[<< wikibooks] LMIs in Control/pages/Optimal Output Feedback Hinf LMI
== Optimal Output Feedback ==
  
    
      
        
          H
          
            ∞
          
        
      
    
    {\displaystyle H_{\infty }}
   LMI
Optimal output feedback control is a problem which arises from not knowing all information about the output of the system. It correlates to the state feedback situation where the part of the state is unknown. This issue can arise in decentralized control problems, for example, and requires the use of an "observer-like" solution. One such method is the use of a Kalman Filter, a more classical technique. However, other methods exist that do not implement a Kalman Filter such as the one below which uses an LMI to preform the output feeback. The 
  
    
      
        
          H
          
            ∞
          
        
      
    
    {\displaystyle H_{\infty }}
   control methods form an optimization problem which attempts to minimize the 
  
    
      
        
          H
          
            ∞
          
        
      
    
    {\displaystyle H_{\infty }}
   norm of the system.


== The System ==
The system is represented using the 9-matrix notation shown below.

  
    
      
        
          
            [
            
              
                
                  
                    
                      
                        x
                        ˙
                      
                    
                  
                
              
              
                
                  z
                
              
              
                
                  y
                
              
            
            ]
          
        
        =
        
          
            [
            
              
                
                  A
                
                
                  
                    B
                    
                      1
                    
                  
                
                
                  
                    B
                    
                      2
                    
                  
                
              
              
                
                  
                    C
                    
                      1
                    
                  
                
                
                  
                    D
                    
                      11
                    
                  
                
                
                  
                    D
                    
                      12
                    
                  
                
              
              
                
                  
                    C
                    
                      2
                    
                  
                
                
                  
                    D
                    
                      21
                    
                  
                
                
                  
                    D
                    
                      22
                    
                  
                
              
            
            ]
          
        
        
          
            [
            
              
                
                  x
                
              
              
                
                  w
                
              
              
                
                  u
                
              
            
            ]
          
        
      
    
    {\displaystyle {\begin{bmatrix}{\dot {x}}\\z\\y\end{bmatrix}}={\begin{bmatrix}A&B_{1}&B_{2}\\C_{1}&D_{11}&D_{12}\\C_{2}&D_{21}&D_{22}\end{bmatrix}}{\begin{bmatrix}x\\w\\u\end{bmatrix}}}
  where 
  
    
      
        x
        (
        t
        )
        ∈
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle x(t)\in \mathbb {R} ^{n}}
   is the state, 
  
    
      
        z
        (
        t
        )
        ∈
        
          
            R
          
          
            p
          
        
      
    
    {\displaystyle z(t)\in \mathbb {R} ^{p}}
   is the regulated output, 
  
    
      
        y
        (
        t
        )
        ∈
        
          
            R
          
          
            q
          
        
      
    
    {\displaystyle y(t)\in \mathbb {R} ^{q}}
   is the sensed output, 
  
    
      
        w
        (
        t
        )
        ∈
        
          
            R
          
          
            r
          
        
      
    
    {\displaystyle w(t)\in \mathbb {R} ^{r}}
   is the exogenous input, and 
  
    
      
        u
        (
        t
        )
        ∈
        
          
            R
          
          
            m
          
        
      
    
    {\displaystyle u(t)\in \mathbb {R} ^{m}}
   is the actuator input, at any 
  
    
      
        t
        ∈
        
          R
        
      
    
    {\displaystyle t\in \mathbb {R} }
  .


== The Data ==

  
    
      
        A
      
    
    {\displaystyle A}
  , 
  
    
      
        
          B
          
            1
          
        
      
    
    {\displaystyle B_{1}}
  , 
  
    
      
        
          B
          
            2
          
        
      
    
    {\displaystyle B_{2}}
  , 
  
    
      
        
          C
          
            1
          
        
      
    
    {\displaystyle C_{1}}
  , 
  
    
      
        
          C
          
            2
          
        
      
    
    {\displaystyle C_{2}}
  , 
  
    
      
        
          D
          
            11
          
        
      
    
    {\displaystyle D_{11}}
  , 
  
    
      
        
          D
          
            12
          
        
      
    
    {\displaystyle D_{12}}
  , 
  
    
      
        
          D
          
            21
          
        
      
    
    {\displaystyle D_{21}}
  , 
  
    
      
        
          D
          
            22
          
        
      
    
    {\displaystyle D_{22}}
   are known.


== The LMI: Optimal Output Feedback ==
  
    
      
        
          H
          
            ∞
          
        
      
    
    {\displaystyle H_{\infty }}
   Control LMI
The following are equivalent.
1) There exists a 
  
    
      
        
          
            
              K
              ^
            
          
        
        =
        
          
            [
            
              
                
                  
                    A
                    
                      K
                    
                  
                
                
                  
                    B
                    
                      K
                    
                  
                
              
              
                
                  
                    C
                    
                      K
                    
                  
                
                
                  
                    D
                    
                      K
                    
                  
                
              
            
            ]
          
        
      
    
    {\displaystyle {\hat {K}}={\begin{bmatrix}A_{K}&B_{K}\\C_{K}&D_{K}\end{bmatrix}}}
   such that 
  
    
      
        
          |
        
        
          |
        
        S
        (
        K
        ,
        P
        )
        
          |
        
        
          
            |
          
          
            
              H
              
                ∞
              
            
          
        
        <
        γ
      
    
    {\displaystyle ||S(K,P)||_{H_{\infty }}<\gamma }
  
2) There exists 
  
    
      
        
          X
          
            1
          
        
      
    
    {\displaystyle X_{1}}
  , 
  
    
      
        
          Y
          
            1
          
        
      
    
    {\displaystyle Y_{1}}
  , 
  
    
      
        Z
      
    
    {\displaystyle Z}
  , 
  
    
      
        
          A
          
            n
          
        
      
    
    {\displaystyle A_{n}}
  , 
  
    
      
        
          B
          
            n
          
        
      
    
    {\displaystyle B_{n}}
  , 
  
    
      
        
          C
          
            n
          
        
      
    
    {\displaystyle C_{n}}
  , 
  
    
      
        
          D
          
            n
          
        
      
    
    {\displaystyle D_{n}}
   such that

  
    
      
        
          
            [
            
              
                
                  
                    X
                    
                      1
                    
                  
                
                
                  I
                
              
              
                
                  I
                
                
                  
                    Y
                    
                      1
                    
                  
                
              
            
            ]
          
        
        >
        0
      
    
    {\displaystyle {\begin{bmatrix}X_{1}&I\\I&Y_{1}\end{bmatrix}}>0}
  

  
    
      
        
          
            [
            
              
                
                  A
                  
                    Y
                    
                      1
                    
                  
                  +
                  
                    Y
                    
                      1
                    
                  
                  
                    A
                    
                      T
                    
                  
                  +
                  
                    B
                    
                      2
                    
                  
                  
                    C
                    
                      n
                    
                  
                  +
                  
                    C
                    
                      n
                    
                  
                  
                    B
                    
                      2
                    
                    
                      T
                    
                  
                
                
                  
                    ∗
                    
                      T
                    
                  
                
                
                  
                    ∗
                    
                      T
                    
                  
                
                
                  
                    ∗
                    
                      T
                    
                  
                
              
              
                
                  
                    A
                    
                      T
                    
                  
                  +
                  
                    A
                    
                      n
                    
                  
                  +
                  (
                  
                    B
                    
                      2
                    
                  
                  
                    D
                    
                      n
                    
                  
                  
                    C
                    
                      2
                    
                  
                  
                    )
                    
                      T
                    
                  
                
                
                  
                    X
                    
                      1
                    
                  
                  A
                  +
                  
                    A
                    
                      T
                    
                  
                  +
                  
                    B
                    
                      n
                    
                  
                  
                    C
                    
                      2
                    
                  
                  +
                  
                    C
                    
                      2
                    
                    
                      T
                    
                  
                  
                    B
                    
                      n
                    
                    
                      T
                    
                  
                
                
                  
                    ∗
                    
                      T
                    
                  
                
                
                  
                    ∗
                    
                      T
                    
                  
                
              
              
                
                  (
                  
                    B
                    
                      1
                    
                  
                  +
                  
                    B
                    
                      2
                    
                  
                  
                    D
                    
                      n
                    
                  
                  
                    D
                    
                      21
                    
                  
                  
                    )
                    
                      T
                    
                  
                
                
                  (
                  
                    X
                    
                      1
                    
                  
                  
                    B
                    
                      1
                    
                  
                  +
                  
                    B
                    
                      n
                    
                  
                  
                    D
                    
                      21
                    
                  
                  
                    )
                    
                      T
                    
                  
                
                
                  −
                  γ
                  I
                
                
                  
                    ∗
                    
                      T
                    
                  
                
              
              
                
                  
                    C
                    
                      1
                    
                  
                  
                    Y
                    
                      1
                    
                  
                  +
                  
                    D
                    
                      12
                    
                  
                  
                    C
                    
                      n
                    
                  
                
                
                  
                    C
                    
                      1
                    
                  
                  +
                  
                    D
                    
                      12
                    
                  
                  
                    D
                    
                      n
                    
                  
                  
                    C
                    
                      2
                    
                  
                
                
                  
                    D
                    
                      11
                    
                  
                  +
                  
                    D
                    
                      12
                    
                  
                  
                    D
                    
                      n
                    
                  
                  
                    D
                    
                      21
                    
                  
                
                
                  −
                  γ
                  I
                
              
            
            ]
          
        
        <
        0
      
    
    {\displaystyle {\begin{bmatrix}AY_{1}+Y_{1}A^{\text{T}}+B_{2}C_{n}+C_{n}B_{2}^{\text{T}}&*^{\text{T}}&*^{\text{T}}&*^{\text{T}}\\A^{\text{T}}+A_{n}+(B_{2}D_{n}C_{2})^{\text{T}}&X_{1}A+A^{\text{T}}+B_{n}C_{2}+C_{2}^{\text{T}}B_{n}^{\text{T}}&*^{\text{T}}&*^{\text{T}}\\(B_{1}+B_{2}D_{n}D_{21})^{\text{T}}&(X_{1}B_{1}+B_{n}D_{21})^{\text{T}}&-\gamma I&*^{\text{T}}\\C_{1}Y_{1}+D_{12}C_{n}&C_{1}+D_{12}D_{n}C_{2}&D_{11}+D_{12}D_{n}D_{21}&-\gamma I\\\end{bmatrix}}<0}
  


== Conclusion: ==
The above LMI determines the the upper bound 
  
    
      
        γ
      
    
    {\displaystyle \gamma }
   on the 
  
    
      
        
          H
          
            ∞
          
        
      
    
    {\displaystyle H_{\infty }}
   norm. In addition to this the controller 
  
    
      
        
          
            
              K
              ^
            
          
        
        (
        
          A
          
            K
          
        
        ,
        
          B
          
            K
          
        
        ,
        
          C
          
            K
          
        
        ,
        
          D
          
            K
          
        
        )
      
    
    {\displaystyle {\hat {K}}(A_{K},B_{K},C_{K},D_{K})}
   can also be recovered.

  
    
      
        
          D
          
            K
          
        
        =
        (
        I
        +
        
          D
          
            K
            2
          
        
        
          D
          
            22
          
        
        
          )
          
            −
            1
          
        
        
          D
          
            K
            2
          
        
      
    
    {\displaystyle D_{K}=(I+D_{K2}D_{22})^{-1}D_{K2}}
  

  
    
      
        
          B
          
            K
          
        
        =
        
          B
          
            K
            2
          
        
        (
        I
        +
        
          D
          
            22
          
        
        
          D
          
            K
          
        
        )
      
    
    {\displaystyle B_{K}=B_{K2}(I+D_{22}D_{K})}
  

  
    
      
        
          C
          
            K
          
        
        =
        (
        I
        +
        
          D
          
            K
          
        
        
          D
          
            22
          
        
        )
        
          C
          
            K
            2
          
        
      
    
    {\displaystyle C_{K}=(I+D_{K}D_{22})C_{K2}}
  

  
    
      
        
          A
          
            K
          
        
        =
        
          A
          
            K
            2
          
        
        −
        
          B
          
            K
          
        
        (
        I
        +
        
          D
          
            22
          
        
        
          D
          
            K
          
        
        
          )
          
            −
            1
          
        
        
          D
          
            22
          
        
        
          C
          
            K
          
        
      
    
    {\displaystyle A_{K}=A_{K2}-B_{K}(I+D_{22}D_{K})^{-1}D_{22}C_{K}}
  where,

  
    
      
        
          
            [
            
              
                
                  
                    A
                    
                      K
                      2
                    
                  
                
                
                  
                    B
                    
                      K
                      2
                    
                  
                
              
              
                
                  
                    C
                    
                      K
                      2
                    
                  
                
                
                  
                    D
                    
                      K
                      2
                    
                  
                
              
            
            ]
          
        
        =
        
          
            
              [
              
                
                  
                    
                      X
                      
                        2
                      
                    
                  
                  
                    
                      X
                      
                        1
                      
                    
                    
                      B
                      
                        2
                      
                    
                  
                
                
                  
                    0
                  
                  
                    I
                  
                
              
              ]
            
          
          
            −
            1
          
        
        
          [
          
            
              
                [
                
                  
                    
                      
                        A
                        
                          n
                        
                      
                    
                    
                      
                        B
                        
                          n
                        
                      
                    
                  
                  
                    
                      
                        C
                        
                          n
                        
                      
                    
                    
                      
                        D
                        
                          n
                        
                      
                    
                  
                
                ]
              
            
            −
            
              
                [
                
                  
                    
                      
                        X
                        
                          1
                        
                      
                      A
                      
                        Y
                        
                          1
                        
                      
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                  
                
                ]
              
            
          
          ]
        
        
          
            
              [
              
                
                  
                    
                      Y
                      
                        2
                      
                      
                        T
                      
                    
                  
                  
                    0
                  
                
                
                  
                    
                      C
                      
                        2
                      
                    
                    
                      Y
                      
                        1
                      
                    
                  
                  
                    I
                  
                
              
              ]
            
          
          
            −
            1
          
        
      
    
    {\displaystyle {\begin{bmatrix}A_{K2}&B_{K2}\\C_{K2}&D_{K2}\end{bmatrix}}={\begin{bmatrix}X_{2}&X_{1}B_{2}\\0&I\end{bmatrix}}^{-1}\left[{\begin{bmatrix}A_{n}&B_{n}\\C_{n}&D_{n}\end{bmatrix}}-{\begin{bmatrix}X_{1}AY_{1}&0\\0&0\end{bmatrix}}\right]{\begin{bmatrix}Y_{2}^{T}&0\\C_{2}Y_{1}&I\end{bmatrix}}^{-1}}
  for any full-rank 
  
    
      
        
          X
          
            2
          
        
      
    
    {\displaystyle X_{2}}
   and 
  
    
      
        
          Y
          
            2
          
        
      
    
    {\displaystyle Y_{2}}
   such that

  
    
      
        
          
            [
            
              
                
                  
                    X
                    
                      1
                    
                  
                
                
                  
                    X
                    
                      2
                    
                  
                
              
              
                
                  
                    X
                    
                      2
                    
                    
                      T
                    
                  
                
                
                  
                    X
                    
                      3
                    
                  
                
              
            
            ]
          
        
        =
        
          
            
              [
              
                
                  
                    
                      Y
                      
                        1
                      
                    
                  
                  
                    
                      Y
                      
                        2
                      
                    
                    
                      B
                      
                        2
                      
                    
                  
                
                
                  
                    
                      Y
                      
                        2
                      
                      
                        T
                      
                    
                  
                  
                    
                      Y
                      
                        3
                      
                    
                  
                
              
              ]
            
          
          
            −
            1
          
        
      
    
    {\displaystyle {\begin{bmatrix}X_{1}&X_{2}\\X_{2}^{T}&X_{3}\end{bmatrix}}={\begin{bmatrix}Y_{1}&Y_{2}B_{2}\\Y_{2}^{T}&Y_{3}\end{bmatrix}}^{-1}}
  .


== Implementation ==
This implementation requires Yalmip and Sedumi.
https://github.com/eoskowro/LMI/blob/master/OF_Hinf.m


== Related LMIs ==
Optimal Output Feedback H2


== External Links ==
LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013.


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