---
title: "Ackermann's formula"
chunk: 3/3
source: "https://en.wikipedia.org/wiki/Ackermann's_formula"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T13:34:54.532302+00:00"
instance: "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