CL}}^{n-1}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/671b59d20749c162274643e967e0490d2c9ea5a8)
![{\displaystyle {\begin{aligned}\Delta (\mathbf {A} _{\rm {CL}})&=\overbrace {(\mathbf {A} ^{n}-\mathbf {A} ^{n-1}\mathbf {Bk} ^{\rm {T}}-\mathbf {A} ^{n-2}\mathbf {Bk} ^{\rm {T}}\mathbf {A} _{\rm {CL}}-\ldots -\mathbf {Bk} ^{\rm {T}}\mathbf {A} _{\rm {CL}}^{n-1})} ^{(\mathbf {A} _{\rm {CL}})^{n}}+\overbrace {\ldots +\alpha _{2}(\mathbf {A} ^{2}-\mathbf {ABk} ^{\rm {T}}-\mathbf {Bk} ^{\rm {T}}\mathbf {A} _{\rm {CL}})+\alpha _{1}(\mathbf {A} -\mathbf {Bk} ^{\rm {T}})+\alpha _{0}\mathbf {I} } ^{\sum _{k=0}^{n-1}\alpha _{k}\mathbf {A} _{\rm {CL}}^{k}}\\[4pt]&=(\mathbf {A} ^{n}+\alpha _{n-1}\mathbf {A} ^{n-1}+\ldots +\alpha _{2}\mathbf {A} ^{2}+\alpha _{1}\mathbf {A} +\alpha _{0}\mathbf {I} )-(\mathbf {A} ^{n-1}\mathbf {Bk} ^{\rm {T}}+\mathbf {A} ^{n-2}\mathbf {Bk} ^{\rm {T}}\mathbf {A} _{\rm {CL}}+\ldots +\mathbf {Bk} ^{\rm {T}}\mathbf {A} _{\rm {CL}}^{n-1})+\ldots -\alpha _{2}(\mathbf {ABk} ^{\rm {T}}+\mathbf {Bk} ^{\rm {T}}\mathbf {A} _{\rm {CL}})-\alpha _{1}(\mathbf {Bk} ^{\rm {T}})\\[4pt]&=\Delta (\mathbf {A} )-(\mathbf {A} ^{n-1}\mathbf {Bk} ^{\rm {T}}+\mathbf {A} ^{n-2}\mathbf {Bk} ^{\rm {T}}\mathbf {A} _{\rm {CL}}+\ldots +\mathbf {Bk} ^{\rm {T}}\mathbf {A} _{\rm {CL}}^{n-1})-\ldots -\alpha _{2}(\mathbf {ABk} ^{\rm {T}}+\mathbf {Bk} ^{\rm {T}}\mathbf {A} _{\rm {CL}})-\alpha _{1}(\mathbf {Bk} ^{\rm {T}})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bde111460a740147e2de5ae235eec4dee104eb3)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c96024b5daa89aff38dcd1f940fc32703263740)
![{\displaystyle \mathbf {A} ^{2}=\left[{\begin{smallmatrix}2&3\\3&5\end{smallmatrix}}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c41bfb3f1e22579ee7dc85c13484269adc603cc8)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b06076cd373d8e11e7e2d231b49a7660c21a67d)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3d952143085c8611820adb7e3148cb8501f55dc)
